Handbook of Measure Theory
Edited by: E. Pap
Copyright © 2002 Elsevier
ISBN: 978-0-444-50263-6

Table of Contents
Vol.1
Part 1
Ch. 1
Ch.2
Ch.3
Ch.4
Ch. 5
Ch. 6
Ch.7
Preface
List of Contributors
Classical Measure
Theory
History of Measure
Theory
Some Elements of the
Classical Measure
Theory
Paradoxes in Measure
Theory
Convergence Theorems
for Set Functions
Differentiation
Radon-Nikodym
Theorems
One-Dimensional
Diffusions and Their
Convergence in
Distribution
Endre Pap
Djura Paunic
Endre Pap
Miklos
Laczkovich
Paolo de Lucia
and Endre Pap
Brian S.
Thomson
Domenico
Candeloro and
Aljosa Volcic
James K.
Brooks
v-vi
vii-viii
1-26
27-82
83-123
125-178
179-247
249-294
295-343

Part 2
Ch. 8
Ch.9
Ch. 10
Part 3
Ch. 11
Ch. 12
Ch. 13
Ch. 14
Part 4
Vector Measures
Vector Integration in
Banach Spaces and
Application to Stochastic
Integration
The Riesz Theorem
Stochastic Processes and
Stochastic Integration in
Banach Spaces
Integration Theory
Daniell Integral and
Related Topics
Pettis Integral
The Henstock—
Kurzweil Integral
Set-Valued Integration
and Set-Valued
Probability Theory.. An
Overview
Topological Aspects of
Measure Theory
Nicolae
Dinculeanu
Joe Diestel and
Johan Swart
James K.
Brooks
M. Diaz
Carrillo
Kazimierz
Musial
Benedetto
Bongiorno
Christian Hess
345-399
401-447
449-502
503-530
531-586
587-615
617-673

Ch. 15
Ch. 16
Ch. 17
Ch. 18
Vol. II
Part 5
Ch. 19
Ch.20
Ch.21
Ch.22
Ch.23
Density Topologies
FN-Topologies and
Group-Valued Measures
On Products of
Topological Measure
Spaces
Perfect Measures and
Related Topics
Order and Measure
Theory
Riesz Spaces and Ideals
of Measurable Functions
Measures on Quantum
Structures
Probability on MV-
Algebras
Measures on Clans and
on MV-Algebras
Triangular Norm-Based
Measures
Wladyslaw
Wilczynski
Hans Weber
Stratos Grekas
Doraiswamy
Ramachandran
Martin Vath
Anatolij
Dvurecenskij
Beloslav
Riecan and
Daniele
Mundici
G. Barbieri
and H. Weber
D. Butnariu
and
E.P. Klement
675-702
703-743
745-764
765-786
787-825
827-868
869-909
911-945
947-1010

Part 6
Ch.24
Ch.25
Part 7
Ch.26
Ch.27
Ch.28
Ch.29
Ch.30
Part 8
Geometric Measure
Theory
Geometric Measure
Theory.. Selected
Concepts; Results and
Problems
Fractal Measures
Relation to
Transformation and
Duality
Positive and Complex
Radon Measures in
Locally Compact
Hausdorff Spaces
Measures on
Algebraic—Topological
Structures
Liftings
Ergodic Theory
Generalized Derivatives
Relation to the
Foundations of
Mathematics
Miroslav
Chelbik
Kenneth J.
Falconer
T.V. Panchaan
Piotr
Zakrzewski
W. Strauss,
N.D. Macheras
and K. Musial
Frank Blume
Endre Pap and
Arpad Takaci
1011-1036
1037-1054
1055-1090
1091-1130
1131-1184
1185-1235
1237-1260

Ch.31
Ch.32
Part 9
Ch.33
Ch.34
Ch.35
Ch.36
Ch.37
Real Valued
Measurability; Some Set-
Theoretic Aspects
Nonstandard Analysis
and Measure Theory
Non-Additive Measures
Monotone Set Functions-
Based Integrals
Set Functions over Finite
Sets.. Transformations
and Integrals
Pseudo-Additive
Measures and Their
Applications
Qualitative Possibility
Functions and Integrals
Measures of Information
Author Index
Subject Index
Aleksandar
Jovanovic
Peter A. Loeb
P. Benvenuti,
R. Mesiar and
D. Vivona
Michel
Grabisch
Endre Pap
Didier Dubois
and Henri
Prade
Wolfgang
Sander
1261-1293
1295-1328
1329-1379
1381-1401
1403-1468
1469-1522
1523-1565
1567-1585
1587-1607

Preface
The main goal of this Handbook is to survey measure theory with its many different
branches and its relations with other areas of mathematics. Mostly aggregating many
classical branches of measure theory the aim of the Handbook is also to cover new fields,
approaches and applications which support the idea of the measure in a wider sense,
e.g., the ninth part of the Handbook. Although chapters are written as surveys to special
areas they contain many special topics and challenging problems valuable for experts
and reach sources of inspiration. I hope that mathematicians from other areas as well
as physicists, computer scientists, engineers, econometrists will find useful results and
powerful methods for their research. The reader may find in the Handbook many close
relations to other mathematical areas: real analysis, probability theory, statistics, ergodic
theory, functional analysis, potential theory, topology, set theory, geometry, differential
equations, optimization, variational analysis, decision making and others.
Measure theory, as a classical mathematical area, is treated in many textbooks and
monographs and new results are widespread in many different journals. In the last 30
years, traditional conferences on measure theory in Germany (Oberwolfach) and Italy
(CARTEMI - Capri, Ischia, Maiori, Grado) were reach sources for new results and
further development of many specific areas of measure theory. The increasing interest in
measure theory (theory and applications) initiated the creation of the GEM working group
(GEnerealized Measures) in the framework of the international EUSFLAT association, and
parts of this Handbook are related to the GEM work. I mention also the specific approach
by Bourbaki A965) and the recent work of D.H. Fremlin B000) with a systematic approach
to measure theory of which volumes 1 and 2 are already published, and drafts of most
parts of the other three volumes are available on web page at www.essex.ac.uk/maths/staff/
fremlin/mt.htm. Encyclopaedia of Mathematics, Kluwer Academic Publishers (especially
the very useful CD-ROM version, 1997) contains also many measure theoretical items.
The reader of this Handbook may see here one place where new results are obtained and
new areas are developed.
The Handbook preparation started in 1998 with many difficulties. Many discussions on
the content and possible authors were undertaken with the measure theoretical community,
well connected thanks to the previously mentioned conferences. I am obliged to the
authors who agreed to contribute to the Handbook. In the cooperation with them it
is the encouragement I was given and the nice personal relations with many of them
not only through e-mail (more than 2000 messages), but also often in direct contacts,
that have brought this project to the final stage. Although there was a great pressure
on authors to make some unification, first of all because of common subject index and
author index, they have succeeded in preserving their own scientific styles and approaches.

VI
Preface
Many mathematicians were contacted and involved, but several of them because of other
obligations were not able to deliver their contribution themselves. The reader will also note
that some areas are missing, or some areas are under-represented. Some previous issues of
the Handbook series already cover some missing parts as Measure Algebras (D.H. Fremlin)
in the Handbook of Boolean Algebras (J.D. Monk, ed.), Borel Measures (R.J. Gardner,
W.F. Pfeffer) in the Handbook of Set-Theoretic Topology (K. Kunen, J.E. Vaughan, eds).
For the readers convenience, the subjects covered by the Handbook in 37 chapters are
organized in nine parts although there are close interactions between them.
In editing of the Handbook, I received much help from the contributors, as well as
many useful advices from Professors D.A. Fremlin, D. Kolzow, W.A.J. Luxemburg, and
W.F. Pfeffer. I am grateful to the Johannes Kepler University, Linz, Austria, where I have
managed a lot of related research. The project was supported during the visit of the editor to
the University Federico II, Naples, as visiting professor for PhD students, by INdAM, Italy,
in the period May-June, 2000, and by numerous visits to Naples, supported by MURST.
I want to thank for the partial financial support of the Project in the Fields of Basic Research
"Mathematical models of nonlinearity, uncertainty and decision" A866) supported by
Ministry of Science, Technology and Development of Serbia. I gratefully thank for the
fruitful cooperation and support of Dr. A. Sevenster, B. Lightfoot, A. Deelen from Elsevier
Science Publishers. Finally, I would like to thank VTEX Typesetting Services and in
particular Dr. Z. Kryzius for their fine work in converting the Handbook to its final typeset
form.
End re Pap

List of Contributors
Barbieri, G., Universita di Udine, Udine (Ch. 22)
Benvenuti, P., Universita degli Studi "La Sapienza ", Roma (Ch. 33)
Blume, R, John Brown University, Siloam Springs, AR (Ch. 29)
Bongiomo, В., Dipartimento di Matematica ed Applicazioni, Palermo (Ch. 13)
Brooks, J.K., University of Florida, Gainesville, FL (Chs. 7, 10)
Butnariu, D., University of Haifa, Haifa (Ch. 23)
Candeloro, D., Dipartimento di Matematica, Perugia (Ch. 6)
Carrillo, M.D., Universidad de Granada, Granada (Ch. 11)
Chlebik, M., Comenius University, Bratislava (Ch. 24)
de Lucia, P., Universita "Federico II", Napoli (Ch. 4)
Diestel, J., Kent State University, Kent, OH (Ch. 9)
Dinculeanu, N., University of Florida, Gainesville, FL (Ch. 8)
Dubois, D., IRIT- UPS, Toulouse (Ch. 36)
Dvurecenskij, ?., Slovak Academy of Sciences, Bratislava (Ch. 20)
Falconer, K.J., University of St Andrews, Fife, Scotland (Ch. 25)
Grabisch, M., University of Paris VI, Paris (Ch. 34)
Grekas, S., University of Athens, Athens (Ch. 17)
Hess, C, Universte Paris Dauphine, Paris (Ch. 14)
Jovanovic, ?., University of Belgrade, Belgrade (Ch. 31)
Klement, E.P., Johannes Kepler University, Linz (Ch. 23)
Laczkovich, M., Eotvbs Lorand University, Budapest (Ch. 3)
Loeb, PA., University of Illinois, Urbana, IL (Ch. 32)
Macheras, N.D., University of Piraeus, Piraeus (Ch. 28)
Mesiar, R., Slovak Technical University, Bratislava
and Systems Research Institute PAN, Warszawa (Ch. 33)
Mundici, D., University of Milan, Milan (Ch. 21)
Musial, K., Wroclaw University, Wroclaw (Chs. 12, 28)
Pap, E., University ofNovi Sad, Novi Sad (Chs. 2, 4, 30, 35)
Panchapagesan, T.V., Universidad de los Andes, Merida (Ch. 26)
Paunic, D., University of Novi Sad, Novi Sad (Ch. 1)
Prade, H., IRIT- UPS, Toulouse (Ch. 36)
Ramachandran, D., California State University, Sacramento, CA (Ch. 18)
Riecan, В., Slovak Academy of Sciences, Bratislava (Ch. 21)
Sander, W., Technical University of Braunschweig, Braunschweig (Ch. 37)
Strauss, W., Universitat Stuttgart, Stuttgart (Ch. 28)
Swart, J., University of Pretoria, Pretoria (Ch. 9)
vii

viii List of Contributors
Takaci, ?., University ofNovi Sad, Novi Sad (Ch. 30)
Thomson, B.S., Simon Fraser University, ВС, Canada (Ch. 5)
Vath, M., University ofWurzburg, Wiirzburg (Ch. 19)
Vivona, D., Universita degli Studi "La Sapienza", Roma (Ch. 33)
Volcic, ?., Dipartimento di Scienze Matematiche, Trieste (Ch. 6)
Weber, H., Universita di Udine, Udine (Chs. 16, 22)
Wilczynski, W., University of Lodz, Lodz (Ch. 15)
Zakrzewski, P., University of Warsaw, Warsaw (Ch. 27)

CHAPTER 1
History of Measure Theory
Djura Paunic*
Institute of Mathematics. University of Novi Sad. Trg D. Obradovica 4. 21000 Novi Sad. Yugoslavia
E-mail: djura®im.ns.ac.yu
Contents
Introduction 3
1. Beginnings 3
2. The Greeks 4
3. Archimedes (>
4. Infinitesimal methods 8
5. Loss of measure 1*
6. New beginning 1 &
7. Newly found measure 21
References 25
*The author wants to thank for the financial support of the Project in the Fields of Basic Research supported by
Ministry of Science, Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1

This page intentionally left blank

History of measure theory
3
Introduction
At the beginnings of civilization mathematics could be differentiated from science and
technology as "rational art of solving abstract problems with numbers and geometrical
figures". The accent in this definition is on "rational", "abstract", and "problem solving".
It means that the solution has to be obtained from initial data with some kind of rational,
logical mental process.
To calculate the size of a geometrical figure was always regarded as one of the most
basic aims of mathematics. Mathematically, size calculation of an object was divided into
tree problems which were considered different: finding lengths of lines, areas of surfaces,
or volumes of bodies. It was tacitly supposed that size of an object, which will be called
measure, has the following properties:
• Same objects have equal measure.
• Part of an object has smaller measure than the whole.
• If an object is divided into nonoverlapping parts, the measure of the whole object is
equal to the sum of the measures of the parts.
• Points have 0 measure in measuring lengths, lines have 0 measure in measuring areas,
and surfaces have 0 measure in measuring volumes.
The exposition of all results in this article will be in contemporary mathematical
language and notations. It introduces the problem of interpretation of what many authors
actually wanted to say since often their mode of expression is not sufficiently clear, or the
results are not explicitly stated but only implied, or the original exposition does not exist
any longer.
1. Beginnings
The first civilizations rose in the fertile river valleys of China, India, Mesopotamia, and
Egypt in the third millennium ВС. Unfortunately, in China and India very perishable
material was used for writing so it is very little known of their mathematical knowledge
at early times. The dryness of climate in Egypt preserved fair number of documents, so
that two papyri are preserved, Rhind papyrus and Moscow papyrus, devoted entirely to
mathematics. Even more is preserved in Mesopotamia since they used to write on clay
tablets which are almost indestructible, especially by fire.
In Egypt and Mesopotamia a lot of empirical rules were discovered for solving various
mathematical problems. Some of them turned out to be mathematically correct, other only
approximate.
Rhind papyrus contains 84 mathematical problems. Approximately 20 problems are
devoted to computations of areas and volumes. The area of parallelogram (product of the
base by height), triangle (half product of the base by height), and trapeze (product of half
sum of the bases by height) was computed correctly. For quadrangles it was used incorrect
formula (product of half sum of one pair opposite sides by the half sum the other pair
of sides). Very intriguing is the 50th problem which says that the area of round field of
diameter 9 is equal to the area of square whose side is 8. This is equivalent to

4
D. Paunic
Even more interesting are 10th and 14th problems in the Moscow papyrus. In the 14th
problem it is drawn trapeze with base 4, height 6. and upper base 2. From the solution it
follows that volume of truncated square pyramid is to be calculated. Volume is calculated
as sum of 16,4, and 8 multiplied by 2. Is it application of
V = -(a2+ab + b2)?
It is very difficult to say yes, and equally difficult to say no. Much more unclear is the 10th
problem. It is to be calculated the surface area of figure which resembles basket. It seems
that the formula
( ?2 ?
? = I 1 - - I Bx)-x, where ? =4-,
is applied. Since A - gJ is Egyptian equivalent for тг/4 one gets calculation of surface of
a body related to circle. Half sphere or half cylinder or something else?
In Mesopotamia there were many different civilizations but all of them used cuneiform
writing on clay tablets so all will be considered as one civilization. Most mathematical
tablets come from two libraries: Tiglatpilasars I in Assur and Assurbanipals in Ninevah.
The sexagesimal system for writing numbers was used (without zero), so the calculations
were simple to perform. They could solve much more difficult problems than Egyptians and
were more interested in algebraical problems then geometrical. In Mesopotamia appear
solution of quadratic equations and Pythagorean theorem. The area of regular polygon
or circle was calculated that the length of the perimeter was multiplied by corresponding
constants. For circle the constant was 1/12 which corresponds to 3 for ?.
2. The Greeks
Greek philosophers and mathematicians made the great discovery that mathematical rules
could be proved and that the whole of mathematics can be organized in axiomatic theory.
The Greek geometry started with Thales of Miletos, but Pythagoreans made the initial
development in VI and V centuries ВС. They investigated the properties of numbers,
geometrical figures; especially regular polygons, regular solids and circle, and created the
theory of rational proportions.
One of the most important discoveries Pythagoreans made was the existence of
incommensurable lengths, i.e., that there are segments whose lengths cannot be determined
by the process which is today known as Euclidean algorithm. It seems that this discovery
was made ca. 430 ВС, and there are two hypothesis how it was done.
First hypothesis is that Pythagorean tried to measure the length of diagonal of regular
pentagon by its side, since Pythagorean sign was regular pentagon with all diagonals
drawn. The problem of measuring the diagonal with the side reduces to the problem
of measuring the diagonal of smaller pentagon, formed from diagonals of the original
pentagon, to its diagonal, and one gets infinite regress. This explanation is not documented
in ancient literature.
The other hypothesis is based on the parity of numerical values for sides of right angled
triangle and Pythagorean theorem. It is the consequence of easy observations that square

History of measure theory
5
of even number of the form 4k, and that the square of odd number is of the form 4k + 1
(Knorr, 1975, Theorems 7, 8, p. 151). If not all sides are even, then hypotenuse cannot
be even since its square is of the form 4k and the sum of two odd squares is of the form
4n + 2. So hypotenuse must be odd. Of the remaining sides one must be even the other
odd. If hypotenuse is the diagonal of the square, then one side of the square is even and
the other odd, so the length of the side of the square is number which is at the same time
even and odd, which is absurd. This explanation can be much more easily defended with
ancient sources.
Theodoros of Kyrene (V century ВС) studied the problem of irrationality of square roots
of numbers and proved that square roots of number 3, 5, etc. to 17 are irrational, "but in
this one [17] for some reason he encountered difficulties" (Knorr, 1975, p. 62). The more
complete theory of quadratic irrational numbers was created by Theaetetos of Athens (V-
IV century ВС) and was included as Book X into Euclid's Elements.
Democritos of Abdera (ca. 460-ca. 370 ВС) was good mathematician and proponent
of atomic theory. He was pupil of Leuccipos of Miletos, and in philosophical circles
it is understood that atomic theory originated from Leuccipos. But it is almost nothing
known of Leuccipos. Atomic theory is very useful for mathematical applications since
from it follows that any geometrical figure is made of atoms, and that one can use some
kind of limiting process to determine lengths, areas or volumes. Archimedes attributes to
Democritos discovery that a cone is one third part of the cylinder, and the pyramid one
third of the prism, which has the same base and equal height, but that he did not prove it.
The works of Democritos are lost.
The discovery of incommensurable magnitudes showed that all lengths, all areas or all
volumes cannot be measured by rational numbers, and that the existing Pythagorean theory
of integral proportions is insufficient. The Greeks did not create the notion of real number
but used complicated definition of proportionality of ratios of geometric magnitudes so
that so called geometric algebra could be applied. It was rather cumbersome system of
calculations with geometric magnitudes but which was, or could be made, completely
rigorous. It can be regarded precursor of the definition of real numbers by Dedekind cuts,
but they had no intention of developing it further. The originator was Eudoxos of Cnidos
(first half of IV century ВС). Essentially his theory of proportions was presented in Book
V of Euclid's Elements.
In Book XII of Euclid's Elements are presented rigorous Eudoxean proofs of the
following results of Democritos and Hypocrates of Hios in which two geometric
magnitudes are compared:
A) Areas of two circles are in the same ratio as the squares of their radii. This result
was discovered by Hypocrates of Hios.
B) Volumes of two cylinders with the same height are in the same ratio as the squares
of their radii.
C) The volumes of two pyramids with the same height are in the same ratio as the ratio
of the areas of their bases.
D) A cone and the cylinder with the same height are in the ratio of 1 to 3.
E) The volumes of two spheres are in the same ratio as the cubes of their radii.
For the proofs of these theorems the so-called method of exhaustion was used. It consists
in showing that the area or volume of the difference between the given figure, and the

6
D. Paunic
sequence of inscribed polygons (Р„)/16ь: or prisms, cylinders etc. {Уп)пчп can be made
smaller than any magnitude of the same kind if ? is sufficiently large.
3. Archimedes
The greatest mathematician of antiquity was Archimedes of Syracuse B877-212 ВС). Of
his extant works 6 are devoted to calculations of lengths, areas, and volumes. These are:
• Measurement of a Circle. It consist of two theorems and one corollary, and is only
fragment of the original work.
• On the Sphere and Cylinder.
• Quadrature of Parabola.
• On Conoids and Spheroids.
• On Spirals.
• Method.
Only the first two works were very popular and well-known throughout the history of
mathematics. The last one was discovered only in 1906, and it had no influence at the
development of mathematics.
The first Greek who is known to have tried to square the circle was Anaxagoras of
Clasomenae. While he was in prison he occupied himself with this problem. Nothing
further is known. Next, Hypocrates of Hios tries to divide circle into lunes (gr. meniskoi.
lat. lunnulae), geometric figure bounded by two arcs. He succeeded to square 5 lunes, some
greater than the half circle, some smaller, but this approach to the determination of the area
of the circle was apparently dead end.
Dinostratos tried to square circle using special mechanical curve which was used by
Hippias of Elis to trisect an angle. This curve is obtained in the intersection of the upper
side of square when it moves uniformly parallel to itself towards the lower side, and the
left side when it rotates uniformly around lower vertex to the right for 90° till it coincides
with the lower side. Both sides start moving at the same moment and reach lower side at
the same time. The Nicomedes showed that the side of the square is the geometric mean
of the quarter of the circumference (side of square is radius), and the segment of the side
between left lower vertex, and the intersection point of the curve with the lower side of the
square. This solution was objected that uses point on the curve which does not exist, since
when both sides reach lower side of the square they coincide and there is no intersection
point of the curve and the side.
The first Greek who tried to square the circle by inscribing regular polygons in it was
Antiphon, sophist from Athens, and contemporary of Socrates. His work is not extant,
and there are only vague description of it by commentators. It seems that his idea was
successively to double the number of sides of the inscribed polygons so that after finite
number of iterations the polygon will coincide with the circle. Every polygon can be
transformed into square so circle can be squared. The reason for coincidence is that
after finite number of steps sides of the polygon will coincide with the circle (atom will
be reached). This procedure criticized that the side of the polygon will never coincide
with the circumference since the line can only touch circumference in one point. His
idea, as presented by commentator, can be regarded more as a thought experiment than

History of measure theory
7
mathematical proof. If one understands it that circle is limiting case of inscribed polygons
then it becomes true. Anyway Antiphon was the first to propose an idea that could be
transformed into mathematical proof.
One generation later Bryson, pupil of Socrates or Euclides of Megara, proposes that one
has to inscribe square in the circle, and to circumscribe square around circle, and that the
circle will be equal to the an intermediate square. It very unclear what he actually meant,
but the most important in his construction was that area of the circle is bounded from
above and from below. Eudoxos was the first who transformed vague ideas of Antiphon
and Bryson into rigorous proofs, but it was Archimedes who succeeded to apply them to
the circle.
In Measurement of a Circle are proved two important facts: First the area of the circle
is the same as the area of right-angled triangle in which one of the sides about the right
angle is equal to the radius, and the other to the circumference of the circle. It is equivalent
to the statement that the ratio of the diameter to the circumference is the same as the ratio
of the area to the square of the radius. The second is that 3 Щ < ? < 3 j. This inequalities
were determined by approximating the length of the circumference with the perimeters of
inscribed and circumscribed regular polygons of 96 sides. These perimeters were obtained
starting with regular hexagon and in each step doubling the number of sides of polygons
and calculating their perimeters.
The second work On the Sphere and Cylinder is the most interesting, and is divided in
two books. The main result is that the surface area of the sphere is four times as the area
of the greatest circle Dлт2, proposition 33 of the first book) and the volume of the sphere
is for times of the volume of the cone which has its base equal to the greatest circle of the
sphere and its height equal to the radius of the sphere (^??\ proposition 34 of the first
book).
In Quadrature of Parabola he calculated that the area of the segment of parabola is four
thirds of the area of the triangle with the same base and equal height. Here the sum of
geometric series was used to obtain the result.
Conoid is solid which is obtained by rotation of parabola around its principal axis or
hyperbola around its major axis. Spheroid is ellipsoid of revolution. On Conoids and
Spheroids is devoted to the computation of volume of these solids and the Archimedean
constructions for it are very similar to the modern definition of definite integral. To prove
that the volume of paraboloid of revolution is half of the volume of the cylinder in which
it is inscribed the paraboloid is divided into ? slices in which ? — 1 cylinders are inscribed
and around which ? cylinders are circumscribed. In this work he computed that the area of
an ellipse is abn, where a, and b are greater, and smaller semiaxis.
The work On Spirals is devoted to the investigation of Archimedean spiral. It is very
interesting because in it transcendental curve is investigated mathematically for the first
time. The first twenty propositions are devoted to determination of the tangent line to
the spiral, and played great role in the development of differential calculus. Last eight
propositions are devoted to area computation. The main result is that the area of the figure
bounded by the first turn of the spiral and the straight line segment joining its beginning
to its end is equal to one third of the area of the circle whose radius is that line segment.

8
D. Paunic
Archimedes succeeded to calculate the area of spiral sector bounded by radius vectors r\,
and Г2 and which has central angle ?
? ( 1 Д
a(S)= -I Г|Г2 + -(? -r\)-\.
The problem is not important but his skill of the computation is fascinating.
In the Method Archimedes showed how he has found some of his results with ingenious
use of lever principle. For the sake of discovery he supposes that plane figure is composed
of the parallel straight segments. Then these segments are balanced with another set of
parallel segments which belong to known figure at the other end of the lever so that area
of unknown figure can be determined. The decomposition of plane figure into straight line
segments was extensively used in the XVII century. Method ends with volume computation
of two new bodies. In proposition 14 Archimedes computes that the volume of the wedge
cut from cylinder of diameter d, and height d/2 by a plane through a diameter of the
base and one point on the circumference of the other base is d^/\2. In proposition 15
is computed that the volume of the body, which is obtained as common region of two
cylinders with equal diameter d whose axes are intersecting and perpendicular, is 2t/3/3.
In antiquity there was no new ideas in measure theory after Archimedes. Only Pappos
in his Collection gives areas of some interesting surfaces. The most interesting is the
area bounded by the spiral on the hemisphere. If the hemisphere is generated so that
quadrant makes one revolution about vertical axis, and the point during that rotation moves
uniformly from the pole to the equator then the path of the point describes spiral on the
hemisphere. Pappos proves that the spiral, and quarter of the circumference which connects
its starting, and end point bound the surface which has the same ratio to the hemisphere
as has the sector of the quadrant to the quadrant. The other result is the theorem which
anticipates Guldin's theorem. He says: "Figures generated by complete revolution of a
plane figure about an axis are in a ratio compounded A) of the ratio of the areas of the
figures, and B) of the ratio of the straight lines similarly drawn to (i.e., drawn to meet at
the same angles) the axes of rotation from the respective centers of gravity" (Heath, 1921,
II, p. 403).
4. Infinitesimal methods
During middle ages interest for mathematics in Europe was very low, although some dim
knowledge of mathematics was present, but without proofs and almost empirical. So, for
instance, in the tenth century Adelbold von Utrecht knew that the volume of the sphere is
obtained as 11/21 parts of the cube of the diameter (? = 22/7), but did not know how the
area of equilateral triangle whose side is 7 can be 28 or 21. Gerbert explained to him that
he has confused triangle with triangular number.
Only in Islamic countries from VIII to XIV century appeared some interest in
mathematics. They were not very interested in measure theory computations, but have
obtained some very interesting results. The most interesting result obtained al-Haitham
(ca. 965-1039). Islamic mathematicians admired Archimedes and studied On the Sphere

History of measure theory
9
and Cylinder but On Conoids and Spheroids was unknown. Thabit ibn Qurra (ca. 835-901)
calculated the volume of paraboloid of revolution in a new way, in a rather complicated
manner. His proof was simplified by al-Kuhi (X century), but al-Haitham succeeded to
solve the problem in "full generality", i.e., he found that the solid obtained by rotation of
segment of parabola around its chord has volume 8/15 of the volume of circumscribed
cylinder whose height is the cord. The proof was standard exhaustion argument in the
manner of Archimedes. Basis for his calculation was recurrent formula for sum of powers
of numbers
?*/+1 +? (?*') = ("+ !)?*'·
*=l j=\ \k=\ / k=\
Using this formula it was possible to obtain the fundamental limit
?—>oo ri 5
Archimedes used this limit for / = 1 and / = 2.
The Renaissance men were interested in all things from antiquity and soon wanted to
master ancient science and philosophy in which mathematics played prominent role. The
level of mathematical knowledge gradually rose so that in the second half of the XVI
century was high enough that discoveries of new mathematical results became possible.
The mastery of Elements of Euclid and Conies of Apollonios prepared scientist for study
of Archimedes. The Archimedean book On the Equilibrium of Planes on determination of
centers of gravity of plane figures seems to be the work which spurned the imagination
of mathematicians to apply infinitesimal methods to determination of centers of gravity of
solids. Archimedes uses various results on center of gravity of solids but at that time no
systematic work of his that contained proofs was known.
Francesco Maurolico A494—1575), improved on Archimedean determination of center
of gravity of solids in De momentis aequalibus from 1548, where the proofs were
systematized, but it was not printed till 1685 so its influence was limited. Federigo
Commandino A509-1575) in his work Liber de centro gravitatis solidorum, Bologna
1565, determines the center of gravity of conoid, but uses Archimedean proofs strictly.
His work was continued by Simon Stevin A548-1620) who tried to simplify proofs.
Stevin's book Beghinselen der Weeghconst, Leyden 1583 (Introduction to art of weighing)
was written in Dutch so its influence was not so great till the beginning of 17th century
when it was translated into Latin and French. He deliberately tried to simplify and modify
Archimedean proof structure, but he did not try to make general theory of it and in every
theorem repeated reasoning from the beginning. Luca Valerio A552-1618) continued of
Commandino's work in Italy. Valerio published the book De centro gravitatis solidorum
Rome 1604. He was very skillful geometer, and expounded his proofs in Euclidean manner.
He had great influence especially upon Galileo and his pupils.
The most influential work appeared at the beginning of the 17th century: Nova
stereometria doliorum vinariorum, Linz 1615 (New solid geometry of wine barrels) by
Johann Kepler A571-1630). Kepler has become famous few years earlier especially after

10
D. Paunic
the publication of "Astronomia nova" (New astronomy), 1609, in which the motion of
planet Mars from Tycho Brahe's observations was analyzed, and Kepler's first two laws of
planetary motion were expounded. In Stereometria Doliomm he tries to calculate volume
of wine barrel since he was fascinated that wine merchants in Linz, Austria, used single
measuring rod to determine volume of the cask without respect to its size, and without
any calculation. In order to explore the validity of this procedure he studies the solids
of revolution. In the first part he presented Archimedean theorems on area of circle and
volume of sphere but gave proofs based on intuitive application of indivisibles. He said:
"We could obtain absolute and in all respects perfect demonstrations from these books of
Archimedes themselves, were we not repelled by the thorny reading thereof" (Edwards,
1982, p. 103). Circle is regarded as infinitude of isosceles triangles whose infinitesimal
base is on the circumference, and the third vertex in the center. Sphere is regarded as
infinitude of cones whose base is on the surface of it and all vertexes in the center. Kepler
found that if a segment of conic section rotates around an axis 92 different bodies can be
formed. Dissecting torus with planes which pass through its axis of rotation he reduced
calculation of volume of torus to that of cylinder; Any ring with circular or elliptic cross
section is equal to a cylinder whose altitude equals the length of the circumference which
the center of the rotated figure describes, and whose base is the same as the cross section
of the ring, this theorem is found in Heron's Metrica where it is attributed to Dionysodoros
whose works are lost. If the solid is formed by rotation of a segment of circle around its
chord, he calls it apple if segment is greater then half circle or lemon if it is less, then he
showed that its volume can be reduced to that of the wedge of cylinder whose base is the
given segment of the circle, and height equal to the length of circumference whose radius
is height of the segment.
Kepler's work was much studied, and soon appeared mathematical books in which the
infinitesimal methods were used to calculate areas and volumes. Around 1630 there was
a lot of interest among mathematicians for finding areas and volumes, mostly using some
sort of subdivision method. The ablest were in Italy Cavalieri and Torricelli, in France
Fermat, Descartes, and Roberval, and Gregoire of St. Vincent, Guldin and Tacquet. They
mostly communicated by letters or in direct contact and had very varied interests so it is
very difficult to established the priorities.
Galileo Galilei A564-1642) was interested in infinite and infinitesimal. He made some
applications of them in his mechanics but it remained to his pupils and assistants Cavalieri
and Torricelli to develop more fully their application to mathematics.
Bonaventura Cavalieri A598-1647) tried systematically to apply infinitesimals to
geometrical problems since approximately 1626, and wrote two books Geometria
indivisibilibus continuorum nova quadam ratione promota (Geometry deduced by new
calculations using indivisibles of continua) 1635. and Exercitationes geometricae sex (Six
geometrical exercises) 1647. Cavalieri differed in two ways from Archimedes and Kepler.
Firstly, his method consists in effort to connect two different objects, one known and one
unknown. Then both objects are decomposed in the same way into indivisibles so that there
is one to one correspondence between indivisibles and that indivisibles are in certain ratio.
Then the area or volumes of the figures are in the same ratio as the ratio of indivisibles.
Secondly, the indivisibles in which the figures are decomposed always have one dimension
less then the figures. Using this method the limit processes were hidden.

History of measure theory
11
Cavalieri had still another technique, more arithmetical, and at the first sight not so
general. If the lengths of the lines could be expressed arithmetically then he used "sums of
power of lines" which is equivalent to integral
Cavalieri's exposition was far from rigorous and very verbose so it was difficult to read
him and understand what he exactly meant.
Evangelista Torricelli A608-1647) fully recognized the advantages and disadvantages
of Archimedean rigor and indivisible methods and presented his ideas much more clearly
than Cavalieri. He published his account of indivisible methods in Opera geometrica
(Geometrical works) in 1644 which was illuminating to many who found the work of
Cavalieri too obscure and difficult to follow. Although he admired Cavalierean method and
gave full account of it, demonstrations according to the methods of the ancients were also
given. Torricelli's most interesting results are the transformation of arc of higher spiral
rm =квп to the arc of higher parabola xm+" = k{,J^f-y)n. This reduction is equivalent to
the transformation of the integrals
for this curve. He was the first who succeeded to determine the volume of the infinite solid
which is obtained by rotating hyperbola xy = a around jc-axis. For this he used cylindrical
indivisibles.
Gregoire de Saint-Vincent A584-1667) had very interesting ideas on calculating
volumes geometrically. He developed his ideas during 1622-1629 but because of war his
results were not published until 1647. Then many of his results were not new, and his
geometric approach was not so promising as algebraic one proposed in 1637 in Geometry
of Descartes. In Opus geometricum (Geometrical work) Gregoire established that the area
under hyperbola xy = 1 has the property that A(at, bt) = A(a,b), a, b, t > 0, where by
A(a,b) the area between hyperbola and .v-axis from ? = a to ? = b is denoted. His friend
A. A. de Sarasa noticed that from it immediately follows that the area under hyperbola has
logarithmic property since
A(l,xy) = A(l,x) + A(x,xy) = A(l,x) + A(l,y),
and published it in Solutio problematis a Mersenne propositi, 1649. Gregoire's geometric
method for finding volumes was called "ductusplani in planum", and is presented in the 7th
part of his Opus. It is applicable to the solids whose cross section are parallelograms, since
they can be constructed by means of two plane surfaces standing on the same ground line,
and parallel normals to that ground line determine cross sections. In that way he reduced
the volume calculation to the properties of plane figures. His best results are application of
that method to the volume calculation of wedges, formed by cutting right circular cylinder

12
D. Paimic
by means of an oblique plane through a diameter of the base. Opus geometricum was
carefully studied by B. Pascal, Chr. Huygens, and G. Leibniz.
In France at the beginning of XVII century the center of mathematical life was around
M. Mersenne A588-1648) who was interested in natural sciences and mathematics,
who organized regular meetings, and corresponded with mathematicians and scientist
throughout Europe. Thanks to his correspondence Galileo, Cavalieri, and Torricelli in
Italy were acquainted with work of Roberval in Paris, Fermat in Toulouse, Descartes in
Holland, and vice versa just to mention the most famous scientist which were kept in
touch through him. One of problems posed was to investigate the path of a point at the
rim of a rolling wheel. The investigation of this curve, called cycloid, was very popular
in the XVII century. Origin of the cycloid can be traced to Bouvelle (XVI century) but
Galileo posed it as a problem to Mersenne in 1630, and the first investigation was done
by Roberval after it. Gilles Personne de Roberval A602-1675) was professor at College
Royal in Paris since 1634. This post was renewable every three years. The professor had
to propose problems to the candidates and author of best solutions becomes new professor.
Roberval did not publish anything about his methods but succeeded to remain professor till
his death. He solved many problems with infinitesimal methods which he developed around
1628. He formulated his ideas in the book Traite des indivisibles which was written 1634
but was not published till 1693. The indivisibles there were used intuitively and without
rigorous proofs, and were presented as narrow rectangles. Roberval found that the cycloid
can be geometrically presented in a way which is in modern notation ? = at — ? sin?.
у = a — a cos t and to find the area under an arc of the cycloid he calculated the area under
accompanying curve (in this case sinusoid ? = at, у = a — a cost) on which he has to
add one circle and finally obtained that the area under one arc of the cycloid is 3 times
the area of generating circle. His success provoked Descartes and Fermat to find their own
solution to the same problem. In 1638 Descartes used for it an ad hoc method similar to
the Archimedean use of triangles in the quadrature of parabola and infinitesimal reasoning.
Fermat uses direct decomposition of the area into horizontal rectangles and compares all
the lines of the cycloid to the all the lines of rectangle which contains it. Roberval also
found the volume of the solid which is generated by rotation of sinusoid. He geometrically
found following integrals
2a2/ sin21d(at) =??\ a2 I A + costJd(at) = —-a\
Jo Jo 2
In the next generation of geometers the influence of Descartes algebra begins to be felt
and many results were discovered arithmetically. The greatest influence had the second
edition of the Latin translation of his Geometry which appeared in 1659, since it contained
many additions on arithmetical application of infinitesimal methods written by various
authors.
John Wallis A616-1703) published in 1655 his most famous book Arithmetica
infinitorum. His aim was to present his own method of investigation instead of proving
things, so that in it there is practically no proofs. After verifying some statement for 1,

History of measure theory
13
2 and 3 or so Wallis claimed that it holds for any rational k. He introduced fractional
exponents and claimed that for any rational к the results equivalent to the
Jo k+ 1
holds. This result was proved earlier by Fermat and Torricelli but they published
later. Fermat commented about Arithmetica infinitorum: "all these propositions could be
demonstrated via ordinaria, legitima et Archimedea (in ordinary, correct, and Archimedean
way) with much less words then there is in his book".
About 1655-1660 the determination of arc length was done. First W. Neil determined
the arc length of semicubic parabola (y = x^l1) and after him Ch. Wren succeeded to
determine the arc length of cycloid. In all arc length determination the calculation was
reduced to the quadrature of auxiliary curve. This is clear since the length of the curve
у = f(x) is calculated geometrically as the integral
Chr. Huygens A629-1695) and R.F. de Sluse A622-1685) studied cissoid (y =
хъ/2{2а - jc)~1/2). They found that area between the cissoid and the asymptote is 3?2?,
that the volume of the solid which is obtained by rotating the curve around its asymptote
is 2??2?3, and most surprising that the volume of the solid which is obtained when
the curve and its asymptote rotate about y-axis is 10??2?3. Surprising about it was that
this solid enclosed volume which was infinite. This result was the consequence that the
center of gravity of the area between cissoid and the asymptote is in the point T( ja, 0).
Huygens could not make up his mind how to present his results. For him indivisibles were
not rigorous enough but Archimedean proofs were too long, so in his most important
mathematical book Horologium oscilatorium which he wrote about 1656, and finally
published 1673, he gave only results.
The next three mathematicians Blaise Pascal A623-1662), James Gregory A638-1675),
and Isaac Barrow A640-1677) succeeded to make a synthesis of earlier infinitesimal
methods but they presented it in a geometrical language so that their presentations had very
limited influence to the further development of calculus. Pascal wrote on cycloid Histoire
da la roulette where he presented most things known about it, but his writing Lettre de
Amos Dettonville had great influence on the development of analysis, especially on Leibniz.
In the first letter in the part Traite des sinus du quart de cercle he very successfully applied
the technique of characteristic triangle in solving geometrically
r ??? d{r9) = r2(cosa - sin/3),
and some generalizations to the higher powers of sine. Leibniz repeatedly stated that he
was lead to the invention of calculus by a study of the works of Pascal especially his
application of characteristic triangle. This is the use of similarity of infinitesimal triangle
L

14
D. Paunic
in the point (x, y) of the curve у = f(x) formed by ??, ?}\ and As with triangle formed
by y, sn (subnormal-segment of Jt-axis between ? and intersection with normal in the
point (x, y)), and ? (normal-segment of the normal to the curve between (x, y) and
praxis), and the triangle formed by st (subtangent-segment on jc-axis between intersection
of the tangent in (x, y) and x), y, and t (tangent-segment on the tangent between ? -axis
and the point (x, y)). Pascal never seriously studied algebra so he missed the possibilities
which algebraic suggestive symbolism offers.
Gregory presented his version of general method in Geometriae pars universalis
(Universal part of geometry) published in 1668. There he proves that determination of arc
length is reduced to the quadrature of auxiliary curve, but also asks the converse question:
to determine a curve (u(x)) whose arc length s has constant ratio to the area of another
given curve (/(*)). He solves this problem in Proposition VI and uses the ordinate of the
curve to determine the tangent to the second curve which represents the area of the first
curve. So he comes to the Fundamental theorem of the calculus, but his proof are totally
verbal and geometrical in the Archimedean manner. So to determine result equivalent to
fQ dr/ cos t he had to make six transformations. This process can be fascinating to some
but algebraically it is obtained in a couple of lines.
Similar result obtained Barrow in his Lectiones geometricae (Geometrical lectures)
published in 1670. This book is very similar to Gregory's. It can be explained that both
were in Italy for considerable time so that they were well versed in infinitesimal methods
of Cavalieri, Torricelli, and Roberval. Barrow's book is more profound than that of Gregory
but he also used geometrical language, correct geometrical proofs so that it was even
more difficult to understand. In the lecture X, Proposition 11 is explained how to construct
tangent to the curve which represents area of another curve using ordinate of another curve.
In lecture XI, Proposition 19 he proves the other part of Fundamental theorem, in modern
notation
ь
f'(x)dx = f(b)-f(a),
showing that large rectangle R(f(b) - /(a)) (R is unit of measure) is equal to the
area formed by all infinitesimal rectangles under the curve Rf'(x)dx. But he was rather
disappointed that his book passed practically without much influence in scientific circles.
The ease of calculation in algebraic method was paramount and nobody was much
interested in long chains of geometrical reasoning. Everybody wanted to apply mathematics
as fast as possible to obtain explanations and discoveries of the natural laws. Rigorous
geometrical methods went out of fashion in mathematics for more then a century and a
half.
5. Loss of measure
Between 1665 and 1685 Isaac Newton A643-1727) and Gottfried Wilhelm Leibniz A646-
1716) created calculus. It were two different versions of general approach to the problem
solving using infinitesimal technique. The novelty consisted mainly in two things: First,
,

History of measure theory
15
discovery that integration and differentiation are inverse operation to each other so that
integration can be done by the formula (in modern notation)
ь
F'(x)dx = F(b)- F(a).
Second, making algorithmic procedure out of it so that it can be applied systematically
and generally. Their procedure, specially Leibniz's, were very similar to ours but it should
be remembered that both of them never used notion of function in modern sense since it
was created much later. They used "quantities", "magnitudes" or "variables" which change
with one another, one independently and the other depends of the first, or both depend on
time.
Newton made his most important discoveries in 1664-1666, his "anni mirabiles"
(wonderful years). His first systematic work on calculus was the so-called October 1666
tract on fluxions, the next were De analyst per aequationes numero terminorum infinitas
(On the analysis by equations with infinitely many terms) written in 1669, spurned by the
publication of series development of logarithm by Mercator in 1668, and De methodis
serierum et fluxionum (On the methods of series and fluxions) written in the winter
1670-1671. He tried to publish De methodis, but it did not go easily and later he was
no more interested. The second work, De analyst, had limited circulation in the English
mathematical community since Newton sent it to the Royal Society in order to establish
himself as a scientist which had found powerful new method for solving mathematical
problems.
In De analyst he presented his discovery in three rules:
A) \iy = axm'n then the area under ? is ? r('"/"')+'.
4 ' J J (m/n)+\
B) If у = y\ + у2 + ¦ ¦ ·, were the sum can be finite or infinite, then the area under у is
the sum of the areas under every term.
C) If the area under the curve f(x, y) = 0 is to be determined then у has to be
developed into the sum of terms of the form ax'"/" and the first two rules applied.
To solve f{x, у) = О Newton developed "Newton method" of successive approximations
/(*„)
Xn-\-\ —Xn — ,
f'(Xn)
"Newton parallelogram" for solving implicit equations, and used reversion of series.
In De methodis in the first part he developed his ideas from De analyst and applied them
to 12 general problems. This paper remained unfinished and unpublished till 1711. Newton
used this manuscript to send two letters to Leibniz in 1676 with review of his results, but
Newton was not interested to continue mathematical correspondence with Leibniz.
Leibniz did not publish any book on calculus but presented his ideas in papers and letters
so he changed his point of view with time. Around 1680 he wrote "I represent the area of a
figure by the sum of all the rectangles contained by the ordinates and the difference of the
abscissae", and that he "obtains the area of the figure by finding the figure of its summatrix
or quadratrix; and of this indeed the ordinates are the given figure in the ratio of sum of
differences".
L

16
D. Paunic
He started with publishing on differentiation and determination of extrema in a paper
with long name. The name is abbreviated as Nova methodus... (New method) which was
published in 1684. After it he published papers on solution of various problems. In 1693
he made his idea of integration more explicit, and in a paper Supplementum geometriae...
Leibniz says: "Now I shell show that the general problem of Quadratures can be reduced
to the finding of a curve whose inclinations (declivitas) have a given law" (Leibniz, 1995,
p. 263).
Application of calculus used three facts which were supposed to be true:
A) The so-called Fundamental theorem of calculus. For any function holds
ь
F'(x)dx = F(b)-F(a).
B) Any function can be developed into a series of functions
f(x) = u0(x) + u\(x) + ui(x)-\ ,
particularly into power series (Taylor's formula) and later into trigonometric series.
C) If / is presented by a series then one can differentiate and integrate it term-wise
f'(x) = u'0(x) + u\(x) + u'2(x) -\ .
pb pb pb pb
? f(x)dx= I uo(x)dx+ I u\(x)dx+ I U2(x)dx-\
Ja J a Ja Ja
Since Leibniz published his results, and his method was more easily understandable and
applicable many able mathematicians soon became his followers. The most prominent were
brothers Jacob and Johann Bernoulli, Jacob Hermann, marquis de I'Hopital, etc. After
death of Leibniz (in 1716) the most famous representative of his calculus became Johann
Bernoulli A667-1748). For analysis was important his introduction of the notion of the
function in 1718: "The function of a variable magnitude is called the magnitude which is
composed in some way or another from this variable magnitude and constants" (Struik,
1969, p. 368). Under "in some way or another" is to be understood as an expression in
which algebraic operations are applied finite or infinite number of times. His most famous
pupils were marquis de I'Hopital, L. Euler, M. de Maupertuis, G. Cramer, and A. Clairaut.
This notion of function was used by Leonhard Euler A707-1783) in the most influential
book on analysis in the XVIII century Introductio in analysin infinitorum. In it the function
is defined: "A function of a variable quantity is an analytical expression composed in any
way from this variable quantity and from numbers or constant quantities" (Edwards, 1982,
p. 271).
The next important moment for further development of function concept was discovery
of partial differential equations by Jean le Rond d'Alambert A717-1783), i.e., his analysis
of vibrating string. In 1747 he published that the "general" solution of the equation
/
Э2.У _ тд2у
dt2 ~a дх1-'

History of measure theory
17
is y(t,x) = \<p(at + ?) + \?(?? — ?), where ? and ? are arbitrary functions. The
only problem for him was whether any continuous curve which can be drawn with free
hand ("libera manu ducta") over a finite interval can be expressed by a single analytical
expression, his idea of function. In 1749 Euler also published on vibrating string and found
essentially the same solution, but he was confident that any continuous curve "libera manu
ducta" over a finite interval can be represented by analytical expression if necessary by
discontinuous one. By this Euler understood that curve is defined arc-wise by different
analytic expressions, in modern language it is continuous over the interval but its derivative
is discontinuous in finite number of points. He also notes that special solution can be
??? ?? ???? 2?? 3??? . 3??
y(t,x) = acos sin Ь/Scos sin h ? cos—-— sin—- l· ¦ ¦ ¦,
which is obtained when initial condition is
. ?? . 2?? . ??
у (О, ? ) = a sin— + /3sin—- \- ? sin— ? ,
without any determination whether the sum is finite or infinite.
In 1753 Daniel Bernoulli A700-1782) published his analysis of the vibrating string
problem. Bernoulli based his work on previous analysis of Brook Taylor from 1713.
Taylor found that any sinusoidal curve whose half period goes whole number of times
into the length of the string can be solution. Bernoulli's solution was that every solution
can be obtained by superposition of these Taylorean "elementary" solutions. According to
Bernoulli any solution is
????? ???
b„cos—— sin-—,
n = \
since any possible initial curve over the interval [0, /] can be represented by
oo
*-~» ???
f{x) = 2_ansm~—,
n=\
because "there is enough constants to make this series fit any curve".
Neither Euler nor d'Alambert admitted Bernoulli's solution as general.
In 1759 Joseph Louis Lagrange A736-1813) studied the same problem in the similar
fashion as Daniel Bernoulli. He supposed that string is approximated by massless string
loaded by ? equal and equally spaced masses and then passed to the limit. But in the
subsequent development he at one moment interchanged sum and integral and missed
possibility to discover the law of formation of the Fourier coefficients. After long and
complicated calculations which had some gaps he obtained result of Euler and d'Alambert.
The only consequence of all these discussions was that Euler improved his definition
of function in preface of his Institutiones calculi differentialis (Lectures on differential
calculus) from 1755. "If some quantities so depend on other quantities that if latter are

18
D. Paunic
changed the former undergo change, then the former quantities are called functions of
the latter. This denomination is of broadest nature and comprises every method by means
of which one quantity could be determined by others. If, therefore, ? denotes a variable
quantity then all quantities which depend upon ? in any way or are determined by it are
called functions of it" (Edwards, 1982, p. 271).
6. New beginning
Fundamental step, which had overwhelming influence on development of analysis, was
taken by Jean-Baptist-Joseph de Fourier A768-1830). In his paper on propagation of heath
submitted to the French Academy of Science in 1807, he noticed that coefficient of the
trigonometric series can be easily calculated using integrals.
Fourier supposed that the arbitrary function / defined over interval [—?. ?] has the
development into trigonometric series
1 x
f(x) = -ao + \] (an cos ? ? + bn sin nx).
n=\
It remains to determine coefficients. He proposed two proofs. In the first proof he solved
infinite system of equations with infinite number of unknowns to obtain coefficients. The
second proof was more formal, and the same idea was used earlier by Euler and Lagrange.
In order to obtain coefficients one integrates f(x) or f(x)cosnx or f{x)smnx. If the
exchange of sum and integral is permitted then one obtains
ao
1 fn 1 ?
= — I f(x)dx, a„ = — I f(x)cosnxdx, n = l,2 ,
? J-? ? у_.т
1 Г
bn = — I f(x)sinnxdx, n = l,2
? J_„
Fourier's work was amplified and finally printed in 1822 in his book: Theorie analytique
de la chaleur. It is interesting that in it Fourier obtained three different expansions for \x
on the interval ]0, ?/2[:
1
1
ix
1
ix
=
=
=
sinjc
2
-
— sinjc
?
?
4 ~
2
?
1
2
"
sin 2x + - sin 3.v -
3
2
3^
COSJC -
2
sin 3?· ?——
5-?
2
- -^—cos 3x -
32?
. + ...
sin5.v
2
^
_ + ..
cos 5 .?
This result strongly opposed the notion of the function as analytic expression.
The question is what are the integrals for an and bn in case of "arbitrary" function?
Fourier did not answer that question, but some years later Augustin Louis Cauchy A789-

History of measure theory
19
1857) faced the same problem when started to lecture on calculus in the Polytechnic school.
In 1823 he defined that
/'
J a
f(x)dx is the limit of the sum S = }f(xj-\)(x, — .r,--i),
i' = l
over every division of the interval [a,b], a = xo < x\ < ¦ ¦ ¦ < *n-\ < x„ = b such that
the maximum length of the subintervals tends to 0 as ? -> oo. He introduced the modern
definition of continuity (?-?) and proved that for continuous functions, this limit exists.
But "arbitrary" function obviously does not have to be continuous.
Peter Gustav Lejeune Dirichlet A805-1859) proved in 1829 that a function / is
represented by its Fourier series over the interval ]—?, ?[ "if the function f(x), which
is assumed to be bounded and single valued ("finite and determined"), has only a finite
number of discontinuities between limits —? and ?, and if besides it has only a finite
number of maxima and minima between these limits..." (Birkhoff, 1973, p. 146). He
expanded this article in 1837 and there defined function in modern sense.
Dirichlet's pupil Bernhard Riemann A826-1866) in his Uber die Darstellbarkeit einer
Function durch eine trigonometrische Reihe (On the possibility of representation of a
function by trigonometric series) from 1854, but published in 1868 asks the same question
and answers:
"First of all: What is to be understood by
Ja
f(x)dx1
In order to establish this, we take the sequence of values x\.x2 xn-\ 1У'пе between a and b
and ordered by size, and, for brevity, denote x\ — a by A \, .vi — x\ by Л1 b — xn~\ by ?„,
and proper positive fractions by ?,. Then the value of the sum
S= Л;/(а+?|Л|) + Л2/(?? + ???2) ? + Л„/(л„_| +?»Л»)
will depend on the choice of the intervals Л, and the quantities ?,. If it has the property that,
however the Л, and the ?, may be chosen, it tends to a fixed limit A as soon as all the Л, become
infinitely small, then this value is called
r.
b
f(x)dx"
(Birkhoff, 1973, p. 22).
After it Riemann gives criterion when function is integrable.
. Secondly, let us determine the extant of the validity of this concept, and ask: in what cases is a
function integrable and in what is not?
We first consider the concept of integral in the strict sense, that is. we assume that the sum S
converges when all the Л, become infinitely small. Thus let us denote the largest variation of the
function between a and x\, that is. the difference of its largest and smallest value in this interval,
by D ?, that between x\ and jtt by Di tnat between x„ _ ? and b by D„; then
Л;0| + ЛтОт ? + Л„0„
must become infinitely small with the quantities Л," (Birkhoff, 1973. p. 22).

20
D. Paunic
After this Riemann constructs his famous example of the function which is integrable
but has an dense set of discontinuities in the set of real numbers.
Riemann's ideas were soon developed in Germany, France and Italy. Especially lucidly
they were developed in France by Gaston Darboux A842-1917). Darboux in his paper
Memoire sur les fonctions discontinue from 1875 analyzed only bounded functions over
[a,b].lf f is bounded function then for every partition ? = xq < x\ < ¦ ¦ ¦ < x„~\ < xn=b
he formed the expressions
M(n) = M\S\ ? h M„S„, m(n)=m\S\-\ \-mn&„,
? = ?\&\ +--- + ?„8„,
where ?,- = jc, — *,¦_ ?, ?,- the least upper bound for / over <5,, w, the greatest lower bound
of / over <$,-, and ? = ?,- - w, the so-called "oscillation" of the function in the interval.
He showed that M(n), m(n), and ?(?) converge to the unique limit which depends only
on a, /), and /, as ? —> oo and <5, —> 0. Later the suggestive notation
/ f(x)dx= lim y^rriiSi and / f(x)dx= lim У^М,-5,-,
Ja_ /=, J a ( = 1
was introduced by V. Volterra in 1881. With this preparation it easily follows that for
Darboux sum
?
S(n,S,e) = YjSif(xi-l+0lSi).
1 = 1
which depend only on ?, 8 = (<S|,.. ..<$„), ? = (?\, ...,??), 0 sC #, sC 1, fori = l,...,n,
holds
m(n)^S(n,S,e)^M(n),
since w,- ^ f(xi-\ + ??&?) ^ Mi. This is his
THEOREM 2. Darboux sums S(n, S, ?) tend to the unique limit iff ?(?) tends to zem or
equivalently if
Jfb rb
I f(x)dx= / f(x)dx.
a J a
In that case Darboux denotes the limit by / f(x) dx. As consequences he proves:
• Every continuous function is integrable in Darboux sense.
• Mapping ? (->¦ f* f{t) dt is continuous in ?.

History of measure theory
21
• Let F(x) = f* f(t) dt. If / is continuous in xo then F'(jto) = /C*o)·
• If the function F, defined over [a,b], has bounded and integrable derivative / = F'
then
F(x)-F(a)=l f(t)dt, for every ? e ]a.b[.
Ja
Similar construction of Riemann's integral were presented in the same year by
J.K. Thomae A840-1921), G. Ascoli A843-1896), P. du Bois-Raymond A831-1889),
and H.J.S. Smith.
Georg Cantor A845-1918) in sequence of papers between 1879 and 1884 investigated
the properties of infinite linear point sets. There he introduced notion of the first species
set, set whose «th derived set is finite, and investigated dense sets, nowhere dense sets etc.
He actually gave these names although these sets were recognized as important earlier.
Most mathematicians of the period were impressed with Riemann's idea of integration
and tried to developed them but soon turned out that knowledge of topology was
insufficient to make much further progress. Dirichlet's example of the characteristic
function of rational numbers
f(x)= lim ( lim cos2"m\nx ),
which is not Riemann integrable prompted the analysis of dense and nowhere dense sets,
but these are topological notions which do not have great influence on measure theory since
dense set (topologically large) can have zero measure and nowhere dense set (topologically
small) can have positive measure.
So H. Hankel, A. Harnack, and R du Bois-Raymond published some interesting
examples but made wrong conclusions. One of their mistakes was that in their proofs it
was used as fact that nowhere dense sets are equivalent to first species sets. Ulisse Dini
A845-1918) showed 1878 that the first species sets are of zero content, so that they cannot
have influence on the integrability of functions.
English mathematician Henry John Stanley Smith A826-1883) was the first who, in
his paper On the integration of discontinuous functions from 1875, constructed nowhere
dense set, he called it "in loose order", of positive measure. But his paper remained without
influence, probably since his main interest was in number theory, and the reviewer noticed
only that in the paper the sharper version of Riemann's integration theory was presented.
Vito Volterra A860-1940) published practically the same construction in 1881 but it also
did not have much influence. When the same fact was discovered independently for the
third time by R du Bois-Raymond in 1880 its importance was finally recognized.
7. Newly found measure
In 1881 appeared the outer content of a set. It was introduced by Otto Stolz A842-1905)
for subsets of an interval and bounded sets of the plane. Independently from him G. Cantor
introduced equivalent definition of content and extended to the и-dimensional space, but

22
D. Paunic
he made no distinction between a set and its closure. In 1885 Harnack proposed his version
content but he was so fascinated with dense sets that he could not admit that dense set could
have measure 0. He introduced infinite covers but had no clear idea of boundary of a set so
his results remained incomplete.
Giuseppe Peano A858-1932) using upper and lower integrals in 1883 introduced most
natural definition of area by inscribed and circumscribed polygons. This theory was
developed in Applicazioni geometriche del calcolo infinitesimale from 1887. In it he
defined that if the greatest lower bound of the area of circumscribed polygons is equal
to the least upper bound for the area of inscribed polygons then the set has an area.
Although Peano introduced the notion of measurable set their importance for the
integration theory was fully recognized by Camille Jordan A838-1922). He studied it
in 1892 in the context of double integrals and applicability of Fubini's theorem, i.e.,
when double integral is equal to the iterated integrals and when the order of integration
of iterated integrals can be interchanged. Jordan remarked that notion of function of two
variables was clear enough but the notions of its domain of definition remained intuitive
and not well-developed. So he carefully analyzed point sets in plane and defined interior,
exterior, boundary, limit points, and closed set. He succeeded to use measurability of sets
to construct double and multiple integrals in the Riemann sense with clarity, generality,
and suggestiveness they never had before. This theory he included in the second edition of
his celebrated Cours d'Analyse from 1893-1896 which was in three volumes.
Further advances in measure theory were made in the work of Emil Borel A871-1956).
In his thesis he studied the problem of analytical continuation, particularly the problem
of analytical continuation over the boundary on which the singularities are dense, and
analyzing them he used theorem which later got the name Heine-Borel theorem. Instead to
approximate the size of bounded sets from outside by covering them by intervals he used
Cantor's result that every open set of the real line is union of denumerable family of disjoint
open intervals he simply took for its measure the sum of the lengths of the components. In
his book on theory of functions from 1898 he writes:
"When a set is formed of all the points comprised in a denumerable infinity of intervals which do
not overlap and have total length s, we shall say that the set has measure s. When two sets do not
have common points and their measure are s and s'. the set obtained by uniting them, that is to say
their sum, has measure s + s'.
More generally, if one has a denumerable infinity of sets which pairwise have no common point
and having measure s |, si, s„ . their sum ... has measure s ? + si + ¦ ¦ ¦ + sn ?
All that is consequence of the definition of measure. Now here are some new definitions: If a set
? has measure s and contains all the points of a set E' of which the measure is s', the set ? — ?',
formed of all points of ? which do not belong to E', will be said to have measure s —s'... .
The sets for which measure can be defined by virtue of the preceding definitions will be termed
measurable sets by us ..." (Hawkins, 1970, pp. 103. 104).
He described the family of sets which is obtained from open sets by infinitely iterating
countable unions and difference and showed that over that family (later called Borel sets)
completely additive measure can be defined, but nowhere in his book the notion of measure
is connected to the integration.
Henry Lebesgue A875-1941) announced his work on measure theory and integration
in five research announcements from June 1899 to April 1901 in Comptes Rendu, journal
of the French Academy of Science. This work is contained in his thesis which appeared

History of measure theory
23
1902. In it he developed Borel's ideas on measure with greater clarity and generality.
He introduced Lebesgue measure, axiomatically as nonnegative function ? defined on
bounded sets of the real line such that
A) Two equal sets have the same measure.
B) The measure of the set which is the sum of finite or countable infinite number of
sets, which are pairwise disjoint, is the sum of measures.
C) The measure of all points in ]0, 1 [ is 1.
After it he introduced Lebesgue integral which is his totally new invention. In order to
define integral Lebesgue partitioned range of the function / instead its domain. If m
denotes the greatest lower bound and the ? the least upper bound of / over [a, b], ?
denote the partition
m = ко < k\ < ¦ ¦ ¦ < fc„_| < k„ = ?,
of the interval [m, M], and
Ei = {xe[a,b]: ?,_, </(*)<*,·}, / = 1,2,... .
Let || ? || denotes the maximum of the differences fc, — fc,_i then
rb def " '<
/ /= lim VV|M(?,)= lim Vfc;M(?,).
J a II1-0^ IIPIHO^
Lebesgue's integral solved a lot of problems in analysis. Many results got their natural
formulation and easy proofs since generalization to any set ? on which the measure is
defined is immediate. In his thesis Lebesgue spent much effort to determine connection
between integral and primitive function. He proved that if /' exists and is bounded on
[a, b], then /' is summable and /д* /' = f(b) — f(a). He gave some conditions in the
case /' is not bounded.
The most useful result in Lebesgue's thesis was that if (/„),1€ц is a sequence of
measurable functions defined on a measurable set ? such that |/«(-*) I ^ В f°r aU x ш
?, and all n, and
if lim f„(x) exists, then / lim /„(*)= lim / f„(x).
In 1908 Lebesgue generalized it and obtained his well-known "Dominated convergence
theorem".
This theorem came at the end of a long search for the solution of the problem whether
it is permissible to integrate series term by term. Cauchy and earlier mathematicians
thought that it is permissible. First counterexample gave Niels Henrik Abel A802-1829),
but he died too young to pursue these ideas further. That additional sufficient condition
is uniform convergence became clear to Karl Weierstrass A815-1897) around 1841, and
he emphasized it in his lectures when he became the professor of mathematics at Berlin
University in 1856. Since it is also the sufficient condition for the continuity of the sum

24
D. Paunic
of continuous functions there was not much research in this direction. That the sum of
a series of continuous function is continuous if the convergence of the series is uniform
was discovered independently by Philip Seidel A821-1896) in 1850 and George Stokes
A819-1903) in 1848. It became clear in the seventies of the XIX century that uniform
boundedness is sufficient with some additional condition to insure integrability. P. du Bois-
Raymond attacked the problem in 1886 and gave complicated and wrong additional
conditions. Finally, the problem was solved by William Fogg Osgood A864-1943) in 1897
and Cesare Arcela A847-1912) in 1885 (Arcela result remained not widely known until
republication in 1900), in a very complicated way and with some additional conditions
which are difficult to check. Arcela's conditions are slightly more general. Lebesgue solved
the problem efficiently, more generally and with easy proof.
Measure-theoretic ideas of Jordan and Borel stirred other mathematicians to further
developed ideas on measure. In Italy Giuseppe Vitali A875-1932) created theory of
measure similar to Lebesgue's but without integration. When in 1904 became acquainted
with Lebesgue's work Vitali improved on it and in 1905 proved that a function is an integral
iff it is absolutely continuous. Using axiom of choice Vitali proved in the same year the
existence of nonmeasurable set on real line, but Lebesgue did not recognize axiom of
choice because it is too idealistic.
William Henry Young A863-1942) was doing research on measure and integration
independently of Lebesgue from 1902 to 1905. He defined the integral based on countable
additive measure which was similar to Lebesgue's but not so general and Young did not
prove all the important consequences which follow from it.
Lebesgue's thesis came in very important moment in development of analysis. In a short
note published in 1900 Ivar Fredholm A866-1927) has introduced his methods for solving
integral equation
f(s) = <p(s)+ / K(s,t)<p{t)dt,
J a
for unknown function ?, which showed that general theory of all integral equations can be
made very simple. This work was completed in his subsequent paper published in 1903.
Fredholm's theory was further developed by David Hubert A862-1943) who published six
papers between 1904 and 1906 on integral equations. The fundamental was the fourth paper
from 1906 where, in modern terminology, he presented spectral theory for bounded linear
operators in Hubert space. Hubert's work on integral equations and Lebesgue's book on
trigonometric series stimulated Fridyes Riesz A880-1956) in 1906 to analyze if (а„)п€щ
is a square summable sequence whether there is a function such that a„ = f^ ??„ for some
complete orthogonal system of square-summable functions (&>„);ieN· Не found that the
answer is positive. The same theorem was independently discovered in the same year by
Ernst Fisher A875-1959). This discovery opened the road to the generalizations which led
to Lp spaces.
F. Riesz also directed the development of analysis in another direction. In 1909 he proved
that Stieltjes integrals / i-> /' / ?? are the most general continuous linear functionals
on C[a,b]. Thomas Jan Stieltjes A856-1894) introduced his integrals in 1894. Stieltjes

History of measure theory
25
studied the convergence problem of continuous fractions and for its solution introduced
generalization of Riemann integral
rb "
/ fd<p = ,J}m У]/(c<)M*<)-?(¦*!-1)).
Ja ||Р||->-ос*—'
i = l
where ? is partition a = xq < x\ < ¦¦¦ < jc„_i < x„ = b, f is continuous, jc, _ ? ^ c, ^ x,.
Stieltjes noticed that his integral could be extended to larger class of functions, "but it
is of no interest to accord complete generality to the function /(к)" (Hawkins, 1970,
p. 181). In 1913 Johann Radon A887-1956) used Lebesgue's procedure to obtain an
integral using arbitrary completely additive set function instead of Lebesgue's measure.
This paved the road to generalization of measure theory to arbitrary sets via ?-rings and to
general representation of measure by Otton Nikodym A887-1974) obtained in 1930.
Translational invariance of Lebesgue integral has generalization with important
application in the representation theory of groups. In 1897 Adolph Hurwitz A859-1919) used
integrals to generate invariants for SO(n) and SU(n). In 1924 Issai Schur A875-1941),
while studying rotation group in «-dimensional space, recognized that much of character
and representation theory of finite groups remains valid if, instead of summation over group
elements, suitable integration is applied over the compact manifold formed by the elements
of the rotation group. Hermann Weyl A885-1955) generalized this work and obtained
explicit expression for irreducible characters of compact Lie groups. This research
culminated in Peter-Weyl theorem, perfect analog of decomposition of the regular characters
in irreducible components for finite groups. It became clear that the Peter-Weyl theorem
could be proved for a class of topological groups for which left invariant measure exists. In
1914 Felix Hausdorff A868-1942) proved that there is no translational invariant positive
finite additive set function defined for all sets in R3. In 1923 Stefan Banach A892-1945)
solved positively the same problem for R and R2, and in 1924 together with Alfred Tarski
A902-1983) gave his famous counterexample for R3 that any two balls B\, Bi can be
decomposed into finite number of congruent sets, so ? ? = (J"= \S,, ?? = U/'= ? d G?)'where
C, are congruent transformations. In 1933 Alfred Haar A885-1933) constructed left
invariant measure on separable locally compact topological group. His result was completed by
John von Neumann A903-1957) who proved its uniqueness for compact groups in 1934
and in general case in 1936, and independently from him by Andre Weil A906-1998).
The great success of measure theory was that it can be used as foundation for the
axiomatization of probability theory, which was done by Andrei Nikolaevich Kolmogoroff
A903-1987) in 1933.
References
Aaboe, A. A964), Episodes from the Early History of Mathematics, Random Hause, New York.
Andersen, K. A985), Cavalieri's method of indivisibles. Arch. Hist. Exact Sci. 31, 291-367.
Baron, M.E. A969), The Origins of the Infinitesimal Calculus, Pergamon Press, Oxford.
Birkhoff, G. (ed.) A973), A Source Book in Classical Analysis, Harvard University Press, Cambridge.
Bourbaki, N. A969), Elements d'Histoire des Mathe'mathques, Hermann, Paris.

26
D. Paunic
Boyer, C.E. A959), The History of the Calculus and its Conceptual Developments, Dover, New York.
Boyer, C.E. and Merzbach, U.C. A989), A History of Mathematics, Wiley, New York.
Edwards, C.H., Jr. A982), The Historical Development of the Calculus, Springer, New York: I st edn 1979.
Elstrodt, J. A996), Mass- und Itegrationstheorie, 2nd edn. Springer, Berlin.
Grattan-Guinness, I. A970), The Development of the Foundations of Mathematical Analysis from Euler to
Riemann, MIT Press, Cambridge.
Grattan-Guinness, I. (ed.) A980), From the Calculus to Set Theory 1630-1910, Duckworth, London.
Grattan-Guinness, I. A997), The Fontana History of Mathematical Sciences, Fontana Press, London.
Hausdorff, F. A914), Grundzuge der Mengenlehre, Veit & Co., Leipzig.
Hawkins, T. A970), Lebesgue's Theory of Integration, University of Wisconsin Press, Madison.
Heath, T. A921), A History of Greek Mathematics I, II, Clarendon Press, Oxford. (Republished in 1981 by Dover.)
Jahnke, H.N. (ed.) A999), Geschichte der Analysis, Spektrum AV, Heidelberg.
Katz, V.J. A998), A History of Mathematics, 2nd edn, Addison-Wesley, Reading.
Knorr, W.R. A975), The Evolution of the Euclidean Elements, Reidel PC, Dordrecht.
Kolmogoroff, A.N. A933), Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer, Berlin.
Lebesgue, H. A904), Legons sur ? Integration et la Recherche des Functions Primitives, Gauthier-Villars, Paris.
Bnd edn, 1927.)
Lebesgue, H. A966), Measure and the Integral, Holden-Day, San Francisco.
Leibniz, G.W. A995), La Naissance du Calcul Differentiel, Vrin, Paris. B6 articles from Acta Eruditorum.)
Medvedev, F.A. A974), Development of the Integral Notion, Nauka. Moscow (in Russian).
Medvedev, F.A. A975), Lectures on the History of the Functions of Real Variable, Nauka, Moscow (in Russian).
Paplauskas, A.B. A966), Trigonometrical series from Euler to Lebesgue, Nauka, Moscow (in Russian).
Pesin, I.N. A966), Development of the Integral Notion, Nauka, Moscow (in Russian).
Pier, J.-P A994), Integration et mesure 1900-1950, J.-P Pier (ed.). Development of Mathematics 1900-1950,
Birkhauser, Basel, 517-564.
Saks, S. A930), Theorie de Vlntegrale, Warszawa.
Serres, M. (ed.) A989), Elements d'Histoire des Sciences, Bordas, Paris. (German translation: Elemente einer
Geschichte der Wissenschaften, Suhrkamp, Frankfurt A998).)
Struik, D.J. (ed.) A969), A Source Book in Mathematics 1200-1800, Harvard University Press, Cambridge.
(Republished in 1986 by Princeton University Press, Princeton.)
van der Waerden, B.L. A961), Science Awakening, Oxford University Press, Oxford.
Weil, A. A940), I'Integration dans les Groupes Topologiques et ses Applications, Hermann, Paris.
Whiteside, D.T. A960-1962), Patterns of mathematical thought in the later 17th century. Arch. Hist. Exact Sci.
1 179-388.
Zaanen, A.C. A967), Integration, North-Holland, Amsterdam.

CHAPTER 2
Some Elements of the Classical Measure Theory
Endre Pap*
Institute of Mathematics, University ofNovi Sad, Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia
E-mail: pap@im.ns.ac.yu, pape@eunet.yu
Contents
Introduction 29
1. Measurable functions 30
1.1. Classes of sets 30
1.2. Set functions 32
1.3. Step functions 34
1.4. Measurable functions 34
2. Measures 35
2.1. Positive measures 35
2.2. Measure spaces. Extension of measures 36
2.3. Completion of a measure space 38
2.4. Measures with finite variation 39
2.5. The variation of real-valued measures on a ring 40
2.6. Measurability with respect to a positive measure 41
2.7. Examples of measures 43
2.8. Lebesgue measure on K1' 47
3. Integration 50
3.1. The immediate integral 51
3.2. Integral of positive step functions 53
3.3. Integral of positive functions 54
3.4. The classical integral 55
3.5. Lebesgue and Riemann integrals 59
3.6. Integration with respect to a real-valued measure 61
3.7. Density theorem 63
3.8. Absolutely continuous functions 63
3.9. Absolute continuity of measures 65
3.10. The Radon-Nikodym theorem 65
3.11. Conditional expectation 67
3.12. The Lebesgue-Stieltjes integral 68
*The author wants to thank for the partial financial support of the Project in the Fields of Basic Research
"Mathematical models of nonlinearity, uncertainty and decision" A866) supported by Ministry of Science,
Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
27

28 ?. Pap
3.13. Transformation of coordinates 68
4. Function spaces 69
4.1. Essentially bounded functions 69
4.2. The space LP (?) 70
4.3. Convergence of sequences of functions 71
5. Product measures and the convolution 72
5.1. Product measures 72
5.2. Repeated (iterated) integral 73
5.3. Infinite products 75
5.4. Convolution 77
5.5. Integral transforms 77
6. Regular measures 79
6.1. Topological measures 79
6.2. Convolution of measures 81
References 81

Some elements of the classical measure theory
29
Introduction
In this chapter we shall give some basic notions and related results from the classical
measure theory in a very condensed form. The main purpose is to make it easier to follow
the whole Handbook.
A measure space is a set in which some subsets may be assigned a measure. The usual
notions of length or area or volume can be applied only to reasonably regular sets. Today
measure theory is remarkably powerful and allows that a large class of sets are adequately
regular to be measured. The class of sets which it measures have to be firstly specified.
The classical abstract integral presented here is closely related to the main particular case
- the Lebesgue integral based on the Lebesgue measure. The standard integral as defined
here is an 'absolute' integral, i.e., if / is integrable so is |/|. This means that although
the Lebesgue integral extends the 'proper' Riemann integral, there are functions with finite
'improper' Riemann integrals which are not Lebesgue integrable.
We want to emphasize that the natural domain of a ?-additive measure is <5-ring or a
?-algebra - but not a ?-ring. In fact, the measurability and the integral involve only sets
of finite measure, which form a <5-ring (or a ?-algebra, if the measure is finite). A finite,
?-additive measure on ?-ring can always be extended to a finite, ?-additive measure on
the ?-algebra generated by the ?-ring.
One of the goals is also the identification of those real functions which are indefinite
integrals. One of the central results of measure theory is relating countably additive
measures to indefinite integrals. The objective is to give a complete description of the
set functions which can arise as indefinite integrals of integrable functions. These can be
characterized as the 'truly continuous' additive set functions (Fremlin, 2000a), or more
commonly used concept, is that of 'absolutely continuous' additive set function.
One of the most important properties of the Lebesgue integral is related to the following
problems. Given an integrable function /: [a, b] —» R, we can form its indefinite integral
F(x) = /ax f(t)dt for ? e [a, b]. Two questions arise:
(a) Can we expect to have the derivative F' of F equal to /?
(b) Can we characterize those functions F which will appear as indefinite integrals?
There are reasonably satisfactory answers for both of these questions: F' = f almost
everywhere and indefinite integrals are the absolutely continuous functions.
We present some important function spaces as L° (the space of equivalence classes
of all real-valued measurable functions, in which all the other spaces of the chapter are
embedded), L1 (equivalence classes of integrable functions), L°° (equivalence classes
of bounded measurable functions) and V (equivalence classes of pth-power-integrable
functions).
One of the most important feature of the concept of product measure is the fact that we
can use it to discuss repeated integrals. We give versions of Fubini's theorem and Tonelli's
theorem.
In the preparation of this auxiliary chapter there were used many different sources. The
author wants to stress the useful Fremlin B000a, 2000b), where one can find the proofs
and additional explanations. I thank N. Dinculeanu for his useful advice.

30
?. Pap
1. Measurable functions
1.1. Classes of sets
1. Throughout the paper, S is a non-empty set, V, 7?, A, V, «S, ? are, respectively, a
semiring, a ring, an algebra, a <5-ring, a ?-ring, a ?-algebra of subsets of S.
A semiring of sets is a non-empty class 73 of subsets of 5" closed under intersection ??) ?
and having the property that for any pair (A, B) of sets from V with А с S. there is a finite
family (Ci)o^i^n of sets from V such that
A = Co С C\ С · · · С C„ = ?
and
C,\C,-ie-p, forlsC;iC«.
An important example of a semiring is the class of intervals of the form ]a,b].
A rwg is a non-empty class of subsets of S closed under union A U В and difference
A\B.
An algebra is a ring containing the whole set S.
A S-ring is a ring closed under countable intersections.
A ?-ring is a ring closed under countable unions. A ?-algebra is a ?-ring containing S.
A monotone class is a non-empty class ? of subsets of S, closed under unions of
increasing sequences and under intersections of decreasing sequences.
Every ring is a semiring; every <5-ring is a ring; every ?-ring is a <5-ring and a monotone
class.
For any class ? of subsets of S there is a smallest semiring (respectively, ring, algebra,
<5-ring, ?-ring, ?-algebra, monotone class) containing T, denoted by sr(T) (respectively,
г{Т),а{Т),&г{^),аг(Т),аа(Т),М{Т)), and called the semiring (respectively, ring,
algebra, <5-ring, ?-ring, ?-algebra, monotone class) generated by ?.
Examples.
(a) The ?-algebra of subsets of S generated by 0 is {0. S].
(b) The ?-algebra of subsets of N generated by {{«}: ? e N) is V(N) (the power set
ofN).
(c) The class V\ (respectively, V\ [Щ) of intervals of the form ]a, b] with a, b e R
(respectively, a, b e R) is a semiring on R (respectively, R).
(c') The class V\ of intervals ]a. b] С R with a. b rational, is a countable semiring on R.
(d) The class Рт (respectively, P2M) of intervals of the form [a, b[ with a, b e R
(respectively, a, fo e R) is a semiring on R (respectively, R).
(d') The class V-, of intervals [a,b[cR with a, b rational, is a countable semiring on R.
(e) If S, ? are non-empty sets and if V, Q are semirings on S, T, respectively, then the
class V ? Qof rectangular sets ? ? ? with A eV and В е Q is a semiring on the
product set S xT.lfV and Q are rings, the class PxQis not necessarily a ring.

Some elements of the classical measure theory
31
(f) If S is a topological space, the ?-algebra generated by the class of open sets is called
the Borel ?-algebra of S and is denoted by B(S). The sets of B(S) are called Borel
subsets of S. The ?-algebra B(S) is also generated by the class of closed subsets
of 5".
(g) Let D с К be a dense set (e.g., the set of rational numbers). The Borel ?-algebra
В(Ш) is generated by each of the following classes: the class of intervals ]a, b] with
a,b e D; the class of intervals [a,b[ with a,b e D; the class of open intervals ]a, b[
with a,b & D; the class of closed intervals [a, b] with a,b e D. In particular, B(R)
is generated by any one of the semirings V\,V\,V^,V~,.
(g') Let D с Ш be a dense set. The ?-algebra В(Ш) is generated by each of the following
classes: the class of intervals ]a, b] with a e D and b e D U {+00}; the class of
intervals [a, b[ with ? e DU{- 00} and/? e D; the class consisting of intervals ]a, b[
with a,b e D and {+00}; the class of intervals [a, fo] with a e D and fo e DL)[+oo].
In particular, B(R) is generated by the semiring V\ [M], or by the semiring PifR].
(h) Let S, ? be two non-empty sets and <S, ? ?-rings (respectively, ?-algebras) on
S, T, respectively. The ?-ring (respectively, ?-algebra) generated by the semiring
of rectangular sets ? ? ? with A e <S and В е T, is called the product ?-ring
(respectively, product ?-algebra) of S and ? and is denoted by S <g> T. When no
confusion is possible, <S ® ? is also denoted S xT.
(h') One can define similarly the ?-ring (respectively the ?-algebra) ® |<,<„ «S, of ?
?-rings (respectively, ?-algebras) 5, on the set 5,-, 1 ^ /' ^ /;.
Properties. If S is a countable class, then the ring r(S) generated by S is countable. The
ring r{V) generated by a semiring V, consists of all finite, disjoint unions of sets from V.
The ?-ring ar{T>) generated by a S-ring T>, consists of all countable, disjoint unions of
sets from V. Let ?? be a ring of subsets of S and (A,),e/ a finite family of sets from 1Z.
Then there is a finite family (Bj)j€j of disjoint sets from TZ such that each set A, is the
union of a certain subfamily of (Bs )j€j.
Let S be a class of subsets of 5" and A e 5". Denote 5ПА = {СПА: Се S]. Then
гEПА) = гE)ПА and агEПЛ) = агE)ЛЛ.
Definition. Let (?„)„ем be a sequence from T. Then we define
DC DC DC DC
limsup?„ = p| [J Ek and liminf?„ = [J P| ?ь
n=\k=n n=\ k=n
whenever the right parts are meaningful. If
lim sup ?n = lim inf E„
? »
holds we denote this set by lim,, E„.
MONOTONE CLASS THEOREM. Let 1 be a class of subsets ofS such that ? ? J e 1 for
all I, J el. Let M. с V(S) be a family of sets (monotone class) such that

32 ?. Pap
(a) SeM,
(b) ? \ A e ? whenever А, В e ? and А с ?,
(c) UneN ^« e -^ w/?e/iever (A„),ie[i й ? non-decreasing sequence in M,
(d) I CM.
Then ?4 includes the ?-algebra of subsets of S generated by I.
Specially, let TZ be a ring on S. The monotone class M(TZ) generated by TZ is equal to
the ?-ring ??(??) generated by TZ.
EXAMPLE (Dynkin system). A family ? с V(S) is called a Dynkin system on S if
(a) SeT,
(b) Ac e ? whenever Aef,
(c) UneN A" e -F whenever (А„)„е^ is a sequence of pairwise disjoint sets in T.
A Dynkin system is a monotone class.
For any class ?, we denote by Т\ж the class of subsets А с S that are locally in T. i.e.,
such that А П ? e J" for every В e T. If 4 is a ring, 7?.|OC is an algebra. If ? is a <5-ring
and <S is a ?-ring, then T>\oc and <S|0C are ?-algebras.
The characteristic function of a set А с S is denoted by хд, 1 д or /д.
1.2. Set functions
Let J" be a class of subsets of a set 5 with 0 e f. We give now a list of basic notions
related to the set functions.
2. Definition. A function m:T -> [-oo, +oo] is called extended real-valued set
function. We say that (we suppose further on that w@) = 0)
A) m is (finite) real-valued set function if —oo < m{A) < +oo for all Aef.
B) m is monotone if А С ? implies m( A) ^m(B) for every А, В е Т.
C) m is normalized if m(S) = 1 for S e .F.
D) w is non-negative if it is finite and m(A) ^ 0 for all A ef.
E) w : ? -> [0, +oo] is positive.
F) w : 7? -> R is supermodular (convex) if for every ?, ? e Я we have
m(AL)B)+m(AnB) ^m(A) + m(B).
G) w : 4 -> R is superadditive if for every А.ВеЯ with ? ? ? = 0 we have
w(AUfi) ^m(A) + m(fi).
(8) m : TZ -> R is submodular (concave) if for every ?, ? e TZ we have
»i(AUB)+m(Anfi)^m(A)+m(fi).
(9) m : 72. -> R is subadditive if for every А, В eTZ with ? ? ? = 0 we have
w(AUfi) iCm(A)+m(fi).

Some elements of the classical measure theory 33
A0) т:И-+ [0,+oo[ is totally monotone if for every ? e N and A\,...,A„ e 7? we
have
-(UA·)^ ? (-1)|/|+|-(ПА<)·
\?=? / {/:/С{1 и)) Me/ '
A1) т : .F -> ? is additive (signed finitely additive measure) if we have
m(AUS) = m(A)+m(S),
for all A,B eF with ? U ? e ? and ? ? ? = 0. Specially for w : 11 -> [0, ooL
and 5 = U^eT^ ^ we ca^ m a^so content.
A2) m :T —* ? is finitely additive if we have
In \ ?
U?i)=5>(?i),
w ?
for every finite family of pairwise disjoint sequences (?j)"=i from .F, i.e.,
?„ П ?m = 0 for и ^ w, and (J"_, ?, e J".
A3) w : ? -> ? is ?-additive (signed measure) if we have
oo
m I
u = l / i = l
|j?, )=][>(?,¦),
for all pairwise disjoint sequences (?,-),-efj from F, i.e., ?„ ? ?„, = 0 for /; ? m,
andU",?ie^·
A4) m :?? -> ? is a measure if it is nonnegative and ?-additive. E, .?7, ?), is aproba-
bility space if ?E) = 1. In this case ? is called a probability or probability
measure.
A5) w: ? -> ? is continuous from below if for every sequence (?„),ieN such that
?i С Е2 С ···,?„ e ?, ?^=? ?n e F we have
/ ОС
m I
[JE„ I = lim m(E„).
/1—>OC
Mi = l /
A6) w:? -> ? is continuous from above if for every sequence (?„)„eN from ?
such that ?| Э ?2 Э ···,?„ e 7?, ?,%? ?» e -^ and tnere exists  such tnat
\m(E„0)\ < 00 we have
/ 00
m I
ОБ,, ) = lim w(?„).
/1—>DC
wi = l /

34
?. Pap
A7) m is exhaustive (s-bounded) if lim,,^^ m(E„) = 0 for every sequence (E„)n€n of
pairwise disjoint sets from T.
A8) A positive set function m is called ?-finiteon ? if for every set ? e ? there exists
a sequence (E„ )„€щ such that
DC
[J En = ? and m(E„) < do, for every /; e N.
»=i
m:T -* [0, oo] for S e ? is totally finite if /n(S) < oo.
1.3. Step functions
3. Let J" be a non-empty class of subsets of S and F a vector space (for example, E",
C, E). We denote by Sf(!F ) the vector space of JF-step functions /: 5 -> F of the form
/ = ?"=l xAjXi, with A, e J" and x, e F. If F = ? we write S(T) instead of <S;r(.F).
Remark. There is no need to consider step functions with values in E.
Any constant function is an .F-step function. We remark that if V is a semiring and
Tl = r(V) is the ring generated by V, then Sr(V)= S/-(K).
If ? is a ring, in the definition of .F-step functions the sets A, can be chosen mutually
disjoint.
LEMMA. Let H be a ring and f: S -> F an TZ-step function f = ???? ??,?" with
A, e TZ and Xj e F. Then f can be written in the form f = J2jeJ XS/-vy· w'1^ J finite,
Bj e TZ mutually disjoint and v, e F.
Moreover f can be written uniquely in the form f = ^2j€j ??,??- w'th )'j ? 0 for
every j e J and y-; ? y^ for j ? к in J.
Then
?/? = ???,?>??·
where for any function /: 5 -> F or E, we denote by |/| the function defined by
|/|(?)=|/(?)|, fori e S.
1.4. Measurable functions
Measurability is defined with respect to a ?-algebra. We shall consider, mainly, measura-
bility of real-valued functions, with values in E.
Let ? be a ?-algebra of subsets of S. The pair (S. ?) is called a measurable space. The
elements of ? are called ?-measurable sets. In the sequel, E". ?) is a measurable space.

Some elements of the classical measure theory
35
4. Definition. A function /: 5" -> ? is said to be ?-measurable, if for every Borel set
В С ? we have/-1 (A) e Г.
If S is a topological space and Г = ^E), the Borel ?-algebra of S, a ?-measurable
function /: S -> ? is called a Borel function. Any continuous function /: 5 -> ? is a
Borel function.
Examples.
(a) Any constant function / :S -> ? is .^-measurable.
(b) If А с 5, then the characteristic function хд : S -> ? is ?-measurable if and only
if Л е 2Г.
(c) Every iT-step function /: S -> ? is .^-measurable.
5. THEOREM. Let f :S -*Ж be ? ?-measurable function. Then there is a sequence
(/«)neN of ?-step functions f„ : S -> ? .ягс/г ?/??? /„ -> f pointwise and |/„| ^ |/|/or
еас/г и.
//¦/ й positive (with values in [0, +oo]), ?/ге sequence (/„)„е[,- саи foe chosen to be
increasing.
If f is real-valued and bounded, the sequence (/„)„ек can foe chosen to be uniformly
convergent.
The following theorem gives a characterization of iT-measurability.
.T-measurability is preserved by pointwise convergence:
6. THEOREM. //(/„)//eN is a sequence of Ж valued, ? -measurable functions converging
pointwise to a function f, then f is also ? -measurable.
2. Measures
2.1. Positive measures
7. A positive measure m:TZ-* [0, +oo] has the following additional properties:
If ? is positive and additive on TZ, then ? is increasing.
If ? is positive and additive on TZ, then ? is subadditive.
If ? is positive and ?-additive on TZ, then ? is ?-subadditive, i.e., for every sequence
(A„ )neN °f sets from TZ with union in TZ, we have
?? |JA„ U]TM(A„).
\/ieN / neN
Conversely, if ? is positive, additive and ?-subadditive on TZ, then ? is ?-additive.
A positive additive measure ?: TZ -> [0. +oo] on a ring 72. has the ^ш;'7е measure
property (FMP) on TZ, if for every set A e TZ we have
?(?) = 5??{?(?): Ве7гПА, ?(?) < oo).

36
?. Pap
If ? does not have the FMP on Tl, we can replace it with the measure ? : TL -> [0, +00]
defined for every set A e 7? by
m'(A) = sup{m(B): ? е7гп ?. ?(?) < oo}.
Then ? has the FMP and ? (A) = ?(?) if ?(?) < oo. If ? is ?-additive then ? is ?-
additive.
2.2. Measure spaces. Extension of measures
8. The framework for this section is a measure space (S. ?, ?), consisting of a
nonempty set 5", a ?-algebra ? of subsets of 5" and a positive, ?-additive measure, with finite
or infinite values, ?: ? -> [0, +oo], i.e., such that
(a) ?@) = ?;
(b) if (?n)„eN is a disjoint sequence in ?. then
Members of ? are called measurable sets, and ? is called a measure on S. ?, or (S1. ?1, ?),
is totally finite if ?E) < oo.
Remark. The '00 of measure' corresponds to the notion of infinite length or area or
volume. The basic operation we need is addition: ?? + ? = ? + ?? = ?? for every
a e [0, oof, and oc + 00 = 00. This makes [0, 00] a semigroup under addition. We take
oc¦ — a = 00 for every a e E; but we avoid to interpret the formula 00 — 00. As for
multiplication, it turns out that it is usually right to interpret 00 · 00, a ¦ 00 and 00 · a
as 00 for a > 0, while 0 · 00 = 00 · 0 can generally be taken as 0.
9. Example. Let 5" be a set. If /: 5" -> [0. 00] is any function, then for every ?CS
we define ?(?) = ??^? /(*) (taking Х]хец/<*) = 0), discrete measure. For infinite sets
? we have X](e?/U) = sup{]Tve/ /(¦*): / <? ? is finite), because every f(x) is non-
negative. (S, V(S), ?) is a measure space. A special case of the above is when f(x) = 1
for every ? e S. Then ?(?) is the number of points of ? if ? is finite, and is 00 if ? is
infinite. We will call this the counting measure on S. Another special case is with S = N,
/(л) = 2-" for every и. Then ?E") = ?,^?2"" = 1.
10. Our purpose is to extend a positive, ?-additive measure ? from a semiring V to
the ?-algebra ? = aa(V) and to obtain a measure space E, ? .?). This will be done in
several steps. We consider first extensions of an additive measure from a semiring to the
ring generated by it.
THEOREM. Let V be a semiring, ?? = r(P) the ring generated by V and m:V —* F
or [0, +00] a finitely additive measure. Then m can be extended uniquely to an additive
measure in' :1?—> F or [0. +00]. Ifm is ?-additive on V. then m is ?-additive on TL.

Some elements of the classical measure theory
37
Example. Let /: R -> F (or R) be a function, and V\ be the semiring of the intervals
]a, b] с R. Define the additive measure m : V\ -> F (or R) by m]a, b] = /(b) - /(a), for
]?,?)]?7?. Then m can be extended to an additive measure m': 1Z -> F (or R) on the ring
П = г(Р\) generated by P,.
In this example we can replace V\ by the semiring Vi of the intervals [a, b[ с R.
11. Next, we consider extensions of a positive ?-additive measure from a <5-ring to the
?-ring generated by it.
THEOREM. Let V be a S-ring, S = ar{V) the ?-ring generated by V and m : V ->
[0, +oo] a ?-additive measure. Then m can be extended uniquely to a ?-additive measure
m': <S -> [0, +oo]. /fm is finite and bounded on V, then m is finite and bounded on S.
12. The most important step of the extension is that from a ring to a ?-ring. We consider
first the uniqueness of such an extension.
THEOREM. Let TZ be a ring, S = or(R) the ?-ring generated by 1Z. Let m,n:S ->
[0, +oo] be two ?-additive (finite) measures. Ifm = ? on ??, then m=n on S.
The above theorem remains valid if we replace S by the <5-ring V = 8r(lZ) generated by
П.
13. Finally we consider the extension of a positive, ?-additive measure from ?-ring to
a ?-algebra.
Let again ?? bearing, S = ог(Я) the ?-ring generated by TZand ? =??A?) = ??E)
the ?-algebra generated by Tl or S. Let ?: TZ -> [0. +oo] be a ?-additive measure and
?': S -> [0, +00], the ?-additive extension of ? stated in Theorem in 12.
In most cases, ? is finite on 7Z, hence, in this case, the extension ? is unique.
If S is not a ?-algebra, we consider a second ?-additive extension ?": ? —> [0, +oo].
Namely, for every set A e ? we define
?"(?) = $\}?{?'(?): BeS, В С A}.
The measure ?" is not necessarily the only ?-additive extension of ?' to ?, even if ?' is
finite; but ?" is the smallest among all these extensions.
14. Example. Let S— {a, b], a set consisting of two elements, S = {0,{a}} and ? : S ^
[0, oof defined by ?'{?) = 1. The ?-algebra ? =aa(S) is ? = {0. {a}, {b}, S]. For any
a e [0, +oo[ the measure ? : ? -> R defined by ?{?) = ?{?] = 1 and v{fo} = a, is a ?-
additive extension of ?'. In this example ?"{^} = 0.
15. To distinguish ?" from other ?-additive extensions of ?', we call ?" the canonical
extension of ?' (and of ?). In the sequel we shall continue to denote ?' and ?" by ?. We
obtained a measure space (S, ?, ?).

38
?. Pap
The idea of the 'outer' measure of a set A is that it should be some kind of upper bound
for the possible measure of A. It can happen that this is just the measure of A; but this is
likely to be true only for sets with adequately smooth boundaries.
16. DEFINITION. Let S be a set. An outer measure on S is a function ? : V(S) -> [0, oo]
such that
(a) 4@) = 0,
(b) if А С ? then ? (A) iC ?(?),
(c) for every sequence (Е„)п€н of subsets of S, we have v({Jll€r:E„) ^ ?^=??(?„)-
17. THEOREM (Caratheodory's method). Let S be a set and ? an outer measure on S. Let
? = {E: ?CX, i)(A) = /)(An?) + f)(A\?)/oreven-ACS).
Then ? is ? ?-algebra of subsets of S. If ? : ? -> [0. oo] is defined by ?(?) = ?(?) for
? e ?, then (S, ?, ?) is a measure space.
2.3. Completion of a measure space
Let (S, ?, ?) be a measure space.
18. We define first the ?-negligible sets. A set А С S is said to be ?-negligible (shortly
negligible) if there is a set В e ? with А С В and ?(?) = 0. A set А С 5" is ?-conegligible
(shortly conegligible) if S \ A is ?-negligible.
The following properties are immediate:
19. Every subset of a ?-negligible set is ?-negligible.
20. A countable union of ?-negligible sets is ?-negligible.
21. The measure space (?,?,?) is said to be complete if ? contains all the ?-
negligible sets.
If the measure space (S. ?, ?) is not complete, we can replace it with the complete
measure space (S, ?(?). ?), which is obtained in the following way: ?(?) is the ?-
algebra generated by ? and all the ?-negligible sets. ?(?) consists of all the sets of
the form A = В U N with В e ? and N a ?-negligible set; we can take В and N disjoint.
? (?) consists also of the sets of the form A = B\N with В e ? and N. a ?-negligible set.
A set A belongs to ? (?) iff there are sets B.C &? with ? с А с С and ?{0 \ ?) = 0.
From the definition of ?(?) it follows that a set А с 5" belongs to ?(?), iff there is a
set ? e ? such that the symmetric difference AAB - (A \ ?) U (? \ A) is ?-negligible.
Then we extend the measure ? from ? to ?(?) and we denote the extension by the same
letter ?. Namely, for every set A = В U N with В e ? and N a ?-negligible set, we set
?(?) = ?(?).

Some elements of the classical measure theory
39
It follows that if N is a ?-negligible set, then ?(?) — 0. The definition of ?( A) depends
only on A.
The measure space (S, ?(?),?) is called the completion of the measure space
(S, ?, ?). The sets of ?(?) are called ?-measurableor simply measurable if it is no doubt
with respect to ?. We denote by ?/ (?) the <5-ring of the sets A e ?(?) with ?(?) < oo.
The sets of ?/ (?) are called ?-integrable.
22. THEOREM. Let TZbea ring generating the ? -algebra ?. Then for every set A e ?(?)
and TZ is a ring generating ?(?). Then TZ is dense in ?(?) for the semidistance
p(A, ?) = ?(???),^?, В е ?(?).
This, in turn, is used to prove the extension theorem for real-valued or vector-valued
?-additive measures m, absolutely continuous with respect to a finite, positive, ?-additive
measure ?.
The above theorem is also used to prove that the set of 7^-step functions is dense in the
space L' (?) of ?-integrable functions.
2.4. Measures with finite variation
Let V be a semiring of subsets of S (in particular, V can be a ring, a ?-ring or a ?-algebra).
Let ? be a Banach space and m : V —> ? (or R) a finitely additive measure.
We shall define the variation of the vector-valued measure m, although, in this chapter,
we shall be concerned especially with real-valued measures.
23. DEFINITION. For every set А с S we define the variation |w|(A) of m on A, by the
following equality
|w|(A) = sup/]w(A,-)|.
where the supremum is taken for all finite families (A, ),e/ of disjoint sets from V contained
in A.
If V is a ring and A e V, in the above definition we can take the supremum for all finite
families (A,),e/ of disjoint sets from V with union equal to A.
We say m has finite (respectively, bounded) variation if |w|(A) < oo for every A eV
(respectively, \m\(S) < oo).
Properties.
24. (K |w |(A) ^+oo, and |w(A)K \m\(A), for A e V.
25. |m|(A) = Oifandonlyifm(B)=0, for every set В e V with В С A.
26. |/я | is finitely additive on V.

40
?. Pap
27. If m is ? -additive on V, then | m | is ? -additive on V.
28. EXTENSION THEOREM. Let m :TZ-* ? {or Ж) be ? ?-additive measure with finite
variation \m\ on a ring TZ and let V — Sr(TZ) be the ?-ring generated by TZ. Then m can
be extended to ? ?-additive measure m' :T> —> ? with finite variation \m'\ and \m'\ is the
unique ?-additive extension of\m\ from TZ to V.
Ifm has bounded variation \m \ on TZ, then m can be extended uniquely to ? ?-additive
measure m : ? -> ? (or ?) with bounded variation \m'\ on the ?-algebra ? = ar(TZ)
generated by TZ, such that \m'\ is the canonical extension of\m\.
2.5. The variation of real-valued measures on a ring
Let TZ be a ring of subsets of S and m : TZ -> ? an additive measure.
29. Ifm ^0, then \m\(A) =m(A). for A eTZ.
30. THEOREM. For every set Ac S we have
sup |w(S)| ^ |w|(A) ^ 2 sup |w(B)|.
BcA.BtTl BcA.BzTl
Ifm is complex valued, we replace 2 by 4.
31. m has finite (respectively, bounded) variation \m\, ifandonlyifw is locally bounded
(respectively, bounded) on TZ. We say m is locally bounded on TZ if for every set A eTZ,
m is bounded on TZ ? A.
32. Assume m has finite variation \m\. We define the positive part m+ of m and the
negative part m~ of m by:
m+ = -(|m|+m), m~ = -(\m\— m).
Then m+ and m~ are positive, additive measures on TZ. If m is ?-additive, then \m |, m+
and m~ are also ?-additive on 7?. We have
w=w+—w~ and \m\= m+ + m~.
33. Every ?-additive, real-valued measure on a <5-ring is locally bounded and has finite
variation. Every ?-additive, real-valued measure on a ?-ring is bounded and has bounded
variation.
34. THEOREM (Hahn decomposition). Let S be a set, ? a ?-algebra of subsets ofS, and
m: ? -> ? ? ?-additive measure. Then m is bounded and there is a set ? e ? such that
m(F) ^0, whenever F e ? and F с Р,

Some elements of the classical measure theory
41
m(F) ^ 0, whenever F e ? and F П ? = 0.
35. PROPOSITION (Jordan decomposition). Let S be a set, ? ? ?-algebra of subsets ofS,
and m: ? —» ? a real-valued measure. Then m can be expressed as the difference of two
finite measures with domain ?.
36. Let m : V -> ? be a ?-additive measure with finite variation \m\ on a <5-ring P. Let
5 = ar{V) be the ?-ring generated by V and ?1 = ??(?>) the ?-algebra generated by V.
Consider the ?-additive, positive measures \m\, m+, m~ on V. They have canonical
extensions to ?1, denoted by the same letters. We still have |w| =w++w~.
2.6. Measurability with respect to a positive measure
Let E, ?, ?) be a measure space and ?(?) the completion of ? with respect to ?.
37. We say that a property P(s) defined for every s e S is true ?-almost everywhere
(?-a.e.) if the set of points s e S for which P(s) is false is ?-negligible.
38. A function / : S -> ? is said to be ?-negligible is /(s) = 0, ?-a.e.
39. A function /: S -> ? is called ?-measurable, if it is ? (?)-measurable, i.e., if for
every Borel set В с! we have f~](B) e ?(?).
The i7^)-step functions are also called ?-measurable step functions.
40. A function /: S -> ? is ?-measurable iff there is a .^-measurable function g:S^>
? such that f = g, ?-a.e. If /| = /?, ?-a.e. and if f\ is ?-measurable, then /2 is ?-
measurable. It follows that / is ?-measurable iff there is a sequence (/n)„eN of iT-step
functions such that /„ -> /, ?-a.e.
41. We extend the ?-measurability for functions defined ?-a.e. An ?-valued function
defined ?-a.e. on S is said to the ?-measurable, if it has a ?-measurable extension
g : S -> ? on the whole space S. Then any extension of / to S is ?-measurable.
42. PROPOSITION. Le? (S, ?, ?) foe ? measurable space, and D a subset of S. If ?? is
the subspace ?-algebra of subsets of D, then for any function f: D —> ? the following
assertions are equivalent, in the sense that if one of them is true so are all the others:
(a) {x: f\x) < a] e ?D for even,· a e E;
(b) {x: f(x) ^ a] e ?? far every a el;
(c) {*: f(x) > a] & ?? for every a e E;
(d) {jc: /U)^a}e ?? for every a e E.
DEHNITION. Let Z) be a subset of S. A function /: D -> ? is called measurable (or
?-measurable) if it satisfies any of the conditions (a)-(d) of the preceding proposition.

42
?. Pap
43. THEOREM. Let f and g be real-valued functions defined on domains Dom/,
Dom g с S.
(a) If f is constant it is measurable.
(b) If f and g are measurable, so are f + g, f ¦ g, f/g, where (/ + g)(x) = f(x) +
g(x), (/ · g)(x) = f{x) ¦ g(x), (f/g)(x) = f(x)/g(x)forx e Dom /ODomg and
additionally g(x) ? Ofor f/g.
(c) If f is measurable and eel, then cf is measurable, where (cf)(x) — с ¦ f(x)for
? e Dom/.
(d) If f is measurable and ? Cl is a Borel set, then there is an F e ? such that
f~\E) = {x: f(x)e E) is equal to FODom/.
(e) If f is measurable and h is a Borel measurable function from a subset Dom/? of Ж
to R, then hf is measurable, where (hf)(x) = h(f(x))for ? e Dom(/?/) = {v: у e
Dom/, /(у) е Dom/?}.
(f) If f is measurable and A is any set, then /|д is measurable, where Оот(/|д) =
А ПDom/ and (f\A)(x) = f(x)forx e А П Dom/.
44. The main point of the theory of measure and integration, since Lebesgue, is that we
can deal with limits of sequences of functions, and the set on which lim„^oc fn(x) exists
can be irregular, even for a well-behaved functions /„, see Fremlin B000a, Vol. 1).
THEOREM. Let (/„)„eN be a sequence of ?-measurable real-valued functions with
domains contained in S.
(a) Define a function lim,,^^ /, by (lim,,^^ fn){x) = \\mn^oc fn(x) for all those
? e Ц[екПт>иО°т/и for which the limit exists in R. Then lim,!en /, is ?-
measurable.
(b) Define a function supneN /, by (sup„et:/»)(·*) = sup„et;/»(-*) for all those
? e P|neNDom/„ for which the supremum exists in R. Then sup„et;/« '¦* ?-
measurable.
(c) Define a function inf„eK /„ by writing (inf„e;; /„)(*) = inf„ei>; fn(x) for all those
? e P|,!eNDom/„ for which the infimum exists in R. Then inf„ef; fn is ?-
measurable.
(d) Define a function limsup,,^^/,, by (limsup,,^^ fn)(x) = Hmsup,,^^ fn(x)
for all those ? e U„eN Пт>„ Dom/,,, for which the limsup exists in R. Then
limsup,!e^/„ is ?-measurable.
(e) Define a function liminf,,^^ /„ by (liminf,,^^ f„)(x) = liminf,,^^ fn(x) for all
those ? & UneN Пт>и Dom /„, for which the lim inf exists in R. Then lim inf„e^ /,
is ?-measurable.
PROPOSITION. Let f, g and /„, for ? e N, be ?-measurable real-valued functions whose
domains belong to ?. Then all the functions
f + g, f-g, f/g< sup/,, inf/,, lim /„, limsup/,, lim inf/„
,,6f: net·. n — -x. n^-oc и-»зс
have domains in ?. If h is a Borel measurable real-valued function defined on a Borel
subset of Ш, then Dom/?/ e ?.

Some elements of the classical measure theory
43
45. For two ?-measurable functions / and g we write / ~ g if / = g a.e. Then ~ is an
equivalence relation in the set of all ?-measurable functions ?° and we shall write L , or
L°(m), for the set of equivalence classes. We denote by the same symbol / the equivalence
class corresponding to the measurable function /. The set L° is a vector space with respect
to the naturally transferred addition and multiplication with constant of functions.
Let / be a measurable real-valued function defined almost everywhere on Er. Then
there is a sequence (/n)„eN of continuous functions converging to / almost everywhere.
This is a version of Lusin 's theorem.
46. DEnNlTlON. Let (S, ?, ?) be a measure space.
(a) ?, or (S, ?, ?), is strictly localizable or decomposable if there is a disjoint family
(Xi)iel of measurable sets of finite measure such that S = U/e/ ^' ??(^
? = {E: ? с X, ? ? ?,¦ е ?, for every i e /},
?(?)=^?(???,), for every E e ?.
;e/
Such a family (X/),e/ is called a decomposition of 5.
(b) ?, or (S, ?, ?), is semifinite if whenever ? e ? and ?(?) = oo there is an F с Е
such that Fel and 0 < ?^) < oo.
(c) ?, or E, ?, ?), is localizable or Maharam if it is semifinite and, for every fcr,
there is an // e ?1 such that
(i) E\H is negligible for every ? eJ7,
(ii) if G e ? and ? \ G is negligible for every F e JF, then // \ G is negligible.
We call such a set // an essential supremum of .F in ?.
(d) ?, or (S, ?, ?), is locally determined if it is semifinite and
? = {?: F с 5, EOF e ? whenever F e ? and m(F) < oo},
i.e., for any F e ?E) \ ? there is an F e ? such that ?^) < oo and F ? F ? X\
(e) A set F e ?1 is an atom for ? if ?^) > 0 and whenever F e ?, F ? ? one of F,
F \ F is negligible, ?, or E, ?1, ?), is atomless if there are no atoms for ?.
(f) ?, or (?, ?, ?), is purely atomic if whenever ?еГ and F is not negligible there
is an atom for ? included in F.
2.7. Examples of measures
47. 77?e counting measure. Let 5 be an uncountable set, e.g., S = Ш, and let ? to be
counting measure on S. Then ? is ?-additive, complete, it is not ?-finite, it is not a
probability measure nor totally finite, it is strictly localizable, and ? is purely atomic.
Obviously, ? is not atomless.
48. The countable-cocountable measure. Let S be any set. Let ? be the family of those
sets ?CS such that either F or S \ ? is countable. Then ? is a ? -algebra of subsets

44
?. Pap
of S. ? is called the countable-cocountable ?-algebra of S. Let ? : ? -> {0, 1} be defined
by ?(?) = 0 if ? is countable, ?(?) = 1 if ? is not countable. Then ? is a ?-additive
measure, ? is called the countable-cocountable measure on S. If S is any uncountable set
and ? is the countable-cocountable measure on S, then ? is a complete, purely atomic
probability measure.
49. The Stieltjes measure. Let ? e R and / an interval, closed to the left, of the form
[a, b[ with a < b sC +00 or of the form [a, b], with a < b < +00. We denote by V[I] the
semiring of the subintervals of/, of the form [а, r],or]s, t] with ? < s. The ring generated
by V[I] consists of finite, disjoint unions of sets from V[I]. The semiring V[I] generates
the Borel ?-algebra ??(/).
DEnNlTlON (Variation of a function). Let / be a real-valued function and D a subset
of E. VarD(/), the variation of f on D, is defined by: if D ODom/ = 0, VarD(/) = 0,
and otherwise, Varo(/) is
sup|^|/(a,) - /(a,_|)|: a0,a\,... ,an e DnDom/. a() ^<3| ^ · · · ^ an \,
allowing VarD(/) = 00. If VarD(/) is finite, we say that / is of bounded variation on D.
We write also Var / for VarD0m f(f).
The basic properties of the variation are collected in the following (see, e.g., Fremlin
B000b), Vol. 2).
Proposition.
(a) If f, g are two real-valued functions then
VarD(/ + g) <: Var0(/) + VarD(S).
(b) If с e ? then Varp(c/) = |c| VarD(/).
(c) Ifx e ? then
Var0(/) ^ Varoni-ac.A-ii/) +??·?)?[?.?[(/),
with equality ifx e D ? Dom /.
(d) //?> cD'cl г/геи VarD(/) ^ Var/r(/).
(e) |/(jc)-/()')I ^ VarD(/)/ora//.v, ? e DODom/. Consequently if f is ofbounded
variation on D then f is bounded on D ? Dom / a/id (if D ?? Dom / ^ 0)
sup |/(v)| si |/(.v)|+Var/>(/)
veDnDom/
/or ever}' ? e D ? Dom /.

Some elements of the classical measure theory
45
(f) If f is monotone and Del meets Dom /, then
VarD(/) = sup /(*)- inf f(x).
.teDnDom/ .veDnDom/
THEOREM. The following are equivalent:
(a) 77?ere are fwo bounded non-decreasing functions f\, /2: ? -> R skc/? ?/??? / =
/? - /2 оя ?) ? Dom/;
(b) / is of bounded variation on D;
(c) There are bounded non-decreasing functions f\, /2 : R -> R smc/? rftaf / = /? — /2
ои ?) П Dom / a/id" VarD(/) = Var/, + Var/2.
Let g : I -> R be a function. We assume g is r/'g/?i continuous at every point ? ^ a and
that g has finite variation Vary (g) on every bounded interval J. The variation function of g
is the function \g\:I -> R+ defined by |g|@ = Var[a.,j(g),forf e /.Then |g| is increasing
and right continuous at every point t ? a.
To the function g we associate the finitely additive measure mg : V[I] -> R+ defined by
mg[a,t]=g(t) - g(a)
and
mg]s,t]=g(t)-g(s), ifa<s.
The measure mg is positive if and only if g is increasing.
Then to the variation function |g| we associate the positive measure m\g\, and we have
\mg(A)\^m{g{(A), for A e ИЛ·
Since |g| is right continuous at every t ? a, the measure m\g\ is ?-additive on V[I] and
can be extended uniquely to a ?-additive, positive, finite measure on the ring TZ(I), still
denoted by m\g\.
From the above inequality, it follows that тц is ?-additive and with finite variation
\mg\ ?; m\g\ on V[I], hence mg can be extended to a ?-additive measure on the ring7?(/),
still denoted by mg, and we still have
\mg\(A)^mlg](A), for A eTZ(I).
In fact, we have the equality \mg\ =m\g\.
The positive measure m\g\ can be extended uniquely to a ?-additive measure ?: ??(/) ->
[0,+oo]. Moreover, ? is finite on the <5-ring T>(I) of bounded subsets of /. It follows
that the measure mg can be extended to a ?-additive measure m :T>(I) -> R with finite
variation \m\ = ?.
If g has bounded variation /, then |g| is bounded on /, the measure m\g\ is bounded on
TZ(I) and its extension ? is bounded on ??(/). It follows then that mg can be extended to

46
?. Pap
a ?-additive, finite measure m : B(I) -> R on the Borel ?-algebra, with bounded variation
\m\ = ?.
We shall continue to denote ? by /И|<,| and m by m4. Then we still have
\mg\(A)=mlgl(A), forAeP(Z),
or for A e ??(/), if g has bounded variation. The ?-additive measure ms on D(/) or B(I)
is called the Stieltjes measure on / associated to the function g.
50. The Lebesgue measure. The Lebesgue measure on / is the measure mg
corresponding to the continuous, increasing function g(s) = s, for s e /. In this case we have
mg]s, t] = mg[s, t] = mg[s, t[=m4]s, t[= t — s for every s < t in /.
51. Let —oo ^ a < +oo and consider an interval /, open to the left, of the form ]a, b]
with a < b ? +oo or of the form ]a,b[, with a < b < +oo. In particular we can have
/ = ]-oo, +oo[. This time we denote by V[I] the semiring of the subintervals of /, of the
form ]s, t] with a < s. We still denote by Щ1) the ring generated by V[I]. The semiring
V[I] generates the Borel ?-algebra B(I).
Let g: / —> R be a function. We assume g is right continuous at every point t e / and
that g has finite variation Var/(g) on every interval ]a,t] С /. The variation function of g
is the function \g\:I -> R+ defined by \g\(t) = Varj(,.,j(g),forr e /.Then |g| is increasing
and right continuous at every point t ??.
To the function g we associate the finitely additive measure m4: V[I] -> R defined by
mg]s,t]=g(t) - g(s).
The measure m? is positive if and only if g is increasing. Then to the variation function
|g| we associate the positive measure т\ц\. As in the previous case, the measure т\я\ is
?-additive, mg is ?-additive, with finite variation |/я?| = m\g\. We can extend m\g\ to
a ?-additive ?:??(?) -> [0.+oo] and then we can extend mg to a ?-additive measure
m : T>(I) -> R with finite variation |w | = ?. If g has bounded variation on /, then mg can
be extended to a ?-additive measure m : B(I) -^Ron the Borel ?-algebra, with bounded
variation \m \ = ?. As in the previous case, we continue to denote ? by W|?| and m by mg.
The ?-additive measure mg on T>(I) or B(I) is called the Stieltjes measure on / associated
to the function g.
52. The Lebesgue measure on / is the positive measure associated to the increasing,
continuous function g(s) = s, for s e /. In this case, we still have
mg]s, t] =mg[s, t] = m?[i, t[ =mg]s. t[ = t—s.
53. Remark. If / is an interval closed to the left or open to the left and if g : I -> R is
a function with finite variation, we can consider the right continuous function g+: I —> R
defined by g+(a) = g(a), if ? e / and g+(t) = g(t+), if a < t, and proceed as above.

Some elements of the classical measure theory 47
54. As a consequence of the Monotone Class Theorem we have the following result: if
?, ? are two measures on B(W), where r ^ 1, both defined, and agreeing, on all intervals
of the form
]-oo,a] = {x: ? ??} = {(?\,...,??): ?? ????, for every/ ^r),
for ? = (? ?,... ,ar) eW, ап<1д(Кг) < oo, then ? and ? are equal on all theBorel subsets
ofRr.
55. THEOREM (Image measures). Let (S, ?,?) be a measure space, ? any set, and
?~] : S -> ? a function. Set
T={F:Fcy,i)-|(F)ei|, v(F) =m(<p~'(F)), foreveryFeT.
Then (Y, T, v) is a measure space, ? is called the image measure ??~'.
2.8. Lebesgue measure on W
One of the most important examples of measure is the Lebesgue measure ?' on the
Euclidean space W for some r e N. Therefore we consider in more details this measure,
with a few of its basic properties.
Let a = (? ?,..., ar), b = (fi\,..., ?,-), ? = (?\,..., ?,-) e W. Half-open interval in W
is a set of the form
[a, b[= {?: ?, ^ ?, < ?,, for every / $5 r),
where a, be W. If / = [a, b[ с Rr, then either / = 0 or
a, = inf{?,-: ? el}, ?,¦ = sup{|,: ? el],
for every / ^ r, so that the expression of / as a half-open interval is unique.
56. DEHNITION. The r-dimensional volume ??(?) of a half-open interval / is defined by
/·
??@)=?, br{[a,b[) = Y\(fr-a,),
;=i
if ?, < ?/ for every /'.
57. LEMMA. If I cKr is a half-open interval and (//)yeN " a sequence of half-open
intervals covering I, then
ОС
?'?/???'?/;).
7=0

48
?. Pap
For the proof see Fremlin B000a, Vol. 1, Lemma 115B).
58. DEFINITION. We define ?: V(Rr) -> [0. oo] by
?(?) =mf{^2kr(Ij): (Ij)jef; of half-open intervals with А с [J I}
4=0 yef.
Every set A can be covered by some sequence of half-open intervals, e.g., А с
и„ем[-п'п[' where n=(n,n, и) е R''. We have a non-empty set A that ?(?) is
always defined in [0, oo].
This function ? is called Lebesgue outer measure on R''.
PROPOSITION, ? is an outer measure on W such that ?(?) = ?'"(/) for every half-open
interval I cRr.
59. Using the Caratheodory's method we construct by Lebesgue outer measure a
measure, denoted by kr, which is called Lebesgue measure on Rr. The sets ? for which
??(?) is defined, i.e., for which ?(???) + ?(?\?) = ? (A) for every А с R', are called
Lebesgue measurable. Sets which are negligible for ?' are called Lebesgue negligible
(these are just the sets A for which ?(?) = 0).
Suppose that t e W. Then ?(? + ?) — ?{ A) for every A cR', where A + t = {x+t: ? e
A}. If ? с W is measurable so is ? + r, and ?'(? +t) = У (Е).
LetoO. Then/j(cA)=c/j( A) for every А с R', where cA = {c.v: ? e A}. If ? cR'
is measurable so is cE, and ?' (с?) = crkr{E).
Example. All Borel subsets of R' are Lebesgue measurable; in particular, all open sets,
open intervals }a, b[, closed intervals [a,b], together with countable unions of them. We
have for Lebesgue measure ??:
/'
кг(]а,Ь[)=кг([а,Ь]) = 1\№-сц).
i=\
whenever a ^ b in W. Therefore every countable subset of W is measurable and of zero
measure.
?? is complete, ?-finite and therefore is strictly localizable, localizable, locally
determined and semifinite. ?'' is atomless. Remark that for kr as the restriction of Lebesgue
measure to В the measure space (R, ?, ?') is atomless, ?-finite and not complete.
60. If S is R'\ and ? is its Borel ?-algebra B(W), a S(R')-measurable function is
called Borel measurable. If S is R'\ and ? is the ?-algebra of Lebesgue measurable sets,
a ?-measurable function is called Lebesgue measurable.

Some elements of the classical measure theory·
49
PROPOSITION. Let S be Rr for some r ^ 1, D a subset of S, and f: D -> R a function. If
f is Borel measurable it is Lebesgue measurable. If f is continuous it is Borel measurable.
Particularly, if r = 1 and f is monotonic it is Borel measurable.
Let / be ?1-measurable real-valued function whose domain belong to ?. If h is a Borel
measurable real-valued function defined on a Borel subset of R, then Dom h о f e ?.
PROPOSITION. Let S be a set and ? ? ? -algebra of subsets of S. Let f\, ¦¦¦, /,· be
measurable functions defined on subsets of S. Set D = C\i<r Dom f and for ? e D set
f(x) = (Mx),...,fr(x))eW.Then
(a) for any Borel set ? С Rr, there is an F e ? such that /~' (?) = ?) ? F;
(b) // h is a Borel measurable function from a subset Dom h of Rr to R, then the
composition h о f is measurable.
61. Not every subset of Rr is Lebesgue measurable. The construction of an example
appeal to the Axiom of Choice, see Fremlin B000a, Vol. 1). It is in fact the case that very
large parts of measure theory can be worked out without appealing to the full strength of the
Axiom of Choice. The significance of this is that it suggests the possibility that there might
be a consistent mathematical system in which enough of the Axiom of Choice is valid
to make measure theory possible, without having enough to construct a non-Lebesgue-
measurable set. Such a system has indeed been worked out by R.M. Solovay A970), see
Fremlin B000a, Vol. 5).
We remark that while we need a fairly strong form of the Axiom of Choice to construct a
non-Lebesgue-measurable set, a non-Borel set can be constructed without using any form
of the Axiom of Choice (Fremlin, 2000b, Vol. 4).
62. Proposition.
(a) If А с R' is any set, then for Lebesgue outer measure ?
?(?) = mf{k'(G): G is open, G 3 A}
= min{kr(H): ? is Borel, НЭ A}.
(b) If ? с Rr is measurable, then
??(?) =sup{k' (F): F is closed and bounded, FC?),
and there are Borel sets H\, № such that H\ с ? с //2 and
kr(H2 \ //|) = kr(H2 \E)= ?' (? \ //|) = 0.
(c) If А с W is any set, then A has a measurable envelope, i.e., a set ? e ? such that
А с ? and k(F ? ?) = /j(F ? A) for all F e ?, which is a Borel set.
(d) If f is a Lebesgue measurable real-valued function defined on a subset ofR', then
there is a conegligible Borel set ? с R'~ such that /|я is Borel measurable.

50
?. Pap
63. Examples, (a) Cantor set. The Cantor set С с [0, 1] is obtained by removing from
[0, 1] first its middle third ]\., |[, then from the remaining two intervals [0. j] and [|, 1]
again their open middle thirds ]g, f [ and ]? |[, respectively, and so on. We obtain a
sequence (C„)»eN of sets, Co = [0, \],C\ = [0. ^] U [^, 1] C„ consists of 2" disjoint
closed intervals each of length 3~". We haveX(C„) = B/3)" for each n. The Cantor set is
C = n«eNC»-Then
?@= lim k(Cn)= lim ( - ) =0.
n—*oc n—>зс\ 3/
We remark that С is uncountable. С itself is the set of numbers expressible as ??=\ ^'cj^
where every cj is either 0 or 2.
(b) The Cantor function. We define a sequence (/„)„et: of functions /„ : [0, 1] -> [0, 1]
by
/»W = (j) Л(С„П[0,лг]),
for each ? e [0. 1]. Since C„ is a finite union of intervals, /„ is a polygonal function,
with /„ @) = 0, /„ A) = 1; /„ is constant on each of the 2" - 1 open intervals comprising
[0, 1] \ C„, and rises with slope C/2)" on each of the 2" closed intervals comprising C„.
We have \f„+\(x) - fn(x)\ ? \2~n for every ? e ?, ? e [0, 1]. Therefore (/„)„eN is
uniformly convergent to a function /: [0, 1 ] -> [0, 1 ], which is continuous. / is the Cantor
function or Devil's Staircase.
Since every /„ is non-decreasing, so is / and the derivative f\x) exists and is 0
for every ? e [0, 1] \ C; so /' is zero almost everywhere on [0, 1]. /: [0, 1] -> [0, 1] is
surjective and we have /(C) = [0, 1].
The Cantor function can be used to prove that the composition of a measurable function
with a continuous function need not be measurable.
3. Integration
There are numerous ways to construct the classical integral. Each approach has its
advantages and disadvantages. Some of them are more laborious than others.
We present first the "immediate integral", of bounded, real-valued functions with respect
to a bounded, real-valued, additive measure m on a ring ??. This integral is used in the Riesz
representation theorem and in the integral representation of the dual of 1°°(?).
For the classical integral we shall sketch an approach as it is presented in W Rudin
A973), which seems to be simpler and to lead faster to the definition of the integral. For
the proof of some of the theorems the reader is referred to Halmos A950), Fremlin B000a,
Vol. 1, 2000b, Vol. 2), Rudin A973). The framework for this section is a measure space
(?,?,?).

Some elements of the classical measure theory
51
3.1. The immediate integral
Let 1Z be a ring of subsets of S and SAZ) the set of 7^-step functions
?
f = 2_] XAjXi, with A, e 7? and jc/ e E.
i=\
We can always assume that the sets A, are disjoint, and then |/| = ?'!=\ ??, ?*? ?-
64. Definition. A function /: S -> ? is said to be totally 7^-measurable, if it vanishes
outside a set A e 7? and if there is a sequence (/„)„ек from S(TZ) converging uniformly
to/.
If S e 7?, the condition that / vanishes outside a set of TZ is superfluous. The set of
totally 7^-measurable functions f.S -> ? is denoted by TMAZ). It is a vector space
consisting of bounded functions. If for every / e TMAZ) we set ||/||SUp = supveS |/Cs)|,
then ll/Hsup is a norm on TM(TZ), defining the topology of uniform convergence. The
space S(R) is dense in TM{K) for this topology. If ? is a ?-algebra, then ??{?) is the
space of real-valued, bounded, .?7-measurable functions /: S -> E.
Consider now a real-valued, additive measure m : 7? -> ? with finite variation |w |. This
means that m is locally bounded on ??, i.e., for each set A eTZ, m is bounded on the ring
7гпА.
Proposition. ?/??=\ ??,?; = 0, with A, e TZ and ?, e E, г/геи ?"=l ш(Аг).хг = 0.
Corollary. ?/?'!=\ ??,?\ = ?)=\ Хв,У] with Аг, В} е ?? andxi,ys е Е, г/геи
? к
]Г /я (Аг) хг¦ = ]Г т (В j¦) у> ·
;=? у=|
This corollary leads to the definition of the integral of step functions.
65. Dehnition. For every 7?-step function / = ?'!=\ Хл,х* with A, e К and *,· е Е,
we define the integral J f dm by the equality
/ Sdm = Y^m(Ai)Xi.
J i=\
The definition of the integral is independent of the particular representation of / as
a step function. Since the variation \m\ is additive on ??, we can define the integral

52 ?. Pap
f fd\m\ with respect to \m\. If / = X]"=, XA,Xi with A, e ? disjoint and jc, e R, then
I/1 = ?"=? ??, \xi\, hence
У |/|^|m| = ^|m|(A,)|.r,
i = l
We have the following immediate properties of the integral of x-step functions:
66. f(f+g)dm=ffdm+fgdm.
67. / cf dm =cf f dm, for с el.
68.
U fdmU J \f\d\m\^\\f\\sup\m\{{f?0}).
69. If (fn)neN is a sequence of functions from <S(X), vanishing outside a set A e ?
and converging uniformly to a function / e ???(?). then the sequence (/ /„ dm)n€n is
Cauchy in R. In fact,
\f /„dm - J fhdm\ <: I \f„ - fk\d\m\ ? ||/„ - /*||5UpM(A).
If (gn)neN is another sequence from 5(x) vanishing outside A and converging uniformly
to /, then
lim / /„ dm = lim / gn dm.
n-»oc J n^ac J
This property leads to the definition of the integral of totally measurable functions:
70. DEnNlTlON. For every function / e ???(?) we define the integral f f dm by the
equality
// dm = lim / /„ dm.
n-*xj
where (/„)neN is any sequence from «S(x) vanishing outside a set A e X and converging
uniformly to /.
From property 69 we deduce that the integral / /„ dm does not depend on the sequence
(/n)neN- Properties 66, 67 and 68 remain valid for functions from ???(?).
We have the following representation of the dual of TM{TZ), in terms of the immediate
integral.

Some elements of the classical measure theory
53
71. THEOREM. There is an isometric isomorphism L <-> m between the continuous linear
junctionals L e TM(TZ)* and the measures m e ba{1Z) = the space of real-valued,
bounded, additive measures m : ?? —* R, endowed with the variation norm \\m\\ = \m\(S).
The correspondence L +> m is given by
and we have \\L\\ — ||w||.
The measure m corresponding to L is defined by m(A) = L(xa), for A &1Z.
3.2. Integral of positive step functions
72. For a positive, ?-measurable step function / : S -> R+ of the form
/ = ]P ??,?-?, with A, e ?(?) and jc, e R+
we define unambiguously the integral / / ??. by the equality
?
/
i=l
with the usual convention 0 · oo = 0.
The definition of the integral / / ?? is independent of the particular representation of /
as a step function. The integral of positive, ?-measurable step functions has the following
properties:
73. 0^ffd?^+oo.
74. ?(/ + 8)?? = ?/?? + ?8??.
75. / cfd? = cf fd^, if с ^ 0.
76. H0sifsig,then0<:ffd?<:fgd?.
77. If / is a positive ?-measurable step function and A e ?(?), then /хд is a positive
?-measurable step function. We denote
//^?=//??^?, ??????(?).

54
?. Pap
78. If / is a positive, ?-step function, the indefinite integral v(A) = fAf ?? defined
for A e ?(?) is ?-additive.
3.3. Integral of positive functions
79. We define now the integral f f ?? for positive, ?-measurable functions /: S
[0, +00], by the following equality:
/f ?? = sup / $??.
where the supremum is taken for all positive, finite, iT-step functions s with 0 ^ s ? f. If
/ is a positive, ?-step function, the integral defined above is equal to the integral defined
in the preceding section.
In the above definition we can take the supremum for all positive, finite ?(?)-$???
functions. In fact, for each positive, finite ?(?)-$??? function s there is a positive ?-step
functions' such that s' ? s everywhere and s =s', ?-a.e., hence f ??? = f s' ??.
The integral of positive, ?-measurable functions, with finite or infinite values, has the
following properties. The additivity of the integral has to be postponed until after the
monotone convergence theorem is given. All functions considered below are positive, ?-
measurable functions, with values in [0, 00].
80. 0<: f /?????.
81. $cf ?? = ?$???, ifc^O.
82. If / ^g, ?-a.e., then// ?? ? f %??. If f = g, ?-a.e., then f /?? = f gd^.
83. If/^O, then/ /??=0 iff / = 0, ?-a.e.
84. If / > 0 and / f ?? < oo, then f < 00, ?-a.e, and the set {/ ? 0} is contained in
a countable union of sets of finite measure.
85. If / ^ 0 and A e ?(?), we denote
/ ???= / ?????.
Then
? /?? = sup / s??,
JA O^s^fJA
where the supremum is taken for all iT-step functions s with 0 ^ s ? f.

Some elements of the classical measure theory
55
86. If / ^ 0, the set function ?: ?(?) -> [0, +oo] defined by
v(A) = f
J A
f ??, for A e ?(?),
A
is ? -additive.
87. MONOTONE CONVERGENCE THEOREM. If(fn),i€K is an increasing sequence of ?-
measurable, positive functions f„: S —> [0, +00]. ?/геи
/lim /„ ??? = lim / /„ ??
?—>oc ?—*·? J
or, equivalent!}/
/ sup /„ <*? = sup / /„<*?.
88. ?/ + 8)?? = ?/?? + ????.
89. If O^g ^ /, and if $ f ?? < oo, then /(/ -g)<*M = J f ?? - $ ???. This
property, is used in the proof of Lebesgue's theorem for the Bochner integral.
90. If (/,,)„eN is a sequence of positive, ?-measurable functions /„ : S -> [0. +00],
then
/OO ОС -
????? = ? ????.
П= I !!= I
91. FATOU'S LEMMA. If (fn)„€N is a sequence of positive, ?-measurable functions
f,:S -> [0, +00], ?/геи
/lim inf/„ ?? ^ lim inf / /„ ??.
л-»оо /г —> эс у
We extend the definition of the integral j f ?? for R+-valued, ?-measurable functions
/ defined ?-a.e. on S. For any two extensions g\, gi: 5 -> K+ of / we have g\ = gz, ?-
a.e., hence f g\ ?? = §g-???. We define then unambiguously f f?? = Jg??, where
g: 5 -> R+ is any extension of /. The above properties remain valid for positive, ?-
measurable functions defined ?-a.e. on S.
3.4. The classical integral
At this stage we can define the classical integrability of numerical functions. This definition
of the integral does not use the Lebesgue's dominated convergence theorem, or the density
of the step functions in ?' (?).

56
?. Pap
Remark. An alternative way to define the classical integral is to wait until we define the
Bochner integral of vector-valued functions / :S -> F, for a Banach space F, and deduce
the classical integral as a particular case for F = R. But in order to define the Bochner
integral we have to define first the space ?^ (?) of Bochner ?-integrable functions f :S —>
F, equipped with a seminorm ||/||| = /\/\??, then prove the Lebesgue dominated
convergence theorem and the density of the step functions in ?^(?), and then extend by
continuity the integral f f ?? from the space of step functions to the whole space ?^(?).
This will be done in the paragraph devoted to the Bochner integral in Dinculeanu B002)
(Chapter 8 in this Handbook).
We define first the integrability.
92. Dehnition. An R-valued function defined ?-a.e. on S is said to be ?-integrable if
it is ?-measurable and f \f\ ?? < oo.
From / l/l ?? < oo we deduce that |/| < oo, ?-a.e., hence, a ?-integrable function /
is finite, ?-a.e. If / is ?-integrable, there is a function g: S -> E, defined everywhere and
finite everywhere such that f = g, ?-a.e. Any two such functions g\, gi: S -> ? are equal
?-a.e.
Example. If ? is the counting measure on N then a function /: N -> E, i.e., a sequence
(f(n))n€N, is ?-integrable iff it is absolutely summable, and
[ /?? = f /(?)??(?) = ?/(?).
J Jn „=o
(?, ?>(?),?) is invariant under permutations, i.e., ?(?(?)) = ?(?) for every А с N and
every bijection ? : N -> N. Therefore every definition of integration which depends only
on the structure (N, V(N), ?) have to be invariant under permutations, i.e.,
j /(?(?))??(?) = j /(?)??(?),
for every integrable function / and every permutation ?. Consequently, a series such that
??=? /(?(?)) = ??=? /(") e K for апУ permutation ? have to be absolutely summable.
Therefore for an integral on an abstract measure space (S, ?, ?) depending only on ? and
?, is obligatory to be an absolute integral.
93. For each ?-integrable function / we denote ||/|| ? = / |/| ??.
94. DEFINITION. We denote by ?'(?). or ?'(.?, ?), or C](S, ?,?), the set of real
valued functions /: S -> ? which are ?-integrable.
It follows that if / e ?' (?), then |/| e ?' (?), and then /+, /" e ?' (?).

Some elements of the classical measure theory 57
?' (?) is a vector space and ||/|| | is a seminorm on ?' (?). The topology defined on
?' (?) by the seminorm ||/|| | is called the topology of the convergence in the mean. To
say that /,, -> / in the mean, means that
|/e-/Hi-*0, i.e., f\fa-f\dii^Q.
95. To define the integral of a ?-integrable function / from ?'(м), we remark that
/ can be written as a difference of two positive, ?-integrable functions, for example,
/ = /+-/-¦
If / = /? — fi = g\ — gi with /|, /2, g\, g2, positive, ?-integrable functions from
?'(?), then /, + g2 = /2 + g\. We have
/ ?\?? + / gid[i= / /2?? + / ?\??.
hence
/ /? ??. - \ /2?? = / g\ ?? - / ^2^?-
We can now define unambiguously, the classical integral j f ?? by the equality
/ f??= / ?\??- / /2??,
where f = f\ — f2 with f\, /2 positive ?-integrable functions. Then
J ??? = f f+??- J ???.
The above definition of the integral / / ?? is valid for functions / defined ?-a.e. on S and
finite ?-a.e.; in particular for functions of ?' (?). We remark that if / ^ 0, then/ = / — 0,
and the new definition of the integral coincides with that of positive functions defined in
Section 3.2.
The integral of ?-integrable functions has the following properties:
96. ?/ + 8)?? = ?/?? + /8??.
97. /?/?? = ??/??,{???&?..
98. If/^0, ?-a.e., then// ??^?.
99. If / ? g, ?-a.e., then j f ?? ^ f gd?. If / = g, ?-a.e., then j f ?? = jgd^.

58 ?. Pap
100. If / is ?-integrable, then \ j f ??\^ j \f\??=\\f\\\.
From this property, we deduce that the integral is continuous for the topology of the
convergence in the mean: if /„ -> / in the mean, then / /„ ?? -> f f ??.
101. If / is ?-integrable and A e ?(?), then f ?? is ?-integrable. We denote
/ f ?? = / ?????.
102. If / is ?-integrable, then the set function ?.?(?) -> ? defined by v(A) =
JA f ?? for A e ?(?), is ?-additive.
For ?-integrable functions, the monotone convergence theorem and the Fatou lemma
have a special form.
103. The monotone convergence theorem for ?-integrable functions. If (/H)neN is an
increasing sequence of ?-integrable functions and if sup / \f„ \ ?? < oo, then the function
sup /„ is ?-integrable (hence finite ?-a.e.) and we have /„ -> sup,, f„ in the mean and
/ sup fn ?? = sup / ?„??.
104. Fatou's lemma for ?-integrable functions. lf(fn)„eN is a sequence of ?-integrable
functions and if liminf,,^^ f \^\?? < oo, then the function liminf,,^^//! is ?-
integrable and we have
/lim inf/„ ?? ^ lim inf / /„ ??.
? —»ос ? —»эс J
105. Countable convexity. If (/„),ген is a sequence of ?-integrable functions and if
??=\ II/"II? < °°' men the series Y^=] fn is absolutely convergent ?-a.e., its sum
?-integrable and we have
is
?/-
я=1
I n=l
106. LEBESGUE'S DOMINATED CONVERGENCE THEOREM. Let (/я)яеР? be a sequence
of ?-integrable functions, g a positive, ?-integrable function ana f : S —> Ш a function.
Assume that,
(a) fn -> /, ?-a.e.
(b) |/h| ^ g, ?-a.e., for each n.
Then f is ?-integrable, /„ —> / in the mean ana
\fnd^\
/??.
For the proof, we apply Fatou's lemma to the sequences of positive functions (|/„
/I + 2g)„eN and Bg - |/„ - /|)„6N·

Some elements of the classical measure theory
59
107. The set S(Ef) of finite ?1/--step functions is dense in ?' (?).
We use the fact that for every function / e ?' (?) there is a sequence (/n)neN of finite
.?/-step functions such that /„ -> / pointwise and |/„ | ^ |/|, for each n; we apply then
Lebesgue's theorem.
108. The space ?' (?) is complete. More precisely: If (/„)„eN is a Cauchy sequence in
?' (?), then there is a function / e ?' (?) and a subsequence (/,uheN such that /„ -> /
in the mean and f„k -> /, ?-a.e. and in the mean.
From property 108 one can deduce the following properties:
109. For each function / e ?' (?) there is a sequence (/„) of IT/-step functions such
that /,, -> /, ?-a.e. and in the mean.
110. If (/„)„6N is a sequence from ?'(?) such that /„ -> / in ?'(?) and /„ -> g,
?-a.e., then f = g, ?-a.e.
111. / e ?' (?) iff there is a sequence (f„)n€n of IT/ -step functions which is Cauchy
in the mean and such that /„ -> /, ?-a.e.
112. Remark. The statement of property 11 lean be taken as a definition of integrability;
then we define the integral f fd?Ъy the equality f /?? = lim,,^^ / /« ??. However, to
prove that the integral J f ?? is independent of the sequence (/n)neN is more complicated.
113. Definition. Let (S, ?, ?) be a measure space.
(a) If D с 5 and /: Z) -> С is a function, then / is measurable if its real and imaginary
parts Re /, Im / are measurable.
(b) If / is a complex-valued function defined ?-a.e., / is integrable if its real and
imaginary parts are integrable, and then
j /??= ???/?? + ? j Im/??.
(c) Let ? e ? and / be a complex-valued function defined on a subset of S. Then fH f
is /(/Ih) ??? if this is defined in the sense of (b), taking the subspace measure ?#,
i.e., the restriction of ? to ??-
3.5. Lebesgue and Riemann integrals
114. When ? in Section 3.4 is the Lebesgue measure ?'' on W we say that J f ?? is
the Lebesgue integral of /, and that / is Lebesgue integrable if this is defined.
115. Riemann integral. There are many ways of describing the Riemann integral; see
Fremlin B000a, Vol. 1), Halmos A950). If [a,b] is a non-empty closed interval in R, then
a partition (dissection) of [a, b] is a finite list ? = (ar>,a\,..., an), where ? ^ 1, such that

60
?. Pap
a = ?0 ^ ? ? ^ . · · ^ ?? = b. If / is a real-valued function defined and bounded on [a, b],
the Darboux upper sum and lower sum of / on [a, b] depending on ? are defined by
я
Sp(f) =]?(?; -?;-?) sup /(л),
/I
¦sp(/) = y^Sai -a,-\) inf /(*).
/ = l
If ? and P' are two partitions of [a, b] and if ? с ?', then sp(f) ? Sp'(f). The upper
Riemann integral and /ower Riemann integral of / are defined by
J[a.h}(f) = inf{Sp(/): ? is a partition of [a, b]],
ha.b\(f) =suP{sP(f)- ? is a partition of [a, b]},
respectively. We say that / is Riemann integrable over [a, b] if J[u.b\{f) = I{«-b}(f)' 'dn<^
in this case take the common value to be the Riemann integral /((' / of / over [a,b].
If /: [a, b] -> ? is Riemann integrable, then it is also Lebesgue integrable, with the
same integral.
Remark. For unbounded functions and unbounded intervals, one uses various forms of
'improper' integral. For example, the improper Riemann integral f^° ^ dx is defined by
Ит.,_>оо /J ^T5" dx, while /J In ? dx is defined by lim.5 ю /,.' In* dx. The second exists as
a Lebesgue integral, but the first does not, because f^° | ^ | rfjc = oo.
There is a characterization of the Riemann integrable functions, as follows.
THEOREM. A bounded function f : [a, b] -> ? й Riemann integrable iff it is continuous
almost everywhere with respect to the Lebesgue measure on [a, b].
116. As a consequence of Lebesgue dominated convergence theorem we have the
following result.
COROLLARY. Let (W, ?, kr) be a measure space and ]a,b[a non-empty open interval
in E. Let f : S ? ]a, b[ —> ? be a function such that
(a) the integral F(t) = f f(x, t) dx is defined for every t e]a, b[;
(b) the partial derivative jr of f with respect to the second variable is defined
everywhere in Er ? ]?, b[;
(c) there is an integrable function g : Er -> [0, oof such that \ ^jj)'" I ^ g(x) far every
xeW,te]a,b[.
Then the derivative F'(t) and the integral f d'f^'n dx exist for every t e ]a,b[, and are
equal.

Some elements of the classical measure theory
61
3.6. Integration with respect to a real-valued measure
Let ? be a <5-ring of subsets of S and m:T> -> ? a real-valued, ?-additive measure
with finite variation |w|. Let ? be the ?-algebra generated by V. The variation |w|
is a positive, finite, ?-additive measure on V and has a canonical ?-additive extension
? : ? -> [0, +oo], defined by
M(A)=sup{|m|(D): DeP, О С A}, for A e Г.
We shall continue to denote ? by |w|. We obtained a measure space (S, ?, \m\). Instead
of |w |-a.e. we shall write w-a.e.
117. ?????????. We say that an ?-valued function /, defined w-a.e. is w-negligible
(respectively, w-measurable, w-integrable) if it is |m|-negligible (respectively, |w|-
measurable, |w|-integrable).
All considerations in these sections hold also for Banach space valued functions.
It follows that an w-integrable function is finite w-a.e.
We denote by C\m) the space of w-integrable functions /:S -» E, i.e., ?' (w) =
Lx(\m\). We consider on ?'(w) the topology of convergence in the mean of ?'(|w|),
defined by the seminorm
\\f\\\= f\f\d\m\, for/e?'(w).
Any w-integrable function is equal w-a.e. to a function from ?' (w). All properties stated
before for the space ?' (?) remain valid for ?' (w).
118. We now define the integral j f dm with respect to w. For step functions / e
S(Ef) of the form / = ]?'.'=| ????, with A, e ?( and x, e ? we defined the integral
{ "
/ /dw = ]Tw(A,)jt, el.
We obtain a linear mapping L:8{?]) -> ?, defined by
L(f)=jfdm, for feS(Zf).
If we consider above the sets A/ mutually disjoint, then
/1
l/l = ][>4,l-v;l

62
?. Pap
and
\f fdm ^Yi\m(Ai)\\xi\^Yi\m\iAi)\xi\= j \f\d\
hence
|адКн/Ц|.
It follows that L is continuous on S(Ef) for the topology of ?'(m). Since <S(.?/) is dense
in ?' (w), we can extend L to a continuous linear mapping L': ?' (w) -> R, and we still
have
|?'(/)|<ll/lli, for/e?'(m).
119. Dehnition. For any function / e ?' (m), the value L'(/) of L' at / is denoted by
f f dm and is called the integral of / with respect to m.
If / is m-integrable, with values in R and defined m-a.e., and if g is any function
from ?'(w) such that f = g, w-a.e. we define the integral f f dm by the equality
/ /dm = f g dm, and the integral depends only on /.
120. For/,ge?'(m) and с е R we have
/ (f + g)dm = / fdm + / gdn
I cf dm = с I f dm,
and
\j fdm Uj\f\d\m\ = \\f\U.
From the last equality it follows that if /„ -> / in the mean, then / f,dm -> / / dm.
The Lebesgue's dominated convergence theorem remains valid for real-valued
measures m.
121. Remark. For real-valued m an alternative definition of integrability and of the
integral with respect to m is obtained by means of the positive measures m+ and m~,
and does not use the density of the step functions in ?' (m). Namely, we remark that a
function /: S -> R is ?-integrable if and only if it is w+-integrable and m~-integrable
and define the integral / / dm by the equality
/ fdm = I fdm+— ? fdm

Some elements of the classical measure theory
63
3.7. Density theorem
122. VlTALI'S THEOREM. Let A be a bounded subset of R and I a family of non-
singleton closed intervals in R such that every point of A belongs to arbitrarily small
members ofX. Then there is a countable set Iq с X such that
(a) Io is disjoint, i.e., ? Л /' = 0 for all distinct I, Г е Io,
(b) k(A\{JI0)=0.
123. THEOREM. Let I с R be an interval and /: / -> R ? monotone function. Then f is
differentiable almost everywhere on I.
124. THEOREM. Let f be a real-valued function which is Lebesgue integrable over [a, b].
Then F(x) = f(x / exists in R/or every ? e [a,b], and the derivative F'(x) exists and is
equal to f(x)for ?-almost every ? e [a, b].
125. LEBESGUE'S DENSITY THEOREM. Let I be an interval in R, and let f be a real-
valued function which is ?-integrable over I. Then
1 Cx+h 1 Г 1 fx+h
/U)=lim-/ fdk = lim- fdk = \im— fdk
for almost every ? e /.
As a consequence we obtain
THEOREM. Let I be an interval in R, and let f be a real-valued function which is
Lebesgue integrable over I. Then
lim— / \f(y)-f(x)\dy=0
for almost every ? e /.
3.8. Absolutely continuous functions
For more details in this section see Fremlin B000b), Vol. 2.
126. THEOREM. Let I (ZRbean interval, and f : / -> R afunction of bounded variation.
Then f is differentiable almost everywhere in I, f is integrable over I, and we have
УИ^УагД/).

64
?. Pap
127. THEOREM. Let (S, ?.?) be any measure space and f an integrable real-valued
function defined ?-a.e. Then for any ? > 0 there are a measurable set ? of finite measure
and a real numberS > 0 such that jF \f\dp ? ? whenever F e ? ? ? and m(F) ^ S.
128. DEHNITION. If [a, b] С R and / : [?, b] -> R we call / absolutely continuous if for
every ? > 0 there is a 5 > 0 such that ]T'!=| |/(fo;) - /(?,)| ^ ? whenever ? ???\ ?? ?
аг ^ Ьг ^ · · · ^ a„ ^ b„ ^ fo and ?"_, (fo, - ?,) ^ 5.
129. The basic properties of absolutely continuous functions are contained in the
following
Proposition.
(a) An absolutely continuous function on [a, b] is uniformly continuous.
(b) If f: [a,b] -> R is absolutely continuous it is of bounded variation on [a,b].
Consequently is differentiable almost everywhere on [a.b], and its derivative is
integrable over [a,b].
(c) Iff, g : [a, b] -> R are absolutely continuous, so are f + g and cf.for every eel.
(d) If f, g : [a, b] -> R are absolutely continuous so is f ¦ g.
(e) If g : [a, b] -> [c, d"] a/id" / : [c, d"] -> R are absolutely continuous, and g is non-
decreasing, then the composition f о g : [a, fo] -> R /s absolutely continuous.
130. The fundamental theorem of calculus is a part of the following
THEOREM. Let [a, b] С R a/id" F : [a. fo] -> R foe a function. Then the following assertions
are equivalent:
(a) 77?ere is an integrable real-valued function f such that F(x) = F(a) + /(/l f dk for
every ? e [a, b].
(b) f* F' dk exists and is equal to F(x) — F(a)for every ? e [a. b].
(c) F is absolutely continuous.
As a consequence we obtain
THEOREM (Integration by parts). Let f be a real-valued function which is integrable
over an interval [a,b] с R, and g:[a.b] -> R an absolutely continuous function. If
F(x) = f* fdk forx e [a,b], then
b pb
fgdk = F(b)g(b)-F{a)g{a)- I F ¦ g'dk.
J и
PROPOSITION. Let [a,b] С R· Let f:[a,b] -> R be a continuous function which is
differentiable on the open interval ]a, b[. If its derivative f is integrable over [a, b], then
f is absolutely continuous, and f(b) - f(a) = fj f'dx.
Example. Continuous function of bounded variation which is not absolutely continuous:
Let С с [0, 1] be the Cantor set. Recall that the Cantor function is a non-decreasing
,

Some elements of the classical measure theory
65
continuous function /: [0, 1] -> [0, 1] such that /'(-*) is defined and equal to zero for
every ? e [0, 1] \ C, but /@) = 0 < 1 = /A). Then / is of bounded variation and not
absolutely continuous.
3.9. Absolute continuity of measures
131. ?????????. Let (S, ?, ?) be a measure space and v:I^Ra finitely additive set
function.
(a) v is absolutely continuous with respect to ? if for every ? > 0 there is a ? > 0 such
that \v(E)\ ^ ? whenever ? e ? and ?(?) ^ <5. We write ? <<C ?.
(b) ? is truly continuous with respect to ? if for every ? > 0 there are ? e ?, S > 0
such that ?(?) < oo and \v(F)\ ^ ? whenever F e ? and ?(? ? F) ^ <5.
(c) If ? is countably additive, it is singular with respect to ? if there is a set F e .?7 such
that m(F) = 0 and v(E) = 0 whenever ? e ?, ? с 5" \ F. We write write ?_?_?.
The basic properties of notions introduced in the previous definition are contained in the
following
Proposition.
(a) Ifv is countably additive, it is absolutely continuous with respect to ? iffv(E) =0
whenever ?(?) =0.
(b) ? is truly continuous with respect to ? iff
(i) is countably additive,
(ii) is absolutely continuous with respect to ?,
(iii) whenever ? e ?, ?(?) ? 0 there is an F e ? such that m(F) < oo and
v(EHF) ?0.
(c) If(S, ?, ?) is ?-finite, then ? is truly continuous with respect to ? iff is countably
additive and absolutely continuous with respect to ?.
(d) If(S, ?, ?) is finite, then ? is truly continuous with respect to ? iff it is absolutely
continuous with respect to ?.
For more details see Fremlin B000b), Vol. 2, 232 A, B.
3.10. The Radon-Nikodym theorem
132. PROPOSITION. Let (S, ?, ?) be a measure space, and f a ?-integrable real-valued
function. For ? e ? set
v(E) = j fdti.
Then ? : ? -> R is a countably additive measure and is truly continuous with respect to ?,
therefore absolutely continuous with respect to ?.

66
?. Pap
The mapping ? н» fE f is called the indefinite integral of /.
133. The Radon-Nikodym Theorem. Let (S, ?, ?) be a measure space and ?: ? -+
? a set function. Then the following are equivalent:
(i) there is a ?-integrable function f such that v(E) = jE f ?? for every ? e ?;
(ii) ? is finitely additive and truly continuous with respect to ?.
COROLLARY. Let (S, ?,?) be ? ? -finite measure space and ?: ? -> ? a set function.
Then there is a ?-integrable function f such that v(E) = fE f for every ? e ? iff ? is
countably additive and absolutely continuous with respect to ?.
COROLLARY. Let (S, ?, ?) be a finite measure space and v. ? -> ? a set function. Then
there is a ?-integrable function f on S such that v(E) = fL f ????? every ? e ? iff ? is
finitely additive and absolutely continuous with respect to ?.
Remark. By the Radon-Nikodym Theorem, the question immediately arises: for a given
v, how much possible variation is there in the corresponding /? The answer is: two
integrable functions / and g give rise to the same indefinite integral iff they are equal
almost everywhere.
134. Theorem (The Lebesgue decomposition of a signed measure, see Fremlin B000b),
Vol. 2, 323 I).
(a) Let (S, ?, ?) be a measure space and ?: ? -> ? a signed measure. Then ? has
unique decomposition as
? = us + v.dC = us + цс + ve,
where цс is truly continuous with respect to ?, vs is singular with respect to v, and
vac is absolutely continuous with respect to ? and ve is zero on every set of finite
measure.
(b) If S = Rr, ? is the algebra ofBorel sets in W and ? is the restriction of Lebesgue
measure to ?, then ? can be uniquely represented as vp + vcs + vac where v-aC is
absolutely continuous with respect to ?, vcs is singular with respect to ? and zero
on singletons, and vp(E) = X]vei vp{x] for every ? e ?.
135. For the next notion the following result is important.
THEOREM. Let (S, ?. ?) be a measure space, and f a non-negative ?-measurable real-
valued function defined on a conegligible subset of S. If v(F) = J f ¦ ?/.- ?? whenever
F с 5" is such that the integral is defined in [0. oo], then ? is a complete measure on S, and
its domain includes ?.
Definition. Let (S, ?, ?) be a measure space, and ? another measure on S with domain
?. ? is an indefinite-integral measure over ?, or sometimes a completed indefinite-integral
measure, if it can be obtained from some non-negative, finite measurable function /
defined ?-almost everywhere on S. In this case we will call / a Radon-Nikodym derivative
of ? with respect to ?.

Some elements of the classical measure theory
67
3.11. Conditional expectation
136. One of the important applications of the Radon-Nikodym Theorem is related to
the conditional expectation.
Let S be a set and ? a ? -algebra of subsets of S. A ?-subalgebra of ? is a ? -algebra
? of subsets of S such that Tcr.
PROPOSITION. Let (S, ?, ?) be a measure space and ? a ?-subalgebra of ?. A real-
valued function f defined on a subset of S is ?\?-integrable iff (a) it is ?-integrable,
(b) Dom/ is ^\i-conegligible, (c) / is ?\?-measurable; and in this case f fd^\x) =
ffdfi.
137. Dehnition. Let / e L'(m)- A conditional expectation of f (defined on S) with
respect to ? is a function g defined ?-a.e. on 5* which is ? ??-integrable and T-measurable
such that fEgd(?\?) = ff. f ?? for every ? e T.
By the preceding lemma we see that for such a g we have
/ gd(p\T)= / g-XEd^\T)= / g- ????= gd?,
for every ? e T. Then g is almost everywhere equal to a T-measurable function defined
everywhere on 5" which is a conditional expectation of / on T, too.
138. The following two statements are closely related to the convex functions, which
will give an important result related the conditional expectation.
PROPOSITION. Let I c R be a non-empty open interval (bounded or unbounded) and
?: I -> R a convex function (?: I -> R, for an interval I с R, is convex if (p(ab + A —
a)c) ^ oup(b) +A — <*)<p(c) w/?e/iever fo, с e / a/ida e [0. 1])
(a) For every a e I there is b e R лис/г ?/??? <р(лт) > <?(?) + fo(* - a) for every x e I.
(b) If we take by (a), for each q e I nQ, fov e R iwc/г г/шг <p(;t) ^ <p(<7)+ Ь(/(.х —q)for
every x e /, ?/геи
<pU) = sup (<р(?) + ^и-?)).
^e/nQ
(c) <p ? Bore/ measurable.
PROPOSITION (Jensen's inequality). Let (S, ?, ?) be a measure space and ?-.Ш^- R a
convex function.
(a) Suppose that f, g are real-valued ?-measurable functions defined ?-almost
everywhere on S and that g >0 а.е., f gd?= 1 and f ¦ g is ?-integrable. Then
<pl g- /??.)? I g-(<pof)dp,
where we may need to interpret the right-hand integral as oo.

68
?. Pap
(b) Specially for ?(?) = 1 and f a real-valued function which is ?-integrable over S,
we have (p(f fd?)^f?of ??.
139. PROPOSITION. Let (S, ?, ?) be a probability space, and ? a ?-subalgebra of ?.
Let f be a ?-integrable function and let h be ? (?\?) -measurable real-valued function
defined (?|?)-?/№??? everywhere on S. Let g, go be conditional expectations of f and \f\
with respect to T, respectively. Then f ¦ h is ?-integrable iff go · h is ?-integrable, and
then g ¦ h is a conditional expectation of f ¦ h with respect to T.
3.12. The Lebesgue-Stieltjes integral
140. Let / be an interval and let g : I -> R be a function with finite variation |g|, and
right continuous on /, except, possibly, at the left end point of /. Let B(I) be the Borel
?-algebra of / and V(I) the ?-ring of bounded Borel subsets of /. Let ms : T>(I) -> R be
the ?-additive measure with finite variation |/ms| = m|4], associated to g (see Section 2.7).
Consider the space ?' (mg) = ?' (\т^\) of m^-integrable functions /: S -> R. The space
C\mg) is also denoted by С[(g) and for any function / e ?'(ffls). the integral f f dmx is
also denoted by f fdg and is called the Lebesgue-Stieltjes integral of / with respect to g.
3.13. Transformation of coordinates
141. We will give a generalization of the basic formula of calculus Jg(y)dy =
j g(<p(x))<p'(x)dx in the context of a general transformation^ between measure spaces.
THEOREM. Let (?, ?, ?) and (?, ?, v) be measure spaces, and ?:?? -> ?, J :Dj ->
[0, oof functions defined on conegligible subsets ??, Dj ofX such that
whenever left side exists and F e ? and v(F) < oo. Then
/ J -^??)??= / gdv
whenever left side exists, for every v-integrable real-valued function g, where we put Ofor
(J ¦ (g ? ?)){?) when J (?) = 0 and g((p(x)) is undefined.
See Bauer A990), Elstrodt A996), Fremlin B000b), Vol. 2, for the proof and further
generalizations.
142. Change of variables. Let Q С R' be a bounded open set with a boundary dQ of
measure zero which is mapped with a function ? : Q -> R'\ ? = (??,..., ?,-), of class

Some elements of the classical measure theory·
69
C1 as a bijection on a bounded open set ? с К' with a boundary 3? of measure zero with
the Jacobian
9(?|,...,?,.)
J = —— ^—i^O.
Э(Л|,...,ЛГ)
Then the following holds.
(a) The functions ? and ?^1 map every set of measure zero on a set of a measure zero;
(b) The function /: ? -> R is (Lebesgue) integrable on the set ? if and only if the
function (/ о <t>)\J\ is (Lebesgue) integrable on the set Q. Then the following
equality holds
? f(y)dy= f (fo0)(x)\J\dx,
Jn Jq
where dx — dx\ ¦ ¦ ¦ dx, and dy = dy\ ¦ ¦¦ dyr.
4. Function spaces
4.1. Essentially bounded functions
143. Let (S, ?, ?) be any measure space. We recall that we denote by ?°, or ?°(?)
the space of real-valued functions / defined on conegligible subsets of S which are ?-
measurable, that is, such that f\n is measurable for some conegligible set ?CS, see 45.
We define a partial ordering relation ^ on L° (see 45) by saying that / ^ g (as classes
of equivalence) iff / ^ g a.e. as measurable functions. L° is a Riesz space or vector lattice,
that is, a partially ordered linear space (a (real) linear space with a partial ordering ^ such
that if и ^ ? then и + w ^ ? + w for every w, if 0 ^ и then 0 ^ cu for every с ^ 0) such
that и ? ? = sup{w, ?}, и л ? = inf{w, и) are defined for all u, ? e L°.
We introduce the normed (complete, Banach) vector space L'(m) as the set of
equivalence classes of members of ?' (?), which is a Riesz space (even it is Dedekind
complete, i.e., every non-empty subset which is bounded has a least upper bound in L' (?)).
144. Let (S, ?,?) be a measure space. Let Cx = ?:?(?) be the set of functions
/ e ?° = ?°(?) which are essentially bounded, i.e., there is some ? ^ 0 such that
l/l ^ ??, ?-a.e., and we define
L°° = L°°(M) = {/ (as class): / e ?^(?)} с ?°(?).
THEOREM. Let (S, ?, ?) be any measure space. Then
(a) L°° = ?°°(?) ? a linear subspace of L° = ?°(?).
(b) Ifue Lx, ? e L° and \v\ ^ \u\ then ? e Vе. Then \u\, и ? ?, и л ?, и+ = и ? 0
an<i и~ = (-и) ? 0 belong to Lx for all и. ? e L^.
(c) Taking e = 1, ?/?e equivalence class in L° of the constant function with value 1, ?/геи
аи element и of I? belongs to L°° iff there is an ? > 0 ??/с/г f/гаг |и| ^ Me.

70
?. Pap
(d) If и, v e Lx then и ¦ ? e Lx.
(e) IfueLx, veL] = L'(m), then и ¦ ? e L1.
The property of L^ to be a Riesz space is dominated by the fact that it has an order unit
e = 1 with the property (c).
145. Let (S, ?, ?) be any measure space. For / e Cx = ?:?(?), say that the essential
supremum of |/| is
esssup|/| =inf{M: ? ^ 0. \f\^M, ?-a.e.}.
Then l/l ^ esssup |/| ?-a.e.
We may define a norm || ¦ \\? on Lx = ?^(?) by setting ||м||зс = esssup |/| whenever
и = f (equivalence class). We have for any и e Lx, \\u\\x = min{y: \u\ ? ye], where
e = leL°°.
If u, ? e L00 and \u\ ? \v\ then ||м||зс ^ IIf Hoc an(J
11^ * ^lloc ^ II и || эс || I'll oc , 11^ ^ ^ll^c
^тах(||м||зс, ||и||зс).
for all u,v & L°°, i.e., Lx is a commutative Banach algebra and a Banach lattice.
146. If и e L> = ?'(?) and ?)?[? = ?^??), then и · и e L1 and |/и ¦ ?|<*? ^
II" II ? II I'll ос · Therefore, we have a bounded linear operator ? from Lx to the normed space
dual (L1)* (space of continuous linear functionals) of L1, given by
(Tv)(u) = J и ¦ ???,
for every и e L], ? e L00.
4.2. The space Lp(?)
147. DEnNlTlON. Let E, 2?, ?) be any measure space, and ? e ]l,oo[. Then we define
???) = {?; MeL°, iHl'^i'tM)).
THEOREM. Let (S, ?, ?) be a measure space, and ? ? [1. oo].
(a) ?7{?) is a linear subspace of ?/'(?).
(b) //и e ?,''(?). ? e ^°(?) ?«? ? ^ |и|, then ? e ?/(?)- Therefore \u\, и ? ? and
и ли belong to Lp^)forall и, ? e ?,''(?)·
(c) The partial ordering inherited from ?°(?) makes ?/(?) ? /?/??: space.

Some elements of the classical measure theory
71
148. We define the norm ||и||р = (/ \u\p???/? for every и е L''(m)·
PROPOSITION (Holder inequality). Suppose (?,?,?) isa measure space, and that p,q e
]1, oof are such that - + ^ = 1. Then и ¦ ? e Ll (?) and
I/
и ¦ ???
for и eL'4M), ??^(?).
149. PROPOSITION. Let (?, ?, ?) be a measure space and ? e ]1, oof. V is a Banach
space under the norm \\ ¦ \\. If q = p/(p — 1), then for every и e L''(M), ||h||p =
max{/и ¦ ???: ? e ^(?), \\?\\^ ?. 1}.
Specially, for ? = 2 is a Hubert space with the inner product f f ¦ gdp.
150. THEOREM (Duality in V spaces). Let (S, ?,?) be a measure space, and ? e
]l,oo[. Ifq = ?/{p- 1), then uv e ?-'(?) and \\u -v\\\ ^ ||?||?||????/ whenever и е ??{?)
and ? e Lq (?). Therefore we have a bounded linear operator ? from L'1 (?) to the normed
space dual (?''(?))* of ?7(?), given by
(Tv)(u) = I u ¦ ???
for all и е ^(?), ? e L''(m) and the canonical map ? : ?>(?) -> Z/(M)* is a normed
space isomorphism.
4.3. Convergence of sequences of functions
151. A sequence (/„),„=n of functions, f,:S-> E, is said to converge in measure on
the set S to a measurable function /: S -> R if, for every ? > 0, we have
lim ?({?: \f„(x) - f(x)\ ^ ?}) = 0.
If the sequence (/„)„ем converges ?-a.e. to a function / and ?E) < oo, then it converges
to / in measure as well, while if the sequence (/„),„=n converges to / in measure, then
there exists a subsequence of (/„)„еь that converges to / ?-a.e.
152. The following theorem states the relation between almost-everywhere convergence
and uniform convergence.
THEOREM (Egorov). Let ? be ? ?-additive measure defined on ? ?-algebra ?, let
? e ?, ?(?) < oo, and let a sequence (/„);,er; of ?-measurable almost-everywhere finite
functions f„: ? -> R converge almost-everywhere to a function f. Then for every ? > 0

72
?. Pap
there exists a measurable set ?? С ? such that ?(?\?() < ?, and such that the sequence
(/u)/ieN converges to f uniformly on Ee.
For the case where ? is the Lebesgue measure on the line this was proved by D.F. Egorov
A911). Egorov's theorem has various generalizations, e.g., for a sequence of measurable
mappings of a locally compact space into a metrizable space.
5. Product measures and the convolution
5.1. Product measures
153. Dehnition. Let (S\, ?\,?\) and (Si. ??,??) be two measure spaces. For А с
5? ? S2 we take
/j(A) = inf
]???(?„)·?2(^): EneZi, F„ eb(neN), А с (J
l;i=0 ueK
(using the convention 0 · oo = 0).
? is an outer measure on S\ ? Si.
154. Dehnition. Let E|,2?|,??) and (Si. ?2. ??) be two measure spaces. The
primitive product measure on S\ ? Si is the measure ? derived by Caratheodory's method
from the outer measure ? from 153.
155. An alternative definition of the product of measures is given in the following
Dehnition. If S\ and Sj are sets with ?-algebras ?\ с V(S\) and ?? с V{Si),
respectively, we denote by ?\ § ?? the ?- algebra of subsets of 5? ? Si generated by
the semiring of rectangular sets {? ? F: ? e ?\, F e ??].
156. PROPOSITION. Lei E?, ?\, ??) and (S2. ?2, ??) be two measure spaces; let ? be
the primitive product measure on S\ ? Si, and ? its domain. Then ?\®?? с Т and
?(? ? F) = ?\{?) ¦ V2{F) for all ? e ?\, F e ??.
157. The third type of product of measures is given by the following definition.
?????????. Let E?,2?|,??) and (Si. ??. ??) be two measure spaces, and ? the
primitive product measure. The c.l.d. product measure on 5? ? Si is the function
?: Dom# -> [0, 00] defined by
v(A)=sup{e(Af)(E ? F)): Eeli.Feb. ?\(?) < oo. ?2(F) < 00}
for A e Dom ?.

Some elements of the classical measure theory
73
158. THEOREM. Let (S\, ?\, ?\) and (Si, ??, ??) be two measure spaces; let ? be the
c.l.d. product measure on S\ ? Si, and ? its domain. Then
(a) ?\ <g) ?? с ? and v(E ? F) = ?\(?) ¦ ??(^) whenever ? e ?\, F e ?? and
?\(?)-?2(?) < oo; ^
(b) for every AeT there is а В е ?\® ?2 such that ? с A and v(B) = v(A)\
(c) (S\ ? S2,T, v) is complete and locally determined, and in fact is the c.l.d. version
of(S\ ? Si,T,9); in particular, v(A) =?(?) whenever ?(?) < oo;
(d) if AeT and v(A) > 0 then there are ? e ?\, F e ?? such that
?\(?)<??, ?2{?) < oo and ?(? ? (? ? F)) > 0;
(e) if A e ? and v(A) < oo, then for every ? > 0 there are Eq, ..., En e ?\,
Fq, ..., F„ e ??, all of finite measure, such that
v(aa|J(?, xF;)W.
Example. Let r, ? ^ 1 be integers. Then there is a natural bijection ?: W ? Es -> W+s.
This bijection identifies Lebesgue measure on W+s with the product of Lebesgue measure
on Rr and Lebesgue measure on W.
For more details see Fremlin B000b), Vol. 2.
5.2. Repeated (iterated) integral
159. Let E?, ?\,?\) and (So, ??,?2) be two measure spaces, and / a real-valued
function defined on a set Dom/ с 5? ? S2. The repeated (iterated) integral
11 ??,?)???(?)??2(\·)= f(f ??.\)???(?))??2(}·).
is the integral j^(?)??2(?) (if this is defined), where
h(y)= ? /(?,?)??\(?),
у e ? ?: ? e 5, / f(x,z)d?\(x) is defined in R\.
Analogously, reversing the roles of 5? and S2, we can define a repeated integral
Jf /(?,?)??2?)???(?)= {({ /(?,?)??2(}'))<1??(?).
We can connect these repeated integrals to each other by connecting them both with the
integral of / itself with respect to the (primitive) product measure on 5? ? S2.

74
?. Pap
160. THEOREM (Fubini). Let (S\, ?\, ??) ???? (So, ??, ??) be two measure spaces, and
? the primitive product measure on S\ ? Si. Let f be ? ? -integrable real-valued function.
Then ff f(x, ?)??\(?)???(?) and ff f(x, \)???(?)??\{?) exist and are both equal to
ff(x,y)d0(x,y).
161. THEOREM (Tonelli). Let (S\, ?\, ??) and {Si, .???, ?:) be two measure spaces, ? the
c.l.d. product measure on S\ ? Si, and ? its domain. Let f be a T-measurable real-valued
function defined on a member of ?, and suppose that either ff \f(x, }>)\??\(?)???(?) or
ff !/(¦*> У)\ац.г(у)ац.\(х) exists in R. Then f is v-integrable.
162. We have used the primitive product measure in Fubini's theorem and the c.l.d.
product measure in Tonelli's theorem. When these two measures are the same we have the
following
COROLLARY. Let (S\, ?\,?\) and (Si, ??. ??) be two ?-finite measure spaces, ? the
c.l.d. product measure on S\ ? Si, and ? its domain. Let f be a T-measurable real-valued
function defined on a member ofT. Then if one of
? |/(*.
JS, xSt
y)\dv(x,y),
f f \f(x,y)\dpdx)dp2(y),
Js^ Js,
? ? ^(?^)\???(?)?\(?)
Js, Js^
L
S, xS->
exists in Ш, so do the other two, and in this case
f(x,y)dv(x,y)= ? ? f(x.x)d?](x)d?2(y)
Js2 Js{
= 11 ?&.>·)<??2(}·)<1?\(?).
J St JS2
Example. Let E?, ?\,?\) be [0, 1] with theLebesgue measure, and let (Sb, ??, ??) be
[0, 1] with the counting measure. Take the set A = {(;, ?): ? e [0, 1]} с 5? ? Si. We have
neNjt=0
k + l
n + l и + 1
k + l
П + 1 /7+1
e ?\ ®?2-
Then
// ??(?.?)??\{?)???(?)= / 0??2(?)=0.
// ??(?.?)???^)??\{?)= / ???(*) = 1.
We see that the two repeated integrals differ although both repeated integrals exist and are
finite. We remark that ?(?) ? v(A). For more details see Fremlin B000b), Vol. 2, 252 J.

Some elements of the classical measure theory
75
5.3. Infinite products
For more details see Fremlin B000b), Vol. 2, 254.
163. Let {{Si, ??, M,)},e/ be a family of probability measure spaces. Let S = f]ie/ 5,-
be the family of functions ? with domain / such that x(i) e 5,- for every i el. Denote by
С the family of subsets of S of the form
с = Пс>,
ie/
IS
where C, e IT,- for every /' e / and the set {/: C, ^ 5,-} is finite. For a non-empty С е С thi
representation is unique. Members of С are called measurable cylinders.
164. We define a set function щ : С -> [0, 1 ] by
40(C) = f[/if (С,-)
ie/
whenever C, e 2Г,- for every / e / and {/: С\Ф S,} is finite. Since only finitely many terms
in the product can differ from 1, so that it can be treated as a finite product. If С = 0, one
of the Cj must be empty, so щ(С) is surely 0, although the representation of С as F](.g/ Q
is no longer unique.
Now define ? :V(X) -> [0, 1] by setting
»j(A) = inf
ОС
? /?o(C„): C„ e С for every n e N, А с (J C„
,u=0 ueN
The set function ? is an outer measure on S.
165. Let {(Sj, ??, ?/)},-e/ be a family of probability spaces, and 5 the Cartesian product
П,-е/ и The product measure on 5 is the measure defined by Caratheodory's method from
the outer measure ? from 17.
166. Let {S,-};e/ be any family of sets, and 5" = П/е/ S- If -??? is a ?-subalgebra of
subsets of Si for each/ e /, we denote by ®,-е/2Г,· the ?-algebra of subsets of 5 generated
by
{{x: xeS, x(i)eE): i el, Ее ??}.
THEOREM. Let {(Si, ??,??)}?€? be a family of probability spaces, andlet ? be the pmduct
measure on S = П/е/ ^ defined as in 165 vv/fft the domain T. Then
(a) v(S) = l.

76
?. Pap
(b) // ?/ e ?? for every i e /, and {/: ?, ? S,} is countable, then J~I(.g/ ^' е ^~>
and v(Y\j€l Ei) = Y\j€l /J-i(Ej). In particular, v(C) = щ(С) for every measurable
cylinder C, and if i e / then ? н» ? (i):: S -> S/ /? inverse-measure-preserving.
(c) ®,.е/Г,-сТ.
(d) ? /? complete.
(e) For every A e ?, ? > 0 ?/геге /? a finite family Co C„ of measurable cylinders
such that ?(?? (J^,, Q) ^ ?.
(f) For ever}' AeT ?/геге are A\, Ai e ^j€lEj such that A\ С А с At and
?(?2\?,)=0.
167. EXAMPLE (Theproduct measure on {0. 1}'). Let Sj ={0, l},/e/, and let each ?,
be the 'fair-coin' probability, i.e., ?,({0}) =?,·({1}) = 1/2.
The product S = {0, 1}' has a family {?,},,= / of measurable sets such that
v|P|?,-|=2"|У|, if У с / is finite.
where ? is the product measure on {0. 1}'.
This ? is called the usual measure on {0. 1}'. If / is finite then v({x}) = 2_l" for each
? e S, and if / is infinite, then v({*}) = 0 for every ? e S.
168. There is a natural bijection between {0. 1}' and V(I), matching ? e {0, 1}' with
Av = {/: / e I, x(i) = \]. In this way we obtain a standard measure ? on V(I), which is
called the usual measure on V(I). We remark that for every finite Sc/, every С с ? we
have
v({A: АПВ = С}) = v{x: x{i) = 1 for/ e C. .*(/) =0for/ e B\C]
_ 2-i«i
169. If / is countably infinite, then there is a very important relationship between the
usual product measure of {0, 1}' and Lebesgue measure on [0, 1].
PROPOSITION. Let ? be the usual measure on S = {0. 1 }'\ and let ? be Lebesgue measure
on [0, 1]. Let ? be the domain of ? and ? for the domain of ?. For ? e S set ?(?) =
?,??2~'~'-*(')¦ Then we have ?~\?) e ? and ?(?^(?)) = ?(?) for every ? e ?,
and <p(F) e ? and X(<p(F)) = v(F) for every F e T. There is a bijection ?:?—* [0, 1]
which is equal to ? at all but countably many points, and any such ? is an isomorphism
between (S,T,v) and ([0, 1], ??,?).
170. If ? is the usual measure on {0. 1}'. then L1 (v) has a countable dense subset, for
its norm topology, iff / is countable.

Some elements of the classical measure theory
77
5.4. Convolution
171. Definition. Let / and g be measurable complex-valued functions defined almost
everywhere in R''. We denote by / * g the function defined by
*g)(x)= /
(f*g)(x)= I f(x-y)g(y)dy
for every set for which the integral (with respect to Lebesgue measure) is defined. Then
/ * g is called the convolution of the functions / and g.
172. Basic properties. Linearity of the integration implies
((/l +f2)*g)(x) = (fl*g)(x)+(f2*g)(x),
(cf * g)(x) = (f * cg)(x) = c(f * g)(x)
whenever the right-hand sides of the formulae are defined.
If /, g are measurable complex-valued functions defined almost everywhere in W, then
/ * g = g * /. in the strict sense that they have the same domain and the same value at
each point of that common domain.
If /?, /2, g\, g2 are measurable complex-valued functions defined almost everywhere
in R'', and /1 = /2 a.e., g\ = gi a.e., then for every ? e W we have (/1 * g\)(x) =
(/2 * gi)ix) in the sense that if one of these is defined so is the other, and they are then
equal.
If /, g are complex-valued functions which are integrable over W, then / * g is
integrable, and we have
j f*gdx=j fdx j gdx. j\f*g\dx^j\f\dxj\g\dx.
173. THEOREM. Let f, g, h be complex-valued measurable functions defined almost
everywhere in W. If ? eW is such that one of (\f\ * (\g\ * \h\))(x), ((|/| * |g|) * \h\)(x)
is defined in R. Then f *(g*h) and (/ * g) * h are defined and equal at x.
174. PROPOSITION. Suppose that f, g are measurable complex-valued functions defined
almost everywhere in R, and that f e C~, g e Ci where p,q e [1, 00] and jy + - = 1
(writing-^ =0asusual). Then f * g is defined everywhere inR, and supveK|(/ *g)(x)\ ?
WfWpWgWq-
5.5. Integral transforms
175. Dehnition. The Fourier transform ? of a function / e L1 (R') is defined by
(rf)(x) = f(x) = —L- f e-i:xf(z)dz {xeW),

78 ?. Pap
where ? ? = ?'*=??***-
176. Theorem. Let f eL'(R'). Then:
A) /eC(R');
B) ifzk ¦ f(z) is also a function from l} (R''), then there exists Dkf(x) and we have
Dkf(x) = -i?f(x);
more generally, if z"k ¦ f(z) for к = 1, ..., r are also functions fют L'(R'"), then
there exists Da f(x) and we have
Daf(x) = Dtrf(x)
for a = (<*],... ,<*,), ai-eNLJ {0};
C) ifDkfeLl(W)UC(Rr),then
L\f(x)=ixkf(x);
more generally, if D"k f e L' (Rr) U C(R') for к = 1,..., r, then
D^f(x) = (-ix)af(x)
for a = (<*],... ,<*,-), ai-eNU {0}.
177. THEOREM. For a linear differential operator
L(D)= ? aaDa
and a function f e C'"(R') with D" f e L] (W)for \a\ ^ m the following equality holds
L(D)f(x) = L(ix)f(x).
178. The following exchange formula allows us to transfer the convolution in the usual
product.
Theorem. lff,geLl(W),then
T7g(x) = Bn)"'2f(x)g(x).
179. Dehnition. The inverse Fourier transform T~l of a function / e L'(IR') is
defined by
•F-1 (/)(*) =/(-*)¦

Some elements of the classical measure theory
79
THEOREM. /// e L'(E'),/ e L'(R') and f(z) is a continuous function at the point
z = z°, then f(z°)=T-l(f)(z°), i.e.,
f(z°) = Bn)-"'2 [ e-^f(y)dy.
180. Let / be a complex-valued function defined almost everywhere on [0, oof,
endowed as usual with Lebesgue measure. Its Laplace transform is the function F defined
by writing
poo
F(s) = e-"f(x)dx
Jo
for all those complex numbers ? for which the integral is defined in С Then we have
(a) if ? e Dom F and Res' ^ Res then s' e Dom F;
(b) F is analytic on the interior of its domain;
(c) if F is defined anywhere then limRe.v_».x; F(s) = 0;
(d) if/, g have Laplace transforms F, G then the Laplace transform of / + g is F + G,
at least on Dom F П Dom G.
6. Regular measures
6.1. Topological measures
181. Dehnition. A measure ? defined on the Borel ?-algebra B(T) of a Hausdorff
topological space ?, such that г с ? (? is the family of all open sets), is called regular if
for any Borel set В and any ? > 0 there is an open set G С ? containing В, В С G, and
such thatM(G\ ?) <?.
An equivalent definition is as follows: For any В e B(T) and any ? > 0 there is a closed
set F С В such that ?(? \ F) < ?. For example, the Lebesgue measure is regular.
182. An additive set function ? defined on a family of sets in a topological space is
regular if its total variation \?\ satisfies the condition
\?\(?) = inf |m|(G) = sup|/i|(F), Fc?cG,
where G denotes the interior of a set G and F the closure of a set F (and E,G, F, are in
the domain of definition of ?).
THEOREM (Aleksandrov). Every bounded finitely additive regular set function, defined on
a semiring of sets in a compact topological space, is countably additive.
183. We restrict now to the case ? = W.

80
?. Pap
Definition. Let ? be a measure on R'", where r > 1, and ? its domain, ? is a topological
measure if every open set belongs to ?. ? is locally finite if every bounded set has finite
outer measure. If ? is a topological measure, it is inner regular for the compact sets if
v(E) = sup{v(K): К с ? is compact}
for every ? e ? (because ? is a topological measure, and compact sets are closed, v(K)
is defined for every compact set ?), ? is a Radon measure if it is a complete locally finite
topological measure which is inner regular for the compact sets.
If ? is a Radon measure on B(W). and ? its domain, then ? is ?-finite, and for any
? e ? and any ? > 0 there is a closed set F с ? such that v(E \ F) ? ?. For every
? e ? there is a set ? с Е, which is the union of a sequence of compact sets, such that
v(E\H)=0.
THEOREM. A measure ? on W is a Radon measure iff it is the completion of a locally
finite measure defined on the ? -algebra В ofBorel subsets ofW'.
THEOREM. Let ? be a Radon measure on W, with domain ?, and f a non-negative ?-
measurable function defined on a v-conegligible subset ofW. Suppose that f is locally
integrable in the sense that fEf< oo for every bounded set ? e ?. Then the indefinite-
integral measure ? on W defined by
v'(E) = I fdv whenever {?. ? e E, f(x) > 0} e ?
is a Radon measure on W .
Examples.
(a) Lebesgue measure on W is a Radon measure.
b) Let (f„)„epj be any sequence in R'\ and (а„)яек any summable sequence in [0, oof.
For every ? с R set
?(?) = ]?{?„: t„eE].
Then ? is a (totally finite) Radon measure on R'.
(c) Cantor measure. Recall that the Cantor set is a closed negligible subset of [0, 1],
and that the Cantor function is a non-decreasing continuous function /: [0, 1] ->
[0, 1] such that /@) = 0, /A) = 1 and / is constant on each of the intervals
composing [0, 1]\C. It follows that if we set g(x) = ^(x+f(x)) for ? e [0, l],then
g : [0, 1 ] -> [0, 1 ] is a continuous bijection such that the Lebesgue measure of g(C)
is j; consequently g_1 : [0, 1] -> [0, 1] is continuous. Now extend g to a bijection
h : R -> R by setting h(x) = ? for ? e R \ [0. 1]. Then h and /г-1 are continuous.
Note that h(C) = g(C) has Lebesgue measure 5.

Some elements of the classical measure theory
81
Let v\ be the Radon measure on R obtained by applying the method in the last
Theorem to Lebesgue measure ? on R and the function 2?(/,(?)- Then v\ (h(C)) =
v\(R) = 1. Let ? be the measure v\h, that is, v(E) = v\(h(E)) for just those
? с R such that h(E) e Dom v\. Then ? is a Radon probability measure on R,
and v(C) = 1, v(R \ С) = м(С) = 0.
For more details see Fremlin B000b), Vol. 2.
If ? is a Radon measure on R' then it is outer regular, i.e.,
v(E) = inf{ v(G): ? с G is open}
for every set in the domain of v.
THEOREM (Lusin). If ? is a regular Borel measure onW , ? is a Borel set offinite measure
on R', and f is a Borel measurable function on E, then, for every ? > 0, there exists a
compact set К С ? such that ?(? \ ?) < ? and such that f is continuous on K.
6.2. Convolution of measures
184. ?????????. Let v\, vi be two finite Radon measures on R'. Let ? be the product
measure on W ? R'; then ? is also a (totally finite) Radon measure. Define ? : R' ? R' ->
R' by ?(?, y)=x+y; then ? is continuous, therefore measurable. The convolution v\ * i>2
of ?>? and i>2 is the image measure ??~?. This is a Radon measure.
Remark that if v\ and i>2 are Radon probability measures, then ? and ?>? * vi are also
probability measures.
185. THEOREM. Let v\, vi be two finite Radon measures on R'; let ? = v\ * i>2 be their
convolution, and ? their product on R' ? R'. Then for any real-valued function h defined
on a subset ofW, we have
/ h(x+y)dv(x,y)= / h{x)dp(x)
if either integral is defined in [—oo, oo].
186. If ?>? and i>2 are finite Radon measures on R', then v\ * i>2 = V2 * v\ ¦ If vn v2 and
V3 are finite Radon measures on R', then (?? * V2) * vj = v\ * (^2 * V3)-
References
Bartle, R.G. A966), The Elements of Integration. Wiley. New York.
Bauer, H. A990), Mass- und Integrationstheorie. W. de Grayter & Co.. Berlin.
Bhaskara Rao. K.P.S. and Bhaskara Rao. M. A983). Theory of Charges. Academic Press. London.

82
?. Pap
Bongiomo, B. and Dinculeanu, N. B001). The Riesz representation theorem. Extension of additive measures.
J. Math. Anal. Appl. 261. 106-132.
Bourbaki. N. A952-59). Integration, Chapitres I-VI. Hermann. Paris.
Caratheodory, C. A917), Vorlesungen tiber reelle Funktionen, B.G. Teubner. Leipzig-Berlin.
Cohn. D.L. A980). Measure Theory, Birkhauser. Basel.
Diestel, J. and Uhl, J.J.. Jr. A977). Vector Measures. Amer Math. Soc.. Providence. RI.
Dinculeanu. N. A967), Vector Measures. Pergamon Press. Oxford.
Dinculeanu. N. B000). Vector Integration and Stochastic Integrations in Banach spaces. Wiley. New York.
Dinculeanu. N. B002). Vector integration in Banach spaces and application to stochastic integration. Handbook
of Measure Theory. E. Pap. ed„ Elsevier. Amsterdam. 345-399.
Dudley. R.M. A989), Real Analysis and Probability. Wadsworth & Brooks/Cole.
Dunford. N. and Schwartz. J. A958). Linear Operators. Part I. General Theory. Wiley, New York.
Egorov. D.F. A911). Sur les suites de functions mesurables. C. R. Acad. Sci. Paris 152. 244-246.
Elstrodt. J. A996). Mass- und Integrattonstheorie. Springer. Berlin.
Engelking, R. A989). General Topology. Sigma Series in Pure Mathematics, Vol. 6. Heldermann. Berlin.
Feller. W. A966). An Introduction to Probability Theory and its Applications. Vol. II, Wiley. New York.
Fremlin. D.H. B000a). Measure Theory. Volume I. Torres Fremlin.
Fremlin. D.H. B000b), Measure Theory, Volumes 2-5. Internet: www.essex.ac.uk/maths/staff/fremlin/mt.htm
George. C. A984), Exercises in Integration, Springer. Berlin.
Halmos. PR. A950). Measure Theory. Van Nostrand. New York.
Halmos. PR. A960). Naive Set Theory. Van Nostrand. New York.
Ionescu Tulcea. A. and Ionescu Tulcea, С A969). Topics in the Theory of Lifting. Springer. New York.
Kelley, J.L. A955), General Topology. Van Nostrand. New York.
Lebesgue. H. A904), Lefo/?i sur /'integration et la recherche des functions primitives, Gauthier-Villars. Paris.
Lebesgue, H. A966). Measure and the Integral, Holden-Day.
Munroe, M.E. A953). Introduction to Measure and Integration. Addison-Wesley. Reading. MA.
Renyi, A. A970), Probability Theory, North-Holland. Amsterdam.
Royden. H.L. A963). Real Analysis, Macmillan. New York.
Rudin, W. A973), Real and Complex Analysis. McGraw-Hill, New York.
Shiryayev, A. A984), Probability, Springer. Berlin.
Solovay. R.M. A970). A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math.
92, 1-56.
Widom, H. A969). Lectures on Measure and Integration, Van Nostrand, Reinhold.
Williamson, J.H. A962). Lebesgue Integration. Holt. Rinehart & Winston.

CHAPTER 3
Paradoxes in Measure Theory
Miklos Laczkovich
Department of Analysis. Eotvos Lorand University, Pdzmany Peter setany I/C. Budapest 1117. Hungary
E-mail: laczko@renyi.hu
Contents
Introduction 85
1. Paradoxical sets 86
2. Paradoxes in Ж" for ? > 3 and in non-euclidean spaces 89
3. Invariant measures and amenable groups 92
4. Decompositions and perfect matchings 97
5. The type semigroup 98
6. Nonamenable actions and local commutativity 101
7. Marczewski's problem 108
8. Tarski's circle-squaring problem 110
9. The problem of equidecomposability with measurable pieces 115
10. Countable equidecomposability and countably additive invariant measures 117
11. The nonconstructive element in the paradoxes 119
References 120
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
83

This page intentionally left blank

Paradoxes in measure theory
85
Introduction
The term 'paradox' has two different meanings. The first sense of the word refers to
a contradiction or antinomy without any apparent solution. Examples of paradoxes of
this kind were the contradictions concerning the values of some infinite series in the
seventeenth and eighteenth century (before the precise definition of convergence of infinite
series was given by Cauchy), and the set theoretical antinomies, such as Russel's paradox,
before the creation of axiomatic set theory. The other meaning of the word 'paradox' is
a correct mathematical statement contradicting our intuition. In the sequel we shall use
the term in this second meaning. The earliest, and one of the most important example
of such a paradox is the 'paradox of infinity' stating that an infinite set can be mapped
injectively into a proper subset of itself. This fact was first discovered and clearly explained
by Galileo Galilei A638). About 250 years later R. Dedekind A932) realized that this
property characterizes infinite sets and used it as the definition of infinity.
In the period 1914—1929 several geometric variants of the paradox of infinity were
discovered by Felix Hausdorff, the Polish school, and John von Neumann. Among these
discoveries the most spectacular is the famous Banach-Tarski paradox: it states that in
E3 any two bounded sets with nonempty interior are equidecomposable. Paradoxes are
always appealing, but the special charm and beauty of these statements stems from the fact
that they represent a mixture of set theory, measure theory, geometry and even discrete
mathematics. In fact, some of these paradoxes were motivated by the set theoretical
approach itself; that is, by the freedom of forming arbitrary subsets of R" and treating
them as geometric objects. By imitating the procedures of geometric dissections, they also
share the appeal of recreational mathematics.
However, the importance of these paradoxes is due to their close connection to
invariant measures, as discovered by Hausdorff, Banach, Tarski, and von Neumann. Hausdorff's
paradox, for example, was created in order to prove the nonexistence of finitely additive
invariant measures on the sphere. The exact connection is expressed by Tarski's famous
theorem A929) stating that, under very general conditions, the nonexistence of paradoxes is
equivalent to the existence of invariant measures. In this way a theory was taking shape that
unified the investigations of paradoxical sets, invariant measures and equidecomposability.
The paper of Banach and Tarski A924) is the first systematic exposition of this
theory. Tarski's book A949) was mainly motivated by these investigations, and contains
many important results of the topic. The next book that was devoted to the subject was
Sierpinski's A954). The best and most comprehensive account of the topic was given
by Stan Wagon in his book A986). Wagon's work immediately became the standard
monograph of the subject, and the interested reader is urged to consult it concerning the
results up to 1986. The survey papers by Laczkovich A991b) and A994) review some of
the later developments; part of the present article is covered by these surveys. Since this
area also involves amenable groups, we also mention the monograph by Paterson A988),
and the recent article of Ceccherini-Silberstein et al. A999). In this article we shall not
touch on the Ruziewicz problem asking the uniqueness of finitely additive probability
measures on the Lebesgue measurable subsets of the sphere S". We refer to the monograph
of A. Lubotzky A994), where the complete solution of the Ruziewicz problem is discussed.

86
?. Laczkovich
1. Paradoxical sets
The simplest geometric form of the paradox of infinity is exhibited by any set that is
congruent to a proper subset of itself. Every half-line has this property. Also, the set N
of nonnegative integers is congruent to N-, the set of positive integers. In this example
only one element of N can be deleted in such a way that the remaining set is congruent
to N. This is true in general: for every ? с R there is at most one point ? e ? such that
? \ {x} is congruent to H; see Sierpinski A954) andWagon A986, p. 92). E.G. Straus
A957) proved the same for plane sets. In R3 the situation is different. It was proved by
J. Mycielski A954, 1955) that there are infinite and bounded sets ? с R3 such that ? \ A
is congruent to ? for every finite subset of ? (see also Wagon A986, pp. 92-95)).
In the plane we may find bounded sets that are congruent to some proper subsets. Let
С denote the set of complex numbers and select a number с е С such that \c\ = 1 but с
is not a root of unity. Then the sets A = {c": ? e N) and В = {с": neN1") are bounded
(subsets of the unit circle), congruent (a rotation maps A onto ?), and ? is a proper subset
of A. These sets are Fn sets (countable unions of closed sets). We can also find Gs sets
(countable intersections of open sets) with the same property. Namely, [x eC: |дг | = l} \ A
and {x e C: \x \ = l} \ В are bounded Gs sets, they are congruent and the first is a proper
subset of the second. These examples are simplest possible in the topological sense. Indeed,
in R" no bounded and ambiguous set can be congruent to a proper subset of itself. (A set is
called ambiguous, if it is simultaneously F„ and Gs.) This is a special case of the following
theorem proved by A. Lindenbaum A926).
THEOREM l. I. Let A be an ambiguous subset of a compact metric space. If В С A and
B = Athen B = A.
(A = ? denotes that the sets A and В are congruent; that is, there is a distance-preserving
bijection from A onto ?.) The situation in R is much simpler: no bounded linear set can be
congruent to any proper subset of itself. Indeed, if ?, ? are bounded linear sets, A z> В and
A = B, then the isometry mapping A onto В must be a reflection and this easily implies
that A = B. However, we can find bounded subsets of R exhibiting the paradox of infinity
if we replace the notion of congruence by that of piecewise congruence.
We say that the sets А, В с R" are equidecomposable if there are finite partitions
A = (JL ? Aj and ? - (jf= ? ?/ such that A, = ?, for every /' = l k. We shall denote
this fact by A ~ B. If we want to indicate that A and В are equidecomposable using к
pieces, then we write A ~* B.
(We remark that Banach, Tarski and Sierpinski used the term 'equivalent by finite
decomposition', while other mathematicians used "piecewise congruent'. After Wagon's
monograph A986) the term 'equidecomposable' became prevalent.)
As Sierpinski observed A954, Theorem 16), we have [0, 1] ~.ч ]0, 1]. Indeed, let a e
]0, 1[ be an irrational number and put A = {{na}: ? e N), В = {{па): ? e N+), where
{x} denotes the fractional part of the number x. Denoting ? ? = А П [0, 1 — a[. Ai =
АП[1 -?, ?], ?| = ? ? [?, 1], ?2 = ? ? [0, ? [, one can easily check that B\ = A\ +a
and ?2 = A2 + a - 1. Then, putting A3 = B?, = [0, 1] \ A = ]0, 1] \ B, we obtain the

Paradoxes in measure theory
87
partitions [0, 1] = (J,3=i A, ]0. 1] = ULi Bi such that Ai = Bi (' = 1-2, 3) and thus we
have[0, l]~3]0, 1].
A stronger form of the paradox of infinity states that an infinite set can be decomposed
into two subsets, each of which is equivalent to the original. As S. Mazurkiewicz and
W. Sierpinski A914) realized, this paradox can also be exhibited geometrically.
THEOREM 1.2. There are nonempty sets A, A\, A2cR2 such that A = A\ U A2, A\ П
At = fl and A = A| = At.
Proof. Let ? denote the set of polynomials with nonnegative integer coefficients
(including the identically 0 polynomial), and let с be a transcendental complex number
with \c\ = 1. We put A = {p(c): ? e P], A\ = A + 1, A: = с A. Using the fact that с is
transcendental, it is easy to prove that ? ? ? A? = 0, A = ? ? U At and A = A\= ??. ?
A. Lindenbaum A926) proved that a bounded plane set cannot have this property; for
a simple proof, see Wagon A986, p. 196). The situation changes, however, if we replace
congruence by equidecomposability.
A set А с К" is called paradoxical if there are subsets ? ?, At с A such that A =
? ? U ??, A\ П ?? — 0 and A ~ ? ? ~ At. If ? ? ~„ A and At ~,„ A, then we say that A
is (n, т)-paradoxical.
Thus the Mazurkiewicz-Sierpinski theorem says that there are A, 1 )-paradoxical plane
sets, and, as Lindenbaum showed, such a set cannot be bounded. But it was not known for
more than 60 years whether or not bounded paradoxical plane sets exist. Stan Wagon A986,
p. 232) listed this question under Problem 18 in his catalogue of 19 unsolved problems in
the area of paradoxical sets, equidecomposability and invariant measures. The solution
was given by W Just A988a). He proved that there exists a nonempty, bounded and A,3)-
paradoxical plane set. It was shown by G.A. Sherman A990a) that Just's construction is
best possible, in that a nonempty bounded plane set cannot be A.2)-paradoxical. Sherman
also constructed a bounded B,2)-paradoxical plane set. Sherman proves in A991) that
every paradoxical plane set has empty interior, moreover, has inner Lebesgue measure
zero, and asks whether there is a paradoxical plane set with positive outer measure. The
following example given by M.R. Burke B000) shows that the answer to this question
is positive. Let с and A be as in the proof of Theorem 1.2. Using transfinite recursion
it is easy to prove that there exists a set ? с С of positive outer measure such that the
elements of ? are linearly independent over the field Q(c). Let В = {с"? +a: л e ?, ? e
?, a e A}. Then ? с В and thus В has positive outer measure. It is easy to check that
В = (сВ) U (? + 1) and (с?) ? (? + 1) = 0, therefore ? is paradoxical. Burke B000)
also constructs bounded paradoxical plane sets of positive outer measure.
The situation in R1 is, again, much simpler: there are no nonempty paradoxical sets in
E1. This was proved by Tarski A938a) and, independently, by Sierpinski in A946) and
A954, Theorem 19). We shall present their proof in a more general setting.
If X is a nonempty set then ?? will denote the symmetric group on X; that is, the group
of bijections mapping X onto itself. If G is a subgroup of Sx then we say that G acts on X.
' Our definition slightly differs from Wagon's. The equivalence of the two definitions follows from Theorem 2.5.

88 ?, Laczkovich
The sets А, В С X are said to be G-equidecomposable, if there are finite partitions
A = U/=| A, and В = U/=i Я/ and gr0UP elements g,¦ e G (/' = 1,... ,k) such that
S, = gi (A,) for every / = 1,... ,k. We shall denote this fact by A~S.
The set А с X is called G-paradoxical, if there are subsets A\, A 2 с A such that
A = A, UA2, A, HA2=0and A~A|~A2.
In the sequel we shall denote by G„ the group of isometries of R". Thus two sets in R"
are equidecomposable if they are G„-equidecomposable, and a set А с R" is paradoxical
if it is G„ -paradoxical.
If G is a group and Я is a subset of G then ун(п) will denote the set of elements
gi ·' -gn, where gi e ? U ?~' for every / = 1,..., ?. The group G is called exponentially
bounded, if lim,,-*;*, |yw(n)|''" = 1 for every nonempty finite ? с G. In other words, G
is exponentially bounded if, for every finite ? с G and for every ? > 0, there is an щ such
that the number of distinct group elements that can be represented as a word of length ?
with letters from ? U H~' is less than A + ?)" for every ? > щ.
Tarski and Sierpinski realized that G ? is exponentially bounded. (Proof: Each isometry
of ? is of the form g(x) = ax + b (x e R), where a = ±1. If g\ gr e G\, gi(x) =
a,x + bi, and g = g'"{ ¦¦¦g'"', then g(x) =ax +k\b\ -\ + krbr, where a = ±1 and
\kj\ < |w,-| for ever /'. If the length of the word g is и, then |w,| -? ? for every / and thus the
number of different b's is at most (In + 1/. Thus the number of different isometries of the
formg' ¦¦¦g'"r with 53,- |w,| =и is at most 2 (In + l)r < A +?)" if ? is large enough.)
The same argument shows that if G is Abelian or has an Abelian subgroup with finite
index then G is exponentially bounded. Now the statement that R1 has no nonempty
paradoxical subsets is a special case of the following theorem.
THEOREM 1.3. If an exponentially bounded group G acts on X then X does not contain
nonempty G-paradoxical subsets.
PROOF. Suppose that 0 ? А с X is paradoxical, and let A - В U С, В П С = 0,
С С
А^ В^ С. Then there are partitions
r s r s
/=1 j=\ i=l y=l
and maps У), gj e G such that //(A,) = S,, gj(A'-) = Cj (i = 1 г, у = 1 ,s).
Lei F\(x) = fj(x) (x e A,, / = 1,..., r), and F2(x) =gj(x) (x e A'y, у = 1,. ..,?), then
F| (A) = ?, /*2(A) = C. It is easy to see that the images of a fixed ? e A under the maps
FhFh...FiH (i iw = 1,2)
are distinct. Each of these images is of the form h\ ...hn(x), where h\,...,h„ e
{/? fr,g\ gs}- Thus the words of length ? with the letters f.gj define at least
2" different maps of G, which contradicts the fact that G is exponentially bounded. D

Paradoxes in measure theory
89
Every group acts on itself by left multiplication. That is, if G is a group then we may
identify every element g e G with the map ? н> gx (? e G), and thus we identify G
with a transformation group acting on the set G. According to this identification, the
sets А, В С G are equidecomposable (denoted by A ~ ?), if there are finite partitions
A = UjL, ?,, ? = Uf= ? ?< and group elements g\ , gu such that B, = g, A,- for
every / = 1 к. The set А С G is paradox, if A = A | U At, where ? ? ? ?? = 0 and
A ~ ? ? ~ A2- A group is called supramenable, if does not contain nonempty paradoxical
sets.
By Theorem 1.3, every exponentially bounded group is supramenable. It is not known
if the converse is true or not. (This is Problem 12 in Wagon's list A986, p. 231) of
unsolved problems). However, in Theorem 1.3 we may replace 'exponentially bounded'
by 'supramenable'. In other words, if a supramenable group G acts on X then X does not
contain nonempty G-paradoxical sets. Indeed, it is easy to see that if А с X is a nonempty
G-paradoxical set, then for any ? e A the set {g: g(x) e A} is a nonempty paradoxical set
in G, which is impossible.
Returning to E1 and R2, recall that (i) E1 does not have nonempty paradoxical subsets,
and (ii) every paradoxical plane set is small in that its inner Lebesgue measure is zero
(Sherman A991)). We shall see in the next section that in R" for ? ^ 3 and in non-euclidean
spaces there are 'large' paradoxical sets.
2. Paradoxes in E" for ? ^ 3 and in non-euclidean spaces
G. Vitali A905) proved that there are no nontrivial isometry-invariant and ?-additive
measures defined on all subsets of R. Motivated by Vitali's theorem, F. Hausdorff raised the
question whether or not finitely additive invariant measures exist in Ш". In 1914 he showed
that the analogous problem on the sphere S2 = {x e ??: |jc| = 1} has a negative answer.
He proved this by means of the following theorem (the so-called Hausdorff paradox).
THEOREM 2.1. There are decompositions S2 = A | U A2 U C\ and S = A3 U A4 U A5 U Сг
such that the sets A, (/' = 1,..., 5) are congruent to each other, and C\, Сг are countable.
This implies that there is no finitely additive isometry-invariant measure m defined
on all subsets of S2 satisfying m(S2) = 1. Indeed, such a measure vanishes on every
countable set С С S2, since S2 contains ? disjoint congruent copies of С for every ? e N.
Thus S2 = ?, U A2 U C\ implies w(A,) = 1/2, while S1 = ?? U A4 U A5 U C2 gives
w(A|) = 1/3, a contradiction.
An essential ingredient of Hausdorff's proof is the discovery that S03, the group of
rotations of S2 contains a subgroup isomorphic to the free product ?? * Z.v Since ?? * ? у
contains a free subgroup of rank 2 (if a and b are free generators of ?? * ?^ then bab and
ababa freely generate such a subgroup), it follows that SO3 contains a free subgroup of
rank 2. This fact has many direct proofs: see Wagon A986, p. 20). We may add the proof
given in Laczkovich A991b, Theorem 3.2) that uses the stereographic projection and the
fact that the group of linear fractional transformations contains free subgroups. In this way

90
?. Laczkovic/?
we even find free subgroups of SO3 with continuum many generators, a fact first proved
by Sierpinski in A945b); see also Wagon A986, pp. 94-95).
We remark that Hausdorff's paradox can be constructed without the countable sets
C\, Ci, as J.F. Adams showed in A954). However, Adams has to use orientation-changing
isometries, while Hausdorff only used rotations.
The problem of the existence of invariant finitely additive measures in R' and R- was
solved by Banach A923).
THEOREM 2.2. For ? = 1 and « = 2 the Lebesgue measure can be extended to all subsets
o/R" as a finitely additive and isometry-invariant measure.
(These extensions are called Banach measures.) Banach's theorem also gives the
following important corollary.
THEOREM 2.3. Let ? = 1 or ? =2. If the Lebesgue measurable sets А. В С R" are
equidecomposable, then ?„(?) = ?„(?).
Indeed, let A = (J*=, A, and В = (J*_, S, be partitions such that A, = ?, for every
/ = \,...,k.lfm is a Banach measure then we have
к к
X„{A)=m{A) = Y^m(Ai) = Y^m(Bi)=m(B)=k,AB).
i = \ i=\
The nonexistence of Banach measures in R3 was proved by Banach and Tarski A924),
by making use of the following theorem, the so-called Banach-Tarski paradox.
THEOREM 2.4. If А, В С R3 are bounded sets with nonempty interior then A ~ B. In
particular, every bounded subset ofM? with nonempty interior is paradoxical.
The same is true in R" for every ? ^ 3. We can see, using the argument of the proof of
Theorem 2.3 that in R" (? ^ 3) there is no Banach measure.
The proof of Theorem 2.4 given by Banach and Tarski uses Hausdorff's paradox and the
following result.
THEOREM 2.5. Suppose that G acts on the set X.IfAcX is G-equidecomposable to a
subset of В С X using к pieces, and В is G -equidecomposable to a subset of A using ?
pieces, then A ~ В using к -\-n pieces.
This is an immediate consequence of the following statement, the so-called Banach-
Schroder-Bernstein theorem: if f:A —> В and g: В —*¦ A are injections, then there are
decompositions A = A\ U At and В = B\ U S: such that f{A\) = B\ and g(Bi) = Aj.
See Banach A924) and Wagon A986. Theorem 3.5).
The proof of the Banach-Tarski paradox can be slightly simplified if - instead of
Hausdorff's paradox - we use the following statement, which is an easy consequence of

Paradoxes in measure theory
91
the fact that SO3 contains a free group of rank 2: There is a subset ? С S2 such that (i) S~
can be covered by four congruent copies of E, and (ii) 5"" contains infinitely many disjoint
congruent copies ofE. (See, e.g., Laczkovich A991b, Theorem 3.3).) Now let А, В с R3
be bounded sets with nonempty interior. We may assume that both A and В contain the unit
ball D = [x e R3: |jc| ^ 1). Suppose that A can be covered by к congruent copies of D
and that В can be covered by ? congruent copies of D. Let ? be a set with the properties
(i) and (ii) above, and let E* = \J{rE: 0 < r < 1). Then D is covered by 5 congruent
copies of E* (one of them is needed to cover the center), and also contains infinitely many
disjoint congruent copies of E*. From the fact that A is covered by к congruent copies
of D it follows that A is equidecomposable to a subset ofDcS using 5k pieces, and,
similarly, В is equidecomposable to a subset of D с A using 5« pieces. Therefore, by
Theorem 2.5, we obtain that A ~ В using 5(it + /;) pieces. In particular we find that if D\
and Z>2 are disjoint balls of radius 1 then D ~ D\ U Di using 15 pieces. This number can
be reduced to 11, since D is equidecomposable to D\ using one single piece, and thus, by
Theorem 2.5, we only need 1 +5-2=11 pieces. In other words, D is paradoxical using
11 pieces. In Section 6 we will see that, in fact, 5 pieces suffice.
Similar paradoxes exist in non-euclidean spaces, too. Let the elliptic and hyperbolic 11-
space be denoted by L" and H", respectively. It is easy to see that the group of rotations
of L"_l around a fixed point acts in the same way as SO,,, the special orthogonal group,
acts upon R". This implies that L" is paradoxical for every ? > 2. Also, the action of the
group of rotations of H" around a fixed point is isomorphic to the action of SO,, on R"
and hence, for ? ^ 3, every bounded subset of H" with nonempty interior is paradoxical.
It was shown by J. Mycielski and S. Wagon that H2 is paradoxical using Borel sets; see
J. Mycielski and S. Wagon A984) and Wagon A986, Theorem 5.9). They used the fact
that the isometry group of //- contains a subgroup isomorphic to the free product ? 2 * ? ?
(Note that the same group appeared in the original proof of the Hausdorff paradox.) Later
Mycielski proved A989) that the bounded subsets of H2 with nonempty interior are also
paradoxical.
Bounded paradoxical subsets of H2 and S2 were also investigated by G.A. Sherman
A990b). In this paper Sherman determines those pairs (in. 11) for which there exists an
(m,n) -paradoxical subset of a disc of given radius in either ? - от S-.
The paradoxes discussed in this section depend on the fact that the transformation groups
in question contain F2 (the free group of rank 2) as a subgroup. It was proved by von
Neumann A929) that every such group is paradoxical.
THEOREM 2.6. If a group G contains Fj as a subgroup then G is paradoxical.
PROOF. First we show that Fi is paradoxical. If a, b are the generators of Ft, then every
element of F2 is represented by a reduced word in the letters a, a~', b, b~'. Let A denote
the set of those reduced words that begin with the letters a or a-1. Since Fj = A U a~ A
and abA, a~]bA are disjoint subsets of A, it follows that Ft is equidecomposable to a
subset of A. Therefore, by Theorem 2.5, F2 ~ A. Also, bA and b~] A are disjoint subsets
of F2 \ A, therefore F2 ~ Fi \ A, and thus F? is paradoxical. (One can prove that F2 is
B, 2)-paradoxical; see Wagon A986, Theorem 4.2).)

92
?. Laczkovich
Now suppose that Fj is a subgroup of G, and let U с G be a set containing exactly one
element of each right coset of Fi. Then every g e G can be written uniquely in the form
hu, where h e F2 and и eU. Let F> = A U ? be a decomposition such that Ft ~ A ~ B.
Then G = (AU) U (Si/) is a decomposition with G ~ ??/ ~ BU, showing that G is
paradoxical. D
A similar argument shows that if a group G contains a free subsemigroup of rank 2,
then G contains nonempty paradoxical subsets. Indeed, if S С G is a free subsemigroup
generated by the elements a and b, then a S. bS are disjoint subsets of S, and thus
S ~2 (S \ aS). Therefore 5 is A. 2)-paradoxical. (Note that the proof of Theorem 1.2
exhibits a free subsemigroup of G: generated by a rotation and a translation.)
3. Invariant measures and amenable groups
It was realized by John von Neumann A929) that most of the previous results can be put
into a more general context, which also explains the different properties of the action of
G„ in the cases of ? ^ 2 and /; ^ 3. The central notions of this theory are finitely additive
invariant measures and amenable groups.
Let А С V(X) be a ring of sets. A set function m : A -*¦ [0. oo] is said to be a finitely
additive measure if w@) = 0 and m(A U ?) = m(A) + m(B) whenever A and В are
disjoint elements of A. If G acts on X then we say that m is invariant under G if g( A) e A
and m(g(A)) =m(A) for every g e G and A e A.
A group G is said to be amenable if there is a finitely additive probability measure on
V(G) that is invariant under the action of G on itself. In other words, G is amenable, if
there is a finitely additive measure m on V(G) such that m(G) = 1 and m is left invariant;
that is, m(gA) — ?(?) for every А С G and g e G.
John von Neumann discovered that the contrast between Theorems 2.2 and 2.4 is due
to the fact that the transformation groups G\ and G: are amenable, while G3 is not.
In fact, G\ and G: are solvable and, as von Neumann proved, every solvable group is
amenable. (G\ is solvable because that group T\ of translations of R is an Abelian and
normal subgroup of G1 such that the factor group G\/T\ is the two-element cyclic group.
The group G: is also solvable. Indeed, let Г: denote the group of translation of R-, and
let SGi denote the group generated by the translations and rotations. Then in the sequence
of groups {identity}, Г2. SG2. G: each group is a normal subgroup of the next, and the
factor groups are Abelian.)
Since G| and Gi are amenable, the existence of Banach measures in R1 and R2 is a
special case of the following result, called the invariant measure extension theorem.
THEOREM 3.1. Suppose that G is an amenable group acting on X, and let m be a G-
invariant finitely additive measure defined on a G- invariant field А С V(X). Then m can
be extended to V(X) as a G-invariantfinitely additive measure.
This important theorem, in this general form, was first stated and proved by Jan
Mycielski A979) although in some special cases it was known to Banach A923) and von

Paradoxes in measure theory
93
Neumann A929). Since the group of translations is commutative and commutative groups
are amenable by von Neumann's theorem, we obtain the following corollary.
THEOREM 3.2. For every ? there exists a translation-invariant extension of the L^ebesgue
measure ?„ toV(R").
PROOF OF THEOREM 3.1. First we need the measure extension theorem stating that if m
is a finitely additive measure on the ring Л С V(X), then m can be extended to V(X) as
a finitely additive measure. (Proof: Let ? denote the set of functions 53!'= ? c'Xa,· where
c\,..., c„ e Ш and A\,. ..,A„ are pairwise disjoint elements of ? of finite m-measure.
Then ? is a linear space, and Lf = 53!'= ? c,w(A,) defines a positive linear functional on
M. We put
M* = {f:X^R: 3/ieM, |/Ю}.
and define p(f) = miLh, where h runs through all functions from ? satisfying / ^ h.
It is easy to check that M* is also a linear space and p(f + g) ^ p(f) + pig) and
p(cf) = cp(f) for every /. g e M* and с > 0. Also, if / e ? then Lf sC p(f). By
the Hahn-Banach theorem, L can be extended as a linear functional to M* satisfying
Lf ^ p(f) for every / e M*. Now we define v(H) = LxH if хн e M* and v(H) = oo
otherwise. It is easy to see that у is a finitely additive extension of m to V(X).)
We remark that the measure extension theorem is actually equivalent to the Hahn-
Banach theorem; see Wagon A986, Chapter 13).
We shall also need the notion of integral with respect to finitely additive measures.
In the sequel we shall denote by B(H) the linear space of bounded real valued functions
defined on the set H.
Let m be a finitely additive measure defined on V(X). Let А с X be a set of finite m-
measure. If /: A —*¦ ? only takes a finite number of values (that is, if / is a so-called step
function), then there is a partition A = A \ U · · · U A» and there are real numbers с \...., ctl
such that / = J3!'= ? ciXa, ¦ In this case we define
f "
/ fdm = УЗс;/я(А,-).
Ja ы\
If / e B(A) is arbitrary then we choose a sequence ft'-A -» R of step functions
converging uniformly to /, and define
/ f dm = lim / fidm.
J a k-*^JA
It is easy to check that the limit exists and is independent of the choice of the sequence
(fk). It is also easy to prove that integral defined in this way has the following properties,
(i) inf/ · m(A) ^ fA f dm ^ sup/ · m(A) for every / e B(A);
(ii) For every А с X with finite m-measure the map / н> fA f dm defines a linear
functional on the space B(A);

94
?. Laczkovich
(iii) for every / e B(X), the integral Д f dm defines a finitely additive set function on
the ring {А С X: m(A) < oo}.
Now the proof of Theorem 3.1 is as follows. Suppose that m is a G-invariant finitely
additive measure on the G-invariant ring Л С V(X). By the measure extension theorem,
there is a finitely additive extension ? of m to V(X). Since G is amenable by assumption,
there is a finitely additive and left invariant probability measure ? on V(G). If А с X then
let /д(#) = v(g~f (A)) (g e G). Now we define ?(?) = fG fA dy if /д is bounded and
?( A) = oo otherwise. It is easy to show that ? is a G-invariant extension of m. D
Let m be a G-invariant finitely additive measure on V(X). If A~Z? then necessarily
m(A) = m(B). Indeed, let A = (jf=| A and ? = (J*=i ?/ be partitions, and let g,- e G
be transformations such that ?, = g, (A,) for every /' = 1 k. Then we have m(A) =
??=? w(A,) = ?*=| m(B,)=m(B).
С С
If А с X is G-paradoxical, then A = ? ? U A?, where ? ? П At = 0 and A~ ? ? ~ At.
Therefore m(A) = m(A\) + w(A:) = 2m(A) and thus either m(A) = 0 or w(A) = oo.
If G is amenable and m is a left invariant finitely additive measure on V(G) satisfying
m(G) = 1, then every paradoxical subset of G must be of measure zero. In particular, the
group itself cannot be paradoxical. In Section 5 we will see that, by a theorem of Tarski,
nonamenable groups are paradoxical and thus a group is amenable if and only if it is not
paradoxical.
Next we shall briefly review the basic facts about amenable groups.
(i) If G is amenable then there is a finitely additive probability measure ? on V(G)
which is bilaterally invariant; meaning that v(gA) = v(Ag) = v(A) for every
А С G and g e G. (Proof: if m is a left invariant finitely additive probability
measure on V(G) then we put v( A) = fG f:\ dm for every А с G, where fA(g) =
m(A~]g) (g e G). It is easy to check that ? satisfies the requirements.)
(ii) If G is amenable then so is every subgroup and homomorphic image of G. (Proof:
Let m be a finitely additive invariant probability measure ? on V(G). If ? is a
homomorphismof G onto ? then v(A) = т(ф~] (А)) (А с Н) defines a finitely
additive invariant probability measure onV(H).
Next suppose that ? is a subgroup of G, and let U с G be a set containing
exactly one element of each right coset of H. Then ? (A) = m(AU) defines a
finitely additive invariant probability measure on V(H).)
(iii) If Я is a normal subgroup of G. and if ? and G/H are both amenable then
so is G. (Proof: Let ? and ? be nonnegative finitely additive and bilaterally
invariant probability measures on V(H) and V(G/H), respectively. We extend
? by ?(#?) = ?(?) (А с ?. g e G). This definition makes sense, since
A\. A2 С Я, g|A| = #2^2 implies g^'gi e Я and thus, by the invariance of
?, ?(?|) = ?(g^?g\A\) = ?(?2). This extension defines ? on the power set of
each coset of H.
Let ?-.G^- G/H be the natural homomorphism, then 0_l (y) is a coset of ?
for every у e G/H. Let А с G be fixed. Then g(y) = ?(? ? </>"' (У)) defines a
bounded function on G/H. We put у (A) = fG/H gdv; it is easy to check that ?
is an invariant finitely additive probability measure on V(G).)

Paradoxes in measure theory
95
(iv) Let G be a group. We say that L: B(G) ->· R is a left invariant mean, if L is a
linear functional satisfying inf/ ^ Lf ^ sup/ and L/ = LTKf for every / e
B(G) and g e G, where the function ?<,/ is defined by TKf(x) = f(g.x) (x e G).
Then the group G is amenable if and only if there is a left invariant mean
on B(G). (Proof: If L is a left invariant mean then m(A) = L/a defines a left
invariant finitely additive probability measure on G, and thus G is amenable.
On the other hand, if G is amenable and m is a left invariant finitely additive
probability measure on G, then Lf = fG f dm is a left invariant mean on B(G).)
We note that the existence of a left invariant mean on B(G) is sometimes used as
the 'Bourbaki style' definition of amenability.
(v) Let G be a group, and let N denote the set of functions of the form ]T"=, [THj f —
f], where /?,..., /„ e B(G) and g\ g„ e G. Then G is amenable if and only
if inf / ^ 0 ^ sup / for every / e N ('Dixmier's condition'). (Proof: If L is a left
invariant mean on B(G) then Lf =0 for every /?? and thus inf/ ^ 0 ^ sup /
by the definition of means. If the condition is satisfied then an easy application of
the Hahn-Banach theorem gives a linear functional L on B(G) such that Lf = 0
for every f e N and Lf ^ sup / for every / e B(G). It is easy to check that L is
a left invariant mean.)
It is a remarkable fact that if G is nonamenable then not only inf / > 0 holds
for some / e N, but actually N = B(G). In other words, G is amenable if
and only if ? ? B(G). This was proved by G.A. Willis A988). For ? > 1 let
lp(G) = {/: G -+ R: ? G |/(g)|'' < oo), and let N,, be the linear span of the
functions Tsf — f (/ e lp(G). g e G). Willis also proves that G is amenable if
and only if Np ? lp(G). For a generalization for graphs see G. Elek A998).
(vi) Every Abelian group is amenable. The 'classical' proof runs by checking that
Dixmier's condition holds: if / = ?"=, [??, / - / ] then the average of the values
of / at the points g'^ ¦ ¦ ¦ g,'," @ ^ /1 i„ < ?) tends to zero if N ->· oo due to
cancellations, and thus inf/ ^ 0 ^ sup /. (The same argument proves that every
exponentially bounded group is amenable.) A nice functional analytic proof runs
as follows: Let ? = LX(G)' with the weak* topology. Then
К = \L e LX(G)': inf/ ^ Lf ^ sup /(/ e L^G))}
is a compact convex subset of E. The translations Tg (g e G) form an Abelian
semigroup of continuous affine transformations mapping К into itself. Then, by
the Markov-Kakutani fixed point theorem, they have a common fixed point, which
is a left invariant mean on B(G). See Wagon A986, p. 159) and Paterson A988,
p. 14).
(vii) Every solvable group is amenable; this is clear by (iii) and (vi). It is also easy to
see that if G has a normal solvable subgroup of finite index then G is amenable.
(viii) A group G is amenable if and only if every finitely generated subgroup of G
is amenable. (Proof: if G is amenable then so is every subgroup by (ii). If G is
not amenable then, by (v), there is a function / e N satisfying sup / < 0. Let
/ = ??=? lTx, fi ~ Я, where / /„ e B(G) and g, gn e G. and let

96
?. LaczMorich
? denote the subgroup generated by gi gn. Since sup f\H ^ sup/ < 0, it
follows from (v) that ? (a finitely generated subgroup of G) is not amenable.)
(ix) A group G is amenable if and only if any of the following statements is true
('F0lner's conditions').
(F|) For every nonempty finite set А с G and for every ? > 0 there exists a nonempty
finite В CG such that \(??)??\ <?\?\.
(F?) For every nonempty finite set А с G and for every ? > 0 there exists a nonempty
finite В с G such that \(??)??\ < ?\?\ for every ? e A.
(Here ? denotes symmetric difference.) A particularly simple proof of the equivalence
(F|) «=> (Fi) «=> G is amenable
was found recently by T. Ceccherini-Silberstein et al. A999) using the result of W.A. Deu-
beret al. A995) on paradoxical metric spaces. See also Laczkovich B001).
There are several other characterizations of amenability. One can use recurrence of
random walks on the group, or certain growth conditions (akin to the notion of exponential
boundedness), and there are cohomological characterizations as well. For the details we
refer to Chapter 12 of Wagon's book A986), to Paterson's monograph A988), and to
the paper by Ceccherini-Silberstein et al. A999). All these notions and results can be
generalized to topological groups. A topological group is called amenable if there is a
finitely additive left invariant probability measure on the Borel sets of the group. Here we
only mention Paterson's theorem A986): a locally compact group is amenable if and only if
it is not paradoxical using Borel pieces. As for the theory of amenable topological groups,
we again refer to Paterson's book.
As we saw in Theorem 2.6, if a group G contains Ft as a subgroup then G is paradoxical
and, consequently, is not amenable. It was not known for 50 years whether the converse
is true or not. The problem was solved by A.Yu. Ol'shanskii A980), who constructed a
paradoxical and periodic group. (A group is periodic if each of its elements has finite
order.) It is obvious that a periodic group cannot contain Ft, and thus Ol'shanskii's group
is nonamenable, but does not contain Ft. Later S.I. Adian proved A983) that the Burnside
group B{m,n) is also paradoxical (and, of course, also periodic) if m ^ 2 and ? ^ 665 is
odd. (The group B(m,n) has m generators and is defined by the relations w" = e, where
w runs through all words.) Recently Ceccherini-Silberstein et al. A998) showed that for
m ^ 2 and ? ^ 665 odd the group B(m. n) is paradoxical with at most 14 pieces.
However, in some important classes of groups, amenability is equivalent to the
nonexistence of free subgroups. By a theorem of J. Tits A972), for every subgroup G
of GL„ (the group of nonsingular linear transformations of R") either G contains a normal
subgroup of finite index, and thus it is amenable, or contains F> as a subgroup. The same
is true for every subgroup of Gn. See Wagon A986, p. 151).
We note that a group G contains Fn as a subgroup if and only if there is an infinite subset
? С G such that for every finite set D с ? there is an element g e G with gE = ? \ D
(theorem of EG. Straus); see Wagon A986, pp. 92-94).
A periodic group does not contain free subsemigroups either, and thus B(m,n) (for
m ^ 2 and ? odd, ? ^ 665) is also an example of a non-supramenable group without free

Paradoxes in measure theory
97
subsemigroups. However, by a theorem of Rosenblatt A974), every solvable group is either
supramenable or contains a free subsemigroup of rank 2. Rosenblatt conjectured that the
same is true for amenable groups; this is still open. As we mentioned in the first section,
it is also open, whether or not every supramenable group is exponentially bounded. See
Wagon A986, Problems 11 and 12on p. 231), and PatersonA988, Chapter6).
4. Decompositions and perfect matchings
It was discovered by D. Konig (the founder of modern graph theory) that
equidecomposabiUty of sets can be formulated in terms of perfect matchings of bipartite graphs: see
Konig A926). Tarski A938a) also used graphs in questions of equidecomposabiUty.
Recently W. Just A988b) described this approach explicitly.
Let А, В be arbitrary sets. By a bipartite graph on the pair of sets (A. B) we shall
mean a multiset of unordered pairs (л-, у) such that ? e A and у e B. The pairs (x, y) are
called lines connecting ? and y. Note that we allow multiple lines. Also, if А П ? ? 0
and ? e ? ? ?, then the loop (x,x) is allowed. We shall use the terms degree, walk, path,
cycle and perfect matching in the usual sense; see Lovasz and Plummer A986, pp. xxix-
xxxii). Thus Г is a perfect matching, if there is a bijection / of A onto В such that
Г = ((х,/(х)):хеА].
Let G be a family of functions mapping subsets of X into X, and let А. В С X. Wc.
define a bipartite graph on (A, B) by
Гс(А,В) = {(х,у): хеА. у e B. 3/e G, ? eDom/, f(x) = y].
The connection between equidecomposabiUty and perfect matchings is explained by the
following statement; it is an immediate consequence of the definitions.
THEOREM 4.1. Let G act on X. For every А, В e X we have A ~ В if and only if there is
a finite subsystem ? CG such that Гн(А, В) contains a perfect matching.
By this theorem, in order to prove the equidecomposabiUty of two sets, we have to find
perfect matchings in some bipartite graphs. The most important tool in the search of perfect
matchings is the following theorem due to M. Hall A948) and R. Rado A949). A graph
will be called locally finite, if the degree of each point is finite. If У is a subset of the
points of a graph Г then we shall denote by ?(?) the set of those points of ? which are
connected to at least one point of ?.
THEOREM 4.2. A locally finite bipartite graph ? contains a perfect matching if and only
if\r(Y)\ ^ |У| holds for every finite set ? of points of Г.
This theorem is an immediate consequence of the Banach-Schroder-Bernstein theorem,
and the following result (called marriage lemma): Let Г be a locally finite bipartite graph
on the pair of sets (?, ?) and suppose that for every finite ? С A we have \?(?)\ ^ |У|.
Then there is an injective map f: A —> В such that (x, f(x)) e Г for every ? e A.

98
?. Laczkovich
For a proof see Hewitt and Ross A979, p. 248) or Laczkovich A991b, p. 157).
An important application of Theorem 4.2 is the following classical result of D. Konig
and S. Valk<5( 1925).
THEOREM 4.3. Let G act on X. Let ? ? Al. B\ Bk С X, and suppose ?? ?) A j =
BjnBj=Q, Ai-Aj, ?,- Bj (i ? j). flndUi=i Aj?\Jki = \Bj.Then A\~ B\.
PROOF. There is a finite subsystem ? с G such that the graphs Гц(A\. A,), ?)/(?|. В,)
(/' = 1,—к) and Гн(и?=| ^bULi ?') contain the perfect matchings M,, N,- (/ =
\,...,k) and />, respectively. We define a graph ? on (A\. B\) as follows: we put (x,y)
in ? if and only if there is an /' and there are points xj e A,, y, e S, such that (x, .r,) e ?,,
U/,}·,) e i" and (у,-, y) e N/. Then Г с Twi(A|, ?|) and ? is regular (the degree of each
of its point is к). This easily i mplies that Г satisfies the Hall-Rado condition | Г (?) | ^ | У |
and hence, by Theorem 4.2, Г contains a perfect matching. Obviously, Я3 is a finite subset
of G and thus, by Theorem4.1, A| ~ ?|. ?
In the applications of Theorem 4.1 the following simple facts are often useful; cf.
Laczkovich A991b, 1992c).
PROPOSITION 4.4. Let ? be a connected and locally finite bipartite graph.
(i) If ? is a tree, and there is at most one point of ? of degree 1, then ? contains a
perfect matching.
(ii) If ? contains at most one cycle, and if the degree of each point of ? is at least two,
then ? contains a perfect watching.
5. The type semigroup
Now we come to the fundamental theorem of Tarski A929, 1938b) establishing the exact
relation between paradoxical sets and invariant measures.
THEOREM 5.1. Suppose G acts on X and ? С X. Then there is a G-invariant finitely
additive measure defined on V(X) such that ?(?) = 1 if and only if ? is not G-
paradoxical.
As an immediate corollary we obtain that a group is amenable if and only if it is not
paradoxical.
Tarski's proof is based on the notion of the type semigroup: the free Abelian group
generated by V(X) factorized by the equivalence relation of equidecomposability. The
exact definition is the following. Let G act on X. Put X* = ? ? ? and let R be the ring of
sets U"^0 A, x {/'} (n e N, A,¦ с X (i = 0 «)). If g e G and ? is a permutation of N
then we define the map (g. ?) by
(g,n)(x,n) = (g(x).n(n)) (xeX. neN).

Paradoxes in measure theory-
99
Obviously, the set G* of all these maps (g, ?) forms a group of bijections of X* onto
itself, and R is a G*-invariant ring. The semigroup types are defined as the equivalence
classes with respect to the equivalence relation ~ in R. If [A] and [?] denote the classes
containing the sets А, В e R, respectively, then [A] + [?] is defined as the class containing
the set g\(A) U gi(B), where g\, gi e G* are such that g\(A) ? gi(B) = 0. It is easy
to check that this operation is well-defined and makes the set of types a commutative
semigroup denoted by S. If we identify X with ? ? {0} then for every А, В с X we
have A ~ В if and only if A ~ B. That is, [A] = [?] if and only if A ~ ?.
Since 0 = [0] is a neutral element of S, it follows that S is a monoid (semigroup with a
neutral element). We can introduce a partial order in S by defining
x ^y «=> 3; e 5", jc + ; = y.
If А, В с X then [A] ^ [?] if and only if A is equidecomposable to a subset of B. The fact
that ^ is a partial order (reflexive, transitive, and satisfies ? ^ у. у ^ ? =>¦ ? = у) is an
immediate consequence of Theorem 2.5. It is easy to see that ? ^ у implies ? + ? ^ у + ?
for every ? e S.
In the language of the type semigroup, the Konig-Valko theorem (Theorem 4.3)
becomes the statement nx = ny =>¦ ? = у (х. у е S). That is, the cancellation law
holds in the type semigroup S. One can also prove (using the marriage lemma instead
of Theorem 4.2) that nx ^ иу =>¦ ? ?. у for every x, ye 5.
It is clear that a set А с X is paradoxical if and only if [A] = 2[A]. We prove that А с X
is paradoxical if and only if there is an ? e N such that (n + 1 )[A] ^ n[A]. The 'only if
part is clear. Let [A] = x, and suppose (/; + 1 )л ^ nx. If я = 0 then ? = [A] ^ 0. A = 0
and then A is paradox. If ? > 0 then (n +k + \)x ^ (« + k)x for every A; e N, and thus
2nx ^ Bи — 1)jc ^ - - -^ (n + 1 )x ^ nx. Since я.г ^ 2«.v is also true, we have ? ? = 2nx.
Then ? = 2x by cancellation, and thus A is paradox.
Let m be a G-invariant finitely additive measure on V(X). As we saw in Section 3,
?~? implies m{A) = m(B) for every ?, ? с X. Therefore ? ([?]) = m(A) (А С X)
defines a map ?-.S^- [0, oo]. It is clear that ? is a homomorphism from S into the
additive semigroup [0, oo]. The converse is also clear: if ? is a homomorphism from S into
the additive semigroup [0. oo], then m(A) = ф[А] (А С X) defines a G-invariant finitely
additive measure on V(X).
Therefore Tarski's theorem is equivalent to the following statement.
THEOREM 5.2. Let S be an Abelian semigroup and a e S. Then either (n + 1 )a ^ na for
some ? e N or there is a homomorphism ? of S into the additive semigroup [0, oo] such
????(?) = 1.
PROOF (Sketch). Suppose that (n + \)a ^ na holds for no ? e N. Then the elements
na (n e N) are distinct. Let Sa = [na: ? e N) and F = (x e S: ? + у = na for some у е
S, ? e N). Then Sa, F are subsemigroups of S. Let ?(??) = ? (n e ?), then ? has the
property that whenever ? ? л„, у ? jie Sa and x\ -\ \-x,i+z. = y\ ? \~Ук
for some ? e F then ?(?\) -\ +?(?„) ?. ?(\\) -\ \-ф(ук). Then one proves, using

100
?. Laczkovich
transfinite induction, that ? can be extended to F preserving this property. Finally we
define ?(?) = oo for every ? e S \ F. ?
Tarski's theorem can also be deduced from general 'sandwich theorems' for monotone
additive functions, see P. Plappert A995).
As we saw above, if S is a type semigroup then (S, ^) is a partially ordered monoid
satisfying ? ^ у => ? + ? ^ у + ? for every ? e S. The following, more general structure
was introduced by Friedrich Wehrung. We say that (A. ^) is a positively ordered monoid
(POM) if A is a commutative monoid, and ^ is a quasiorder (reflexive and transitive
relation) on A satisfying 0 ^ ? and ? ^ у =? ? + ? $5 у + ? for every ?, у, ? e S. The
investigations of Wehrung A992) were motivated, to a large extent, by Tarski's theorem.
If a POM can be embedded into a type semigroup then it has to satisfy the axioms
? ^ у, у ^ ? =>¦ ? = у and nx ^ иу =>¦ д ^ y. Wehrung discovered three more axioms
that, together with these two axioms, characterize those POMs that can be embedded into
a type semigroup. An important consequence of this result is the fact that the set of all
universal formulas of the language (+. ^) that hold in every type semigroup is decidable.
See Wehrung A994a, Theorem 5.3 and Corollary 5.5).
The following is an example of a formula that holds in every type semigroup:
(x +z =y + z and ? ^ д. ? ^ y) =>- ? = }'¦
(Proof: Let ? = и + ?, y = v+z. Then ?+? = ?+ ?=? u+2z = ? + 2?=? 2(u+z) =
u+u+2z = u + v + 2z = ? + и +2z = ? + ? + 2z = 2(v + z) => ? = u + z = v+z = }'-)
In general ? +z =y +z does not imply л = у. For example, if X = K, G is the group of
translations, ? = [N] = [N+], л· = 0, у = [{0}], then ? + ? = [Щ = [{0}] + [?+] = у + г,
but obviously ? ? у. The implication л- + ? = у + ? => ? = у may fail even if ? < }'·
Indeed, if a is the type of a nonempty paradoxical set, then 0 + a = a + a, but 0 ??.
Moreover, it can happen that ? + ? = ? + ?- but д· ^ ? does not hold (see Wehrung A994a,
Proposition 5.1)). However, Tarski proved that if the transformation group is exponentially
bounded then (x + ? = у + ? and ? ^ у) =>· x = у. See Tarski A949, pp. 224-229). This
theorem is a generalization of the fact that if the transformation group is exponentially
bounded, then there are no nonempty paradoxical sets.
As we saw above, if А, В с X are G-equidecomposable then m(A) = m(B) for
every G-invariant finitely additive measure. The converse is not true, as the following
example shows. Let X = Q be the set of rationals, and let G denote the group of all
translations by rational numbers. Let m be any G-invariant finitely additive measure on
Q. If m(]0, 1] П Q) = oo then obviously m([0. 1] П Q) = oo. If m(]0, 1] ? Q) < oo,
then m(\x}) = 0 for every ? e Q and hence w([0. 1] П Q) = m(]0, 1] П Q). That is,
/w([0. 1] П Q) = w(]0, 1] П Q) holds for every G-invariant finitely additive measure ?.
On the other hand, it is easy to see that ]0.1 ] П Q and [0, 1 ] ? Q are not equidecomposable;
see Sierpinski A954, Theorem 17, p. 48).
It can also happen that ?(?) = ?(?) holds for every G-invariant finitely additive and
finite valued signed measure without A and В being equidecomposable. Indeed, let X — Ъ
and let G be the group of translations of ? by integers. If ? is a finite valued G-invariant

Paradoxes in measure theory
101
signed measure on ? then ?(?) = ?(?+) and hence /?({0}) = 0 = /j@). On the other hand,
{0} and 0 are not equidecomposable.
The situation changes, however, if the value oo is also allowed. By a G-invariant finitely
additive signed measure we shall mean a map ?: V{X) ->-RU {oo} such that /j@) = 0,
?(? U ?) = ?(?) + ?(?) whenever ?, ? are disjoint subsets of X, and ?^(?)) = ?(?)
for every А с X and g e G. The following theorem is proved in Laczkovich A991a).
THEOREM 5.3. For every A,BcXwe have A~ В if and only ifn(A) = ?(?) holds for
every G-invariant finitely additive signed measure ?.
An equivalent formulation of this result is that every type semigroup can be embedded
into a power ofM. U {oo}. Now we consider the more general situation where the sets A, B
and the pieces used in the decompositions are restricted to be in a prescribed ring of subsets
ofX.
By a space we shall mean a triple (X. G, A), where X is a nonempty set, G is a group
of bijections of X onto itself, and Л is a G-invariant ring of subsets of X. We say that the
sets А, В e A are G-equidecomposable in A, if they are G-equidecomposable in such a
way that the pieces used in the decompositions belong to A. The type semigroup of the
space (X, G, A) is defined similarly (with the obvious modifications) as in the case of
A = V(X). Laczkovich A991a) also proves the following generalization of Theorem 5.3:
Let (X, G, A) be a space and let А, В е A. Then ?(?) = ?(?) holds for every G-invariant
finitely additive signed measure on A if and only if there is a positive integer n such that
n[A] = n[B]. Therefore the statement of Theorem 5.3 is true in a space if and only if the
cancellation law holds in the type semigroup of the space.
The problem, whether there are spaces for which the cancellation law fails, was
formulated by Wagon A986, Problem 14, p. 231). The first examples were given in
Truss A990) and in Gardner and Laczkovich A990). The transformation groups of these
examples were noncommutative. Examples with Abelian transformation groups were
constructed in Laczkovich A991a). For example, let X = E, let G be the group of
translations, and let ?, ? be positive real numbers such that ?/? is irrational. It is shown in
Laczkovich A991a) that the cancellation law fails in the space (X, G,A), where A is the
translation-invariant field generated by the sets
OO "V
(J [na,Bn + \)a/2[ and |J [??, Bn + \)??[.
ll= — OC П — — ОС
Wagon also asks if the cancellation law holds for Borel equidecomposability in a locally
compact topological group. This difficult and important problem is open even in E.
6. Nonamenable actions and local commutativity
Suppose that G acts on the set X, and let ло e X be fixed. It is easy to check that the map
Ah A = {geG: g(.x0)e A] (ACX)

102
?. Laczkovich
has the following properties: ? ? ? = 0 =>¦ АПЙ = 0. and ? ~ ? =>¦ A ~ B. From this
observation it follows that (i) if X contains a nonempty paradox subset then so does G;
and (ii) if X is paradox then so is G. In other words:
(i) if G is supramenable then X does not contain nonempty paradox subsets;
(ii) ifG is amenable then X is not paradox.
The converse of these statements is not true. Moreover, as the following example shows, it
can happen that X does not contain nonempty paradox subsets, but G is paradox, moreover,
contains Fi as a subgroup.
Let X„ (n = 1,2,...) be pairwise disjoint finite sets with \X„\ = n, and put X =
\JT=\ X»· ^et ^ be ше set °f those elements of Sx that map each X„ onto itself. If
А, В С X and A ~ В then | А П Xn \ = \ ? ? X„ | for every «, and it follows that X does
not contain nonempty paradox subsets. Now we define two maps /, g e Sx such that they
generate a free subgroup of G. Let ?\. ??, - - - be an enumeration of all reduced words in
the letters /, /-l,g, g~\ and let «? < in < ¦¦¦ be an increasing sequence of positive
integers such that ? ? is greater than the length of ?? (к = 1.2 ). It is easy to define
/ and g on the set X„k in such a way that they both map X„k onto itself, and ??, as a
composition of/, /~'.g. g_l, is not the identity on X,u. If « ^ «^ (?=1.2—), then
we may define / and g on X„ as the identity. It is obvious that the maps / and g generate
a free subgroup of G.
Nevertheless, under some extra conditions the converses of the statements (i) and (ii)
will be true. Suppose that x$ e X is not a fixed point of any nonidentity element of G. It is
easy to see that if ? с G is paradox then so is {g(.vo): g e E). Therefore, in this case G is
supramenable if and only if X does not contain nonempty paradox subsets.
The converse of (ii) is true if G acts without nontrivial fixed points; that is, if no
nonidentity element of G fixes a point of X. Indeed, the orbits [g(x): g e G] (x e X)
constitute a partition of X. Let ? С X contain exactly one point of each orbit. If G acts
without nontrivial fixed points, then the map
? ^ H* = \J{g(E): geH] (HcG)
has the following properties: #ПЛ' = 0=>#*ПЛ'* = 0, and Я~А'^Я*~
К* for every H. К с G. Then we obtain the following statement; cf. Wagon A986.
Proposition 1.10).
THEOREM 6.1. Suppose that G acts without nontrivial fixed points. Then G is amenable
if and only if X is not paradox.
In the actual applications the condition of Theorem 6.1 proves to be too strong:
usually the fixed point are in abundance. However, the converse of (ii) can be obtained
under substantially weaker conditions. Let G v denote the group [g e G: g(x) = .x) (the
'stabilizer' of x) for every л е G. The following theorem is due to Rosenblatt A981); see
also Wagon A986, Theorem 11.25).

Paradoxes in measure theors
103
THEOREM 6.2. Suppose that G, is amenable for every ? e X. Then G is amenable if and
only if X is not paradox.
The special case when the groups G.v are Abelian is particularly important. The action of
G is called locally commutative if this happens. In other words, the action of G is locally
commutative if the following condition is satisfied: whenever two elements of G have a
common fixed point then they commute.
The role of local commutativity in the theory of equidecomposability was discovered by
R.M. Robinson A947). In this paper Robinson finds the minimal number of pieces which
are needed to duplicate a ball. Banach and Tarski did not specify the number of pieces
to obtain a paradoxical decomposition of the ball. In 1929 von Neumann remarked that
9 pieces suffice. Sierpinski A945a) used 8 pieces; more precisely he proved that the ball
is E,4)-paradoxical. Finally, Robinson A947) showed that the minimal number is 5; that
is, the ball is B,3)-paradoxical. He also proved that S2 is B.2)-paradoxical. Robinson's
proof is based on the fact that the group of the rotations of a sphere is locally commutative:
if two rotations have a common fixed point then they have the same axis and hence they
commute.
Actually, Robinson proved a much more general statement concerning the existence of
sets satisfying a prescribed system of congruences; see Wagon A986, pp. 43-51). If our
only aim is to prove that S2 is B, 2)-paradoxical then a simpler proof is available using the
following result of Laczkovich A992c).
THEOREM 6.3. Let G bea locally commutative group acting on a set X, and suppose that
G is freely generated by the transformations f\ /„. Let A. B. H\ H„ be subsets
ofX such that
(i) for every ? e A there are indices 1 ^ ;'. j ^ n. i ? j such that ? e Я, П Hj and
f(x)eB, fj(x)eB;and
(ii) for every у e В there are indices 1 ^ /. j ^ ?. ?? j and points x, e Я, П A. x} e
Hj П A such that f (x,) = fj (x,-) = у.
Then there are partitions A = (J"_| A; and В = (J' = i Bi sucn tnat &i c Д' an^
fi (A,) = B, for every i = 1 n.
PROOF (Sketch). LetF= [fi\Hr. i = 1 «}andr = />( ?. ?). We have to prove that
? contains a perfect matching. It is enough to prove that there is a perfect matching in any
connected component ?\ of Г. We show that Г\ satisfies condition (ii) of Proposition 4.4.
Since the degree of each point of Г is at most ?, ?\ is locally finite. The conditions (i)
and (ii) imply that the degree of each point of Г\ is at least two. Finally, the fact that Г\
contains at most one cycle, can be deduced from the local commutativity of G. ?
The following theorem is due to T.J. Dekker A956): see also Wagon A986, Theorem 4.5
and Corollary 8.6). The proof is taken from Laczkovich A991b).
THEOREM 6.4. if the action ofG on X is locally commutative, where G contains Fj as a
subgroup, then X is B, 2)-paradoxical.

104
?. Laczkovich
PROOF. Suppose that / and g generate a free subgroup of G, and let g, = fg' f ' (/' =
1, ...,4).Thengi, ...,g4 generate a free group of rank 4. Let X; = ? ? {/'}(/' = 1,2), and
define/; (/"= l,...,4)on ? = X\ UX, by
M*J)={gi(x)J) (x eX. /=1,2. 7=1,2), and
MxJ)=(giix).3-j) ixeX, /=3,4. y= 1,2).
It is easy to check that f (/' = 1,..., 4) generate a free and locally commutative group
on Y. Now we apply Theorem 6.3 with A=Y. B = X\. H^=H2 = X\, #з = Щ = X2.
Thus we obtain partitions X\ = A\ UAi and X2 = A3 U A4 such that f\(A\) U /т(Аг) U
/з(Аз) U /4(Ад) is a partition of X \. Taking the projections to X we obtain the partitions
X = B\ UB2andX = ?3??4 such that g\ (B\) U g2(B2) U ?з(В3) U #4(?4) is a partition
of X. D
It is a remarkable fact, proved also by Dekker in A956), that the converse of Theorem 6.4
is also true: ifG acts on X and if X is B. 2)-paradoxical, then G contains two independent
elements such that the action of the generated subgroup is locally commutative. See also
Wagon A986, Theorem 4.8).
Since SO3 contains independent rotations and its action on S2 is locally commutative,
we find that S2 is B,2)-paradoxical.
Let D = (x e R3: |лг| ^ 1} be the unit ball and let D° denote its interior. The map
? ?* Ял = \J{rH: 0 < r < 1} (H с S) commutes with the elements of SO3 and
thus E2)? = D° \ {0} is B,2)-paradoxical. More precisely, we find a decomposition
D° \ {0} = D\ U D2 U D3 U DA such that g\ (D\) Ug2(D2) = ЫД0 U g4(U4) = D° \ @),
where g\,...,g4 are independent rotations. Now, in order to prove that D is B,3)-
paradoxical, we have to 'steal a point' from S2. One can prove, using the argument of
Theorem 6.4, that if ? e S2 is not the fixed point of any of the nonidentity elements of the
group generated by g \,..., #4, then there is a decomposition S2 \ {?} = С \ U C2 U Cj U C4
such that gi(C|)Ug2(Ci) = gT,(Ci)U g^iC^) = S2. Then we obtain the decomposition
D = (D\ UC| U{0})U(D2UC2)U(D3UC3)U(D4UC4)U{p}
such that
g\ (D| U Ci U {0}) U g2{D2 U C2) = g3(D3 U C3) U g4(U4 U C4) U {0} = D.
Note that the point ? is mapped into 0 by an isometry (say, a translation), and thus we
obtain that D is B, 3)-paradoxical.
Local commutativity proved to be useful in other topics of equidecomposability as well.
One of these originates in von Neumann's paradox concerning piecewise contractions.
A function / defined on А С R is a contraction, if there is a q < 1 such that
\f{x) — f(y)\ < q\x — y\ for every x,y e A. A map /: A -> R is called piecewise
contractive if there is a finite partition A = A\ U ·· · U A„ such that the restriction /| A, is
a contraction for every / = 1 , n. The following theorem - the so-called von Neumann
paradox - was proved by von Neumann A929). See also Wagon A986, Theorem 7.12).

Paradoxes in measure theors
105
THEOREM 6.5. For arbitrary intervals I and J there is a piecewise contractive bijection
from I onto J.
The linear fractional transformations are the functions (ax + b)/(cx +d) (a,b,c,d e
R, ad — be ? 0) mapping R U {oo} onto itself. The set of linear fractional transformations
will be denoted by LFT. It is easy to see that LFT forms a group under the operation of
composition of functions.
The proof of Theorem 6.5, as given by von Neumann in A929), uses the fact that LFT
contains free subgroups. For example, the maps a(.x) = ? + 2 and ?(?) = ? /B? + 1)
generate a free subgroup of LFT. Indeed, let ? be a nonempty reduced word in the letters
?, ?-1, 0.0-'.Since
a±\x) = ±2+x and 0±l(.v)= , ' ,, ,
±2+ \/x
it follows that ? can be expressed as a continued fraction whose value tends to a finite
number as ? —> oo. Therefore ? cannot be the identity map.
Von Neumann also proves that if the numbers a*, bk. Ck,dk (к е /) are algebraically
independent over the rationals, then the linear fractional transformations c*k defined by
otk(x) = (akx + bk)/(ckx + dk) generate a free subgroup of LFT. (Proof: Suppose that
the reduced word ? = a' ¦ ¦ -акПг represents the identity map. It is easy to check that
p(x) = (Ax + B)/(Cx + D), where A, B. C, D are polynomials of a*·,, bk, ¦ c*,, dk, with
integer coefficients, and AD — ВС ^ 0. If ? is the identity map then В = С — 0
and A = D. Since ak,bk.Ck,dk are algebraically independent, these equations must be
identities. Therefore, if ? denotes the number of different indices among k\,... ,kr, then
for arbitrary 0| , 0„ e LFT, 0"'1 · · · 0^"' is the identity map. However, if a(x) = ? + 2
and ?(?) =x/Bx + 1) and ?? =</0</ (k = 1 и), then 0™' •••0™' is a nonempty
reduced word in the letters a±i and 0±! and, consequently, is not the identity.)
Theorem 6.5 easily implies the following: if ?. ? are bounded subsets of №. with
nonempty interior, then A can be mapped, using a piecewise contractive map, onto B.
Suppose that the set А с R is mapped, using a piecewise contractive map, onto B.
If ?(?) and the number of pieces, n, are given, then ?(?) cannot exceed ??(?). The
next theorem by Laczkovich A992c) gives the sharper estimate ?(?) < »?( ?)/2, and also
shows that this bound is the best possible.
Theorem 6.6.
(i) Let А, В CK be measurable and suppose that there is a map f: A —> Ш. and a
partition A = A\ U · · · U A„ such that В = f(A) and /|A, is a contraction for
every i = ?,.,.,?. ThenX(B) <n ·?(?)/2.
(ii) Let AcRbe measurable and let J be an interval with \J\ <n ¦ ?( A)/2, where ? is
a positive integer. Then there is a map f : A —* R and a partition A = A \ U · · · U A„
such that f(A) — J and f\ A, is a contraction for every i = 1 n. If A is an
interval, then f can be chosen to be a bijection between A and J.

106
?. Laczkovich
Let J be an interval with 1 < |7| < 3/2. Then (ii) of Theorem 6.6 implies that there is
a bijection from [0, 1] onto J which consists of three contractions; that is, von Neumann's
paradox can be realized using three pieces. On the other hand, such a paradoxical
decomposition does not exist if only two pieces can be used, as (i) of Theorem 6.6 shows.
The action of LFT on R U {oo} is not locally commutative. For example, the functions
1 /x and 2x — 1 have a common fixed point, but do not commute. However, it was proved
in Laczkovich A992c) that if ал . Ьл, ?. ^ (к e /) are algebraically independent over the
rationals, then the linear fractional transformations (a^x + bk)/(cux + dk) generate a free
subgroup of LFT with a locally commutative action. The proof of Theorem 6.6(ii) is based
on this result together with Theorem 6.3.
The statement (i) of Theorem 6.6 is actually a special case of the following theorem
proved in Laczkovich A988a). Let Lip / denote the Lipschitz constant of the map /: A ->·
R"; that is, let
Lip/ = sup{|/(.v)-/(y)|/|.r- v|: .v.veA. хфу\.
Suppose that A = A \ U · · · U Ал is a partition of the set А С К", and f: A ->· R" is a map
such that /| A, is a Lipschitz function with Lip(/|A;) ^ M, for every i = 1 к. Then
the inner Lebesgue measure of f (A) is at most ? ¦ ?„(?), where
М = тах(м!',...,М^]Гм;'У
If the restrictions /|A, are contractions, then we may apply this estimate with M\ = ¦¦¦ =
M„ = 1 - ?, and obtain (i) of Theorem 6.6, even in higher dimensions. On the other hand,
it is not known whether or not Theorem 6.6(ii) can be generalized to R". The heart of the
matter is the following question posed in Laczkovich A991b).
PROBLEM 6.7. Is it true that every measurable subset ofR" of positive measure can be
mapped by a Lipschitz map onto a ball?
For ? = 1 the answer is yes, as the following simple argument shows. Let К be a
compact subset of A having positive measure. Suppose К с [a. b] and ?(?) = d. Then
the function
f0 ifx^a.
/(?)=\?(??[?,?]) ifxe[a.bl
id ifx^b
maps both К and A onto [0, ii], and satisfies the Lipschitz condition \f(x) - /(v)l ^
\x-y\(x,yeR).
For ? = 2 the problem was solved by D. Preiss A992) in the affirmative. A different
proof was found by J. Matousek A997). For ? ^ 3 the problem is open.
Now we return to nonamenable actions and paradoxical sets. Let SL„[R] and SL„[Z]
denote the groups of ? ? ? matrices of determinant 1 with real and integer entries,

Paradoxes in measure theory
107
respectively. For ? ^ 2 these groups contain free subgroups. Indeed, it is easy to check
that
\ ax +b
J ex +d
defines a homomorphism of SL2[R] into LFT. Since the maps <*(*)=¦*+ 2 and ?(?) =
x/Bx + 1) generate a free subgroup of LFT, it follows that the matrices [,', ]] and [\ , ]
generate a free subgroup of SL2[Z]. If // > 2 then SL„[R] and SL„[Z] contain SL2[Z] as a
subgroup, therefore they also contain free subgroups.
It was proved by von Neumann A929) that the unit square is paradoxical under affine
transformations with determinant 1. Wagon proved the following generalization: Let ?. ?
be two independent elements o/SL2[Z], and let G denote the group generated by ?. ? and
by all translations. Then any two bounded subsets ofR2 are G-equidecomposable. See
Wagon A986, Theorem 7.3).
Wagon A986, Question 7.4, p. 101) asked if the interior of the unit square is paradoxical
under SL2M, or even under SL2[Z] (translations are not allowed).
Recently J. Mycielski A998) proved the following theorem. There exists a finitely
additive measure m over all bounded subsets ofR" which is an extension of the Jordan
measure and is invariant under SL„[Z], moreover, satisfies m(rH) = \r\"m(H)for every
r eRand bounded ? с R".
Mycielski's theorem implies that no bounded subset of R" with nonempty interior can
be paradox under SL„[Z]. Mycielski A998) also proved that the interior of the unit square
is paradoxical under SL2[R], subject to the following conjecture (C): there is a free group
F acting on the set D = {x e R2: 0 < |.r| < 1} without nontrivial fixed points such that
each transformation of F is the union of finitely many elements of SL2[R], restricted to
subsets of D.
In fact, Mycielski proves that conjecture (C) implies the following more general result:
if A and В are bounded subsets of R2 \ {@.0)} with nonempty interior, and either they
contain triangles with one vertex at @.0) or their distance from the origin is positive, then
A and В are SL2[R]-equidecomposable.
This result was proved, without using conjecture (C), in Laczkovich A999). The proof is
based on the fact that the action of SL2[R] on the set R2 \ {@.0)} is locally commutative;
see Wagon A986, p. 39). The question, whether conjecture (C) is true or not, remains open.
We conclude this section with the following question motivated by Theorem 6.1: find
conditions on the nonamenable action G that are necessary and sufficient for there to be
no finitely additive, G-invariant measure on V(X) having total measure 1. (Greenleaf's
problem; see Wagon A986, Problem 6, p. 230).)
We mention that the following analogue of Dixmier's condition works. Let N denote the
linear span of the functions (fog) — /, where / e B(X) and g e G. Then there is a G-
invariant finitely additive probability measure on V(X) if and only if inf / ^ 0 ^ sup/
holds for every f e N. Indeed, if m is a measure with the required properties then
fx f dm = 0 for every / e N and thus the condition is necessary. The sufficiency follows
by an easy application of the Hahn-Banach theorem.
a b
с d

108
?. Laczk/nich
7. Marczewski's problem
When Banach proved that the Lebesgue measure can be extended to all subsets of R2 as a
finitely additive invariant measure, E. Marczewski realized that a modification of Banach's
construction yields a measure m with the following properties: m is a finitely additive
invariant measure defined on all subsets of R-, m is an extension of the Jordan measure,
and m vanishes on meager sets. (A set is meager, or first category, if it is the union of
countably many nowhere dense sets.) Around 1930 Marczewski asked whether a measure
with these properties exists on all Borel subsets of R-\ See Mycielski A979) and Wagon
A986, Problem 1, p. 229). This problem remained unsolved for 60 years until, in 1991,
R. Dougherty and M. Foreman presented a most surprising solution.
A set is said to have the Baire property, if it differs from an open set in a meager
set. A variant of Tarski's theorem (Theorem 5.1) shows that the following statements
are equivalent: (i) Marczewski's measure does not exist in R3; and (ii) the Banach-
Tarski paradox can be realized with pieces having the Baire property. Since (ii) sounded
very unlikely, it was generally believed that Marczewski's measure exists in R3; see
Wagon A986, p. 30). However, as R. Dougherty and M. Foreman proved in A992) and
A994), (i) and (ii) are true.
THEOREM 7.1. For every ? ^ 3, the ball B" = {.v e R": \x\ ? 1} is paradoxical with
pieces having the Baire pmperty. Consequently, Marczewski's measure does not exist in
R" for ? > 3.
Dougherty and M. Foreman proved that ?3 is paradoxical using six pieces having
the Baire property. This is best possible by the following theorem of Wehrung A994b):
every paradoxical decomposition of a compact metric space with pieces having the Baire
property contains at least six pieces.
Theorem 7.1 is an easy consequence of the (classical) Banach-Tarski paradox and the
following result called 'the main lemma' in Dougherty and Foreman A994).
THEOREM 7.2. Suppose X is a separable metric space and G is a countable group of
homeomorphisms of X acting without nontrivial fixed points. Suppose that {ri.g,: 1 ^
/' ^3)CG generate a free subgroup of G of rank 6. Then there are disjoint open sets
Rj, G,¦ A ^ / ^ 3) such that U,3= ? П (/?/) = U/L ? Si (Gi) is a dense open subset ofX.
Since Theorem 7.1 uses the Banach-Tarski paradox, its proof is nonconstructive. On
the other hand, the proof of Theorem 7.2 does not use the axiom of choice. In fact, the
proof of Theorem 7.2 runs like this. Fix a countable dense G-invariant subset D in X, and
endow it with the subspace topology. It suffices to construct disjoint relatively open subsets
Ri, G, (i = 1. 2, 3) of D such that (JiLi /·,(/?,) = U, = i gi(Gi) is dense in D. Indeed, if
we let R- be the interior of the closure of /?, in X and G ¦ be the interior of the closure of
d in X then /?,·, G, will satisfy the requirements of Theorem 7.2.
Now, for the countable subspace D. the open sets /?,. G, are given by an explicit
construction based on an intricate induction argument.

Paradoxes in measure theory
109
If A, Set" are open sets then we shall write A « B, if there is a pairwise disjoint
collection {A \,..., А к} of open subsets of A whose union is dense in A and a collection
{/?. · · ·. Л-} of isometries of R" such that [f\ (A\) /*(Ajt)} is a pairwise disjoint
collection of open subsets of В whose union is dense in B. It is easy to see that ъ is
an equivalence relation. The following theorem is a striking corollary of Theorem 7.2.
THEOREM 7.3. Let ? > 3, a/id A a/id ? foe nonempty bounded open subsets ofR". Then
A ^B.
PROOF. Let n, g-, (/' = 1. 2. 3) be independent rotations of the open unit ball U = {x e
R": \x\ < 1}, and let G denote the group generated by r, and g,. It is easy to see that the
set F = {x e U: f(x) = ? for some / e G. f ? identity) is G-invariant and consists of a
countable union of line segments. Thus X = U \ F is a dense G-invariant subset of U. By
Theorem 7.2, there are disjoint relative open subsets /?,-. G, С X such that Ui = i r>'(Ri) =
U/=i SiiG;) is a dense relatively open subset of X. Let /?(' and G- denote the interior of
the closure of /?, and G,, respectively, and put C/| = Uf=i K('. ^ = U/=i ^?· ^ 's c'ear
that U\C\U2 = 0, and U\^U2^ U.
This easily implies that if an open set V can be covered by two open balls of radius 1,
then V % ? for some open subset ? с U. Then an obvious induction argument shows
that this is true for every open set that can be covered by any (finite) number of balls of
radius 1; that is, for every bounded open set.
Now let A and В be nonempty bounded open sets. We may assume that they both
contain the unit ball. Then A « H\ с В and В ~ Н2С A and hence there are collections of
disjoint open sets {A\, ?*}, {?| Bm} and isometries f\ fk. g\ gm such
that Uf=i At is a dense subset of A. U'Li ^j is a dense subset of B. and the systems
{/?(A|),..., fk(Ak)}, {g\(B\) gni(Bm)} consist of pairwise disjoint subsets of В
and A, respectively. Let A' = ljf=| A,. B' = (J'j'=i Bj, and /(.v) = /,(л) (? e A,, /' =
1,...,k), g(x) = gj(x) (x e Bj, j=l m). Now we apply a modification of the
argument proving the Banach-Schroder-Bernstein theorem.
We put Co = int(A \ g(B')) and define by induction Q+? =(?o /)(Q·) for every
A: = 0, 1,.... Putting С = (Jjtto C^ and D = int<A \ C)- one can check that C U D
is a dense open subset of A, /(C) and g~] (D) are disjoint open subsets of B, and
/(C) U g~l (D) is dense in B. Thus A^B, which completes the proof. D
We gave this proof in order to make it clear that this amazing theorem is proved entirely
constructively. As R. Dougherty and M. Foreman point out, we have a theorem, proved
without the axiom of choice, and asserting that there is a collection of disjoint open subsets
of the sun, that fill the sun (in the sense that there are no 'holes' of positive radius) and that
can be rearranged by rigid motions to remain disjoint and fit inside a pea.
When Marczewski constructed his measure, he also asked if the linear measure (one-
dimensional Hausdorff measure) can be extended to all subsets of R2 as a finitely additive
invariant measure; see Mauldin A981, Problem 169). J. Mycielski A979) showed that
the answer is yes. Moreover, as Mycielski noted, for every 0 ^ s ^ 2, the s-dimensional
Hausdorff measure has an invariant extension to ^(И2) by the invariant measure extension
Theorem 3.1. The analogous statement in R3 is false: the 2-dimensional Hausdorff measure

по
?. Laczkovich
does not have an invariant extension for all subsets of R3, since the existence of such an
extension would make the Hausdorff paradox Theorem 2.1 impossible. On the other hand,
it is not known whether or not the s -dimensional Hausdorff measure for 0 < s < 2 can be
extended to all subsets of R3 as a finitely additive invariant measure.
8. Tarski's circle-squaring problem
Tarski A925) asked whether a disc and a square of the same measure are equidecom-
posable. (By Theorem 2.3, if two Lebesgue-measurable plane sets are equidecomposable,
they must have the same measure.) The answer to Tarski's problem is negative if we
impose some restrictions either on the pieces of the decompositions or the isometries of the
rearrangement. L. Dubins et al. A963) proved that the disc is not 'scissor-congruent' to the
square; that is, if the pieces are restricted to be Jordan domains (topological discs), then
the disc and the square are not equidecomposable.
The other negative result is due to R.J. Gardner A985). Gardner proved that the 'circle-
squaring' is impossible, if the pieces are to be moved by isometries generating a locally
discrete group; there are no restrictions on the pieces themselves. (A group G of isometries
of R" is called discrete if for every compact subset С С R", we have С П g(C) = 0 for
all but a finite number of g e G. The group G is locally discrete, if each finitely generated
subgroup of G is discrete.) This negative statement is a corollary of the following positive
theorem. See Gardner A985, Corollary 21). A generalization is given in Gardner and
Laczkovich A990).
THEOREM 8.1. Let G be a locally discrete group of isometries ofR". Suppose that K\ is
a convexpolytope and Ki is a convex body in R". IfK\ and K2 are G-equidecomposable,
then Кг is a convex polytope and К \ and K2 are G-equidecomposable with convex pieces.
Tarski's original question was answered affirmatively in Laczkovich A990). As it turns
out, the disc is equidecomposable to a square of the same area using translations alone.
Here we shall discuss a generalization of this result to arbitrary dimensions and more
general figures as given in Laczkovich A992b).
For ?, ? с R" we shall write ?~?, if A and В are equidecomposable using
translations; that is, if there are finite decompositions A = U'/ = i Aj- В = \J'j=\ Bj an(^
vectors ? ?,..., xd e R" such that Bj = Aj +xj (j = 1 d). By Theorem 3.2, if ?, ?
are Lebesgue measurable and A~ ?, then ?„(?) = ?„(?). We shall prove the following
partial converse of this statement: if A. В с R" are bounded Lebesgue measurable sets
with ?„(?) = ?„(?) > 0, then A ~ B, provided that A and В have small boundaries. As
it turns out, the relevant notion of smallness here is the box dimension defined as follows.
Let Q" denote the set of cubes
??-1
m
a\~
—
m
? ·
• X
a„ - 1 a»
m m
(a, eZ, /' = 1 и).

Paradoxes in measure theory
111
and let N (m, E) denote the number of cubes Q e Q", with Q ? ? ? 0. The box dimension
of the bounded set ? с R" is defined as
\ogN(m, E)
?(?) = limsup .
hi->зс logw
It is well-known that
?(?) = limsup 1-й,
?^?+ -log ?
where U{E,s) = (x: dist(jr, ?) ^ ?). The following theorem is proved in Laczkovich
A992b).
THEOREM 8.2. Suppose that H\ and H2 are bounded measurable sets in R" such that
k„(H\) = k„(H2) >0andA(dH\) <??. ?C?2) <?. Then H\~H2.
It is well-known that if ? is bounded and convex then there is a constant С > 0 such
that k„(U(dH, ?)) ^ С · ? for every 0 <? ^ 1; see Eggleston A963, Theorems 41 and 42,
p. 86), Thus A(dH) ^ ? - 1 for every convex HcR". Also, if A is a Jordan domain in R2
with a rectifiable boundary, then Л(Э A) = 1. Therefore we obtain the following corollary.
Corollary 8.3.
(i) If A, B cR" are bounded convex sets with ?„(?) = ?„(?)> 0, then A ~ ?.
(ii) If A, BcM2 are Jordan domains of the same area and with rectifiable boundaries,
then A ~ B.
Clearly, each of (i) and (ii) settles Tarski's question. C.A. Rogers asked whether or not
the set
" 1 2
. 3' 3
и
7 8
9' 9
U
25 26'
27' 27 ,
is equidecomposable to @, 1/2). See Wagon A986, pp. 119 and 230). It is easy to check
that Л(Э А) = О and hence, by Theorem 8.2, the answer to Rogers' question is affirmative.
More generally, if [a*, bk ] is a sequence of intervals such that 0 ^ au < bk <k~e for every
к with a positive ?, then В = (Jbli [fl*> ^*] is equidecomposable to an interval. Indeed, it
is easy to see that in this case Л(дВ) ? 1 /A + ?) < 1 and hence Theorem 8.2 applies. On
the other hand, it is proved in Laczkovich A993) that there exist intervals converging to 0
such that their union is not equidecomposable to an interval.
The proof of Theorem 8.2 is based on a sufficient condition for the equidecomposability
of sets in terms of the discrepancy of some special sequences.

112
?. Laczkovich
We shall denote the unit cube {(r,,.... t„): 0 ^ t,¦ < 1 (/ = 1...., n)} by /". If F с /"
is finite, \F\ = N, and ? с /" is measurable, then the discrepancy of F with respect to ?
is defined as
D(F: H) =
¦^\FnH\-k„(H)\
N
If a e R then {a} denotes the fractional part of a, that is, {a} = a - [a]. For every
z = (z\,...,Zk)eR" we denote (z) = ({? ?}...., U,,}) (i.e., U) e/" and с - (?) e ?").
If и, jti ,..., Xd e R" and N is a positive integer, then we put
FN(u\x\, ....xti)
= {(u+n\x\ ? VihiXdY- »/ =0 N- 1 (/' = 1 d)\.
The following theorem was proved for plane sets and for d = 2 in Laczkovich A990), for
subsets of R" and for arbitrary d, in Laczkovich A992b).
THEOREM 8.4. Let H\, H2 be measurable subsets of I" with k„(H\) = ?„(?2) > 0 and
suppose that there are vectors x\,..., xj eR" such that
(i) the unit vectors e, = @, 0. 1,0 0) (/ = 1, n)andx\ , ?</ are linearly
independent over the rational numbers, and
(ii) there are positive constants ?. ? such that
D{FN(M\x\,...,xd)\Hj)^K-N-
¦ \-t
for every ueR". N =1,2 and j= 1.2.
Then H\ ~ H2.
This theorem can be formulated in a slightly simpler way if we identify the unit cube
/" with the torus R" /Ъ". Indeed, in this case we do not have to bother with the fractional
parts in the definition of Fn(u; x\ , x^) and with the unit vectors e, in condition (i).
It is easy to see that this identification does not affect the equidecomposability of
subsets of /" if we use only translations. To be precise, let ?: R" -> R" /Z" be the natural
homomorphism. Then, for every А, В с I". A~S in R" if and only if ? (?) ~ ?(?) in
R"/Z". (See Laczkovich A994, p. 172).)
In this form Theorem 8.4 is not restricted to R"/Z", but can be formulated in every
Abelian group as follows. Let G be an Abelian group with the group operation written
additively.
Let m be a finitely additive and invariant probability measure on V(G) (since Abelian
groups are amenable, such a measure exists). If F с G is finite, \F\ = N and А с G is
arbitrary, then we denote
D(F;A) =
-\FDA\-m(A)
N

Paradoxes in measure theory
113
If u, x\,..., xd e G and NeN then we shall write
FN(u;x\, ...,Xd)
= {u+n\x\ ? \-ndXd'- nt = 0 ? - 1 (/' = 1, —d)}.
The elements ? ?,..., Xd e G arc caWcd independent, ifn\x\ -\ YndXd = 0(ni , n</ e
Z) implies n\ = ¦ ¦ ¦ = nd = 0. Then we have the following generalization of Theorem 8.4.
THEOREM 8.5. Suppose that H\, H2 С G. m(H\) = m(H2) > 0, and there are
independent elements x\,..., лг</ е G and positive constants ?, ? such that
D(FN(u;xi,...,xd):HJ)<:K-N~i-F
for every и e G, N= 1,2,..., and j = 1.2. Then H\ ~ H2.
The core of the proof of Theorem 8.5 is a combinatorial statement proved in Laczkovich
A992a, Remark 3.3). We shall say that a set S с M.d is discrete, if every bounded subset of
S is finite. By a lattice cube we mean a set of the form Q = [a\,a\ + N [ ? · · · ? [?(/, ?</ +
?[, where ? e N+ and a, e ? (/' = 1 d). The length of the side of the cube Q is
denoted by s(Q).
THEOREM 8.6. Let 5?, S2 be discrete subsets ofW1, and suppose that there are positive
constants ?, ?, ? such that
\\SjnQ\-aXd(Q)\^K-s(Q)d-]~F
for every lattice cube Q С W1 and j = 1,2. Then there is bijection ? from S\ onto S2 such
that \?(?) —x\^ ? for every· ? e S\, where the constant ? only depends on d, ?, ? and
a.
Supposing Theorem 8.6, Theorem 8.5 can be proved as follows. We put ? =m{Hj) (j =
1, 2). If a = (a\,..., ad) e M.d then we shall abbreviate the linear combination <3|jti +
h ad Xd by a ¦ ?. Suppose that ? \,..., Xd satisfy the conditions of the theorem and let ?
denote the subgroup of G generated by jr 1, xcj. Let U с G be a set containing exactly
one element of each coset of H, then every element g e G has a unique representation of
the form g = и + ? ¦ ? where и e U and ? e Zd. Putting
Sj(u) = [n eZd: u+n-x e Hj) (и е G. j= 1,2),
it is easy to see that the condition of Theorem 8.5 implies that
НЗД П Q\-aXd{Q)\ ? Ks{Q)d^~F

114 ?. Laczkovich
holds for every lattice cube Q с К'' and j — 1. 2 (see the proof of Theorem 1 in Laczkovich
A992b)). Then, by Theorem 8.6, there is bijection ф„ from S\ (и) onto S2(u) such that
\<Pn(z) — z\^M for every с e S\ (/<).
where the constant ? only depends ond. ?.? and a. The important point here is that ?
does not depend on u.
If g = u+nxeH\ (u eU. ? e Z''), then // e 5|(/<). Let <?„(") = "'· As "' e 5%(и),
we have и + н' · лг е Я2. Let ?(#) = /< + ?' · .v. Then ? is a well-defined map from H\
into H2. As 0„ is a bijection from 5? (и) onto S2 (и), it is easy to see that ? is, in fact, a
bijection from H\ onto #2 such that \n' — n\ ^ M. Thus for every g e H\ there is a vector
a = (a\,... ,aj) eZ'1 such that |a, | ^ ? for every / = 1 d, and ? (g) = g + ? · ?.
Let {^??,^? be an enumeration of the elements a ¦ л. where a e Z'1 and |?,| ^ ? for
every / = 1,..., d. Let
A, = [geHr. X(g) = g+d,} (t=\ ?).
Since ? is a bijection from #i onto H2, it follows that Ur = i A/ and Ui=i(^' + ^')
are disjoint decompositions of H\ and H2, respectively, and this completes the proof of
Theorem 8.5.
Now Theorem 8.2 can be deduced from Theorem 8.4 by making use of some results of
discrepancy theory. If F с /" is finite then the (absolute) discrepancy of F is defined as
D(F) = supD(F; J).
J
where the sup is taken over all subintervals J = [a\.b\[x- ¦¦ ? [a„ .b„[c I" ¦ The quantity
D(F) can be estimated using exponential sums by a formula due to Erdos, Turan and
Koksma (see Kuipers and Niederreiter A974, p. 116)). Using this estimate and some results
of W. Schmidt A964), one can show that for almost every .? ? xct e /" there are positive
constants С and К such that D (FN( u: ? ? ?,/)) ^C (logN)K ¦ N~cl for every и е R"
and N = 2.3, Also, by a theorem of Niederreiter and Wills A975, Korollar 4, p. 133),
Л(ЭА) < ? implies that the discrepancy D(F: A) can be estimated by a power of D(F).
Putting these results together we obtain that if ? (9 H\) <n. А(дН2) <п, then, for dlarge
enough, condition (ii) of Theorem 8.4 is satisfied for almost every choice of x\...., x<i,
and hence we have H\ ~ H2.
Theorem 8.2 leaves several interesting questions open. The first concerns the number
of pieces in the decompositions. Let N be the minimal number such that D ~,v Q,
where D is a disc, Q is a square and arbitrary plane isometries can be used. The proof
of Theorem 8.2 gives a very large upper bound for N: a rough estimate is 104(). (The
proof of Laczkovich A992b) gives a smaller number than that in Laczkovich A990); see
Section 10 of the latter.) It would be desirable to give a reasonable estimate for N, not only
for its own sake, but also because such an estimate would require new ideas. We remark
that it is not known whether or not N > 3. By an unpublished (and nontrivial!) result of
R.J. Gardner, we have N > 2. If only translations are allowed, then N > 3 follows from

Paradoxes in measure theon
115
Gardner's Theorem 8.1. Indeed, suppose that D and Q are equidecomposable using the
translations t\,r2,r3. Replacing D by t\(D) we may assume that t\ is the identity. Then
t\, t2, i3 generate a discrete group, which contradicts Theorem 8.1.
Another important question concerns the measurability of the pieces. We shall discuss
this problem in the next section.
9. The problem of equidecomposability with measurable pieces
The following two problems are posed by S. Wagon A986, p. 229) as 'variations on
Tarski's circle-squaring problem'.
PROBLEM 9.1. Is a disc equidecomposable to a square of the same area using Borel
measurable pieces!
PROBLEM 9.2. Is a regular tetrahedron in R·1 equidecomposable to a cube using Lebesgue
measurable pieces!
We may add the following question posed in Section 10.2 of Laczkovich A990).
PROBLEM 9.3. Is every polygon equidecomposable to a square using translations and
Lebesgue measurable pieces!
All these questions are open. Theorem 8.2 does not settle these problems, because its
proof uses the axiom of choice and does not give any information about the measurability
of the pieces. A more general conjecture was formulated by R.J. Gardner A989, p. 54).
CONJECTURE 9.4. Suppose A and В are Lebesgue measurable sets in R". If A and В are
equidecomposable under isometries from an amenable subgroup G of G„, then they are
equidecomposable with measurable pieces.
The condition of amenability is important, since, by the Banach-Tarski paradox,
two balls of different radius are equidecomposable under the (nonamenable) group of
isometries of R3, but are not equidecomposable with measurable pieces. If Conjecture 9.4
is true, it provides a positive answer to Problem 9.2. Indeed, by Theorem 8.2, the
regular tetrahedron is equidecomposable to a cube of the same measure using translations.
Since the translation group is amenable (even Abelian), Conjecture 9.4 applies. Clearly,
Conjecture 9.4 would also imply that a disc is equidecomposable to a square of the same
area with Lebesgue measurable pieces. If, in the conclusion of Conjecture 9.4 we insisted
that A and В are equidecomposable with measurable pieces under the same isometry group
G, then it would imply that the answer to Problem 9.3 is also positive.
Motivated by Theorem 8.1, Gardner also formulates the following conjecture.
CONJECTURE 9.5. Let ? be a polytope and К a convex body in R". If ? and К are
equidecomposable with Lebesgue measurable pieces under the isometries g\ gk from

116
?. ^czkovich
an amenable group, then ? and К are equidecomposable with convex pieces under the
same isometries g\,..., gk·
However, at most one of Conjectures 9.4 and 9.5 can be true. Indeed, the disc and
the square are not equidecomposable with convex pieces, and hence, if Conjecture 9.5
is true then the disc and the square cannot be equidecomposable with measurable
pieces. Thus Conjecture 9.5 contradicts Conjecture 9.4. As Gardner remarks A989,
p. 58), of Conjectures 9.4 and 9.5, probably the latter should be discarded. And, in fact,
Conjecture 9.4 is true if we replace the pieces of the decompositions by functions.
Let ? ? denote the characteristic function of the set H. The sets ?. ? с R" are said to
be continuously equidecomposable, if there are functions f : R" —> [0, 1 ] and isometries
g,¦ (/ = 1,... ,k) such that ?? = ?*=l f and ?? = ?., = \ f> ° Si- (This notion was
introduced by F. Wehrung in A990).)
Obviously, A ~ В implies that A and В are continuously equidecomposable (let
/?,..., fk be the characteristic functions of the pieces in the decomposition of A). It is
not difficult to show that the converse is also true, so that continuous equidecomposability
is actually equivalent to the 'classical' equidecomposability; see Wehrung A994a,
Lemma 1.3). However, the following result was proved independently by F. Wehrung and
by the author; see Laczkovich A996).
THEOREM 9.6. Suppose A and В are Lebesgue measurable sets in W. If A and В are
equidecomposable under the isometries g\ gk /ют an amenable group, then A and В
are continuously equidecomposable with Lebesgue measurable functions f\ fk and
with the same isometries g\,.... gk.
In addition, F. Wehrung proved a theorem containing the following result as a special
case. See Wehrung A992, Corollary 3.23) and A993).
THEOREM9.7. Assume that there exists a medial measure. IftheBorelsetsA, В С ?" are
equidecomposable under the isometries g\, ¦. ¦. gk from an amenable group, then A and В
are continuously equidecomposable with universally measurable functions f\...., fk and
with the same isometries g\,.... gk·
A medial measure is a universally measurable finitely additive measure on V(N)
vanishing on singletons and normalizing N. It is known that the continuum hypothesis
implies the existence of medial measures and thus the conclusion of Theorem 9.7 is
consistent with ZFC.
One is tempted to consider Theorems 9.6 and 9.7 as evidence for the truth of
Conjecture 9.4. Moreover, these results even suggest the following: if the Lebesgue measurable
sets A, В с ?" are equidecomposable under the isometries g\, ...,gk from an amenable
group, then A and В are equidecomposable with measurable pieces under the same
isometries g\,...,gk· Unfortunately, this is not true even for ? = 1, as the following
example by Laczkovich A988b) shows.
Let и be an irrational number, and let g\ (x) = x + u. gj{x) = x — u, gj(x) = и — x
and g4(дг) = 2 - и - x (.t e R); then g\ g4 are isometries of E. It is shown in

Paradoxes in measure theory
117
Laczkovich A988b) that [0, 1] is equidecomposable to itself under g\,..., gA, but it is
not equidecomposable to itself under these isometnes with Lebesgue measurable pieces.
The group generated by g\,..., #4 is amenable, since it is a subgroup of the isometry group
of R. We do not know if such an example exists with commuting isometnes, so we pose
the following question.
PROBLEM 9.8. Let A and В be Lebesgue (Borel) measurable sets in R" and suppose
that A and В are equidecomposable under the commuting isometnes g\,..., gk- Is it true
that A and В are equidecomposable under the same isometnes with Lebesgue (Borel)
measurable pieces'?
Since translations commute, a positive answer to Problem 9.8. would also provide
positive answers to Problems 9.1, 9.2 and 9.3.
10. Countable equidecomposability and countably additive invariant measures
We start with the following theorem of K. Ciesielski and A. Pelc A985).
THEOREM 10.1. The Lebesgue measure in R" does not have a maximal, countably
additive and isometry-invariant extension.
This solves an old problem of Sierpinski. For a short proof and related results see
Ciesielski A990) and Zakrzewski A990). Zakrzewski A995) also gave a somewhat
improved version of the Ciesielski-Pelc construction. For more details see Section 6 of
Zakrzewski B002) (this Handbook).
Let G be a group acting on a set X. We say that the sets А, В с X are countably
equidecomposable, A ~oo B, if there is a partition of A into countably many sets A; (/' =
1,2,...), and there are transformations g; e G such that U/?? gi(Ai) is a partition of B.
This notion was introduced by Banach and Tarski A924). They proved, among other
results, that any two subsets of R" with nonempty interior are countably equidecomposable.
(For this topic, see also Wagon's book A986, pp. 135-145).)
A set А с X is called countably paradoxical, if there are subsets A ?, ?? с A such that
A = A1 U A2, ? ? П At = 0 and ? ~x ? ? ~x A?.
In this section by the term measure we shall mean a countably additive, nonnegative
extended real valued set function.
It is natural to ask whether the analogue of Tarski's Theorem 5.1 holds for countably
additive invariant measures and countably paradoxical sets. The exact analogue would say
that for every ? с X the following statements are equivalent: (i) there exists a G-invariant
measure that normalizes E\ and (ii) ? is not countably paradoxical. R.B. Chuaqui A969)
conjectured that this generalization holds, but later he found the following counterexample
(see Chuaqui A973) or Wagon A986, p. 136)).
Let X be a set of cardinality ? ? and let G be the group of those permutations of X which
fix all but a finite number of points of X. By a well-known theorem of Ulam (see Oxtoby
A980, Theorem 5.6)), there is no nontrivial measure that vanishes on all singletons. Since

118
?. Laczkovich
all singletons are congruent under G's action, a G-invariant measure must vanish on all
singletons. Thus, by Ulam's theorem, (i) of Chuaqui's conjecture does not hold for ? = X.
However, X is not countably paradoxical. Indeed, suppose that there are disjoint subsets
? ?, Хг of X such that X ~x X \ ~x X2. Since the transformations involved in ? ~? ? ?
and X ~зо Xi can move only a countable set of points, there is ? e X such that ? is a fixed
point of all these transformations. This implies ? e X \ and ? e Xi,a contradiction.
With this example in mind, Chuaqui modified his conjecture as follows. Let Nc denote
the family of all sets А с X such that there are infinitely many pairwise disjoint sets
А,- с X with ?, ~? A. Then the following are equivalent: (i) there exists a G-invariant
measure that normalizes X; and (ii) there exists a measure that normalizes X and vanishes
on Nq. (Note that a measure vanishes on No if and only if it vanishes on all countably
paradoxical set. Indeed, it is easy to see that every countably paradoxical set belongs to Nc
and every element of Nc is a subset of a countably paradoxical set.) This conjecture was
proved by P. Zakrzewski A991). More exactly, Zakrzewski proved the following stronger
result.
THEOREM 10.2. Let the group G act on X and let ? С X. Then the following are
equivalent.
(i) There exists a G-invariant measure that normalizes E.
(ii) There exists a measure that normalizes ? and vanishes on all countably paradoxical
subset of E.
Zakrzewski A993) also proved the following generalization. A set А с X is called
weakly wandering under G if there are transformations g„ e G such that the sets
gn(A) (n = 1, 2,...) are pairwise disjoint.
THEOREM 10.3. Let G be a group of measurable transformations of a ?-finite measure
space (?, ?, m). Then the following are equivalent.
(i) There exists a G-invariant probability measure ? defined on ? such that m is
absolutely continuous with respect to ?.
(ii) There does not exist any set A e ? of positive measure which is weakly wandering
under G.
As an immediate corollary, Zakrzewski obtains the following. Let G act on X, and let ?
be a G-invariant ?-algebra. Then there is a G-invariant probability measure on ? if and
only if there is a probability measure on ? that vanishes on Nc-
For further results on invariant measures see Zakrzewski B002) in this Handbook.
It was proved by Becker and Kechris A993) that for Borel actions of a Polish group
on a standard Borel space the perfect analogue of Tarski's theorem holds: There exists an
invariant Borel probability measure if and only if the space is not countably paradoxical.
See also Becker and Kechris A996, Chapter 4) and Sections 4 and 5 of Zakrzewski B002).
We also mention the paper by Penconek A991) which contains several interesting
results about countably paradoxical sets in Ш.". Penconek proves, among others, that every
measurable set А С Ш." contains a countably paradoxical set В such that A\ В is null.

Paradoxes in measure theory
119
11. The nonconstructive element in the paradoxes
The axiom of choice, formulated by Zermelo A904), aroused much controversy from the
very beginning. As early as in 1905, the Bulletin de la Societe mathematique de France
published a debate among Baire, Borel, Hadamard, and Lebesgue on Zermelo's axiom
and, in the same year, several articles of the Mathematische Annalen were also devoted to
this topic. Among those who opposed to the axiom most vehemently were Baire, Borel
and Lebesgue. They did not realize that the theory itself they had developed contained
nonconstructive elements in that the ?-additivity of the Lebesgue-measure cannot be
proved in ZF alone. (For this early opposition to Zermelo's axiom see Sierpiriski A918),
Moore A982) and Dauben A990, pp. 250-259).)
The existence of nonmeasurable sets and the paradoxes intensified the criticism, and
these discoveries were used as 'evidence' against the axiom of choice. Strictly speaking
this is not justified, since either one has fundamental and philosophical objections against
the axiom of choice (as the early opposers had) and then it does not really matter what
actual consequences the axiom has, or one accepts the axiom and then has to accept
the consequences as well. Justified or not, this criticism motivated the investigations to
determine the exact relation between the Banach-Tarski paradox and the axiom of choice.
A comprehensive account of these investigations (up to 1986) is given in the last chapter
of S. Wagon's book A986). Here we only mention the most important facts. First, by a
theorem of R.M. Solovay, the existence of the Banach-Tarski paradox is independent from
the Zermelo-Fraenkel axioms. More exactly, Solovay proved that it is consistent with ZF
that there exists an isometry-invariant and countably additive extension of the Lebesgue
measure to all subsets of Ш."; moreover, supposing the axiom of choice for countable
families of sets (or even a slightly stronger statement called the axiom of dependent
choice), the existence of such a measure still remains consistent. Obviously, an isometry-
invariant extension of the Lebesgue measure makes the Banach-Tarski paradox impossible.
Thus the 'consistency-strength' of the Banach-Tarski paradox over ZF is not lower than
that of the axiom of dependent choice.
As M. Foreman, F. Wehrung and J. Pawlikowski showed, this consistency-strength
can be estimated from above as follows. The axiom of choice, in its full strength, is
almost never used in analysis. Some weaker statements, however, are needed in several
applications. Such statements are, for example, the Boolean Prime Ideal theorem and the
statement that the product of any number of compact Hausdorff spaces is compact. It is
known that these statements are equivalent in ZF and are strictly weaker than the axiom of
choice. An even weaker statement is the Hahn-Banach extension theorem which, in turn,
is equivalent to the statement that all Boolean algebras bear a [0. 1 ] valued finitely additive
measure; see Wagon A986, table on p. 214). M. Foreman and F. Wehrung A991) proved
the following.
THEOREM 11.1. The axioms of ZF together with the Hahn-Banach theorem imply that
there exist non-Lebesgue measurable sets.
Using some of their ideas, J. Pawlikowski A991) gave the following improvement.

120
?. Laczkovich
THEOREM 1 1.2. The axioms of ZF together with the Hahn-Banach theorem imply the
Banach-Tarski paradox.
These results say that rejecting the Banach-Tarski paradox we would also have to reject
the Hahn-Banach theorem and the Boolean Prime Ideal theorem, and thus large parts of
topology and functional analysis as well.
But even the complete negation of the axiom of choice could not save us from paradoxes.
Indeed, the existence of paradoxical plane sets (Theorem 1.2) and the Dougherty-Foreman
paradox (Theorem 7.3) are proved without using the axiom of choice.
The author is pleased to acknowledge the permission of Birkhauser to use portions of
the material in the reference Laczkovich A994) for this chapter.
References
Adams, J.F. A954), On decompositions of the sphere. J. London Math. Soc. 29, 96-99.
Adian, S.I. A983), Random walks on free periodic groups. Math. USSR Izv. 21, 425^434.
Banach, S. A923), Sur le probleme de la measure. Fund. Math. 4, 7-33.
Banach, S. A924), Un theoreme surles transformations biunivoques. Fund. Math. 6, 236-239.
Banach, S. and Tarski, A. A924), Sur la decomposition des ensembles de point en parties respectivement
congruents. Fund. Math. 6, 244-277.
Becker, H. and Kechris, A. A993), Borel actions of Polish groups. Bull. Amer. Math. Soc. 28, 334-341.
Becker. H. and Kechris, A. A996), The Descriptive Set Theory of Polish Group Actions. London Math. Soc.
Lecture Note Ser., Vol. 232, Cambridge Univ. Press, Cambridge.
Burke, M.R. B000), Paradoxical decompositions of planar sets of positive outer measure, submitted.
Ceccherini-Silberstein, Т., Grigorchuk. R.I. and de la Harpe. P. A998). Decompositions paradoxales des groupes
de Burnside. C. R. Acad. Sci. Paris 327. 127-132.
Ceccherini-Silberstein, Т., Grigorchuk. R.I. and de la Harpe. P. A999). Amenability and paradoxical
decompositions for pseudogroups and discrete metric spaces, Trudy Mat. Inst. Steklov. 224. 68-111 (in Russian). The
web-site http://www.unige.ch/math/biblio/preprint/1998/ contains a summary in French including also a one
page abridged English version. The file's name is cgh98.ps.
Chuaqui. R.B. A969), Cardinal algebras and measures invariant under equivalence relation. Trans. Amer. Math.
Soc. 142. 61-79.
Chuaqui, R.B. A973), The existence of invariant cotmtably additive measures and paradoxical decompositions.
Notices Amer. Math. Soc. 20. A636-A637.
Ciesielski, K. A990), Isometrically invariant extensions of Lebesgue measure. Proc. Amer. Math. Soc. 110. 799-
801.
Ciesielski, K. and Pelc. A. A985). Extensions of invariant measures on Euclidean spaces. Fund. Math. 125, 1-10.
Dauben. J.W. A990), Georg Cantor, his mathematics and philosophy of the infinite. Princeton University Press,
Princeton. NJ.
Dedekind, R. A932), Was sind und Has sollen die ZahlenT. Gesammelte Mathematische Werke, Braunschweig.
English translation: The nature and meaning of numbers. Essays on the Theory of Numbers, Dover. New York.
Delcker, T.J. A956), Decompositions of sets and spaces. I and II. Indag. Math. 18, 581-589, 590-595.
Deuber, W.A.. Simonovits, M. and Sos. V.T. A995). A note on paradoxical metric spaces. Studia Sci. Math.
Hungar. 30, 17-23.
Dougherty, R. and Foreman, M. A992). Banach-Tarski paradox using pieces with the property of Baire, Proc.
Nat. Acad. Sci. USA 89 B2), 10726-10728.
Dougherty, R. and Foreman. M. A994). Banach-Tarski decompositions using sets with the property of Baire.
J. Amer. Math. Soc. 7 A), 75-124.
Dubins. L., Hirsch, M. and Karush. J. A963). Scissor congruence. Israel J. Math. 1, 239-247.
Eggleston, H.G. A963). Convexity. Cambridge Univ. Press, Cambridge.
Elek, G. A998). Amenability. I?,-homologies and translation invariant junctionals, J. Australian Math. Soc. 65,
111-119.

Paradoxes in measure theory
121
Foreman, M. and Wehrung. F. A991), The Hahn-Banach theorem implies the existence of a non-Lebesgue
measurable set. Fund. Math. 138. 13-19.
Galileo Galilei A638), Discorsi e dimostrazioni matematiche. English translation: Dialoges Concerning Two New
Sciences, Macmillan, New York A914).
Gardner, R.J. A985), Convex bodies equidecomposable by locally discrete groups ofisometries. Mathematika 32,
1-9.
Gardner, R.J. A989), Measure theory and some problems in geometry, Atti Sem. Mat. Fis. Univ. Modena 39,
39-60 (Proceedings of the Conference on Real Analysis and Measure Theory held in Capri, 1988).
Gardner, R.J. and Laczkovich, M. A990), The Banach-Tarski theorem on polygons, and the cancellation law,
Proc. Amer. Math. Soc. 109, 1097-1102.
Hall, M. Jr. A948), Distinct representatives of subsets. Bull. Amer. Math. Soc. 54, 922-926.
Hausdorff. F. A914), Bemerkung iiber den Inhalt von Punktmengen. Math. Ann. 75, 428^433.
Hewitt, E. and Ross, K.A. A979), Abstract Harmonic Analysis. 2nd edn. Springer, Berlin.
Just, W. A988a), ? bounded paradoxical subset of the plane. Bull. Polish Acad. Sci. Math. 36, 1-3.
Just. W. A988b), Equivalence under decomposition. A graph-theoretical approach, Jahrb. Kurt-Godel-Ges., 111-
117.
Konig, D. A926), Sur les correspondances multivoque des ensembles. Fund. Math. 8, 114-134.
Konig, D. and Valko. S. A925), Uber mehrdeutige Abbildungen von Mengen, Math. Ann. 95. 135-138.
Kuipers, L. and Niederreiter, H. A974), Uniform Distribution of Sequences, Wiley, New York.
Laczkovich, M. A988a), Von Neumann's paradox with translations. Fund. Math. 131. 1-12.
Laczkovich, M. A988b). Closed sets without measurable matching, Proc. Amer. Math. Soc. 103. 894-896.
Laczkovich, M. A990), Equidecomposability and discrepancy; a solution ofTarski's circle-squaring problem,
J. Reine Angew. Math. 404, 77-117.
Laczkovich, M. A991a), Invariant signed measures and the cancellation law, Proc. Amer. Math. Soc. Ill, 421-
431.
Laczkovich, M. A991b), Equidecomposability of sets, imariant measures, and paradoxes, Rend. Istit. Mat. Univ.
Trieste 23, 145-176.
Laczkovich, M. A992a), Uniformly spread discrete sets in K''. J. London Math. Soc. 46, 39-57.
Laczkovich, M. A992b), Decomposition of sets with small boundary. J. London Math. Soc. 46, 58-64.
Laczkovich, M. A992c), Paradoxical decompositions using Lipschitz functions, Mathematika 39, 216-222.
Laczkovich, M. A993), Decomposition of sets of small or large boundary, Mathematika 40, 290-304.
Laczkovich, M. A994), Paradoxical decompositions: a sur\ey of recent results. First European Congress of
Mathematics (Paris, July 6-10, 1992), Vol. II, Progr. Math.. Vol. 120, Birkhauser, Basel. 159-184.
Laczkovich, M. A996), Decomposition using measurable functions, C. R. Acad. Sci. Paris 323, 583-586.
Laczkovich, M. A999), Paradoxical sets under SLt(R), Annales Univ. Sci. Budapest 42. 141-145.
Laczkovich, M. B001), On paradoxical spaces. Studia Sci. Math. Hungar. 38, 267-271.
Lindenbaum, A. A926), Contributions а Г etude de I'espace metrique I. Fund. Math. 8, 209-222.
Lovasz, L. and Plummer, M.D. A986). Matching Theory. Akademiai Kiado, Budapest.
Lubotzky, A. A994), Discrete Groups, Expanding Graphs and Invariant Measures, Progr. Math., Vol. 125,
Birkhauser, Basel.
Matousek, J. A997), On Lipschitz mappings onto a square. The Mathematics of Paul Erdos, Vol. II, R.L. Graham
and J. NesetFil, eds, Springer, Basel, 303-309.
Mauldin, R.D. (ed.) A981), The Scottish Book. Birkhauser, Basel.
Mazurkiewicz, S. and Sierpinski, W. A914), Sur un ensemble superposables avec chacune de ses deux parties,
C. R. Acad. Sci. Paris 158, 618-619.
Moore, G.H. A982), Zermelo's Axiom of Choice, Its Origins, Development, and Influence, Springer, Berlin.
Mycielski, J. A954), On a problem of W. Sierpinski concerning congruent sets of points. Bull. Acad. Pol. Sci. CI.
Ill 2, 125-126.
Mycielski, J. A955), About sets with strange isometrical properties (I), Fund. Math. 42, 1-10.
Mycielski, J. A977), Two problems on geometric bodies, Amer. Math. Monthly 84, 116-118.
Mycielski, J. A979), Finitely additive invariant measures I, Colloq. Math. 42, 309-318.
Mycielski, J. A989), The Banach-Tarski paradox for the hyperbolic plane. Fund. Math. 132, 143-149.
Mycielski, J. A998), Non-amenable groups with amenable action and some paradoxical decompositions in the
plane, Colloq. Math. 75, 149-157.

122
?. Laczkovich
Mycielski, J. and Wagon, S. A984). Large free groups of isometrics and their geometrical uses, Enseign. Math.
30, 247-267.
von Neumann, J. A929), Zur allgemeinen Theorie des Masses. Fund. Math. 13, 73-116.
Niederreiter, H. and Wills, J.M. A975). Diskrepanz und Distanz von Massen heziiglich konvexer und Jordanscher
Mengen, Math. Z. 144, 125-134.
Ol'shanskii, A.Yu. A980), On the problem of existence of an invariant mean on a group. Russian Math. Surveys
(UspekhiK5, 180-181.
Oxtoby, J.C. A980), Measure and Category. Graduate Texts in Math., Vol. 2, 2nd edn.. Springer, Berlin.
Paterson, A.L.T. A986), Nonamenability and Borel paradoxical decompositions for locally compact groups. Proc.
Amer. Math. Soc. 96, 89-90.
Paterson, A.L.T. A988), Amenability, Math. Surveys Monographs. Vol. 29, Amer. Math. Soc.. Providence. RI.
Pawlikowski, J. A991), The Hahn-Banach theorem implies the Banach-Tarski paradox. Fund. Math. 138, 21-22.
Penconek, M. A991), On nonparadoxical sets. Fund. Math. 139. 177-191.
Plappert. P. A995), A sandwich theorem for monotone additive functions. Semigroup Forum 51, 347-355.
Preiss, D. A992), Manuscript.
Rado, R. A949), Factorization of even graphs. Quart. J. Math.. Oxford Ser. 20, 95-104.
Robinson. R.M. A947). On the decomposition of spheres. Fund. Math. 34, 246-260.
Rosenblatt, J.M. A974), Invariant measures and growth conditions. Trans. Amer. Math. Soc. 193. 33-53.
Rosenblatt, J.M. A981), Uniqueness of invariant means for measure-pre ser\-ing transformations. Trans. Amer.
Math. Soc. 265, 623-636.
Schmidt, W. A964), Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110, 493-518.
Sherman, G.A. A990a), On bounded paradoxical subsets of the plane. Fund. Math. 136. 193-196.
Sherman, G.A. A990b), Paradoxical subsets of hyperbolic and spherical discs, Geom. Dedicata 34. 125-138.
Sherman, G.A. A991), Properties of paradoxical sets in the plane, J. Geom. 40. 170-174.
Sierpinski, W. A918), L'axiome de M. Zermelo et son role dans la theorie des ensembles et /'analyse. Bull. Intem.
Acad. Sci. Cracovie A, 97-152 (Oeuvres Choisies, Vol. II. PWN, Warszawa. 1975, 208-255).
Sierpinski, W. A945a), Sur le paradoxe de MM. Banach et Tarski. Fund. Math. 33. 229-234 (Oeuvres Choisies.
Vol. Ill, PWN, Warszawa, 439-444).
Sierpinski, W. A945b), Sur la paradoxe de la sphere. Fund. Math. 33. 2.35-244 (Oeuvres Choisies, Vol. III. PWN.
Warszawa. 445^453).
Sierpinski, W. A946). Sur la non-existence des decompositions paradoxales d'ensembles lineaires. Actas Acad.
Nac. Ciencias Lima 9, 113-117.
Sierpinski, W. A954), On the Congruence of Sets and Their Equivalence by Finite Decomposition. Lucknow.
Reprinted by Chelsea, 1967.
Straus. E.G. A957), On a problem ofW. Sierpinski on the congruence of sets. Fund. Math. 44, 75-81.
Tarski, A. A925), Probleme 38, Fund. Math. 7. 381.
Tarski, A. A929). Sur les fonctions additives dans les classes abstraites et leur applications au probleme de la
mesure. С R. Seances Soc. Sci. Lettres Varsovie. CI. Ill 22. 114-117.
Tarski. A. A938a), liber das absolute Mass linearer Punktmengen, Fund. Math. 30, 218-234.
Tarski, A. A938b), Algebraische Fassung des Massproblems. Fund. Math. 31. 47-66.
Tarski, A. A949), Cardinal Algebras, Oxford Univ. Press. Oxford.
Tits, J. A972), Free subgroups in linear groups. J. Algebra 20. 250-270.
Tricot. С Jr. A982). Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91, 57-74.
Truss, J.K. A990), The failure of cancellation laws for equidecomposability types. Canad. J. Math. 42, 590-606.
Vitali, G. A905), Sul problema delta mesura dei gruppi dipunti di una retta. Bologna.
Wagon, S. A986). The Banach-Tarski Paradox, 2nd edn. Cambridge Univ. Press. Cambridge. First paperback
edition, 1993.
Wehrung. F. A990), Theoreme de Hahn-Banach et paradoxes continus ou discrets. С R. Acad. Sci. Paris Ser. I
310, 303-306.
Wehrung. F. A992), Injective positively ordered monoids I. II. J. Pure Appl. Algebra 83, 43-82, 83-100.
Wehrung. F. A993). La quadrature geometrique continue de cercle. Seminaire d'Initiation a l'Analyse Exp..
No. 6, Publ. Math. Univ. Pierre et Marie Curie, Vol. 104. Univ. Paris VI. Paris.
Wehrung, F. A994a). The universal theory of ordered equidecomposability types semigroups. Canad. J. Math. 46,
1093-1120.

Paradoxes in measure theory
123
Wehrung, F. A994b), Baire paradoxical decompositions need at least six pieces. Proc. Amer. Math. Soc. 121.
643-644.
Willis, G.A. A988), Continuity of translation invariant linear fiinctionals on Cq(G) for certain locally compact
groups G, Monatsh. Math. 105. 161-164.
Zakrzewski, P. A990). Extensions of isometricallv invariant measures on Euclidean spaces. Proc. Amer. Math.
Soc. 110. 325-331.
Zakrzewski, P. A991), Paradoxical decompositions and invariant measures, Proc. Amer. Math. Soc. 111. 533-
539.
Zakrzewski, P. A993). The existence of invariant probability measures for a group of transformations, Israel J.
Math. 83, 343-352.
Zakrzewski, P. A994), When do sets admit congruent partitions. Quart. J. Math. Oxford Ser. B) 45, 255-265.
Zakrzewski, P. A995), Extending isometricallv invariant measures on K" -a solution to Ciesielski's query. Real
Analysis Exchange 21, 582-589.
Zakrzewski, P. B002), Measures on algebraic-topological structures. Handbook of Measure Theory, E. Pap. ed..
Elsevier. Amsterdam, 1091-1130.
Zermelo, E. A904), Beweis, dassjede Menge wohlgeordnet werden kann. Math. Ann. 59. 514-516.

CHAPTER 4
Convergence Theorems for Set Functions
Paolo de Lucia
Universita "Federico 11", Dipartimento di Matematica e Applicazioni "Renato Caccioppoli".
80126 Napoli, Italy-
E-mail: padeluci@unina.it
Endre Pap*
Institute of Mathematics, University o/Novi Sad Trg D. Obradovica 4. 2I0O0 Novi Sad, Yugoslavia
E-mail: pap@im.ns.ac.yu; pape@eunet.yu
Contents
1. Introduction 127
2. Classical results 129
2.1. Exhaustive set functions 129
2.2. Cafiero uniform exhaustivity theorem 130
2.3. Nikodym convergence theorem 132
2.4. Vitali-Hahn-Saks theorem 133
2.5. Nikodym boundedness theorem 135
2.6. Drewnowski lemma 137
2.7. Cafiero theorem for additive set functions 138
2.8. Brooks-Jewett theorem and related results 139
2.9. Vitali-Hahn-Saks theorem for additive case 139
2.10. Counter-examples on algebras '43
2.11. Algebras with SCP and SIP 144
3. Related convergence theorems 146
3.1. Integration with respect to vector measures 146
3.2. Orlicz-Pettis theorem 146
3.3. Schur lemma and Phillips lemma 149
3.4. Rosenthal's lemma 150
3.5. Hewitt-Yosida theorem 152
*This chapter was written during the visit of the second author to the university "Federico II". Naples, as visiting
professor for Ph.D. students, supported by INdAM, Italy, in the period May-June. 2000, and finished during the
visit to Naples, supported by MURST. in March. 2001. The second author wants to thank for the partial financial
support of the Project in the Fields of Basic Research "Mathematical models of nonlinearity, uncertainty and
decision" A866) supported by Ministry of Science, Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
125

126 R de Lucia and E. Pap
3.6. Biting Lemma 153
4. The relation between boundedness and exhaustiveness '54
5. Diagonal theorem for triangular set function 156
5.1. ?-boundedness 157
5.2. Diagonal theorems 160
5.3. SCP and SIP algebras 162
6. Dieudonne type theorems '63
6.1. Triangular set functions 164
6.2. Convergence theorem 168
7. Convergences of measures related to partial order '69
7.1. Nikodym type theorems for lattice-valued measures '69
7.2. Convergences of measures on orthomodular posets '70
7.3. General Nikodym type theorems '71
References '74

Convergence theorems for set functions
127
1. Introduction
Nikodym boundedness and convergence theorems belong to the most important theorems
in measure theory. First of them says that a family ? of countable additive signed
measures ?, defined on a ?-algebra ? of subsets of a set S, which is pointwise bounded,
i.e., for each ? e ? there exists Me > 0 such that
\?(?)\<?? (?? ?),
is uniformly bounded, i.e., there exist ? > 0 such that
|?(?)| <? (??? ?&?)
(see, e.g., Dunford and Swartz A958, Section 2.5)). Nikodym convergence theorem,
Section 2.3, says that a pointwise convergent sequence {?„}„,=[.: of countable additive
measures defined on ana-algebra ?. i.e., ???,,^,? ?,,(?) = ?(?), ? & ?,
(i) converges to a countable additive measure ?.
(ii) {?,,?/ieN is uniformly ?-additive.
The evolution of these theorems reflects the development of today integration theory.
Namely, Vitali proved in 1907, that if {/„}„ei; is a sequence of functions from L](k) (? is
the Lebesgue measure) such that lim/,-,.^ /„ = /, then lim„^.xfA f„ dk = fAfdk for all
measurable set A if and only if {/ /„ dk}„€\; is an equi-X-continuous sequence.
Lebesgue, in 1909, proved that if {fn}n€\; is a sequence of functions from ?-'(?) such
that (*): НгПп^эс/д /„dk = 0 for every measurable set A, then {/ f„dk}„€i; is an
equi-X-continuous sequence. In 1922 Hahn improved Lebesgue theorem removing the
assumption (*) by the condition that lim,,^^ fA f„ dk exists for every measurable set A.
O. Nikodym A931, 1933) extends this convergence result in an abstract settings as
formulated above (the part (i)). Saks A933) has given a Baire Category proof, since till that
the proofs were made by sliding hump method. The part (ii) was observed by Doubrovski
in 1947 and Pettis in 1951 as a part of Saks proof from 1933.
Further generalization of Brooks and Jewett A970) (stated first by Ando A961)) says
that the limit of the sequence of finitely additive scalar and exhaustive set functions on ?-
algebra is again additive and exhaustive, Section 2.8. The method of proof is going back to
sliding hump method, since the Baire category methods are unsuitable for finitely additive
case.
As a generalization of finitely additive vector measures of bounded variation and
countably additive measures on ?-algebra Rickart A942) introduced the notion of a
strongly bounded vector measure (s-bounded), which we shall call exhaustive. This notion
is closely related to countably additivity, Section 2.1. and boundedness, Section 4. Cafiero
result A952) is related to uniform exhaustivity, Sections 2.2 and 2.7, as a very useful matrix
type lemma.
It is known that for additive set functions the Brooks-Jewett theorem is equivalent
with Nikodym convergence theorem, and even more with Vitali-Hahn-Saks theorem
(Section 2.4), Drewnowski A972b), which says that the limit of sequence of signed

128
P. de Lucia and E. Pap
measures which are continuous with respect to a measure is also continuous with respect
to this measure.
As it is well-known, the Nikodym convergence theorem for measures in general fails
for algebras of sets, Section 2.10. But there are convergence theorems in which the initial
convergence conditions are on some subfamilies of a given ?-algebra; those subfamilies
may not be ?-algebras. The considered algebras are with the Sequential Completeness
Property (SCP) and the Subsequentional Interpolation Property (SIP) (see Constantinescu
A981), Schachermayer A982), Freniche A984) and H. Weber A986, Section 2.11)).
Adding the topology to the basic set S, a famous theorem of Dieudonne A951) states
that for compact metric spaces the sequence of regular measures which converges on open
sets, converges on all Borel sets, see Brooks and Chacon A980), Section 6.
Many important convergence results are closely related to Nikodym convergence and
boundedness theorems (Section 3): Orlicz-Pettis theorem, Shur lemma, Phillips lemma
and Hewitt-Yosida theorem. In Section 3 we present also Rosenthal's lemma A968) and
Biting lemma as very useful methods in convergence problems.
The Nikodym boundedness theorem for measures in general also fails for algebras
of sets, Section 2.10. But there are uniform boundedness theorems in which the initial
boundedness conditions are on some subfamilies of a given ?-algebra; those subfamilies
may not be ?-algebras. As a related result to the theorem of Dieudonne A951) we have
that for compact metric spaces the pointwise boundedness of a family of Borel regular
measures on open sets implies its uniform boundedness on all Borel sets. We shall present
a generalization of Dieudonne's theorem on a wider class of set functions. The class
of finitely additive regular Borel set functions gives nothing new, because each finitely
additive regular Borel set functions (also in the case of vector measures) is necessarily
countable additive. We present a generalized Dieudonne type theorem for the class of
^-triangular set functions (Pap, 1987), see also Pap A995), Section 6.
Nikodym boundedness and convergence theorems were generalized not only with
respect to the domain (Sections 2.11, 5.3, 7.2 and 7.3) of the measure but also with
respect to the range and the properties of the considered measures. So there are vector,
group and even semigroup valued measure versions of Nikodym boundedness theorem,
Sections 5.2 and 7.1. On the other side, the interest for non-additive set functions
(submeasures, ^-triangular set functions (Section 5.1), decomposable measures, null-
additive set functions) is growing up mainly of the need of many different applications.
There are recent investigations of set functions with values in sets endowed only with
some topological structures without any algebraic operations: metric space (Pap, 1988),
special topological spaces (Klimkin, 1989), uniform spaces (Pap, 1991), see Pap A995),
Section 7.3.
We shall present in this chapter many of the mentioned convergence theorems together
with the related important convergence results and giving their quite elementary proofs
based on diagonal theorems (matrix methods), Section 5. Diagonal theorems as a further
extensions of sliding hump method were initiated by Mikusinski A970). Many results in
measure theory and functional analysis can be found in the monographs by Pap A982),
Antosik and Swartz A985), Pap A995), Swartz A996) (see also Diestel and Uhl A977)),
where the method of diagonal theorems is used instead of the usually used Baire Category

Convergence theorems for set functions
129
Theorem (as, e.g., in Dunford and Swartz A958)), which is unsuitable for obtaining more
general results.
2. Classical results
2.1. Exhaustive set functions
The notion of a strongly bounded vector measure was introduced by Rickart A942) as
a generalization of finitely additive vector measures of bounded variation and countably
additive measures on ?-algebra.
Let ?? be a ring and ? a family of finitely additive set functions defined on 1Z. A finitely
additive set function m :7? —>¦ R is exhaustive (strongly additive or strongly bounded) if
lim^oo m{Ej) = 0 for every sequence {Ej}j€n of pairwise disjoint elements from ??.
A family ? of additive set functions is uniformly exhaustive if lim^oc m{Ej) = 0
uniformly in m e ?4 for every sequence {Е;}/^ of pairwise disjoint elements from ??. It
is obvious that a finite measure on a ?-ring is exhaustive.
PROPOSITION 2.1. Let ? be a family of finite additive exhaustive set functions on 7?..
The following statements are equivalent
(i) M. is uniformly exhaustive,
(ii) for every increasing sequence {?/heH the sequence {w(?'i)}ieN is a convergent
sequence uniformly for m e ?4,
(iii) for every disjoint sequence {?/}/ек of TZ the series ?,¦»!(?,¦) 's convergent
uniformly for m e Ai.
PROOF, (i) => (ii). Suppose that ? is not uniformly exhaustive, then there exists an
increasing sequence {f/Ьек of 7^ such that
lim m(Ej) =0
not uniformly in M. Then there exists ? > 0 such that for every r, q in N there exists ? ^ r
and m eM such that
\m(Ep) -m(Eq)\ > ?.
By induction it is the possible to construct a subsequence (?Рл}/Fщ of {?,-},-ек and a
sequence {w„}„epj of ? such that
|m„(?p„ \ ?,,„_,) | > ? for every и е N,
and this is a contradiction because {EPn \ El>n_i }„ер; is a disjoint sequence of 7?.
(ii) => (iii). It is enough to observe that if {?,-};ек is a disjoint sequence of TL then
?/}/eN is an increasing sequence of ??. Therefore {53y=i w(^y))ieK f°r m ? Л4
are convergent sequences uniformly in m e Л4.
(iii) => (i) is obvious. ?

130
? de Lucia and E. Pap
PROPOSITION 2.2. Let ? be a family of finite additive exhaustive set functions on К.
Then
(i) M. is uniformly exhaustive,
implies
(ii) for every decreasing sequence {?,-},-er: the sequence {m(Ei)}i€f: is a convergent
sequence uniformly for m e ?4.
If TZ is an algebra then (i) and (ii) are equivalent.
PROOF. It is enough to observe {E\\ ?,-},· er; is an increasing sequence of 1Z an to refer to
the Proposition 2.1.
In a similar way we prove the second part of the proposition. ?
Let ? be a ?-algebra and ? family of ?-additive set funciions defined on ?. A family
? of countable additive measures ? : ? —>¦ ? is uniformly countable additive if
lim Vm(?;)=0
>DC
uniformly in ? e ? for every sequence {Ej}j€r. of pairwise disjoint elements from ?.
We have by Propositions 2.1 and 2.2.
PROPOSITION 2.3. Let ? be a ?-algebra and ? family of ?-additive set functions
defined on ?. The following statements are equivalent
(i) ?4 is uniformly exhaustive.
(ii) ?4 is uniformly countable additive.
(iii) for every decreasing sequence {?;};ef: of ? the sequence {?(??)},??; is a
convergent sequence uniformly for ? e ?.
2.2. Cafiero uniform exhaustivity theorem
We present here a result of Cafiero A952) related to the uniform exhaustiveness. First we
shall give an elementary matrix type lemma.
LEMMA 2.4. Let (a,,,),,.,^· be an infinite matrix of real numbers such that
A) for every ? e N and every subset I of ? there exists X];e/ a,u;
B) for every sequence {//theN of pairwise disjoint subsets of ? and for every ? > 0
there exists к e N and «oeN such that
????
< ? for every ? > «о.
Then
lim a„i = 0 uniformly in ? e N.

Convergence theorems for set functions
131
PROOF. We note that if the matrix (a„,-)„.jeN has the properties A) and B) then also every
its submatrix has the same properties. By A) we have
lima,„-=0 (neN). A)
/—>00
Then we need to prove that: for every ? > 0 there exist k,s eN such that |?„,-| < ? for
every /' > к and every ? > s. Suppose that this is not true. Then there exists ? > 0 such that
for every k,s eN there exist /' > к and ? > s such that \а,„ | > ?. By induction we can then
construct two strictly increasing sequences {цк-ен an(J {i/theN of natural numbers such
that \a„Kik | ^ ? for every к eN. Therefore there exists a submatrix of the starting matrix,
which we denote by the same symbol (a„/)„.ier; such that it satisfies A), B) and for some
? >0.
C) |?«| > ? for every к е N.
Let {ar}repj be a decreasing sequence of real numbers such that
reN
from B) we have that for every r e N there exist /', and sr such that
\a,ur\ <°r
for every ? ^ mr. We can suppose that the sequences {/VlreN and {ir}reN are strictly
increasing and so to construct a new submatrix of (am)w.ieN. which we denote by the
same symbol (a„;),M-epj such that it satisfies A), B), C) and
for every j e ?, |?„, | < ?, for all ? ^ st. B)
Now we construct by induction two strictly increasing sequences {p;,b,eN and {г„}„ем of
natural numbers with the property
p„ > r„ > max{/o„_ ?, ???_,} for every ? e N,
where po = so = 0, such that we have for every ? e N,
\<>rki I < ?,„ for every / > p„ and к = 1 ?. C)
Suppose that Г|,... ,r„_i; у0| , ??-? are determined and let be r„ an element of N such
that
r„ > max{pn-\,srn_l},
by A) we can find p„ such that p„ > r„ and C) is true.
Consider now the submatrix (ar„r,)n./sH. for и e N and / = l,...,n — 1, we have
r„ > ??; and then, by B) \аГпГ \ < ??, and for /' ^ ? + 1 it results r, > p,-_i, л ^ / — 1,

132
P. de Lucia and E. Pap
and then, by C) \аГпП | < ???_,. Hence for an infinite subset / of N and ? e / we have
byC)
Y^ar„r, > KrJ- ? \??,?\>?-??? = 4-
ie/
/e/. <??
/eK
If {h }k€N is a disjoint sequence of infinite subsets of N we have for every ieN
??^
> — for all ? & h
2
and this contradicts B).
Let ? be a ?-algebra in the whole chapter.
D
THEOREM 2.5 (Cafiero). A sequence {?„}„?? of countable additive real measures is
uniformly exhaustive if and only if the following condition holds
(a) for every sequence {En}„€^ of pairwise disjoint elements of ? and every ? > 0
there exist k, no e N such that
\??(?0\ < ? for every ?^??-
PROOF. Let {Еп}п€ц be a sequence of pairwise disjoint elements of ?. Then {??{??))„?€?
is an infinite matrix of real numbers. By countable additivity of every ?„ we have that for
every « e N and / с N there exists J2i€l ??(??). If {Ik)keN is a sequence of pairwise
disjoint subsets of N, then {[Ji€l ?,-}/tei; is a sequence of pairwise disjoint elements of ?
and then by (a) for every ? > 0 there exist k. «0 e N such that
< ? for every и ^ no.
Therefore the conditions A) and B) of Lemma 2.4 are satisfied and we have that
lim ?,,(??) = 0 uniformly in ? e N.
D
2.3. Nikodym convergence theorem
THEOREM 2.6 (Nikodym convergence theorem). Let {?„}/<?? be a pointwise convergent
sequence of countable additive measures defined on an ?-algebra ?, i.e.,
lim ?„(?) = ?(?). Ее ?.
D)
then

Convergence theorems for set functions
133
(? {?????? converges to a countable additive measure ?,
(ii) {?/,}??? is uniformly ?-additive.
PROOF. We consider first a special case. If {?„}/<?? is pointwise convergent to zero then
condition (a) in Theorem 2.5 are satisfied. Then {??},?€? is uniformly exhaustive and by
Proposition 2.3 it is uniformly ? -additive.
The general case for (ii), i.e., under the condition D), easily follows by the fact that
by D) [??(?)}„€^ is a Cauchy sequence. Namely, suppose that (ii) does not hold for
{M/iL?eN, ie., that there is a sequence of pairwise disjoint sets {?,,?,,^? from ?, a
subsequence {?*„}„?? of {?„}/,?? and ? > 0 such that |?*„(?/?·„)| > 2?. By exhaustivity
of ??? there exists a subsequence {р„}„€н of {fc,ibieN such that \????(??>?^?)\ ^ ?. Taking
m„ = ??„+? — ??„ we obtain a sequence {т„},1€^ of countable additive set functions which
is pointwise convergent to zero and therefore by the previously proved part it is uniformly
countable additive, but this is in a contradiction with
\m„(EPn+i)\ > \???+?(???+?)\-\?,)?(?,,?+])\^?
for all ? e N.
To prove (i), we have to use (ii). Namely, by (ii) we have
? ?j ) = lim ?, ( ? ?j ) = lim lim ][>,(?,)
D
2.4. Vitali-Hahn-Saks theorem
Absolute continuity of functions was introduced by Vitali A905) who proved that a real
valued function on the unit interval is absolutely continuous if and only if it is the integral
of its derivative. Absolute continuity of set functions was introduced by Radon A919).
Let ? be a ?-algebra. Let ?: ? -> [0, +oo] be a non-negative set function and ? a
signed measure defined on ?. We say that ? is absolutely continuous with respect to ?,
denoted by ? <<C ?, if for every ? > 0 there exists S > 0 such that \v(E)\ < ? whenever
? e ? and ?(?) < S. A sequence {v„}ll€i-: is uniformly absolutely continuous with respect
to ? if for every ? > 0 there exists ? > 0 such that \v„(E)\ < ? whenever ? e ?, ? e N,
and ?(?) < S.
THEOREM 2.7 (Vitali-Hahn-Saks theorem). For any sequence {?„},?€? of signed
measures v„ which are absolutely continuous with respect to a measure ?, i.e., v„ «д, and for
which lim^oc v„(E) = v(E) exists for each ? e ?. hold
(i) the limit ? is also absolutely continuous with respect to this measure, i.e., ?«?;
(ii) {l7,}„i=n is uniformly absolutely continuous with respect to ?.

134
P. de Lucia and E. Pap
As we already mentioned this theorem is closely related to integration theory. Namely, if
{/wbieN is a sequence of functions from L'([0. 1]), where ? = ? is the Lebesgue measure,
and
lim / /„
dk = v(E)
exists for each measurable set E, then the sequence {/ f„ dk},,^ is uniformly absolutely
?-continuous and ? is absolutely ?-continuous.
Proof of Vitali-Hahn-Saks theorem. We shall prove (ii). First we consider a
special case and we prove that if {Ay}yei; is a decreasing sequence with ?(?> ^y) = 0
then we have uniformly in /'
lim vj(Aj) = 0. E)
Let A — ?] ¦ Aj by the ?-absolute continuity of every v, we have v,(A) = 0. Then E),
uniformly in /', follows from the Nikodym convergence Theorem 2.6.
Consider now the general case for (ii). Suppose that (ii) is not true. Then there exists
? > 0 such that for each S > 0 there is ? e N and ? e ? such that
|?„(?)|^? and ?(?)<8.
Take <5| > 0, n\ = 1 and E\ = 0. Then we can define by induction a subsequence {/;,},ек
and sequence {?,-};epj from ? and {5;};еь:. <5, > 0 such that
<5, + i <<5,Д \v,,,+,(Ei+i)\>?. ?(?-? + \)<8?/2
and
\vni+l(A)\<e/2 ????(?) <Si+[.
Take A, = Utli ^ · Then by countably subadditivity
?(??\??)<??(?? + ?)<: ? ?(?? < ? | < ? у^ш = S"
k=i + \ k=i k=i
i.e.,
?(?,\?,)<5, F)
and therefore
? ? ?
K,+ |(A'+I \^i + l)| < у

Convergence theorems for set functions
135
Then we have
III I ?
K+l(?/+l)| = K,*i<Ai + l) - v„,ri(Ai + \ \ ?/ + 1I > 2
and therefore
\vn,{Ai)\>\- G)
But {A/},-epj is decreasing and for A = ?); ?< we obtain by F) that ?(?) = 0. Then by the
special case we have E) uniformly in /'. Contradiction with G).
Obviously (ii) implies (i). ?
Phillips A940) and Rickart A942) extend Vitali-Hahn-Saks theorem for measures with
values in a locally convex topological vector space.
There are further generalizations for functions defined on orthomodular lattices and with
more general properties (see Antosik and Swartz A985), Pap A995)).
2.5. Nikodym boundedness theorem
Nikodym boundedness theorem was obtained for countably additive scalar measures on
?-algebras by Nikodym A931, 1933). The proof by Baire category was given by Saks
A933). It has been generalized by Grothendieck A957) for bounded vector measures.
Theorem 2.8 (Nikodym boundedness theorem). A family ? of countable additive
signed measures ?, defined on ? ?-algebra ?. which is pointwise bounded, i.e., for each
? e ? there exists Me > 0 such that
\?{?)\<?? (?? ?).
is uniformly bounded, i.e., there exist ? > 0 such that
|?(?)| <? (? e ?, ? e ?).
PROOF. It is clear that it is enough to prove the theorem for sequences of measures. We
start to prove that for every disjoint sequence {E,},¦ e[; of ? the set
{??(??): neN, /eN}
is bounded. Then we need to prove that there exists ? ^ 0 and h such that
|?„ (?,) I ^ ? for every ? ^ m and for every / ^ h.

136
P. de Lucia and E. Pap
Otherwise there exist two strictly increasing sequences of ?: {?,,?,^? and {/,;},;i=n such
that
\????(???)\ ^ ? for every ? e N. (8)
Let {a„}„epj be an infinitesimal sequence of positive real numbers. Clearly the sequence
{?„?„}?€^ verifies Nikodym theorem 2.6 hypothesis then it is uniformly exhaustive. It
follows that the set
{?„?„(?,): ???,?'??)
is bounded. By (8) it follows that the sequence {—} »eN is bounded for every infinitesimal
sequence of positive real numbers. A contradiction.
Now we consider the general case and suppose that the theorem is not true. Then
sup{^„(?)|: ? e ?, ? e ?} =oo.
Therefore for any ? > 0 there is a partition (A. B) of X and ? e N such that
????{|?„(?)|, |?„(?)|} > ?.
Namely, by
|?„(?)| > ? + sup^;(X)|
we have also
\?„(? \ A)| > |?„(?)| - \?„(?)\ > ?.
Therefore there exists ? ? and a partition (? ?. B\) such that
гтп{|мл,(А|)|, |m„, (??)|} > M-
Then either
sup{^„(Cn A|)j: ? eN. С е ?} = ??
or
sup{|M„(CnB|)|: ne N. Ce ?) = oo.
Taking whichever ? ? or ? ? satisfies one of the preceding condition and denote it by F|
and put ?| = X \ F\. Taking now F\ as X in the preceding procedure we obtain a partition
(Eo, Fi) of F\ and m > n\ such that
|?„2(?2)|>2 and sup{^„(C ? F2)\: ? e N, С е Г} = oo.

Convergence theorems for set functions
137
Continuing this procedure we obtain by induction a subsequence {и,-},·ек and a sequence
{?/};eN of pairwise disjoint sets from ? such that
|м«,-(Е/)| > '·
Contradiction with previously proved boundedness on disjoint sequences. D
2.6. Drewnowski lemma
To extend the previous results to the finitely additive case we will prove a very useful
lemma obtained by Drewnowski A972b).
Lemma 2.9 (Drewnowski lemma for monotone set functions). If m:lZ—>- R is an
exhaustive monotone set function with m@) =0 and {Е„}„ец is a sequence of pairwise
disjoint sets from TZ, then there exists a subsequence {E^ }n€i; о/{?'„}„ек such that m is
order continuous on the ?-ring 1Z\ generated by {Е^„ }„ем. i.e., limm(A„) = 0 for every
sequence {An}„^ifrom TZ\ such that A„ \ 0.
PROOF. Let {./,}/eN be a sequence of pairwise disjoint infinite subsets of N. Then
{Utev, ?/tL'eN is a sequence of pairwise disjoint subsets of ?? and by the exhaustiveness
of m there exists an infinite subset ? ? of N such that
™(U?*)<i
In the same way, if {У,},ен is a sequence of pairwise disjoint infinite subsets of N\ \
{minN|}, we can find an infinite subset N2 of N\ \ {minN|} such that
m(liEk)<^·
Xk€N2 '
By induction we construct a decreasing sequence {N, },ек of infinite subsets of N such that
W/+I ? Nj \ {minNj}, mi [J Ek\ < — forevery/' e N.
Hew, '
Let kj = minNj and let TZ\ be the ?-ring generated by {E/t„}Hei·:· Let {A„},iei; be a
decreasing sequence of elements of 7Z| such that С\11€^Ап = 0. If there exists m e N
such that A,, = 0 for every ? ^ m then the theorem is proved. Therefore we suppose that
A„ ? 0, и е N. For any и e N let /„ be a subset of N such that
A„=\jEki.

138
P. de Lucia and E. Pap
Obviously /,,+i с /„ and because none of the Ekj is enclosed in ПпеК^" we nave a'so
????,,^?? /„ = oo. For arbitrary ? > 0 we choose peN such that 1/2'' < ?. Then we have
for ? ^ ? that
A„ = (J Ek, с (J Ekl
i'e/M ieN/,
and therefore m(A„) < 1/2" < ? for every и ^ p. ?
Lemma 2.10 (Drewnowski lemma for additive set functions). // ?,:7? —>¦ R is a
sequence of exhaustive additive set functions and {E„}„ef. is a sequence of pairwise
disjoint sets from 1Z, then there exists a subsequence {?*„}„еь: of' {E„}„€u suc^ that ?,
is countable additi ve on the o-ring 7Z\ generated by {Ekfl }nei··
PROOF. Let for every ? e N denote by |?„ | the total variation of ?„. We introduce the set
function m : 7? —>¦ R by
?1 IMhI
.
„??2+??"
where M„ = sup{^„|(A): A e1Z]. Since the function m satisfies the conditions of
Drewnowski Lemma 2.9 there exists a ?-ring 7?| generated by a subsequence {??„}«eK
of {??}?,6? such that the restriction of m to Я ? is order continuous. Then easily follows
that the restrictions of |?„| to TZ \ are order continuous and therefore every ?„ is countable
additive on ?? \. ?
2.7. Cafiero theorem for additive set functions
We will generalize Theorem 2.5 to the additive case.
THEOREM 2.11 (Cafiero). A sequence {?„}??? of finite additive bounded set functions is
uniformly exhaustive if and only if the following condition holds
(a) for every sequence {E„}„€t: of pairwise disjoint elements of ? and every ? > 0
there exist к, щ e N such that
| ?? (Ej.) | < ? for every ? ^ iiq.
PROOF. Suppose that the {?„}„,=?; do not are uniformly exhaustive. Then there exists a
disjoint sequence {?, };eN of ? such that lim,,^^ ?,,(??) =0 but not uniformly in и. Then
there exists ? > 0 such that we can construct a subsequence of {?,},?=? and a subsequence
°f {M/ilneN. that for simplicity, we will denote yet by the same symbols such that
?„ (?„) | > ? for every ? e N.
(9)

Convergence theorems for set functions
139
By Drewnowski Lemma 2.10 there exists a subsequence {Ejk }^€щ of {?;},ек such that if
?\ is a ?-algebra generated by {?,t }^еь' the restriction of ?„ to ?\ are ?-additive. From
Theorem 2.5 it follows that these restrictions are uniformly exhaustive but by (9) we have
| ?,;. {Eik) | > ? for every к eN.
A contradiction. ?
2.8. Brooks-Jewett theorem and related results
A generalization of the Nikodym convergence theorem was obtained by Brooks and Jewett
A970).
THEOREM 2.12 (Brooks-Jewett). A pointwise convergent sequence {m„}„eN of finitely
additive scalar and exhaustive set functions (strongly additive) defined on an ?-algebra
?, i.e., lim,,
^oc "i«(?) — w(E), ? e ?',
(i) converges to an additive and exhaustive set function m,
(ii) {w„}„epj is uniformly exhaustive.
PROOF. If {ш„}„ем is pointwise convergent to 0 then the condition (a) of Theorem 2.11 is
verified and so in this special case (ii) is true. The general case for (ii) follows in the same
as in the proof of Nikodym theorem.
Then (i) follows by (ii)
lim m(Ej)= lim lim mj(Ej)= lim lim >tij(Ej) = 0. D
j —* DC j —> DC ? —> DC I—> DC j —* DC
Theorem 2.13 (Nikodym boundedness theorem for additive case). A family ? of
finitely additive bounded set functions m. defined on an ?-algebra ?, which is pointwise
bounded, i.e., for each ? e ? there exists Me > 0 such that
\m{E)\ < Me (m eM),
is uniformly bounded, i.e., there exist ? > 0 such that
\m(E)\ < ? (m eM, ? e ?).
The proof is same as the proof of Nikodym boundedness Theorem 2.8 only using the
Brooks-Jewett Theorem 2.12 instead of Nikodym convergence Theorem 2.6.
2.9. Vitali-Hahn-Saks theorem for additive case
For the proofs of Brooks-Jewett Theorem 2.12 and Nikodym boundedness Theorem 2.13
it was crucial Drewnowski Lemma 2.10. To prove finitely additive version of Vitali-Hahn-
Saks theorem we need another useful lemma.

140
P. de Lucia and E. Pap
THEOREM 2.14 (Vitali-Hahn-Saks for finite additive case). For any sequence {т„}п€ц of
finite additive bounded set functions m„ which are absolutely continuous with respect to a
non-negative finite additive set function ?. i.e., m„ <*C ?, and for which lim,,^^ m„(E) =
m(E) exists for each ? e ?, hold
(i) the limit m is also absolutely continuous with respect to ?, i.e., m <5C ?;
(ii) {m„}nepj is uniformly absolutely continuous with respect to ?.
The proof of Vitali-Hahn-Saks Theorem 2.14 will be based on the following two lemma.
LEMMA 2.15. Let M. be a family of finite additive set functions uniformly exhaustive in
the ring TZ. Then for every ? > 0 there exists ?4', finite subset ofM, and ? > 0 such that if
\m(E(~)F)\<8 for every E. F e TZ and m e ?4'
then
\m(E)\<s foreverymeM.EeTZ.
PROOF. If the thesis is not true there exists ? > 0 such that for every S > 0 and every finite
subset M! of M. there exist ? e TZ and m e M. such that
\m(E П F)\ < ? for every m eM' and F eTZ
and
m'(E) > ?.
By induction it is so possible to construct a sequence {»u-}jteN of ? and a sequence
{EjtheN ofTZ such that
\mp(Ff)Ek)\< -^ for every ? ? к and F e 7г. A0)
\mk+i(Ek)\ > ?. A1)
If h e N, let
к
E[= [J ?, fork>h.
i=h+\
{E^UeN is an increasing sequence of TZ; by the uniform exhaustivity {m(F П E'k)}k>h is a
Cauchy sequence uniformly in m e ? and F eTZ. Then there exists /?'>/? such that
|w(F П ?^)-w(F П ?;,.)|<-^?- for every m e M. F e 7г. ? ^ h'. A2)

Convergence theorems for set functions
From A2), taking F П ? к instead of F, we obtain
141
Fn?t\ \J ЕЛ
\ i=h+\ I
< o/T+T fore\ery m eM, F eTl, k^h'. A3)
Let {к„}п€щ be a strictly increasing sequence of natural numbers such that
FDEk\ \J ?,
V /=*„ + ! /
<-?— forevery meM, F eK, k^k„ + 1. A4)
Put for every ? e N
*4. = (*4.n( (j ?,))u(?,„\ Q' E,\
\ \i=k„+\ II \ i=k„+\ I
and denote ?*„ \ U, ="!', +1 ?, by T„, so we have
mk, + i(Ek„) = m,„ + 1 ? Ek. П ? Q ?, ) ) + mf„ + l ( T„ П |J 7]
»-i
A5)
i=l
We will estimate all three summandson the right side of the equality A5).
We have for the first summand, for every ? e N,
?*»П( U E'))= ? ffl^l№n?i)·
where F,¦, i = k„ + 1,..., k„+\, are suitable elements of 7? mutually disjoint. Therefore
observing that in our case is /' ^ k„ + 1, by A0) we have
|hu„ + i(F,· П Ej)\ < -—j forevery/ = k„ + l,... ,kn+\.
and so we obtain
k„
\ \ ;=*„+i
< Г -
/=*„ + !
2' +1
A6)

142 P. de Lucia and E. Pap
We observe for the second summand that it is
»u„+i \Tn? (J?? = ?^,+???,,\ U?? ???
??-\ ?/ *,-1 \ \
= ]>>*„ + , [[Ekn\ (J eAdF,
where F-, is a suitable element from TZ. and then by A3)
n-\
r»k„ + \
T„ n\jT,
i=l
и—\ k„
?? ^-Ц ?
i=l ?' = ?
(?)
We obtain for the third summand by the uniform exhaustivity of ? that there exists «o e N
such that
H-l
[?,? U ?,
By A5)-A7) we have for every ? ^ no
< — for every m e ?4 and /j ^ no.
A8)
l^„+i(?i„)| <^^ + |<^
and this contradicts to A1).
D
LEMMA 2.16. Let TZ be a ring. ? a family of finite additive set functions on TZ and ?
a non-negative finitely additive set function on TZ. If ? is uniformly exhaustive and every
element of ? ?-absolutely continuous, then ?4 is uniformly absolutely ?-continuous.
PROOF. By Lemma 2.15 for every ? > 0 there exist S > 0 and M', finite subset of M,
such that if
\m(EC\F)\<8 for every m e ?',
A9)
then
\m(E) \< ? for every m e ?.
B0)
But the elements of M! are ?-absolutely continuous, then there exists ? > 0 such that
?(?) <? implies A9) and B0). ?

Convergence theorems for set functions
143
Proof of Vitali-Hahn-Saks Theorem 2.14. By Brooks-Jewett Theorem 2.12 we
have that the sequence [тп}п€щ is uniformly exhaustive. The thesis then follows by
Lemma 2.16. D
2.10. Counter-examples on algebras
We shall give three examples showing that generally Nikodym convergence, Nikodym
boundedness and Vitali-Hahn-Saks theorems are not true on an algebra.
Example 2.17. Let ?? be the ring generated by the right half open subintervals of [0, 1[,
and let for every ? e N, be /„ : [0. 1 [ -> [0, 1 [ the function given by /„(x) = x". Let ?„ be
the set function defined on the right half open intervals [a, b[ by the formula
4n{[a,b[) = fn(b) - /„(a) = b" -a"
and extended on ?? in the natural way. We note that, for every /; e ?, ?„ is countable
additive. It is clear that for every a, b with 0 ^ a < b < 1 we have lim,,^^ ?„([?. b[) = 0,
and for every a such that 0 ^ a < 1 we have lim,,^ x ?„ ([a, 1 [) = 1. Then for every ? e 1Z
.· //74 10 ifl??'
hm ?„(?)= ?
n—>oc I
1 if 1 e ?¦'.
Then {?„}„?=? is a sequence of countable additive measures defined on ring, pointwise
convergent to a function ?? = lim,,^^ ?„, but ?? is only finitely additive not countably
additive.
Example 2.18. Let J be the ring of the finite subsets of E, and let v„ for every /; e N
be the function
M?) = (
? if ? e ?,
0 ????.
Clearly vn is countably additive. If ? is the algebra, generated by J, of finite and cofinite
subsets of R, we extend v„ to ? by the formula
_iv„(E) if Ее J.
?"(?) -\-?„(?') ifE'eJ.
Clearly {?„}„?? is a sequence of finitely additive measures. We observe that if {?, },ен is
a disjoint sequence of elements of A such that U,eh' ^< e -^ tnen a" e'ernents of {?|},-eN
excepts a finite number are empty and therefore we can say that every ?„ is countable
additive. We have for every ? e ?
. . |{0}U(Nn?) if Ее J,
\?„(?): neN С '
1 ' ~ @)U{/eR: -teNDE] ifE'eJ.

144
P. de Lucia and E. Pap
Then, for every ? e A, the set {?„(?): ? e ?} is bounded and clearly ?„ is bounded, but
M"({n}) = ? for every /; e N and therefore {?„(?): ? e ?, ? e A] is not bounded.
Example 2.19. Let ? be the algebra from the preceding example. We define
v„(E) =
? if ? e ? finite,
—? if ? e ?' finite,
0 otherwise.
Then every u„ is bounded, finitely additive and lim,,^^ v„(E) exists for each ? e A.
Taking
?(?) =
?,,??? iff is finite,
.?+????? if ?¦' is finite,
we have that ? is bounded, finitely additive and v„ <K ? but {y„}„eN is n°t absolutely
?-continuous.
2.11. Algebras with SCP and SIP
Let ? be an algebra of subsets of a nonempty set S.
We have the following definitions by Constantinescu A981), Freniche A984) and
H.Weber A986).
Dehnition 2.20. An algebra Л has the Sequential Completeness Property (SCP) if each
disjoint sequence {?„)„6pj from Л has a subsequence {ЕП] }jSn< whose union is in Л.
Dehnition 2.21. An algebra Л has the Subsequentional Interpolation Property (SIP)
if for each subsequence {Ау„}„ек of each disjoint sequence {А}}]€п from ? there exist a
subsequence {Aju };.ещ and a set В е Л such that
А]ч с В (* e ?)
and A j П В = 0 for ; e N \ {;„t :JteN).
Example 2.22. Let J be the ring of the finite subsets of N and let I be a maximal ideal
of V(N) containing J.
We note that by a well-known characterization of maximal ideals, we have
Eel if and only if Ec il. (*)
The ideal I is not a ?-ideal, because I is a proper ideal and therefore N = U„eN{n) ^ ^·
Let {?H}„6N be a disjoint sequence of elements of I such that [Jn(z^ E„ does not belong

Convergence theorems for set functions
145
to I and let {N\, N2} be a partition of N such that N\ and N2 are infinite. We claim that
U(ieyV] E„ or Une№ ?" belongs to I. Suppose the contrary. Then we have by (*)
=((u44U?"))eI·
This is a contradiction because I is a proper ideal. Then I is a ring which is not ?-
complete, but with SCP.
Example 2.23. Let N0 = {0} U N. Obviously if I is a ring of P(N0), then
Л={?е-р(М0): ?eIorN0\?eI)
is a subalgebra of V(Nq) and it is not a ?-algebra. We can observe that if ? e ? \ I then
0e? and we have also No \ F = ? \ F el and then ? П ? ? I. Therefore we have
EeA\I if and only if ? = @)UF with F с N and F ? I. (**)
Let {?n}„eN be a disjoint sequence of elements of A. We consider two cases:
(i) There exists ? e N such that 0 e Ep. Then we have by (**) Ep = {0} U F with
F с N and F i I, F„ e I. Then F„ с N for every ? ? p. Therefore
|Jf„ = {0}u[fu(|Jf„)Y
II ? ?
where F U (Ц,^о En) is a subset of N that does not belong to I. By (**) we have
(ii) Let0? E„ for every и e N. Then {F„}„eN is also a disjoint sequence of elements of
I. We know by Example 2.22 that I has SCP and therefore there exists an infinite
subset N\ of N such that \Jn€N E„ e Л
We have by Pap A987).
Theorem 2.24. Let A satisfy SCP and {m,,},,^;, m„ : A -+ [0, +oo[ (и e N), be a
sequence of additive exhaustive set functions. If exists
lim mn(E) = mo(F)
n—>oo
for each ? e A, then [m„}^t0 is uniformly exhaustive.
Using Theorem 2.24 we can obtain more general statement.

146
P. de Lucia and E. Pap
THEOREM 2.25. All assumptions and statement as in Theorem 2.24 only the algebra ?
satisfy SIP instead o/SCP.
3. Related convergence theorems
3.1. Integration with respect to vector measures
Related the integral of scalar valued measurable function with respect to a vector measure,
see chapter written by N. Dinculeanu B002), Section 2.3, two crucial properties are related
to Vitali-Hahn-Saks Theorem 2.7 vector case.
First we recall the definition of the integral of a simple function / = ?"=\ XiXE, · where
E\,..., E„ are sets in ?.
I f(s)d?(s) = J2x:?(Er\E,).
Je tt
Now we have that a scalar valued measurable function / is integrable if there exists a
sequence {/„}»eN of simple functions such that {/„}„ем converges ?-almost everywhere
to / and the sequence {jE f„(x)d?(.x)}n€?; converges in norm in X for each ? e ?. Then
the limit of this sequence of integrals is the integral of / with respect to ?.
The following two crucial properties are strongly related to Vitali-Hahn-Saks theorem
(see Dunford and Swartz A958)).
If ? e ? and / is a scalar valued ?-integrable function, then the integral with respect
to ? over ? is an uniquely defined element of X.
Second, if / is a scalar value ?-integrable function, then v(E) = fE f(.x)d?(x) is a
countable additive set function on ?.
3.2. Orlicz-Pettis theorem
One of the basic tool of Banach space theory is given by Orlicz A929) and Pettis A938).
THEOREM 3.1 (Orlicz-Pettis). Let X be a normed vector space and let ?„ ?? be sub-
series convergent in X with respect to the weak topology. Then ?„?" 's subseries
convergent with respect to the norm topology.
Consequently, a weakly countably additive vector measure on ? ? -algebra is (norm)
countably additive.
We shall give now an elementary proof of Theorem 3.1 which is based on the following
simple Diagonal theorem (see Antosik A976), Pap A982) and Pap A995)).
Theorem 3.2 (Diagonal Theorem for non-negative matrix). Let x\s e E+, /', ;' e N, such
that
lim xjj = 0, lim ;c,y = 0, lim .v,, = 0.
7—>OC j—>3C i—>TC

Convergence theorems for set functions
147
Then for every ? > 0 there exists an infinite subset I o/N such that
? X'J < M-
,.j?l
Moreover, the elements of the set I can be ordered in a increasing sequence {р,},€ы such
that
lim ^P-Xp,^ =0 and lim У~\????? = 0.
PROOF. Let ;'o = 0. We shall choose a sequence {/,,},,^,' of natural numbers such that
(i) /,,_i < /'„ for every и е N,
(ii) ximik < 2~k~s for every l^s^n,l^k^n and « e N.
The proof goes by induction. Since lim,--»^.*,·,- = 0 there exists an index r such that
хц < 2~~2 for /' ^ r. Let /| = r. Then (i) and (ii) hold for ? = 1. Suppose now that we
have already find /? ,ip such that (i) and (ii) hold for 1 ^ s ? ?, 1 ^ к ^ р. Since
*,-,,¦ —> 0 and *,¦,¦, —>¦ 0 as /' —>¦ oo for ? = 1,..., ? and .?,, —>¦ 0 as / —>¦ ??, there exists an
index ip+\ > ip such that
x
l\'p+\
<2-''-J-|M and .v,„ „¦<2-p-s-lM
for /n = 1,..., ? + 1. Therefore (i) and (ii) hold for 1 ^ s ^ ? + 1 and 1 ^ к ^ ? + 1. So
we have constructed a sequence {/„}„eri with the properties (i) and (ii). By (ii) we obtain
OO ОС
??1'·''<?-
ш=1к=\
Taking / = {/'?, /?,...} we obtain the desired conclusion. Since the sequence of summands
of a convergent series converges zero, we obtain
oo
lim y^Xi,ik =0.
Take pk = ? for к eN. D
Proof of Orlicz-Pettis theorem. We can suppose without loss of generality that
X is separable (replacing X by the closed subspace spanned by the sequence {-х^Ьем)· Let
{xnj },epj be an arbitrary subsequence of {*„}/, eN· Taking 5, = Y^'k=\ x,n we nave t0 Prove
that {Sj }iepj is a norm Cauchy sequence. We take an arbitrary increasing sequence {p, }iepj
and n = J2k=Pi + \ х>ч- So we have t0 prove that lim,^^ ||z,|| = 0. Suppose that this is
not true, i.e., for some ? > 0 there exists a subsequence of {?i}„eN (denoting it some)
such that Hzill ^ ?. As a consequence of Hahn-Banach theorem for every ? there exists

148
P. de Lucia and E. Pap
z* e X* such that ||z*|| = 1 and z*(z„) = \\z„\\. Then by Banach-Alaoglu theorem (for a
neighborhood V of 0 the set {x* e X*: \x*(x)\ ^ 1 (x e V)} is weak* compact, see Swartz
A992, 16.12) {z*L,eN has a subsequence (denote it same) such that it converges weak* to
some z* e X*. Since {z„}„eN is weakly convergent we have \z*(z„)\ < ?/2 for large ? and
therefore
\(?*-?*)(??)\>?-\?*(?„)\^?/2
for large n. Taking a subsequence of {г*}„ек (again denoting it same) we have
|(?;-?*)(?„)|>?/2
for all n. Taking
v. i|(s*-z*)(zy)| for/*;.
0
for/ =;,
we obtain by Diagonal Theorem 3.2 (take ? = ?/4) that there exists a subsequence {s, },ep
such that
? ? x^j<e/4.
/=1./= ?. ]??
We have by Diagonal Theorem 3.2
?«-;*)^)
^ ?/2-?/4.
?«-^)
?'
Hence
7=1
^ ?/4 for every / e N.
Contradiction with the weak subseries convergence of ?? zn ¦
a
Theorem 3.1 implies directly the locally convex version, i.e., if X is a locally convex
vector space and if the series ?? x" IS subseries convergent with respect to the weak
topology, then ?? ?? ls subseries convergent with respect to the Mackey topology of X.
Further generalization is obtained by Tweddle A970): if ???" IS subseries convergent
with respect to the weak topology, then ?? x» IS subseries convergent with respect to
the Mackey topology ?(?, G*), where G* is the vector space of all linear functionals y*
on X such that ?*(????) = ?,,)'**-*") f°r a" series ?„?» which are weak subseries

Convergence theorems for set functions
149
convergent. Moreover, ?(?, G*) is the strongest locally convex topology on X such that
every weak subseries convergent series is subseries convergent. As a consequence of
Tweddle's result it can be given an interpretation of Nikodym convergence theorem in
the spirit of Orlicz-Pettis theorem, see Antosik and Swartz A985), where also other types
of Orlicz-Pettis theorems can be found.
3.3. Schur lemma and Phillips lemma
In this part we present the classical lemmas of Schur and Phillips which have many
applications in measure theory and functional analysis.
One version of the classical Schur lemma (see Dunford and Swartz A958)) states that a
sequence in i' converges weakly iff it converges strongly. We present here a generalization
of Schur lemma to the group case obtained by Antosik and Swartz A985).
Let G be commutative group endowed with a quasi-norm | · |, i.e., | · |: G —>¦ K+ such
that |0| = 0, |-дс| = |*| and \х+у\Ц \x\ + \y\.
THEOREM 3.3. Let Xjj e G for /', j e N. Assume that the rows of the matrix (-*,, ),.,,=?
are subseries convergent and lim,^^ xjj = Xj exists for each j e N. If {?];<=o x4 beN IS
convergent in G for each DcN, then
(a) the series ]T ¦ Xj is subseries convergent;
(b) lim,^oo Y^j€DXij = T,j€Dxj uniformly for D с N.
The proof is based on Basic Matrix Theorem 5.10, see Antosik and Swartz A985).
Theorem 3.3 is a generalization of the version of Schur lemma for Banach spaces
obtained by Brooks A974, Corollary 2). We remark that Brooks' method cannot be
transferred to the previous case since it is based on duality methods.
The classical Schur lemma is contained in the following (for tj = 0).
COROLLARY 3.4 (Schur lemma). Let (i,y)i.yeN be a real infinite matrix. If
lim У tjj exists for each DCN
JtD
andiftj = lim/^oo i,y, then {iy}yeN belongs to l] and
oo
.lim ??'?-';? = °·
7—>?? *
7 = 1
Phillips lemma was obtained by Phillips A940). We present here a generalization of
Phillips lemma to the group case obtained by Antosik and Swartz A985).

150
? de Lucia and ?. Pap
THEOREM 3.5. Let G be a sequentially complete normed group and let Vj: ? —>¦ G,
/ e N, be exhaustive. If\imj^x Vj(E) = v(E) exists for each ? e ?, then for each disjoint
sequence {Ej}j€fqfrom ?
lim ]P vi (Ej) = ]P v(Ej) uniformly for D с N.
'^°°jeD j€D
The proof presented in Antosik and Swartz A985) is based on Theorem 3.3, Basic
Matrix Theorem and Drewnowski Lemma.
As an easy consequence of Theorem 3.5 follows the classical Phillips lemma (for
v(E) = 0).
Corollary 3.6 (Phillips lemma). Let v, : V(N) -> R be bounded and finitely additive.
Iflimj^oo Vj(E) = v(E) exists for each ?CN, then
oo
lim ?\?>№)-?(?)\=°·
?—>oc '—'
We can interpret the preceding classical Phillips lemma in the following way. The dual
space of ix is the space ba of all bounded finitely additive set functions on P(N) with the
total variation as norm. Then for each ? eba, the series ? ¦ v({j}) is absolutely convergent
and ? h*- MO'Db'eN gives a projection ? from ba onto Iх. By Phillips Lemma 3.6
for each Cauchy sequence {у,-},-ещ from ba with respect to the topology a(ba,mo) the
sequence {Ри,},ем is norm convergent in €', where wo is the subspace of Iх containing
all sequences with finite range.
3.4. Rosenthal's lemma
One useful method close to diagonal theorems and a sharpening of Phillip's Lemma 3.6
was given by Rosenthal A968). We shall prove in this part in a quite elementary way a
countable version of the important Rosenthal's lemma. It is used as the basis for a deep
study of the behavior of operators on spaces of continuous functions, see Diestel and Uhl
A977). The presented proof follows Antosik and Pap A981), where it is used an idea of
KupkaA974).
Lemma 3.7 (The countable case of Rosental's lemma). Let v„ for ? e N be additive
non-negative set functions defined on V(N). If there exists a number ? > 0 such that
v„ (?) < ? for each n= 1,2,
then for each number ? > 0, there exists an infinite subset N\ ofN such that
Vn(N\ \ {«}) < ? foreachneNi.

Convergence theorems for set functions
151
PROOF. Assume that, for some ? > 0, no such subset N\ exists. Let M\, ЛЬ.... be a
sequence of infinite disjoint subsets of N such that (J Mp = N. For each ? there is an
index к ? e Mp with vk (Mp \ {kp}) ^ ?. Otherwise there would exist an index ? such that
v„(Mp \ [n]) < ? for all ? e Mp which is impossible. Let К be the set of all members of
the sequence {кр}р€ц, i.e., К = {kp: ? e N}. We note that for each ? e К we have
?„(?) + ?„(?\?)? 1
(without loss of generality we may take 1 instead of ? > 0). Since Mp \ {kp\ с ? \ ? for
all p, we have vn(K) + ? ^ 1, for all « e A-. Hence, for all ? e A-. vn(K) < 1 — ?. Next,
apply the same argument to v„ with и е К and A-, by replacing v„ for « e N by v„ for
и е К and N by A". If the process does not stop, then there is a new subset S ?? ? such that
v„ (S) ^ 1 — 2? for each /; e 5.
It now becomes apparent that the process must come to end before ? iterations, if /; is the
smallest positive integer such that 1 — «? < 0. ?
Let T(N) be the family of all finite subsets of the set N.
COROLLARY 3.8. Let {9„}пе^ be a sequence of additive non-negative set functions
defined on the family .F(N). If there exists a number ? > 0 such that
?„(?) < ? for each ? = 1.2 and each set A ef(N),
then for each ? > 0 there exists an infinite subset N\ o/N such that
e„(D\{n})<?
for each finite subset D of N\ and each ? e N.
PROOF. We introduce the functions v„ defined on V(N) whose values are given by
v„(A) = J^ ?„({;}) for each ? = 1, 2,... and each A e V(N)
and apply Lemma 3.7 to the sequence {v„}n€i;. ?
Now it is easy to obtain the countable case of Rosenthal's lemma formulated in Diestel
and Uhl A977).
COROLLARY 3.9. Let {вп}п€ц be an uniformly bounded sequence of additive real valued
set functions defined on an algebra Л of subsets of a set S. Then for each sequence {En }п(-н

152
P. de Lucia and E. Pap
of pairwise disjoint members of ? and each ? > 0 there exits a subsequence {ЕП] }j^n of
{?,,?,,^?? such that
i^i( U Е'ч)<?
for all finite subsets D ofN\ and all j e N.
If in addition A is a ?-algebra, then the subsequence {E„ }y epj may be chosen such that
i^i(U?'u)<?
for all j= 1,2,....
PROOF. To prove the first part of the corollary we consider the functions v„ defined on
.F(N) whose values are given by
Vn(A) = Wn\(\jEj\
for each ? = 1, 2,... and each A e .F(N), and apply Corollary 3.8.
To prove the second part, we consider the functions v„ defined on 7?(N) whose values
are given by
vn(A) = W„\(\jEj\
???
for each ? = 1, 2,... and each A e V(N). and apply Lemma 3.7. ?
3.5. Hewitt-Yosida theorem
In this part using the Nikodym convergence Theorem 2.6, we shall obtain an important
decomposition theorem of finitely additive set function.
For a real additive set functions v\ and vi on an algebra A we write v\ ^ vi if
v\ —vi^-O. This ordering is a partial order on the family of all real valued finitely additive
set functions on A, and they form a lattice with respect to this ordering.
Dehnition 3.10. A bounded finitely additive set function ? : A -> Ш+ is purely finitely
additive if 0 ^ ? ^ ? and ? is countable additive imply that ? = 0.
We shall need the following lattice property of special spaces of set functions, see
Dunford and Swartz A958).

Convergence theorems for set functions
153
THEOREM 3.11. The lattices ba(S.A) (bounded finitely additive set functions) and
ca(S, A) (countable additive set functions) are complete lattices.
Now we have the following version of Hewitt-Yosida decomposition theorem, Hewitt
and Yosida A952), see also Dunford and Swartz A958) and Diestel and Uhl A977).
THEOREM 3.12. If ? is a bounded finitely additive non-negative set function then there
exists a unique decomposition
? = v\ + V2, Vj ^ 0,
where v\ is countable additive and vi is purely finitely additive bounded set function.
PROOF. Let С = {? e ca(S\ A): 0 ^ ? ^ v]. Take a sequence {??/??? from С such that
lim ?„E) = 5ирдE) < ??.
% Mi ^ ?'!=? My, » = 1,...,«, it follows by Theorem 3.11 that there exists sup^i,...,
/J-n} = ~?~?~ in ca(S, A). We have ?^ ^ ?,,+ ?, ? e N. Let ?\ be the ?-algebra generated
by A and denote the unique extensions of measures {д^},,ем on ?\ by the same symbol.
The extensions are also non-decreasing (extension of non-negative set function from A
to ?\ is also non-negative) and therefore for each ? e ?\ there exists \im„^0CJia(E)
and we denote it by v\ (E). Then by Nikodym convergence Theorem 2.6 v\ is countable
additive on ?\, and so also its restriction to ?. We introduce vi by vi(E) = v(E) — v\ (E)
for ? eA. Then by the definition of v\ and {??? we have that vj ^ 0. If we suppose that
V2 is not purely finitely additive then there would exists a non-zero countably additive ?
such that ?' -?,? — v\. This would imply v\ ^ v\ + ?' ^ ? and therefore supMeC ?(?) =
v\(S) < v\(S) + v'(S). Contradiction. Uniqueness follows by Theorem 3.11. D
The Hewitt-Yosida theorem was extended by Uhl A971) to the vector measures with
Stone space argument, and with a direct proof by Huff A973). Drewnowski A973) and
Traynor A972) have extended Hewitt-Yosida theorem for group-valued measures.
3.6. Biting Lemma
One useful method close to diagonal theorems are given by Brooks and Chacon A980).
Biting Lemma is applied by Brooks and Chacon A980) for a proof of Vitali-Hahn-Saks
theorem, in the proof (the part (b)) of Biting Lemma) of Dieudonne convergence theorem
A978), and in a short proof of a result of Akcouglu and Sucheston A978) related the
existence of an exact dominant for a superadditive process.
LEMMA 3.13 (Biting Lemma). Suppose {v„b<eN is a uniformly bounded sequence of
additive set functions defined on the measurable space (S, ?), each absolutely continuous
with respecttoO, where ? is a positive, bounded and additive set function on (S, ?). Then

154
P. de Lucia and E. Pap
(a) If ? and ? are positive, then there exists a set Cf.? and a subsequence {v,u}i€n
{depending on ? and ?) such that: 0(CF^) < ?, and there exists S > 0 such that
|u„,(A)| < ? for every i eN whenever ?(?) < ? and А С S\Cf^.
(b) If all the v„ and ? are measures, then there exists a subsequence {v„; },<=pj such that
for every ? > 0 there exists a set BF such that 0(BF) < ? and {vHl },ек is uniformly
absolutely continuous with respect to ? on S\ Bf.
(c) If all the v„ and ? are measures, then there exists a subsequence {vUl}i€fq that
converges w, i.e., there is a sequence B„ \ ?. ?(??) -> 0 such that if А С S \ ?,
for some i eN, then v„(A) -^ v(A), and \v\(B) = 0.
PROOF. We only give a proof of (b). Suppose that the statement is not true. Then there
exists ? > 0 and a subsequence {v,h },€\; such that for A e ? such that ?(?) < ?, there is
no further subsequence of {v,u }ier; which is uniformly absolutely continuous with respect
to ? on S \ A. Let D \ denote the set of all с > 0 such that for any к} > 0 there exists a set ?,?
such that ?(?„) < ? and \vni(En)\ > с for infinitely many /'. Since D\ is nonempty and
bounded there exists sup D\ which we denote by 2c\. Take a set C\ such that 9(C\) < ?/22
and |i>i.i|(C|) > c\, where {vuhep; is a subsequence of {vtli },-e^·. Let Di denote the
set of all с > 0 such that for any ? > 0 there exists a set ?? such that ?,, с S \ C\,
?(??) < ? and | ?\?\(??) > с for infinitely many /'. Take a set Ci С S\C\,e(Ci) < ?/23, a
subsequence {t^.iheN of {vuheN such that |i>2.j|(Ct) > ci for all /', where c? = ^supZK
Continuing this procedure we obtain {^./}iei:, Q-.c; such that {Q}jteN forms a sequence
of pairwise disjoint sets such that 0(Q) < ?/2*-1, |i>ju1(Q) > ck-' e N. If the sequence
{cjtUeN would converge to zero, then the diagonal sequence {i^.jtheN would be uniformly
absolutely continuous with respect to ? on S \ (J* Q · wnat 's a contradiction, since
^(Ujt Cjt) < ?. If the sequence {с;-}л-еь- would not converge to zero, this would contradict
the boundedness of {i>n }„ ep,- ¦ ?
4. The relation between boundedness and exhaustiveness
Let X be a Banach space. A set function m: ? -> X is exhaustive (strongly bounded,
shortly s-bounded) if lim,,^^ m{En) = 0 for each sequence {?„}„ек of pairwise disjoint
sets from ?-algebra ?. A sequence {w/},ef,- of set functions m^-.? —* X, i e N, is
uniformly exhaustive if lim,,^^ ш,(?„) = 0 uniformly in /' for each sequence {?,??,???
of pairwise disjoint sets from ?-algebra ?. It is obvious that exhaustivity implies
boundedness. As we have seen a real additive set functions on a ring is bounded if and
only if it is exhaustive. For Banach space valued additive set functions this is no more true.
Example 4.1. Let ? be the ?-algebra of Lebesgue measurable subsets of [0, 1] and
let Lx be the Banach space of essentially bounded functions with usual essentially sup
norm II · II. Define m '. ? —> Ly^ by m{E) = ?^, ? 6 ?. Obviously, m is additive but
not ?-additive. It is bounded by ||№(?)|| = ||??|| ^ 1. But it is not exhaustive, e.g., for
En = [j^,~[ we have ||w(?„)|| = 1 for all n e N.
With some additional suppositions on the range we have for vector valued measures on
a ring, see Diestel and Uhl A977).

Convergence theorems for set functions
155
THEOREM 4.2 (Diestel and Faires). Let TZ be a ring of subsets of a set S. A bounded
finitely additive measure m : TZ -> X is exhaustive if and only if X does not contain a copy
of cr,.
We shall need in the proof the following theorem.
THEOREM 4.3 (Bessaga-Pelczynski). A Banach space X is such that every weakly
unconditionally convergent series in X is subseries convergent in norm if and only if X
contains no copy ofcc,.
PROOF. Suppose that the theorem is not true. Then for a weakly unconditionally
convergent series ?^ *, there is a subseries J2?Li xn, which does not converge. Then
there is a ? > 0 and an increasing sequence {p, },-е^ such that
Pi+i
k=p, + \
> ?.
Take n = J2k='Pl + i xm- The series ]T. zi is weakly unconditionally convergent ({]Tief c,·:
F с N finite} is norm bounded since J2kel·' x»k 's norm bounded).
We can suppose without loss of generality that X is separable. Then as a consequence
of Hahn-Banach theorem for every ? there exists c* e X* such that ||г,*|| = 1 and
z*(zn) = \\z,i\\- Then by Banach-Alaoglu theorem {г*},/ек has a subsequence (denote it
same) such that it converges weak* to some z* e X*. Then
\(?*-?*)(?„)\>?-\?4?„)\>?/2
for large n. Taking a subsequence of {г*}„ек (again denoting it same) we have
\(?*„-?*)(?„)\^?/2
for all n. Taking
\(zf - Z*)(Zj)\ fOTi ? j.
0 for/=y,
we obtain by Diagonal Theorem 3.2 (take ? = ?/4) that there exists a subsequence {A:,-},-epj
such that
? ? ?^<??.
Define ?: со -> X by
OO
i = l

156
P. de Lucia and E. Pap
Then we have by Diagonal Theorem 3.2
2\\T({ti}ieN)\\^ \\(??-?*)?{{^]???)\\
^|?,|?/2-||{?,},??||?/4.
Taking supremum over / we obtain
2||Г({г,}/еК)|| ^ 2||{i,}jeN||?/4.
Hence ? has a bounded inverse. ?
PROOF OF Theorem 4.2. Suppose m is not exhaustive. Then there exist a disjoint
sequence {Е„}„€щ and ? > 0 such that ||m(?„)|| > ? for every n. Then for every x* e X*
the scalar measure x*m is bounded on К and therefore it has a bounded variation. Hence
II
^|jc*w(?,)| <sup{|jt*m|(A): AeTZ} <oo.
Therefore the series ?"_| x*m(Ei) is unconditionally convergent and by Theorem 4.3 X
contains a copy of со. ?
THEOREM 4.4. Let ? be a ?-algebra of subsets of a set S. A bounded finitely additive
measure m : ? —* X is exhaustive if and only if X does not contain a copy ofix.
PROOF. Repeating the part of the proof of Theorem 4.3 from the part "We can suppose
without.. ."taking now ц =m(Ei) and applying Drewnowski Lemma 2.9 before defining
the operator T, now in the form ?: ex(J) -> X, where J = [n,¦: /' e N) and {«, },ек is the
sequence from Drewnowski Lemma 2.9, by
ОС
T({ti}ieji) = YltiZ*m(Ei).
i = l
we obtain that ? has a bounded inverse. ?
5. Diagonal theorem for triangular set function
In this section we shall present a general version of very useful Diagonal theorem for
triangular set functions.
The first Diagonal Theorem was proved by J. Mikusinski A970), and first Diagonal
Theorem for triangular set functions by Pap A976), then by H. Weber A986) and Saeki

Convergence theorems for set functions
157
A992). We present here Theorem 5.3, the version of Diagonal Theorem obtained in
Antosik and Saeki A995), see Pap A995). Weber has proved Theorem 5.3 in H. Weber
A986) only for the case
lim lim m„(Bm) = 0.
in—>DC It —>DC
Thus the second possibility in Theorem 5.3 cannot occur in his setting unless lim,^x sup
(d„ — a) ^ 0. On the other hand, in Saeki A992) this result was obtained only for a = 0
and d„ — Ш/;(В/7)/2. Corollary 5.5 with a = у = 0 is a generalization of the nontrivial part
of Theorem 2.4 in H. Weber A986). Actually, a careful examination shows that his proof
works if у =0.
Corollary 5.7 (Diagonal Theorem) was proved for group case in Antosik A971) and
for semigroup case in Pap A974). Diagonal Theorem 5.9 denoted by DT is proved by
P. Antosik and the Basic Matrix Theorem 5.10 was obtained in Antosik and Swartz A985).
5.1. a-boundedness
Let 1Z be a ring subset of a set S. For a given к e [ 1, oo[ the set function m: 7Z -» [0, oo[
is a ^-triangular (triangular for к = 1) set function if m @) = 0 and
m(A)-km(B) ? m{A U В) ^ m{A) + km{B).
for A, BeTZ, ??? = 0.
Let m : ?? —* [0, oo] be a triangular set function. For each set А с S, let
in(A) = sup{m(B): В е П, В С A].
DEHNITION 5.1. We say that m is ?-bounded @ ^ a < oo) if inf„ m(Bn) ^ a for each
disjoint sequence {В„}„ек in ??.
If (A„}„epj, is a sequence of sets and / С ?, we write A/ for Une/ ^"·
LEMMA 5.2. Let m:Tl^> [0, oo] be ?-bounded. Then in is a-bounded on the subsets
of X. Moreover, if [ B„ }n€n, is a disjoint sequence of subset of S and if ? > a. then there
exists an infinite set I С N such that w(Une/ ^") < ?-
PROOF. The first assertion is obvious. For the second, we represent N as an infinite disjoint
union of infinite sets. ?
THEOREM 5.3 (Diagonal Theorem for triangular set functions). Let {т„}„ен be a
sequence of?-bounded triangular set functions on the ring ?? @ ^ a < oo), let {Sn},ieN
be a disjoint sequence in TZ. and let {Sn}n€f>s be a sequence in [0, oo]. Then either there
exists an infinite set I С N such that
(i) mll(\Ji€,Bi\Bll)<S„ (nel),
or there exists a finite set J С N which satisfies (i) with I = J and
(ii) m,i{Bj) ^ <5„ —a for infinitely many ? e N.

158
? de Lucia and ?. Pap
PROOF. If <5„ ^ a for infinitely many ? e N, we can take 7 = 0. Suppose <5„ > a for all
? e N large enough. Let us call / с N a nice set if it satisfies (i).
Suppose that no finite nice sets / с N satisfy (ii). We shall construct a sequence {/„}„,=?
of infinite sets and a sequence {/цкеь; in N as follows. Let /o = N, and let n\ be any
natural number with <$„, > a. Note that \n\) is a nice set. Let к &N, and suppose that
infinite sets IqD ¦¦ ¦ D h-\ and a nice set {n \ in} have been chosen in such a way
that
m
for l^j^k. B1)
"j ( U ?»< ) < S»j ~a for К ./ < *. and B2)
ч ^ 4-
'y'-i
li' = l
Я,, ? (J ?») < &nj - mnj I \J B„, \ for 1 <: ; ^ к - 1
B3)
(ie/, \i = l
By B2), the right-hand side of B3) with; = к is larger than ?. since each mn is ?-bounded,
it follows from Lemma 5.2 that h-\ contains an infinite set h which satisfies B3) for
j = k. Moreover, no finite nice sets J satisfy (ii) and {n\ и*} is a nice set. So
/„ П]и*,оо[ has an element m+i which satisfies B2) with j = к + 1. The arguments
in the next paragraph will show that {щ >n+1} is a nice set.
Define / = {/«: к e N). To show that / satisfies (i), pick any j e N. Then к > j implies
nk e /*-1 С I j by B1). It follows by the subadditivity of w„( that
m„j ( (J B„ \ Bnj J < тП] i [J ВП] \ + mH] ( [j B„ J < Snj,
which completes the proof. ?
Corollary 5.4 (Webers Version of the Nikodym Boundedness Principle). LetH be a
a-algebra, and let ? be a set of?-bounded triangular set functions on ?? for a < oo, such
that
sup{w(S): m e T} < oo
for each В e ??. Then ? is uniformly bounded.
PROOF. Suppose that the theorem is not true. Then there exists a disjoint sequence
{?/ilneN ш "R- and a sequence {w„}„ef.; in S such that m„(B„) > 2n for each ? e N.
Replacing ?? by a smaller ring, we may suppose ?? = {Une? ^»: ^ С N and U»e? ^" e ^)·
If У С N is a finite set and m„(\JneJ B„) > ? - a for infinitely many ? e N, then J
contains a set F such that w„(U,ieF ?„) > ? - ? for infinitely many ? e N. Hence
sup,, w„(U„e/r S») = oo, a contradiction. It follows from the Theorem 5.3 with <$„ = ?

Convergence theorems for set functions
159
that there exists an infinite set / с N such that w»(U»e/\(n) ?») < n ^or eacn " e ^-
Since ft is a ?-algebra, we may suppose U,,e/ ^" e ^- ^ut tnen " e ^ implies
1" ( U ?") ^ m"(?")-"'»( U ?» )
\ie/ ' \ie/\(/i) '
> 2n — ? = ?.
/ie/\H
Contradiction. ?
COROLLARY 5.5. Let ?, ?,? e [0. oo]. and let {m„}nef; be a sequence of a-bounded
triangular set functions on the ring ft.
(i) Suppose that for each disjoint sequence {Z?„}„ef; in ft and each subsequence
\т,ч кем of{m„}n(zn, we have
infIm„k ({jB,\.keEcNand |J ?„ e ft ^ /S, and
lim I lim w,u(Sy) ^ylimw„(By).
j —»зс ' к —> эс '
Г/геи ?/ге wn are uniformly (a + ? + y)-bounded.
(ii) //"ft й а ?-ring and if
? (A) = lim m„(A)
G—>3C
is ?-bounded, then the sequence {m„}n(.\\ is uniformly (a + 2fi)-bounded.
PROOF. For (ii), we may suppose ? = ?. Suppose to the contrary that m'n are not uniformly
(?+? + y)-bounded. Then there exists a disjoint sequence {б„}„ек in ft, ? > a + ? + ?,
and a sequence {n*heN in N such that m,4(Bk) > ? for each к eN. Since each m„ is
?-bounded and ? > a, we may suppose n\ ^ in ^ · · ·
Case l. Suppose N contains an infinite set / such that w,u(Une/\jjt) ^") < (P + a +
? — ?)/2 for each к e /. Then к e ? с / implies
i'u(Us"Msw"<(S'0~w"<( U ?")
\ie? ' \ie?\(M '
> ?-(? + ?+?-?)/2
= (?-?-? + ?)/2>?. B4)
So, {m„<,}jteN does not satisfy the first requirement in (i). Also B4) shows that ? (Be) >
(p - ? - у + /3)/2 whenever ? с / is an infinite set with \Jn(zE B„ e ft. Therefore, if ft
is a ?-algebra, then V is not /3-bounded.

160
P. de Lucia and E. Pap
Case 2. Suppose that N contains no infinite sets / as in Case 1. Then, by Theorem 5.3
with 8k = S = (? + ? + ? — ?)/2, each infinite subset of N contains a finite set J and an
infinite set / such that к е / implies
т'ч ( U B" ) > (P + a + У - ?2 -? = {?-?+?- ?)/2 > ?.
An inductive application of this fact yields a disjoint sequence {F/}yeN °f finite subsets
of N, a sequence {Ay}yepj in 1Z, and a subsequence [mSj}j€n of {m^heN such that
Aj с Bfj and
m4(Aj) ^ {? — ? + ? — ?)/2 > ? for every к ^ j ^ 1.
Hence the second requirement in (i) is not satisfied, and ? is not y-bounded. D
5.2. Diagonal theorems
Let S be commutative semigroup with a neutral element 0 which is endowed with a
triangular functional /, i.e., /: S -> [0, oof such that
fix) - f(y) < fix + У) < fix) + /0')
and /@) = 0 (Pap A974), see also H. Weber A976), Pap A982)).
Remark 5.6. Let d.S -> [0, +oo[ be a pseudometric such that satisfies the following
condition
d{x +x\,y + y\) ^ d(x,y) +d(x\,y\) forevery x,x\, y, vi e S. B5)
There was proved by H. Weber A976) that the uniformity in each commutative uniform
semigroup is induced by a family of pseudometrics which satisfy B5), see Pap A982,
1995). If S is endowed with a pseudometric d which satisfies the inequality B5) then the
pseudometric d induces a triangular functional / in the following way: f(x) = d(x,Q)
(x e S). Specially, if S is a commutative topological group (it is a uniform semigroup) its
topology is generated by a family of quasi-norms. Quasi-norm is a functional defined on
a group S such that it satisfies the conditions: (i) |0| = 0; (ii) |jc| = |—*|; (iii) \x + y\ <
1*1+ M.
Let (JCiy)i.yeN be an infinite matrix (indexed by ? ? ?) with entries in S. For each
sequence {xj]jsm in S and each /cN, write
Vye/ 7 \е/П[|.п] 7

Convergence theorems for set functions
161
Corollary 5.7 (Mikusinski-Antosik-Pap Diagonal Theorem). Suppose
(i) linwoo/(*,-,-) = 0(/eN).
Then there exists an infinite set I and a set J С I such that
(a) ?/e/ f(x'j) < oo (/' e N), and
(b) П%^хф>ПхнI2ЦЫ).
PROOF. By (i), we may suppose that the U,;),.yeR, satisfy ?,??/<-*<'./') < °° f°r eaun
/' e N. Define
(?) = /(??,? (i€N,?cN).
Then each w, is a continuous (hence exhaustive) triangular set function on V(N). If / С N
is any set such that w,(/\{/'}) < /и,-({/})/2 whenever/ e /, then /' e / implies
т/(/)^т,({/})-т,(/\{/})^т,({/})/2.
Therefore the desired result follows by Diagonal Theorem 5.3 for triangular set functions
with Si = mi ({/ })/2 and a = 0. ?
Let G be a quasi-normed group, i.e, a commutative group endowed with a quasi-
norm | · |.
Corollary 5.8 (Antosik Diagonal Theorem). Let ?, ? e [0, oof, and suppose that
(i) limy^ooU,y| = 0(/eN),_
(ii) for each finite set F С ?, limy-^ | ?/??^?'? ^ ?· and that each principle infinite
submatrix of(xu)l, yepj //as a further principle submatrix („v/y) smc// //га/
(iii)
Шп^зо??^?у/;Кути «i
l'-»00
PROOF. Taking infinite submatrix of (-X/y)/.yeN- we may suppose that ? у era l-^/y I < °°
for each /' e N and lim,-^^ \хц\ exists in [0, oo]. Let m-, be the continuous triangular set
function on T-4N) defined as in the proof of last theorem.
Fix any ?' > ?. Then (ii) implies that there exists no finite set У с N such that
irij(J) ^ ?' for infinitely many i e N. So, by Diagonal Theorem 5.3 with a = 0 and
Si = ?', there exists an infinite set / с N such that /и,-(/\{/}) < ?' f°r each i e /. Then,
/' e /' с / implies
\xii\=mi({i})^mi{l')+mi(l'\{i})<mi(l')+P'.
Then by (iii) we obtain that lim \хц \??+?'. ?
As an immediate consequence we obtain a Diagonal Theorem (which we will denote by
DT) given in the next theorem.

162
P. de Lucia and ? Pap
THEOREM 5.9. Let (G, | · |) be a quasi-normed group, and(xij)i,j€n an infinite matrix in
G such that for every increasing sequence [kj }, <=t; in N there exists a subsequence {и/ },-epj
°/{^iheN such that
DC
lim jc„,„ =0 for all j e N. and lim / .v,,,„ =0.
i —> ОС 1 —> ЭС i-~l
7 = 1
Then lim,-^ ос хц = 0.
As a consequence we obtain the Basic Matrix Theorem.
THEOREM 5.10. Let (G, | · |) be a quasi-normed group, and (.viy)/.yen an infinite matrix
in G such that
lim Xi. = Xj (I)
I—>OC
exists for all j e N and for every increasing sequence {шу}уек /и ? ?/геге eiisii ?
subsequence {иу}уем o/{wy}yeN i/zc/г ?/??? {?/1? ·??"; )iet·: '5 ? Cauchy sequence.
Then liirii^jcxtj = Xj. uniformly with respect to j e N.
PROOF. If we suppose that the theorem is not true, then there exist ? > 0 and a
subsequence {s,}, ем of natural numbers such that sup \xSlJ —x} | > ?. By (I) we can choose
sequences {uheN and {улЬек such that the matrix (ya-.s)л-.лег; for }\s = -*/u\ - Xh+u<
satisfies the conditions of DT, but |vu| > ? (к e ?) (taking eventually a submatrix).
Contradiction. ?
There are many other types of diagonal theorems (see Antosik A971, 1978), Pap A982),
Swartz A996)). There are many different applications of diagonal theorems in measure
theory and functional analysis (see Pap A982, 1985a, 1985b), Antosik and Swartz A985),
Pap A995), Swartz A996)) as Banach-Steinhaus theorem, Banach uniform boundedness
theorem, Bourbaki theorem on joint continuity, Orlicz-Pettis theorem, Schur lemma,
Phillips lemma, Swartz kernel theorem, Adjoint theorem, Closed Graph Theorem, etc.
5.3. SCP and SIP algebras
THEOREM 5.11. Let ? satisfy SCP and [m„}ll€i;, m„ :A -> [0, +oo[ (/; e N), be a
sequence of k-triangular exhaustive set functions. If exists
lim m„(E) = mo(E)
n—>oc
for each ? e ? and wo is exhaustive, then {тп}^={) is uniformly exhaustive and wo is
k-triangular.

Comergence theorems for set functions
163
Using Theorem 5.11 we can obtain more general statement, see Pap A987).
THEOREM 5.12. All assumptions and statement as in Theorem 5.11 only the algebra ?
satisfy SIP instead of SCR
6. Dieudonne type theorems
We shall present a generalized Dieudonne type theorem A951) for the class of triangular
set functions, see Pap A986).
Let ? be a locally compact space. Then К and ? denote the family of all compact and
open sets, respectively.
Definition 6.1. Let Я be a ring of subsets of T. A set function m :TZ^]-oo, +oo[ is
regular if for every AeTZ and every number ? > 0 there exist a compact set К с A and an
open set V D A such that for every set В e 1Z with В с V \ К we have \m{B)\ < ?.
To the rest of this section let m be always non-negative monotone set function. Then a
set function m is regular if for each set A e В and each ? > 0 there exist К е К and VeO
such that К с А с V and
m(V\/0 <?.
Remark 6.2. Obviously, that a non-negative monotone set function m is regular iff
inf{w(V\^): KcAcV. К е fC, VeO}=0
for each set A e B.
PROPOSITION 6.3. Regular positive monotone set function m is O-exhaustive, i.e.,
lim m(O„)=0
n—>oc
for each sequence {Оп}п€ц of open sets from О which are pairwise disjoint.
Proof. Since
ЭС
U 0„ e О
and m is regular, we have that for any ? > 0 there exists a compact set К such that
ЭС / ЭС \
KG [JO,, and m j (J 0„ \ К 1 < ?.

164 P. de Lucia and E. Pap
{?,???? is an open cover of К and therefore exists /?o e N such that К с U"li ^»· The
monotonicity of m implies
m(Ok) < m ? Oi. U ( Q 0„ \ К j j ^ in ( Q 0„ \ ?? J
for it > no + 1. ?
6.1. Triangular set functions
Let В be the collection of all Borel sets of a Hausdorff locally compact topological space T.
The disjoint variation in of m is defined by in(A) = sup/ J2iei lw(A')l. '4 С Т, where the
supremum sets such taken over all finite families {Д }ie/ of pairwise disjoint sets such that
Uie/ A = ?, see Pap A995). It is obvious that a triangular set function m with regular
variation is itself regular.
We formulate the main theorem.
THEOREM 6.4. Let ? be a family of triangular set functions defined on В with regular
disjoint variations. If the set {m{0): m e M} is bounded for every open set O, then
{m(By. me ?, Be В]
is a bounded set.
We shall assume in the following proofs that ? is a compact Hausdorff space. Namely,
we can replace ? with an Alexandrov one point ? compactification ? U {?}, taking
m(a>) =0 (m e ?4).
COROLLARY 6.5. Let ? be a family of regular scalar measures defined on B. If the set
{\m@)\: m e ?4] is bounded for every open set O, then
{\m(B)\: meM. В е В)
is a bounded set.
PROOF. Let v(B) = \m(B)\ (B e B, m e M). It is obvious that the family of all such
set functions ? satisfies the conditions of Theorem 6.4 since ? = m is also regular (for real
valued this follows from \m\ = m+ +m~ by Jordan decomposition and for complex valued
from the inequality \m\ ^ |»i|| + |»i2|, where m =m\+imT). So we apply Theorem 6.4. D
In the proof of Theorem 6.4 we shall use the following lemma.

Convergence theorems for set functions 165
LEMMA 6.6. Let m be a triangular set function defined on В with regular variation.
Then m is ? -subadditive on each sequence of disjoint open sets {0„},ieN- '¦<'¦,
@0 \ 00
PROOF. First we will prove that m is order continuous on open sets, i.e., for each sequence
{tMneN of open sets such that Uj D Uj+\ (j e N) and Hyli ui = ^· we have
lim m(Uj)=0.
For each ? > 0 there exists a sequence of compact sets {?„b<eN such that Kj С Uj and
m(Uj\Kj)<e/2J (j e N). B6)
Then there exists «o e N such that ?'/=? Kj = 0 f°r aH n ^ "o· Let и > no- Then we have
?
m(U„) = mlu„\f>\Kj\=ml \J(U„ \ Kj) 1 < m I |J(i/,,\^))·
Hence, since m is subadditive (i.e., m(A U S) ^ w(A) + w(B) for every pair A, B of not
necessarily disjoint sets from B), and nondecreasing. we obtain by B6)
?
m(U„)^Yim(Uj\Kj)<e
7 = 1
for all ? ^ «o- Now, let {0„}„e^ be a sequence of disjoint open sets. Then we have
(ОС \ /7 / 'yZ \
U°mE^)+m U °j\
7 = 1 / y=l \y=H+l /
Taking n^oowe obtain
/ ОС \ ОС
(ОС
7=1 / 7=1
Proof of Theorem 6.4. It suffices to prove that every point in ? belongs to an open set
О on which holds
sup{m(A): А с О (A e B). m eM] < oo.
B7)

166
P. de Lucia and E. Pap
Suppose that this is not true. Then there exists a point ? e ? such that B7) does not hold for
every open set О such that xeO. We shall prove that there exists a sequence of pairwise
disjoint open sets {?/,}„ещ andasequence {/и„}нег; fromAI such that №,(?;) > /' (/' e N).
For any open set О such that ? e О there exists a Borel set ? с О and m \ e ? such that
W|(B)>4 + 2 sup m({x}). B8)
It is easy to prove that the preceding supremum is finite. Since m \ has regular variation,
there exists a compact set К с В and an open set О' С О, В с О' such that m\(B') < 1
for each В' с О' \ К. We have by the subadditivity of m \
ml(K)+ml(B\K)^m[(B).
Using the preceding inequality, the inequality m \ (В \ K) < 1 and B8), we obtain
mi(?)>3+2 sup w({jc}).
Let ^i = KU{x]. Then the last inequality implies (directly for.? e К)Ъу the triangularity
of m ? (for ? ? ?) that
m\(K\)>3+ sup w({.v}).
By the regularity of m \ there exists an open set U such that О D U Э K\ and m \ (B") < 1
for every B" cU\ K\. The preceding inequality together with the inequality
mi(U)>mi(Ki)-mi(U\Ki)
implies
m\(U) >2+ sup m({x}). B9)
Again by the regularity of m \ there exists an open set W such that {x} С W с U and
w,(S'")<l C0)
for every S'" с W\{;t}. _ _
Let ? be an open set such that ? e ? с Я с W (where H is the closure of the set H).
Then we have
m\{H)^ sup wi(A) +wi((.r}) ^ sup w i(S) + wi ({*}).
ACtt\(.v| ВсИ'\(л-|
Hence by C0) we obtain
miG7) < 1 + sup iw({*}). C1)

Com'ergence theorems for set functions
167
Let E\ =U\~H. Then we have E\ С О and E\ ?77 = 0. By the inequality
wi(?|)+w|G7) >wi(t/).
B9) and C1) we obtain wi(?|) > 1. Using the preceding procedure, taking in the
inequality B8) the constant 5 instead of 4 and taking into account the facts that ? e ? and
the family ? is not bounded on H, we obtain open sets Ei, H\ (H\ с ?) and mj e ?
such that ?? ? #? = 0, jc e H\ and ???(??) > 2. We have ?| П ?2 = 0· Continuing this
procedure we obtain a sequence {w,-},-ei.; from ?4 and a sequence {?,},ef: of pairwise
disjoint open sets such that
«?,(?,)>/ (ieN). C2)
We shall prove that m, (i e N) are exhaustive on the sequence {?„}„en of disjoint open
sets, i.e.,
lim mj(Ej)=0 (i e N). C3)
У — ЗС
Since U/li ^У i-s an °Pen set and "»/ are regular, for ? > 0 there exists a compact set
К' с UJli Ej such that w,(C) < ? for each / e N and each С С UjLi ?/ \ *'¦ Since
{E/lyeN is an open cover of K\ there exists uo e N such that ?' С U"li Ej- Then we
have for m ^ «o + 1
mj(E,„) ? supm,(C') ^ supw,(C) < ? (/' e N)
с с
where С С Em U ((J!, Ey \ K') and С с Uyli Ej \ K'¦ So we obtain <33)·
Let
m
xn =
(Ej)/y/i fori ? j,
4 JO for/=;·.
We have by C3) lim,^^ ,vi; =0 (/' e N). We obtain by the boundedness assumption of
the theorem that lim,^^ xt] = 0 (j e N). Applying Diagonal Theorem 3.2 for the infinite
matrix (Xij)i.jeN we obtain a sequence {/„},ie?; from N such that
DC
11тЕ*<л=о C4)
k = \
for every ? e N. Using the triangularity of пц„ (? e ?) and Lemma 6.6 we obtain
C0 \ DC
(J Eik I > min (?,„) - ? '"'" (?'^) <" e N)·
)t = l / )t=l. Ып

168
P. de Lucia and E. Pap
Hence by C2)
, / 00 \ . . DC
V'^1 min I |J Eik 1 > y/i~] mhl(Eiu)- ??? ]? min(Ek )
U-=l I к=\.кфп
Let ? -> oo. Then by C4) we obtain
V'^VJ U?iJ ^°°
but since Ut^i ?;* is an °Pen set we obtain a contradiction with the boundedness of
{mi„ IneN on open sets. ?
Let S be endowed with a pseudometric d which satisfies the inequality B5). Now we
can extend the definition of the difference regularity of a set function ?: ? -> S taking only
? andd(v(A), ?(?')) < ? instead of m and \m{A) — m(A')\ < ?, respectively.
Now we define the variation ? of a set function ?: ?? -> S with i>@) = 0 in the following
way:
v(E)=sup]T/(v(A)) (???).
Лея
where the supremum is taken over all partitions ? of ? into a finite number of pairwise
disjoint members of ??. It is easy to see that ? is superadditive.
THEOREM 6.7. Let ? be a family of semigroup valued triangular set functions with
difference regular variations defined on B. If the set [f(v@)): v e M] is bounded for
every open set ?, then
\f(v(B)): veM, BeB]
is a bounded set.
PROOF. We take m(B) = f(v(B)) (B eB.veM) and we apply Theorem 6.4. D
6.2. Convergence theorem
THEOREM 6.8. Let [тп}„€^ be a sequence of k-triangular set functions defined on В
with regular disjoint variations. If
lim m„@) =mo@)
n—>oc

Convergence theorems for set functions
169
exists for all open sets О and wo is exhaustive with respect to the family О of all open sets,
then {mn}^_0 is uniformly exhaustive on the whole O.
PROOF. We have to prove that jm,,}^,, is uniformly exhaustive on the family О of all
open sets in T. We note that m, (/' e N) are exhaustive on the family O, i.e., if {?n}neN is
a sequence of disjoint open sets then
lim m,(?.)=0 (/ eN).
у— oc
This follows by the regularity of w, (see the proof of Theorem 6.4).
Suppose that theorem is not true. Then there exist ? > Oand a disjoint sequence {Еп}„€щ
of open sets such that
m„(E„)>2s (neN).
Then repeating the proof of Theorem 5.11 (Pap A987)), taking Lemma 6.6 and the fact
that arbitrary union of open set is again open set, we obtain that {m„}„€p- is uniformly
exhaustive on the family O. ?
The assumption on the regularity of variation is not so restrictive, because it turns out
that in the case of regular additive set functions (then they must be also regular measures)
variations are also regular. So we have as a consequence the classical Dieudonne's theorem.
COROLLARY 6.9. Let {т,,}„€щ be a sequence of regular scalar measures defined on B. If
lim m„@) = mo(O)
11 —>DC
exists for all open sets O, then {w„}„ef. is uniformly exhaustive on the whole В and mo is
exhaustive.
PROOF. Let v„(B) = \m„(B)\ (B e ?, ? e N). The sequence {vn}„eK satisfies the
conditions of Theorem 6.8, since |v| = \m \ is also regular. Applying Theorem 6.8 we obtain
that {w„}„eK is uniformly exhaustive on O. ?
Instead of the family of open sets it can be considered a general class so-called Wells
class, KupkaA980).
7. Convergences of measures related to partial order
7.1. Nikodym type theorems for lattice-valued measures
Let X be a real vector lattice (see Luxemburg and Zaanen A971)). If и is a positive element
of X, then a sequence {.х„ЬеК in X is м-convergent to ? e X if there exists a scalar

170
P. de Lucia and E. Pap
sequence {t„]ne^ such that t„ -> 0 and \x„ — x\ ^ t„u for all ? e N. A sequence {х,,},1€®
is relatively uniformly convergent to ? e X if {jc„},,epj is м-convergent for some и е X.
The sequence is relatively uniformly ^-convergent to ? e X if every subsequence has a
subsequence which is relatively convergent to ?, and then we shall write lim,,^ ? xn = x
(we shall consider here only such convergence). We say that X has the property (Y)
(Antosik and Swartz A992)) if:
(Y) If (xij ),-.yeN is an infinite matrix in X such that lim,--^ Xij = 0 for each jeN and
limj-*ooXij = 0 for each /' e N, then there exists a principal submatrix (>·,,),., eN
of (xij)i.jeN such that lim/^эс X\-e/li v</ = 0 for any sequence {A,},^ of finite
subsets of N with /' ? A,.
Example 7.1. The following vector lattices X satisfy (Y).
(a) X is complete metrizable topological vector lattice.
(b) X has the property ? (Luxemburg and Zaanen A971,70.1)).
Now we have the following version of Nikodym convergence theorem, Antosik and
Swartz A992).
THEOREM 7.2. Let X be with the property (Y). Let ?„ : ? -> X be countable additive
measures such that
lim ?„(?) = ?(?). ? e ?.
?—>эс
exists. Then
(i) {^nbieN converges to a countable additive measure ?,
(ii) {/x„L?i=n is uniformly ?-additive.
We have the following version of Nikodym boundedness theorem, Swartz A989c).
THEOREM 7.3. Let X be Archimedean and have an order unit u. Let ??:? -> ?, ? e /.
be a family of finitely additive, order bounded set functions. If {?,(?): i e /} is order
bounded for each ? e ?, then {?,(?): i e /, ? e ?] is order bounded.
7.2. Convergences of measures on orthomodular posets
Nikodym convergence and boundedness theorems are extended for measures defined on
orthomodular posets (see d'Andrea and de Lucia A991), de Lucia and Morales A988),
Guarigla A990, 1991)). Theorem 5.12 was used in the proof of the following theorem
(Guariglia, 1991).
THEOREM 7.4. Let L be an orthomodular lattice with the (SIP), {w„}„er; a sequence of k-
triangular and exhaustive functions from L to [0. +oo[ such that ???,,^+,? m„ (a) = mo(a)
for every a eh and wo is an exhaustive function then mo is k-triangular and [m„}ll€^' is
uniformly exhaustive.

Convergence theorems for set functions
171
7.3. General Nikodym type theorems
Chovanec and Корка A994) have introduced the notion of difference poset (Z)-poset)
equivalent with the notion of effect algebra introduced by Foulis and Bennett A994), see
for more details the chapter written by Dvurecenskij B002).
Definition 7.5. A D-poset (difference poset) is a partially ordered set L with a partial
ordering ^, maximal element 1, and with a partial binary operation 9:LxL^L, called
difference, such that, for a, b e L, b ? a is defined if and only if a ^ b, for that the
following axioms hold for a, b, с е L:
(DP|) bQa^b;
(DP2) bQ{bQa)=a\
(DP3) a^b^c =3- cQb^cQa and (c ? ?) ? (с ? b)=bQ a.
The previous axioms implies that there exists also a minimal element 0 (= 1 ? 1)·
For an arbitrary but fixed element a e L we define a± = 1??. We have:
(i) ?±?- = ?\
(ii) a^b =>· bL ^aL.
The elements a and b from L are orthogonal iff a ^ bL (or b ^ a±). We define a partial
binary operation ?: L ? L -> L for orthogonal elements ? and b such that
b ^a®b and ? = (? ? b) ? b.
This operation ? is commutative and associative.
Let {? ?,..., a„} с L. We define
???···??,, =0 for«=0.
? ? ? · · · ? ?„ = a \ for ? = 1.
? ? ?···??„ =(?? ?···??„-?)??„ for» ^3.
supposing that ? ? ? · · · ? ?„_ ? and ? ? ? · · · ? ?„ exist in L. We have
DEHNITION 7.6. A finite subset {a\ a,,} of L is ©-orthogonalif a\ ? · -·??„ exists
in L.
We say that an ©-orthogonal subset {? ? a„} of L has an ©-sum, ?"=, ?,. defined
by
/?
??; =?? ?··??„.
? = ?
The preceding ©-sum is independent of any permutation of elements.
Definition 7.7. A subset G of L is ©-orthogonal if every finite subset F of G is ©-
orthogonal.

172
P. de Lucia and E. Pap
We say that an ?-orthogonal subset G = {a,: i e /} of L has an ©-sum in L, ф,е/ at
if in L there exists the join
? ?, = sup ? ? щ: F finite subset of /
i€l ^i?F
Every subset of ?-orthogonal set is again ?-orthogonal.
Dehnition 7.8. An D-poset L is complete D-poset (a(©)-D-poset) if, for every ©-
orthogonal subset (every countable ?-orthogonal subset) G of L, there exists the ?-sum
in L.
Dehnition 7.9. A D-poset L is quasi-?-complete if for every ?-orthogonal sequence
{a;}ieN m L there exists a subsequence {а,},ем such that ф(е/ ?( e L for each / с М.
Let ? be an uniform space with the uniformity U.
Definition 7.10. A subset ? of У is bounded (^-bounded) if for every U eU there
exist a finite set К с В and a natural number ? such that
BcU"[K],
where t/1 = U, U" = U ? ?/"-1 (o-composition of the relations) and U[K] is the set of
all ? e ? such that (x, y) e U for some у е К.
A subset ? of a metrisable uniform space {Y,U) is ^-bounded if and only if it is
d -bounded for every metric d generating the same uniformity U. The following known
characterization of W-boundedness will be often used.
THEOREM 7.11. A set В С ? is U-bounded if and only if it is d-bounded for every
uniformly continuous pseudo-metric d defined on Y.
We denote with Vm the family of all uniformly continuous pseudometrics defined on
(УМ).
Let L be a quasi-?-D-poset.
Dehnition 7.12. For d e Vm the d-semivariation of a function m-.L-* Y. with respect
to a point x0 & ? is
m*"(b) = sup{d(m(c), xn): с ? b. с e L} (be L).
We define for d e Vm and x„ еУ,а function m:L·—* ?
aj"(a, m) =\imsup\d[m(a ? b).x„): m*,"(b)< -. b
11^ ОС [ "
eL
(oeL).

Convergence theorems for set fimctions 173
Dehnition 7.13. A function m : L -> ? is said to be x„-exhaustive, for x„ e Y, if for
each d e Vm,
lim d(m(an),x0) = 0
/l-»00
for each ?-orthogonal sequence {a„ }„е^ of elements from L.
We have the following general Nikodym boundedness type theorem (de Lucia and Pap
A995); see Pap A995)).
THEOREM 7.14. Let ? be a family of xo-exhaustive functions m:L-> Y, where L is a
quasi-?-D-poset and ? is a uniform space. Then the set
\m{a): m e M, a eh]
is U-bounded if and only if the following conditions hold
(i) For each d e Vm, m e M. and each r e N there exists s(r) e N such that
d(m(a), m{b)) > s(r)
implies either for b ^ a
d(m{a Qb),xo) > r.
or for a ^ b
d(m{b ??),??) > r:
(ii) For each d eVm the set {ay (a, m): m e ?4] is bounded for each a e L;
(iii) For each d e Vm
{а(т(ац),хо): m e ?, ? e N}
is bounded for every orthogonal sequence {a„ }/ieN from L.
For a special important case, orthomodular lattice, we can relax the conditions in the
previous theorem.
THEOREM 7.15. Let L be ? ? -orthomodular lattice. Let ? be a family of functions
m:L^> Y. Then the set \m{a): m e M, a eL) is U-bounded if and only if the following
conditions hold:
(i) For each m e ? and each r e N there exists s(r) e N such that for each a, b e L
d(m(a),m(b))>s(r) (d eVm)

174
P. de Lucia and E. Pap
implies
either mx/{{b ? a')') > r or m'j0{(a ? b')') > r;
(ii) For eachd e Vm theset {m{an): m e ?, ? e N} is d-boundedfor every orthogonal
sequence {ап)Н€щ from L.
We have the following general Nikodym convergence type theorem (de Lucia and Pap
A994a); see Pap A995)).
THEOREM 7.16. Let ? be an uniform space and L a quasi-?-complete D-poset. Let
{"biLieN be a sequence of ?,,-exhaustive functions m,,, m„ :L -> Y, for an element x„
from Y, such that they satisfy the following conditions
(i) for each d e Vm and for each ? > 0, there exists ? > 0 such that
d(mn(a),x() < S andd(m,i(b),x„) < S for a ^b, a,b eL (n e N)
implies d(m„(b ? a), x(>) < ?'.
(ii) for each d e Vm and for each S > 0, there exists ? > 0 such that
d(mn(a),x„) <?, ueL(«eN) implies aXJ' (a, m„) < S (n e N);
(iii) for eachd e Vm we have \imn^ocd{mn{a),m{a)) =0 for each a e L.
Then m is ?,,-exhaustive if and only ifmn (n e N) are uniformly ?,,-exhaustive.
We remark that Dvurecenskij A996) have obtained a Nikodym boundedness type
theorem for a family of completely additive measures defined on the system L(H) of all
closed subspaces of a real or complex infinite-dimensional Hilbert space ?, see the chapter
written by Dvurecenskij B002).
References
Abraham, P. A992), Saeki's improvement of the Vitali-Hahn-Saks-Nikodym theorem holds precisely for Banach
spaces having cotype. Proc. Amer. Math. Soc. 116. 171-179.
Akcouglu, M. A. and Sucheston. L. A978). A ratio theorem for superadditive processes. Z. Wahrscheinlichkeitsth.
verw. Geb. 44, 269-278.
Aleksjuk, V.N. A968), Two theorems on existence of quasibase for a family of quasimeasures. Izv. Vyssh.
Uchebn. Zaved.. No. 6 G3), 11-18 (in Russian).
Aleksjuk, V.N. A970), On the weak compactness of a family of quasimeasures, Siberian Math. J. 11. 723-738,
(in Russian).
Ando, T. A961), Convergent sequences of finitely additive measures. Pacific J. Math. 11, 395-404.
Antosik, P. A971), On the Mikusinski diagonal theorem. Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. 19.
305-310.
Antosik, P. A976), A diagonal theorem for nonnegative matrices. Bull. Acad. Polon. Sci. Ser. Math. Astronom.
Phys. 24. 955-959.
Antosik, P. A978). Sur les suites d'applications. С R. Acad. Sci. Paris A 287. 75-77.

Convergence theorems for set functions
175
Antosik. P. and Pap. E. A981), ? simplification of the proof of Rosenthal's lemma for measures on fields. Proc. of
the Conference on Convergence, Szczyrk. 25-30.
Antosik, P. and Saeki, S. A995). A lemma on set functions and its applications. Different Aspects of
Differentiability. Warsaw, 1993. Dissertationes Math. 340, 13-21.
Antosik. P. and Swartz, C. A985), Matrix Methods in Analysis. Lecture Notes in Math.. Vol. 1113, Springer,
Heidelberg.
Antosik, P. and Swartz, C. A992). The Nikodym convergence theorem for lattice-valued measures. Rev. Roumaine
Math. Pures Appl. 37, 299-306.
Berberian, S.K. A965), Measure and Integration. New York. London.
Bhaskara Rao. K.P.S.. and Bhaskara Rao, M. A983). Theory of Charges. Academic Press. London.
Birkhoff, G. A967), Lattice Theory. 3rd edn.. Amer. Math. Soc. Colloq. Publ.. Vol. 25, Amer. Math. Soc.,
Providence, RI.
Brooks, J.K. A969), On the Vitali-Hahn-Saks and Nikodym theorems. Proc. Nat. Acad. Sci. USA 64, 468^471.
Brooks, J.K. A973), Equicontinuous sets of measures and applications to Vitali's integral convergence theorem
and control measures. Adv. Math. 10. 165-171.
Brooks. J.K. A974). Sur les suites uniformement convergentes dans un espace de Banach. C. R. Acad. Sci. Paris
A 274, 1037-1040.
Brooks, J.K. A980). On a theorem of Dieudonne. Adv. Math. 36. 165-168.
Brooks. J.K. and Dinculeanu, N. A995), Integration in Banach spaces. Application to stochastic integration. Atti
Sem. Mat. Fis. Univ. Modena 43. 317-361.
Brooks. J.K. and Chacon, R.V A980), Continuity and compactness of measures. Adv. Math. 37, 16-26.
Brooks, J.K. and Jewett, R.S. A970). On finitely additive vector measures. Proc. Nat. Acad. Sci. USA 67, 1294—
1298.
Cafiero, F. A952), Sulle famiglie difunzioni additive d'insieme nnifonnemente continue. Rend. Accad. Lincei (8)
12, 155-162.
Chang, C.C. A958), Algebraic analysis of many valued logics. Trans. Amer. Math. Soc., 467^490.
Cook, A.T. A978). The Nikodym-Hahn-Vitali-Saks theorem for states on a quantum logic, Proc. of Mathematical
Foundations of Quantum Theory. Academic Press, New York.
Constantinescu, С A981), On Nikodym's boundedness theorem, Libertas Math. 1, 51-73.
Constantinescu, С A984), Spaces of Measures. Walter de Gruyter, Berlin.
Constantinescu, С A991), Some properties of spaces of measures. Atti Sem. Mat. Fis. Univ. Modena Suppl. 35,
1-286.
Darst, R.B. A967), On a theorem of Nikodym with applications to weak convergence and von Neumann algebras.
Pacific J. Math. 23, 1297-1298, 473^477.
Darst, R.B. A970), The Vitali-Hahn-Saks and Nikodym theorems for additive set functions. Bull. Amer. Math.
Soc. 76, 1297-1298, 123-138.
Darst, R.B. A973), The Vitali-Hahn-Saks and Nikodym theorems for additive set functions. Bull. Amer. Math.
Soc. 79, 758-760.
Dashiell, F.K. A981), Non weakly compact operators for order-Cauchy complete C(S) lattices, with applications
to Baire classes. Trans. Amer. Math. Soc. 266, 397^413.
d'Andrea, A.B. and de Lucia, P. A991), The Brooks-Jewett theorem on an orthomodular lattice. J. Math. Anal.
Appl. 154, 507-522.
d'Andrea, A.B.. de Lucia, P. and Morales, P. A991), The Lebesgue decomposition theorem and the Nikodym
convergence theorem on an orthomodular poset, Atti Sem. Mat. Fis. Univ. Modena 34, 137-158.
de Lucia, P. A985), Funzioni Finitamente Additive a Valori in un Gruppo Topologico, Pitagora Editrice, Bologna,
de Lucia, P. A998), Convergence theorems in orthomodular posets. Atti Sem. Mat. Fis. Univ. Modena Suppl. 46,
171-179.
de Lucia, P. and Morales, P. A986). Equivalence of Brooks-Jewett. Vitali-Hahn-Saks and Nikodym convergence
theorems for uniform semigroup-valued additive functions on a Boolean ring, Ricerche Mat. 35, 75-87.
de Lucia, P. and Morales, P. A988), Some consequences of the Brooks-Jewett theorem for additive uniform
semigroup valued functions, Conf. Sem. Mat. Univ. Bari 227. 1-23.
de Lucia, P. and Morales, P. A992), A noncommutative version of a theorem of Marczewski for submeasures.
Studia Math. 101 B), 123-138.

176
P. de Lucia and E. Pap
de Lucia, P. and Morales. P. A994), Noncommutative decomposition theorems in Riesz spaces. Proc. Amer. Math.
Soc. 120, 193-202.
de Lucia, P. and Pap, E. A994a), Nikodvm convergence theorem for uniform space valued functions defined on
D-poset. Math. Slovaca 45. 367-376.
de Lucia, P. and Pap, E. A994b), Diagonal theorems and their applications. Preprint.
de Lucia, P. and Pap, E. A995), Noncommutative version of Nikodvm boundedness theorem for uniform space-
valued functions. Intemat. J. Theoret. Phys. 34. 981-993.
de Lucia, P. and Traynor. T. A994), Non-commutative group valued measures on an orthomodular poset. Math.
Japon. 40, 309-315.
de Lucia, P. and Salvati, S. A996), A Cafiero characterization of uniform s-boundedness. Rend. Circ. Mat.
Palermo B) Suppl. 40, 121-128.
de Maria, L.J., and Morales. P. A994), A non-commutative version of the Nikodvm boundedness theorem, Atti.
Sem. Mat. Fis. Univ. Modena 42, 505-517.
Diestel, J.J. and Uhl, J.J. Jr A977), Vector Measures. Math. Surveys. Vol. 15. Amer. Math. Soc. Providence. RI.
Dieudonne, J. A951), Sur la convergence des suites de measures de Radon. Anais Acad. Brasil. Ci. 23, 21-38,
277-281.
Dinculeanu, N. A967), Vector Measures. Pergamon Press. New York.
Dinculeanu, N. B002), Vector integration in Banach spaces and application to stochastic integration. Handbook
in Measure Theory. E. Pap, ed., Elsevier, Amsterdam. 345-399.
Dobrakov, I. A972). On subadditive operators on Q,(T). Bull. Acad. Polon. Sci. 20, 561-562.
Dobrakov, I. A974), On submeasures -1, Dissertationes Math. 112.
Dobrakov, I. and Farkova, J. A980), On submeasures II. Math. Slovaka 30, 65-81.
Drewnowski, L. A972a), Topological rings of sets, continuous set functions, integration, I, II, III. Bull. Acad.
Polon. Sci. Ser. Math. Astronom. Phys. 20, 269-276. 277-286. 439-445.
Drewnowski, L. A972b), Equivalence of Brooks-Jelwett, Vitali-Hahn-Saks and Nikodym theorems. Bull. Acad.
Polon. Ser. Sci. Math. Astronom. Phys. 20, 725-731.
Drewnowski, L. A978). On the continuity of certain non-additive set functions. Colloq. Math. 38, 243-253.
Dunford, N. and Schwartz. J.T. A958). Linear Operators, Part I. Interscience. New York.
Dvurecenskij, A. A991), Regular measures and completeness of inner product spaces. Contributions to General
Algebras, Vol. 7, Holder-Pichler-Tempski Verlag, 137-147.
Dvurecenskij, A. A993), Gleason 's Theorem and Applications. Kluwer Academic, Dordrecht. Ister Science Press.
Bratislava.
Dvurecenskij, A. A996), A note on the Nikodvm boundedness theorem. Tatra Mt. Math. Publ. 8, 229-238.
Dvurecenskij, A. B002), Measures on quantum algebras. Handbook of Measure Theory, E. Pap, ed.. Elsevier,
Amterdam, 827-868.
Faires, B. A974), On Vitali-Hahn-Saks type theorems. Bull. Amer. Math. Soc. 80, 670-674.
Faires, B. A976), On Vitali-Hahn-Saks-Nikodym type theorems. Ann. Inst. Fourier (Grenoble) 26, 99-114.
Foulis, D.J. and Bennett M.K. A994), Effect algebras and unsharp quantum logics. Found. Phys. 24. 1325-1346.
Freniche, F.J. A984), The Vitali-Hahn-Saks theorem for Boolean algebras with the subsequential interpolation
property, Proc. Amer. Math. Soc. 92, 262-266.
Garcia, F. A997), Convergence theorems for topological group valued measures on effect algebras. Depart-
tamento de Matematica Aplicada, E.U. Informatica, Universidad Politecnica de Madrid, Technical Report,
1-15.
Grothendieck, A. A953), Sur les applications lineares faiblement compactes d'espaces du type C(K). Canad. J.
Math. 5, 129-173.
Grothendieck, A. A957), Un resultat sur le dual d'une C*-algebre. J. Math. Pures Appl. (A) 36, 97-108.
Guariglia, E. A990a), On Dieudonne's boundedness theorem. J. Math. Anal. Appl. 145, 447^454.
Guariglia, E. A990b), Uniform boundedness theorems for k-triangular set functions. Acta Sci. Math. 54, 391-
407.
Guariglia, E. A991), k-triangular functions on an orthomodular lattice and the Brooks-Jewett theorem. Radovi
Mat. 7, 241-251.
Guselnikov, N.S. A975). On the extension of quasi-Lipshitz set functions. Mat. Zametki 17, 21-31 (in Russian).
Habil, E.D. A993), Orthoalgebras and noncommutative measure theory. Ph.D. Thesis, Manhattan. KS.
Hahn, H. A922), Ober Folgen linearer Operationen, Monatsh. Math, und Phys. 32. 3-88.

Convergence theorems for set functions
177
Halmos, PR. A950), Measure Theory, Springer, New York.
Haydon, R. A981), A non reflexive Grothendieck space that does not contain l^_. Israel J. Math 40, 65-73.
Hejcman, H. A959), Boundedness in uniforni spaces and topological groups, Czehoslovak Math. J. 9, 544-563.
Iseki, K. and Tanaka, S. A978), An introduction to the theory of BCK-algebras, Math. Japon. 23, 1-26.
Kalmbach, G. A983), Orthomodular Lattices, Academic Press, New York.
Kalton, N.J. and Roberts, J.W. A989), Uniformly exhaustive submeasures and nearly additive set functions. Trans.
Amer. Math. Soc. 278, 803-816.
Kalton, N.J. and Montgomery-Smith, S.J. A993), Set-functions and factorization. Arch. Math. 61, 183-200.
Klimkin, V.M. A989), Uniform boundedness of a family of nonadditive set functions. Mat. Sb. 180 C), 385-396
(Russian).
Kli§, С A978), An example of noncomplete normed (K)-space, Bull. Acad. Polon. Sci. 26. 414-420.
Корка, F. and Chovanec, F. A994), D-posets. Math. Slovaca 44, 21-34.
Kupka, J. A974), A short proof and generalization of a measure theoretic disjointization lemma, Proc. Amer.
Math. Soc. 45, 70-72.
Kupka, J. A980), Uniform boundedness principles for regular Borel vector measures, J. Austral. Math. Soc. Ser.
A 29, 206-218.
Landers, D. and Rogge, L. A971a), The Vitali-Hahn-Saks and the uniform boundedness theorem in topological
groups, Manuscripta Math. 4, 351-359.
Landers, D. and Rogge, L. A971b). Equicontinuity and com-ergence of measures, Manuscripta Math. 5, 123-131.
Luxemburg, W. and Zaanen, A. A971), Riesz Spaces. Vol. I, North-Holland, Amsterdam.
Maharam, D. A947), An algebraic characterization of measure algebras, Ann. of Math. 2 D8), 154-167.
Mikusinski, J. A970), A theorem on vector matrices and its applications in measure theory and functional
analysis. Bull. Acad. Polon. Sci. Ser. Math. 18, 193-196.
Molto, A. A981), On the Vitali-Hahn-Saks theorem, Proc. Roy. Soc. Edinburgh Sect. A 90, 163-173.
Nikodym, O. A931), Sur les suites de functions parfaitement additives d'ensembles abstraits, C. R. Acad. Sci.
Paris 192, 727.
Nikodym, O. A933), Sur les suites convergentes de functions parfaitement additives d'ensembles abstraits,
Monatsh. Math. 40, 427^432.
Orlicz, W. A929), Beitrage zur Theorie der Orthogonalentwickliingn II, Studia Math. 1, 241-255.
Orlicz, W. A968), On spaces L*^ based on the notion of a finitely additive integral, Comment. Math. 12, 99-113.
Pap, E. A974), A generalization of the Diagonal theorem on a block-matrix. Mat. Vesnik 11 B6), 66-71.
Pap, E. A976), Uniform boundedness of a family of exhaustive set functions. Mat. Vesnik 13 B8), 319-326.
Pap, E. A982), Functional Analysis (Sequential convergences. Some principles of functional analysis), Novi Sad
(with English Summary).
Pap, E. A985a), Functional analysis with K-convergence, Proceedings of the Conference on Convergence,
Bechyne, Czech, 1984, Akademie-Verlag, Berlin, 245-250.
Pap, E. A985b), The adjoint operator and K-convergence, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser.
Mat. 15B), 51-56.
Pap, E. A986), A generalization of a theorem Dieudonne' for k-triangular set functions. Acta Sci. Math. 50,
159-167.
Pap, E. A987), The Vitali-Hahn-Saks theorems for k-triangular set functions, Atti Sem. Mat. Fis. Univ. Modena
26,21-32.
Pap, E. A988), Nikodym type theorem for metric valued XQ-exhaustive set functions, Univ. Novom Sadu Zb. Rad.
Prirod.-Mat. Fak. Ser. Mat. 18, 101-109.
Pap, E. A991a), On non-additive set functions, Atti Sem. Mat. Fis. Univ. Modena 39, 345-360.
Pap, E. A991b), The Brooks-Jewett theorem for non-additive set functions, Univ. Novom Sadu Zb. Rad. Prirod.-
Mat. Fak. Ser. Mat. 21 A), 75-82.
Pap, E. A995), Null-Additive Set Functions, Kluwer Academic, Dordrecht. Ister Science Press, Bratislava.
Pap, E. and Swartz, С A990a), The closed graph theorem for locallv convex spaces. Boll. Unione Math. Ital. G)
4-B, 109-111.
Pap, E. and Swartz, С A990b), On the closed graph theorem, Proc. of the Conference on Generalised Functions
and Convergence, Katowice 1988, World Scientific. Singapore, 355-360.
Pap, E. and Swartz, С A994), A locally convex version of adjoint theorem. Univ. Novom Sadu Zb. Rad. Prirod.-
Mat. Fak. Ser. Mat. 24 B), 63-68.

178
P. de Lucia and E. Pap
Pap, E. and Swartz, C. A996), The adjoint theorem on ?-spaces. Novi Sad J. Math, (former Univ. Novom Sadu,
Zb. Rad. Prirod.-Mat. Fak. Ser. Mat.) 26 A). 63-68.
Pettis, B.J. A938), On Integration in vector spaces. Trans. Amer. Math. Soc. 49, 277-304.
Phillips, R.S. A940), Integration in a convex linear topological space. Trans. Amer. Math. Soc. 47, 114-145.
Preston. C.J. A971), A theory of capacities and its applications to some convergence results. Adv. Math. 6A).
78-106.
Ptak, P. and Pulmannova, S. A991). Orthomodular Structures as Quantum Logics, Kluwer Academic, Dordrecht.
Radon, J. A919), Uber partielle und totale Differentiziebarkeit von Fiinkionnen mehrerer Varieblen und uber die
Transformation der Doppelintegrale, Math. Ann. 79. 340-359.
Randall, С and Foulis, D. A981). Empirical logic and tensor products. Interpretations and Foundations of
quantum Theory, Vol. 5, H. Neumann, ed.. Wissenschaftsverlag. Bibliographisches Institut. Mannheim, 9-
20.
Rao, M.M. A987), Measure Theory and Integration. Academic Press. New York.
Rickart, C.E. A942). Integration in a convex linear topological space. Trans. Amer. Math. Soc. 52. 498-521.
Rickart. C.E. A943). Decomposition of additive set functions. Duke Math. J. 10, 653-665.
Rosenthal, HP A968), On complemented and quasi-complemented subspaces ofС(S) for Stonian S, Proc. Nat.
Acad. Sci. USA 60. 1165-1169.
Saeki, S. A992), The Vitali-Hahn-Saks theorem and measuroids. Proc. Amer. Math. Soc. 114. 775-782.
Saks, S. A933), Addition to the note on some functionals. Trans. Amer. Math. Soc. 35. 967-974.
Sazenkov, A.N. A982), Uniform boundedness principle for topological measures. Mat. Zametki 31. 263-267 (in
Russian).
Schachermayer, W. A982). On some classical measure-theoretic theorems for non-sigma complete Boolean
algebras, Dissertationes Math. 214. 1-33.
Swartz, С A987), The closed graph theorem without category. Bull. Australian Math. Soc. 36, 283-288.
Swartz, C. A989a), The uniform boundedness principle for order bounded operators. Intemat. J. Math. Math.
Sci. 12 C), 487^492.
Swartz, C. A989b), A generalization of Mackey's theorem, and the uniform boundedness principle. Bull.
Australian Math. Soc. 49. 123-128.
Swartz, C. A989c). The Nikodym boundedness theorem for lattice-valued measures. Arch. Math. 53. 390-393.
Swartz, С A992), Introduction to Functional Analysis, Marcel Dekker, New York.
Swartz, С A994). Measure, Integration and Function Spaces. World Scientific, River Edge, NJ.
Swartz, С A996), Infinite Matrices and the Gliding Hump. World Scientific, River Edge. NJ.
Traynor, T. A992), A diagonal theorem in non-commutative groups and its consequences, Ricerche Mat. 41.
77-87.
Tweddle, I. A970), Unconditional convergence and vector-valued measures. J. London Math. Soc. 2. 603-610.
Vitali. G. A905), Sulle funzioni integrali, Atti R. Accad. Sci. Torino 40. 753-766.
Vitali, G. A907), Sull' integrazione per serie. Rend. Circolo Mat. di Palermo 23, 137-155.
Wang, Z. A984), The autocontinuity of set function and the Fuzzy integral. J. Math. Anal. Appl. 99, 195-218.
Weber. H. A976), Fortsetzung von Massen mit Werten in unifonnen Halbgruppen. Arch. Math. 27, 412^423.
Weber, H. A984), Topological Boolean rings. Decomposition of finitely additive set functions. Pacific J. Math.
110.471^495.
Weber, H. A986), Compactness in spaces of group valued contents, the Vitali-Hahn-Saks theorem and Nikodym's
boundedness theorem. Rocky Mountain J. Math. 16. 253-275.

CHAPTER 5
Differentiation
Brian S. Thomson
Mathematics Department, Simon Fraser University. ВС, Canada V5A IS6
E-mail Thomson @ cs. sfii. ca
Contents
1. Preface 181
2. Introduction 181
3. Differentiation in K" 183
4. Some motivation 184
5. Derivation bases 186
5.1. Covering relations 186
5.2. Derivation bases 187
5.3. The limit operation 188
5.4. The dual basis 190
5.5. Some properties of the dual 192
5.6. The variation 193
5.7. Growth estimates 194
5.8. Differentiation under strong Vitali assumptions 198
5.9. Topological considerations 200
5.10. Differential equivalence 202
6. Standard examples 203
6.1. A general construction 203
6.2. Derivation bases in a metric space 204
6.3. Measure contraction 204
6.4. Set contraction 205
6.5. Busemann-Feller bases 205
6.6. Federer's scheme 205
6.7. Net bases 206
6.8. Bases associated with a lifting 206
6.9. Self dual bases 207
7. Derivation bases in a metric space 207
7.1. Differentiation of the integral 208
7.2. The density property 212
8. Differentiation under strong Vitali assumptions 218
9. Strong Vitali conditions 222
9.1. Strong Vitali property 223
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V All rights reserved
179

180 B.S. Thomson
9.2. Classical proofs of the Vitali theorem 224
9.3. (Q)-Property 226
9.4. Besicovitch-Morse property 228
10. Weak Vitali covering properties 230
10.1. Weak halo properties 231
11. Derivation bases in К 231
11.1. The ordinary derivative 232
11.2. The symmetric derivative 237
11.3. The sharp derivative 237
11.4. Density bases on К 238
11.5. The approximate derivative 238
12. Derivation bases in R" 239
12.1. The cube basis 239
12.2. The basis of centered balls 240
12.3. The interval basis 240
12.4. Rectangle basis 241
12.5. Regular bases 242
12.6. Star bases 242
13. De La Vallee Poussin theorem 243
14. The Radon-Nikodym theorem 244
15. Some further remarks 244
References 245

Differentiation 181
1. Preface
At the outset let us state the scope of this presentation. The topic is a large one,
embracing many different, related ideas. We choose to focus only on the interaction
between differentiation properties and covering properties. For example, all mention of
the Hardy-Littlewood maximal operators and related inequalities are left out, although
the reader should have little trouble finding numerous extensive accounts of such topics
elsewhere, as they play a dominant role in many parts of modern analysis.
The general presentation is limited to a sketch of an abstract framework within which
the main ideas can be expressed, and an exploration of many of the main themes. Other
authors have attempted to construct an elaborate abstract theory encompassing all the
special applications. To this date none of this seems to have attracted the following needed
to generate, what could be called, a "theory" of abstract differentiation. The monographs
of Hayes and Pauc [31] and Kenyon and Morse [41], each in their own way, require a
serious devotion to a viewpoint and an elaborate language in order to enter their universe.
The merits of the monographs are thus easy to overlook. The former is accessible, with
many rewards, but still requiring careful and detailed study that few might wish to commit
to. The latter is apparently impenetrable.
Thus there is no common structure that all would agree amounts to a theory of abstract
differentiation in the way that most of us agree as to what constitutes metric space theory,
topology, measure spaces, etc. The scheme offered here is meant to communicate the
essentials of the various themes that have emerged in this area, but may well, too, find
few followers who are willing to devote the necessary effort to mastering the terminology.
In Sections 2-4 we give some motivating discussion. In Section 5 a general framework
with the main concepts is introduced. Section 6 gives a number of instances of this general
theory and how it may be applied. Sections 7 and 8 shows how the general theory may be
applied to obtain the natural differentiation results desired in the theory. Section 9 discusses
the strong Vitali property and standard methods for obtaining it. Sections 10 and 11 discuss
some of the main derivation bases in R and R".
2. Introduction
The fundamental theorem of the calculus, presented to elementary students, relates
integration and differentiation as inverse processes. Specifically, in the setting of the
Riemann integral, one proves that for any integrable function /,
-f- f f(t)dt = f(x) A)
ax Ja
at any point of continuity of /, and for any function / with an integrable derivative /',
h
f\t)dt = f(b)-f(a). B)
L

182
B.S. Thomson
These two results are known collectively as the fundamental theorem of calculus and are
proved by entirely elementary means.
Much more remarkable and much deeper is the version for more advanced integrals. In
Lebesgue's theory one shows that for any integrable function /,
— / f(t)dt = f(x) C)
dx Ja
at almost every point jc, even for badly discontinuous functions. The integrability of the
derivative /' in general requires going beyond Lebesgue's theory (there are derivatives
that are not integrable) and leads to the integration theories associated with the names
Denjoy, Perron, Henstock and Kurzweil.
If we express this in the correct form the path to a generalization outside the setting of
the real line suggests itself. The derivative statement in C) can be visualized, directly from
its definition, as asserting that
I'm- / f(t)dt = f(.x) a.e.
More picturesquely, by thinking of / = [?, ? + h] or [x + h, x] as an interval whose length
|/| = \h\ shrinks to zero, we could write C) in the more suggestive form
lim — /
'-* I/I J ?
f(t)dt = f(x) a.e..
where / -> ? assumes some appropriate meaning (here, for example, merely that / is
an interval, ? e / and length |/| -> 0). Indeed, expressed this way, we more clearly see
the differentiation theorem as an averaging result. The reader should know that a simple
measure-theoretic argument (cf. [24, pp. 156-157]) allows one to extend this to the sharper
statement that
lim — /
/-* I/I J ?
\f(t)-f(x)\dt=0 a.e.
(The set of points ? at which this holds is called the Lebesgue set of /.)
We might now ask whether, for any other measure ? on the real line, one should have
also
lim —— / /(?)??(?) = f(x) ?-a.e.
/—.r ?A) J/
and, moreover, whether some such statement might be true in any appropriate measure
space. For the latter, however, we must now agree on how to assign some meaning to the
notion / -> ? of some sets / that are required to "shrink" down to a point x.

Differentiation
183
The other half of the fundamental theorem of calculus also suggests some problems in
this more general setting. If ? and ? are measures and there is a reasonable interpretation
of the assertion
v(I)
lim-—-=/(*) ?-a.e.
I^x ?(?)
then can one recover ? from an integral
v{E)= j /(?)??(?)?
Thus the fundamental theorem of calculus suggests, if only vaguely stated, some
important problems in measure theory. Is there an analogue of differentiation in more
general settings that will allow a generalization of familiar theorems on the real line
connecting differentiation and integration. This is briefly our motivation for the subject
that follows. We begin with some history of the early results in this topic.
3. Differentiation in R"
To carry the Lebesgue differentiation theorem over to higher dimensions we could begin
by considering it in the form of a symmetric derivative statement,
,limni f f(t)dt = f(x) a.e.
The natural and most geometrically appealing analogue in R" would be to take sequences
of open Euclidean balls B(x,rk) centered at ? and with radii rk tending to zero. Lebesgue
proved in 1910 that for any integrable function / on R" and almost every point ???",
lim ,D, ' ч, ? fiy)dy = f{x)
JB(x.?
k^oc \B(x, rk)\
n)
for every sequence {B{x, rk)} of balls contracting to the point x. (Here |?| denotes the
? -dimensional Lebesgue measure of a set ? с R" ¦) The argument depends on a geometric
covering property of Euclidean balls: If every point ? in a set ? С R" has associated with
it some sequence B(x, rk) of balls centered at ? and with radii rk tending to zero then one
may extract a collection of disjoint balls that can be used to approximate the «-dimensional
Lebesgue measure of E. This Vitali covering theorem, it was realized in time, was the
crucial property to having the differentiation theorem.
The question then arose as to whether the balls shrinking to their center could be
replaced by cubes, or intervals, or other geometric objects. This, in turn, leads to the
question of whether the Vitali covering theorem (the key tool to proving the differentiation
theorem) would remain valid with such replacements. Harold Bohr showed that the family
of intervals (rectangles with sides parallel to the axes) in E2 behaved much worse with

184
B.S. Thomson
regard to this Vitali property. In 1934 Saks [70] showed that the differentiation theorem
fails for some integrable functions if the balls are replaced by intervals. A construction of
Nikodym [65] showed that situation was even worse if the balls are replaced by sequences
of rectangles shrinking in diameter.
It is now realized that the Lebesgue differentiation theorem as stated for sequence of
balls B(x,rk) shrinking to ? may not be true if the balls are replaced by other simple
geometric objects. Thus any attempts to extend this theorem to a general measure space
and to develop a differentiation theory in an abstract setting must confront this fact, that
the theory will depend on the underlying geometry of the situation.
To capture and express these ideas in a more abstract setting requires us to be familiar
with the Vitali covering theorem and the way that it is used to prove the differentiation
theorem. Almost all of the abstract theory is an attempt to use these tools and ideas in a
more general way.
Our goals then are this. In a fairly general setting of a measure ? on a space X we wish
to assign some meaning to the limit
lim /
/(?)??(?· D)
We wish to capture the notion of a Vitali cover relative to this limit, and we wish to express
the essential ingredients of the arguments needed to make the Vitali covering theorem work
so as to answer the question as to whether D) is equal to f(x) at ?-almost every point
? e X. Conversely we hope, too, to discover some conditions under which a relation
v(I)
,hm -77T = /<*> E)
would allow ? to be reconstructed as the integral of / with respect to ?.
4. Some motivation
To express the notion of a derivative in an abstract setting, consider how the ordinary
derivative might be viewed. If F'(x) = f(x) then for all sequences of intervals [ak,bk]
with ak ^ ? ^ bk and bk -aj^Owe have
F(bk)-F(ak)
—z * f(x^-
bk - ak
Vitali arguments require just that there is given some sequence of such intervals at a point
x. We can formulate this by considering collections of pairs (/, x) where / is an interval
and ? e /. In particular, we expect to encounter the collection
fie = {([a,b],x): xe[a,b], b-a <?\.

Differentiation
185
Let us define V to be the collection
?={??: ?>0].
Then note that
¦nev ч · r F(b)-F(a)
DF(x)= inf sup
and
n*v ч ¦ r F(b)-F(a)
IJF(x) = sup inf
^e-p([«.fr]..v)e/j fo-a
expresses the upper and lower derivatives of F at any point x.
How does the notion of a Vitali cover enter into this scheme? If we consider a Vitali
cover as consisting of pairs (/, jc) (and not just intervals as is common) then we could say
? is a Vitali cover on R if for every ? e R there is at least one sequence ([a*, ?>*], .v) e /3
for all A: with a* ^ jc ^ ?>*, a*: -> jc, and bt -> x. Now let P* be the collection of all Vitali
covers on R. This plays the role of a dual basis (and hence the notation) and captures the
essence of the Vitali covering notion.
The collection V* expresses the same derivative, but in a different orientation:
¦nir, ч · f F(b)-F(a)
DF(x)= sup inf
?*&?>* (l«-ft]..T)e/i* b — a
and
n/r^ ¦ f F(b)-F(a)
DF(x)= inf sup .
^*eu*([„.ft)..v)e/j. fo-a
To grasp these two statements is to grasp the nature of the Vitali cover and to see a way to
express these ideas more abstractly.
The fundamental concept we use to express these ideas in a general setting is the notion
of a covering relation on a set E.
DEnNiTiON 1. Let X be an abstract set and let ? с X. We say that a collection
? с 2х ? ?
is a covering relation on ? if corresponding to each ? & ? there is at least one pair
In the definition we do not assume that ? e I, although for most applications that will
be the case. Thus the phrase "covering relation" is a bit of a misnomer since there is no
implied covering in general.

186
B.S. Thomson
5. Derivation bases
We now develop an abstract framework for discussions of differentiation processes in very
general settings.
5.1. Covering relations
Our general setting is to assume we are given a space X (frequently a metric space, but for
the moment no additional structure is imposed) on which we wish to construct a derivation
basis. We always assume that there is given to us a covering relation a on X relative to
which the basis is to be defined. Thus ? is a collection of pairs (I.x) with ? e X and
/ с Х- This describes, as a first step the objects within the derivation basis, but does not
describe how the convergence is to be defined.
On the real line R we might naturally, and almost invariably, start with the relation
a = {([a,b],x): ? e [a. b], a <b\.
To discuss, instead symmetric derivatives, we would prefer to narrow this to
a = {([a,b],(o+b)/2): a <b\.
In a metric space X a natural place to begin is with
a = {(B(x,r),x): xaX, r > 0}. F)
which relation associates each point ? with balls B(x,r) centered at x. (This is the
analogue in a metric space of the symmetric derivation basis covering relation above.)
The elements / in the pairs (I,x) e a play the role of the intervals in the one-
dimensional case and we might think of them as "generalized intervals".
Definition 2. A collection ? с a is said to be a covering relation on a set ? С Х
provided for every point ? e ? there is at least one pair (/, x) e ?.
The terminology was originally used with the additional requirement that the set /
contains x, but this an unnecessary addition for most of our theory. Loosely the phrase
"covering relation" without any mention of set simply means a collection, even possibly
empty, of pairs (I,x) ea.
From a covering relation ? we sometimes extract subsets that play a special role. These
have been called "prunings" since they do, indeed, remove some unnecessary elements.
DEnNiTiON 3. For any ? с X we write
?(?) = {(?,?)??: ICE] G)

Differentiation
187
?[?] = {(?,?)??: xeE}. (8)
5.2. Derivation bases
Definition 4 (Derivation basis). By a derivation basis relative to the pair X and a we
mean a nonempty collection of covering relations ? с a on X. Thus ? is a derivation basis
if every element ? e ? is a covering relation on X. We shall always assume the following
properties of a derivation basis:
(B|) If ? e X and /3 e В then there is at least one pair (/, x) in ?.
(B2) If /51 e ? and /3| cftCa then necessarily /32 e ?·
Frequently we shall need also the following properties, which will be specified as
needed:
(B3) If for every ? \, ?? e В it is true that ? \ ??2 eB then В is said to be filtering.
(B4) The basis В is said to be decomposable or to have pointwise character if whenever
there is given for every ? e X an element ?? e ? then it must follow that the
collection
xeX
belongs to B.
(B5) The basis ? is said to be ?-decomposable if whenever there is given a sequence of
elements ?„ e В and a denumerable partition {X,,} of the space X it must follow
that the collection
00
\Jfi„[X„]
n=\
belongs to ?.
Finally in the presence of a topology on X we shall usually require the following
property (see Section 5.9 for a discussion):
(Вб) We say a derivation basis ? on a space X furnished with a topology is compatible
with the topology if for every ? с X and every open set G D ?
/3(G) eB[E] for every ? e B[E].
Note. The condition (B|) just reasserts that each element of В should be a covering
relation on X, thus no point in the space is excluded from discussion. The general scheme
in Hayes and Pauc [31] fixes a subset ? с X in advance and all derivation statements are
made relative to that set. This is typical of the subject: Each attempt to introduce greater
generality adds a sometimes cumbersome level of detail.

188
B.S. Thomson
The conditions (B2) and (B3) together assert that ? is a filter on a. This is the most
natural setting for our theory and allows limits to be taken in the sense of the filter.
If ? is a derivation basis on X then it convenient to consider also the collections
?(?) = [?(?): ???] and B[E] = {?[?]: ? e В}
defined by means of the prunings in G) and (8). These play the role of restrictions of the
basis В although we will not insist on interpreting them as derivation basis themselves,
although the language could be tailored to allow such a usage.
Note, too, that (B4) =>· (B5). The terminology "decomposable" comes from Hen-
stock [36] and "pointwise character" from McShane [53] in a related context.
5.3. The limit operation
There are essentially two basic operations we wish to take relative to a derivation basis for
any extended real-valued function h defined on a. The first is a limit operation.
Dehnition 5. If h :a -> lU {±эо} then we write
??limsup/2(/,x) = inf sup h(I.x)
1-х PeB{i.x)ep
and
Bliminfh(I,x) = sup inf h(I.x).
?-** ?(???.?)€?
As usual in such theories, if the limit superior and limit inferior agree we can write the
common value as #lim/_».( h(I,x).
Since we assume so little in general, there is no assumption that the limit superior must
exceed the limit inferior. Indeed, it is an essential feature of the theory that we do not
impose any requirements implying this. We can prove the following lemma, but it should
be considered special to the case where ? is a filter.
LEMMA 6. If В is filtering [i.e., satisfies (B)), (B2), and (B3)] then
Slim inf h(I,x) ? Slim sup h(I.x). (9)
'^* l^x
PROOF. If there are numbers t and s with
t > Blimsuph(I, x) and s < Slim inf /?(/,. v),
l^x '^v
then, using the definition, there is a ?\ e В so that h(I, x) < t for all (/, x)e ?? and there
is a ?2 e В so that h(I, x) > s for all (/, л) e ?2. By (B-,) we have ?\ ? ?? e В. But this

Differentiation
189
means, by (B|) that there is at least one pair (/, x) with the properties that h(I, x) < t and
/?(/, x) > s. In particular, s < t. From this inequality (9) follows. ?
In working with limits taken relative to a derivation basis, the following device should
be noted. In the proof of Lemma 6 we observed that, for each ? e X, if it is the case that
f(x) > Slim suph(I.x),
then, using the definition, there is a ?? e В so that h(I,x) < f(x) for all (/.x) e ??. If
the derivation basis is decomposable [i.e., satisfies (B4)] then there is an element ? so that
h(I,x) < f(x) for all (/, x) e ?. By assumption (B2) then the collection
{(/,*): h(I,x)<f(x)}eB.
We express this observation in the form of three lemmas.
LEMMA 7. Assume that В is decomposable (B4). Suppose that
Blimsuph(I,x) < f(x) for all ? eE.
Then
{(I,x)ea: xeE, h(I,x) < f(x)} e B[E].
LEMMA 8. Assume that В is decomposable (B4). Suppose that
Bliminf h(I.x) > /(л) for all ? e ?.
/-».v
Then
{(I,x)ea: ? e E, h(I,x)> f(x)}eB[E].
LEMMA 9. Assume that В is filtering and is decomposable [i.e., satisfies (B|), (B2), (B^),
and (B4)]. Suppose that
Slim h(I,x) = f(x) for all ? eE.
/-».v
Then for all ? > 0
{(/,jt)ea: jc e ?. |/?(/..v) - /(.v)| <?} eB[?].

190
B.S. Thomson
5.4. The dual basis
The notion of a Vitali covering enters our theory as a duality.
Dehnition 10. Let В be a derivation basis on X. The family B* of all collections ? с ?
with the property that
????{?]]??>
for every element ? \ e В and every ? e X is called the dual of ?.
Observe that the dual of a derivation basis is itself a derivation basis; this requires
checking the two properties (B|) and (B2), both of which are immediate. The dual basis
B* always is decomposable [i.e., satisfies the property (B4)]. In general when В is filtering,
the dual basis may not be.
The fundamental relationship between a derivation basis and its dual is expressed in the
next lemma, which shows the essential role the dual plays in the study of limits.
LEMMA 11. Let В be a derivation basis and B* its dual. Then
B* lim sup h(I. x) = В lim inf /?(/. x)
l—.x '—¦»'
and
B* lim inf h(L x) = B\im sup h(I. x).
/^¦v /-л-
In particular,
B* lim h(Lx) = В lim /;(/..x)
/-».V /-».?
when the limit exists.
PROOF. Fix x. Suppose that
s < Slim sup /?(/..x) < t A0)
/—.V
if possible. Then there must exist an element ?\ of В with the property that h(I,x) <t
for all (/, x) e ?\ and also for every element ?2 of В there must exist at least one element
(?,?) (??2 withs <h(I,x).
Let /S3 be the collection of all (/. y) e a for у ? ? and also all pairs {1.x) with
h(I,x) > s. Note that, for every element ?2 of ?, ?? ? &[{*}] ? 0 because of our
observation above. In particular, /З3 e B* so
?Tliminf/2(/.x) = sup inf /?(/..x)>s.

Differentiation 191
On the other hand if ?4 e B* then Д; ? ?\[{?}] ^0 so some pair (I.x) in ^4 must exist
for which h(I,x) < t. Thus
B*\im'mfh(I, x) = sup inf h(I.x)<t.
?^? ?€?^?.?)€?
From these two inequalities, together with the fact that s and t are any numbers
satisfying A0), it follows that
B* lim inf/?(/, jc) = Slim sup h(I. x)
as required. The other assertions follow similarly. ?
Note. The lemma illustrates the role of the dual basis in capturing the limit. Some readers,
encountering this for the first time, experience a small shock that should be lessened if one
recalls the simple fact that an ordinary limit superior lim supv_T+ /?(y) on the real line can
be written as an "inf-sup"
lim sup h(y) = inf sup h(x)
y-*.\+ 5><v-.v<S
or equivalently, by taking a supremum over all sequences tending to x, as as a "sup-inf"
lim sup/2(j) = sup infh(x„).
? —-V+ A„\.V "
We can use the dual basis and Lemma 11 to offer some variants on Lemma 7 that will
be useful in the sequel.
LEMMA 12. Assume that В is decomposable (B4). Suppose that
g(x) < B\imsuph(I, x) < f(x) for all ? e E.
/—.V
Then
{(/,*): xe E, I el. h(I.x) <f(x)\ e B[E]
and
{(I,x): xeE, I el. g(x) < h(I.x)} e B*[E].
LEMMA 13. Assume that В is decomposable (B4). Suppose that
g(x) <BYimmih(I,x) < f(x) for all ? e E.

192 B.S. Thomson
Then
{(/,*): xeE, I el. h(I.x) < f(x)} e B*[E]
and
{(I,x): xe E, /el, g(x) <h(I,x)} e B[E].
5.5. Some properties of the dual
For ease of manipulation and notation the dual basis allows us to express the limit
properties of a basis with some economy. Here are several useful properties.
LEMMA 14. // В is filtering [i.e., satisfies (B|), (B2), and (Вт,)] then for every ? \ eBand
every ft e B*, it follows that ?\ ? ?2 e В*.
PROOF. Let ? e В and ?2 e B* and ? e X. Take any ?3 e В and note that
081 П ft) П ?3[{?}] = (ft П ft) П &[{*}] ? 0,
since/3, nfteB and ?2 e ?*. Thus /3, ??2 e ?*. ?
LEMMA 15. //? is filtering [i.e., satisfies (B|), (B2), anii (B3)] then В С ?*.
PROOF. If ? is filtering and ?\, ?2 e ? then ft = (ft ? ft) e ? by (fo,)
?\??2[{?}] = ?3[{?}]??.
It follows that ft e ?*. ?
Lemma 16. В с В** = (В*)*.
Proof. If ft е ? and /3 е ?* then
?\??[{?]]?91
for all ? e X. But this holds for all ? e B* and so ft e (?*)*. ?
Lemma 17. в*** = В*.
PROOF. If ft e ?* then ft e ?*** by Lemma 16. Conversely if ft e B*** then
ftnft{*}]^0
for all ? e X and all ? e ?**. In particular, this same statement holds for all ? e В since
? С ?**, again by Lemma 16. This proves that ft e ?*. ?

Differentiation
193
5.6. The variation
In addition to the limit operation in a derivation basis we require also an operation that
can be used to reconstruct the measure. We choose to do this by taking estimates that use
sequences {(/*,&¦)} where the {Ik} are disjointed; this is not the only possible way, but it is
the way that is associated with strong Vitali conditions which is one of our main concerns
in this exposition.
Suppose that h is a real-valued function on a.
DEnNiTiON 18. If ? is a covering relation then by V(h. ?) we denote the supremum of
the sums
II
J2\h(Ik,xk)\
k=\
where {h, jc^)} is any finite (or, equivalently, denumerable) subset of ? for which // and
Ik are disjoint if j фк.
Dehnition 19. If ? is a family of covering relations (in particular if it is a derivation
basis) then
V(h,B) = inf V (/?,?).
???
We refer to this as the variation of the function h over the basis.
Some properties are immediate.
V(ch,fi) = \c\V(h,fi). A1)
У(Й|+Й2,0КУ(Й|,0) + У(Й2,0). A2)
If В is filtering [i.e., satisfies (B|), (B2), and (B3)] then
V(hi+h2,B)^V(hi,B) + V(h2.B). A3)
(This is not true, in general, unless В is filtering.)
THEOREM 20. If В is decomposable or a -decomposable [i.e., satisfies (B4), or (Bj)] then
the set function
E-> V(h,B[E])
is an outer measure on X.

194
B.S. Thomson
PROOF. Let ? С Ц?| Ek. Write X0 = X \ Ц^, Ek and Xk = Ek \ U*=! E, for * > 1.
Note that {Xk\ is a partition of X. Let ? > 0. Choose /5* e ? so that
V(h,pk[Ek])^V(h,B[Ek])+2ks.
By the ?-decomposable property (or by the decomposable property which is stronger still)
there is a ? e В with ? [Xk ] с fik [Xk ] for all к = 0. 1. 2 It is easy to check that
DC
jt=l
and consequently
ЭС DC
V(h, B[E]) ? V(h,fi[E]) <: ? V(h. pk[Ek]) ? J2(V(h, B[Ek]) + 2ke).
k=\ k=\
This property establishes that the set function in the lemma is an outer measure. D
5.7. Growth estimates
Let us establish now some connections between the limit operation and the variational
measure. The following lemmas are fundamental to our study, and entirely elementary. In
a less abstract setting [14, pp. 281-286] we have called these growth conditions in order to
motivate a presentation of some applications of the classical Vitali theorem.
We assume throughout this section that h and к are nonnegative real-valued functions
defined on the fundamental covering relation a and that we are interested in the studying
the limits
h(I,x) h(I.x)
ulimsup and Sliminf .
/^.v k(I,x) i^x k(I.x)
We shall use the following short notations for any set ? с Х '¦
h*{E)=V(h,B[E]),
k*(E) = V(k,B[E]),
h*(E) = V(h,B*[E]),
and
k+(E) = V(k,B*[E]).
We recall that h*, k*, h* and k* are outer measures on X when В is assumed to satisfy
(B4)or(B5).

Differentiation
195
LEMMA 21. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B|), (B2), and (B3), and (B4)]. Suppose that s > ?, ? С X and that
? lim sup < s
for all x e E. Then
h*(E)<isk*(E).
Proof. If
иг h(I'x)
? lim sup <s
,^/k(I,x)
for all ? & ? then
?\ ={(I,x)ea: xe E, h(I.x)^sk(Lx)\ e B[E].
Let ?2 e ?· Then since the basis is filtering ? = ?\ ? /Ь е ?[?] and so
?(?,?[?])< V(M)OV(MKjV(*,ft>[?]).
As this holds for all ?2 e ?, it follows that V(h. B[E]) <; s V(k, B[E}). D
LEMMA 22. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B|), (B2), and (B3), and (B4)]. Suppose that s > 0, ? С X and that
о lim mi > s
for all ? & E. Then
h*(E)^sk*(E).
PROOF. The proof is similar to the proof of the preceding lemma. D
LEMMA 23. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B|), (B2), and (B3), and (B4)]. Suppose that s > 0, ? С X and that
В lim sup > s
l^/k(I,x)
for all ? e E. Then
h*(E) ^sk^(E).

196
B.S. Thomson
PROOF. If
? lim sup > s
,^/ k(I,x)
for all ? e ? then
?\ ={(I,x)ea: ? e E. h(I,x) ^ sk(I,x)\ eB*[E].
(Note the use of the dual basis here; in every other respect the proof is following closely
theproofof Lemma21.)Let02 e ?. Then by Lemma 14/3 = ?\ П ?2 e B*[E] and so
sV(k,B*[E]) ? V(sk^) <: V(h.fi) ? V(h,p2[E]).
As this holds for all ?2 e B, it follows that V(h. B[E]) > s V(k, B*[E]). D
LEMMA 24. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B|), (Вг), and (B3), and (B4)]. Suppose that s > 0, ? С X and that
hi· · fW·*)
oliminf < s
l^x k(I,x)
for all ? & E. Then
h*(E)^sk*(E).
LEMMA 25. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B,), (B2), and (B3), and (B4)]. Then ifh*(E) < 00 it follows that
hi· йG·*)
В lim sup < 00
for all ? e ? except for a set N С ? for which k*(N) = 0.
PROOF. By Lemma 23 we have that the set N of points ? in ? at which
В lim sup = 00
1^/ k(I,x)
must satisfy
sV(k,B*[N])^V(h.B[N]) <эо
for alb >0. It follows that V(k, B*[N]) = 0. ?

Differentiation 197
LEMMA 26. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B,), (B2), and (B3), and (B4)]. Then ifh*(E) = 0 it follows that
„ . h(I,x)
В lim = 0
l^x k(I,x)
for all ? e ? except for a set N С ? for which k*(N) = 0.
PROOF. By Lemma 23 we have that the set Ns of points ? in ? at which
? lim sup > s
must satisfy
sV(k,B*[N])^V(h,B[N])=0
for all s > 0. It follows that V(k, B*[NS]) = 0. Since the set N of ? in ? at which
„ . h(I,x)
В lim ' ? 0
is the union of the set Ns for s > 0 and s rational it follows, from Theorem 20, that
V(k,B*[N]) = 0. ?
LEMMA 27. Suppose that the derivation basis В is filtering and decomposable [i.e.,
satisfies (B|), (Вг), and (Вз), and (B4)]. Suppose that
В im-^- = f(x)
exists finitely for every point ? in a set E. Then, ifk*(E) < 00,
V(h-fk,B[E])=0.
PROOF. Under these hypotheses
? = {(I,x)ea: \h(I.x) - f(x)k(l,x)\ <c\k(I,x)\] e B[E]
for every с > 0. Consequently, for every ?\ e B[E],
V(h-fk,B[E])iZV(.h-fk.finfi\)^cV(k.pnP\)^cV(k,P\).
From this we obtain
V(h - fk, B[E]) ? cV(k, B[E]) = ck*(E).

198
B.S. Thomson
Since this is valid for all с > 0 and k*(E) < oo, it follows that
V(h-fk,B[E]) = 0
as required. ?
5.8. Differentiation under strong Vitali assumptions
We note in the preceding section that many limiting properties of the quotient h/k
of the two nonnegative functions h and it on a have been expressed in terms of the
variational measures associated with the derivation basis В and its dual B*. Now we add
some additional hypotheses on these measures. These can be considered as "Vitali-like"
assumptions. See Sections 7.1 and 9 for more detailed accounts of the relation between
this and the Vitali covering theorem. For now we just exploit what seem to be natural
assumptions in this theory.
As before we require a derivation basis on a space X supplied with a fundamental
covering relation a. We assume throughout this section that the derivation basis В is
filtering and decomposable [i.e., satisfies (B|). (Вт), and (B?), and (B4)]. The functions
h, к: a —> R are assumed to be nonnegative valued.
The needed "Vitali" assumptions are
h*(E) = V(h. B[E\) = V(h. B*[E]) = h*(E) A4)
and
k*(E) = V(k,B[E]) = V(k,B*[E])=k*(E) A5)
for all sets ? с X that arise in the discussion.
Recall that h* and k* are outer measures on X generated by the functions h, and к and
the derivation basis (because of Theorem 20). The force of the assumptions are that the
basis В and its dual B* generate precisely the same outer measures.
Lemma 28. //
. h(I,x)
В hm mi < с
1->? k(I,x)
for all ? e ? then
h*(E)sick*(E).
PROOF. This follows immediately from Lemma 24 and the assumed identities A4)
and A5). ?

Differentiation 199
Lemma 29. //
В hm sup > с
1^/ k(I,x)
for all ? e ? then
h*(E) >ck*(E).
PROOF. This follows immediately from Lemma 23 and the assumed identities A4)
and A5). D
Lemma 30. //
hi· h{I'x)
В hm sup = oo
1^/ k(I,x)
for all ? e ? andh*(E) < oo then k*(E) = 0.
PROOF. This follows immediately from Lemma 29. ?
Lemma 31.//
hi- · fh{Lx) ?
В hm inf = 0
/^.v k(I,x)
for all xe ? and k*(E) = 0 then h*(E) = 0.
PROOF. This follows immediately from Lemma 28. ?
LEMMA 32. Suppose that k*(E) < oo. Then
hi· h{I'x)
В hm
l^x k(I,x)
exists finitely or infinitely (i.e., it may be +oo)for h*-a.e. ? e ? and k*-a.e. ? e E.
PROOF. The set of points where the identity
hi- · *h{I<x) иг h{Lx)
В hm mi = В hm sup
l->x k(I,x) ,_l* k(I.x)
is not valid is the union of the countable collection of sets
[ h(I.x) „ h(I.x)
Rah = ?? e E: Slim inf <a <b < ohm sup—-—-
l /-л k(L.x) i^/k(Lx)

200
B.S. Thomson
where a < b are positive rational numbers.
Combining Lemmas 28 and 29 we must have
a"*h*(R«i,) ^ k*(R„b) ^ b'lh*(R„i>)
which, since a < b and k*(R„h) ^ k*(E) < oo, is only possible if
h*(R„b) = k*(R„b) = 0.
From this the lemma evidently follows. ?
Lemma 33. Ifh*(E) < oo then
В hm
l->x k(I,x)
exists and is finite for k* -a. e. ? e E.
Proof. It follows immediately from Lemma 30 that
h(I,x)
В lim sup < oo
1^/ k(I,x)
except for a set of points in ? of it*-measure zero. Thus the result follows from
Lemma 32. ?
5.9. Topological considerations
In most applications of a derivation basis ? on a space X the space X is already furnished
with a topology.
A case in point is if X is a metric space. In this situation we would expect that the
convergence in the sense of the derivation basis is also the same as convergence in the
sense of the metric, that is that
Slim{diameter(/U{.v})}=0. A6)
This idea can also be captured by expressing this in terms of pruning by open sets. (This
repeats condition (Вб).)
Dehnition 34. (B6) We say a derivation basis ? on a space X furnished with a topology
is compatible with the topology if for every ? С X and every open set G D ?
?@)??[?]
for every ? e B[E].

Differentiation
201
Note that if the topology is given by a metric and the basis is compatible with the
topology generated by that metric then A6) would hold at every point x.
An alternative approach has been suggested by a number of studies. Instead of assuming
that a topology is given in advance and that the derivation basis pays some attention to the
topological structure, we can obtain a topology directly from the basis itself.
Let us suppose that ? is a derivation basis on a space X. Let ? с X. We say that a point
? e X is a B-interior point of ? if for every ? e ??[{.v}] it is true that ?(?) e ??[{.*}]. We
say that a point ? e X is in the B-closure of ? if it is false that for every ? e B[{x}] we
must have ?(? \E) e B[{x}].
Thus in this setting we have the notions of ?-interior, ?-closure, ??*-interior, and
.Enclosure. These can be used to develop topological ideas and concepts useful in discussing
a derivation basis when no natural topology is at hand. Ideas of this kind can be found in
Hayes and Pauc [31 ] and in the papers of A.P. Morse [57] and [58], and in a different, but
related, setting in Henstock [34].
We shall not pursue these notions as they add a level of complexity and detail that is not
necessarily rewarding.
Here is an example just to illustrate the particulars that can enter into the subject. In the
space R2 let a be the covering relation consisting of all pairs (/. x) where / is an interval
(i.e., rectangle with sides parallel to the coordinate axes) and ? & I. The derivation basis В
is defined in the most natural way relative to the metric in R2:
Let 8A, x) = diameter(Z) for every (I.x) e a and then a covering relation ? da
belongs to В if for every ? e R2 there is a ? > 0 so that (I.x) e ? if ?(?,?) < ?.
It is not difficult to check the topological notions for this example. Without giving the
details, let us just give an illustration. Let ? = [0. 1 ] ? [0. 1 ] be the closed unit square in R2.
The ?-interior points of ? are exactly the ordinary interior points of E, namely the points
in the open square ?° = @. 1) ? @, 1). The ?-closure of ? is ? itself. The ?*-interior
points of ? are exactly the points in ? itself. The ?*-closure of ? is also exactly ?.
Contrast this with the situation for the open square ?° = @. 1) ? @. 1). The ?-interior
points of ?° consists of ?° itself. The ?-closure of ?° is again merely the set ?° (and
contains no points of the ordinary closure). The ??*-interior of ?° is also ?°. The
.Enclosure of ?° is the closed square ?.
Bruckner and Rosenfeld [15] approach the subject from a different viewpoint. If ? is a
derivation basis on a measure space (?.?.?) then consider the class of functions / with
the property that for every ? > 0 there is a ? e В so that
J \/(?)-/?\??(?)^??(?)
for all ? с /, ? e ?, (?,?) e ?. We consider such functions "continuous". The
motivation comes from the case in R where continuous functions can be described as those
/ for which for every ? and ? > 0 there is a S > 0 with
I \f(x)-f(y)\dy^ek(E)

202
B.S. Thomson
for all measurable ? с (x — S,x + S). One then studies the smallest topology on X for
which this class of functions are continuous in the sense of that topology. They obtain
numerous natural results, among them an analogue of the Lusin theorem that measurable
functions can be approximated by continuous functions.
In the sequel we make no use of topological concepts obtained directly from the basis,
but we do assume frequently that the basis being studied is compatible with the metric in
the sense of Definition 34 (or equivalently condition (Вб)). The following properties then
play a role. Let us suppose that X is a metric space.
LEMMA 35. If a derivation basis В on X satisfies (Вб) then so too does the dual basis B*.
LEMMA 36. Suppose that the derivation basis В on X satisfies (B|), (B:), (B4), and
(Вб). Suppose two sets E\ and ?4 ore separated in the sense that d(E\. ?2) > 0. Then
there exists ?\, ?? e В with
?\[?\]??2[?2] = 1».
LEMMA 37. Let В be filtering and compatible with the metric on X [i.e., satisfies (B|),
(B2), (B3), and (Вб)]. Suppose two sets E\ and Ei are separated in the sense that
d(E\, ?2) > 0. Then there exists ? ? В with
/*[?,] П/3[?:] = 0.
LEMMA 38. Let В satisfy (B|), (B2), and (Вб) and suppose that h is a real-valued
function on a. Define the set functions h* and /7* by h*(E) = V(h. B[E]) and /j*(?) =
V(h, B*[E]) for all ? С X. Then h* and h* are metric outer measures on X.
(Although the notation is suggestive of inner and outer measures, that is not the intention;
both are outer measures and, if В is filtering, then In^h*.)
5.10. Differential equivalence
Let ? be a derivation basis on X satisfying properties (B|)-(B4). For any functions
h ,k : a -> R let us write h = k and say that h and к are differentially equivalent relative to
the derivation basis В if
V(h-k,B) = 0.
Under these hypotheses this is an equivalence relation and if h = к then h and к behave
very much alike with respect to many properties. For example, the outer measures h* and
k* would be identical, as would /г* and it*. Also if g: a -> ? then the limits
h(I,x)
В lim and В lim
l^xg(fx) l^x

Differentiation
203
would be closely related up to a set of g*-measure zero. Write dh for the equivalence class
of all functions that are differentially equivalent to h. Then one can assign meaning to
statements such asdh = fdk.
For example if ? and ? are measures on X then the "differential notation"
?? = fdv
now has a significance defined relative to the derivation basis.
The method derives from Henstock's "variational equivalence" which he exploits in
many arguments, and that in turn is derived from earlier ideas of A.J. Ward. S. Leader [46-
48] has shown how, in some settings, the classical notion of a differential and its traditional
properties can be handled by this procedure.
6. Standard examples
In order to provide some perspective we give a number of concrete constructions of
derivation bases. A later section will explore some of these examples in greater detail.
6.1. A general construction
In any space X let there be given a covering relation a and suppose that there is also given
a positive, real-valued function S on a. We can express the limit
lim h(I,x)
S(/..t)—0
as a limit relative to a derivation basis as follows:
By В we mean the collection of all covering relations ? С a with the property
that for each ? e X there exists a positive ? (depending in general on ?) so that
(/, jc) e ? whenever (I,x) ea and ?(?. ?) < ?.
The dual basis B* can be described as containing all ? С a with the property that for
each ? e X and for any positive ? there is at least one pair (/,x) e ? for which it is true
that <$(/,-*) < ?.
Hayes and Pauc [31] refer to <5 as a numerical index of contraction and call such bases
D-bases (although defined in their scheme, not as here, using covering relations). In order
for the method to work (so that В and B* are derivation bases in our sense) there must be
for every ? and every ? > 0 at least one pair (/. x) e a with <5(/. ?) < ?.
Perhaps the most natural instance of such a scheme occurs in a metric space where we
would use
<S(/,Jt) = diameter({;t}U/)

204
B.S. Thomson
as an index of contraction (Section 6.2). It might also seem natural in the setting of a
measure space to use instead
?(?,?) = ?(?).
The discussion in Section 6.3 includes a simple example illustrating that this might not
have nice properties.
6.2. Derivation bases in a metric space
In the special case where X is a metric space one most frequently desires that the limit
taken is in the sense of sets contracting in the metric towards the point x. This can be
achieved in general by selecting, as usual, a covering relation a on X and then using the
numerical index of contraction
<5(/,;t) = diameter({jt}U/)
to generate a derivation basis as in Section 6.1. Most derivation bases in metric spaces will
be found to be of this type.
6.3. Measure contraction
In some studies it is useful to consider a limit
lim h(I,x)
/i(/)^0. .re/
where ? is a measure on the space X and, so, the limit is understood in terms of shrinking
values of ? (I). Here we can capture this by taking as a fundamental covering relation a
on X such that for each pair (I,x) e a, x e I. I is in the domain of the measure ? and
0 < ?(/). Then an appropriate numerical index of contraction is
?(?,?) = ?(?).
The reader should be immediately aware of some limitations in such systems. At first
sight it might seem like an intuitively appealing notion of limit.
Consider for example, however, X = R2, ? planar Lebesgue measure and let a be the
covering relation of all pairs (/, x) where / is an interval (i.e., a rectangle with sides parallel
to the axes) and ? e /. If ? denotes the derivation basis generated by taking <$(/,.*) = ?(/)
as the numerical index of contraction then observe an unpleasant limit computation. Let ?
be the upper half plane. It is easy to check that
В lim sup = +эо and В lim inf — = 0
/^v M(/) /--v flU)

Differentiation
205
at every point ?. Thus this system does not have the density property, and should be lacking
in most reasonable properties.
6.4. Set contraction
In some cases some might wish to study a limit
\im h(I,x)
/-».v
where the sets / are contracting in the sense of set inclusion. For this define an appropriate
covering relation a on X such that for each pair (?,?) ea,x e / and take ? С a to belong
to В if for every ? e X there is an element (J.x) e a so that every pair (I.x) e a for which
J С I belongs to ?.
It should be noticed that this notion seldom has very nice properties unless the class of
sets involved has been tailored properly. (We shall see that in the presence of a net structure
or a lifting this can be done quite successfully.)
6.5. Busemann-Feller bases
In the euclidean space Ш" take I to be a fixed family of open bounded sets that has the
property that if / e I then every set homothetic to / is also in I. The fundamental covering
relation is simply a = I x Ш". Then the derivation basis is defined by using, as in the
preceding discussion, the numerical index of contraction
<S(/,;t) = diameter({.v}U/).
The early study of Busemann and Feller [17] used, essentially, this basis.
6.6. Federer's scheme
Federer [24] sets his study in a metric space X equipped with a locally finite metric outer
measure ? for which all bounded sets have finite ? measure. The fundamental covering
relation a is presumed to have some strong properties:
(i) each (?,?) ea has ? e I and / a bounded Borel set;
(ii) a is fine at each point ? in the sense that
inf diameter(Z) = 0;
(/..t)ea
(iii) if ? С a is a fine cover of a set ? с X in the sense that ? is fine at each ? e ? then
there is a countable family {(/,-,?,-)} С ? for which {/,·} is disjointed and so that
v(z\\Ji,)=o.

206
B.S, Thomson
The limit now is taken in the sense of the metric so that one can use the numerical index
of contraction
<5(/, x) = diameter(Z)
as before. Note that the dual of the derivation basis here would be, in this language, the
collection of all ? с a that are fine at each point ? e X. The extra assumptions on the
covering relation a assure that the usual methods of the Vitali covering theorem can be
applied.
6.7. Net bases
The space R" can be carved up geometrically in a fashion that is useful in many
investigations. By a dyadic cube in R" we mean a set of the form
[(/? - lJ-*,/|2-*[ ? [(/2 - lJ-*./22-*[ ? ··· ? [(/„ - lJ-*,/n2-*[
where k, i\, /'2,..., /„ are integers. The collection of all dyadic cubes forms a net structure
on R" that has the essential property that if N\ and N2 are any elements of the collection
then one of the following must be true:
N\ С N2, or N2CN\, or N\ ? ?2 = 0.
Note, too, that (i) at most a finite number of sets of the net contain a given net set, (ii) the
sets are simple Borel sets, in this case of type Fa, and (iii) each point is in sets of arbitrarily
small diameter. Any family in a metric space with such properties is called a net.
Besicovitch made much use of this in his investigations of Hausdorff measures and, in
a more general form, the study of "comparable net measures" plays an important technical
role. (SeeEdgar [20, pp. 58-61] and Rogers [69, pp. 101-122].)See also Saks [72, pp. 152-
156] and Hayes and Pauc [31, p. 110] in the setting of abstract derivation theory. The way
in which the net structure carves up the space allows, as in the preceding section, for the
usual methods of the Vitali covering theorem to be applied to any appropriate measure on
the space.
A similar structure is available in many measure spaces. The actual requirement here
is that the measure space be separable. For measure spaces this means roughly that the
space can be carved up into a finite or infinite sequence of small pieces; it is also the same
as saying that L2(X), the square integrable functions on the space, is a separable Hubert
space. A brief account, but with a large citation list, is given in Bruckner [13, pp. 34—35].
6.8. Bases associated with a lifting
In a general measure space (?, ?, ?) one cannot hope to carve up the space as neatly as
does the net structure. In the presence of a lifting, however, this can be done. A lifting is

Differentiation
207
just an operation on measurable sets that picks out one for every family of sets with the
same ?-measure and does it in a special algebraic way, preserving essential features.
Let L be a lifting on the measurable sets (roughly L(M) picks out a measurable set with
the same measure as ? in a way that preserves intersections and unions). Then interpret
/ =?> ? to mean that ? e / and / is lifted (i.e., L(I) = I) and "shrinking" is just meant
as set inclusion shrinking. Thus the basis В is constructed by first taking a as all pairs
(/, x) where / is a lifted set and ? e /. The basis is constructed using set contraction as in
Section 6.4.
This basis can be proved to have the strong Vitali property for ?. Kolzow [43] in 1968
showed more or less that there is a lifting exactly when the Radon-Nikodym theorem is
available. Then, because of this fact, if the Radon-Nikodym theorem holds for the measure
space there must be a differentiation basis for the space that has the strong Vitali property
and so that the Radon-Nikodym derivative is a genuine a.e.-derivative for that structure.
We omit any further discussion leaving the topic of liftings to the chapter prepared by
W. Strauss and N.D. Macheras in this volume.
6.9. Self dual bases
In any space X with a covering relation a on X let U be an ultrafilter of subsets of a. This
means that U is filtering and that for every subset ? с a either ? e U or else a \ ? e U.
Given any filtering derivation basis В there must be, by the maximal principle, an ultrafilter
U that contains B; thus such objects are easy to come by, if rather difficult to conceive.
The derivation basis U has the fundamental property that it is self-dual, that is that
U = W.
To see this suppose that ? e U. Then, since U is filtering, ? e U* by Lemma 15. If ? e U*
then either ? e U or else a \ ? e U. But the latter is impossible since
??\(?\?) = 8
which is not allowed for an element ? e U*. Thus U = U*.
Note that, because of Lemma 11 and the self-duality, for every function h : a -> R,
Uhmsup h(I,x) = Uhm'mfh(I.x)
(possible infinite) must hold at every point ? e X. Thus limits are particularly simple in
such settings.
7. Derivation bases in a metric space
For this section let us suppose that X is a metric space. We require that there is a Borel
regular outer measure ? on X; thus all Borel sets are measurable and we need too that that
all bounded sets have finite ? measure.

208
B.S. Thomson
For the fundamental family a that gives the covering relation on which the derivation
basis is defined we require in advance only that each pair (/, x) e a has / a bounded Borel
set for which 0 < ?(?) < oo. It is not generally needed that ? e I.
Let ? be a derivation basis on X. Although many results have versions under weaker
hypotheses, for our exposition here let us assume that В satisfies all of the properties (? ?),
(Вг), and (В-,), and (B4) and we ask also for a compatibility with the metric by assuming
the property (B^).
Associated with ? (which remember is itself an outer measure) are two further metric
outer measures ?* and ?* defined for all ? с X by
?*(?)=?(?,?*[?]) and ?*(?) = ?{?. ?[?]).
Because of our assumptions we can show that
?* ^?* ^ ?.
(Note that while these two additional outer measures are generated by ? and the derivation
basis they may not be related to the usual outer measures constructed in such situations.)
7.1. Differentiation of the integral
We now return to one of our central problems, the differentiation of the integral. We do
this in the setting of a metric space with all the metric and measure assumptions needed to
produce a reasonable theory (as outlined already).
Thus we seek conditions on a derivation basis ? on a metric space X furnished with a
locally finite, Borel regular outer measure ? so that
В lim / f du= fix) ?-a.e.
/-,?(/)? Jy
for any ?-integrable function /.
We begin with a variational characterization of the integral that leads immediately to a
way to estimate the measure of the Lebesgue set.
THEOREM 39. Let f be ? integrable on X and write
&(/,?)= ?\/(?)-/(?)\??(?)
for all (?, ?) ea. Then
V(h,B) = 0.

Differentiation
209
PROOF. Tosimplify the proof and clarify the arguments we present only the case f(x) ^ 0
for all ? e X. (To handle the set {x e X: f(x) < 0} requires a repeat of the same
arguments.)
Let ? > 0. Using the absolute continuity of the integral, we can determine ? > 0 so that,
whenever A is a measurable subset of X with ? (A) < ?, then
?/??<?-. A7)
Ja 4
Choose 1 < t < oo so that
A-0 ? ???<- A8)
Let Z = {x & X: f(x) = 0} and form a denumerable, measurable partition of the set X\Z
(i.e., the set of points where / is positive) by writing
Xm = {xeX: Г </(*)<;?'"+'}
form = 0, ±l,±2, ±3,....
Note that each ?(?,„) < oo because of the inequality
fi(Xm)^ ? ?/(?)??(?)??'" ? /??.
JXm JX
Thus we may, by the assumption on the measure ?, choose open sets G,„ D Xm so that
each ???,? \ Xm) is as small as we please. We do this so as to satisfy the two inequalities
oo
]? ?@,„\?,?)<? A9)
and
]T r+V(G„;\^»)<J· B°)
;n=—oo
Finally, again by our assumptions on the nature of the measure ?, we can choose an open
set W D ? so that ?(? \?) <?.
Take any element ?\ e В and, using the properties (B2) and (B4), construct from it an
element ? e ?, ? С ?\ with the property that for all ? e ?, ?[{?\] С fi(W), and for all
xeX\Z,
?[{x\\c?{Gml?)) B1)
where m(x) is an integer selected so that ? e Xmu> С G,n(.V). In order to estimate V(h, ?)
we now consider now a collection {(/,, ??)} contained in ? and with the pairs /,, // disjoint

210
B.S Thomson
for i ? j. Corresponding to each ?, select an integer m(i) so that ?? e XB1|,) С Gm(,) and
consider the sums
T\h(Ii^,)\ = T ?\/(?)-/(?,)\??(?). B2)
? ^
The sum in B2) can be dominated by the four separate sums:
? = ?? \/(?)-/(?,)\<1?(.?),
? J /,???,,?,)
2 = ?? \?(?,)\<??(.?),
? = ?( \/(?)\??(?)
I Jli\Xmii)
all of which sums are taken only for points ?? e X \Z, and
? = ? ? \?{?)-?{?,)\??{?)
which latter sum is taken only over points ?? e Z. The sum ? is simplest to handle since
f(x) = 0 for jr e ? and /(?,) = 0 for each element in the sum. Thus
jJi,\z Jw\z 4
because of A7).
For ? note that if both points ?, ?? are in /, ? X,„(,) then, writing simply m = m(i),
tm < f(x) <: r'"+l, t'" < /(??) <: r'"+l, and so
|/U) - №-)\ < t'"+] - t'" = t'"(t - 1) iC (t - I)f(x).
Consequently
?^?? (\-?)/(?)??(?)^(\-?) ? ???<-
jJhnx,„U) J ? 4
by A8).
For Q note that if ?, e XmU) then 0 < /(?,¦) ^ r'"(',+ l and /, \ Xm(i) С G„ni) \ ?,„(,,.
Consequently
DC
?<; ? ?'"+]?(?,„\?,„)<?-
m=—oc
by B0).

Differentiation
211
Finally for R, define
A = (J(/,-\ *«(/))
and observe that the union is disjoint. Thus
ОС
?(?) = ]??(/,\Xm{l))ii ? ?(?„\??,)<4
/ /я=—зс
because of A9). Thus we have
R = y[ /(?)??(?)? ( fixUfiM^-
because of A7).
Thus we have proved that V(h, ?) < ?. Since ? is arbitrary it follows that V(h, B) = 0
as required. ?
From this theorem we can deduce the differential equivalence of ? and fdv (in the
language of Section 5.10). In particular, we have the following relation for the associated
measures.
COROLLARY 40. Write v(I, x) = /, / d^for every pair (I, x) e a. Then
?(/?,?[?])=?(?,?[?])
/oral! ? CX.
PROOF. The corollary follows from the theorem along with the variational
property A3). ?
Finally we have our main result showing that the Lebesgue set of an integrable function
/ has zero measure in a certain special sense.
THEOREM 41. Suppose that the derivation basis В satisfies properties (B|), (Вт), (В4),
and (Вб). Let f be integrable with respect to ? on X. Then there is a set N С X with
???) = 0 so that
Blim-^- i|/U)-/(y)|^(y) = 0
/^.Гд(/)У/'
forallxeX\N.
PROOF. This follows directly from Theorem 39 and Lemma 26.
D

212
B.S. Thomson
Finally now we can express our fundamental problem in this setting. What conditions
on the derivation basis give us
/^л ?(?) J,
for any ?-integrable function /? We evidently require that ?*, ?* and ? are identical on
the exceptional set. This identity is intimately related to what are known as strong Vitali
conditions and, as we shall see in the sequel, can often be proved in important settings.
7.2. The density property
The preceding section demonstrated that the integral in this setting (metric space under
natural assumptions) can be differentiated outside an exceptional set of ?^-measure zero.
It is possible, however, in some examples that the measure ?* is trivial so that the statement
has no content. (The interval basis in R" [see Section 12.3 below] with ? the «-dimensional
Lebesgue measure is such a case.)
Using ideas of De Possel [64] we can refine the arguments. Rather than employing the
variation, which uses estimates involving disjointed sequences, we can allow some overlap
of the sets. To do this we must, though, give up the chance of differentiating the integrals
of all integrable functions, reducing ourselves now to discussions of bounded functions.
In particular, note that applying the differentiation problem to a function f = ??, the
characteristic function of a measurable set A, we can ask more narrowly for conditions
under which we can be assured that
?(???)
B ,lim 777~ = *aW B3)
?-*.* ?(/)
for ?-almost every x e X. Following common usage we say that the derivation basis has
the density property for ? if B3) holds for every measurable set A.
The analysis that addresses the problem of finding conditions under which the density
property holds is due to De Possel and constitutes mainly a shift of viewpoint on the usual
Vitali arguments, by considering approximating sequences that are not disjoint but have
a small overlap. Fix s > 0 and let ? be a covering relation. By V(h,fi) for a function
h : ? -> Ш the reader will recall that we computed
SUp]P|/2(//,?/)|
/
where the supremum is taken over all collections {(/,-,?;)} С ? with the {/,} disjointed.
Now we define \??.5(??, ?) to be the supremum of the same sums, taken overall collections
{(//,?;)} С ? for which
]? ?(/,) - ? ((J/,)<*. B4)

Differentiation 213
It is immediately clear that if {/,} is disjointed then the condition B4) does hold. Thus
?^,?)^\??.?^,?).
Finally we define, for any family В of covering relations,
???
Again it is clear that V(h,B)^ Wtl,s(h, B).
LEMMA 42. Let В be any collection of covering relations. IfG is open then
\?,5(?,?@)?;?@ + ?.
PROOF. Let ? e B(G) and suppose that {(/,-. ?,)} с ? satisfies B4). Then, since also each
/; С G,
]??(/,-) <C mMJ/,J +s ??@ + ;
Thus ??.,?,?) iC m(G) + s and the result follows. ?
Let now ? be a derivation basis on a metric space under all the same assumptions as
before. By the lemma we see that
?(?) + 5>???.1(?,?[?])
for all s > 0 and all measurable sets. This is because, under our assumptions,
?(?) = inf{M(G): G open, G D E\.
Thus in general we have \??.4(?, ?[?]) as an underestimate of the measure ?(?). For
the density property we shall see that it is an accurate measure. We shall show that, under
appropriate hypotheses, the derivation basis В has the density property relative to a measure
? if and only if
?(?)<?\??..,(?,?*[?])
for all measurable sets ? and all s > 0.
Let us first show the following lemma which contributes half of this result.
LEMMA 43. Suppose that 0 < ?(?) < oo. //
mim^^Ul B5)
/-* ?(?)

214
B.S. Thomson
?-a.e. in A for every measurable subset Ac E, then
?(?)<?\??.,(?,?*[?])
for each s > 0.
PROOF. Fix ? e B*[E]. Choose 0 < a < 1 so that
(? -1)?(?)<?.
By our hypotheses
В hm = 1
/-* ?(/)
for most points ? of ? so
0| ={(/,*) e0: ?(/??)>??(/)}
cannot be empty and, hence, in particular
r|=sup{M(/): (?,?)??]}>0.
(Recall we are assuming throughout these sections that ?(/) > 0 for all (/, x) e a.) Choose
(/|,?i) e/3, 5???3??(/|)> 3r,/4. We have that ?(/, ? ?) >??(?\).
Now write E\ = ?, ?2 = ?| \ I\. By our assumptions ?2 is a measurable subset of ?
and so the hypotheses allow us to repeat these arguments applied to ?2 if ? (?2) > 0. If
?(?2) = 0 the process must stop. Thus, continuing inductively we shall find sequences
En = E„-\ \ /,,_|,
?„ = {(?,?)??: ?(??)?„)>??(?)}.
rn = sup^(/): (/, лг) e ?„} > 0,
and
(?„,?„)??„ withM(/„)>3r„/4.
As long as ?(?„) > 0 the process continues to obtain the element (/„,?„); if ?(?„) = 0
then the process stops there.
We arrive at a sequenced 1, ?1), (/2,?>), (/i,?0,... of elements of ? that is either finite
(if ?(?„) = 0 at some stage) or is infinite. In the event of a finite sequence evidently
?(?\\^??=0.

Differentiation 215
But this equality is also true for the infinite sequence. To see this observe that
]??(/,) ^fl-' ]??(/,¦ n?,-)=fl-V(U(/,¦ П?,-Л
= fl-V(?niU/iJJ^/i(?)<oo.
In particular, ]?. ?(/,) < oo and so ?(/,) -> 0 as / -> oo. Since r, < 4?(/,)/3 this means
r,- -> 0 as / -> oo too. If contrary to what we want to prove
?(?\\^??>0
then write Ex = ? \ (J,- /; with ?(??) > 0. Applying the density property to ?Oc as a
subset of ? we would obtain, following the same line of reasoning as used above,
?? = {(?,?)??: ?(/??^)>??(/)},
and
roc = sup^(/): (/, лг) e ?^ \ > 0.
But then /Ззс с ?? for all / and it follows that
0 < rx ^ r,¦ -> 0
which is impossible. Thus we have established that
?(?\\^??=0.
Now we argue that the sequence (?\.?\), (h,b)Ah-b)'¦¦¦ of elements of/3 satisfies
condition B4). To see this we use the facts that
?(?M??]??(/,) and ?(?)^?([}??
to compute
? ? i
which is precisely the condition B4) that we wished to verify.

216 B.S.Thomson
Since
it follows that
and the lemma follows. ?
Now we turn to the other direction, by establishing the density property under the
measure hypothesis.
LEMMA 44. Assume that for every measurable set ? that
?(?)<?\??.,(^?*[?])
for all s > 0. Then В has the density property with respect to ?.
PROOF. Let ? > 0. Take any ? measurable and write
? = \x?M: Slim sup — >0 ,
( T ,^/ ?(?) J
Mk = \x<?M: Slim sup —-—- > \/k\.
{ ?^? ?(?) J
Note that the sequence of sets Nt increases to ?'. By our regularity assumptions we may
select Gk open so that Gk D Nk and ?(Gk \ Nk) < ?.
Let
Pk = {(I,x): ?(??)?)>?(?)/?<}
and notice that & e B*[Nk]. Note also that ft(G*) e ?*[W*]. Take any {(/,·,&)} С fik(Gk)
that satisfies condition B4)
J>(/,-)-/*Aj7'"
Note that this can also be written as
у (??'·--??/,)^<?·

Differentiation
217
Then
]Гм(/,)<;*]Гм(/,пмк*[у (???, -xu/,W + ?(,?n^J/,^
<ks + k^(Gk\Nk) <ks + ks.
From this it follows that
W^s(h,pk(Gk))<ks + ke
and hence that
Evidently then ?(?) ^ ^ M(Njt) = 0 and we have proved that
?(??)?)
В hm = 0
?^? ?(?)
at ? almost every point ? that is not in ?. Applying this observation to the complement of
M, the set X \ M, and using the trivial fact that
?(?) = ?(? П /) +?((? \ ?) ? /)
we obtain that
кг М(/ПМ) .
В hm = 1
/-^ м(/)
at ? almost every point ? that is in ?. Since this is the desired density property the proof
is complete. ?
Thus the results of this section (due to De Possel, but in a different language and setting)
show that the density property is equivalent to there being a weak Vitali estimate for the
measure from the dual basis B*. In fact under fairly general hypotheses the density property
is equivalent to the differentiation property that the basis satisfies
Slim / /??= fix) ?-a.e.
for any bounded ?-integrable function /.

218
B.S. Thomson
8. Differentiation under strong Vitali assumptions
Let us generalize the setting by supposing merely that (?, ?, ?) is a measure space. We
suppose that we are given a covering relation a so that / e ?4 and 0 < ? (/) < oo for every
pair (/, jt) e a.
Suppose that
-/.
<p(E)=J fdiL
for all ? e ? where / is a nonnegative, ?-integrable function. If a filtering, derivation
basis В is given on a, then what hypotheses will allow us to conclude that / is a genuine
pointwise derivative in the sense that
«Г Ф{1) ft ч
/—* ?(/)
for ?-a.e. point дг е X?
The needed assumptions are
?*(?)=?(?,?[?])=?(?.?*[?]) B6)
and
0*(?) = ?@,?[?]) = ?@.?*[?]) B7)
for all ? с X to begin with. For now we do not assume that ?* and ?* are the actual outer
measures generated by the measures ? and ? by measure-theoretic methods. For the first
several lemmas the fact that ? and ? are related by ?(?) = JE f ?? plays no role.
The assumptions B6) and B7) (along with the refinements B8) and B9)) are called
strong Vitali properties. This repeats the development in the metric space setting of
Section 7.1, but now in a measure space with these assumptions in force. Now without the
topology present we seek a different method than the direct variational one given before.
Lemma 45. If
hi· · гфA)
uliminf < с
/-* ?(?)
for all ? & ? then
?*(?)???*(?).
PROOF. This follows immediately from Lemma 28 and the assumed identities B6)
and B7). ?

Differentiation 219
Lemma 46. If
? lim sup > с
for all ? e ? then
?*{?)>??*{?).
PROOF. This follows immediately from Lemma 29 and the assumed identities B6)
and B7). ?
Lemma 47. //
В lim sup = oo
?^? ?(?)
for all ?? ?????*{?) < oo then ?*(?) = 0.
PROOF. This follows immediately from Lemma 30. ?
Lemma 48. //
Sliminf— =0
for all ? e ? and ?* (?) = 0 then ?* (?) = 0.
Proof. This follows immediately from Lemma 28. ?
LEMMA 49. Suppose that ?*(?) < oo. Then the identity
„.. . ?(?) ?(?)
В lim ??? = В lim sup
/-.t ?(?) ?^, ?(?)
holds for ?* -a.e. ? e E. and ф*-а.е. ? e ?.
PROOF. The set of points where the two limits are unequal is the union of the countable
collection of sets
[ </>(/) ~ ФИ)
Rah = \ x e E: Sliminf <a <b < ? lim sup
?^? ?(?) ?^? M(/)
where a < b are positive rational numbers.

220
B.S. Thomson
Combining Lemmas 45 and ?? we must have
0~ V (Rab) < fl*(Rah) < &~ V(/U)
which, since ? < b and everything is finite, is only possible if both <p*(Ruh) = ?*(?«/>) =
0. From this the lemma evidently follows. ?
Lemma 50. 1/ф*(Е) < оо ?/?еи
В hm
/-*/*(/)
emfs and is finite for ?*-?.?. ? e ?.
PROOF. It follows immediately from Lemma 47 that
? hm sup < oo
/^/ M(/)
except for a set of points in ? of ?*-measure zero. Thus the result follows from
Lemma 49. ?
There remains now only the problem of determining whether
/^.r ?(/)
at most points. For this we need to add a few assumptions to B6) and B7) that allow the
arguments to be used.
The extra needed assumptions are
?(?)=?(?,?[?]) = ?(?.?*[?]) B8)
and
?(?) = ?(?,?[?])=?(?,?*[?]) B9)
for all ? e ? so that the outer measures ?* and ?* are indeed closely related to the usual
outer measures generated by the measure space. Also we require that the measures are
complete so that sets of measure zero that enter into the discussion are also covered by the
assumptions in B8) and B9). Finally we shall assume that the limit
g(x) = В lim is measurable C0)
?^? ?(/)
so that we can use some recognizable measure-theoretic arguments on g to show it is
precisely / almost everywhere. (Measurability can, in some settings, be obtained directly
from strong Vitali assumptions; see [24, p. 155].)

Differentiation
221
THEOREM 5 1. Under these additional assumptions, if ? С ?, ?(?) < oo, and ?(?) <
oo, then
Slim ? /?? = /(?)
for ?-a.e. ? e E.
PROOF. We have established already, in our more abstract version of this (Lemma 33), the
existence of this limit g(x) outside a set of д*-теа5иге zero in E. We shall show that
?(?) = ?/?? = ?8??.
Since this shows this identity for all such sets the functions / and g can differ only on a
set of measure zero.
Let
E\ = {x e E: g(x)=0} and E2 = {x e E: 0 < g(x) < oo}.
We deduce from our usual growth estimates that ф(Е\) = 0 and that
?(?\(?]??2))=0.
Thus we can focus on Ег.
Fix 1 < t < 00 and partition Ei into a denumerable collection of disjoint measurable
sets
A„ = {xeE2: t" ? g(x)<tn+i}
where ? is an integer (positive, negative or zero).
Using Lemmas 45 and 46 along with B8) and B9)
?(?2) = ? ?(?»^ ? '" + ??(?„? ? 4 ?<1? = ? &??.
?=-? ?=-? ?=-? Ja" jEl
Similarly
ЭС Г^С -л- л г*
?(?2)= ? ?^?^> ? г"^А^> ? ?_' / &?? = ?? / &??.
J A,, J E-
? = — ? н=— эс ??=—??. "
Since ?(?\) = ?(?\) = 0 and ?(? \ (?, U ?2)) =?(? \ (?, U ?2)) = 0 it follows that
/ 8?? = ?(?) = j ???
and the theorem follows. ?

222
B.S. Thomson
9. Strong Vitali conditions
The assumptions
?(?) = ?(?,?[?]) = ?(?,?*[?]) C1)
and
?(?) = ?(?,?[?])=?(?,?*[?]) C2)
that played a role in our theory are known as strong Vitali conditions. We now turn to the
methods that have been used to establish such conditions.
To keep the discussion to its most basic level and avoid abstractions and generalities
that might obscure the main themes let us work in a metric space X. We assume as before
that ? is a locally finite, Borel regular outer measure on X. The derivation basis В under
consideration must be compatible with this metric in the sense of (Вб). As usual we require
that В is filtering [i.e., satisfies (B|), (B2), (B.i)].
We need to recall some expression of the classical Vitali theorem. It asserts that, for
the usual derivation basis on the real line and Lebesgue outer measure ?, the elements
of the dual (i.e., the so-called Vitali covers) can be used to estimate the measure of sets.
Specifically let ? be a set of finite Lebesgue outer measure. Let /3 be a fine covering
relation on a set ? с К so that /3 consists of pairs ? e E, I an interval of ? with there
being, for each x, at least one sequence {(h.x)} with ? e h ? ? and |h \ —* 0 as к —> oo.
Then, for every ? > 0 the Vitali theorem provides the existence of a sequence
{(/*,&)} С 0
such that
A) The {/;·} are disjointed.
B) The measure
??|/?<?(?) + ?.
[Here ? is Lebesgue outer measure and there is no prior assumption on the
measurability of E.]
C) The measure
k(E\\Jlk\=0.
Expressed this way it is easy to fail to appreciate the role of item B). By the regularity of
Lebesgue outer measure we can find an open set G D ? so that k(G) < HE) + ?. Recall
that the pruned covering relation /3(G) is also a Vitali covering relation on E. Then any
selection of {(/*,&-)} С /3(G) must always satisfy B). Since for any Vitali cover /3 it is also

Differentiation
223
true that /3(G) is a Vitali cover, item B) can be settled in advance. Thus one needs, in order
to prove this theorem, only to obtain items A) and C). Moreover if ? is a measurable cover
of ? then by choosing G D Ё so that X(G \ ?) < ? we can arrange for B) to be replaced
by a sharper version
B') The measure
?(?>\^)<?·
Some versions of the theorem are expressed this way instead.
Since we are faced with the need to find an abstract expression of this theorem in the
setting of an abstract derivation basis we have a number of choices. We might simply, as do
many authors, define a strong Vitali property by exactly these three conditions, expressed
in an obvious way relative to elements of the basis B[E] or B*[E]. Alternatively we can
do as many other authors have done and take only the conditions A) and C) for an abstract
expression of the strong Vitali property and place our assumptions in a context which
guarantees that B) will hold automatically. The latter is simpler and better expresses the
nature of the theory, although there is some formal merit in taking all three properties as a
starting point for developing a theory.
Note, in the setting of the real line, this expression of the Vitali theorem shows that
?(?,?(?))!??@ and ?(?./8)>]??(/*)>?(?)-?
к
so that
?(?? ?(?, ?*[?]) <;?@<?(?) +?
and it follows that
?(?) = ?(?, ?>[?]) = ?(?. ?>*[?])
giving our Vitali condition in the form C1) expressed at the beginning of the section.
9.1. Strong Vitali property
As our definition we merely take the classical Vitali theorem expressed previously as a
model.
Dehnition 52. Let ? be a derivation basis on a space X. Then В is said to have the
strong Vitali property with respect to ? if the following property holds for all subsets ? С
of finite ? outer measure: For every /3 e B[E] there exists a sequence
{(/*,&)} С/8

224
B.S. Thomson
such that
A) The {(h)} are disjointed.
B) The measure
?(?\???=0.
Since we are assuming throughout that ? is a Borel regular outer measure on a metric
space X we have immediately that, for any derivation basis B,
?(?,?[?])???(?). C3)
This follows from the fact that for any ? > 0 and any ? e B[E] there is an open set G D ?
with m(G) ^ ?(?) + ? and so
?(?,???])??(?,?(?)??(?) + ?.
Thus the strong Vitali property leads to the following two conclusions.
LEMMA 53. Let В be a derivation basis on a space X that has the strong Vitali property
with respect to ?. Then
?(?,?[?])=?(?). C4)
LEMMA 54. Let В be a filtering derivation basis on a space X such that the dual B* has
the strong Vitali property with respect to ?. Then
?(?,?[?]) = ?(?,?*[?]) = ?(?). C5)
9.2. Classical proofs of the Vitali theorem
There is now a large literature devoted to covering theorems of the strong Vitali type, that
is coverings consisting of countable disjointed families covering almost all points in some
set. One theme that has emerged in the general theory is to take some standard technique
that can be used to prove the classical theorem and then tailor an abstract version. The most
popular versions can be traced back to Banach's [3] proof of the Vitali covering theorem.
Here is one theme that we might follow. Let ? be a ball in R". By В we denote the ball
with the same center but 5 times the radius of B. Then one can prove the following:
Let ? be a collection of nondegenerate closed balls in E" with uniformly
bounded diameters. Then there is a countable disjointed family To С Т so
that
??? ????

Differentiation
225
For a simple proof and further discussions see Mattila [51, pp. 23-25]. Proofs using the
Hausdorff maximal principle instead can be found in Ziemer [91, pp. 7-8], Gariepy [22,
pp. 27-28], and Federer [24, 2.8.4-2.8.6]. On the basis of this geometric covering theorem
one can then prove, using only measure-theoretic arguments, the Vitali theorem, where
here ?" denotes the и-dimensional Lebesgue measure on Ш":
Let ? be a collection of nondegenerate closed balls in K". Suppose that ? is a
Vitali cover of a set E. Then there is a countable disjointed family iicf so
that
X"U\ [J в)=0.
The theorem is special to Lebesgue measure, although not restricted to that measure.
The key feature in the proof is that
?"(?) = 5"?"(?)· C6)
In the discussion above let us refer to the set В \ В as the "halo" around the ball B. In order
to apply these methods to a measure ? on R" other than ?" we would need to have some
control on the ?-measure of the halo around a ball in proportion to the measure of the ball
that would play the role that is played by C6) for Lebesgue measure.
To generalize to measures on an arbitrary metric space takes some caution^since
troubling details emerge. Since balls do not have unique centers the definition of В and
hence the definition of the halo for a ball В requires some thought. Also something
analogous to C6) for a general measure would be needed.
A careful analysis of the conditions needed to extend these arguments to general setting
was carried out by Morse [57] and [58]. He expressed his ideas as "halo properties",
because of the terminology just used for В \ В. An extensive account of halo properties
in relation to strong Vitali properties can be found in Hayes and Pauc [31, pp. 41-77].
See also Federer [24, pp. 141-145] for a discussion, but not using Morse's language. The
expository account in Bruckner [13, pp. 32-34] also describes other treatments of these
ideas.
Here is a brief sample to illustrate the kind of language. The methods are technical
adaptations of the classical methods.
Let X be a metric space and ? a measure on X. Let ? be a nonnegative, bounded
function defined on the class of generalized intervals. For some fixed positive number ?
and any generalized interval J we denote by J the set
J = \J{I: (I,x)ea, inS^O, 8(I)^tA(J)\
and refer to J as the ?-? enlargement of J. This enlargement is to play the same role as
the enlargement obtained in the classical version by multiplying the radius by a factor of 5.
We now need an assumption to play the role of C6). We assume, for some ? < ??, that
?G)????G).
C7)

226
B.S. Thomson
Under these assumptions any derivation basis В that is compatible with the metric on
X will have the strong Vitali property with respect to ?. For an elegant proof (and full
statement of hypotheses) see Federer [24, pp. 141-145]. Morse's original version appears
in [57]. Hayes and Pauc [31, p. 42] call ? a disentanglement function and prove this and
several other versions. These methods are known popularly as "halo methods". The most
common choice of disentanglement function is to take ?(?) as the diameter of / but other
interesting choices can be found in Federer [24, pp. 145-146] and Morse [57, p. 296].
In the spirit of abstract derivation theory we should ask whether there is anything that
can be said in the converse direction: If a basis has the strong Vitali property or does
differentiate integrals must there also be some kind of halo property as above? The key is
the requirement in the strong Vitali property that the sets extracted be disjoint. By adding a
countable "cloud" to the sets / in the pairs (/. .v) we can interfere with the Vitali property
without interfering with the differentiation properties of the basis. (See Hayes [30] and
de Guzman [28, pp. 32-33].) As we shall see, however, weak halo properties and weak
derivation properties do go in both directions.
9.3. (Q)-Pmperty
The strong Vitali property can be proved in some settings by establishing a variant which
involves a proportional cover. To illustrate some aspects of the technique let us use the
following definition.
Definition 55. We say that a derivation basis В has property (Q) with respect to an outer
measure ? on X if there is a constant q > 0 so that for every ? С X with ?(?) < oo and
every ? e B[E] there is a collection
(?\,?\)??2.?2)??3,?3) (/„.?„) e0
with {/,} disjointed so that
????\^??^??(?). C8)
Conditions of this type are satisfied by many bases. For some applications the constant q
can be made to be dependent on the choice of ? and ?. Also one usually needs some kind
of estimate in the other direction, requiring the measure of the set in the inequality C8)
not to exceed ?(?) by a small amount. In our development we normally take this from the
topological assumptions.
Let us assume that X is a metric space, that the basis В is compatible with the metric
structure (Вб), and that ? is a Borel regular outer measure on X. To illustrate the simplest
version, we assume that all sets / appearing in the pairs (/, x) e a are closed.
Then under these assumptions we can prove the following implication.

Differentiation 227
THEOREM 56. If the derivation basis В has property (Q) with respect to ?, if ?(?) < oo,
/3 e B[E], and G С X is open, then there is a a collection
(/|,?l),(/2,fc),(/3.?0,···
from ? with {/,} disjointed so that
ОС
\JliCG C9)
and
??(? HG) \\Jli\=0. D0)
PROOF. Let apply the property (Q) to the set G ? ?. Since /3(G) e B[G ? ?] there is a
finite collection
(?\,??)??2.?2)??3.?3) (??,.??,)
from /3(G) so that the {/,·} are disjointed and
?? (EDG)r)[jl,j >qp(EC\G). D1)
Our general assumptions always require that the set Ui'ii ^ ^e ?-measurable so
?(? П G) = ?? (? ? G) ? \J I, \ + ? ? (? ? G) \ |J /,-J .
Together with D1) this shows that
/Л (E П G) \ U /,¦ W A - ?)?(? П G).
Now let G2 = G \ (J"=| A· Since we are assuming that the sets /, are closed,1 it follows
that G2 is open. We apply the same arguments just used to the set ? ? G2 and obtain in
/HG2) a collection
(/,,I + l,?n, + l), (??,+?-??,+2). (/??,+3.^!,+.?) (??,2,??2)
'it is here that the closed assumption is used, to allow the pruning /JiGi) by the open set Gj. There are ways
around this in more abstract settings; see, for example. Alfsen [ I ].

228
B.S. Thomson
with the property that the {/,} are disjointed and
д (?лс2)\ Q /;?(?-?)?(???2).
\ ! =/! 1 + I
But that means that
??(???\?/,)=??(???2)\ (j /,J < A - qJ ?(? ? G).
Continuing this process inductively we would obtain
? ? (? П G) \ Q I,? ? A - «7)V(? П G).
Note that the entire sequence {I\,h, 1щ} is disjointed. From this the theorem evidently
follows since
9.4. Besicovitch-Morse property
In the period 1945-47 Besicovitch [5,6] and Morse [58] independently generalized the
classical covering ideas of Vitali as part of an investigation of derivation in more general
setting than those in which the usual Vitali arguments (of Caratheodory and Banach) could
be used.
Here is one expression of Besicovitch's original theorem.
There is a constant N depending only on ? so that if ? is a collection of
nondegenerate closed balls in the space R" with uniformly bounded diameters
and ? is the set of all centers of the balls in ? then there exist subcollections
Т\,Тг,.-ч ?'?, of ?, so that each T-, is a countable collection of disjoint balls
from ? and
EcUU*·
/=l Be^",
Because the theorem is an entirely geometric one, depending not at all on a given
measure, it can be used to obtain strong Vitali properties of all Radon measures on R"
relative to a derivation basis consisting of balls centered at points.
For recent discussions of the constants N = N(n) appearing here see Boyvalenkov [11],
Fiiredi and Loeb [25], Loeb [49,50], and Sullivan [73]. Proofs of the classical theorem
abound in the literature and should be easy to find. For example try first Mattila [51,
pp. 28-34], de Guzman [28, pp. 2-7], Ziemer [91, pp. 9-12] or Gariepy [22, pp. 30-37].

Differentiation
229
An abstract presentation of the Besicovitch-Morse and Vitali covering theorems can be
found in Bliedtner and Loeb [7].
Note, in this covering theorem, the geometric position of the associated points at the
center of the balls does play a role. A simple example (given in Mattila [51, p. 28])
illustrates that there may be no nice covering property or nice differentiation results. In that
example the associated points are taken rather far from the centers, leading to a collapse of
the properties.
We give an abstract expression of the Besicovitch-Morse ideas that can be used in many
settings:
Dehnition 57. Let ? be a covering relation on a space X. We say a has the Besicovitch-
Morse property if there is a positive constant N so that whenever ? с X and ? с a is a
covering relation on E then we may select at most N subsets ?\, /Зг, · · ·, ? ? of ? so that
?cU U '
1=1 (I.X)?fij
and each ?\ consists of countably many pairs {(/;.;,?;.;)} with the {/,.7} disjointed.
We say a derivation basis В has the Besicovitch-Morse property if the underlying
covering relation a on which В is based has that property. For example, any derivation
based on the covering relation expressed in F) in R" has this property because of the
classical Besicovitch covering theorem expressed previously. Also the net derivation bases
of Section 6.7 have the Besicovitch-Morse property.
We can very quickly relate this property to the property (Q) and hence to strong Vitali
properties relative to appropriate measures under additional hypotheses.
THEOREM 58. Let В be a derivation basis on space X and suppose that В has the
Besicovitch-Morse property. Then В has the property (Q) with respect to any outer Borel
measure ? on X.
PROOF. Let ? С X and ? e B[E], where ?(?) < ??. Let N be the constant in the
Besicovitch-Morse property. Then ? is a covering relation on ? so that we may select
at most N subsets ?\,?? ? ? of ? so that
N
Ec\J [j I D2)
; = l (/..t)e/j,
and each ?-, consists of countably many pairs {(/,.?,)} with the {/,} disjointed. Let
q = ?/?. For at least one value 1 ^ /' ^ N it must be true that
^ (/..OeA '

230
B.S. Thomson
which is exactly the property (Q) that we wish to establish. If not then, because of D2),
N I \
?(?)???????) \J A < ???(?) = ?(?)
/ = l ^ (/..t)e/i, '
which is not possible. ?
10. Weak Vitali covering properties
In an elegant and highly readable paper De Possell [64] initiated the abstract theory of
differentiation by expressing the notion of an abstract differentiation and then seeking
to find an analogue of the strong Vitali property that would supply instead the density
property. Suppose ? is a derivation basis on a space X furnished with a measure ?. The
density property asserts that for a ?-measurable set ? of finite ?-measure,
В hm <^± D3)
/-·* ?(?)
is ?-a.e. equal to 0 or 1 depending on whether ? e ? or ? ? ?.
Certainly the strong Vitali property relative to ? implies the density property merely
because
/
?(???)= / ????
and the strong Vitali property implies the differentiation theorem. A weaker version
suffices. We have already used some closely related ideas in Section 7.2.
Dehnition 59. A derivation basis В is said to have the weak Vitali property relative to
a measure ? if whenever ? с ?, ? D ? is a measurable cover of E, and ? e B[E] and
? > 0, there is a countable collection {(/,, ??)} с ? for which
m(e\ [J/,-) = ().
m(U7'\M)<?
and
?>(/,)-? (jj/,)<*. D4)

Differentiation
231
There are other expressions of this possible and the conditions can vary if the setting
changes. "Countable" can be replaced by "finite"; the second condition is often dropped
in settings where, for example, an appropriate open set G D ? э Е can be chosen and
? replaced by the pruning /3(G). If the sum ?? M(^) is not finite some other expression
will be needed. The main point to observe is that the only essential change from the strong
Vitali condition is that the generalized intervals {/,} that appear are not necessarily disjoint,
but are instead required to satisfy the condition D4) which controls the amount of overlap
allowed.
What is most remarkable is that De Possel was able to prove not merely that if the
basis B* has this property this implies the density property for B, but conversely that it is
equivalent to it. The proof is just measure-theoretic (we have used the ingredients of the
proof already in Section 7.2). De Possel proved under fairly broad assumptions that five
properties are equivalent for a derivation basis ?, including (i) the density property, (ii) that
the derivation basis differentiates the integrals of all bounded measurable functions, and
(iii) that this weak version of the Vitali covering theorem must hold.
10.1. Weak halo properties
Not too surprisingly there are weak versions of the halo methods that can be used to
establish that a basis has the weak Vitali property. It is possible to describe a halo property
that is exactly equivalent to the weak Vitali property, and hence equivalent in turn to the
density property and the property of differentiating all integrals of bounded functions.
There is also an equivalent formulation in terms of properties of the maximal function. This
should be considered one of the remarkable successes of the methods of abstract derivation
theory, that in the right setting such a variety of different techniques and methods can be
identified as equivalent.
Variants on weak Vitali and weak halo methods can be adapted to answering the
following questions for a basis that is assumed to have the density property:
A) If / is a given integrable function what condition on the basis is equivalent to the
basis differentiating the integral of /?
B) What condition on the basis is equivalent to the basis differentiating the integral of
all functions in Lp(X)?
The answers can be given as special kinds of weak Vitali properties where the overlap is
measured in a different but related way to D4).
Any discussion of these ideas would require much apparatus and time for development.
The interested reader can consult the following for accounts of weak Vitali methods,
including weak versions of halo methods: Alfsen [1], Bruckner [13], Busemann and
Feller [17], de Guzman [28], Hayes and Pauc [31], H. Kenyon and A. Morse [41],
Morse [57,58], De Possel [64], and Thomson [79]. The bibliographies in [13,28,41] also
contain many further sources.
11. Derivation bases in №
The study of derivatives on the real line is a huge one with a large and deep history. This
might be a bit of a surprise to the novice reader who expects that a typical undergraduate

232
B.S. Thomson
and graduate education has exposed pretty well all that is to be known about derivatives.
The three monographs Saks [72], Bruckner [12], and Garg [26] should give a good idea of
the nature of the problems and methods and accounts of many of the highlights.
The process of differentiation on the real line can be studied from a more general point
of view and, in some aspects, allows a clarification of ideas. In this section we outline some
of the ideas that be developed.
11.1. The ordinary derivative
The ordinary derivative
fix) = iim /<>>-/(*) D5)
_v->-.t У — X
can be captured by any of several derivation bases. Let us define the following covering
relations:
A) ?, = {([?,/>[,?): a < b and ? = ? ?? ? = b).
B) a2 = {([a, b[, ?): a < b and ? ?? ? ?? b).
C) ?3 = {([?, b[, ?): ? < b and a < ? < b).
Note that the use of the half-open interval [a, b) rather than [a, b] does not impose any
lack of symmetry since the points ? for the pairs ([a,fo),?) are, in all cases, handled
symmetrically. The choice is made only to allow disjointed collections of intervals to be
more general.
Any one of ? ?, «4, or ат, can be used to express the ordinary derivative; slight differences
arise in application. Let us use the first covering relation a\ and define V to be the
derivation basis generated by a\ using the real metric (as in Section 6.2). It can be checked
that V satisfies all of the properties (B| )-(Вб)·
11.1.1. The Riemann integral. As a preliminary to studying V consider the following
basis that describes the ordinary derivative in a more elementary fashion and lacks some of
the properties of T>.
By ?? we mean the collection of all ? с a \ with the property that for some ? > 0 all pairs
([a, b[, ?) e a\ for which b - a <S necessarily belong to ?. The basis 11 satisfies (B|),
(B2), and (B3) but does not satisfy (B4), (B5), or (B6). It can be checked that K* = V*
and that П** = V.
This basis allows an expression of the ordinary derivative, the Riemann integral, and
the Peano-Jordan content. If we pass instead to the basis V which does satisfy all of
the properties (В|)-(Вб) we shall have expressions of the ordinary derivative, but also
more powerful measure-theoretic tools. The key difference can be considered to be the
property (B4). Over the years there have developed many slogans by which one can express
why the Lebesgue theory of measure and integration surpasses the Riemann theory. Within
the admittedly narrow setting of abstract derivation theory we might claim it rests on the
property (B4) that is enjoyed by V but not by H.

Differentiation
233
11.1.2. Measure properties. For any function h:a\ —* ? we define the two outer
measures
h*(E) = V(h,T>[E]) and /?*(?)= V(h, V*[E]) (?cR).
In general /2* ^ h*. As we have seen several times already, the identity /г* = И* is
considered as a strong Vitali property.
Let ?? denote the function defined on ? ? by
Ax([a,b[^) = b-a.
Then ??* and ??* are identical and are equal to the Lebesgue outer measure ?. This
is a consequence of the classical Vitali covering theorem. It also can be obtained by
establishing directly the property (Q) of Section 9.3. (Had we used instead ?(??, Tl[E])
we would capture the Peano-Jordan content of the set E; the expression ?(??, TV [?])
using the dual is, however, the Lebesgue outer measure of E.)
An interesting variant is obtained by considering
k*(E) = ?((???.?>[?]) and
???) = ?((???,?*[?])
for 0 < s ? 1. The case s = 1 we have seen. For 0 < s < 1 the two outer measures ?,? and
?? play a role in many studies. The latter (the smaller of the two) is closely related to the
Hausdorff s-dimensional measure and the former to the s -dimensional packing measure
of Tricot [89] (see also Taylor and Tricot [74,75]). An account, together with applications
to the study of Lipschitz numbers (as first defined by Besicovitch [4]), can be found in
Thomson [76] (but using the slightly modified basis that would arise from аз above).
What is the situation for the variational measures associated with a function /: R -> R?
As above, write
?/([?,?,?) = /?)-/(?).
It is easy to show that the variation of the function / on an interval [a, b] (denoted as usual
as V(/, [a, b])) can be expressed by the variation relative to the derivation basis as
V(Af,T>([a,b[)) = V(f,[a.b]).
More generally there are two outer measures which carry the information as to the total
variation of/:
kf(E) = V(Af,T>[E]) and
kf(E) = V(Af,T>*[E]).
If / is continuous and has locally bounded variation then it can be proved that these are
identical and are the Lebesgue-Stieltjes measure associated with the total variation of /.

234
B.S. Thomson
(This property, too, can be obtained from the property (Q) or from conventional Vitali
arguments.) Thus again, this can be considered a strong Vitali property. The differentiation
theory for functions of bounded variation can be based on the identity of ? -? and kf.
For functions / that are not locally of bounded variation there is now an extensive
literature studying these measures. We can anticipate that the identity of ?' and ?/ on
all subsets of a set А с R (i.e., a strong Vitali property) should be intimately related to the
differentiation properties of / on A. Indeed this is the case. Functions with this property
were originally investigated by Saks and Denjoy in the 1920s and 1930s but by much
different techniques. An account of their ideas can be found in Saks [72] expressed in the
language of VB* and VBG* functions. It is now recognized that the most fertile expression
of these notions is in the language of the variational measures instead. A sampling of some
of the theory can be found in B. Bongiorno et al. [8,10], Ene [21], Pfeffer [62,63], and
Thomson [78,79,81-84,86,87]. Note especially that these measures form an interesting
subclass of the ?-finite measures on R, a subclass that has important implications in the
study of derivatives and nonabsolutely convergent integrals on R.
One other interesting theme that should not be overlooked is the relation between the
classes of AC* and ACG* functions and the absolute continuity of the measures ?-1 and
?/ with respect to Lebesgue measure.
11.1.3. Differentiation of integrals. Let ? be any Lebesgue-Stieltjes measure on R (i.e.,
? is a locally finite Borel measure). If / is locally integrable with respect to ? then
}']?
1
?[?,?]
for ?-almost every jr. There are numerous accounts of this in the literature. Federer [24,
pp. 163-168] has a clear exposition, obtained from some very general and useful principles.
The result also follows in the setting of Section 7 above. The key in most presentations
is that there is a strong Vitali property associated with all the measures that arise in the
discussion.
It should be remarked that the local finiteness of ? (that is, the fact that ? is finite on all
intervals) cannot be replaced in general by ?- finiteness.
For a certain class of ?-finite measures there is a strong Vitali property and hence there
would be a differentiation theorem. See Thomson [86] for an account of the nature of these
measures.
11.1.4. Integration of derivatives. The recovery of a function from knowledge of its
derivative is one of the aspects of the fundamental theorem of calculus. One version of this
can be obtained as a simple application of our variational methods. If F has a derivative at
every point of a measurable set ? then the methods of Section 5.7 can be used to prove that
kF(E) = V(AF. ?>[?])= V(F'Ax.T>[E]) = j \F'(x)\dx.

Differentiation
235
From this we see that we can recover the variational measure associated with F from the
derivative. The classical expression of a special case of this can be found in many analysis
books in the form
fh
V(F,[a,b])= / \F'(x)\dx
Ja
for an everywhere differentiable function F (where V (F, [a, b]) denotes the total variation
of F).
On the basis of this connection between the variation of a function F and the values of
the integral of | F'\ one can, in many cases, recover F from F'. This is essentially the basic
ingredient in all arguments in Lebesgue's measure theory.
A problem arises for everywhere differentiable functions F for which
h
\F'(x)\dx = oo
(and such functions do exist). For such functions the measure-theoretic arguments can not
provide a method to recover F from the values of F'. e Historically this led to a number of
different approaches. The simplest in concept (although arguably not a "method" of
inversion of derivatives) was given by Perron using collections of major and minor functions.
Since the collection of major and minor functions cannot themselves be constructed any
more easily than the function F itself, this concept received harsh criticism from Denjoy,
who was the proponent of another method. By analyzing the structure of differentiable
functions Denjoy was able to give a countable, but transfinite, series of extensions of the
Lebesgue integral that can recover the primitive F (up to a constant) from its derivative F'.
In our present context of derivation bases there is a simpler presentation of a method
(which cannot be considered constructive in the sense in which Denjoy's is). One notes
first a very special and elementary covering property of the derivation basis V:
COUSIN'S LEMMA. Let ? e T>[a, b). Then there are points
a = xq < ? ? < x2 < ¦ ¦ ¦ < x„ = b
and points ? e [.v,_i, лг,-[ so that eachpair ([*,-_ |..r,-[, ?,) e ?.
We consider this a "covering theorem", although of a considerably more elementary nature
than the Vitali covering theorem.
Expressed more popularly, elements in V have "partitions" of every interval [a,b).
Cousin's lemma can be made into a convenient method of proving compactness statements
in real analysis (as we do in [88, Section 4.5.3]). Indeed it was periodically rediscovered
as a method (see the account in Thomson [76]). More interesting for our purposes is the
fact that it allows for a Riemann type integral to be defined (relative to the basis T>) that
inverts derivatives. The idea is nearly trivial: If F' = f everywhere then for every ? > 0
the collection
L
? = {([*,?[,?): \F {y)-F (?) - f {$){y-x)\ <?{?-?), ? = ? or ? = у}

236
B.S. Thomson
belongs to V. Using now the partition using elements from ? promised in Cousin's lemma,
F{b) - F{a) = ? F(Xi) - F(xi-i) = ]T/(?)(*, - *,·_,) + R
;' = l ; = l
where certainly
и
\R\ < 2^?(дг/ — дг/_|) = ?? -?).
i = \
Thus if one defines a Riemann-type integral as a limit of Riemann sums
II
i = \
taken in the sense of the filter V then that integral will have the feature that it generalizes
the ordinary Riemann integral and satisfies
h
F'(x)d.x = ??)- F(a)
for every differentiable function F. (Had we used instead the basis H in place of V then
we would have defined exactly the classical Riemann integral; the advantage of properties
(В4)-(Вб) becomes clear.)
The integral was discovered and exploited independently by R. Henstock and J. Kurzweil
around 1960. There is a relatively large literature devoted to the study of this integral and
to a number of variants. In settings where a derivation basis can be constructed that has a
property analogous to the Cousin lemma such an integral can be constructed. Mostly this
is not the case.
Pedagogical issues abound also around this integral. It is curious indeed that an
extremely general integral on the real line should be so easily defined (in the "natural"
Riemann manner) and yet have been overlooked until half a century after Lebesgue. Is
it easier to learn than the Lebesgue theory? No, ultimately one develops the measure-
theoretic tools needed in any case. And measure theory is more important as a general
subject than this particular method, which applies only in very special situations.
Still it is an interesting chapter in the study of derivation bases and it has led to many
techniques and problems of interest on their own. Elementary accounts of the basic integral
can be found in McLeod [52] and Gordon [27]. Henstock has written a number of accounts
himself: see [32,35,36]. The broader perspective in W. Pfeffer [62] is recommended to the
serious reader. The survey by W. Pfeffer on the Henstock-Kurzweil integral in this volume
should be consulted first.
,

Differentiation
237
11.2. The symmetric derivative
The ordinary derivative can be generalized in a simple and natural manner by replacing the
familiar expression D5) by one that demands instead:
t^o It
Certainly if the ordinary derivative /'(x) exists so too does fs (x) and with the same value.
The fact that symmetric derivatives arise in certain problems in trigonometric series should
guarantee their study to some extent. From the point of view of measure theory we might
take the viewpoint of Khintchine [42]: He proved that for a measurable function / the
set of points ? where fs(x) exists and f'(x) does not is a set of Lebesgue measure zero.
For that reason he felt that the symmetric derivative could be discounted as an object of
study. Since pointwise considerations (as opposed to a.e. pointwise) do play a role in many
problems this is not a completely fair comment.
The symmetric derivation process has been studied in many aspects. From the point of
view of Vitali covering properties there is nothing new to say, but there are some other
interesting covering properties. An account of many aspects and a large bibliography can
be found in Thomson [84].
11.3. The sharp derivative
The ordinary derivative can be narrowed by replacing D5) by one that demands more of
the derivative:
f*(x)= Hm /?~/(:). D7)
This is known as the sharp or unstraddled derivative (the latter because the interval [y, z]
shrinks to ? but need not straddle or contain x). Certainly when f#(x) exists so too does
f'(x) and with the same value. For a differentiable function / to have a sharp derivative at
a point requires that /' is continuous at x.
The derivation basis that expresses this can be defined by using as a general covering
relation
a# = {([a,b[,?): a.fe^eBanda <b\.
Again note that the use of the half-open interval [a, b[ rather than [a. b] does not impose
any lack of symmetry since the points ? for the pairs ([a, b[, ?) are, in all cases, handled
symmetrically.
Define T># to be the derivation basis generated by <*# using the real metric (as in
Section 6.2). This basis has some interesting and curious properties. While it expresses
a very narrow kind of differentiation, if a Riemann-type integral is defined using it in

238
B.S. Thomson
a natural way the resulting integral, which we would expect to be narrower than the
Henstock-Kurzweil integral discussed above, is precisely the Lebesgue integral on the
line. This was exploited by McShane [53] in a monograph and developed later by him for
other purposes in McShane [54]. His expository account in [54] should also be consulted.
A number of papers have pursued these ideas in various settings.
11.4. Density bases on R
The basis T># does not have the density property with respect to Lebesgue measure. That is
to say, using the limit in D7),
hm D8)
y.z—x z- у
need not be a.e. equal to 1 for points in ? and to 0 for points outside E. Certainly the
ordinary basis V does have the density property (it differentiates all integrals) as would
any "straddled" basis.
A natural question arises as to what natural unstraddled bases on the real line should
have the density property with respect to Lebesgue measure.
Let us formulate the problem more narrowly. Let [ak, bk} be a sequence of intervals
converging to 0 in the sense that
О <ак+\ <bk+\ <ak <bk <¦¦¦
and ak —* 0. At every point we consider the translation invariant version of this limit,
namely
[x +ak, ? + bk] -> x.
(An expression in terms of derivation bases can be made but would be more cumbersome.)
What conditions on the sequences {ak} and {bk} would require that the corresponding
derivation basis has the density property with respect to Lebesgue measure. The answer
was given by Aversa and Preiss [2] as an application of one of the basic halo properties in
Hayes and Pauc [31, Theorem 2.2]: If
bk+\
hm sup < oo
k^oo bk - ak
then the derivation basis has the density property with respect to Lebesgue measure. For
other interesting variants on this same theme see [2].
11.5. The app roxima te de riva tive
This section can continue to spawn further subsections if we are prepared to account for the
full variety of derivation processes studied on the real line. For example, the reader should

Differentiation
239
by now have little trouble in describing a derivation basis A that captures the approximate
derivative and in describing its dual A*. It will have most of the Vitali properties of the
ordinary basis. More interesting is to ask for which further class of functions / will
there be the strong Vitali relations V(Af.A*[E]) = V(Af.A[E]). This seems not yet
to have been investigated, although it should not require entirely new methods. There are
other covering theorems; the Cousin lemma has an analogue in this setting (see the proof
in [76]). Consequently there is a Riemann-type integral that can be defined which inverts
approximate derivatives.
Let us terminate here, however, and move on to higher dimensions.
12. Derivation bases in R"
12.1. The cube basis
In the space R" let a denote the collection of all pairs (/, ?) where / is an «-dimensional
cube and ? e /. The derivation basis T>(, generated from a using the ordinary euclidean
metric is called the "cube basis" and differentiation relative to this basis is usually called
"ordinary differentiation".
Let ?„ denote Lebesgue measure on R". The basis cube basis V„ has the strong Vitali
property relative to ?„. There are a number of ways to prove this. The methods of Section 9
can be used, and indeed some of the details of the method are given already in that section.
It follows then that the differentiation theorem holds for this. Namely for every locally
Lebesgue integrable function /, the set of points ? e R" at which
V0 Jim —— f \f(x) - f{y)\dX„(y) = 0, D9)
that is the Lebesgue set of /, includes ?,,-a.e. point of R". In particular, for ?,,-almost
every point ? e R"
V0 Jim —— f f(y)dX„(y) = f(x). E0)
Thus, in many ways, this seems to be the most natural and important derivation basis on
R". A variant on this that would be as natural and have these same properties is to take the
balls instead: Thus a consists of all pairs (/, ?) where / = B(y, r) is a ball of radius r and
? e I (which requires that |лг — y\ < r). Note that in both cases the associated point ? in
the pairs (/, x) is in the set / but not required to be in any special geometric position in
that set.
Consider, however, the question as to whether this same result holds true if ?„ is replaced
by any locally finite Borel measure ? on R". An easy example shows that this is not the
case. Let L be the line у = ? in R2 and let ?(?) be the linear measure of ? П L. Then there
is no strong Vitali property for V„ relative to ? and there is no differentiation theorem. An
examination of this example shows that the position of the point ? e / for the pairs (?, ?)
seems to be a critical issue for the strong Vitali property.

240
B.S. Thomson
12.2. The basis of centered balls
The issue raised in the preceding section suggests requiring a natural geometric condition
on the position of the point ? e I for the pairs (/, ?) e a. Let us consider the basis obtained
from centered balls. Now a denotes all pairs (/. x) where / = B(x, r) is a ball of radius r
and x, the associated point, is the center of the ball /.
The basis generated from a from the euclidean metric on R" can be proved to have the
Besicovitch-Morse property. Consequently this basis has the strong Vitali property relative
to any locally finite Borel measure and the analogous statements in E0) and E0) hold for
any such measure.
Thus, perhaps instead of the cube basis, this seems to be the most natural and important
derivation basis on R". Certainly it is used as the main object in many investigations
and many accounts can be found in the literature. Consult, for example, Gariepy and
Evans [22], Mattila [51 J, and Ziemer [91] for full developments with proofs of the
geometric covering properties and differentiation properties possessed by this basis and
the role it plays in geometric measure theory and many important questions in analysis.
As we have remarked, the fact that the associated point ? in the pair (/, ?) here is the
center plays a crucial role: The Besicovitch-Morse property fails if ? is merely taken as
any point in the ball /. But there is room for maneuver. Provided the point is not too far
from the center, the same property can be proved. Also the balls may be replaced by cubes
provided the points are, once again, not too far from the centers of the cubes.
12.3. The interval basis
Having seen the success of the cube basis for the differentiation of Lebesgue integrals in
Section 12.1 one might expect to be able to generalize to a basis consisting of intervals,
rather than cubes. Take a to consist of pairs (?, ?) where / is an interval and ? e /.
The basis Vs generated from a is known as the "interval basis" and the corresponding
differentiation process, since it is clearly more demanding than the cube basis, is known as
"strong differentiation".
The strong differentiation fails to have the strong Vitali property with respect to
Lebesgue measure and the Lebesgue differentiation property fails. The first indication
of this appears in Caratheodory [18] where he presents a counterexample of H. Bohr.
A simplified version of that appears in de Guzman [28, pp. 92-95] where he is able to
use that construction to show both that the strong Vitali property fails and that the interval
basis does not differentiate the integrals of all integrable functions.
It fails somewhat dramatically. Saks [70,71] showed that for all functions / in the
Banach space L \ (R") excepting only a first category set,
P.Hmsup ffdk"=oo E1)
at every point ? e R". This is sometimes known as the "Saks rarity theorem" and is an
early version of what has become a large industry, showing that some curious behaviour of

Differentiation
241
functions holds "typically" or "generically" in a certain space of functions. Full proofs can
also be found in de Guzman [28, pp. 96-99] and Hayes and Pauc [31, pp. 107-109].
If the strong differentiation basis T>s fails to differentiate the integrals of all Lebesgue
integrable functions, what positive properties does it have? Even if the typical integrable
function satisfies E1), for what class of functions does it differentiate their integrals?
The first (and merely partial) answer to this question is obtained by showing that T>s has
the weak Vitali property with respect to Lebesgue measure. This can be proved by weak
halo methods. See Hayes and Pauc [31, pp. 78-79]. Consequently T>s differentiates the
integrals of all bounded measurable functions. (Recall that that this differentiation property
is equivalent to the density property and to the weak Vitali property and to the weak halo
property.)
But this is not the best possible. A remarkable theorem of Jessen, Marcinkiewicz and
Zygmund [40] describes precisely (in a sense) what that class of functions is. It is larger
than the class of bounded functions, but because of the Saks rarity theorem must form a
first category subset of L \ (W). This is a truly wonderful story to recount but perhaps it
is best left to others. There is much to relate and deep connections with other ideas in
analysis. See Hayes and Pauc [31, pp. 83-94] and Guzman [28, pp. 50-51] for proofs of
the theorem.
12.4. Rectangle basis
While the cube basis has the best properties with respect to Lebesgue measure, we have
seen that the interval basis still has very strong and useful properties. What happens, say in
R2, if we replace intervals by rectangles, that is if we no longer require of our rectangles
that the sides be parallel to the axes?
The resulting derivation basis does not have the density property with respect to
Lebesgue measure and so all reasonable Vitali or halo properties fail. An interesting
account of the history and the properties of this basis can be found in Guzman [28,
Chapter 5] along with a proof that the density property fails. There is a brief discussion
also in Hayes and Pauc [31, pp. 104-106]. Other proofs can be found in Busemann and
Feller [17] and Papoulis [61].
For our purposes we should be content with an observation that arises from
considerations of the three bases here (the cube basis, interval basis, and rectangle basis). From
a metric or measure-theoretic viewpoint they seem hardly distinguishable. Had we been
inclined to think that the covering properties and derivation properties of a derivation
basis should be approachable solely by measure-theoretic and topological arguments, these
examples should serve to disabuse us of the idea. Certainly there are geometric arguments
needed at many stages in our theory.
For example, with an appropriate restriction, a derivation basis constructed from
rectangles in different orientations could have the strong Vitali property relative to Lebesgue
measure even though the full interval and rectangle bases do not. For example let the class
of rectangles be restricted to one in which every pair of rectangles in the class has the
property that one of them includes a translate of the other; then the basis constructed from such
a family is an example of what Morse calls a "hive" and it does have the strong Vitali
property relative to Lebesgue measure. See Morse [57] and Hayes and Pauc [31, pp. 111-112].

242
B.S. Thomson
12.5. Regular bases
Intermediate between the cube basis and the interval basis are regular bases that are
designed to supply the strong Vitali property with respect to Lebesgue measure in R". The
key is the shape of the intervals compared to the cubes: Long narrow intervals interfere
with strong Vitali property.
For a set ? с R" define its index of regularity as
р(Н) = %?клп
where the supremum is taken over all cubes С that contain the set ?. This measures how far
? is away from being close to cube shaped, values close to zero indicating the unpleasant
extreme.
We define a variant on the interval basis as follows. For a take all pairs (/, x) where / is
an interval and ? & I. The basis T>,- is defined so as to contain all ? с a with the following
property. For each ? e R" there is a S > 0 and ? > 0 so that ? contains all pairs (/, x) e a
for which diameter(/) < ? and p(I) > <5. (The numbers S and ? can depend on x.)
This basis has the strong Vitali property. One need not use intervals here; any measurable
sets with the same properties would work. The proof just uses a halo argument. See de
Guzman [28, pp. 25-27] for an elementary and direct proof.
For a generalized version of a regular basis and a proof that such bases have strong Vitali
properties see Comfort and Gordon [19].
12.6. Star bases
Generally in any space, such as R" here, we would select a derivation basis to study that
arises from some compelling geometrical consideration. Certainly the bases above, the
cube basis, centered ball basis, and interval basis are entirely natural objects of study.
But from the general viewpoint of abstract derivation theory we likely feel motivated
to ask a problem in a converse direction. Exactly what derivation bases would have some
specified property. Indeed let us ask what bases on R" would have the strong Vitali property
relative to all locally finite Borel measures?
We have seen that the centered ball basis does. The cube basis does not, although if we
modified it by requiring that the associated points are at the centers of the cubes then it
would have this property.
This question was considered by Besicovitch [5] and by Morse [58]. The answer is
suggested by the examples in the preceding paragraph. There must be some geometric
relation between the associated point ? and its set / in the pairs (/, x). A basis is said
to be a star basis if the appropriate condition holds, and for star bases the strong Vitali
property will hold for all locally finite Borel measures. A brief expository account can be
found in Bruckner [13, pp. 12-14] and a full account with proofs in Hayes and Pauc [31,
pp. 114-119].

Differentiation
243
13. De La Vallee Poussin theorem
De La Vallee Poussin obtained a version of the Lebesgue differentiation theorem that
clarifies considerably the nature of the differentiation result for functions of bounded
variation. If F is a function of bounded variation on [a, b] then most students learn that
I
b
F'(t)\dt ? V(F,[a,b])
and that, if in addition F is absolutely continuous, then
J \F'(t)\dt= V(F,[a,b]).
Ja
Elementary textbooks often do not give the best version of this collection of ideas. Let
?? be the signed Borel measure associated with F (i.e., the Lebesgue-Stieltjes signed
measure), let f^and Е-ж denote the set of points at which F'(x) = ooand F'(x) = -oo.
Then, assuming F is continuous and has bounded variation,
= /''
Jb
??(?)= / ?'A)??+??(????)+??(??)?-^)
Jb
for all Borel sets B. A proof appears in [14, pp. 301-302]. One sees that the singular part
of ?? is concentrated on the sets where the derivative exists but is infinite.
How much of this can be expressed in a general setting for a signed measure ? and a
measure ?? Are there general hypotheses that would lead to the relation
?(?)= ? /??+??(????)+??(???-?)
Jb
where
f{x) = В hm ——
?^? ?(?)
and ?oo and ?-? are the sets of points at which this limit exists infinitely as oo and —oo?
Bruckner [13. p. 39] illustrates that this is not true in any generality by showing that for
most natural bases В in E2 that it is possible that ? is singular and yet
? or ¦ ?·?(/) or ?(/)
0 = В hm ??? < В hm sup = oo
/-.? ?(?) /_/ ?(?)
on the set where ? is concentrated. Thus there is no infinite derivative anywhere.
It is not known if there is any general version possible. Saks [72, pp. 155-156] shows
that for a net structure basis in R" the analogue of the De La Vallee Poussin theorem does
hold. See also the proof in [14, pp. 352-353].

244
B.S. Thomson
14. The Radon-Nikodym theorem
In most applications of measure theory one can use the Radon-Nikodym theorem to
express the relationship between a pair of measures ? and ? by
?(?)= ? fdti.
This is true in many measure spaces (?,?,?) for all measures ? that are absolutely
continuous with respect to ? and not too large. Commonly the function / whose existence
is claimed by the theorem is called a Radon-Nikodym derivative of ? with respect to ?,
although it is not obtained as a derivative in the sense of any limit.
From the point of view of the abstract theory it becomes natural to ask whether any
derivation basis В can be imposed on the measure space so that the function / can be
expressed as a genuine pointwise limit,
В lim = f(x)
?-a.e. This question may not be natural from all points of view. In most applications of
a measure space we are more likely to construct a derivation basis from the geometry to
hand and then ask for its properties, rather than ask for the existence of a derivation basis
that supplies some need.
What we require, apparently, is that the derivation basis we want have appropriate strong
Vitali properties. Net structures always supply the strong Vitali properties needed and so
whenever a net structure can be imposed this solves our problem. Recall, from Section 6.7,
that if the measure space is "separable" in the sense given there then a net structure does
exist.
More generally any space that has a linear lifting also provides a derivation basis with
adequate strong Vitali properties. A collection of necessary and sufficient conditions are
given in Kolzow [43] for a measure space to permit this. Without going into details one can
summarize popularly by saying that whenever the Radon-Nikodym theorem is available
so too is a derivation basis that supplies that Radon-Nikodym derivative as a genuine
derivative.
For all the ingredients of a readable and accessible proof of the connection between
linear liftings and bases with the strong Vitali property see the account in Bruckner [ 13,
pp. 35-37]. This material is also developed in [14, pp. 355-362].
For a different perspective on the Radon-Nikodym derivative, this time appearing as a
stochastic derivative, see Hayes and Pauc [31. pp. 177-178].
15. Some further remarks
If there is not a unified theory of abstract derivation, there is certainly a body of knowledge
that deserves that name. The concepts and techniques of the subject (Vitali covering

Differentiation
245
properties, Besicovitch-Morse covering properties, halo properties, Hardy-Littlewood
maximal operator methods, variational measures, liftings, etc.) are part of the general
culture of analysis. The survey in this chapter has hardly served as a tour, merely an
introduction to a selected variety of themes.
The story of the Hardy-Littlewood methods will have to be told by someone else.
There is also a connection between martingale methods and differentiation that should
be explored elsewhere. The monograph of Hayes and Pauc [31 ] is half of it devoted to that
subject. We should remark that martingale methods also seem to be able to replace Vitali
methods in a number of differentiation results. Morayne and Solecki [55] and Morayne [56]
have proved the Lebesgue differentiation theorem and the Jessen, Marcinkiewicz and
Zygmund theorem by using powerful martingale methods in place of appeals to covering
theorems. Bliedtner and Loeb [7] develop connections between martingale convergence
theorems, covering properties, and differentiation of measures.
The high level of technical apparatus needed to develop some of the ideas has been
obscured by our simple presentation. The interested reader will have to go more seriously
into the details of Hayes and Pauc [31] and de Guzman [28, pp. 2-7] to get a deeper view
of the subject. Probably the monograph Ken yon and Morse [41] should be mastered, but
only by the most devoted.
References
[1] E.M. Alfsen, Some covering theorems of the Vitali type. Math. Ann. 159 A965), 203-216.
[2] V. Aversa and D. Preiss, Hearts Density- Theorem. Real Anal. Exchange 13 A) A987-88), 28-32.
[3] S. Banach, Surle theoreme de Vitali, Fund. Math. 5 A924), 130-136.
[4] A.S. Besicovitch, On linear points of fractional dimension. Math. Ann. 101 A929), 161-193.
[5] A.S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions.
Proc. Cambridge Philos. Soc. 41 A945). 103-110.
[6] A.S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions
11, Proc. Cambridge Philos. Soc. 42 A946). 1-10.
[7] J. Bliedtner and P. Loeb, A reduction technique for limit theorems in analysis and probability theory.
Ark. Mat. 30A992), 25^43.
[8] B. Bongiorno, W.F. Pfeffer and B.S. Thomson. A full descriptive definition of the Gage integral. Canadian
Math. Bull. 39 D) A996), 390-401.
[9] B. Bongiorno, L. Di Piazza and V. Skvortsov. A new full descriptive characterization of Denjoy-Perron
integral. Real Anal. Exchange 21 A995/96). 656-663.
[10] B. Bongiorno, L. Di Piazza and D. Preiss, Infinite variation and derivatives in K", J. Math. Anal. Appl. 224
A998), 22-33.
[11] P. Boyvalenkov, On the Besicovitch constant in small dimensions. C. R. Acad. Bulgare Sci. 50 A997).
17-18.
[12] A.M. Bruckner, Differentiation of Real Functions. Lecture Notes in Math.. Springer, Berlin A978). [Second
edn., CRM Monograph Series, Vol. 5, Amer. Math. Soc.. Providence. RI A994).]
[13] A.M. Bruckner, Differentiation of integrals. Amer. Math. Monthly 78 A971) (Slaught Memorial paper
No. 12).
[14] A.M. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall A997).
[15] A.M. Bruckner and M. Rosenfeld, On topologizing measure spaces via differentiation bases. Ann. Scuola
Norm. Sup. Pisa 23 A969), 243-258.
[16] Z. Buczolich and W.F. Pfeffer, Variations of additive functions. Czechoslovak Math. J., in press.
[17] H. Busemann and W. Feller, Ziir Differentiation der Lebesgueschen Integrate. Fund. Math. A934).

246
B.S. Thomson
[18] С. Caratheodory, Vorlesungen iiber reelle Funktionen. Leipzig A927).
[19] W.W. Comfort and H. Gordon, Vitali's theorem for invariant measures. Trans. Amer. Math. Soc. 99 A961)
83-90.
[20] G.A. Edgar, Integral, Probability; and Fractal Measures. Springer. New York A998).
[21] V. Ene, Real Functions - Current Topics. Lecture Notes in Math.. Vol. 1603. Springer, Berlin A995).
[22] L.E. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press. Boca Raton.
FLA992).
[23] K.J. Falconer, The Geometry of Fractal Sets. Cambridge University Press. Cambridge A985).
[24] H. Federer, Geometric Measure Theory. Springer. Berlin A969).
[25] Z. Furedi and PA. Loeb. On the best constants for the Besicovitch covering theorem. Proc. Amer. Math.
Soc. 121A994), 1063-1073.
[26] K. Garg, Theory of Differentiation. CRM Monograph Series, Vol. 24, Amer. Math. Soc.. Providence, Rl
A998).
[27] R.A. Gordon, The Integrals of Lebesgue. Denjoy. Perron and Henstock. Grad. Studies in Math.. Vol. 4,
Amer. Math. Soc., Providence, Rl A994).
[28] M. de Guzman, Differentiation of Integrals in K". Lecture Notes in Math., Vol. 481, Springer. Berlin A975).
[29] M. de Guzman. A general form of the Vitali theorem. Colloq. Math. 34 A975), 69-72.
[30] C.A. Hayes, Differentiation with respect to ?-pseudo strong blankets and related problems. Proc. Amer.
Math. Soc. 3 A952). 283-296.
[31] C.A. Hayes and C.Y. Pauc. Derivation and Martingales. Springer. Berlin A970).
[32] R. Henstock, A Riemann type integral of Lebesgue power. Canad. J. Math. 20 A968). 79-87.
[33] R. Henstock, Generalized integrals of vector-valued functions. Proc. London Math. Soc. C) 19 A969),
317-344.
[34] R. Henstock, Generalized Riemann integration and an intrinsic topology, Canad. J. Math. 32 A980), 395-
413.
[35] R. Henstock, Lectures on the Theory of Integration. World Scientific, Singapore A988).
[36] R. Henstock, The General Theory of Integration. The Clarendon Press, Oxford University Press. New York
A991).
[37] E.J. Howard, Analyticity of almost everywhere differentiable functions, Proc. Amer. Math. Soc. 110 A990),
745-753.
[38] K. Iseki, On relative derivation of additive set-functions, Proc. Japan Acad. 36 A960). 630-635.
[39] K. Iseki, On decomposition theorems of the Vallee-Poussin type in the geometry of parametric cur\es, Proc.
Japan Acad. 37 A961). 169-174.
[40] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals. Fund. Math.
25A935). 217-234.
[41] H. Kenyon and A. Morse, Web Derivatives, Amer. Math. Soc. Memoirs A973).
[42] A. Khintchine, Recherches sur la structure des functions mesurables. Fund. Math. 9 A927). 212-279.
[43] D. Kolzow, Differentiation von Massen, Lecture Notes in Math., Vol. 65. Springer, Berlin A968).
[44] J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter,
Czechoslovak Math. J. 82 A957), 418^*46.
[45] J. Kurzweil and J. Jamik. Differentiability and integrability in ? dimensions with respect to a-regular
intervals, Results Math. 21 A992). 138-151.
[46] S. Leader, What is a differential? A new answer from the generalized Riemann integral,
Amer. Math. Monthly 93 A986), 348-356.
[47] S. Leader, A concept of differential based on variational equivalence under generalized Riemann
integration, Real Anal. Exchange 12 A986-87). 144-175.
[48] S. Leader, Integral and Differential on the Real Line: An Innovative Unified Theory, to appear.
[49] PA. Loeb, An optimization of the Besicovitch covering. Proc. Amer. Math. Soc. 118 A993), 715-716.
[50] PA. Loeb, On the Besicovitch covering theorem, SUT J. Math. 25 A989). 51-55.
[51] P. Mattila, Geometry of Sets and Measures in Euclidean Space, Cambridge University Press. Cambridge
A995).
[52] R.M. McLeod, The Generalized Riemann Integral. Math. Assoc. Amer.. Washington, DC A980).
[53] E.J. McShane, A Riemann-type integral that includes Lebesgue-Stieltjes. Bochner and Stochastic integrals.
Amer. Math. Memoirs. No. 88, Providence. Rl A969).

Differentiation
247
[54] E.J. McShane, A unified theory of integration, Amer. Math. Monthly 80 A973). 349-359.
[55] M. Morayne and S. Solecki. Martingale proof of the existence ofLebesgue points. Real Anal. Exchange 15
A989/90). 401^406.
[56] M. Morayne. to appear.
[57] A.P. Morse, A theory of covering and differentiation. Trans. Amer. Math. Soc. 55 A944), 205-235.
[58] A.P. Morse, Perfect blankets. Trans. Amer. Math. Soc. 61 A947). 418^*42.
[59] M.E. Munroe, Introduction to Measure and Integration. Addison-Wesley A953).
[60] A. Novikov and W.F. Pfeffer, An invariant Riemann type integral defined by figures, Proc. Amer. Math. Soc.
120A994). 849-853.
[61] A. Papoulis, On the density theorem. Proc. Amer. Math. Soc. 2 A951). 709-717.
[62] W.F. Pfeffer, The Riemann Approach to Integration, Cambridge Univ. Press. New York A993).
[63] W.F. Pfeffer. On variations of functions of one real variable. Comment. Math. Univ. Carolin. 38 A) A997).
61-71.
[64] R. de Possel, Surla derivation abstraite des functions d'ensembles A936).
[65] O. Nikod^m. Sur la mesure des ensembles plans dont tons les points sont rectilineairement accessible.
Fund. Math. 10A027) 116-168.
[66] D. Preiss, Gaussian measures and covering theorems. Comment. Math. Univ. Carolin. 20 A979). 95-99.
[67] Ch. de la Vallee Poussin, Sur Vintegrate de Lebesgue. Trans. Amer. Math. Soc. 16 A915), 435-501.
[68] Ch. de la Vallee Poussin, Integrates de Lebesgue. Functions d'Ensemble, Classes de Baire, Gauthier-Villars.
Paris A934).
[69] C.A. Rogers. Hausdorff Measures. Cambridge University Press. Cambridge A970).
[701 S. Saks, Remark on the differentiability of the Lebesgue indefinite integral. Fund. Math. 22 A934). 257-261.
[71] S. Saks. On the strong derivatives of functions of intenals. Fund. Math. 25 A935). 235-252.
[72] S. Saks. Theory of the Integral. Dover A937).
[73] J.M. Sullivan, Sphere packings give bound for the Besicovitch covering theorem. J. Geom. Anal. 4 A994).
219-231.
[74] S.J. Taylor and С Tricot. Packing measure and its evaluation for a Brownian path. Trans. American Math.
Soc. 288 A985), 679-699.
[75] S.J. Taylor and С Tricot. The packing measure of rectifiable subsets of the plane. Math. Proc. Cambridge
Phil. Soc. 99A986). 285-296.
[76] B.S. Thomson. Covering systems and derivatives in Henslock division spaces. J. Lond. Math. Soc. 2 D)
A971), 103-108.
[77] B.S. Thomson, Covering systems and derivatives in Henstock spaces II. J. London Math. Soc. B) 9 A974).
4 pp.
[78] B.S. Thomson. Measures generated by a differentiation basis. Bull. London Math. Soc. 9 A977). 279-282.
[79] B.S. Thomson, On weak Vitali covering properties. Canad. Math. Bull. 21 C) A978). 339-345.
[80] B.S.Thomson, On full covering properties. Real Anal. Exchange 6 A) A981), 77-93.
[81] B.S. Thomson, Derivation bases on the real line (I). Real Anal. Exchange 8 A982-83). 67-207.
[82] B.S. Thomson, Derivation bases on the real line (II), Real Anal. Exchange 8 A982-83), 278^142.
[83] B.S. Thomson, Real Functions, Lecture Notes in Math., Vol. 1170, Springer, Berlin A986).
[84] B.S. Thomson. Derivates of Internal Functions, Amer. Math. Soc. Memoirs, No. 452 A991).
[85] B.S. Thomson, Symmetric Properties of Real Functions. Monographs and Textbooks in Pure and Applied
Mathematics, Vol. 183, Marcel Dekker A994).
[86] B.S. Thomson, ?-finite Borel measures on the real line. Real Anal. Exchange 23 A) A997-98), 185-192.
[87] B.S. Thomson, Some properties of variational measures. Real Anal. Exchange 24 B) A998/99), 845-853.
[88] B.S. Thomson, J.B. Bruckner and A.M. Bruckner. Elementary Real Analysis, Prentice-Hall B001).
[89] С Tricot, Two definitions of fractional dimension. Math. Proc. Cambrigde Philos. Soc. 91 A982), 57-84.
[90] A.J. Ward, On the derivation of additive functions of intenals in m-dimensional space. Fund. Math. 28
A937), 265-279.
[91] W.P. Ziemer, Weakly Differentiable Functions. Springer, New York A989).

CHAPTER 6
Radon-Nikodym Theorems
Domenico Candeloro
Dipartimento di Matematica. via Pascoli. 06123 Perugia, Italy-
E-mail; candelor@dipmat. unipg. it
Aljosa Volcic
Dipartimento di Scienze Matematiche, Piazzale Europa I, 34100 Trieste, Italy
E-mail: volcic@univ.trieste.it
Contents
1. Introduction 251
2. The ? -additive case 255
3. The finitely additive case 259
4. The Banach-valued case 265
5. Finitely additive Banach-valued measures 275
6. Further results 282
Appendix. Singularity and decomposition theorems 290
References 292
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
249

This page intentionally left blank

Radon-Nikodym theorems
251
1. Introduction
Suppose ? is the Lebesgue measure on the real line and / is an integrable function. Then
the measure ? defined for all the Lebesgue measurable sets
v(E)= ? fdk
is called the indefinite integral of / with respect to ?. It is obvious from the definition of
the integral that if ?(?) = 0, then v(E) = 0 for any Lebesgue-measurable set E. This is
expressed saying that ? is absolutely continuous with respect to ? and will be denoted by
? «?.
It is not difficult to check (see [27, Section 30, Theorem B]) that under the assumption
that ? is a finite measure, this condition is equivalent to another one which can be given
in ?-? terms, which will be denoted by ? <gf ?: given any ? > 0, there exists a S > 0 such
that if ?(?) < <5, then v(E) < ?, for any Lebesgue-measurable set E.
We always have that ? <<cf ? implies that ? «?, but the converse implication does
not hold in general, as it can be easily seen considering for instance the measure v(E) =
fE \x\dk(x).
In the ? -additive case, absolute continuity of a measure ? with respect to another
measure ? implies, under mild conditions on ? and v. that ? is the integral measure of
some function f e ? (?): such function is called Radon-Nikodym derivative of ? with
respect to ?, and it is denoted by ^.
Nikodym [44] was the first to prove this result in a quite general setting. If ? and ?
are two finite measures on the same ?-algebra, and if ? <g ?, then there exists a Radon-
Nikodym derivative ?-, i.e., a ?-integrable function / such that
v(E) = J fdiL
holds, for any measurable set E.
Earlier, Radon [46] proved (using Vitali bases) the same implication under the
assumption that the maximal element of the ? -algebra is a measurable subset of the
?-dimensional Euclidean space. For the real line, the corresponding result goes back to
Lebesgue.
However the situation is not so nice, either when the measures are }ust finitely additive,
or when they take values in a Banach space of infinite dimension, and it gets even worse
if both of the above cases occur. We shall outline a survey of the results known so far,
involving the existence of a Radon-Nikodym derivative, in some appropriate sense, and
with respect to suitable types of integrals. In this first section, we deal essentially with a
general Radon-Nikodym theorem for nonnegative finitely additive scalar measures, from
which the basic ideas of further results can be drawn. In the second section, we turn to the
? -additive measures, investigating conditions which permit that even unbounded measures
have Radon-Nikodym derivatives. In the third section we face the finitely additive case for
scalar measures, indicating how the existence of a Radon-Nikodym derivative dv/?? is

252
D. Candeloro and A. Volcic
strictly related to some geometrical properties of the range of the vector valued measure
(?, ?). The fourth section deals with the most beautiful (and difficult) results concerning
those Banach spaces possessing the so-called Radon-Nikodym Property (RNP): the
geometric concepts met in the third section here become essential tools for describing
such Banach spaces. The fifth section is devoted to finitely additive measures taking
values in Banach spaces, and to the research of weaker types of derivatives. Finally, the
sixth section concerns a number of recent results, for locally convex-valued measures, for
multimeasures, for Riesz space-valued measures, and finally also a brief outline of the so-
called fuzzy integration: from this survey one can find that, though hidden by the abstract
settings and the different kinds of integrals involved, the leading ideas of the first section
are always at work.
We end the chapter with an appendix devoted to some decomposition theorems for
measures, which are relevant for the Radon-Nikodym theorem.
We do not give all proofs, some of them being too long and technical: when it is possible,
we give an outline of the main steps, trying to clarify the basic ideas. Also, we realize that
it would be almost impossible to present here a detailed account of all the contributions
given to this problem along almost 100 years, so we simply chose among the results in our
knowledge, sometimes simplifying settings, in order to reach a sufficient variety within a
relatively concise treatment.
The general result, from which we start, is due to Greco [25]. Though the integration
theory and the results by Greco involve more general set functions, we shall give the proof
only for the finitely additive case. It should be noted that the idea to relate the Radon-
Nikodym theorem to the existence of a "scaled" Hahn decomposition goes back, for the
?-additive case, to J.L. Kelley [33].
We need some definitions (see also [19]).
Dehnition 1.1. Let (?, ?) be any measure space, where ? is a ?-algebra of subsets
of ?, and let ? be any monotone nonnegative set function on ?, ?@) = 0. We say that a
measurable function / : ? —>¦ E+ is integrable with respect to ? if the following integral
is finite:
f /??:= ? ?({?: f(x)>t\)dt.
JQ JO
Since {x: f(x) > t] is nonincreasing in t, the second integral has to be understood in
the Riemann sense.
If this is the case, for each set ? e ?, we put:
/?(?):= [ /??:= ? /1???.
Je Jq
The set function /? is often called the integral measure of /, with respect to ?.
The integral defined above is called Choquet integral, and coincides with the usual
integral, in case / is nonnegative and ? is a ?-additive nonnegative measure, or a finitely
additive one.

Radon-Nikodym theorems
253
THEOREM 1.2. Assume that ? and ? are two monotone nonnegative set functions, defined
on the ?-algebra ?, such that ?@) = ?>@) = 0, satisfying:
?E) = ?E) = 0 =>· ?(? U S") = ?(?) (?)
/or ?// ???.
Г/геи, the following conditions are equivalent:
(a) 77?ere exists a measurable nonnegative function f: ? —>¦ К smc/? ?/???
v(E)= ? /?? B)
/ora// ?el
(b) 77?ere emfs a (decreasing) family of sets {Ar}r>o in ?, satisfying:
(bl) v(A) - ?(?) ^ ??(?) - ??(?), ?, ? e ?, ? С А С А,, г > 0;
(Ь2) ?(?) - v(E П Ar) ^ ?(?(?) - ?(? ? A,)), F е Г, г > 0;
(ЬЗ) lim i>(Ar) = 0, as r —> +эо.
It is clear that A) is satisfied, as soon as ? and ? are additive. Moreover, in case ? and ?
are finitely additive, conditions (bl) and (b2) above are equivalent respectively to (b'l)
and (b'2) below:
(b'l) v(F) ^^(F),forall Fc A,., r > 0;
(b'2) v(F) ? ^(F), for all F С ? \ Ar, r > 0.
Moreover, from (b'l) we deduce that lim ? (Ar) = 0 (since ?(??) ^ ?(?) < oo), hence,
in case ? <<cf ?. (ЬЗ) is satisfied, too.
PROOF. As already mentioned, we shall give the proof just in the case of finitely additive,
nonnegative measures, such that ? <ce ?.
We first assume (a) and prove (b). Set: Ar := {x e ?: f(x) > r\ and fix ? с Ar, ? e ?.
As v(E) = f f\l·;d?,ч/egetv(E) ^ r f \??? = r?(F), so (b'l) is proved.
If F is fixed, F e ?, F с ALr, then v(F) = j f\Fd^^ rp(F). so (b'2) holds. As
already observed, (ЬЗ) is satisfied because of the absolute continuity, so the first implication
is completely proved.
Now, we assume that (b) holds, and construct a suitable derivative f.
For all ? e ?, set: f(x) := sup{r > 0: ? e A,}. To see that / is measurable, fix any
re]0,+oo[, and observe that {x: f(x) > t) =)J{Ar: r e D, r > t}, where D denotes the
set of all dyadic positive numbers, i.e., D = {|r, h and к positive integers}.
Now, for each element ^ e D, set B\ := ?,,^???+? }pj.
We see that ? (?*) > ^?(?*), and also ?(?*) < ^-?(?^). For each positive integer ?,
set
? = ?

254
D. Candeloro and A. Volcic
We can easily see that Д. f ?? — liiru fE fk ??, for all ? e ?, and moreover
? /1=1
for all к and E. Therefore, we get:
/????(?).
L
IE
for all E. On the other hand, for each k, we have:
>??{??)-?(?)??
? = ?
As 5??=? v(Bf,) = v(A\/20 ~ v(Ak), from (ЬЗ) we deduce that
lim / fkd?^\?mv{A\l-L^)
from which we also get lim* / ??,?? ^ ?(?), because v(A^2k) ^ 2~V(^i/2*) ^
2"V(«)·
So far, we have seen that the finitely additive measures ? and /? are in this relation:
v(E) ^ /?(?), forall ? e ?, and ?(?) ^ /?(?). From this it follows immediately that
the two measures agree on ? (simply considering the complements of the involved sets),
and therefore the theorem is proved. ?
COROLLARY 1.3. If ? and ? are finite and countably additive, and ? <? ? (or
equivalently ? <<Cf ?), then there exists a Radon-Nikodym derivative ^.
PROOF. Conditions (b'l) and (b'2) mean that there exists a Hahn decomposition for the
measure ? — r?, for all r > 0: this is always the case, for ?-additive measures, and this
concludes the proof. ?
The importance of Theorem 1.2 rests not only on its generality, but also on the
relative simplicity of the involved conditions, from which other similar criteria have been
obtained. We shall see many of them in the sequel, dealing with finitely additive measures,
with Banach- and nuclear-valued measures, and also with different kinds of integrals.
Our dissertation however now focuses the ?-additive case for not necessarily bounded
measures.

Radon—Nikodvm theorems
255
2. The ? -additive case
In this chapter, we mainly deal with countably additive, possibly unbounded measures.
In general, the Radon-Nikodym theorem fails to hold, for unbounded measures, as the
following example shows.
Example 2.1. Let ? = [0, 1], let 0 ^ ;' < j^ 1 and consider the Hausdorff ;'- and j-
dimensional measures ?! and W. Then W «C ?! but dW /dH' does not exist.
The reason for the failure of the Radon-Nykodym theorem in the previous example has
been described already by S. Saks [50] who considered the case / = 0 and j = 1 (see also
Volcic [59]). We will make some additional comments on this example after Theorem 2.12.
However we will see that, under suitable conditions, Radon-Nikodym derivatives do
exist, even if both measures, ? and v, range over [O.+oo]. Such conditions involve
properties of the so-called measure algebra, which we are now going to introduce.
As usual, (?, ?,?) denotes a measure space, where ? is a ?-algebra and ? is any
nonnegative ?-additive measure, taking values in [0, +oo]. We shall assume that the
Caratheodory extension has already been done, and so ? is actually the ?-algebra of all
?-measurable sets. In particular, all subsets of ?-null sets belong to ?. We recall the
definition of the outer measure, ?* :?(?) —> [0. +эо]:
/i*(E) = infljr/i(F„): F„e?. ?c|Jf„ .
Definition 2.2. Given two subsets A and В of ?, we say that A and В are equivalent,
if ?*(???) = 0, and write: А я» ? or A = В ?-a.e.
In a similar fashion, if / and g are real functions defined on ?, we say that / and g are
equivalent (and write / « g, or / = g a.e.) if
?*({???: /(х)фя(х)})=0.
In the quotient ?(?)/^ we can introduce a partial order, as follows:
[A] > [B]
ifandonlyif?*(?\A)=0.
Of course, ?(?)/?, contains ? /я», which is called the measure algebra. One can easily
see that ?>(?)/~ and ? /я» are ?-complete lattices.
We will see that completeness of ?/^ is important for our purposes, however in general
the two lattices are not complete. We have the following facts (more details and related
results can be found in a series of papers by Volcic [58-60]):
Under CH, if ? is the usual Lebesgue measure on ? := [0, 1]. then 'P(i2)/~ is not
complete.
There exist measures such that ?/^ is not complete (see [27, Section 31, Exercise 9]).

256
D. Cande/ою and A. Volcic
DEFINITION 2.3. Given any measure space (?, ?, ?), we say that ? is semifinite if
?(?) = supJM(A): Ac E, AeE, ?(?) < oo},
for all sets ? e ?.
Such measures are called essential measures by N. Bourbaki [8].
PROPOSITION 2.4. Suppose ?<?, suppose that ? is semifinite and suppose moreover
that v(E) = 0 whenever ?(?) < oo. Then ??/?? exists only if ? is identically zero.
PROOF. Suppose dv/?? exists and let ? be any set of finite and positive ? measure. Then
{x e ?: dv/?? > 0} ? ? = [j{x e ?: dv/?? > ^} ? ?.
?
Each set {? e ?: ??/?? > (|} ? ? has finite ? measure and since v(E) > 0, at least one
of them, A„ say, has positive measure ?, a contradiction. This shows that if v{E) < oo,
then v(E) = 0 and from the semifiniteness of the measure we deduce that ? = 0. D
Definition 2.5. A semifinite measure ? on (?, ?) is said to be strictly localizable, if
there exists a family of sets [Еа}аел, such that
(a) 0 < ?{??) < oo
(b) Ea П ?? = 0 when ? ? ?
(c) ?(? П ??) = 0 Vc => ?(?) = 0.
THEOREM 2.6. If ?(?) < oo (or wore w general if ? is strictly localizable), then ?/^
is complete.
PROOF. We will limit the proof to the case of a finite measure. Given any family Q of
measurable sets, we shall show that there exists a measurable set Go such that:
A) ?*(?\?0) = 0?>?3?????, and
B) IfM*(G\G|) = 0forall G e Q, then m*(G0\Gi) = 0.
Without loss of generality, we may assume that Q is a ?-ideal. Now, set:
a:=sup{M(G): GeQ}.
As ? is finite, ? < oo. Now, let (G;,)„e^ beany sequence in Q, such that ? (G„) > or— -,
and put Go := \}G„. As ^ is a ?-ideal. Go e Q- So, we only have to prove that Go
satisfies A) above. If this is not the case, then there exists ? eQ , satisfying ?(#\??) > 0.
This implies that ? U Go e <?, and ?(# U Go) > a, a contradiction. ?
Definition 2.7. A semifinite measure ? is said to be Maharam (or also localizable) if
2?/«? is complete (see [36]).

Radon-Nikodym theorems
257
Remark 2.8. According to Theorem 1, strictly localizable measures are Maharam. It
was a long standing open question whether Maharam measures are strictly localizable.
The problem has been stated explicitly for the first time in [34, Problem 1], and is relevant
in lifting theory. The question has been solved in the negative by D. Fremlin [24].
Dehnition 2.9. Given two measures ? and ? on ? we say that ? and ? are strongly
comparable if ? and ? are the Caratheodory extensions of ?\? and of v\s respectively,
where S is the ideal of all sets ? e ?, such that ?(?) + v{E) < +oo.
The following definition is related to the previous one.
Dehnition 2.10. Given two measures ? and ? on ? we say that ? and ? are weakly
comparable if ? and ? are the Caratheodory extensions of ?|? and of v\z respectively,
where ? is the intersection of the two ?-algebras of ?- and ?-measurable sets (in the
Caratheodory sense).
The measures ?.' and W are not strongly comparable, if i ? j, but they are weakly
comparable.
In 1951, I.E. Segal proved the following version of the Radon-Nikodym theorem [51].
THEOREM 2.11. Given a ?-additive measure ?, the following properties are equivalent:
A) ? is Maharam;
B) for any ?-additive measure v, strongly comparable with ? and such that ? <? ?,
there exists a Radon-Nikodym derivative dv/??. Such function is unique, up to
equivalence.
PROOF. We only prove the implication A) =>· B). So, assume that ? is Maharam, and ?
is any strongly comparable, ?-additive measure, ?«?. We denote by S the ideal of all
sets ? e ?, such that ?(?) + v(E) < oo. For every element ? e S there exists a Radon-
Nikodym derivative of v\e with respect to ? It: we denote by /^ such derivative, and
define fE on Ec as the null function. Now, for every positive real number t, set:
A, :=Y(A,(?): EeS],
where A,(E) = {x e ?: /fU) > t]. (The supremum exists as ? is Maharam.) It is clear
that (A,) is a decreasing family in ?/^. Now, for all .tefiwe define:
f(x) := sup{r > 0: ? e A,}.
As in the proof of Theorem 1.2, one can prove that / is measurable. Let us show that /
is the required derivative. In view of the strong comparability, it is enough to check that
J fd? = v{E)

258
D. Candeloro and A. Volcic
for all F e <S. So, fix any element F e <S, and compute:
J /??= f ?{{?&?: f(x)>t\)dt. C)
Now, ?{{? e F: f{x) > t)) = sup^(F ПЛ(+|): ? e ?). For every positive number r,
we claim that
ЕПАгъАг(Е). D)
Indeed, Ar{E) С ? by definition, and Ar(E) с Ar up to a null-set, hence A,- (F) с ? ? Ar
up to a null-set; conversely, ? ? Ar = v(F ? A,-(F)), where the supremum is taken as
F eS. Now, ? ? A, -(F) = A,(F ? F) because of the essential uniqueness of Д-, hence
F П Ar с A,-(F) a.e. and D) is proved.
From D), we get immediately
suP{m(F П At+l): ? e N} = sup{m(A,+i (?)): neN)= ?(?,(?))
for all positive numbers t, and so from C) we deduce
/??= ?{{???: f(x)>t\)dt= / ?(?,(?))?
./? JO Л)
= ? ???? = ?(?),
by definition of /f. This concludes the proof. ?
For weakly comparable measures, we have the following result.
THEOREM 2.12. Given a ?-additive measure ? on (?, ?), the following properties are
equivalent:
A) ? is Maharam;
B) for any ?-additive measure v, weakly comparable with ? and such that К<Д,
there exists a decomposition ?\, ?? of ?, ?\ e ?, for / = 1,2, such that:
• for any ? e ? and ? С ?\, ?(?) < эо implies that v(E) = 0;
• if we denote by vj the restriction of ? to ? ?) ??, the Radon-Nikodym derivative
'-^ exists.
The decomposition ?\, ?? and the function '-r~ are unique, up to equivalence,
Ju
The proof follows from the combination of Theorem 2.11 and the decomposition
Theorem A.4 from the Appendix.
Remark 2.13. The previous theorem explains why Radon-Nikodym theorem fails
when ? is the counting measure H° on [0. 1], and ? is the Lebesgue measure, in spite

Radon-Nikodym theorems
259
of the fact that ? is localizable. The measures ? and ? are in this case just weakly and not
strongly comparable, and ?;«?. Theorem 3 applies here with ?\ = ?.
We conclude this section presenting two interesting variants of the Radon-Nikodym
theorem.
We need first two definitions.
DEFINITION 2.14. Given a measure space (?, ?, ?), we say that ? admits a monotone
differentiation, if there exists a mapping -f-, defined on the family ?? of all the measures
which are strictly comparable with ? and absolutely continuous with respect to ?, such
that if Vj e ??, for / = 1,2, and ? ? ^ ??, then
—? (?) ? —- (.?),
?? ??
for any ? ? ?.
Definition 2.15. Given a measure space (?, ?,?), we say that ? admits a linear
differentiation, if there exists a mapping j~r defined on the family ?? of all the measures
which are strictly comparable with ? and absolutely continuous with respect to ?, such
that if vj e ??, for / = 1,2, and a \, a^ are two real numbers, then
d(a\v\ +ajvi) dv\ dv2
(x)=a\ —-U)+fl2 —U).
?? ?? ??
for any ? e ?.
The following result is due to D. Kolzow [34, Theorems 4, 7 and 8].
THEOREM 2.16. Given a measure space (?. ?, ?), the following properties are
equivalent:
(i) ? admits a monotone differentiation;
(ii) ? admits a linear differentiation;
(iii) ? /s strictly localizable.
For the long and sophisticated proof we refer to Kolzow's monograph, which also
illustrates in great detail the strong relations between the monotone and linear versions
of the Radon-Nikodym theorem and lifting theory.
The Radon-Nikodym theorem for the Daniell integral is discussed by M.D. Car-
rillo [15], Chapter 11 in this Handbook.
3. The finitely additive case
When the measures (even one of them) are just finitely additive, the Radon-Nikodym
theorem does not hold, in general: (we remark that, in this case, absolute continuity is
to be intended in the ?-? sense).

260
D. Candeloro and A. Volcic
It is appropriate to begin with the oldest result concerning the Radon-Nikodym theorem
for finitely additive measures, attributed to Bochner [7].
THEOREM 3.1. If ? and ? are two finitely additive measures on ?, such that ? <?? ?,
then far any ? > 0 there exists a ?-integrable function fn, such that
????
?
<?.
for any ? e ?.
In order to give an example, we introduce a definition.
Dehnition 3.2. Let ?: ? -> Rj be a finitely additive measure, and let 3 denote the
ideal of null sets. If 3 is a ?-ideal, we say that ? has the property ?.
One can easily see that, if / ^ 0 and ? has property ?. then
fd? = 0
/
implies that / = 0 a.e.
Similarly, one can deduce that, whenever ? has property ?, and / and g are two ?-
integrable functions, then f = g ?-a.e. as soon as fA f ?? = fA gd? for all A e ? (this
entails uniqueness of the Radon-Nikodym derivatives, up to a.e. equivalence).
Example 3.3 ([10]). Let ? be the family of all subsets D с [0, 1], such that the
following limit exists:
,. ?(?)?[0,?[)
hm — -=8(D).
Though ? is not an algebra, the function ? can be extended (using the axiom of choice)
to the whole ? -algebra ? of the Lebesgue measurable sets, in such a way that
?(??[0,?[) ?(??[0,?[)
lim inf ^ ? (?) ^ lim sup ,
f^o+ ? г_0+ ?
for all sets В e ?, and so that ? is finitely additive on ?. Obviously, if ?(?) = 0 then
S(B)=0.
Now set ? := ? + ?. It is clear that ?(?) = 0 if and only if ?(?) = 0, hence ? enjoys
property ?. Moreover, ? <«? ? also in the (?-<5) sense, however one can easily see that
??/?? does not exist. In fact, such a function / should satisfy
?(??[?, 1])= f ???=\ fdk+( fdS= [ fdk,
JBr\\e.\\ Jen[e.\} JBD[fA] J ВП\еЛ]

Radon-Nikodym theorems
261
for all ? > 0, and all В e ?, i.e., / = 1 a.e. in [?, 1] for all ? > 0. Thus, by the property ?,
it should follow that / = 1 ?-a.e., a contradiction.
The property ? has a double face: on one hand, it looks like a strengthening of finite
additivity, in the sense of ?-additivity. On the other hand, it prevents in some sense that
a finitely additive measure might behave like a ?-additive one, at least with respect to the
Radon-Nikodym property. This statement is clarified by the next theorem.
THEOREM 3.4. Let ? : ? —>¦ Rq be any finitely additive measure, defined on the ?-
algebra ?, and enjoying property ?. The following are equivalent:
A) ? is countably additive;
B) for every finitely additive measure ? : ? —>¦ Rj, which is (?-S) absolutely
continuous with respect to ?, there exists ??/??.
PROOF. Of course, since A) and the other assumptions imply that ? is also ?-additive,
just B) =>· A) needs to be proved. Let (A„)„eI: be any increasing sequence of sets in ?,
and denote by A their union. Set: v(B) := 1????(?„ ? ?), for all В e ?. It is clear that ?
satisfies the condition B), hence there exists / = dv/??.
Now, for every В e ?, we have v(B) = v(B ? A) = fB f\Ad?, so / can be replaced
by /1A. On the other hand, it is clear that f\A„ = 1 д„ ?-a.e., hence / = 1 a ?-a.e.
As a consequence, we have lim,,^^ ?(?„) = v(A) = fA f ?? = fA 1 ?? = ?(?). As
the sequence (A„)neN was arbitrary, ? turns out to be ?-additive. ?
In the spirit of the remark preceding Theorem 3.4, we can give an antithetic example.
Example 3.5. Let ? be any infinite set, and let 3 denote the ideal of all finite subsets
of ?. According to the Axiom of Choice, there exists a maximal ideal 3*, including 3.
Define for ? ??(?):
?(?) :=|° ifAe3*·
I 1 otherwise.
According to a well-known theorem by Ulam [57], ? is a finitely additive measure,
lacking the property ?. However, if ?: ?(?) —> R(| is any finitely additive measure, @-0)
absolutely continuous with respect to ?, then the only possibility is that ? = кв, for some
nonnegative constant k. In this case, obviously, к = dv/??.
To find a characterization of the Radon-Nikodym property, we have to introduce the
concept of completeness. In a recent paper by A. Basile and K.RS. Bhaskara Rao [1]
an excellent presentation is given, mainly concerning finitely additive measures defined
on algebras. We shall adapt here one of their results, dealing with the Radon-Nikodym
property.
Definition 3.6. Let ? : ? -> Rj be any finitely additive measure, on a ?-algebra ?.
If A and В are elements of ?, we set: d,t(A, ?) = ?(???). It is clear that (?,??) is a

262
D. Candelom and A. Volcic
semimetrizable space. We say that (?, ?) is complete, if the quotient ?/« is complete as
a metric space, where « is the natural equivalence relation in ?.
The characterization given in [1] can be stated as follows (we recall that in our
assumptions ? is a ?-field, and contains all subsets of sets of measure 0).
THEOREM 3.7. Let ? : ? —> R^ be any finitely additive measure, on ? ?-algebra ?. The
following are equivalent:
A) {?,??) is complete.
B) For every increasing sequence of sets (Fw)„eh' in ?, there exists a sequence of ?-
null sets (#„),>eN in ?, such that
lim M(F„) = lim m(F„ \ H„) = ?( \\{F„ \ H„)\
«—>ос и—«-ос \ v-' /
From Theorem 3.7 one can easily deduce that if ? <Kf ? and (?, d,t) is complete, then
(?, dv) is complete, too.
COROLLARY 3.8. ^(?.??) is complete, and ? is any signed finitely additive bounded
measure such that ? <gf ?, then ? admits a Hahn decomposition.
PROOF. We only have to show that there exists a set ? e ?, such that v(P) =
™??€??(?)-
As ? is bounded, the number S := supAer ? (A) is finite, and there exists a sequence
(^;i)«eN m ?, such that S = lim„^,c v(A„). Setting now
v+(A)= sup ?(??)?) and v~(A) = - inf v(A ? ?), VA e ?,
Ее ? i'er
we readily see that v+(A„AA„t) —>¦ Oand v~(A„AA,„) -+ 0 as both m and// diverge. Now,
i>+and v~ are both (?-?) absolutely continuous with respect to ?, and so is |i>| := v+ +v~,
hence there exists a set ? e ?, such that lim,,^^ |?|(?„??) = 0. The set ? has the
required property. ?
COROLLARY 3.9. The following are equivalent:
A) (?,??) is complete;
B) For every nonnegative finitely aaaitive measure ? <Kf ?, there exists ??/??.
PROOF. We only prove the implication A) =>· B). As ? <^F ?, we see that the signed
measure ? — /·? is ?-? absolutely continuous with respect to ?, and therefore it has a Hahn
decomposition. Due to Corollary 1.3, this implies the existence of ??/??. D
The corollary above has been proved in [1].
A different approach to the problem consists in finding additional conditions on the two
measures ? and v, besides absolute continuity, which ensure the existence of ??/??.

Radon-Nikodym theorems
263
One of the first consequences of Theorem 1.2 and Corollary 1.3 is the following (see
also [3,12,13]):
THEOREM 3.10. Let ?, ? be two nonnegative finitely additive measures, defined on the
same ?-algebra ?, and such that ? <?e ?. If the range of the pair (?. ?) is a closed subset
ofR2, then there exists ??/??.
PROOF. In view of Corollary 1.3, it is enough to show that, for every real number r > 0,
there exists a Hahn decomposition for ? — r\x. But it is clear from the assumption on the
range of (?, ?) that the range of ? — ?? is a closed subset of R, hence there exists an
element A e ? in which ? — r? attains its maximum: so (A, A') is a Hahn decomposition
for ? — ??, and we are done. ?
Of course, closedness of the range of (?, ?) is just a sufficient condition. In [3] and [12]
necessary and sufficient conditions are given on the range of (?, ?) in order that dv/??
exists. In both papers, the condition involves the so-called exposed points of the range,
according with the following definition.
Dehnition 3.11. Let R be any convex, bounded subset of R". Let us denote by R
its closure. We say that a point ? e R", Q e dR, is an exposed point for R if for every
hyperplane H, supporting R at Q, we have RC\ ? = {Q}.
THEOREM 3.12. Let (?, ?) be a pair of nonnegative finitely additive measures, defined
on a a -algebra ? and such that ? <Kf ?, and let R denote the range of (?, ?). Then the
following are equivalent:
A) There exists dv/??.
B) The convex hull of R contains its exposed points.
The theorem above has been proved in [3].
In [ 12] a similar theorem is stated, for the case of continuous measures.
A measure ?: ? —>¦ R^ is said to be continuous if for every ? > 0 it is possible to
decompose ? into a finite number of subsets ? ?, At A„ belonging to ?, such that
? (A,) < ? for all /'.
If ? and ? are both continuous, finitely additive and nonnegative, it is well-known
that the range of (?, ?) is a bounded convex subset of the plane (see also [11]), hence
condition B) of Theorem 3.12 can be expressed in a simpler way. However, in [12] the
following result has been proved, which gives a full description of those bounded convex
sets R CR2 that are the range of some pair (?. ?>) for which dv/?? exists.
THEOREM 3.13. Let R be any bounded convex subset of [0. эо[:. Then the following
properties are equivalent:
A) There exists a pair (?. ?) of nonnegative continuous finitely additive measures,
defined on a suitable ?-algebra, such that there exists dv/??, and such that R
is the range of {?, ?).

D. Candelom and A. Volcic
Fig. 1. Equation of the lower curve: ? = 1 — vO _ ·*")·
0.8
Fig. 2. Equation of the lower curve: ? = A — \/(l - x-))/2.
B) R contains its exposed points, and for every segment L С BR, at least a (possibly
degenerate) closed subsegment I C L is contained in R. (See Figure 5 (a) and (b).)
Usually, if (?, ?) is a pair of nonnegative continuous finitely additive measures, the
range looks like in Figure 1, or in Figure 3, when ? <Kf ?. In Figure 2 a particular situation
is shown, where ? is obtained by adding to ? some measure which is singular with respect
to ?.
We observe that the range is always symmetric with respect to the midpoint (?(?)/2,
?(?)/2): hence in Figure 5 (a) and (b) just half of 37? has been drawn.

Radon-Nikod\m theorems
265
0.2 0.4 ? 0.6 0.8
Fig. 3. Equation of the lower curve: у = л6.
0.6
Fig. 4. Lower curve: у = A - у A - д-2))/2. for .? < 0.5. then linear.
4. The Banach-valued case
The results from the previous sections can be easily extended to bounded signed measures,
since the Jordan decomposition (even in the finitely additive case) always exists. Similarly a
complex measure or function can be studied investigating separately its real and imaginary
part. In the same way we can also extend the previous results to measures or functions
taking values in a finite-dimensional space.
A really different situation occurs with infinite-dimensional measures, as we shall see in
this section. The research on this subject led, mainly in the seventies, to a variety of quite

266
D. Candeloro and A. Volcic
1-
0.8-
0.6-
0.4-
0.2:
1
^
у
X'
!
/
Л,
^~~— Line segment
(a)
0.8
0.6
0.4
0.2-
(b)
Fig. 5. (a) Range closed, (b) Range not closed. Part of the segments is missing.
interesting results, relating the Radon-Nikodym property to topological and geometric
properties of the range space. Here, we shall confine ourselves to the Banach-valued ?-ad-
ditive measures, and with respect to the Bochner integral. (As to absolute continuity, for
the ?-additive case, the (?-?) definition is still equivalent to the @-0) one.) Maybe the
richest survey on this subject is [20], so we refer the reader to that paper, in order to find
the proofs missing here, and an exhaustive historical account.
We start, recalling the definition of Bochner integral [6,21,22].

Radon-Nikodxm theorems
267
Dehnition 4.1. Given a finite measure space (?.?,?), where ? is a ?-algebra,
and a Banach space X, a function /: ? —>¦ X is said to be simple if it is of the form:
/ = ?, = ? -?/ 1 a, , where x/ e X and A, e ?1 for all /' = 1,..., N. If this is the case, then
the integral of / is the element:
Г N
/ /??:=^2???(??)&?.
J i = l
In the same framework, if (/„)„en is any sequence of simple functions, we say that /,
converges in measure to some function / if the following holds:
lim ?*({???: \\f„(o)-f(o)\\ >?\) = 0
П—ЮО II I '
for all ? > 0 (here, ?*{?) is the outer measure defined in Section 2).
Incase /: ? —>¦ X is the limit in measure of some sequence (/,),,ек of simple functions,
then / is said to be measurable (also, strongly measurable).
We say that /: ? —>¦ X is (Bochner)-integrable, if there exists a sequence (/H),ieN of
simple functions, converging in measure to /, and such that
lim [\
/,-/,„ Мм = o.
It can be proved that, if f: ? —>¦ X is integrable, then the sequence (f /???)„?? is
convergent in X, and the limit is independent of the particular sequence (/„ )„ еы ¦ Of course,
the limit lim / /„ ?? is called the integral of /.
Also, as in the real-valued case, one can prove that convergence in measure implies
convergence a.e. for some subsequence, hence a strongly measurable function is essentially
separably valued, i.e., there exists a separable subspace ? С X. such that /(?) e ? for all
but a ?-null set of elements ?. Moreover, a strongly measurable function is also weakly-
measurable, i.e., {x*, f) is measurable for all x* belonging to X*.
Like in the scalar case, it can be proved that integrability of / implies integrability
of 11/11, and that \\//?"?\\ ^ /\\/\\??. When / is integrable with respect to ?, we
also say that / is in ??{?). The latter is a Banach space if we identify functions which
coincide a.e. with the norm ||/|| = f ||/|| ??. Moreover, if / is integrable, then /If is
also integrable, for all ? e ?, and the integral becomes a function of E: we set
"//*-/
/?(?):= ? /??:= / /\???.
One can see that f? is an X-valued measure, which is separably valued, whose variation
is ||/||?. The variation of an X-valued measure ? is denoted by |i>| and is defined as:
|y|(?)=sup^|v(?,)|.

268
D. Candeloro and A. Volcic
where the supremum is taken over all finite partitions of E: one can also regard |v| as the
least upper bound of {| (**, v) \: x* eX*. ||.v*||= 1), in the lattice of measures.
Moreover, /? <«C ?.
The converse question leads, of course, to the problem of the existence of a Radon-
Nikodym derivative. The problem can be stated in various different ways:
A) Given a measure ? : ? —>¦ X, which is absolutely continuous with respect to a scalar
positive measure ?, does there exist a Bochner-integrable function /, such that
?=/??
B) Given a measure ? : ? —>¦ X, with finite variation. |?|. does there exist a function /,
Bochner integrable with respect to |i>|. and such that ? = f\v\?
Of course, a necessary condition for an affirmative answer to A), is that ? has finite
variation. Moreover, if |?| is finite and К<Д. then it is clear that |?>| <? ?: hence, an
affirmative answer to B) would yield an affirmative answer to A). Conversely, assuming
the existence of dv/??, one can decompose ? into the sum ?? + ?:, where ?? <<? |?|
and ?? -L ?· (See Theorem A4.) Hence |?>| <? ??. dv/?? = ??/??\, and dv/d\v\ =
(??/???){???/?\?\).
In conclusion, we can see that A) and B) are essentially equivalent.
Moreover, in [17], Chatterji remarks that in A) above the ?-algebra ? can be
equivalently replaced by the Borel ?-algebra on the unit interval, and (up to irrelevant
renorming) the measure ? by the usual Lebesgue measure.
Now, we give a couple of examples showing that the answer is not always affirmative.
Example 4.2. Let X be the L'space over the unit interval in R. Also, let ([0, 1], ?,?)
be the measure space, where ? denotes the Borel ?-field, and ? the Lebesgue measure.
For all A e ?, set v(A) = 1A, i.e., the characteristic function of the set A.
It is easy to see that ? is an X-valued countably additive measure, defined on ?, and that
| v| = ?. However, if a Radon-Nikodym derivative / := dv/dk existed, we would have:
f g(x)dx=(g, \A)= f (f(x).g)dx
J A J A
for all A e ?, and all g e L°° (= (L1)*). Therefore, for every fixed g e Lx, we would
have:
g(x)= (f(x).g) a.e.
Hence, there would exist a ?-null set N e ? such that, for all ?, ? in [0. 1 ] ? Q, a < ?.
we would have
(/(*). Wi)=i
for all ? e Nl' ? [?, ?]. So. if we fix ? ?_ ?, and choose (a„)„ef,, (/3„)„еь in such a way
that a„ < ? < ??, <*„, ?? e Q, and ?? - a„ —>¦ 0, we get:
|ilimc(/(Jc),lK.ft,])= 1
which is impossible, since f(x) e L'.

Radon-Nikodym theorems
269
Example 4.3. The next example concerns the same measure space, ([0, 1]. ?, ?), but
with a different range: here, X is со, the space of all real sequences, vanishing at infinity.
Let us define:
?(?):=(?,(?),...,?„(?)....).
where
?„(?)= I sinBn nx)dx
J a
for all A e ?, and ? e N.
As |?„(?)| ? ?(?) for all A and all /;. it is clear that ? has bounded variation,
and |?|([0, 1]) ^ 1. Obviously, ?«?, however ??/?? does not exist. Indeed, such a
derivative /,/=(/! /„ ), should satisfy:
?«(?)= / ??B? nx)dx = j f„(x)dx
for all A and n. Thus, it would be /,,(л) = sinB7r//;c) a.e., which is impossible, because
??B???) is not vanishing, as /z —>¦ oo.
The first affirmative answer to the Radon-Nikodym problem in infinite-dimensional
spaces is due to Birkhoff.
THEOREM 4.4 ([4]). IfX is a Hubert space, then every ?-additive measure ?:?^>?
with bounded variation admits a Radon-Nikodym derivative with respect to \?\.
We do not give the proof now, because we will see later some generalizations of this
result.
Remark 4.5. What happens if we try to adapt Theorem 4.4 to the example given in
Example 4.2? One can always think of 1 д as an element of L2, and of ? as an L2-valued
?-additive measure, with ?»«?. However, by the same argument, one can still show
that dv/dk does not exist. What is wrong? A simple calculation shows that ? does not
have bounded variation, when considered with values in L2. This remark also shows the
importance for a measure of having bounded variation. (However, more can be said on this
example, see Remark 4.18.) From now on. we shall write BV to mean "bounded variation".
DEHNITION 4.6. We say that a Banach space X has the Radon-Nikodym Property (RNP)
if A) or B) above holds, i.e., every countably additive X-valued BV measure ?, defined
on the measure space (?, ?), admits a Radon-Nikodym derivative with respect to |?|.
Hence, according to Theorem 4.4. all Hubert spaces have the RNR
Other spaces with RNP can be derived from another important sufficient condition, due
to Dunford and Pettis [23].

270
D. Caiideloro and A. Volcic
THEOREM 4.7. If X is a Banach space, whose dual space X* is separable, then X* has
RNP.
PROOF (Sketch). Let ? : ? -> X* be any BV measure, with ? «; ?. For any ? e X, and
any A & ?, set vx(A) = (v(A), x). It is clear that vx <<C ?. Of course, there exists dvx /??.
The idea now is to set /( = ???/??, and to show that /( defines, as ? varies in X, a linear
continuous functional /; however one must take into account that fx is defined а.е., hence
a "good" function / must be defined carefully, and this can be done, if X* is separable. D
For example, the space L \ is not a dual space, because we know from Example 4.2 that
it has not RNP, however /1 is a separable dual, hence it enjoys RNP.
We will see in the sequel other classes of spaces with that property, but we will introduce
first a very important definition.
Dehnition 4.8. Given a bounded nonempty subset ? of a Banach space X, we say
that В is deniable if for every ? > 0 there exists a point xe e В such that xe does not
belong to the closed convex hull of B\B(xf. ?). In case it is possible to choose the same ?
for all ?, then ? is said to be a denting point of B.
Dentable sets are closely related to the existence of Radon-Nikodym derivatives, as
Rieffel showed in [47,48].
We first remark that В is dentable whenever every countable subset of В has this
property. This was proved by Maynard [39], and can be seen as follows: assume that В is
not dentable; then there exists ? > 0 such that ? e co(B\B(x. ?)), Vjc e B. So, choose any
element ? e B, and points y\ y„ e B, such that ||y,- — jc || > ? for all /.and some convex
combination ? ? of these points belongs to ?(?,?/2). For the initial point x, and each of
these points, say yj, one can choose a finite subset Fj с В, such that \\y - .y/|| > ?, for
any у e Fj, and such that some convex combination Z2 of Fj belongs to B(yj,e/4). Now,
the union of the sets Fj, together with ? and the points \j, gives a finite subset E\ С В.
For each element s & E\ the same construction is possible, giving rise to a finite subset Ei.
We continue by induction. The union of all the sets Ej is then a countable subset of B,
which is not dentable.
Now, let us state a lemma, where the construction of a Radon-Nikodym derivative is
shown, under suitable assumptions.
LEMMA 4.9. Let (?, ?,?) be any finite measure space, and ? : ? —>¦ X be any BV
measure, \v\ <^?. Then dv/?? exists, whenever the following condition holds:
(a) For every ? > 0 there exist sequences (xen)„&i in X, and (Н^)пец in ?, such that
the sets Щ are pairwise disjoint, have positive measure, ?(?) = ]??(#^), and
( ^: Л ? Г. Л С «; I СЯК..).

Radon-Nikodym theorems 271
PROOF. Assume that (a) holds, and fix ? > 0. Consequently, a sequence (xf„)neN exists
in X, and a corresponding sequence (#;f )„ек in ?, according to (a). Now choose ne large
enough, so that ?(?\ \Jj^„f H]) < ?· and set
?·.= ?^4·
It is easy (though tedious) to see that (fe) is Cauchy in ?]<·(?), hence converges to some
function / in this space. Now, since jE f ?? = lim/? fe ?? as ? —> 0, for all ? e ?,
it is enough to compare v(E) with fE fe ??. Of course, up to ?, we can assume ? с
U^„f Щ, and so v(E) = ??,„ ?(? ? Яр, while /? /f ?? = ?^„, X)^E n Яр·
Now, as ? П Я? С Я?, we have ||?(? ? Я") - ?'?(? ? Я?)|| ^ ??(? П Я|) by (a),
therefore \\v{E) - /? /? ??\\ ^ ??{?), and this proves the lemma. ?
THEOREM 4.10 (Rieffel). Let (?,?,?) be any finite measure space, and let ? : ? -> ?
be any BV measure, with \v\ <<C ?. The following are equivalent:
A) There exists ??/??.
B) For еас/г .$« ?el, ?(?) > 0. there exists DcE, D e ?, ?(?) > 0, iwc/г ?/???
Л(О) = I -^-^-: А с О. A e ?, ?(?) > 0 й dentable.
\?(?) J
PROOF (Sketch), A) =>· B) We first observe that the result is obvious, ?? ??/?? is simple:
for every set ? e ?, having positive measure ?. there exists DcE, with D e ?, and
?(?>) > 0, such that ??/?? is constant in D, and hence {-7^: AC D, Ae ?, ?(?) > 0}
is a singleton. In the general case, let / denote the derivative ??/??, and let (s„ )„ек be any
sequence of simple functions, converging to / in the L' -norm, and also almost uniformly.
For any set ? e ?, such that ?(?) > 0, there exists a set D с ?, with D e ?, such that
?(?)) > 0, and s„ —>¦ / uniformly on D. Now, it is easy to apply the previous argument.
B) =>· A) We shall use Lemma 4.9. To prove (a), we proceed as follows: first, we show
that, for every set ? e ?, such that ? (?) > 0, and for any ? > 0, there exists DcE, with
D e ?, and ?{?) > 0, such that the set A(D) is contained in B(x, ?) for some element
xeX.
Once we have proved this, it is possible to replace ? with Dr, and iterate the procedure,
constructing a disjoint sequence as in (a).
So fix ? e ?, with ?(?) > 0, and let ? > 0. We know that there exists ?0 С Е, with
?0 e ?, and ?(??) > 0, such that A(E0) is dentable. Let ? be an element of A(Eo),
such that ? <? co(A(E0)\B(x,e)). Write ? = ?(?H)/?(?)). for suitable Do С ??· If
.4(A)) С ?(?,?), we have finished. Otherwise, there exists E\ С A. with ?(?|) > 0,
such that
y(?i)
? > ?.
?(?\)
As gfi| e Д(А)), we also have g^ e co(.A(Do)\B(*, ?))·

272
D. Candeloro and A. Vollic
Let us denote by k\ the smallest positive integer for which such an element E\ exists,
satisfying ?{?\) ^ y, and choose in this way E\. Set D\ = Dq\E\ . It is impossible that
?(?>?) = 0, because this would imply
= v(Dq) = v(E\)
?(?H)_?(?|)'
a contradiction.
Now, if Л@|) с ?(?,?), we are done. Otherwise, continuing that way, we can
produce a disjoint sequence (?„)„epj in ?1, an increasing sequence (k„)nen in N, such
that ?(?„) < ?1, and ^Щ е гё(Л(^)\А(х,?)). Of course, we have it,, | oo. Setting
\jEn, and ?) = ?H\ ?эс, we can deduce that D is the desired set, i.e., ?(?) > 0
and A(D) с B(x, ?). Indeed, if it were ?(?) = 0, we would have
v(Do) v(Eoc) ?>(?„) ^ ?(?„) ?(?„) _/..„..„. ,ч
x= = = — => eco(A(Eo)\B(x,E)),
?(?H) ?(??) ?(??0) ^ ?(?„) ?(??) V ;
which is impossible.
Now, to show that A{D) с ?(?,?), fix D' с D, with D' e ? and ?(?') > 0. If
gg^^BU.e), then XJfy eco(A(EQ)\B(x.e)), because D' с O0\U"=i Ei for all n.
But then, by the choice of (k„)nem, we must have ?(?') < r- for all n, and hence
?{?') = 0, a contradiction. ?
Let us see some consequences of Theorem 4.10.
For instance, we can deduce that if X has the RNP then every subspace of X has the same
property. Indeed, the construction of Lemma 4.9 shows that, if ? takes values in a subspace
? С X, then the derivative can be constructed as a limit of У-valued simple functions.
Another consequence is that X enjoys RNP as soon as every bounded subset of X is
dentable. However, the converse is also true, as Huff in [29], and Davis and Phelps in [18],
independently proved.
THEOREM 4.11. Let X be any Banach space. The following are equivalent.
A) X enjoys RNP.
B) Every bounded subset ofX is dentable.
We refer the reader to [20] for a proof of the implication A) =>· B).
Another consequence is that Radon-Nikodym Property is separably determined, i.e.,
a Banach space X possesses RNP if and only if every separable subspace ? с X has
RNP. This is now an easy consequence of Theorem 4.11, and the remark following
Definition 4.8: a set is dentable as soon as its countable subsets are dentable.
As weakly compact sets are dentable, it is now clear that every reflexive Banach space
has the RNP.
An interesting consequence of Theorems 4.7 and 4.11 is that X enjoys RNP as soon as
each of its separable subspaces are duals. In [52], Stegall showed a stronger result:

Radon-Nikodvm theorems
273
THEOREM 4.12. The following are equivalent, for any Banach space X:
A) X* has RNP;
B) every separable subspace of X has separable dual;
C) every separable subspace of X is embedded in some separable dual.
A consequence of Theorem 4.12 is that the dual of a separable Banach space possesses
RNP if and only if it is separable.
Some of the previous results are also included in a "martingale-type result", obtained by
Chatterji in [17]. For the sake of completeness, we give some definitions.
Dehnition 4.13. Let (?, ?, P) be any probability space, i.e., any measure space,
with ?{?) = 1. Given any Bochner-integrable function f: ? —>¦ X (here, X is any
Banach space), and given any sub-a-algebra ?? С ?, the conditional expectation of the
function / with respect to ?? is the Bochner-integrable function (defined P-a.e.), denoted
by E(f\?o), which has the following two properties:
A) E(f\Eo) is strongly .To-measurable;
B) fFfdP = fFE(f\Z0)dPfoTiuiyFeZo.
The existence of E(f\Eo) is independent of the RNP, and is proved first directly, for
simple functions /, and then by approximation, in the general case.
DEHNITION 4.14. Let X, and (?, ?, ?) be as above. An X-valued martingale is a
sequence (/„, En)neN of Bochner-integrable functions /„ and sub-?-algebras ?„ с ?,
such that:
A) ?„??„+\, VneN;
B) /„ = ?(/„+, |Г„),УиеМ.
The martingale (/„, ?? )„epj is said to be convergent if there exists a Bochner-integrable
function foo-.?^?, such that /„ = ?(/?\?„) for all ? e N.
A typical way to obtain martingales is the following: assume ? = [0, 1], ? is the Borel
? -algebra, and ? the Lebesgue measure. Construct a sequence of decompositions (D„)«eN
of the unit interval, first splitting [0, 1] into two sub-intervals of the same length, thus
obtaining D\, and then, by induction, dividing each interval from D„-i into two disjoint
sub-intervals of the same length in order to get D„. Let ?„ be the (?-) algebra generated
by D„. Now, if ? is any measure defined on ?, set: f„(x) = ?(?„(?))/?(?„(?)), where
I„(x) is the unique interval of D„ containing x. It is clear that /„ is constant on each
interval of Z)„, hence it is ?1,,-measurable. It is also easy to see that ff fn+\ dk = Д /„ dk
(= v(In)), which implies that (/„, 2Г„)„ещ is a martingale.
If this martingale is convergent, and if ? is ?-additive, then the function fx satisfies the
identity:
J focdk=v(F),
for any F e (J ??, and therefore for any F e ?, i.e., we have fx = dv/dk (and of course
? « ?).

274
D. Candeloro and A. Volcic
In case ? is real-valued, a classical condition for the convergence of a martingale is
particularly simple: as soon as 5ир,1€1;/^ \ft\dk < oo, the functions /„ are uniformly
integrable, and pointwise converging to fx.
When the martingale (or the measure v) takes values in a general Banach space X, this
is no longer true, in general. Chatterji's result clarifies the situation.
THEOREM 4.15 ([17]). Let (?, ?, P) be any probability space, and assume that X is a
Banach space. The following are equivalent:
A) X has the RNP with respect ??(?.?, ?).
B) Every X-valued martingale (f„- ?,,),,^:, satisfying sup,ieH fn \f,\ dP < oo, is
convergent, in the sense that f, converges P-a.e. and strongly in X to a Bochner-
integrablefunction fx : ? -> X, such that E{fx\E„) = /„. V/; e N.
We shall not give the proof here: for the implication B) =>· A), the construction above
gives an idea of the main steps. The converse is less elementary. As a consequence of this
result, one can easily deduce, besides the already mentioned classes of reflexive spaces,
and separable dual spaces, another class of Banach spaces with the RNP, i.e., the weakly
complete spaces with separable dual.
We conclude this section with an analytic result, due to Moedomo and Uhl [41],
connecting "weak" derivatives with "strong" ones. We need a further definition. (See
Chapter 12.)
Dehnition 4.16. Let (?, ?, ?) be any finite measure space, and X any Banach space.
Given a strongly measurable function /: i? —>¦ X (see Definition 4.1), we say that / is
Pettis integrable if for every set A e ? there exists an element J(A) e X, such that
[x\J(A))=f{x\f)dvL
for any x* e X*. The element J (A) is called the Pettis integral of / in A, and is denoted
by{P)fAfdvL.
THEOREM 4.17 ([41]). Let v. ? ^ X be any ?-additive measure, ? «; ?. The following
are equivalent:
A) There exists a Pettis integrable function f: ? —> X such that
(?) ? fd^ = v{A)
J A
for all A e ?. (/ is the Pettis derivative of v.)
B) For every ? > 0 there exists a set He e ?, such that ?(#/) < ?, and the set
{—д|: А С ??, ? (?) > 0} is relatively compact.
C) For every Ae ?, with ?(?) > 0, there exists В e ?, В С A, with ?(?) > 0, such
that {44=}: ? <??, ?(?) > 0} is relatively compact.

Radon-Nikodym theorems
275
{In items B) and C) above, "relatively compact" can be equivalently replaced by "weakly
relatively compact".)
Moreover, f turns out to be Bochner integrable, if and only if ? is BV.
We shall not give a proof of Theorem 4.17, however we emphasize again the fact that
weak compactness of bounded sets ensures that they are dentable, so at least part of this
theorem is a consequence of Theorem 4.10.
Remark 4.18. In some sense, Theorem 4.17 tells us that bounded variation is not an
essential requirement for ? to be an integral measure: if one of the properties B) or C)
of Theorem 4.17 is satisfied, ? is at least the Pettis integral of some function /, with
respect to ?. For example, if X is reflexive, the only requirement is that B) or C) above
hold, with "relatively compact" replaced by "bounded"; however, if ? is not BV, this
is not automatic, even for Hilbert spaces: if we consider the Remark 4.5, the example
outlined there deals with an L2-valued measure v, absolutely continuous with respect to
the Lebesgue measure ?, lacking even a Pettis derivative: indeed, the "averages" y^ fail
to be bounded, as soon as ?(?) decreases to 0.
5. Finitely additive Banach-valued measures
Of course, the problems concerning the existence of a Radon-Nikodym derivative are still
harder, when one allows also finitely additive measures into consideration. As we already
observed, even for real-valued finitely additive measures, there are examples in which the
derivative does not exist, hence it makes no sense to look for spaces with a property which
would take the place of RNP.
This is clarified by the next theorem, which is a Banach-valued version of the
approximate Radon-Nikodym-Bochner Theorem 3.1. We need first a definition.
Dehnition 5.1. If X is a Banach space, we shall say that it has the "approximate finitely
additive Radon-Nikodym property" (AFARNP), if for any measure space (?, ?,?),
where ? is a finitely additive X-valued BV measure, and for any ? > 0, there exists a ?-
integrable function /,; (which may be taken simple), such that
-/.
?(?)- ??
< л-
for any ? e ?.
The following result has been communicated to us by Anna Martellotti.
THEOREM 5.2. A Banach space X has AFARNP if and only if it has RNP.
PROOF. Assume X has RNP and let ? be a BV finitely additive measure on (?. 2?), with
values in X. Since ? is BV, it follows from [11] that it admits a Stone extension, i.e., there
exists a countably additive measure ? on Qs, the Baire ? -algebra of the Stone space S
associated to the quotient of ? with respect to ?? the family of all |?|-???11 sets.

276
D. Candeloro and A. Volcic
It is also known that ? is BV and that
By RNP there exists a Bochner-integrable function f:S->X such that
H(F) = Jfd\H\,
for any F &G$.
A density argument shows that, given ? > 0, there exists a C/a-simple function g:S^>X
such that
j \\?-?\\?\?\<?.
Let g = J2"=l Xj 1g, , with G, pairwise disjoint. Since G, e Qs, there exist pairwise disjoint
sets ? [,..., E„ in ?, such that /?([?,]) = G,, where h is the natural embedding of ?/??
in 5.
Put now ? = ?"=[ X[\ej, and let ? = ?\?\. It is easy to check that у = Jgd\jl\ and
from [ 11 ] we have
??-?| = ^??/-*№|.
and moreover ? — у = ? — ? and |? — ?\(?) = \? — v\(S), so we can conclude that
|?- ?|(?) < /j.
Conversely, assume X has AFARNP. Let ?: ? ->¦ X be a countably additive BV
measure. For each ? > 0 there exists ^:?->? such that
|M-/i)lMl| < »7·
Consider ?„ — ^ and let /„ = f,h. We want to show that (/,, )„ец is a Cauchy sequence in
?'(|?|). Indeed, for any n and w, if we put ?„ = /,,|?|, we have
ll/i. - /». II ? = /" ll/« - /». Ilx rflMl = |v„ - y,„ |.
Since the total variation is a norm in the space of all countably additive BV measures on
(?, ?), we have
\vn -vm\ ^ |y„ -?| + |?- ?„,|.
hence ||/„ - fm \\\ < ?„ + ?,„, and so (/„)„ек is Cauchy.
Since |?| is countably additive, L'(ImI) is complete and therefore there exists / e
L'(ImI) such that /„ -> / (in ?,'(|?|)). By definition, (/„)„eN is a defining sequence

Radon-Nikodym theorems
277
for /, namely / /„ ?? -> J f ??. From the definition of /„ it follows now the conclusion,
namely that ? = f ?\?\. ?
From the second part of the proof above, it follows also that an approximate Radon-
Nikodym-Bochner theorem for Banach-valued ?-additive measures does not hold, unless
the range space has RNP.
From now on, we shall limit ourselves to look for conditions, necessary and/or sufficient,
to ensure the existence of some kind of weaker Radon-Nikodym derivatives.
To be able to do that, besides the Bochner and Pettis-type integral, we shall also deal with
the so-called Gel'fand integral, and later on with the Bartle-Dunford-Schwartz integral.
DEnNlTlON5.3. Let ?: ? -> Rj be a finitely additive measure, and let X beanyBanach
space. Given a function /:?-> ?**, we say that / is Gel'fand integrable with respect
to ?, if:
(a) (/, x*} is ?-integrable, for all x* e X*.
(b) For every Ae ? there exists an element Уд е X such that
(x*,JA)= J (?,?*)??.
for any x* e X*.
(c) The function A -> JA is a ?-continuous finitely additive measure.
(d) For every ? > 0 there exists ? <??, such that ?(??) <e,and sup(„eW ||/(?)|| < ос.
When a finitely additive measure ?: ? -> X is given, with ?<? (throughout this
section we use always the ?-? definition of absolute continuity), we say that ? has a
Gel'fand-type derivative, if there exists a Gel'fand integrable function /:: ? -> X**, for
which J a = v(A) for all Ae ?, i.e.,
j (/,?*)?? = (?*,?(?)),
for any Ae ?, and x* e X*.
An example of such a derivative can be seen in Example 4.3, where the function / takes
values in /-?,, while the space X is cq.
Another interesting situation is described in Example 4.2, where the existence of a
Gel'fand derivative is a consequence of the Lifting Theorem [43,36,30,31] (see also
Chapter 28 in this Handbook [53]), in particular for the Lebesgue measure. Let us denote
by ? the lifting map. For any ? e ? = [0, 1], /(?) can be defined as the element of L^,
which associates 0(/?)(?) to every h e Lx. So we see that (/, h) turns out to be simply
the function <p{h), which is bounded and measurable, for each h e Lx = X*, hence (a)
is satisfied, and fA(f,h)dk = jAhdk= (h, \A) - {h, v(A)), for any A e ? and h e X*.
This proves (b), (c), and the fact that / is the Gel'fand derivative dv/dk.
Dealing with finitely additive measures, the existence of Gel'fand derivatives can be
derived from the following theorems, which include Example 4.3 as a particular case.

278
D. Candeloro and A. Volcic
THEOREM 5.4 ([13]). Let ? be any separable Banach space, and assume that (?, ?, ?)
is any finitely additive measure space, with ? nonnegative (and finite). Let us suppose that
a bounded finitely additive scalar measure T(y) is associated to every у е Y, such that
there exists ??(?)/??, and moreover \T(y)\(A) ?. \\\\\?(?) for every A e ? and у е Y,
where \T(y)\ denotes as usual the total variation ofT(y). Then, there exists a function
f-.?-?-?* such that:
A) f {¦)(}·) = ??(}·)/?? for every у eY.
The proof of Theorem 5.4 is in some sense reminiscent of the sketch of proof given for
Theorem 4.11. We skip the details, because they are too technical.
A direct consequence is the existence of bounded Gel'fand derivatives, under some
conditions.
THEOREM 5.5 ([13]). Assume that X is any Banach space, with separable dual, and let
(?, ?, ?) be as above. Suppose that ? : ? —> X is a finitely additive measure, such that
d{x*, ?)/?? exists, for every x* e X*. Assuming moreover that the set
\?(?)
is bounded, then there exists a bounded Gel fand type derivative ??/??.
PROOF. It is sufficient to apply Theorem 5.4, setting ? : = X*, and T(x*) = (x*, v). D
COROLLARY 5.6 ([13]). Let ?, (?, ?, ?), ? be as above. For the existence of a Gel'fand
derivative dv/?? it is necessary and sufficient that the following two conditions hold:
(i) d(x*, ?)/?? exists, for every ?* e X*, and
(ii) for every ? > 0 there exists ? e ?, ?(?') < ?, such that the set
( v(A)
S(H) := I ——-: ?<??. AcH. ?(?) > 0
\?(?)
is bounded.
PROOF. The necessity is clear, by definition of Gel'fand integrability. To see the converse,
let us take an increasing sequence of sets (Я„)„ек, such that ?(//,') 4- 0* an<J with S(H„)
bounded for each n: then, applying Theorem 5.5 to H„ one gets a Gel'fand derivative /„
on #„, and pasting together the functions /, gives the required derivative. D
It is interesting to compare Corollary 5.6 with the result due to Moedomo-Uhl, quoted
in Theorem 4.17: in some sense, if compactness for the average sets is replaced by
boundedness, then Gel'fand derivatives take the place of Pettis derivatives.
The requirement concerning the existence of d(x*, ?)/?? in the previous theorems may
look too restrictive; however, we can see that it is satisfied under rather mild conditions.
One of these has been outlined in [13], where the following well-known theorem by
Rybakov is used [49].

Radon—Nikodym theorems
279
Let us recall that a measure ? on ? is said to be .s-bounded, if for every disjoint sequence
of sets E„ e ?, we have lim„ ?(?„) — 0.
THEOREM 5.7 ([49]). Given an s-boundedfinitely additive measure ? : ? -» X (here,
? may be just an algebra), there always exists some element x* e X*, such that ? <<C
\{x\v)\.
Definition 5.8. If v. ? -> X is .s-bounded and л* e X* is such that ? « \(x*. v)\,
then the measure \(x*, v)\ is called a Rybakov control for v. We can (and do) assume that
??*?? = ?·
THEOREM 5.9 ([13]). Let X be any Banach space with separable dual, and let (?. ?) be
any measurable space. Assume that ?: ? -> X is any s-bounded finitely additive measure,
and let ? be any Rybakov control for v. If ? has weakly closed range, then a Gel'fand
derivative dv/?? exists if and only if condition (ii) of Corollary 5.6 holds.
PROOF. Having in mind Corollary 5.6, it only remains to show that d{x*. ?)/?? exists, as
soon as x* e X*. Let us denote by y* the element of X*, such that ? = \(y*, v}\. As the
range of ? is weakly closed, we see that the measure {sx* — ry*, v) =s(x*, v) — r(v*, v)
has closed range, hence it admits a Hahn decomposition, for all real numbers s and r,
and all elements x* e X*. Now, an easy consequence of Corollary 1.3 gives the desired
derivative. ?
A more general formulation can be found in the next statement.
THEOREM 5.10 ([13]). Assume that X* has the RNP, and let ?: ? -> X be any s-
bounded measure, with separable weakly closed range. Then (ii) of Corollary 5.6 is a
necessary and sufficient condition for the existence of a Gel fand derivative ??/??, where
? is any Rybakov control for v.
PROOF. If ? denotes the separable subspace of X, generated by the range of v, from
Theorem 4.12 we find that Y* is separable, hence the conclusion follows from the previous
results. ?
In order to obtain Bochner (i.e., strong) derivatives in the finitely additive case, one
has to make different assumptions, as Hagood in [26] showed. The definition of Bochner
integrable function in the finitely additive setting is formally the same as in the ?-additive
one (see also [22]). For the sake of simplicity, we shall give here just a particular form of
Hagood's result, also related to a previous work by Maynard [40].
THEOREM 5.11. Let (?, ?, ?) be any nonnegative finite and finitely additive measure
space, and let ? : ? -> X be any Banach-valuedfinitely additive measure, K</i. Then the
following are equivalent:
A) there exists dv/?? in the Bochner sense:
B) for every ? > 0 and S > 0, there exist С e ?, ?@ > 0, and a e ]0, 1 [ such that:

280
D. Caiideloro and A. Volcic
Bi) ?(?)<?.
Bii) the set S(C) := {~$-}: A e ?, А С С, ?(?) > 0} is bounded,
Biii) for all ? с С, ? е ?, ?(?) > 0, ?/геге ех/ш F с ?, F e ?, such that
M(F) > ??(?) and diam(S'(F)) < ?.
PROOF. First, let us prove that A) =>· B). We denote by / the derivative dv/??, and fix
?, S > 0. By strong measurability of /, there exists a simple function g = ?"=\ ?? 1?,-,
such that the sets ?, form a finite partition of ?, and ?*({? e ?2: ||/(?) - g(o>)|| >
?/4}) < 5. Let us choose a set Я е Г, such that {we ?: ||/(?)-#(?)|| > ? /4} с Я, and
?(#) < ?. Set now: С = Нс. Then, Bi) is satisfied. Moreover, if A e ?, Ac C, we have
||/(?) - g(cu)\\ iC ?/4, for all ? e A, hence ||v(A)|| ^ /д ||g||d/i + |?(?) ^ ???(?),
where ? = тах{||/3,|| + ?/4). So, also Bii) is satisfied. Finally, choose a = ^: if ? is
any fixed element of ?1 contained in C, denote by ?* one of the elements ?, such that
?(? ? ?*) > ^?(?), and put F = ?* ? ?. So, m(F) > ??(?). Now, let us prove that
the diameter of S( F) is less than ?. Let /3 * denote the value of g in ? *, and choose any set
Ae ?, with А С F, and ?(?) > 0. We get
v(A) ^!???? ^?*?(?) | !A(f-g)dn
?(?) ?(?) ?(?) ?(?)
from which we deduce that || —^ - ?*\\ < ?/2, and therefore diamE"(F)) < ?.
We now turn to the converse, i.e., B) => A). Fix ? > 0, and ? = ?. Then, let С and a
be the corresponding elements, obtained from B). Now, apply Biii) to ? = С: we find
a set F| e ?, with F\ С С, and m(F|) > ??@. and such that diamE"(F|)) < ?. If
?№) = ?@ we are finished; otherwise, apply again Biii) to ? = C\F|, thus finding
F2 С C\F|, with F2 e ?, such that ?№) > «m(C\F|), and diamE(F2)) < ?. So, F|
and F2 are disjoint members of ?, both satisfying diamtSXF,)) < ?. By an exhaustion
argument, it is possible to get a (finite or countable) family (F„)„epj of subsets of С of
positive measure, each satisfying diam(S'(F„)) < ?, and such that ]??(^,) = ?@. Now
define fE = ]T/j„ lfi, where /3„ is, for each n, any element of S(F„). It is now easy to
show that fe is Bochner-integrable (indeed, ff is bounded, by Bii)), and
L
v(A)- / ?,??
A
< ?.
for any A e ?, AcC.
A routine argument now gives an increasing sequence (Ck)keK, with ?{€[) i 0, and a
corresponding convergent sequence (/; )??€??, whose limit is the required derivative. D
To conclude the section, we turn to the so-called Bartle-Dunford-Schwartz integral,
which allows to integrate a scalar function with respect to a vector-valued measure. We
shall refer to the paper [37], which in turn was inspired at Musiat's paper [42].
Actually, in [37] locally convex-valued measures were considered, but here we shall
limit ourselves to the special case of Banach-valued ones, for simplicity.

Radoii-Nikodvm theorems
281
For the rest of this section, X will denote a Banach space, ?, у: ? -> X will be s-
bounded finitely additive measures.
Dehnition 5.12. We denote by V the space of all measurable functions /: ? -> R.
such that / e L|(|(jc*,m)|), for any x* e X*.
Given a function / e 73, we say that / is ?-integrable if for every A e ? there exists an
element v(A) e X such that
{x*,v(A))= ? /?·(?*?)
J A
for every x* e X*. We then set: v(A) = fA f??.
In general, when / is ?-integrable, the function у is a finitely additive measure, but
it satisfies a much stronger condition than absolute continuity with respect to ?. This is
evident even in two-dimensional spaces: one can choose the measure space ([0. 1], ?. ?),
X := ?2, and then define: ?(?) = (fAxdx.X(A)). v(A) = (?(?), fA xdx). It is clear
that ? and у are both equivalent to ?, hence у <<c ?, but there is no function /: [0, 1 ] -> ?
such that v(A) = fA f ??: indeed, such a function should satisfy:
?(?)= / xf(x)dx and / .rd.v= / /(jc) ^jc
^A J A J A
for any Ael, thus f(x) — ? a.e. and xf{x) = 1 a.e., which is clearly impossible.
The key property is introduced in the following definition.
DEHNITION 5.13. We say that у is scalarly uniformly absolutely continuous with respect
to ?, and write: у <k< ?, if for every ? > 0 there exists S > 0 such that for every x* e X*
and every A e ? one has the implication:
\(x*, ?)\(?) < S => |(.v*,y)|(A)<e.
We say that у is scalarly dominated by ? if there exists a positive number ? > 0 such
that
|<jc*. v)|(A) ^ Af |<jc*, /i)|(A)
holds, for any x* e X* and any A e ?.
It is easy to see that, in case /: ? -> ? is bounded and ?-integrable, the measure y,
defined in Definition 5.12 above, is scalarly dominated by ?, and satisfies у <sk ?.
The Radon-Nikodym theorem stated in [37] asserts that, under certain conditions,
depending essentially on the finite additivity of the involved measures, scalar domination
is equivalent to scalar uniform absolute continuity, and that each of the two conditions is
also sufficient for the existence of a bounded derivative dv/??. The conditions we shall
mention here are somewhat stronger, for the sake of simplicity.

282
D. Candeloro and A. Volcic
THEOREM 5.14 ([37]). Let ? be any Rybakov control for ?. Assume that for every
x* e X*, the set
1 ?(?) J
is bounded. Assume also that the ranges of the measures ? and ? are closed. Then the
following are equivalent:
A) у <sk ?;
B) у is scalarly dominated by ?;
C) there exists a bounded ?-integrable function f: ? -> Ш, such that
(x*,v)(A)= f /?(?*,?),
J A
for any A e ?, and x* e X*.
6. Further results
In this section, we shall see some recent results, concerning different situations: one of
them concerns Radon-Nikodym derivatives for measures taking their values in locally
convex spaces, and more particularly in nuclear spaces; another situation concerns
multivalued measures, taking values in Banach or locally convex spaces; yet a different
result concerns Riesz space-valued measures; finally, we shall mention some results
concerning Radon-Nikodym derivatives for a different kind of integral, the so-called
Sugeno integral.
As usual, (?, ?, ?) denotes a measure space, where ? is a ?-algebra, and ? is any
nonnegative, finitely additive measure, taking values in [0, +oo[.
We start with some results concerning (finitely additive) measures, taking values in a
locally convex Hausdorff space X. Some definitions are needed, in order to put appropriate
conditions on X. Throughout this section, X* denotes the strong dual space of X.
Dehnition 6.1. Let X be any locally convex space, and let В denote any bounded,
nonempty subset of X. В is said to be bipolar if В = В00, according with the duality
(X, X*). Given any bipolar set В с X, we denote by Хв the subspace of X which can
be absorbed by В (i.e., the space of those elements ? e X such that rx e В for some
positive scalar r); thus the Minkowski functional рв of В can be defined on Хв (we recall
that рв(х) = inf{r > 0: f e B], for ? e Хв)- We can endow Хв with the semi-norm рв
and then consider the normed space Хв/ %, where the equivalence relation is defined by:
x s» ?' «=> pB(x - x') = 0. We shall denote by X(B) the completion of such normed
space. We say that X satisfies the property (SP) if X(B) is separable, for every bipolar set
? CX.
Obviously, a separable locally convex space X satisfies (SP).

Radon-N\kod\m theorems
283
DEnNlTlON 6.2. We say that a locally convex space X has the property (SP)' if X*(B°)
is separable, for all bounded subsets В с X. If X* is separable, then X has (SP)'.
When у takes values in a space X with property (SP)', then a Radon-Nikodym theorem
in the Pettis sense has been proved in [9]. The proof is too long and technical to be
presented here.
THEOREM 6.3. Suppose X has property (SP)', and ?: ? -> X is any ?-continuous
finitely additive measure, satisfying the following two conditions:
A) there exists d{x*. v)/d^,forany x* e X*:
B) the set S := {^щ: A e ?, ?(?) > 0} is weakly relatively compact in X.
Then there exists a bounded weakly measurable function g : ? —> X, such that:
('(?*,8(-))?? = (?\?(?)).
J A
for any x* e X* and all Ae ?.
This theorem is applied in [9] to the case of dual-nuclear spaces. In order to state other
results, we need some more definitions (see also [45]).
Dehnition 6.4. Let X and ? denote two locally convex Hausdorff spaces, and let
?: X -> У be any continuous linear map. We say that ? is nuclear if there exist:
(a) a sequence (?,,),,^ in l\;
(b) an equibounded sequence (x*)„m ш X*·
(c) a bounded sequence (yn)nen in Y, such that
Ф(х) = ^2^Уп(х*,.х)
for any ? e X.
Dehnition 6.5. A locally convex Hausdorff space X is said to be nuclear if every linear
continuous map ?: X -> У is nuclear, for every Banach space Y.
This definition is not the original one, due to Grothendieck, but we have chosen this
equivalent property, because we think it is easier to work with.
Nuclear spaces enjoy very interesting properties: for instance, it follows from the
classical definition of a nuclear space that it is the projective limit of Hubert spaces.
Another important property is that every bounded set in a nuclear space is pre-compact.
A useful condition for nuclear spaces is quasi-completeness, i.e., every closed bounded
subset is complete. Thus, if a nuclear space X is quasi-complete, all closed bounded subsets
of X are compact (hence, X is Montel). If the strong dual X* is nuclear, then the locally
convex space X is said to be dual-nuclear. It turns out easily that a quasi-complete dual-
nuclear space X is semi-reflexive, i.e., the canonical embedding с: X -> X** is onto.

284
D. Candeloro and A. Vollic
From all these remarks, one easily realizes that many interesting results can be found on
measures taking values in spaces of this kind. In [9] there are listed some, for measures
taking values in a dual-nuclear space. We just recall those results, which are strictly
connected with Radon-Nikodym derivatives.
THEOREM 6.6 ([9]). Let X be a quasi-complete and dual-nuclear space. Then every
bounded finitely additive measure ?:?—>? is s-bounded, and admits a Rybakov contml.
THEOREM 6.7 ([9]). Let X be as above, and assume that ?: ? -> X is any bounded
finitely additive measure, such that:
A) (x*, y) «? , and there exists '^ff1 in L^, for all x* e X*;
B) the set S := [j^: ?&?, ?. (A) > 0} is bounded.
Then there exists a Bochner-Гуре derivative j-.
We remark here that "Bochner" means here that the function j^ is the limit in measure of
a sequence of simple functions, and the integral is the limit of the corresponding integrals.
This looks like a strong conclusion, and it is worth mentioning how it is derived: from the
assumptions we see that the mapping ?: X* -> ?.^, defined as T(x*) = ' ")?? , is nuclear.
(For more details we refer the reader to [9], where also completeness of Z-oc(m) is proved.)
Therefore one can write:
T(x*) = J2^(x\e„)y„ ?/ef.
where (A„)„epj e /|, (е„)пец is a bounded sequence in X** = X, and (v„)„ек is a bounded
sequence in L^. Choosing a bounded representative /„ from the class of y„, and putting
g„(o>) := k„e„f„(w) for all ? e ?, the series ]?#„ converges strongly to the desired
derivative.
An even stronger result holds for ?-additive measures.
THEOREM 6.8 ([9]). Let X be as above, and assume that ? is ?-additive. Ifv-.?-* ?
is any measure, К<Д, there exists a Bochner-type derivative ??/??.
We now turn to multimeasures, according with the definitions of [16] and [38]: we will
just recall the notations and the most relevant results in that setting.
DehnitiON 6.9. Given a Hausdorff locally convex space X, the family of all nonempty,
closed, convex, bounded subsets of X will be denoted by CC(X)\ if Q denotes the set of all
continuous seminorms on X, for every ? e Q and every pair of elements А, В е CC(X),
we set:
ep(A, B) = sup inf p(x - y).
.veA.veB

Radon-Nikodym theorems
285
and
dp(A, B) = max{e,,(A. B), ep(B. A)}.
For any A e CC(X), we set also: hp(A) — dp(A. {0}).
As dp is a pseudo-distance, the family {hp: ? e g) defines a topology on C((X), which
is compatible with the following addition: A + В := [a + b: a e A. be B].
We remark that the same topology arises if g is replaced by some absorbing subset
go· Also, if X is complete, so is C((X) with this topology. We shall assume, from now
on, that X is complete. Given an absorbing subset g0 С ?, a subset В с C( (X) is go-
uniformly bounded if sup eg supAeiJ hp(A) < oc.
Any finitely additive measure ? : ? -> C( (X) is said to be a multimeasure in X.
Dehnition 6.10. Given a multimeasure у: ? -> С, (X), for every p e g the p-varia-
tion of у is defined for all ? e ? as:
vp(E)= sup УЧДуСА,)),
(A,-)eP(?)/e/
where P(E) denotes the family of all finite partitions of ? into sets A, e ?1.
We say that у has bounded variation if yp(i2) < oo, Wp e g.
Dehnition 6.11. Given a multimeasure у : ? -> C( (X), and a finitely additive measure
? : ? -> Rj, we say that у is absolutely continuous w.r.t. ? ( у <? ?) if for every ? > 0 and
every peg there exists a <5(?. ?) > 0 such that ?(?) < S implies vp(E) < ?.
We now turn to integration of multifunctions. Our presentation is necessarily concise,
and restricted to the Radon-Nikodym problem; we refer to Chapter 14 in this Handbook
by C. Hess [28], for a more detailed exposition.
Dehnition 6.12. Let ?: ? -> R(| be any finitely additive measure. A map F: ? ->
C( (X) is simple if it can be written as:
where the C, 's are elements of C, (X), and the A, 's are disjoint elements of ?.
The integral of F with respect to ? is defined as:
/^М = ]ГМ(А,)С,.
•^ / = l
As each C, is convex, the definition of integral does not depend on the representation
of F.

286
D. Candeloro and A. Volcic
In case F is not simple, F is said to be totally measurable if there exists a sequence
(Fii)neN of simple multifunctions such that
(al) the function dp(F„, F) is measurable for every ? e N and every ? e Q,
(a2) the sequence (dp(F„, F)) ?-converges to 0 for every ? e Q.
Moreover, we say that F is ?-integrable if there exists a sequence (F„)„eh- of simple
functions, satisfying (al) and (a2) above, such that
(a3) lim„1.,1_3C/d/,(F„,F„1)^=Oforallpe Q.
Such a sequence will be called a defining sequence for F.
In case <2o is any absorbing subset of Q, and the conditions (a2) and (a3) above hold
uniformly with respect to ? e Qq, then F will be said to be strongly integrable (with respect
to <2o, if confusion can arise).
In case F is integrable, and (F„)„ei; is any defining sequence for F, then for every
F e ?1 the sequence (fE F„ ??),,^; is Cauchy in CC(X), and hence convergent to some
element of CC(X), which will be called the integral of F on F, and denoted by /? F ??.
This integral does not depend on the defining sequence (see [38] for details).
PROPOSITION 6.13. If F :? -> С, (X) й ?-integrable, then ? -> Д- Fi/? ife/i/ies ?
multimeasure, which is absolutely continuous with respect to ?.
In order to state a Radon-Nikodym theorem, we need one more definition.
DEFINITION 6.14. Given a multimeasure ?: ? -> C((X). we say that ? hboundedii
sup hp(v(A)) = Mp < oo
for every ? e Q.
If this is the case, set: <2,, := {-jj~: ? e Q. Mp > 0).
If ? : ? -> Rj is finitely additive, for every set ? e ?. and every ? > 0. we put:
5(?):=I^S: Fer· M(F)>o):
lM(F) J
Sp(?. ?) := {С e C< (X): dp(v(F). Cm(F)) ^ e/i(F) VFeT. F с F}:
S(E,e):= P| Sp(?.e).
Finally we can state one of the Radon-Nikodym theorems from [38]. The proof is based
upon the technique of Hagood, see Theorem 5.11.
THEOREM 6.15 ([38]). Let v. ? -> С, (X) be a multimeasure, ? «i ?, where ? is any
nonnegativefinitely additive measure on ?. Assume that
A) ?(?) is Qv-uniformly bounded:
B) for every ? > 0 and every ? e ?, ?(?) > 0. there exists a sequence of pairwise
disjoint subsets (E„)„ey: of E, with E„ e ?, such that ?(?) = ]?„?(?„), and
such that S(E„. ?) фЪ for any п.

Radon-Nikodym theorems
287
Then there exists a Qv-strong!y measurable multifunction G: ? -> CC(X), with Qv-
uniformly bounded range, such that:
/ Gdix = v(E),
for all Ее ?.
A quite different problem arises when the measures are both vector-valued. An example
was given in the last part of Section 5, with Banach-valued measures; but sometimes it
is interesting to consider a richer structure, and to assume that the vector measure takes
values in a Riesz space: this is convenient, when the linear spaces have a natural order
structure, and the "order" convergence is not topological, like in the space Lq, with the a.e.
convergence.
We shall present here a very simplified version of the results, and refer to [5 ] for further
details or generalizations.
Assume that R is an order-complete Riesz space (here, order-complete means that every
nonempty subset F С R which is bounded from above has a least upper bound in R).
Order-completeness allows to define sup, inf. lim sup and lim inf.
We first mention the so-called Riemann-type integral for bounded functions with values
in a Riesz space.
Dehnition 6.16. Given a bounded function /: [a. b] -> R, we say that / is Riemann-
integrable (with respect to Lebesgue measure) if
sup / s(x)dx= inf / t(x)dx
veS/ J teT, J
where Sf (Г/, respectively) is the class of all /?-valued step functions s ?. f (t > /,
respectively), and the elementary integral of a step function is defined in the obvious way.
This integral is well-defined, and is a linear monotone /?-valued functional. Moreover,
one can see (by usual techniques) that monotone functions are integrable.
Definition 6.17. Given a positive finitely additive measure ?: ? -> R, we say that ? is
?-additive if, for every decreasing sequence (En),ie\\ in ?, E„ \. ? implies inf{ ? (?„): ? e
?} = ?(?).
Given a positive finitely additive measure ?:?^> R, and a bounded measurable
function f-.?-* Rq", we say that / is integrable with respect to ? if the following
Riemann-type integral is finite:
f fdii = [ ^ей: f(x)>t})dt.
Jn Jo
This means that the set {/()M ?({? e ?: f(x) > t})dt: ? > 0} is bounded in R, and
fn f ?? is its least upper bound; we observe that the function g(t) = ?({? e ?: /(л) >
t}) is monotone decreasing, and hence Riemann integrable in every interval [0. M].

288
D. Candeloro and A. Volcic
If / is integrable, then /1 ? is integrable too, for every ? e ?, and ? -> / /1 ? ?? :=
§Ef ?? defines a measure ?, which is absolutely continuous with respect to ?, in the
sense that limsupv(A„)=0 as soon as (A„)„€f: is any sequence from ?. such that
limsupM(A„) = 0.
The Radon-Nikodym theorem for Riesz space-valued measures can be formulated
exactly like Theorem 1.2, with (bl) and (b2) replaced respectively by (b'l) and (b'2),
and (b3) replaced by absolute continuity.
THEOREM 6.18. Let ? and ? be positive finitely additive measures defined on ? and
taking values in R, with ?«?. Then the following are equivalent:
(a) There exists a measurable and ?-integrablefunction f : ? —> [0. oof, such that
j fd? = v{E)
for all ? e ?.
(b) there exists a family of sets (A,-) in ?, for r > 0, such that, for any r > 0:
(bl) v(E) > r^(E),forany Ее ?, ? с Аг:
(Ь2) v(E) <; r^(E),forany ? e ?, ? с Acr.
Though the formulation (and the proof) of this theorem is quite similar to Theorem 1.2,
the consequences are not so strong: in fact the same example given before Definition 5.13
shows that even for ?-additive measures, absolute continuity is not sufficient to ensure the
existence of the derivative.
A different approach to the idea of integration was presented by Sugeno in [54]. We
shall give here quite a short outline of the involved concepts, and also a very simple
Radon-Nikodym theorem for this integral. For similar topics, see also Chapter 33 in this
Handbook [2].
Definition 6.19. Let (?, ?) be any measurable space. A fuzzy measure on this space
is a mapping m : ? -> [0, +oo[, such that:
A) m@)=O;
B) m(A)^m(B), whenever А, В e ?, А С S;
C) if (A„) is any monotone sequence in ?, then w(lim F„) = Yimm(F„).
If m is a fuzzy measure on (?, ?), then the triple (?. ?, m) is called a fuzzy measure
space.
Dehnition 6.20. Let (?, ?. m) be a fuzzy measure space, and let h : ? -> Ej be any
measurable function. For each A e ?, define:
(S) hdm :=sup{<* лш(АП Fa)\.
JA a>()
where Fa := [? e ?: h(w) > a].
The number (S) fA h dm is called the Sugeno integral of h on the set A.

Radon-Nikodym theorems
289
(Notice that the definition of Sugeno's integral is reminiscent of the Choquet integral,
one just replaces л with ordinary multiplication, and ? with addition.)
From this definition, it follows immediately that
(S) / hdm<:m(A), УАеГ.
J A
We list a number of results, concerning this integral.
THEOREM 6.21 ([14]). Let (?, E.m) be any fuzzy measure space, and let h-.? -> R(j
be any measurable function.
A) If h is a constant function, h{x)=a. then
(S) / hdm=a/\m(A) for any A e ?.
J a
B) Ifh and ti: ? -> Rj are measurable functions, with h' ^ h, then
(S) / h'dm ? (S) hdm for any AeZ.
C) Ifh = 1 д, for some Ae ?, then
(S) / hdm =m(A).
D) //'(/2„)„epj is any monotone sequence of measurable functions, such that h„ —> h.
then
lim(S) / h„dm = (S) / hdm.
J A J A
for any Ae ?.
From part D) of the previous theorem, it follows that A -> (S)fAhdm is a fuzzy
measure, not greater than m. A useful characterization of the Sugeno integral is the
following:
THEOREM 6.22 ([14]). Given a measurable function h-.? ->¦ Rj, define, for a > 0:
Aa := {we ?: h(a>) > a\. Fa := {we ?: h(w) ><*}.
If(Ba) is any decreasing family in E.fora > 0. such thatAa С Ва С Fa, then
(S) / hdm := sup{<* Лт(А П Ba)}
J A o>()
for all A e ?.

290
D. Candeloro and A. Volcic
Moreover, if I denotes the set of all a e E(|, such that a ^ m(Fa), then
(S) / hdm = max/.
This result allows us to state a Radon-Nikodym theorem.
THEOREM 6.23 ([14]). Let m and ? be two fuzzy measures on (?, ?), with ? ^ m. The
following are equivalent:
(a) There exists a measurable function /7: i? —» R(|, such that
y(A) = (S) / hdm
for any A ? ?.
(b) There exists a decreasing family (Bu) in ?, with a e M.q, such that:
(bl) ?(??\??) ^? лт(АГ)Ва), and
(Ь2) у(А)^т(АПВу(А)).
for all A ? ?, and all a e Rj.
For a richer treatment of Sugeno integral, and its properties, we refer the reader to [54]
and [55].
Appendix. Singularity and decomposition theorems
In this appendix we will present some known facts about singularity of measures, the
Lebesgue decomposition theorem and some related decomposition theorems, which are
relevant in connection with the Radon-Nikodym theorem.
We assume that all the measures have been already completed in the Caratheodory sense.
Definition A.l. We say that a measure ? is degenerate, if it takes no finite, nonzero
values, i.e., its range is contained in {0. со).
In connection to this definition, let us first mention the following result, due to
N.Y.Luther [35].
THEOREM A.2. Let ? be a measure on the ?-algebra ?. Then there exists a unique
decomposition ? = ? \ +?2, where ? \ is semifinite and ?? is degenerate, with the property
that if ?' = ?', + ?'-,, with ?\ semifinite and ?'-, degenerate, then ?\ ^ ?\ and ?? ^ ?-,.
This result and the fact that degenerate measures are not very interesting from the point
of view of their representation as integrals, justify the restriction to semifinite measures in
Section 2 of this chapter. We will make the same assumption in the Appendix.
If ? is a measure on (?. ?), we say that ? is concentrated on A e ?, if ?(? \ A) = 0.

Radon-Nikodxm theorems
291
Definition A.3. Suppose ? and у are two measures on ?, and suppose there exists a
pair of disjoint sets A and В in ?, such that ? is concentrated on A and у is concentrated
on S. Then we say that у and ? are mutually singular (or that у is singular with respect
to ?) and write
?±?. E)
Obviously, this relation between measures is symmetric, so E) holds if and only if
? J_ y.
The following properties are quite obvious:
• If v\ J_ ? and vi -L ?, then v\ + yi J_ ?.
• If У| <<C ? and Ут J_ ?, then v\ J_ ??.
• If у <K ? and у J_ ?, then у = 0.
THEOREM A.4. /f ? and ? are two measures on (?. ?) and ? is ?-finite, then there exist
У0 <K ? and ?\ J_ ? such that ? = vq + ?\. The decomposition is unique.
Let us sketch the proof. Suppose first that ?(?) < oo, and let a = sup{y(S): В е ?,
?(?) = 0). It is easy to prove, using arguments introduced in Section 2. that there
exists С e ?, such that v(C) = a and ?@ = 0. Define now v\ (E) = v(E П C) and
vq(E) = v(E \ C). It is easy to check that uo <?? ? and v\ J_ ?.
If у is ?-finite, we consider a measurable decomposition {A„} of ? such that ?( ?„) <
oo for all и, and apply the argument above to each A„.
If у is not ?-finite, the conclusion of the previous theorem may fail. If for instance ? is
the Lebesgue measure on [0, 1] and у is the counting measure, such a decomposition does
not exist.
We need now two definitions. The first is due to R.A. Johnson [32], the second one is
taken from [59].
DEFINITION A.5. We say that у is S-singular with respect to ?, denoted ?$?, if given
Ее ?, there exists F e ?, F с ?, such that v(E) = v(F) and m(F) = 0.
DEFINITION A.6. We say that у is quasi-singular (Q-singular) with respect to ?, denoted
у Q ?, if there exists Ae ? such that у is concentrated on A and moreover if у (?) < oo
?????(???) = 0.
Singularity implies Q-singularity and the latter implies S-singularity. The three concepts
are not on the other hand equivalent. The counting measure on [0, 1] is Q-singular with
respect to the Lebesgue measure, but it is not singular.
To show that S-singularity does not imply Q-singularity we need a more elaborate
example. Let us consider the measure space defined in [27, Exercise 31.9], which provides
a non-Maharam measure ? as sum of two measures ? and ?. As noted by R.A. Johnson,
ySa anda S y,but it is not у ±?. On the other hand it can not be yQa (or ? Q y) because
of the following proposition [59].

292
D. Candeloro and A. Volcic
Proposition A.7. /f vQm and ?$?, then ? ±?.
R.A. Johnson proved the following decomposition theorem [32].
THEOREM A.8. If ? and ? are two measures on (?. ?), then there exist щ <К ? and
у ? ?? such that ? = vo + ? ?. Always v\ is unique. Moreover, ifv = v'0 + v\ is another such
decomposition, vq ^ v'Q.
The advantage of the previous theorem, compared to Theorem A.2 is that no assumption
is required on ? and у (not even semifiniteness). The disadvantage lies in the weakness of
the concept of S-singularity.
Q-singularity stays in between and shares some good and bad aspects of both concepts.
A reasonable assumption (specially if we deal with the Radon-Nikodym theorem) provides
the following decomposition theorem (Theorem 2.1 in [59]).
THEOREM A.9. If ? + ? is Maharam, then ? admits ?(? + v)-unique decomposition ?,·,
i = 1, 2,3, such that if we put y,(?) = y(? ? ?,) and ?,(?) = ?(? ? ?,) for i = 1,2,3,
then
viQM, F)
M2Qv, G)
V3«M3«V3· (8)
Moreover, щ and ??, are strongly comparable.
The set ?\ is defined as the least upper bound in the measure algebra associated to
(?, ?, ? + у) of the family of sets
{?: ? e ?, y(?) < oo. ?(?) = 0}.
Similarly we define ??. The major difficulty in the proof is to show that ?/, for /' = 1,2
are both ?- and у-measurable.
References
[1] A. Basile and K.P.S. Bhaskara Rao. Completeness of Lp-spaces in the finitely additive setting and related
stories. J. Math. Anal. Appl.. to appear.
[2] P. Benvenuti, R. Mesiar and D. Vivona. Monotone set functions-based integrals. Handbook of Measure
Theory. E. Pap. ed.. Elsevier. Amsterdam B002). 1329-1379.
[3] P. Berti, E. Regazzini and P. Rigo. Finitely additive Radon-Nikodym theorem and concentration function of
a probability with respect to a probability. Proc. Amer. Math. Soc. 114 A992). 1069-1078.
[4] G. Birkhoff. Integration of functions with values in a Banach space. Trans. Amer. Math. Soc. 38 A935).
357-378.
[5] A. Boccuto. On the de Giorgi-Letta integral with respect to means with values in Riesz spaces. Real Anal.
Exchange 21 A995/96). 793-810.

Radon-Nikodvm theorems
293
[6] S. Bochner. Integration von Funktionen deren Werte die Elemente eines Vectorraumes sind. Fund. Math. 20
A933). 262-276.
[7] S. Bochner, Additive set functions on groups, Ann. of Math. B) 40 A939). 769-799.
[8] N. Bourbaki, Elements de nmthematique, Liyre VI, Integration. Hermann. Paris A956-65).
[9] J.K. Brooks. D. Candeloro and A. Martellotti, On finitely additive measures in nuclear spaces. Atti Sem.
Mat. Fis. Univ. Modena 46 A998), 37-50.
[10] D. Candeloro and A. Martellotti, Su alcuni problemi relalivi a misure scalari sub-additive e applicazioni al
caso dell'additivita finita. Atti Sem. Mat. Fis. Univ. Modena 27 A978), 284-296.
[11] D. Candeloro and A. Martellotti. Sul rango di una massa vettoriale. Atti Sem. Mat. Fis. Univ. Modena 28
A979). 102-111.
[12] D. Candeloro and A. Martellotti. Geometric properties of the range of two-dimensional qua si-measures
with respect to Radon-Nikodym property. Adv. in Math. 93 A992). 9-24.
[13] D. Candeloro and A. Martellotti, Radon-Nikodvm theorems for vector-valued finitely additive measures.
Rend. Mat. Roma 12 A992), 1071-1086.
[14] D. Candeloro and S. Pucci, Radon-Nikodym derivatives and conditioning in Fuzzy Measure Theory.
Stochastica 11 A987). 107-120.
[15] M.D. Carrillo. Daniel integral and related topics, Handbook of Measure Theory, E. Pap, ed.. Elsevier.
Amsterdam B002), 503-530.
[16] C. Castaing, A. Touzani and M. Valadier. Theoreme de Hoffman-Jorgensen et application aux a mart
multivoques, Seminaire d'analyse convexe Montpellier A984), Expose No. 7.
[17] S.D. Chatterji, Martingale convergence and the Radon-Nikodvm theorem in Banach spaces. Math. Scand.
22A968), 21-41.
[ 18] W.J. Davis and R.R. Phelps, The Radon-Nikodvm property and deniable sets in Banach spaces. Proc. Amer.
Math. Soc. 45 A974), 119-122.
[19] D. Denneberg. Non-Additive Measure and Integral. Kluwer. Dordrecht A994).
[20] J. Diestel and J.J. Uhl, The Radon-Nikodym theorem for Banach space valued measures. Rocky Mountain
J. Math. 6A976), 1-46.
[21] J. Diestel and J.J. Uhl, Vector Measures. Math. Surveys, Vol. 15. Amer. Math. Soc., Providence. RI A977).
[22] N. Dunford and L. Schwartz. Linear Operators. Vol. 1. Interscience. New York A958).
[23] N. Dunford and B.J. Pettis, Linear operations on summable functions. Trans. Amer. Math. Soc. 47 A940).
323-392.
[24] D. Fremlin, Decomposable measure spaces. Z. Wahrsch. Verw. Gebiete 45 B) A978). 159-167.
[25] G.H. Greco, Un teorema di Radon-Nikodvm per funzioni d'insieme subadditive, Ann. Univ. Ferrara 27
A981), 13-19.
[26] J.N. Hagood, A Radon-Nikodym theorem and Lp-completeness for finitely additive vector measures,
J. Math. Anal. Appl. 133 A986). 266-279.
[27] PR. Halmos. Measure Theory, Van Nostrand, Princeton A956).
[28] C. Hess, Set-valued integration and set-valued probability theory: An overview. Handbook of Measure
Theory, E. Pap. ed., Elsevier, Amsterdam B002). 617-673.
[29] R.E. Huff. Dentability and the Radon-Nikodym property. Duke Math. J. 41 A974). 111-114.
[30] A. Ionescu Tulcea and C. Ionescu Tulcea. On the lifting property I.}. Math. Anal. Appl. 3( 1961), 537-546.
[31] A. Ionescu Tulcea and C. Ionescu Tulcea. On the lifting property II. Representation of linear operators on
spaces L'E, 1 sC r < oo, J. Math. Mec. 11 A962), 773-795.
[32] R. A. Johnson. On the Lebesgue decomposition theorem. Proc. Amer. Math. Soc. 18 A967). 628-632.
[33] J.L. Kelley, Decomposition and representation theorems in measure theory. Math. Ann. 163 A966), 89-94.
[34] D. Kolzow, Differentiation von Mafien. Lecture Notes in Math.. Vol. 65. Springer. Berlin A968).
[35] N.J. Luther, A decomposition of measures. Canad. J. Math. 20 A968). 953-958.
[36] D. Maharam. An algebraic characterization of Measure Algebras. Ann. of Math. 48 A947), 154-157.
[37] A. Martellotti, K. Musial and A.R. Sambucini. ? Radon-Nikodym theorem for the Bartle-Dunford-
Schwartz integral with respect to finitely additive measures. Atti Sem. Mat. Fis. Univ. Modena 42 A994).
343-356.
[38] A. Martellotti and A.R. Sambucini. A Radon-Nikodym theorem for multimeasures. Atti Sem. Mat. Fis. Univ.
Modena 42 A994), 579-599.

294
D. Candeloro and A. Volcic
[39] H.B. Maynard, A geometrical characterization of Banach spaces with the Radon-Nikodfm property. Trans.
Amer. Math. Soc. 185 A973), 493-500.
[40] H.B. Maynard, A Radon-Nikodvm theorem for finitely additive bounded measures. Pacific J. Math 83
A979), 401-413.
[41] S. Moedomo and J.J. Uhl. Radon-Nikodvm tlieorems for the Bochner and Pettis integrals. Pacific J. Math.
38 A971), 531-536.
[42] K. Musiai, A Radon-Nikodvm theorem for the Bartle-Dunford-Schwartz integral. Atti Sem. Mat. Fis. Univ.
Modena 41 A993), 227-233.
[43] J. von Neumann, Algebraische Reprasentanten der Funktionen "bis auf eine Menge vom Mafie Null".
J. Crelle 165 A931), 109-115.
[44] O.M. Nikodym, Surune generalisation des integrities de M.J. Radon, Fund. Math. A930), 131-179.
[45] A. Pietsch, Nuclear ика11у Convex Spaces. Springer. Berlin A972).
[46] M. Radon, Theorie und Anwendungen der absolm additiven Mengenfunktionen, Sitzungsber. der Math.-
Naturwiss. Klasse der Kais. Akademie der Wiss.. Bd. 112. Abt II a/2. Wien A913).
[47] M.A. Rieffel, Deniable subsets of Banach spaces, with applications to a Radon-Nikodvm theorem, in:
Functional Analysis, B.R. Gelbaum (ed.). Washington, Thompson Book Co. A967).
[48] M.A. Rieffel, The Radon-Nikodym theorem for the Bochner integral. Trans. Amer. Math. Soc. 131 A968),
466-487.
[49] V. Rybakov, Theorem of Bartle, Dunford and Schwartz on vector-valued measures, Math. Notes 7 A970),
147-151.
[50] S. Saks, Theory of Integral. New York A937).
[51] I.E. Segal, Equivalences of measure spaces. Amer. J. Math. 73 A95 I). 275-313.
[52] С Stegall, The Radon-Nikodvm property in conjugate Banach spaces. Trans. Amer. Math. Soc. 206 A975),
213-223.
[53] W. Strauss, N.D. Macheras and K. Musiai, Liftings. Handbook of Measure Theory, E. Pap, ed., Elsevier,
Amsterdam B002), 1131-1184.
[54] M. Sugeno, Theory of fuzzy integrals and its applications, PhD Thesis, Tokyo Inst, of Technology A974).
[55] M. Sugeno and T. Murofushi, Pseudo additive measures and integrals. J. Math. Anal. Appl. 122 A987),
197-222.
[56] J.J. Uhl, A note on the Radon-Nikodvm property for Banach spaces. Revue Roumaine Math. 17 A972),
I 13-1 15.
[57] S. Ulam, Zur Mafitheorie in der allgemeinen Mengenlehre. Fund. Math. 16 A930), 140-150.
[58] A. Volcic, Sul teorema di Radon-Nikodvm net caso поп a-finite. Rend. 1st. Mat. Univ. Trieste 2 A970),
42-53.
[59] A. Volcic, Teoremi di decomposizione per misure localizzabili. Rendiconti di Mat. B), Ser. VI 6 A973),
307-336.
[60] A. Volcic, Sulla differenziazione di misure. Rend. 1st. Mat. Univ. Trieste 6 A974), 156-177.
[61] A. Volcic, Localizzabilita, semifinitezza e misure esterne. Rend. 1st. Mat. Univ. Trieste 6 A974), 178-197.

CHAPTER 7
One-Dimensional Diffusions
and Their Convergence in Distribution
James K. Brooks
University of Florida, Gainesville, FL 32611-2082. USA
E-mail: BmokudPinath.ufl.edu
Contents
Introduction 297
1.Brownian motion 297
Introduction 297
1.1. Definitions 298
1.2. Mappings of Brownian motion 298
1.3. Markov times and the strong Markov property 300
1.4. Further properties of Brownian motion 301
2.One dimensional diffusions 304
Introduction 304
2.1. The setting 304
2.2. Definitions and properties 305
2.3. The scale function 306
2.4. Speed measure 307
2.5. Local time for Brownian motion 309
2.6. Time changing Brownian motion 310
3.Weak convergence of diffusions 313
Introduction 313
3.1. Preliminaries 313
3.2. The general convergence theorem 314
3.3. Convergence of diffusions 321
4.Diffusions as a limit of stretched Brownian motions and stretched random walks 330
Introduction 330
4.1. Stretched Brownian motion 330
4.2. Diffusions as a limit of stretched random walks 335
4.3. Natural time 340
References 342
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
295

This page intentionally left blank

One-dimensional diffusions and their convergence in distribution 297
Introduction
The phenomenon of the irregular movement of pollen grains suspended in water was
observed by Robert Brown in 1827. Brownian motion, the name given to this process,
was not given a rigorous mathematical treatment until almost one hundred years later by
N. Wiener in 1923. Then the stochastic processes called diffusions, which possessed some
properties akin to Brownian motion, appeared shortly after Wiener's work on harmonic
analysis in 1930. The real starting point in the theory of diffusions was the series of
fundamental papers by W Feller A952, 1954, 1955, 1957), in which a semi-group
approach was employed. The modern form of the theory of one-dimensional diffusions,
regarded as the most complete chapter in probability theory, was developed in Dynkin
A955, 1959) and Ito and McKean A974). In this Handbook Series we have attempted
to introduce to measure theorists some of the beautiful results in this field; the only
probability prerequisite is the early chapters of Breiman A968). Our treatment in the early
sections leans toward an intuitive approach, which is the only possible approach in this
limited format. Sections 1 and 2, Brownian motion and real valued diffusions, follow the
presentation in the fine book of Freedman A971). Instead of proofs, informal discussions
of the concepts involved are given, although precise definitions are presented so that all
the ingredients are included to reach Feller's theorem which characterizes a regular real
valued diffusion as a space change and time change of Brownian motion. In Sections
3 and 4 proofs are given in detail and the reader has the opportunity of working with
the nuts and bolts of one-dimensional diffusions: scale functions, speed measures, time
changes, Brownian local time and weak convergence. The results in these two sections are
due to Brooks and Chacon A982, 1983) and hopefully will provide some insight into the
structure of diffusions and their weak (distributional) convergence. One path leading to a
study of diffusions is the following: start with Freedman A971) for an elegant introduction,
cover Ito and McKean A974), the bible for the one-dimensional case, and then Stroock
and Varadhan A979) for the multidimensional case (which is far from being a complete
chapter). The reader can construct other paths by consulting the references. In any case,
the advice given in the preface to Loeve A963) is germane: ".. .the reader will have to be
armed permanently with patience, pen, and calculus". The reward? An excursion into one
of the most fascinating areas of Analysis.
1. Brownian motion
Introduction
The botanist Robert Brown in 1827 focused a microscope on a drop of water which had
been trapped in a piece of igneous rock millions of years ago. He observed tiny particles
continually moving with an irregular motion-a motion he had observed before while
studying pollen grains in water. The entrapment of the water in the rock forced Brown
to rule out his earlier explanations causing the phenomenon now referred to as Brownian
motion. Bachelier in 1900 in his doctoral dissertation presented the first mathematical
formulation of Brownian motion as a result of studying stock price fluctuations. Einstein
gave a mathematical description of this phenomenon from the molecular-kinetic theory

298
J.K. Brooks
of heat in 1905. The first existence proof was given in Wiener A923, 1924). The fine
structure of the sample paths of Brownian motion can be found in the deep work of P. Levy
A939, 1948).
1.1. Definitions
First we start off by describing the most basic one-dimensional diffusion: Normalized
Brownian Motion.
Definition 1.1. Normalized Brownian motion ? is a real valued stochastic process
{B(t): 0 ^ t < oo} on a probability space (?, ?, P) with the following properties:
A) For each fixed t, B(t): ? -> R is (?, B\ )-measurable (B„ denotes the Borel subsets
ofR").
B) fi@)=0.
C) ?{·)? is continuous on [0. oof for each ? e ?.
D) For 0 < 11 < · · · < r„_ ? <t„, the increments
?(?|),?(?2)-?(?,) ?(?„)-?(?„_,)
are independent and normally distributed with means 0 and variances t\,tj —
t\, —1„ —t„-\.
In this section, (?, ?, P) will refer to the above probability space. We regarded ? as a
space of sample paths ?, where ?-.t -» ?(?)(?). It is helpful to visualize ? e ? as a small
particle released at the origin in R3 at time t = 0; this particle is constantly buffeted about
by molecular bombardment. To obtain one-dimensional Brownian motion, fix an axis in
R3 and project the motion of the particle ? on this axis. The projection of the position of
? at time t on this axis is ?(?)(?).
Briefly, the process B(t) can be constructed for t running through the binary rationals,
for a suitable probability space. After showing that for almost all ?, the sample paths
are uniformly continuous on bounded time intervals (the hard part), extended these ? to
[0, oof, and define the other ?/s to be the constant function 0. Assume now that normalized
Brownian motion exists and let us list some of the beautiful properties associated with this
fascinating process - a process which has been used in practically every branch of science.
We shall transfer the setting (?, ?, ?) to С (I), the set of continuous functions /on the
interval / с [0, oo). Most of the time / is [0. oc[. [0, 1] or ]0, oof. For such an /, define
the functions ?,, t e /, by ?,(/) = f(t). We then form the ?-field В (or ?(?), which is
the smallest ?-field for which all the ?, : С (I) -> R, t e I, are measurable.
1.2. Mappings of Brownian motion
1. With the measurable spaces (?, ?) and (C(/). B) in hand, we shall examine various
measurable maps from ? to C(I). We set B(t, ?) = ?(?)(?). In the following, ? > 0, s ^
0, and ? e R.

One-dimensional diffusions and their convergence in distribution
299
THEOREM 1.2. The following maps from (?, ?) to (C[0, oof, B) are measurable.
(a) B, which sends ? e ? to the function ?(-,?) e C[0, oof.
(b) —B, which sends ? & ? to —B(, ?).
(c) ?~[ ?(?2, ·), which sends ? e ? to ?~? ?(?2(·), ?).
(d) ? + ?, which sends ? e ? to ? + ?(·, ?).
(e) B(s + ·), which sends ? e ? to the function t -> B(s + t, ?), t e [0, oof.
(f) B(s + ·) - ?(?), which sends ? to the function t -> ?(? + ?.?) — ?(.?.?), t e
[0,oo[.
2. Definitions. Let ? be a measurable map from (?. ?) to (C(/). ?). We define the
P-distribution of ? to be the probability PM~] on B. In particular, for ? e R, set
P, = V(x + ?)". Thus, we can now think of our sample paths as belonging to C(I);
Px handles paths starting from ? at time t — 0. For 0 < ? < oo, define E(s) to be the
?-algebra generated by B(u), for 0 ^ и ^ s that is, the events of Brownian motion up to
time i.
3. We now list some important properties of В.
(a) (symmetry) P(-B)~[ = P0.
(b) (scaling) ?(?-l ?(?2(¦)))-l = PQ.
(c) (Markov property) B(s + ·) - B(s) is independent of E(s) and has P-distribution
equal to Po·
Property (a) is intuitively obvious-remember, the particle (viewed in one dimension) has
equal probability of being pushed to the left or right. Property (b) shows that if we adjust
the time by the factor ?2 and the state space, i.e., the range of B, by the factor ?, we
retain our Brownian motion. Tampering with ? in a slightly different fashion will lead
to "stretched Brownian motion", which is an interesting process different from B, such
processes will weakly converge to a general diffusion-more about this in Section 4.
4. Property (c) is an introduction to a key property belonging to B. Let us examine this
in more detail. Obviously the P-distribution of B(s + ¦) — B(s) gives the full weight
to paths starting from ? = 0; but the interesting aspect is that the particle governed by
B(s + ·) — B(s) forgets about the past times before ? and starts from scratch at t = 0, like
B. Analytically, we have the following. Fix s; then the function Pa,s. .,(·) :flxB-»[0,l]
has the following properties:
A) For each ? e ?, ??(.5.?)(·) is a probability on B.
B) For each A e B, the function ? -> Pb(,.W)( A) is a measurable function on ?.
C) Рвц.шМ) = P(B(s + ·) e ?|2?(?))(?), a.a. ? e ?, A e B,
that is
P(G, B(s + ·) e A) = [ PBls 0))(A)P(da>). G e Г(л)
(write P(G, F) for F(G ? F)).
We say ??(.?)(·) is a regular conditional ?-distribution for ?(? + ·) given ?(.v). This is
the mathematical statement concerning the Brownian particle starting afresh. Another form

300
J. K. Brooks
is, for fixed t > i, P(B(t) e T\E(s))(a>) = PBls.o»(B(t - s) e T), a.a. ?, ? e ?, where
(?(? —s)eT) is the set of functions in C[0. oo) whose range at time t — s lies in T. Thus
the event (B(t) e T), given the information of the process up to time i, is independent of
events prior to time i.
But Brownian motion is even richer - it has this property even when the constant time is
replaced by a random time. Of course, peeking into the future is not allowed in this game,
so we restrict our random times accordingly. To illustrate this point, fix b e E. Let
? (?) = inf{r: B(t,w) = b\, ?&?.
Thus ? (?) is the first time the Brownian path hits b. Note that ? is a random variable with
the property that the event (? ^ t) belongs to ?(?), that is, the random time ? does not
look into the future. (Compare this to the random time ?? (?) = last time the ?-path hits b
for t e [0, 1], which does look into the future). Now let
?(?,?)=?(? + ?(?),?) -?(?(?).?). t ^0
(where we may assume ?(?) < oo on ?). It can be shown that ? is normalized Brownian
motion and that ?(?(?), t > 0) is independent of the "events up to time ?", where a
precise meaning will be given to the last phrase. Thus our Brownian traveler starts afresh
even with respect to path shifts induced by (suitable) random times-this is the so-called
strong Markov property, examined below.
1.3. Markov times and the strong Markov property
Let ? : ? -> [0, oo] be a random variable such that {? < t} e ?(?), for all t > 0. We call
such a ? a Markov time or simply Markov (other terminology-? is optional with respect to
E(t+) := ?\[?(? +?), ? > 0}). Define ?(?+) to be the ?-algebra of all events A e ?
such that А П {? < t} e E(t), for all t > 0. Then ?(?+) is thought of as being the
collection of events prior to time ?.
On {? < oo} define the process ?(? + ·) — ?(?) as in 4.D) in the previous subsection.
This process is a map from [0, oo] ? {? < oo} into R which is measurable with respect
to ??([0, oof) ? ? ? {? < oo}; furthermore, ?(?) on {? < oo} is measurable with respect
to ?(?+). Note that if t is fixed and we let ? =t, then ?(?+) and ?(?+) agree. The
following important theorem says that Brownian motion has the strong Markov property.
THEOREM 1.3. Assume ? is Markov. Given {? < oo}, the ?-algebra ?(?+) is
conditionally ?-independent of the process ?(? + ·) — ?(?), and this process is conditionally
normalized Brownian motion. Furthermore, a regular conditional ? -distribution for ?(? + ·)
given ?(?+) is ????)(·) on ? < oo.
The above theorem says
P(G, ?(? +¦)- ?(?) e D) = P(G)P(B(z + ¦)- ?(?) e D),
for G e ?(?+), G с {? < oo} and D e B.

One-dimensional diffusions and their convergence in distribution
301
Also,
P(G, ?(? + ¦)- ?(?) e D) = ( ??{?.?»@)?(??),
Jg
G e ?(?+), G С {? < oo} and D e B.
Go back to the example where ? is the first time the path hits b. Then Pb(t.w){D) =
Pb(D), that is, starting from time ?, the paths start afresh, and are Brownian motions
starting from b at t = 0.
These results concerning the strong Markov property are due to Hunt and Blumenthal,
but were used by Levy prior to its rigorous establishment; as was often the case. Levy
simply forged ahead with his marvelous intuition.
This property is essentially the defining property of a diffusion as we shall see. To quote
from Dellacherie and Meyer A978) "... the seemingly trivial notion of a stopping time
(due to Doob) is the cornerstone of the 'general theory of processes' ".
1.4. Further properties of Brownian motion
A few results are recorded in this subsection in order to strengthen the readers intuition
concerning Brownian motion. Consult Freedman A971) for proofs of these results.
The first result is the so-called reflection principle of Andre. Consider the simple case
where b is a positive number and a Brownian traveler B(t,co) hits b at the Markov time ?.
Form a new process by defining an associated path after time ? (?) by reflecting the path
?(·, ?) about the horizontal line through b. Because of the symmetry of ?, by now we
should expect this modified process to still be Brownian motion. It is!, and the justification
for this fact lies in the strong Markov property. More generally, let ? be Markov. Define
the process У as ? reflected at ? by defining Y(t,w) = B(t,w) for all t if ?(?) = oo;
otherwise, define Y(t, ?) — ?(?,?), for t ? ?(?) and
?{?,?) = 2?(?(?),?) -B(t.w), fori > ?(?).
Thus В is reflected about the horizontal line through (?, ?(?)). The reflection principle
is that ? is normalized Brownian motion.
Next, let M(t) = ma\{B(s): 0 ^ ? ^ ?}. It turns out that, for у ^ 0 and b ^ 0, we have
P(B(t) <b-y, M(t) >b) = P(B(t) >b + y).
From this, it follows that P(M{t) > b) = 2P(B{t) > b).
Not only do these identities aid our intuition, but they are essential tools in calculations
involving B.
One of the best ways to appreciate the wilderness that is C[0, oof. which is filled with
such exotic forms, is to delve into the study of the sample path properties of Brownian
motion. Perhaps one is initially discouraged when first discovering the futility in trying to
visualize the path of a Brownian traveler:

302
J.K. Brooks
For almost all ?, the sample path ?(·, ?) is monotone in no interval.
The reader is invited to attempt a sketch of continuous function having this property.
If we set ??(?) = {t: B{t,w) = y), then for each у and almost all ?, S\ (?) is closed,
unbounded, of Lebesgue measure zero and dense itself. Finally, by now it should come as
no surprise that almost all paths are nowhere differentiable. Is it any wonder that Brownian
motion has captured the imagination of so many research workers?
A proof of the fact that for a.a. ?, ?? := {t: B(t,co) = 0} has no isolated points
illustrates the use of the strong Markov property. The idea is that the behavior at t = 0
is duplicated every time the path returns to the origin. To handle the action at t = 0, we use
Khintchine's law of the iterated logarithm, which states that
limsupB@(V2ilog(log(l/0))~' = 1 a.s.
no
Note that if we replace sup by inf, and 1 by — 1, equality holds. Hence, due to the fluctuation
of the paths close to t = 0, by the above result, we conclude that t = 0 is a limit point, from
the right, for a.a. ?. Let r be a positive rational and set ?, (?) to be the first t ^ r such that
B(t,w) = 0. Since ?(?, + ·) — ?(?>) behaves like normalized Brownian motion starting at
0, by the strong Markov property, we see that ?, (?) is a limit point, from the right, for a.a.
?, of zeros of ?(·, ?).
Intersecting events over all the positive rationals r, conclude that for a.a. ? we have: the
first zero of the path after every positive rational time is a limit point of ?? from the right,
hence no point of ?? is an isolated point. The case 5?(?), у ф 0, is handled in the same
fashion.
Fig. 1. The above illustration is a close approximation to the plot of a microscopic particle suspended in water
by the French physicist Jean Baptiste Perrin in 1912. The position of the particle was plotted every 30 seconds.

One-dimensional diffusions and their convergence in distribution
303
Let us turn to an example, one among many, which reveals the universal nature of
Brownian motion in the sense that this process can be used to capture random walks; it
is called the Skorokhod representation for a distribution. Assume (?, ?, P) also supports
a random variable which has uniform distribution over [0, 1] and is independent of B.
Let F be a distribution function on R with mean 0 and variance ?2 ^ oo. Then there
exist random variables 0 = ?? < ? < · · · < ос such that ?(?„+\) — ?{?„) are independent
and identically distributed and have common distribution function F, for /; = 0, 1,2,
Moreover the {?„+? — ?„}^() are independent and identically distributed with mean ?~.
This means that {?(??)}^0 is a random walk stemming from random variables having the
given distribution F.
As an aid to feeling more comfortable with Brownian motion, we mention an intuitive
but weak approach to arriving at B. Let (?1, ?', ?') be a probability space that supports
a process (Y„ )^=] of identically distributed independent random variables with mean 0 and
variance ?2 < oo. Set
?
?? = ??>-
i=\
Consider the following construction of functions
?„(·,?) eC[0, 1], we ?',
by linearizing:
1 r-^J- 1
?„(?,?)=—?5/_?(?) + j-= F У,-(?).
? J ? 1 ? J ?
?
ifre[^,i].
ThusX„:i2'^· C[0,1 ]. One can show that the probabilities P' Xn ' onC[0, l]converge
weakly to B, in symbols, P'X~l => Д>, that is
f fd{P'X-')^j fdP0.
for all real valued bounded continuous functions / on C[0. 1].
As a result, the scaled random walk derived from fair coin tossing converges weakly
to normalized Brownian motion. This not only gives us insight into Brownian motion,
but it displays the power of weak convergence: to reach such a rich process by a random
walk generated by such simple random variables. Note that X„ in fair coin tossing has
the property that the path speed is the reciprocal of the distance. In Section 4 we shall
vary the speed in order to weakly approach general diffusions by "stretched random
walks".

304
J.K. Brooks
2. One dimensional diffusions
Introduction
We shall now examine processes of a more general nature called diffusion processes. Such
processes can be observed in common occurrences such as the spreading of color in a
glass of water in which powdered dye has been added. The diffusion of the color is a
consequence of the continuous motion of all the substances involved and it is the process
by which irregularities in the concentration of the solution eventually disappear. There are
many other examples of diffusions: diffusions of gases and liquids through solids, mixing
in turbulent fluids, thermal diffusions, etc. These processes are somewhat reminiscent of
Brownian motion. What are the significant characteristics of Brownian motion to define
a general diffusion? As evidenced in Section 1, to deduce the fundamental properties of
Brownian motion, the property constantly used is the strong Markov property: we want to
retain the property that the past, given a Markov time, is independent of the future-a starting
afresh at these random times. Let us keep this feature and specify that the diffusion traveler
should possess no memory in this sense. What else? Staying in the space of continuous
paths is not only useful analytically, but it is a reasonable requirement to impose on a
particle traveling in what we might imagine to be a diffusion. And that's it! These two
requirements are enough to define a real valued diffusion. Now, can we expect that there is
any relationship between this new, very abstract process and that process endowed with so
much structure and symmetry - Brownian motion? Feller's deep theorem provides a sharp,
strong connection between the two processes. Here is a very brief sketch of this connection.
A general diffusion will not possess the symmetry of Brownian motion, that is, if the
diffusion particle is at a certain point in the state space, we expect it to move to the left
or right with a probability (not 1/2 in general), which will depend upon its location in
space. A function S, called the scale function, will govern this aspect of a diffusion's
movement. Next, if we view Brownian motion as a weak limit of random walks, say,
generated from fair coin tossing, these walks have a certain speed. If we tamper with
this speed at each stage and location, we would expect the weak limit, if it exists.to be a
process which travels at a different rate than Brownian motion, and this rate can vary from
path to path. Surprisingly, pathwise a diffusion behaves like a state-changed Brownian
path which is time changed by a function T(t.w), where the time change is derived from
a "speed measure" m defined on the state space. The existence of such a measure, which
can capture this aspect of a diffusion, requires quite a bit of analysis - but it does exist!
Feller's theorem states that, in distribution, S and m characterize the given diffusion. We
shall begin to sketch these essential components of a general diffusion.
2.1. The setting
From now on, our sample space ? will be C[0. oof. Thus, the sample points ? belong
to C[0, oof, with sample paths t -> co(t). Sometimes we also refer to ? as a sample path
for brevity. Let X be the coordinate process defined on ? by X(t)(w) = w(t), for ? e ?
and 0 <r < oo. Write ?(?,?) for ?(?)(?). Let B, or ?(?), be the ?-algebra of subsets

One-dimensional diffusions and their convergence in distribution
305
of ? such that all the X(t) are measurable. B(t) is the ?-algebra of events generated by
X(s), for 0 ^ ? ^ t. Define B(t+) to be f][B(s), s > t]. We say that the random variable
? : ? -> [0, oo] is a Markov time or Markov if [? < t} e B(t), /or еас/г ?. 77?e ?-field of
events prior to ?, ?(?+), consists of all A eB such that ? ? {? <t} eB(t), for each t. If
? is Markov define the shift operator #: ? -> ? by #?(?) =?(?(?) + ·). that is, ??{?) is
the path whose value at time ? is ?[?(?) + s]. Note that ?? is a measurable map.
2.2. Definitions and properties
For each jc e R, let P, be a probability on (?, ?).
Then {Pv} reK is a diffusion if
(i) Pv(X@)=jc)= 1, for each jc eR;
(?) {?-heR has the strong Markov property, that is, if ? is Markov, the conditional Px-
distribution of ??, given ?(?+) is ??? on {? < oo}. This means that for any ? e R,
A e #(?+), А с {? < oo}, and В е ?, we have
P,(A, ????)= [ ????10)).?)(?)??(??).
J A
The above is the precise statement of the intuitive notion that the diffusion particle starts
afresh at the random times ?. We shall simply write {Px} in place {Px }x€&.
Brownian motion, which of course is a diffusion, is denoted by [Bx] (here S, is the Px
of Section 1.1). If ? > 0, then (??() is called Brownian motion with variance a-. Note
that if {Px} is a diffusion and S: R -> R is continuous, strictly monotone, and onto, then,
if Qx is the Ps-i(v)-distribution of {SX(t), 0 < t < oo}, it follows that [Qx] is also a
diffusion.
Now we shall introduce some definitions and results concerning diffusions, which will
enrich our intuition and cast some light on their behavior. We shall always assume in this
section that {Px} is a diffusion.
A handy tool in establishing properties of a diffusion is the Blumenthal 0-1 law. which
states that the ?-algebra ?@+) is ? -trivial, that is. PX(A) = 0 or 1 if A e ?@+). The
proof follows easily from the strong Markov property applied to the trivial Markov time,
? ? 0. Indeed, for this ?, and A e B@+) since PX(A) = PX(A, ?? e A), and the later is
equal to
? ?????)?)??? = [ PAA)dPx = PX(A)\
Ja J a
since ??(?: ?(?(?)) = ?) = 1. Hence, an event which depends on arbitrarily small t > 0,
happens or does not happen a.s. for each starting position ?.
What about the behavior of the diffusion particle in the vicinity of ? in the state space?
Three events are useful in this situation. The event {stay at ? awhile} is the collection of
paths ? such that w(t) = .x, for t e [0, c]. for some positive с depending on ?; since this
event belongs to B@+), it has Px measure 0 or 1. Call ? instantaneous if this value is 0,

306
J.K. Brooks
and call ? absorbing otherwise; in the latter case, the paths are ? = ?, a.s. Px. The event
{greater than ? near 0} is the collection of paths ? such that ?(?) > ? for i's arbitrarily
close to 0. Define {less than ? near 0} similarly. By the continuity of paths, these events
belong to ??@+), and thus they have Px probability 0 or 1.
The diffusions we shall study here are the regular diffusions, that is, those diffusions
such that P,(hit y) > 0 for all ? and y. Thus these diffusions get to visit any y, starting
from any ?.
From now on in this section, Px is a regular diffusion.
2.3. The scale function
We shall now present a loose argument to show the existence of a continuous, strictly
increasing, onto function S: ? -> R, such that for a < b,
r,(hit b before a) = . for a ^ ? ^ b,
S(b)-S(a)'
where {hit b before a} is the collection of paths ? that take on the value b before taking on
the value a. As in Section 1, if we define ?-(?) to be the least time t such that co(t) = z,
and +oo otherwise, then {hit b before ?} = {?/, < ?(,}; this event belongs to ?(??,+).
Before beginning, what would be a scale function for Brownian motion? Obviously,
for a < b, Ва+ь (hit b before a) = 1/2, due to the symmetry of Brownian motion. More
generally, by partitioning [a, b] by ? equally spaced points a = .x\ < ¦ ¦ ¦ < xn = b, if we
release the particle at any x^, the particle will behave like a symmetric random walk on
x\,..., ;t„, and this allows us to conclude that
?? — a
Bx (hit b before a) = .
b — a
In general, one can show that S(x) = ? is a scale function for Brownian motion. If a
diffusion has a scale function S(x) = x, we say that it is in natural scale.
Back to the general regular diffusion {Px}. The idea is to define, for J = ]a, b[, a < b,
the function
Sj(x) = Px (hit b before a), for ? e J.
By the strong Markov property, for a < ? < у < b,
Px(hit b before a) = Px (hit у before a) PY(hit b before a), A)
which is very intuitive, since you must hit у before ? if you hit b before a, but since
the diffusion starts over again at y, the above equation holds. Hence it follows that
Sj (x) < Sj (y) for such ? and y, after showing 0 < Px (hit у before a) < 1.

One-dimensional diffusions and their convergence in distribution
307
Let ? e ]a, b[; here is an indication why Sj is continuous at .x. If v„ | x, v„ e (a,b),
then A) shows that
Sj(x) = Pv(hit}>, before a)Sj(yn).
Intuitively, P, (hit y„ before a) ->„ 1. hence Sj(yn) ->« Sj(x). This shows right
continuity at ?. Left continuity is similar. So we have at our disposal, a strictly increasing,
continuous function on J, which has the correct properties on J. The next job is to show that if
J\ = }a\, b\ [ D J, then Sj] = Sj on J. After this, we obtain the desired function, which is
denoted by S. Any other scale function for the diffusion is a linear function of S.
We record for future reference the following.
THEOREM 2.1. Let {Px} be a diffusion with scale function S. Let Qx be the Ps-i
^.^distribution of the process {SX(t): 0 ^ t}. Then Qx is a diffusion in natural scale.
In view of this theorem, the characterizations of diffusions reduces to characterizing
diffusions in natural scale. A diffusion in natural scale has the same state space features
as Brownian motion. Now the task - a difficult one-is to show that a diffusion in natural
scale is a time change of Brownian motion. To accomplish this task, we first approach
the problem of describing the speed of a diffusion in natural scale in terms of its "speed
measure" m. Once this measure is in hand, the desired time change is defined in terms of m
and Brownian local time X(y,t,w): the time the Brownian traveler ? spends in (v. y + dy),
up to time t, is ?(>\ t, ?) dy. Local time, worthy of study in its own right, will enhance our
understanding of Brownian motion.
2.4. Speed measure
In order to measure the speed of a diffusion particle, we need the following definitions. In
this subsection, {Px} is always a regular diffusion in natural scale.
Let J denote the interval [a,b], where —oo < a < b < oo. Define
? least time t, if it exists, such that w(t) = a or b, and
?/(?)={
I +oo otherwise.
Then
Ex(tj):= I rjdPx, xeJ,
gives an indication as to the speed of the diffusion, that is. it seems reasonable to expect
that if we know the expected times for a particle at ? to exit J. then this should characterize
the "speed" of our diffusion. It turns out that there exists a unique measure m on B\. called
the speed measure of {Px} with the following properties: m is finite on compact subsets

308
J.K. Brooks
of R, m has strictly positive measure on all non-empty open subsets of R, and for each
bounded interval J = [a, b] we have
"/,
Ex(tj) = / Gj{x,y)m(dy). for.xeJ. B)
where Gj = Gab, is the Green's function defined on [a, b] ? [a,b] by:
(x-a)(/)-y)(fc-a) -'. a
Gab(x,y)=\ ,
l(y-a)(b-x)(b-a)-1, a
? !? у ^х ^ b.
How does the Green's function make its appearance in the definition of speed measure? For
an intuitive look into this, we sketch a proof for the existence of a measure m satisfying
B) from Breiman A968). The proof presented there is for J = ]a, b[, and there is an extra
factor of 2 in that setting, nonetheless, here is what is going on. Partition ]a, b[ by points
a = xq < x\ < ¦ ¦ ¦ < x„ = b, for Xk = a +kS. Let Л = ]хк-1, *к+\ i ¦ Then Еч (?^) tells us
how fast the particle is moving in a neighborhood of Xk- Introduce a process (Z„) only at
the exit times from the ijt's. Then (Z„) is a symmetric random walk moving on the points
xq,...,x„. Set n(xj\Xk) to be the expected number of times the random walk visits Xk,
starting from xj, before hitting Jto or x„. Argue that
?
EX/ (?/) = ^2n(xj;xic)E.4(Tjk).
k=\
which is certainly intuitively obvious. Since nixj'.xk) is equal to \/8Gj(xj,Xk), by
examining this random walk, it follows that
EXj(xj)= / Gj(Xj,y)m(dy),
C)
where in is the measure that assigns mass \E4 (zjk) to x^. Now C) has the approximate
form of B).
Next, construct partitions ?„ of R of successive refinements with mesh Sn -> 0. Define
m„ as in was defined and obtain a subsequence (шП|) which converges weakly to a measure
m on B\ which ultimately satisfies B).
Another defining feature of m is that for any a < a ^ ? < b, we have m(]a, ?]) =
/+(?) — /+(?), where /+ is the right derivative of the function* -> ?, (ry).
Now we have in hand a measure m connected with the diffusion {Px} which does in fact
indicate the speed of the process. Crucial is the result that if {Px} and [Qx] are diffusions
in natural scale, each of which has the common speed measure w, then Px — Qx for each
? e R. Thus, speed measure determines the diffusion, if it is in natural scale. This comes
about by examining the infinitesimal generator ? of the semigroup {Tt}t^o of {Px}, where
T,: Q,(R) -> Q,(R) is given by
(Tlf)(x) = Ex(f(X(t))).

One-dimensional diffusions and their convergence in distribution
309
Here, Q,(R) denotes the space of bounded continuous functions on R. For / in the
domain of ?, we have ? f = h, where
f+(x)-f+(y)=f' hdm.
Thus m determines ?, which determines Tt. which, in turn, determines {Px}.
Now, how does m induce time changes of Brownian motion in such a way that the
resulting process is the given diffusion {P,}? This is where Brownian local time enters.
2.5. Local time for Brownian motion
In this section we shall introduce the fundamental notion of local time for Brownian
motion - a tool used in defining time changes of paths. Let L denote Lebesgue measure; for
A eB\, andi ^ 0, set
?(?,?)(?) = ?(?: 0? ^i. w(s) e A).
Note that ?(?, ?)() is a measure on B\, with ?(?, ?)(?) = t. Since the image law will
be used in the sequel, we mention that ?(?.?) is the image measure of L on [0. t] under
the mapping ? -> a>(s). Trotter's important theorem (Trotter, 1958) states that for each ?,
?(?, ?>)() has a continuous density relative to ?:
THEOREM 2.2 (Trotter). For each ? e ?, there is a jointly continuous nonnegative
function (x,t) -> ?(?, t, ?) such that ? -> ?(?, t, ?) is measurable and
?(?,?)(?)= / k(y,t,a>)dy,
J a
for all t > 0 and AeB\,for every ? e ??, where ?? e В and Bx (??) = 1, for all x.
This says k(y, t, ?) dy is the time Brownian traveler ? spends in (у, у + dy) up to time
t. ? is called Brownian local time.
All the following properties listed below hold for a.a. ?, that is, there is a set ?? e В such
that ??(?0) = 1 for all jc, and the properties hold for ? e ??- These properties provide
additional insight into Brownian motion.
THEOREM 2.3. Properties of local time.
A) ?(?+?,?)(?) = ?(?,?)(?)+?^,?,(?)(?), A e ?,).
B) ?0\ t+s,w) = k(y, t, ?) + k(y, s, ?,{?)).
C) k(y,0,w) = 0.
D) k(y, ·, ?) is nondecreasing.
E) //?@) eJ = [a,b], then k(y. ?/(?), ?) = 0 //у 5$аог .у > b.
F) For each ?, lim k(x,t, ?) := k(x, oo, ?) = ос, а.е. ??;.
t—>oc

310
J. K. Brooks
G) For each ?, ?(?, ?,?) > 0 for all t > 0, a.e. Bx.
(8) For each ? and y, lim,^^ k(y, t. ?) = oo, a.e. Bx.
Again, the Green's function appears in an important context, and will connect m to our
desired time change via ?.
THEOREM 2.4. Let J = [a, b] and assume ? and у are in J.
Then
E,(X(y,Tj,))=2Gj(x,y).
A useful property concerning ? is the following:
Theorem 2.5. Iff e Q>]-oo,oo[ and
F(t)(o):= f fX(s)(w)ds= [ f{yWy,t,o)dy,
J0 J-oc
(the last equality by the image law) then on {??, ?@) e J} we have
F(tj(o)) = j f(y)k(y,Tj((o).(o)dy.
2.6. Time changing Brownian motion
At this point, we are in a position (after glossing over a large number of technical lemmas)
to obtain a diffusion {Px\, which is in natural scale, from a time change of Brownian
motion. Assume {Px\ has speed measure m. Our goal is to define a time change
?:[0,??[?? -> [0,oo[,
which is measurable and such that: T(-, ?) is a strictly increasing, continuous, 7@, ?) =0,
and Hindoo T(t, ?) = oo (of course, all the properties are to hold outside a measurable set
of Bx -measure 0). For an appropriate 7, which should, and does, depend upon m, we
define a process ? by
?(?,?) = ?(?(?,?),?),
that is, the Brownian path ? is time changed into the path t -> ?G(?, ?)). If {Qx} is the
Bx-distribution of ?, we want \QX) to be a regular diffusion in natural scale with speed
measure m, that is, Px = Qx for all x.
How do we define such a Tl Let us back into the definition as follows. Let J — [a,b],
and set rj := the exit time from J of the ? process; Tj(a>) will remain the exit time of

One-dimensional diffusions and their convergence in distribution
311
Brownian motion. Obviously, T~](tj(w),w) = ?) (?). Denote ? [(?,?) by ?(?,?). To
show {Q?} has speed measure m, we must have
Ex(A(tj))= ? Gj{x,y)m{dy), D)
in view of B) in Section 2.4, where expectations are taken with respect to Bx.
If we work with the right hand side of D), we see by Theorem 2.4 in Section 2.5 that
j Gj{x, y)m(dy) = j l-Ex (?(.?. ту, (-))m(dy).
but by Fubini's theorem, the latter integral is
E,\l-Jk(y,Tj.())m(dy)\.
which, in turn, equals, in view of Theorem 2.3E),
eV-J k(y,TjA))m(dy)\.
Thus, for D) to hold we need
Ex(A(tj)) = eA1- ? k(y.Tj.(-))m(dy)\.
Now is clear (cf. Theorem 2.5) what the definition of T(t, ?), or equivalently A(t, ?),
should be. Define
1 ?0
?(?,?)=-? k(yj,w)m(dy), ?&??}. E)
^ J-ос
Then define T(t, ?) = A'1 (?, ?), for ? e ?() such that lim^oc A(t, ?) = oo and ?(·, ?)
is strictly increasing on [0, oof. The set of such ? has В., -measure 1, for each ? eR. Note
that if m is absolutely continuous with respect to L and rh = ^, then, by the image law,
1 f
A(t,w)=- m(X(s)(w))ds. F)
2 Jo
Here is another way to see that F) is the correct definition, in the case m <<C ? and rh is
continuous. If ? < 0 < b and J = [a, b], then for J small, observe that
?0(?])= / Gj@,y)m(dy)= Gj@, y)m(vRv « —\a\b.

312
J. K. Brooks
Since Eq(tj) = \a\b, we see that ??(?] ) ~ i«i@)?o(Ty).
More generally
Thus if it takes Brownian motion time At to exit from J, it takes ? roughly jm(x)At
to exit from J.
Since ? = XT, this means
/ rh(x)At \
7 Г + -—^—,? - T(t,w)^ At.
With smoothness, we have
dT
or
iii w(jc)
rTU-?) rh(X(S)(w))
which yields F).
To further imagine the behavior of ? at time t and positioned at у е Е, let ?? =?(?,?)
and?2 = T(t +At,?).Thus
m(y)
so Y(t + At, ?) - Y{t,a>) is governed by Brownian motion starting at у with the
time increment ?2 - ?i ^ 2At/m(y). Hence y(i + At, ?) - Y(t) is distributed like
у + BBAt/m(y)), that is, normally distributed with mean у and variance 2At/m(y).
Now that we have our expression for T, what remains is to prove that ? (t) has the strong
Markov property. To be precise, call ? Markov for ? if {? < t] e A(t) := a(Y(s): s ^
f); Л(г+) is defined in the obvious way. Then to establish the strong Markov property for
Y, show ??(?} Y~[ is a regular conditional Bx-distribution for ?(? + ·) given ?(?+) on
{? < oo}. On the way to establishing this, a number of technicalities must be dealt with,
for example, for a.a. ?, show that
7" (? (?) + ?,?) = ? (? (?), ?) + T(t, ??{?(?),?)(?)).
But in the end, all is well, and ? satisfies all our expectations.
Finally, consider a general regular diffusion {Qx} with scale function S. Let {P,}
be the associated diffusion in natural scale associated with [Q.x], that is, Px is the
Qs-iurdistribution of {SX(t)},^Q. Let m be the speed measure of {?,? and let ? be

One-dimensional diffusions and their convergence in distribution
313
the associated time change induced by m according to E), where A = ? . Define
?(?)(?) = ?~??(?(?,?),?), and write ? = 5_1X7\ Then \Z{t)) describes the path
functions associated with Qx(t -> 5_1(?G(?,?)))). In other words, Qx is the Ssi.rr
distribution of Z. This gives the promised representation of a diffusion in terms of
Brownian motion.
3. Weak convergence of diffusions
Introduction
In this section we shall examine the relationship between weak convergence of processes
5,7' В T„, where В is Brownian motion, the convergence of the time changes T„ and the
scale functions 5„ (all terms are defined below). In Section 2, we shall show that if the T„ 's
are inverses of functional A„ (not necessarily additive) and A„(t) -^ Ao@ for each
? = Bx, ? eR, then the processes BT„ converge weakly to BTq.
In Section 3, we shall consider real valued regular diffusions in canonical form on
C[0, oo[ and present necessary and sufficient conditions for weak convergence of these
diffusions in terms of pathwise convergence of their time changes, weak convergence of
their speed measures and pointwise convergence of their scale functions. The main theorem
in this section is the following. If (X„),^l() is a sequence of diffusions in canonical form,
then X„ converges weakly to Xo if and only if the diffusion paths converge uniformly on
compact time intervals. All of these results are due to Brooks and Chacon A982).
3.1. Preliminaries
As before, ? = C[0, oo[, and ? will be given the topology of uniform convergence
on compact sets; ? = ?(?) was previously defined as the ?-algebra generated by the
coordinate processes. Again, Bx denotes Wiener measure on ?{?) starting at x. A set
of full measure ?? e В is a set such that ??(??) = 1 for every x. A functional (not
necessarily additive) is a mapping A from [0, oof ? ? onto [0. oof such that there exists
a set of full measure ?? such that A@, ?) = 0 and ?(·,?) is continuous and strictly
increasing for ? e ?. ? will denote the inverse of A. In this section and the next, ? will be
twice the ? in Section 2, to conform to the definition in Ito and McKean A974). B(R) will
denote B\. All diffusions are real valued and regular. Concepts involving weak convergence
can be found in Billingsley A968). Donsker's theorem is proved in Breiman A968).
All additive functionals appearing in this chapter are of the form
A(t,w)= I k(t,y,w)m(dy), G)
where m is a speed measure defined on B(M.) and ? is Brownian local time. The diffusion X
on ?, having 5 as a scale function and m as its speed measure, will be written as 5~' ВТ if
5X has ? = A-1 as its time change relative to Brownian motion, where A is given by G).

314
J. K. Brooks
In general, without usually stating it explicitly, X„, S„. A„,m„ have the same relationship.
The path function Y„(t) associated with X„ is given by Y„(t)(w) = S~l{co(T„(t, ?))}. The
measure Q" associated with X„ is the Z?s;i (v)-distribution of Y„. If A„ is a (not necessarily
additive) functional, then the process Yn, relative to {S,}, is still denoted by S~l BT„. It is
not, in general, a diffusion.
Write X„ =>· Xo if Q"x =>· Q°x, for each ? e E, where =>· denotes weak convergence. If
(m„) is a sequence of measures on В(Ш),т„ =>? wo means j f dm,, —* f fdmK), for every
continuous function on R having compact support. ? will always be a generic symbol for
? ?; write /„ —»p / or lim - ? /„ = / to mean convergence in probability. A sequence of
processes Z„(?) is said to converge in probability uniformly on compacts to Zo@ if
p( sup \Z„(t)- Z()@| >? ->;?0. foreach<i.<S > 0.
Ve[0.</] /
Say Z„(r) converges uniformly on compacts to Zo@ on i?o (almost surely) if ?„(·, ?) ->
??(·, ?) uniformly on compact sets for each ? e i?o (a.a. ?).
The operator У*: ? ->¦ ? is defined by
J," (?) = S„(x) + co.
If jc is understood, write J„. Sometimes J. T, A. Q will be written in place of Jo, To,
Ao, Qo-
3.2. The general convergence theorem
Theorem 3.1.
(a) Let (A„) be a sequence of junctionals (not necessarily additive) such that A„ (t) —» p
A0(t) for each t > 0 and ? = ??, ? e ?. Then ВТ,, =>¦ ВТо, where T„ = ?,?1.
(b) Suppose (S„) is a sequence of scale functions converging pointwise to So- Assume
in addition, that A„(t, JJ) -> A0(i, J*), for each ? e К. Г/геи S S7„ =>· S^1 S70.
PROOF. First we shall prove (b), since (a) follows from the special case S„(x) =x. As
before, Q'{ is the Ss„(.,}-distribution of Yn. We must show that for h : ? -> R bounded
and continuous and for each jc, we have j hdQ'[ -> f hdQ®.. We shall do this in several
stages. We shall need the following lemma.
LEMMA 3.2. Lei {(^„(?,?)},,^ be a sequence offunctionals. IfC„(t) -+p Co(t),for each
t < ??, then lim„ P(sup,^,() |C„(r) - Co(OI ><5) = 0, for each ? > 0.
PROOF. For simplicity, let ?? = 1 · Let
qk(w) = maxCo((; + 1)/*, ?) - Coij/k, ?) > 0.
.К*
Since Co(-,oi) is uniformly continuous on [0, 1], qk(oS) —*k 0 for each ?. Let ? > 0 be
given. By Egoroff's theorem, there exists a measurable set ? such that ?(?') < ?/2 and

One-dimensional diffusions and their convergence in distribution 315
<7„ converges uniformly to zero on A. Let ? > 0 be given. There exists a Icq = ko(S) such
that 0 ^ <7„ (?) < ?/2 for all ? ^ ко and ? e A; hence
/>(?„ > 5/4) <: P(AC) < ?/2
for all n^- ко- Thus for и > &o,
p(max(Co((;· + l)/n) - C{){j/n)) > ?/2] < ?/2. (8)
Let
Ak(l = (max(Co((; + l)/*o) - Co07*o)) > ??).
Since C„(j/ko) -»? Со(;До), j = 1,2,... До» there exists an ? ? = ?\(?,?), ?\ > ко,
such that if
??, = (|C„07*o) - Co07*o)|) > Я/2,
then P(Aj,„) <<5/B?0),for/i >«|. ;' = 1,2 ко- We now assert that
pfmax|C„@-Co@| > s) <?, forn>n\. (9)
To see this, fix t e [;До, 0' + l)Mo] for an appropriate j. Let
*0
Q„ = AkaulijAjA
note that P(Qn) < ?. Fix ? e ?^,'.
Case 1.0^ C„(i, ?) - Co(f, ?). In this case
<КС„(г,<ы)-Со(г,<ы)
< C„@· + l)/*0, ?) - Co07*o, ?)
= С„(ОЧ- 1)/?0,?) -Co@4- 1)/*?,?)
+ Со@' + 1 )/*о, ?) - Со07*о, ?)
?? |С„@" + D/ko,co) - Co((j + 1)/*?,?)
+ |С0@"+ 1)/*?,?) - Со07*о.ш)|
< ?/4 + ?/4,
smccwe(Acj+lM)n(Alo).

316 J. K. Brooks
Case 2. 0 ^ Co (?, ?) — C„(f, ?). In this case
<КСо(г,ю)-С„(г,ш)
<Со(ОЧ-1)/*о,ш)-С„07Ло,ш)
= Co@" + 1 )/*o, ?) - Co07*o. ?) + C007*o. ?) - C„(j/k0, ?)
^\Co((j + l)/ko,a>)-CQ(j/ko.a>)\ + \CoU/ko,(o)-C„(j/kQ,a>)\
< ?/2 + ?/2,
sincewe(/V:„)n(/i<:(]).
As a result, if ? e ??, where ?(?„) < ?, we have \C„(t, ?) - CoO. ?)I < 5. for all
t ^ 1. Hence
pfmax|C„(i)-Co(i)| >г) <?, forn>«i, A0)
which proves Lemma 3.2. ?
We now continue the proof of the theorem. To show convergence in probability
uniformly on compacts of the T„, we need the following lemma.
LEMMA 3.3. Let (A„)„^,0 be a sequence offunctionals such that A„(t) -+p Ao@ for
each t. Then if (Tn)„^o are the associated inverses, T„(t) -+p To(t) for each t\ hence, by
Lemma 3.2, Tn —» 7"n uniformly on compacts.
PROOF. Step 1. Fix t. For each ? > 0 there exists a constant К and sets ?(?,?) such that
?(?(?,?)) ^ 1 - ? and ?,,(?,?) ^ ? for each we ?(?,?), for each n.
To see this, note that for every ? > 0, there exist sets E„ such that P(E„) ^ 1 — ?/2 and
an integer no such that
\?„(?0B?,?))-??(??B?,?),?)\?? (?)
for n > щ and ? e E„. This follows since TBt, ?) is bounded on a set of measure close
to 1, and given the bound large enough, say K, use the fact that A„ -> p Ao uniformly on
compacts, in particular [0, K].
Furthermore, since AoGbBf, ?), ?) = 2?,
|?„(?0B?,?))-2?| <Cjj, ???,,,???- U2)
Hence, 2t - ? ^ A„GbBf, ?), ?) ^ 2? + ? for ? e ?„. n ^ ??· Since /j is arbitrary, we
may conclude that
t ^ ?„G?B?,?),?) foroief,,. //>//().

One-dimensional diffusions and their convergence in distribution
317
Thus
T„(t, ?) <: T„(A„(T0Bt, ?), ?)) = T0Bt, ?).
Since ToBt) is bounded on sets of measure arbitrarily close to 1, there exists a set С
such that P(C) ^ 1 — 5/2 and 70B?. ?) ? ?, for some К, whenever we С. Note that
К does not depend upon n. Let ?(?, ?) = ?„ ? С; hence P(/i(n, 5)) > 1 - 5 and for
? e /}(«,<$), ?,,(?,?) $: ?,?^ no. This completes Step 1, since we can assume ? works
for и ^ «o-
Step 2. Let К be the К appearing in the statement of Step 1. Define
D'n(K) = (\A„(u) - A0(u)\ < ?, и ? ?).
Since А„(и) ->р Ао(и) uniformly on compacts, lim„ P(Df,(K)) — 1. This implies that
|?„G„(?,?))-?0G;,(?.?))| <? for ? e Dj(/f) ? ?(?. ?)
and
P(Dj(i)n4(n,{))> 1 -2?, ????^?(?).
This in turn implies that
|?-??G·„(?,?).?)|<? for ? e Dfn = Dfn(K) ?) ?(?, ?). A3)
Note that P(D^) ^ 1 - 2<$. Now suppose that 7„(f) -^p 70(i). Then there exists an
infinite number of integers (и,-) and /j > 0 such that
?(\???(?)-?0(?)\>?)>?. (?)
In a view of A3), we see that
|Ao(j, ?) - ?| < ?. we D^, ? >/1(?). A5)
for s belonging to an interval having 7b (f, ?) and 7„(f, ?) as its end points, since Aq is
increasing and ??G?(?,?),?) = ?.
Now let ? e Een. = Df,. ? F„,, where F„, = {?: |7"„, (r) - F(,(i)| > /?}. Take ? < ?/4.
Then |?0E,?) - t\ < ? for s ?]70(?,?), 7"„,(?, ?)[ or s ?]7"„,(?,?), ?0(?,?)[ and ? e
?*., where P(E?.) > ?/2. For each ?, let ]af (?). /3f (?)[, &(?) ^ К, be the largest
interval such that \Aq(s, ?) - t\ < ? for s e ]af (?). ?((?)[. The above argument shows
that ??(?) - ??(?) > ? for ? e Qe = U„,>»(f)?», and p(Qe) > l/2> where 1 was
independent of ?, and ? was arbitrary. Take ?(;) { 0 and Q = limsup<2f("; we also have
]a?li)(co),PHi)(w)[l.Let
[?(?),?(?)] = [>\[???)(?),?,0)(?)]

318
J.K. Brooks
and note that ?(?) - ?(?) ^ ?. But ??. ?) s t for s e [?(?), ?(?)] and ? e Q, which
is a contradiction since there is a set Д>, P(i2o) = 1, on which A() is a strictly increasing.
This establishes Lemma 3.3. ?
Next, let h : C[0, 1] -> Ш be uniformly continuous and bounded, where C[0, 1] is given
the metric induced by the uniform norm || ¦ || on C[0, 1]. In the sequel, S~]TJ will denote
the mapping defined on C[0. oof as follows:
(S-lTJ)(a>)(t) = S~l {J(co)(T(t. УМ))} = S~l {S(x)+c(T(t, J (?)))}.
S~' T„ J„ is defined in a similar fashion.
We now need the following lemma.
Lemma 3.4. lim„ -BQh[S~{T„J„] = h[S~^TJ].
PROOF. Step 1. We assert that given ? > 0 there exists a constant К and an integer h such
that for и > й,
?0{(?: \\(T„J„)(w)\\> ?)?(?: \\(TJ)(co)\\>K)}<a. A6)
To see this, find a constant L such that
bJw. sup 17" (r, J(w))\ > L-\) < a/8. A7)
Such a constant L exists since the path T(-, J (?)) is bounded on [0, 1] for each ?. Note
that T„(t, J„) -^-Ba T(t. J) uniformly on compacts. Hence there exists an integer и ? such
that if ? ^ ? ?,
?0(?: sup\T„(t.J(w))-T„(t. J(a>))\ > l) < a/8. A8)
Let
Z, „ = f sup|7(i. ?)| > L - \) U (sup|Г,,(г. Л) - 7"(i, J)\ > l); A9)
then ??(?|.„) < a/4.
Define <Sn:C[0, L]^R by
8,,(?) = sup{|w(i) -co(s)\: \t -s\^h. 0???. s ? L\.
Since ????_???) = 0, by Egoroffs theorem we can find an ? > 0 such that if Zi =
(8e > l),then
B0(Zi) < a/4. B0)

One-dimensional diffusions and their convergence in distribution
319
Next, define Z3.,, = (supi<:| \T„(t, Jn) - T(t. J)\ > ?). There exists an rn_ such that if
?0(?3.„) < ?/4. B1)
Finally, since the curve (TJ)(co) is bounded on [0, 1] there exists a constant К such that
if Z4 = (?: sup,^|G7)(i4)(f) > К - 2), then
S0(Z4) < a/4. B2)
Define
Z„ = Z|.„UZ2UZ3.„UZ4.
Let \S„(x) - S{x)\ < 1 for ? ^ щ. Define й = max(/j|. /г2,«3)- Note that Bq(Z„) < a for
n^ ?. Pick ? e ?',; f ^ 1. Since ? e Z\
|7„(?,?(?))-7(?.7(?))|<;?.
Also, ? e Z\ n implies that T(t, J „(?)) ? L - 1 and, by A9), 7„(f. J „(?)) ^ L. Now use
the fact that ? e Z? П ?'? ;1 and the preceding inequalities to obtain
\j„(u)[Tn(t,J„(w))]-J(cu)[T(t,J(cu))]\
? \S„(x) - S(x)\ + \co(T„(t. ??))) - co(T(t, J(?)))\
? 1 + 1=2.
Thus
|?„./„(?)| <;2 + \\TJ(w)\\ <^2 + K-2=K
for ? ^ ? and ? e Z'ir
Step 1 now follows.
To complete the proof we need to show that given ? > 0. ? > 0 if Z„j is defined by
Z„.s = {\h(S-]T„Jn) - h{S-]TJ)\ > S). B3)
then there exists an n such that whenever n ^ n,
Bo(Z„.s) < ?/?(?„?) < ?. B4)
As we argued earlier, there exists a constant L and integer n \ such that if
Z\.„ = (?: sup7„(f. J„(co)) > L) U (?: sup7(f. J (?)) > LI.

320 J. K. Brooks
then for ? ^ ? ?,
So(Z|.„)</j/8. B5)
Next, by Step 1, there exists a constant К and an integer ni such that if
Z2.„ = (||7-„У„|| > A") U (Ц7-УЦ > A1),
then for и > пг,
Bo(Z2.„) < ?/%. B6)
Note that since 5" is increasing and the 5",, 's are onto E, S~' (y) -> 5""' (y) for each y. The
monotonicity of S~' and S_l, for each и, implies that S~] -> 5" uniformly on [-AT, AT].
Since /г is uniformly continuous, there exists an ? > 0 such that
||a)| — ?>2||<? implies |/г(а>|) — h(coi)\ < S. B7)
There exists an /73 and a ? > 0 such that
\?\-??\??/2, \?[\^?, IjoI^AT, On.i B8)
implies
l^'Ui)-^1^)^?. B9)
Let S/, be defined as in Step 1 on C[0. L]. By Egoroff's theorem there exists a p > 0 such
that
S0(Zo)</?/8, where ?0 = (^ > 0/2). C0)
Define
Z3.„ = (?: sup|7„(f, 7„(?)) - Г(г, 7(?))| > ?). C1)
V ??
There exists an «4 such that
??(?3.„) < /j/8 whenever n > 114.
C2)
Let h = max(/i ?, /72, из, «4, ?.?, where /75 is an integer such that
\S„(x) — S(x)\ <?/2 whenever/j >/is. C3)
Set
?,? = Z|.„ U Zi.„ U Zs.n U Z().

One-dimensional diffusions and their convergence in distribution
321
Note that Bo(Z„) < ?. Fix ? e Zln and n > n. Since ? e Z\ n,
sup17), (?, У,, (?)) - 7"(?, 7(?))| ? ?.
Also note that || 7„У„ (?) || ^ К and \\TJ(a>)\\ ^ ?.Now we Z\' „ implies that T„(t, J„ (?))
? L and T(t, J (?)) ? L, for every t ? 1. Fix t ^ 1; since we Zc0,
\M(o)(T„(t, J„(a>))) - J(o)(T(t. J(a>)))\
? \S„(x) - S(x)\ + \o(T„(t, J„(a>))) - a>(T(t, J(w)))\
??/2 + ?/2 = ?.
Hence
that is,
\?„??)(?)-5"' \TJ(w)(t)\ <?,
|5,7?7"„7(?)-5"?7(?)| ^ ?.
This in turn implies IMS',;1 T„J(w)) - h(S~] TJ(?))| < 8, which completes the proof
of B4), and establishes the lemma. ?
Continuing the proof of the theorem, we now show that, for fixed x, Q" =>· Qx. To
this end, it suffices to show, if h: C[0, ?] -> R, ? < oo, is bounded and continuous,
then f hdQ" -> f hdQx. We may assume /? is uniformly continuous on C[0, M]. For
simplicity, set ? = 1. Observe that
I/hdQ1'- J hdQx\ = \j h{S;]T)dBs„lx)- fh(S-]T)dBSlx]
[h(S-^TJ„)~h(S-]TJ)]dBQ.
/1
By Lemma 3.4, the integrand converges to zero in So-probability. The Lebesgue
dominated convergence theorem implies f hdQ" -> fhdQx; hence Q" =>· Qx. This
completes the proof of the theorem. ?
3.3. Convergence of diffusions
The associated path function Y„ for the diffusion 5~' BT„ was defined in Section 3.1. All
functionals in this section are additive. Since the scale function is determined up to an
affine transformation, for convenience set 5@) = 0 and 5A) = 1. A diffusion 5 ВТ.
where 5 is normalized in this fashion, is said to be in canonical form.

322
J. K. Brooks
THEOREM 3.5. Let (X„)^_0 be a sequence of regular diffusions in canonical form. Then
(X„) converges weakly to Xo if and only if the associated paths converge a.s. uniformly on
compact time intervals.
The proof of the above theorem follows from the following theorem and its corollaries
and will be presented following the remark after Corollary 3.8.
Theorem 3.6.
(A) Let (?„)«5?? be a sequence of regular diffusions such that Xn =>· Xo- Suppose S„ is
a scale function for X„. Then for a ^ .? ^ b,
[Ь,(х)-T„(a)][J„(b)-Jn(a)]~l -»· [J(x)-J(a)][J(b)-J(a)]~l.
Suppose that we chose definite representatives S„ from the equivalence class of
scale functions S„ for X„, say Sn@) = 0 and 5„A) = 1, ? = 0, 1, 2 Then
S„(x) -> Sq(x) for each x. If Xn = S~l BT„, where T„ is the appropriate time
change of S„X„, relative to Brownian motion, induced by the speed measures m„,
then m„ =>·(- wo. Also T„ (t) -> 7b(i) and Y„ (t) -> Yo(t) uniformly on compacts a.s.
Bx, for xeR.
(B) Suppose {Sn)„^o and (T„)n^Q are sequences of scale functions and time changes,
respectively. Assume that S„{x) -> 5?(.?) for each ? and T„(t) -^-p 7b(i) for each
? = Bx. Then mn =>·<. тц. Hence T„(t) -> 7b(i) and Y„(t) -> Yg(t) uniformly on
compacts a.s. Bx, ? e R. Also
S~l BT„ => S~] BT0.
PROOF. Before proceeding with the proof, we shall outline some of the main steps leading
to the fact that weak convergence implies mn =>·< m and the time changes T„ converge
a.s. uniformly on compact time intervals. To avoid technical problems in this discussion,
assume that all the diffusions are in natural scale. Our job is to show BT„ =>· ВТ implies
m„ =>·<. m. First of all note that (m„) is bounded on compact sets. To see this, suppose that
mn([a, b]) -> oo. Argue that there is a point у and an interval /. у e /. and a time ?? such
that the set E\ of all ? such that ? (??. л", ?) > с for some с > 0 and for all ? e / has Bo
measure greater than 1 — ?, where ? is arbitrary. Furthermore, m„(I) -> oo. This shows
that ?„(??, ?) -> oo, for ? e E\. Assume that ?(??, ?) > ? for ? e E\. Then choose a
constant ? large enough so that if Ej is the set of ? such that sups^,() \w(s)\ < M, then
B(Ei) > 1 - ?. Let ? be the set of ? such that sups^ |?(?)| < ? for a fixed p; we then
show that if Q is the measure representing Xq, then Q(dZ) = 0; hence 6(J(Z) -> Q(Z).
But on E\ П ??, ?„(?,?) < ??, which means that X„(n) has its range bounded by ? with
probability greater than 1 — 2?, which contradicts the recurrence of Xo· The contradiction
comes from the fact that on a large set, X„. for large times n, behaves like Brownian motion
up to time ??.
Obtain a measure ? such that m„k =>·< ? for some subsequence (in)', assume пи = к.
Next we show ц.([с,а]) > 0 if с < d, that is. ? is a speed measure. Suppose m„([c.d]) ->
0. Let ? be the midpoint of [c. d]. Obtain a set E\ of B: measure greater than 1 — ? such

One-dimensional diffusions and their convergence in distribution
323
that for a>'s in this set, ? (??, у. ?) < ? for some ? and ??. for all v e [с, d]. Obtain a
number ? such that if
Ei = ? ?: sup|<w(i) — ;| > ?I,
then Bz{Ei) > 1 - ?. Then for ? in ?| П ?2.01 = ?„(??. ?) -> 0. Let L„ be the set
of ? such that sup,,^,, \?{?) - z\ > ?. Choose no large enough so that Q-(L) < 1/2,
where L = Lll(); set ? = l//;o- Consider only /;'s large enough so that c„ < t. Since
Qz(dL) = 0, Q"-{L) -> <2-(L). Note that T„(c„,w) > ?? on a set of measure greater than
1 — 2?, which means that on a large set the processes X„ in a small time interval [0, c„],
behaves like Brownian motion in a time interval at least as large as [0. ??]. This means
that the process X„ travels quite a distance in a very small time interval. On a large set,
sup5^, I X„ (s, ?) | > ? when ? is a large, but the limit of the Q". measure of L must be less
than 1/2, which is a contradiction.
Thus m„ =>·<. ?, where ? is a speed measure. If ? denotes the time change induced
by ?, it follows that ? = m, since ВТ is equal to ВТ in distribution. Hence m„ =>? m,
which, since ? (?. ·,?) is continuous and has compact support, implies ?,,(?.?) -> ?(?.?),
where the convergence is uniform on compact time intervals; this implies the uniform
convergence of the paths. Additional estimates have to be made if the diffusions are not in
natural scale. Now we begin the proof.
To prove the first part of (A), by definition of scale function, we need to show that
Q'l(hit b before a) -> ?°(hit b before a), C4)
where Q" is the measure on ? corresponding to the diffusion X„. Let ? = (?: ? hits b
before ?). The above limit holds if we can prove that Q(](dA) = 0. To show this, let ?? be
the first time ? hits ?; ?/, is similarly defined. Let
A ? = (?: ? hits b before a and max ? = b on [?. ?(/(?)]);
?? — (?: ? hits a before b and mina) = a on [?. ?/,(?)]).
We assert that 3? с A\ U Aj. Suppose ? e (A| U Aj)c and ? hits b before a. Since
? e A,, maxoi = &o > b on [0. ?(/(?)]. Obtain ?? e [0. ?„(?)] satisfying ?(??) = bo. Let
ao = mina» on [0, ??]. Set 0 < ? < min(ao —a, bo - b).To show ?(?,?) С /i, where
S(w,e) = (a/: sup|a/(i) - ?(?)| < ?. ? e [?. ?(,(?)]).
let ?>? e 5"(?,?). Since ??(??) > b, but a>| does not hit a in [0. ??], ?? e /i; hence
? e (dA)c. In a similar fashion, А, С C/i)'. Since 34 С A| U A2, it suffices to show
that <2°(A| U At) = 0. Define D = (?: ?@) = fo and ?(?) ^ b for all .s ^ ? (?) for some
? (?) > 0). Let ? = ?/,. Since Xo is recurrent, ? < oo a.s. Q°. If ft denotes the shift operator,
then A1 С (?? e D). By the strong Markov property
?.?(??) = ??(?<??.?,)< ? Ql(D)dQl = Q[l{D).

324
J. K. Brooks
Since ?0 its regular, Q°b(D) = 0. Likewise Q°f(A2) = 0; hence Q°x(dA) = 0.
If 0 <; ? <; 1, C4) implies 5„U) = [S„(x) - 5„@)][5„A) - ^(O)] converges to 50U),
from the definition of scale function. Using C4) again, and combining either 0 < ? < 1 < у
or у < 0 < ? < 1, we see that S„(у) -> 5o(y).
Next we assert that the (m„) are uniformly bounded on compact subsets of Ш. If not,
suppose that supw„([— 1, 1]) = oo. Using a nested interval argument and passing to a
subsequence (m„{k)), if necessary, we can find a point у e [-1, 1 ] such that for every open
interval / containing у we have mn{k)(I) -*k oo. Since ?(?, y,a>) -> oo a.s. Bo, where ?
is local time, we can find a ??, for any given К and ? > 0, such that ? (??, }', ?) > К for
? e ?(??), where Bq(A(K)) > 1 - ?. By the continuity of ? (??, ·. ?), for ? e A(K), there
exists a symmetric interval /(?) about у such that ?(??. у. ?)^ ?/2 for all * e /(?). Take
ay I 0 and consider
Dj = (?: ?(?0, ?, ?) > .?/2 for all jc e //),
where /y = [y - ay-, у + ay]. Note that Dj is measurable since it can be expressed as
an intersection of sets (?(??. r. ?) ^ AT/2) for rational r e /y by the continuity of ?. If
? e A(AT) and ;' is large enough so that ? (?) D Ij, then ? e D;. Since Z)y \ we have
lim Bq(Dj) > 1 - ?; hence there exists an integer ;'@) such that Bo(?>/@)) > 1 - ?; hence
there exists an integer ;'@) such that /?о@,@)) > 1 - ?. This implies that
AlHk)(to,o))= I k(tQ,z.w)mlHk)(dz)^ / k(t0,z.(i))m,Hk)(dz)
¦I JljM\
> (K/2)m„(k)(Ij@)) -»· oo.
for ? e Dj@). Assume, without loss of generality, that A„(??. ?) > н, н = 1, 2,..., for
we E\ = Dj(Qy This in turn implies that ?„(?,?) < ??. ? = 1.2
Now choose a constant Mo large enough so that Bo(Ei) > 1 — ?. where
; — ? ?: sup|a)(i)| < Mo I
1 -?.
Thus B0(E\ П E2) > 1 - 2?. Choose ? such that ? is larger than \Sq] (-M0) - 11 and
5(y' (M0) + 1. We may and shall assume, since Qq has only countably many atoms, that
2?(?: ?(/?) =-M or M) = 0. for each n.
Let ? be a fixed positive integer and define
? = (?: sup|a)(i)| < ? I.

One-dimensional diffusions and their convergence in distribution
325
Assume for the moment that Q^idZ) = 0. With this assumption, we have Qq(Z) ->
2q(Z). Let и > max(p,/ц), where
[S~l (-Mo), S,;1 (Mo)] С [So1(-Mq) - 1 · Sq1(Mq) +1] for ? > ??.
Fix ? e ?| П ?2· Then 7",,E, ?) < ?? for all s ? p. Since ? e ?2. |?G"„(?, ?))| < Mo, for
s ^ p. Hence
S-l(w(T„(s,w)))e]-M,M[.
Since Qq is the So-distribution of the mapping Y„ :? -> 5~' (?G"„(·. ?))), we have
??(?)^?0(????2)> ? -2?.
This in turn implies that Q®(Z) ^ 1 — 2?. Now let ? -> oo and conclude that
??(?: |?(?| <; ? for alb) > 1 - 3?.
which is a contradiction since Q° is recurrent on R.
To show that Q$(dZ) = 0, define the following sets. A\ is the set of paths ? such that
? reaches ? in the time interval [0. p] and stays less than or equal to ? for a positive
time ? (?) after hitting M. Ai is the set of paths that hit —? in the time interval [0. p] and
stays greater than or equal to -M for a positive time ?(?) after hitting -M. A3 is the set
of paths that hit - ? or ? at time p. We assert that
dZ С A| U A2U A-,.
Choose ???,? A, U A'v If ? hits ? in [0. p] but goes above M, and attains a maximum,
say M\ in [0, p], then if
S(w,e) = la/: sup|ai'(i) — ?(?)| < ? ?.
where 2? = M\ - M, then ^(?, ?) П ? = 0. Hence ? ? 9Z. Similar reasoning applies if ?
goes below — ? in [0, p]. Suppose that ? always stays below ? and above —? in [0, p].
Let 2? be the minimum of ? - M\ and m\ + M, where M| and mi are the respective
max andminof ? in [0, p].ThcnS(w.s)(~)A' = 0; hence ? ? дА. We know ?0'(?;0 — 0
from the choice of M. To show 60'(A|) = 0. let ?(?) be the first time ? hits ? in [0, p],
if it exists, and +00 otherwise. If С is the set of all paths ? such that ? @) = ? and there
exists a ?(?) > 0 such that ?(?) ^ ? for .s e [0. ? (?)], then, if ?, denotes the time shift
operator, we have
Qo(A\)= Q°0(t < 00. A,) ^ ?((>(? < ??.?? e C)
Bм(с)^еоЧем(с)=о.
?
(т<ос)

326
J. K. Brooks
since Q° represents a regular diffusion. Likewise. Q^iAi) = 0. Hence Q^(dZ) = 0. This
completes the proof of the fact that (mn) is bounded on compact sets. As a result, we can
obtain, by a diagonalization process, a subsequence (m„a)) and a measure ?, finite on
compact sets, such that mll(k) =$c ?.
Now we shall show that ?[?, b] > 0, whenever a < b. If we deny this, there exists a
further subsequence m^y, such that mk{j)[a,?>]-»/ 0. For notational convenience, assume
that m„[a,fo] -> 0. Let г = (? + b)/2. Let ? > 0 be given. There exists a io such that
BZ(E\) > 1 — ?, where
??: sup|a)(i) — z\ < (b — a)/3j.
Using the continuity of the distribution of the maximum appearing in ?|, there exists a ? \
such that if \z' - z\ < 8\, then S-*(?|) > 1 - 2?.
For a.a. ?,
sup ?(??, ?.?) < oo;
уe[«.ft)
hence there exists a constant ? such that B-(Ei) > 1 - ?, where
?2= (?: sup ?(??. ?.?) < Ml.
ye[«.fc) ' '
Using the continuity of local time, there exists a <5i such that if |;' - z\ < <5i, then
?:·(?2)> ? -2?.
Finally, choose a p > 0 such that В (?-0 > 1 — ?. where
3 = ??: sup |?(?) — z\ > ?).
Obtain a <$-, such that if \z - z\ < <5ч, then S-*(?;0 > 1 - 2?. Now let ? e ?| ? ?2. We
have
?„??,?) = I k(h),y,w)m„(dy)= ? ?(?(). v.o))m„(dv)
^ Mm„[a, b] = c„ -> 0.
For0>O, ?°_, (L„)->0, where
L„ = ??: sup |?(?) — 5() ' (z)\ > ?).
«<!/;:

One-dimensional diffusions and their convergence in distribution
327
by the continuity ofthe paths ?. Choose an но@) so that Q° . (?„,,(«>) < l/2.Wechoose
S() (,)
? as follows. Let 3? be the minimum of S~' (z) - S~' (z - p) and S(y' (z + p) - 5(y' (z).
Choose и ? so that for ? > и ? we have
min{5,71(z)-5,7l(c-p).5,7l(c + p)-5,7l(c)}>2^.
With this value of ?, let F = 1/но@). Set L = L„o(«). Assume for the moment that
?° (9L) = o. Hence we have ?"_, (L) -> ?0,., (?). Choose и2 > n\ large
¦>„ (.-) ^ S„ (;) 5() (:)
enough so that c„ ^ Ffor all ? > hi. If w e E\ ? ??. ?.ч, we have
sup|wG"„E,ii;)) - z| > sup |?G„(?.?)) - z\ > sup |?(?) - z| > p.
because T„(c„,w) > ??, when ? e ?| ? ??· Hence, for ???|???(? ?.i and /7 > «2, we
have
sup|5,71(wG1,(i.w))-5,7l(z)|
^min{5,7lU)-5,7l(z-p).5,71(c + p)-5,71(;)}>2^.
Since S~l(z) -> 5G' (z), we can find an integer/73 > «2 such that if ? e ?| П ?2 П?3 and
? > из, we have
и <Г
If Y„ denotes the function C[0. oof -> C[0. oof, defined by ^,,(?) = S~l {w(T„(-.w))}, we
see that if w e E\ ? ?? ? ?3 and /7 > «3. then
sup\Y„(w)(u)-S-\z)\>e.
«?
that is, ?| П ?2 П ?-, С У,~' (L).
Choose «4 >/м such that IS^ST1 (;))-;I < min(<S|. S2. 83) for alb; ^«4.Now ?" ,
is the ?,, ,c-i... -distributionof У„; hence
?5|7,<:'?) = ??»(?,<:)>?"??')^?^'\-,'.-''(????2???)^1?·
for и > ид. Hence
1/2 > ?" (L) = lime" , (L)>l-6e.
which is a contradiction since ? was arbitrary.

328
J.K. Brooks
The fact that <2"_, (9L) = 0 is proved in a fashion similar to the one used in the
S() (;)
preceding arguments concerning boundaries of subsets of ?. We omit the details.
Returning now to ?, we have shown that ? is finite on compact sets, and is strictly
positive, that is, on non-empty open sets, ? is greater than zero. Hence ? is a speed
measure.
For notational convenience, assume that mn =>·<- ?. Let
?(?,?) = ?(?,?,?)?(??).
Since ? is a speed measure, A is strictly increasing and continuous on a set of full measure.
Now
A„(t,J„(cu))
= /„(?) + /„(?).
Note that
|/„(?)| < / |?(?, у - S„(x),a>) - k(t, у - SOU), o))\m„(dy).
By Trotter's Theorem 2.2, ?(·, ·, ?) is jointly continuous in t and у for ? belonging to
a set of full measure ?®. Since ?(?, ·, ?) is uniformly continuous on compacts, (w„) is
uniformly bounded on each compact set, and k(t. ·, ?) vanishes outside of a compact set,
we see that ?„(?) -> 0. Since m„ =>·<· ?,
/„(?)-»· A(t,J0(a>)).
Hence An{t, J „(?)) -+ A(t, Jq(co)) for each t and ? e ??-
?? a similar fashion, ?,,(?,?) -> ?(?.?). ? e i?o· Hence by Theorem 3.1, we conclude
that S~lBTn =>· Sq1 ВТ. By the uniqueness of traversal times for a Markov process,
since 5~' BTn =>· Sq1 BTq, we conclude 7 = 7b a.s. P. Alternately, since SQ] ВТ is
distributionally equal to S^1 BTq, and ? is the speed measure for the diffusion S^ ВТ,
m = ? by the uniqueness of speed measures.
Since the above argument was valid for any subsequence of (mn), we conclude that the
full sequence (w„) satisfies mn =>·< w.Thus A„(t) -> Ao@ uniformly on compacts on i?o,
and the same is true for T„(t) -> 7b(f). Since 5,7' -> 5^' uniformly on compact subsets of
E, one can show that Y„(t) -> У(>@ uniformly on compacts on i?o· This establishes (A).
/ k(t,y.S„(x)+w)m„(dy)
J [k(t, у, 5„U) + ?) - ?(?, у, 50U) + ш)]|я„(dy)
+ / k(t,y,S0(x) + cu)m„(dy)

One-dimensional diffusions and their convergence in distribution
329
We now turn to the proof of (B). Assume that T„(t) -*p ??) and S„(x) -> S(x),
? = Bx, for each x. Using Theorem 3.1 and the lemmas appearing in its proof, we see
that BTn =>· BTo- Then by (A) above, m„ =>·(. тд. By arguments used in the proof of (A),
we see that
A„(i,y,f)-»· A0(i.y0v) ???0·
By Theorem 3.1 we conclude 5~' BT„ =>· S,^ BT<). Also wj„ =>¦<· /wo implies 7„(f) ->
7b@ and y„(i) —>¦ Yo(t) uniformly on compacts on i?o· This completes the proof of
Theorem 3.6. ?
Corollary 3.7. // A„(t) -^p Ao@ for each ? = Bx and t ^ 0, then the additive
junctionals A„ converge uniformly on compact sets on a set of full measure.
PROOF. If A„(t) -^p A0(t), then by Lemma 3.2, T„(t) ->p T0(t). Thus by (B) in
Theorem 3.6, mn =^f mo· The proof of Theorem 3.6 shows that A„(t) -+ Ao@ uniformly
on compacts on a set of full measure. ?
Corollary 3.8. S~x BT„ =>· S~[ BT0 if and only ifT„(t) -> 7@ and Y„(t) -> Y0(t)
uniformly on compacts on a set of full measure, assuming the S„ are normalized as in (A).
In particular, BTn =>· BTq if and only if T„(t) -> 7o(i) ?/iii ?(?„(?,?)) -> ?G?(?,?))
uniformly on compacts on a set of full measure.
Remark. Dynkin A965, Chap. VIII) shows that for processes in RUI\ where time
changes, relative to Brownian motion in R(i", are given by measures m„, if mn =>·<¦ m, then
A„(t) —>p Ao(t), for each t. The above analysis shows that for d = 1. A„(t) -> Ao@
uniformly on compacts on a set of full measure. The above results provide a converse to
Dynkin's theorem when d = 1, since if A„(t) -+p A(t), this implies by Lemma 3.2 that
Tn(t) -^-p To(t); hence by Theorem 3.6(B). m„ =>,. w.
Conclusion of the proof of Theorem 3.5. Suppose that S~ ]BT„^S~^BT. Since the
S„ are normalized, (A) of Theorem 3.6 yields the convergence of the paths Y„. Conversely,
suppose we have the stated convergence of (Y„). Note that since У„@) -> Y@) a.s. Bx,
this implies S~^{x) -> 5~'U). which implies uniform convergence of (S„) on compacts.
Consider
\S„Y„(t,w)-SY(t,w)\ <: |5?^,(?.?)-5^,(?,?)| + |5^(?.?)-5^(?,?)|.
The first term goes to zero uniformly on compacts by the uniform convergence on (S„)
and (Y„) on compacts. The second term goes to zero by the uniform continuity of S on
compacts and the convergence of (У„). Hence ?(??(?.?)) —* ?(?(?.?)) uniformly on
compacts a.s. Bx. But then it follows that BTn =>· ВТ. By Corollary 3.8, this implies that
T„(t) -> T(t). By (B) of Theorem 3.6, this implies that 5,7' BT„ =* 5,7' ВТ.

330
J. K. Brooks
4. Diffusions as a limit of stretched Brownian motions and stretched random walks
Introduction
We shall show that every linear regular diffusion is a pathwise limit of processes which are
rather simple in nature. These simple processes, which may be considered to be the basic
building blocks of a general diffusion, are called stretched Brownian motions. They behave
like Brownian motion except that the variance is a constant which depends on the region
of the state space in which the particle is located, and there are a finite number of such
regions. The precise description of stretched Brownian motion, its characteristics, and the
convergence theorem are presented in Section 4.1.
Processes called stretched random walks are defined in Section 4.2. These processes
have the characteristics of a symmetric random walk with variance af when they are
located in ]*,-, *,-+| [, where — oo < ? ? < · · · < лч < oo, and when they reach Xj they have
probability p, of moving to the right of x, and probability q,¦ = 1 - Pi of moving to the left
of Xj. It will be shown that every stretched Brownian motion is a weak limit of stretched
random walks, and this in turn will imply the existence of a sequence of stretched random
walks that converge weakly to a general given diffusion. The last topic deals with the time
change ? (called natural time), relative to Brownian motion, associated with a given scale
function S. A particle undergoing the diffusion X = S~[ ВТ then moves as closely as
possible to a particle undergoing motion which imitates Brownian motion in the sense that
its traversal time is quadratic.
The limit theorem in Section 4.1 complements the deep results obtained by Knight
A962) and Stone A963), who also used local time in considering the problem of obtaining
diffusions as a limit of simple processes.
All the results presented here are due to Brooks and Chacon A983).
4.1. Stretched Brownian motion
First of all, we shall give a loose description of the simplest type of stretched Brownian
motion. Let a > 0 be fixed. Imagine a particle whose movement is governed in the
following manner: If the particle is situated strictly to the left of zero in the state space,
let it undergo ordinary (standard) Brownian motion B{t,a>). When the particle is strictly
to the right of zero, stretch the state space by a and alter the time by the factor 1 /a2, so
strictly to the right of zero, it behaves, due to scaling, like ordinary Brownian motion. Let
Xa denote this process.
At first glance, it appears as though Xa is ordinary Brownian motion, however, this is not
the case, as can be seen by computing its scale function. A moment's reflection shows that
the probability of Xa hitting b before -b. b > 0 starting at zero, is the same as ordinary
Brownian motion starting at zero and hitting b/a before —b. Thus Po(Xa hits b before
-b) = or/(l + a). This shows that Xa, for ? ? 1, is not Brownian motion. Presently, we
will give a rigorous description of X", but from the above, it is clear that zero (any other
point in the state space could have been selected) acts like an internal boundary point. The
scale function of Xa consists of two straight lines with a break or corner at the origin; the

One-dimensional diffusions and their convergence in distribution
331
speed measure is Lebesgue measure on ] —oo, 0] and a multiple of Lebesgue measure on
]0, oo[. Intuitively, Xa is a Brownian traveller to the left and to the right of zero, and when
it hits zero it receives a kick, due to the stretch a, which gives the particle a probability of
a/(l + a) of moving to the right: more precisely (see Corollary 4.3),
P0(Xa(t) >0)=?/A+?). for every t > 0.
Consider now a more general diffusion. Let .vo < ? \ < ¦¦¦ < Xk be given. We can generalize
the above construction to obtain a diffusion whose scale function is linear in the regions
]—oo,x\], [xj,Xj+\], [jtjt, oof and whose time change, relative to Brownian motion, is
obtained by multiplying Brownian time by a constant, where the constant depends upon
the region [*,-,xi+\].
Ito and McKean A974) defined a process called skewed Brownian motion which
has attracted much attention. This process is derived from В by randomly flipping the
excursions of \B\ by means of a biased coin. When the particle undergoing skewed
Brownian motion hits zero, its probability of moving to the right is p, the probability of
obtaining a head from the biased coin on that particular excursion; when away from zero,
skewed Brownian motion behaves like ordinary Brownian motion. However, as pointed out
by Walsh A978), Ito and McKean's argument that skewed Brownian motion is a diffusion
is circular. Walsh mentions that a pathwise construction can be given by using Ito's theory
of point processes, but the construction is intricate.
The convergence theorem stated in this section in a sense justifies the interest in Xa
since, loosely speaking, every stretched Brownian motion is made up of processes Xa.
We believe that the processes X" captures the essence of skewed Brownian motion
since Po(Xa(t) > 0) = or/(or + 1), and it has the advantage of clearly being a diffusion
since it arises from a simple scale and time change of В (?), as will be shown. The special
case Xa of stretched Brownian motion has also been observed, in different contexts, by
Rosenkrantz A975), Portinko A976), Harrison and Shepp A981). Harrison and Shepp
originally arrived at Xa by considering a process which satisfies X(t) = B(t) + al^(t),
where /?(·) is the local time at zero of the unknown process X(t). They discuss X(t) in
terms of a solution of a stochastic differential equation.
We shall first examine the construction of Xa. If we stretch the state space by a when
the particle is to the right of zero, the time must be altered by a factor of \/or for proper
scaling, hence the additive functional we need to obtain the process is
?(?,?)= / f2(B(s,co))ds.
Jo
where f(x) = a if ? ^ 0 and f(x) =1 if ? < 0. ?(-,?) is continuous and strictly
increasing on a set of full measure.
If T(t, ?) — A~~' (t, ?), where all paths are restricted to the above set of full measure,
then it is clear that the process we described above is given by
Xa(t,w) = h{B(T(t,w),w)}.

332
J.K. Brooks
where h(x) — ax if ? ^ 0 and h(jc) = ? for ? < 0. Note that the fact that X" is a diffusion
follows from the general theory of time changes induced by additive functionals (see
DynkinA965)).
In general, let x\ < xi < ¦ ¦ ¦ < Xk- Define
k-\
f = aol]-oc..v,[ + ]Ti*i l[.v,.v, + ,[ + (*k: l[.u.oc[ ,
1 = 1
)t-l
g = A)l]-oc..v,[ + ? &'??*,¦·¦»,*,[ + /М[Ц.зс[ ·
i = \
where ?,, ?, > 0. Next, let
Jo
Finally, define
X(t,w) = h{Bz(T(t,co),co)}.
where T(t, ?) is the inverse of the additive functional A(t, ?), h(x) = fQ f(y) dy, and B-
is Brownian motion starting at z.
X(t) is called stretched Brownian motion (starting at h(z))- Xa corresponds to the case
к = 1, jc? = 0, ?\ = a] = a2, a0 = 1. The scale function of X(t) is /г, whose graph is
the polygonal line having slope \/a-, between h(jc, ) and h(xj+i). When /3, = a-, call X(t)
natural stretched Brownian motion.
We compute the speed measure m of X" as follows: Since the process behaves like
Brownian motion away from zero, the expected time, starting at jc > 0, to leave the interval
(c, d), 0 < с ^ ? ^ d, is cd + ex - x2 - xd, hence the derivative, with respect to h~' (x),
of the expected time is a(—d + c — 2x); this implies the speed measure of ]c,d] is 2a times
Lebesgue measure. Since the particle does not spend any positive time at zero, m({0}) = 0.
Непсет(А) = 2/д/.
To calculate the speed measure of a stretched Brownian motion given by
X = hBT. h= I fdx. T-\t) = A(t)= J g(Bs)ds,
where / and g have been defined earlier, let us first calculate the expected time X(t)
exists from the interval [a,b], given that X is at v. a ^ у ? b. Assume h{x) = у and
jci < jc < JC2. x\ < h~](a) ? h~l(b) < .in. Since scale functions are determined up to
affine transformations, assume that the scale function h~' of X passes through 0 at h{x\)
and has slope \/a\ between h{x\) and h{xi)- Now we are interested in the expected time
?(?/?\) exists from [h~' (a). h~' (b)] starting from h~' (v), which is equal to the expected
time B(t) exists from [?^/?\/?\, Ьу/Щ/а\\ starting at у^/Ща\; this expected time is

One-dimensional diffusions and their convergence in distribution
333
Pi/a2(by — y2 + ay — ab). Taking the derivative of this expression with respect to the
scale function h~[, we obtain (?\/?\)(-2\ + b +a). This function generates 2/5|/ori
times Lebesgue measure in the interval [h(x\). ft(jn)]· The speed measure for X in other
regions are handled in a similar fashion.
Feller's theorem states that if ? e ?>(??), then
Гаф=(а/ат)(а+ф/(аИ-[)),
where Га is the infinitesimal generator of X". If ?" is continuous and bounded on R \ {0},
then (Гаф)(х) = A/2)ф"(х) since the process is locally Brownian at л ? 0. Continuity of
??? implies ф"@+) = ф"@-). Note that the measure induced by ?+??(?\?~') has value
??'(8) - ?'(-?) on ] - 5, <5], and this value is also given by f^s(raf)(x)m(dx). Letting
? -> 0, we conclude that аф'@+) = ф'@-) if ? e ?>(??). Note that reflected Brownian
motion \B\ is, in a sense, the limit of X"asa^ oo; hence for ? to belong to Z?(/]bi),
we must have ф'@+) = 0 and (Г\В\ф)@) = (\/2)ф"@+). The calculations for the scale
function and speed measure for the general stretched Brownian motion are carried out in a
similar fashion.
Suppose now that Z(t) is a regular linear diffusion with the real line as its state space
and scale function S. Then SZ(t) is a regular diffusion in natural scale with a unique
speed measure which we denote by m. Let ? = C[0. oof be endowed with the topology
of uniform convergence on compact sets. Bx is Wiener measure, starting at x, defined on
the Borel algebra of ?. Let ?(?, у, ?) denote local time for Brownian motion. Define the
additive functional
A(t,w) = k(t,y,oj)m(dy), ?&?, t > 0.
and let ? denote the inverse of A.
By Trotter's theorem there exists a set ?? С ? such that ??(??) = 1 for each ? and
?(·, ·, ?) is jointly continuous in t and ? for each ? e i?o; also ?(·, ?) is continuous and
strictly increasing for each ? e ??. Henceforth, all paths ? will be confined to ??- If
Qx is the Bs(X) distribution of the mapping ?(·) -> S~[ {?(?(-, ?))}, then {?,??? is a
diffusion with a scale function S and speed measure m.
There exists a sequence of measures mn on the real line satisfying the following
conditions:
(i) m„(A) = fA f„(x)dx, where/,, = ??<?<>„ < Ь", and ?," > 0, -oo = a'0' < a'{ <
¦¦¦<a": =oo,/;'= ? ,,?"[, with <=?'? =1;
Jn I I — 1 1 1 Jn
(ii) mn =>·? m, that is, J ????? -> J фат, for every continuous function on R having
compact support.
Let S„ be a sequence of scale functions each of which is piecewise linear such that
Sn{x) —>· S{x) for each x. Without loss of generality we may and shall assume that
each S„ is linear on the intervals J',' J", and that these intervals refine the intervals
corresponding to /„. Let Q" be the diffusion which is defined by Sn and w„, that is, if
A„(t,w)= / f„(y)X(t,y,w)dy= I k(t,y,w)m„(dy),

334
J. K. Brooks
and T„ is the inverse of A„, then Q"x is the Z?s„m distribution of the mapping ?(·) ->
S~' {?G„(·,?))}. Note that {<2"}.,eR is a stretched Brownian motion for each n.
To show that F");Ji| converges weakly to Qx, we need to prove that
f gdQ,=\imf gdQ'i
for every bounded function g which is continuous on ?. This is equivalent to showing
lim jg((S;lTnJ„)(w))Bo(dw)= f g((SulT0J0)(o)) B{)(dw),
where J„ :? -> ? is defined by 7„(?) = ?(·) + S„(x) and (S 7„./„)(?) is the curve
whose value at t is S~' {Jn(w)(T„(t, У,,(?))}. Here we set S = SO, 7" = 7b- Since g is
bounded, it suffices to show that g(S~' Tn J„) -> ?(| giS^1 7ЬУо)· However, a much stranger
result holds, namely,
(S-lT„J„)(w)(-)^ (Sq1T0Ji))(o)(-) C5)
uniformly on compact time sets. To see this, observe that
A„(t,J„)(w))= / k(t,y,S„(x) + w)m„(dy)
= J[k(t. y, S„(x) + ?) - k(t, у, So(x)+a>)]m„(dy)
+ / k(t,y,So(x)+a))m„(dy)
= /„(?) + /„(?).
Now
|/„(?)| < ? \k(t, у, S„U) + ?) - k(t, у. Sq(x) + ?) |m„(dy).
Since ?(?, ·, ?) is uniformly continuous on compacts, (m„)n is uniformly bounded on each
compact set, and ?(?, ·, ?) vanishes outside of a compact set, we see that /,,(o>) -> 0. Since
m„ =>·<· m, /,,(?) —* A(t, Jq(co)), hence
T„(t,Ma>))^To(t,Jo(a>)). C6)
Each Tn (·, ?) is continuous and strictly increasing, thus the convergence in C6) is uniform
on compact time intervals. Finally, since each S~' is continuous and strictly increasing,
C5) follows. A similar argument shows that if the processes (X„U))%L0 are defined on ?
by Xn(t)(w) = S~]{w(T„(t,w))}, then ?,,(-,?) -> ??(-,?) uniformly on compact time

One-dimensional diffusions and their convergence in distribution
335
intervals for each ? e ??. Note that Xo(t) is distributionally equivalent to Z(t). With the
convention that we refer all linear diffusions to its canonical representative on ?, we have
THEOREM 4.1. Every regular diffusion is the almost sure pathwise limit of a sequence of
stretched Brownian motions, where the limit is uniform on compact time intervals.
4.2. Diffusions as a limit of stretched random walks
In this section we shall consider a random walk approximation to stretched Brownian
motion. Our detailed description will deal with Xa; once this is done, it will be clear how
the general stretched Brownian motion is to be approximated. Harrison and Shepp A981)
have also obtained a random walk approximation to Xa by methods somewhat different
from those presented here. Walsh A978), under the assumption that Ito and McKean's
skewed Brownian motion is a diffusion, derived the transition probabilities of X", Our
method uses a random walk argument since we do not have Corollary 4.3 as an immediate
consequence of the definition of Xa. As we mentioned in Section 4.1, since
P0(Xa(t) = b before -b)=a/(a+ \) = p,
it is natural to approximate Xa by a random walk which, when away from zero, is
symmetric and when it hits zero it has probability ? to go to the right and probability
q = 1 - ? to go to the left.
Consider a fixed probability space (?, ?, ?) which supports the processes described
below.
?\, ?2, ¦ ¦ ¦ are i.i.d. random variables with ?(?\ = 1) = ?(?\ = -1) = 1/2; ?\, ??,...
are i.i.d. random variables, independent from the first process, satisfying ?(?\ = 1) = ?
mdP(f] =-l)= 1 -?, 0^/?< 1.
Let V be the partition of ? defined by the points
{x,k/n, k=0, ±1,±2,...},
with partition intervals {8k, к = ±1, ±2,...}. where S\, 82, ¦ ¦ ·, are the first, second, ...
partition intervals to the right of x, and <5_i, <5_2, · · ·, are the first, second, ... partition
intervals to the left of ?. \?,¦ | denotes the length of the partition interval <5,.
Define
for
?„«)=? + ?\\??,\+?2\?№2\ + ----
+ <Jk+\pk+\?<4+??' у ~
-I-Pit 141
??4?2 +
+ I4I21
Ok .
t e [|ie, |2/?? + · · · + \Sef/ah, |ie, |2/?, + · · · + |i„t+I |2/<t*+i].

336
J. K. Brooks
where
Ok = p\-\ hpjt,
and
Uk+\ ifx + p]\Sel\ + --- + pk\S0,\^0.
\ Va+? otherwise.
We interpret p\ \??? I + · · · + pi,-\&цК\ = 0 if к = 0. Next, we define the ?* appearing above
as follows: if
Mk+\ = x +pi|5fl,|H hpjt+i|5^+I| >0,
then set au+\ = ??/УДГ; if Mk+\ < 0, then ?*+? = ??/?^· Finally, if M/,+ \ = 0, set
<fk+\ = &k-
This defines the process Xn(t) starting at every ? e E, and X„(t) has continuous
trajectories. When we set <*o = /So = 1 and <*? = ?, ?\ = a2, we obtain an approximation
tor.
In general, given a stretched Brownian motion X = hBT, where h and ? are given
in Section 4.1, one can construct a random walk approximation in a similar fashion-for
example, in the state regions jc,_i < .v, < .v, + i, determine the different variances for X
viewed as a Brownian motion in ]xj,xi+\[ and ]jc, ?,.x,[; when constructing the random
walk, when the particle moves in either of these regions, the speed of the particle should
be the corresponding standard deviation times the reciprocal of the distance traveled, and
the ?? are used. When the particle hits jc,, then ?(,) are to be used, ?(?{-') = 1) = pt
and ?(?['] = — 1) = 1 - pi, where pt is determined by the break in the graph of h at
(Xi ,h(xi)). Call these random walks stretched random walks, and natural stretched random
walks if the variances are all one.
We construct now X„(t) in detail in the case of three space regions (the general case
is similar) given by R\ —]—oo,x\[, /?? =]-??.*2[· ^3 =]*2, °°[, where —oo < x\ <
X2 < oo. Again consider a fixed probability space (?, ?, ?) which supports the following
processes described below. Let ?\,??,... be i.i.d. random variables with ?(?? = 1) =
?(?? = — 1) = 1/2 and let (?^), (?,\) be sequences of i.i.d. random variables independent
of each other and of (?„), with ?(?[„') = 1) = Pj, P(flJ) = -1) = 1 - p}, where p; =
ajl(otj + ?/+?), j = 0, 1. Define the partition V = [x,x\,X2,k/n, к = 0, ±1, ±2,...},
and the 8^ partition intervals indexed with respect to their position to the right and left of ?
as before. Define X,,(t) exactly as before, for t belonging to the appropriate time interval,
where
Pk+i =?k+\ ifx + p\\Stil| ? l·Pk\Sв^\ e R<,
i = 1,2, 3, and where
_ | V°+l if x + P\\8e,\-\ l·Pk\S?^\ =x\,
[?? if ? +?\\??,\-\ V Pk\&t)k\ = хъ

One-dimensional diffusions and their convergence in distribution
337
We still need to indicate the definition of ?*, which is done as follows: if M^+\ e /?,¦, then
ak+\ = otj/y/??, / = 1, 2, 3, and if Mk+\ = x\ or jo, thena^+i = ?*·.
THEOREM 4.2. Every stretched Brownian motion is the weak limit of stretched random
walks. Hence every regular diffusion is a weak limit of stretched random walks.
Proof. Consider first the simple stretched Brownian motion X" with the corresponding
stretched random walks X„(t) defined above. Let (Q")„ be the measures induced on
C[0, 1] by Xn(t) starting at x. Observe, first of all, that (Q"x) is tight. To prove this, consider
|X„(OI· This is the polygonal process associated with a reflected symmetric random walk.
Using Donsker's theorem, we see that |X„(OI =>· \B(t)\, where В is standard Brownian
motion and =>· denotes weak convergence. We shall use the following
THEOREM. (Q"x)„ is tight if and only if (see Billingsley A968))
(i) For every ? > 0 there exists some b such that
(?';)(?: |?@)| >&) ^jj for each n\
and
(ii) For each ?, ? > 0, there exists an h > 0 and an «o such that
Q"(co: Sh(co)^s) !??, form^iiQ.
Here, Sh(w) = sup(\w(s) - ?(?)\: \s -t\ ?. h], h ?. 1.
Obviously, (i) holds in our case. To prove that (ii) holds, simply observe that <5/,(?) ^
2<5/,(|a>|), and apply the above theorem to the measures induced by |X„(?)|.
Next, let S„(t) denote the symmetric random walk, where pu = &· We will use the
following equality: For А С [0, oo[,
Po{X„(t) eA) = P(X„(t) e A, Xn(fl) = 0) = 2pP0(S,,(t) e A). C7)
This can be deduced by observing that, if ? = time of the last visit to 0 in [0, t], then
Po{X„(t)eA, T=l) = 2pP0(S„(t)eA. z = l).
In particular, Po(X„(t) ^ 0) = p, for t > 0, since the zeros of S„ and X„ are the same.
Using Donsker's theorem again, we have
limfb(Xii@ e A) = 2PB0(x(t) e A),
where x(t)(w) = ?(?), and А с [0, oo[. If A c]-oo,0], we have
\imP0(Xn(t) e A) = 2A - p)BQ(x(t) e A).

338
J.K. Brooks
Let <2o be a weak limit point of (Qq),<- The <2o distribution of x(t) can have only
countably many atoms, hence for all but at most a countable number of 0 < a < b,
Qo{x(t) e [a,b]) = 2pB0(x(t) e [a, b]).
A similar result holds for intervals to the left of zero. Hence
^=exp(-y2/Br)Kv. y^O,
6o(*@edy) =
< y.
2л ?
^^ exp(-y2/B0) dy. у < 0.
2.??
C8)
Next, take jc > 0. If ? is the time of the last zero of X„ in [0. t], if it exists, ? = +oo,
otherwise, then if А с ]0, oo[ and Ax is A reflected,
P,(Xn(t) e A) = P,(X„(r) e ?, ? = oo) + Px(Xn(t) e ?, ? < oo).
By the reflection principle,
/\(X„@ e ?, ? = oo) = PtE„(i) e A. S1,, positive on [0, t])
= Pt(S„(t)eA)-Px(S„(t)eA±).
Also
P,(X„@ e ?, ? < oo) = ]Грл(Х„@ e ?, ? =/)
/
= 2р]Г>гE11(г)ед, т = /)
/
= 2pPx(S„(t) e A, S„ hits 0 before f)
= 2??E„(?)??±).
If Qx is a weak limit of (?"), passing to the limit along a subsequence we obtain
Qx(x(t) e A) = 1/^2*7 / (exp(-(y - jcJ/Br)) - exp(-(y +*J/Br))) dy
+ 2p/Vba I exp(-(y+xJ/Bt))dy.
J A
Hence, for jc, у > 0
Q,(x(t)edy)
= l/Vba [exp(-(y - jcJ/B0) + Bp - 1) exp(-(y -xJ/Bt))] dy. C9)
Similar reasoning yields, for jc, у < 0, the above formula with A — 2p) replacing Bp — 1)
in C0).

One-dimensional diffusions and their convergence in distribution 339
If x < 0 and А с ]0, oo[, with ? defined above, then
P,(X„(t) e A) = P,(X„(t) e ?, ? < oo)
= ]??,(?„(?)??|?=/)??(? = /)
/
= J]2pfb(Sll(i-/)eA)Pr(T=/)
/
= 2pP4S„(t)eA).
Hence, if ? < 0 and у > 0,
?*(*? e dy) = Ipjsfba ехр(-(дг - ?J/B?)· D0)
If jt > 0 and у < 0, the formula D0) holds with 2A — p) replacing 2p.
The finite dimensional distributions of x(t) relative to Q can be calculated in a similar
fashion, and it seen that x(t) is a Markov process. The above formulas for the transition
probabilities show x(t) has a Feller-continuous semigroup, hence {Qx} is a strong Markov
process, that is, a diffusion. Also, ?? = ??. For ? ? 0,
(??/)(*)=1/?
if /" is bounded and continuous on R \ {0}. For / described above, /@) = 0,
l-E0(f(x«))) = lE. ? N,{y)dy + ^—^ F Ndy)dy, D1)
' t JO t J-oc
where N,(y) = (\/¦Jbu)exp(-y1 /2t)\~N,(y) = G/Vf )M (у)· Equation D1) is equal to,
for a > 0,
2??-^/'@+)+$? J yH,(.y)dy+j /(}-)'" 4(}') <M
-2A-?)!^/'@-) + ??|"??,(}·)?·-| /(>·)?_??(>')^'|,
D2)
where <5Я -> 0 and Sa -> 0 as ? -> 0. For each a, the second and fourth terms converge as
t -> 0. The sum of the first and third terms in D2) is
/2P/@+)-2(l-P)/'@-)+gfdj,\ r'^yNl{y)dyt
which shows that p/'@+) = A - p)f'@-), or a/'@+) = /'@-), where p = a/(\ +<*),
if (Ге/)@) exists. Hence ?? = TQ.

340
J. K. Brooks
Alternately, we could have argued that since Qx is locally Brownian, ? ? 0, and since
Po(X„(t) hits 1 before - l) = ?
(hence Qo(x(t) hits 1 before - 1) = p), that a scale function for Q is S(x) — ?, ? ^
0, S(x) = (A - p)/p)x, x^O, which is also a scale function for Xa; the speed measure
of Q can be shown to be equal to the speed measure of X" by observing that away from
zero Q is locally Brownian. This shows that Q and Xa have the same scale functions and
speed measures and are therefore equal.
In the case X @ is a general stretched Brownian motion, one can prove in a similar
fashion, tightness and finite dimensional convergence of the distributions of the corresponding
stretched random walks. This proves the first part of the theorem. If X(t) is a regular
diffusion, there exist X„(t), stretched Brownian motions such that X„(t) =>· X(t), by
Section 4.1. But, for each n, there exist W"' =3-,,, X„(r), where W,'"(f) is a stretched random
walk. Since weak convergence is metric, it follows that X(t) is the weak limit of stretched
random walks. ?
COROLLARY 4.3. For any t, P0(Xa (t) ^ 0) = <*/(<*+ 1).
PROOF. Corollary 4.3 follows from C7) of the proof of Theorem 4.2.
D
4.3. Natural time
Let S be a given scale function S: R -> R, where the mapping is onto. If a diffusion has S
as its scale function, what would its time change, relative to Brownian motion, be in order
that a particle undergoing the diffusion move, as much as possible, in a Brownian fashion?
To motivate this, let X = S~' В Т. We want to find conditions on ? so that X(t) locally has
Brownian characteristics. It can be expected, neglecting infinitesimals,
?? - ?? = (X(t\) ~ X(to)J = [S~] B(T(t\ ,?)) - S B(T(t0, ?))?
Dividing both sides of D3) by [B(T(t\, ?)) - B(T(tQ, ?)]2, and using
T(t\, ?) - T{t0, ?) = [B(T{t\ ¦ ?)) - B(T(t0, ?))]2,
we obtain, by allowing ?| -> ??.
dS(z)
dz
z=S-lB(T(i0.a>)).
'¦ _???(?,?) V1 =
"V dt I=J
dA(s)
ds
D3)
D4)
D5)
i=.V()
where ?? = T(to, ?), and A is the inverse of T. Hence, A should satisfy
?(?,?)= / \S'(S~4B(u,w)))] 2du.
Jo
D6)

One-dimensional diffusions and their convergence in distribution
341
Given a scale function S, the time change ? = A-1 defined by D6) is called the natural
time associated with 5".
As a further illustration, we show that for a piecewise linear scale function, the time
change for stretched Brownian motion corresponding, in the notation of Section 4.1, to
the ?? = af case, leads to D6). Let S be piecewise linear scale function with breaks at
У\ < У2 < ¦ ¦ ¦ < Ук with slopes 1/<*?, l/a2 l/<*jt in the corresponding regions.
Define
k-\
/ = aol]-3o.S(y,)[ +^ail[S(y,).s<.v,-,)[ +<**l[S(yt).^[
and
k-\
« = O!01]-3o.S(yI)l + ^Q!i:i[S(.v,).5(v,-I)[+«il[S(yU.3c[ ¦
i = \
Let X = hBT, where h(x) = /Ql /, S = fc-', T~] = A, where
A(t)= f g(B,)ds.
Jo
This gives a process which behaves like Brownian motion in the state space regions
]~oo,y][, ]yit yi+\[,]ykl oof.
Note that
g(S(z)) = J2[S'(z)]-h[SiXi.^l){(S(z)),
or
g@-[5'E-'@)].
Thus A satisfies D6).
Let S be the class of all scale functions S having a second derivative existing such
that there exists a sequence of polygonal piecewise linear approximations S„ converging
pointwise to S satisfying
? ?,? f /о on R, D7)
J(.) J(.)
where /„ = [S'„(S~' )]-2, So — S, and /(, /o is finite on compact sets.
We shall show that if S e S and X = S ] ВТ, where ? is natural time associated with S,
then there exists a sequence of natural stretched Brownian motions and natural stretched
random walks converging weakly to X. Intuitively, we interpret the condition S e S as

342
J.K. Brooks
follows: when S becomes flat in a region, it is harder for the particle under any time change
to move as if it were a Brownian traveller, so one would expect some conditions on S'.
Let 5" = 5? e S and assume that (S„) is the sequence of piecewise linear scale functions
converging to S satisfying D7). As in Section 4.1 we shall work on ??· Define
A„(w,t)= / k(t,y,w)m„(dy).
where mn = f f„, and the /„ are defined by D7). Since m„ =>· w0, we see, as in Section 4.1,
that if
X„(t,co) = S-]{w(T„(t,cu))},
where T„ is the inverse of An, then the natural stretched Brownian motions X„ (t) converge
pathwise to Xo(t) uniformly on compact time intervals, for ? e ??. Since each natural
stretched Brownian motion is the weak limit of natural stretched random walks, by the
results in Section 4.2, we have
THEOREM 4.4. Let S be a scale function belonging to S. There exists a sequence of
natural stretched Brownian motions X„ and natural stretched random walks Y„ such that
Xn converges pathwise almost surely to Xq, uniformly on compact time intervals, and Yn
converges weakly to Xq, where Xq is a diffusion with a scale function S and time change,
relative to Brownian motion, the natural time associated with S.
The author is pleased to acknowledge the permission of Academic Press, Inc. to use
portions of the material in references J.K. Brooks and R.V. Chacon A982, 1983) for this
chapter.
References
Bachelier, L. A900), Theorie de la speculation. These. Paris.
Billingsley, P. A968), Convergence of Probability Measures. John Wiley & Sons, New York.
Blumenthal, R.M. and Getoor, R.K. A968), Markov Processes and Potential Theory, Academic Press, New York.
Breiman, L. A968), Probability, Addison-Wesley. Reading, MA.
Brooks, J.K. and Chacon, R.V. A982). Weak convergence of diffusions, their speed measures and time changes.
Adv. Math. 46 B), 200-216.
Brooks, J.K. and Chacon, R.V. A983), Diffusions as limit of stretched Brownian motions. Adv. Math. 49 B),
109-122.
Brown, R. A828), Philosophical Magazine.
Dellacherie, С and Meyer. P.-A. A978), Probabilities and Potential. North-Holland, Amsterdam.
Dynkin, E.B. A955), Continuous one-dimensional Markov processes. Dokl. Akad. Nauk SSSR 105. 405^408.
Dynkin, E.B. A959), Principles of the Theory of Markov Random Processes. Moscow-Leningrad.
Dynkin. E.B. A965). Markov Processes, Springer. Berlin.
Einstein. A. A905). On the movement of small particles suspended in a stationary liquid demanded by the
molecular-kinetic theory of heat, Ann. Physik 17.
Feller, W A952). The parabolic differential equations and the associated semi-groups of transformations. Ann.
of Math. 55. 133-160.

One-dimensional diffusions and their convergence in distribution
343
Feller, W. A954), The general diffusion operator andpositivity preserving semi-groups in one dimension, Ann.
of Math. 60, 417-436.
Feller, W. A955), On second order differential operator, Ann. of Math. 61, 90-105.
Feller, W. A957), Generalized second order differential operators and their lateral conditions, Illinois J. Math.
1, 459-504.
Freedman, D. A971), Brownian Motion and Diffusion, Holden-Day, London.
Harrison, J.M. and Shepp, L. A. A981), A noted on skewed Brownian motion, Ann. Prob. 9, 309-313.
Ito, K. and Mckean, HP. Jr. A974), Diffusion Processes and Their Sample Paths, Springer- Verlag, New York.
Karatzas, I. and Shreve, S. A991), Brownian Motion and Stochastic Calculus, Springer, New York.
Karlin, S. and Taylor, H. A968), A Second Course in Stochastic Processes, Academic Press, New York.
Knight, F.B. A962), On the random walk and Brownian motion. Trans. Amer. Math. Soc. 103, 218-228.
Knight, F.B. A981), Essentials of Brownian Motion and Diffusion, Math. Surveys No. 18, Amer. Math. Soc.,
Providence, RI.
Levy, P. A937), Theorie de I'addition des variables aleatories, Gauthier-Villars, Paris.
Levy, P. A939), Sur certains processes stochastiques homogenes, Compositio Math. 7, 283-339.
Levy, P. A948), Processus stochastic et mouvement Brownian, Gauthier-Villars, Paris.
Loeve, M. A963), Probability Theory, Van Nostrand Crd edn.).
Mandl, P. A968), Analytical Treatment of One-Dimensional Markov Processes, Springer, Berlin.
Portinko, N.I. A976), General diffusion processes. Lecture Notes in Math., Vol. 550, Springer, Berlin, 500-523.
Rogers, L.C.G. and Williams, D. A944, 2000), Diffusions. Markov Processes and Martingales, Vols 1, 2,
Cambridge University Press.
Rosenkrantz, W A975), Limit theorems for solutions to a class of stochastic differential equations, Indiana
Math. J. 24, 613-625.
Stone, C.J. A963), Limit theorems for random walks, birth and death processes and diffusion processes, Illinois
J. Math. 7, 638-660.
Stroock, D.W and Varadhan, S.R.S. A979), Multidimensional Diffusion Processes, Springer, New York.
Trotter, H.F. A958), A property of Brownian motion paths, Illinois J. Math. 2, 425^433.
Walsh, J.B. A978), A diffusion with discontinuous local time, Asterisque 52-53, 37^45.
Wiener, N. A923), Differential space, J. Math. Phys. 2, I 31-174.
Wiener, N. A924), Un probleme de probabilities enombrables. Bull. Soc. Math. France 52, 569-578.
Wiener, N. A933), Generalized harmonic analysis. Acta Math. 55, 117-258.

CHAPTER 8
Vector Integration in Banach Spaces
and Application to Stochastic Integration
Nicolae Dinculeanu
University- of Florida. Gainesville, FL 32611, USA
E-mail: nd@math.ufl.edu
Contents
Introduction 347
1. Preliminaries 348
1.1. Banach spaces 348
1.2. Measurable functions 348
1.3. Integral of step functions 349
1.4. Measurability with respect to a positive measure 350
2. The Bochner integral 351
2.1. The seminorm 351
2.2. Bochner integrability 352
2.3. The Bochner integral 353
2.4. The spaces ?.?(?) 354
3. Integration with respect to measures with finite variation 356
3.1. Measures with finite variation 356
3.2. Integration with respect to a measure with finite variation 357
3.3. The indefinite integral 359
3.4. The Radon-Nikodym theorem 361
4. Semivariation of vector measures -364
4.1. The semivariation 364
4.2. Semivariation and norming spaces 365
4.3. Semivariation of ?-additive measures 366
4.4. The family nip.z of measures 367
4.5. Extension of measures 367
4.6. Extension of positive measures 368
4.7. Extension of ?-additive measures * 369
4.8. Canonical extensions 369
4.9. Canonical extension of additive measures 370
5. Integration with respect to a measure with finite semivariation 371
5.1. Measurability with respect to a vector measure 371
5.2. The seminorm mf.cif) 372
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
345

346
N. Dinculeanu
5.3. The space ^DimF.c) 374
5.4. The integral 375
5.5. Convergence theorems 376
5.6. The indefinite integral of measures with finite semivariation 377
5.7. Integral representation of linear operations on L''-spaces 378
5.8. The Riesz representation theorem > 379
6. The Stieltjes integral 379
6.1. The variation and the semivariation of a function 380
6.2. Semivariation and norming spaces 381
6.3. The measure associated to a function > · ¦ 382
6.4. The Stieltjes integral 383
7. The stochastic integral 386
7.1. Notations and definitions 386
7.2. The measure ? ?. Summable processes 387
7.3. The stochastic integral 388
7.4. Convergence theorems > 390
7.5. Summability of the stochastic integral 391
7.6. Local summability and local integrability 391
8. Processes with integrable variation or integrable semivariation 392
8.1. Processes with finite variation or semivariation 392
8.2. Optional and predictable stochastic measures 393
8.3. The measure ?? 394
8.4. Summability of processes with integrable variation or integrable semivariation 395
9. Martingales 396
References 398

Vector integration in Banach spaces and application to stochastic integration 347
Introduction
This chapter is devoted to the theory of integration with respect to vector measures with
finite semivaration and its applications. This theory reduces to integration with respect
to vector measures with finite variation, which, in turn, reduces to the Bochner integral
with respect to a positive measure. The Bochner integral, itself, is based upon the classical
integral of real-valued functions with respect to a positive measure. The above presentation
is a description, in reversed order, of the four stages in the development of the integration
theory. We shall present these four stages in their natural order (the first stage in Chapter 2).
Among many approaches to the classical integral we have chosen one which seems to
be simpler and is presented, for example, by W, Rudin A973), see Chapter 2. Any one of
these approaches leads to a vector space L' (?) of integrable functions, equipped with a
seminorm ||/|| ?, for which it is complete and in which Vitali and Lebesgue convergence
theorems are valid.
We shall impose the same requirements to any kind of integration theory. An
integration theory satisfying these requirements is called a "satisfactory" integration. Here,
"satisfactory" refers to the possibility to use the integral to a wide range of applications.
The other three stages of the integration theory yield satisfactory integrals.
The Bochner integrability of vector valued functions / reduces to the classical
integrability of |/|, The Bochner integral is obtained by extending by continuity the
integral of step functions.
There are other types of integrals for vector valued functions, for example the Pettis
integral. These integrals are not satisfactory in the above sense.
The integrability with respect to a vector measure m with finite variation \ m |, is defined
as Bochner integrability with respect to the positive measure \m\. The integral is again
obtained by extending by continuity the integral of step functions. This stage of the
integration theory is presented in detail by Dinculeanu A967),
Finally, integrability with respect to a measure with finite semivariation reduces to
integrability with respect to a family of measures m- with finite variation. The integral
jfdm is defined then as the linear operation си f fdm-. This last stage is very
important. In fact, the most interesting measures do not have finite variation, but may have
finite semivariation.
Among the applications of the last stage of integration theory presented in this
chapter we quote: the integral representation of linear operations on V-spaces, the Riesz
representation theorem, and the Stieltjes integral with respect to a vector valued function
with finite semivariation. The most important application is the Stochastic integral in
Banach spaces, with respect to summable processes: in particular with respect to square-
integrable martingales and with respect to processes with integrable variation or integrable
semivariation.
For a detailed presentation of vector integration and its applications, the reader is referred
to Dinculeanu B000).
In this chapter we give complete statements of definitions and theorems; but no proofs
of theorems are given. For proofs, the reader is referred to different books.
The numbering of definitions and theorems starts anew in each section.

348
N. Dinculeanu
1. Preliminaries
In this section we establish the notations that will be used throughout the chapter,
1.1. Banach spaces
1. Throughout the chapter, E,F,G,D will denote Banach spaces. For any Banach
space ? the norm of an element ? e ? is denoted by |jc|. the dual of ? is denoted
by M* and the unit ball of ? by M\, The duality between ? and M* is denoted by
(x,x*), (x*,x), x*x oreven.vjc*. L (F, G) is the space of bounded linear operations from
F into G. We write ? с L(F, G) to mean that ? is continuously embedded into L (F, G).
i.e., |jcy| < \x\\y\ for* e ? and ? e F. For example. ? = L(R. ?). ? С ?(?*. ?) = ?**.
if ? is a Hubert space, ? = ?(?. R).
We write со ?! ? to mean that ?? does not contain a subspace which is isomorphic to
the Banach space со.
If ? is a Banach space, a subspace ? с ?* is said to be norming for M, if for every
? & ? we have
|*| = sup{|(*,z)|·. zeZ|),
1.2. Measurable functions
2. Throughout the chapter, S is a nonempty set, V, 7?, A P, 5, ? are respectively a
semiring, a ring, an algebra, a <5-ring, a ?-ring, a ?-algebra of subsets of S.
For any class С of subsets of S we denote by r(C), a(C), Sr(C), ar{C),aa{C),
respectively the ring, the algebra, the <5-ring, the ?-ring, the ?-algebra generated by C.
If ?? is a ring, we denote by Sf(R) the vector space of 7^-step functions f:S^> F of
the form / = ?]·'=| Хд,.*,, with A, e 7? and .v, e F. The sets A, can be chosen mutually
disjoint. Then
?
For any function /: S -> F or R, we denote by | /| the function defined by
|/|(?) = |/(?)|, fori e S,
We emphasize that a positive 7^-step function takes only finite values,
Measurability is defined with respect to a ?-algebra. Let ? be a ?-algebra of subsets
of 5".
3. Definition. A function /;S -» F or R is said to be ^-measurable, if there is a
sequence (/„) of iT-step functions /„ :S->F such that /„ -> /, pointwise.

Vector integration in Banach spaces and application to stochastic integration 349
The iT-step functions are .^-measurable. It follows that the set of .^-measurable
functions is a vector space. If / is .T-measurable, then |/| is also .T-measurable.
In the above definition we can choose the sequence (/„) with additional properties,
4. THEOREM. Let f ;S -> F orR be ? ?-measurable function. Then there is a sequence
(/„) of ?-step functions f„\S-> F or R such that fn -> f pointwise and |/„| < l/l for
each n.
If f is positive (with values in [0.+00]). the sequence (/„) can be chosen to be
increasing.
If f is real-valued and bounded, the sequence (/„) can be chosen to be uniformly
convergent.
The following theorem gives a characterization of .T-measurability,
5. THEOREM, A function /: S1 -» F orR is ?-measurable iff it has separable range and
f~\B)e ? for every Borel set В С F or В С I.
iT-measurability is preserved by pointwise convergence:
6. THEOREM. //(/,,) is a sequence ofF or R-valued, ?-measurable functions converging
pointwise to a function f then f is also ?-measurable,
1.3. Integral of step functions
7. Let H be a ring of subsets of S and m ; Tl -> ? с L(F, G) be an additive measure.
For any F-valued. 7^-step function
?
f = 2_] ??,?? with A, e 7? and л, е F,
1 = 1
we define the integral f f dm by the equality
/ fdm = Y^m(A,)x, e G.
J 1 = 1
The definition of the integral is independent of the particular representation of / as a
step function. In fact, one can prove that if ??-\ ??,?? = 0, then ?'?=\ m(^i)xi — 0;
consequently, if
? in
/ = l .,= 1

350
N. Dinculeamt
with A;, Bj e TZ and x,, yy e F, then
/1 HI
]T w (A;,)xi = J^ m (S,)y/.
and J f dm is defined unambiguously.
We have the following immediate properties of the integral of TZ-step functions:
8· f(f + g)dm = f fdm + f gdm,
9. cf dm = c f f dm, for с el.
10. If ? is a positive, finite, additive measure on TZ and if f.S -> F is an 7^-step
function, then f /?? e F and
11. If we want to extend the integral for a larger class of functions, we have to impose
additional conditions on TZ and m, for example, that 7? is a <5-ring or a ?-algebra and m is
?-additive and with finite semivariation. In particular, m can have finite variation or can be
positive.
There are four main stages in the development of the integral / / dm:
(I) The classical integral, with m ^ 0 and / real-valued,
(II) The Bochner integral, with m ^ 0 and / vector-valued,
(III) The integral f f dm, where m is a vector measure with^m/te variation and / is
vector- valued.
(IV) The integral J f dm, where w is a vector measure with^ш/'fe semivariation and /
is vector-valued.
The most important stage is the first one. The others stages can be performed to the
extent that they can be reduced to the first one.
In the following paragraphs we shall present succinctly each of the above stages,
except (I) which was already presented in Pap B002) (Chapter 2 in this Handbook).
1.4. Measurability with respect to a positive measure
The framework for this section is a measure space (S, ?, ?), consisting of a nonempty set
S, a ?-algebra ? of subsets of S and a positive, ?-additive measure, with finite or infinite
values, ? : ? -> [0, +oo], such that ?@) = 0.
We shall assume that ? has the finite measure property (FMP), i.e., for every set A e ?
we have
?(?) = $??{?(?); Be ?, В С ?. ?(?) < ??}.

Vector integration in Banach spaces and application to stochastic integration 351
12. A function / :S -> F or R is said to be ?-negligible is f(s) = 0, ?-a.e,
13. Measurability with respect to the ?-algebra ?(?) is called ?-measurability.
The ?(?)-$??? functions are also called ?-measurable step functions.
A function f:S -> F or R is called ?-measurable if it is .?^)-measurable, i.e., if
there is a sequence (/„) of F or R-valued ?-measurable step functions, such that /„ -> /
pointwise.
14. A function / :S -> F or R is ?-measurable iff there is a .^-measurable function
g:S^> F or R such that f = g, ?-a.e. If f\ = /2. ?-a.e. and if f\ is ?-measurable, then
/2 is ?-measurable. It follows that /is ?-measurable iff there is a sequence (/„) of iT-step
functions such that /„ -> /, ?-a.e.
15. We extend the ?-measurability for functions defined ?-a.e. An F or R-valued
function defined ?-a.e. on S is said to the ?-measurable, if it has a ?-measurable extension
g:S -*¦ F or R on the whole space S. Then any extension of / to S is ?-measurable.
2. The Bochner integral
The framework for this section is a measure space (S, ?,?) and a Banach space F.
The definition of the Bochner integral is very similar to the classical integral, with some
differences.
2.1. The seminorm
1. For every F-valued ?-measurable function / defined ?-a.e. on S we set ||/lh =
/\/\??. For each ?-measurable function / defined ?-a.e., there is a ?-measurable
function g defined everywhere such that f = g, ?-a.e., therefore ||/|| ? = ||g|| ?.
|| /|| ? has the following properties:
2. o< 11/111 <;+oo.
3. 11/111 = 0 iff / = 0, ?-a.e.
4. If / = g, ?-a.e.. then ||/||, = ||g||,.
5. Il/ + g||i<||/||i + ||g||i.
6. ||c/|h = |c| 11/111, forceR.
7. If|/K|g|,then||/||i^||g|||.
8. If 11/111 < 00, then |/| < oo, ?-a.e.

352
N. Dinculeanu
9. Countable convexity. If (/„) is a sequence of ?-measurable functions and if the series
??=? f" IS convergent ?-a.e., then
?/-
77=1
DC
<?«/·?
I 77=1
2.2. Bochner integrability
The Bochner integrability is similar to the classical integrability.
10. Dehnition. We say that an F-valued function / defined ?-a.e. on S is Bochner
?-integrable, if/ is ?-measurable and ||/|h < oo, i.e., / |/|^? < oo.
If II/77 ~ /II1 ""*¦ 0- we say that /„ -> / in the mean.
11. DEHNITION. We denote by Ь\-(ц.) or ^(.??.?) or LF(S, ?,?), the set of all
Bochner ?-integrable functions /: S1 —» F. defined everywhere on S.
The Bochner integrable functions have the following properties:
12. A step function / = ]T"=| xAiXj with A, e ?(?) and *,· e F is ?-integrable iff
A, e ?/(?) for each /'.
13. If feLFfo), then l/l eL1 (?).
14. ?-|?(?) is a vector space.
15. The mapping / ?» ||/|| ? is a seminorm on L\ (?). The topology defined by this
seminorm is called the topology of convergence in the mean.
16. If / e LF{p) and g:S -> F is ?-measurable and satisfies |g| < |/|. ?-a.e., then
geLj.-(/i).
We can prove Lebesgue's theorem, before defining the Bochner integral; at this stage
one can conclude only convergence in the mean.
17. Lebesgue's dominated convergence theorem. Let (/„) be a sequence
from LF(p), /:S-> F a function and g e L] (?). Assume that
(a) /, -> / ?-a.e,
(b) |/n|<g, ?-a.e., for each ?.
Then f & Ьр(ц) and fn -> / ш f/ге wea/г.
For the proof, we apply Fatou's lemma to the sequences (|/„ - /| + 2g) and Bg —
\fn-f\).

Vector integration in Banach spaces and application to stochastic integration 353
After we define the Bochner integral, we can add to the conclusion that / /„ ?? ->
18. The set Sf{Ej) of F-valued, ?/ -step functions is dense in ?^(д).
In fact, for each /??[·(?) there is a sequence (/„) of ^/-step functions such that
fn~*f pointwise, ?-a.e. and |/„| < |/|, ?-a.e. for every n; then we apply Lebesgue's
theorem.
19. LF(?) is complete.
Properties 2.37 through 2.41 in Chapter 2 remain valid for Bochner integrable functions.
2.3. The Bochner integral
20. For a Ef -step function f:S-> F of the form / = ]?"=| ??: jc, with A, e ?/ and
Xi e F, we defined the integral j f ?? by the equality
f "
/ /?? = ^2?(??)??
i = l
Taking the sets A, mutually disjoint we have
/1
\?\ = ????\??\,
i=\
hence
f "
J |/|?? = ]??(?, )|л',|
i = \
It follows that
\j ??? <|?/|?? = |?/???·
This inequality shows that the linear mapping L:Sf-~(Ef)^>F defined by
L(f) = j ???, iorfeSy(Ef),
is continuous for the seminorm ||/|| ?. Since the space Sy(E/) is dense in Lf(m) for the
topology defined by the seminorm || /|| ?. one can extend L to a continuous linear mapping
L*:L]F(u) -> F, and we still have
|?*(/)|<ll/lli. for/eL^(M).

354
N. Dinculeanu
For each function / e L\(?), the value L*(f) e F is denoted by / f ?? and is called the
Bochner integral of / with respect to ?.
The Bochner integral has the following properties:
21. ?/ + 8)?? = ?/?? + ?8??.
22. fcfdp = effdv,ioTceR.
23. |//^?|</|/??=||/|||.
24. If /„ -»¦ / in L[-(M), then / /„ ?? -»· / /??.
This property allows us to add / /„ ?? ^> f /?? to the conclusion of the Lebesgue's
theorem.
If / e LF(u) and A e Г(д). then /?? e L j..(?). We denote, as usual,
/ /?? = / /????.
25. If / e LiF(?), the set function w : ?(?) -> F defined by
m(A) = / /??. for Ae ?(?)
J A
is a ? -additive measure.
26. We can extend the Bochner integral / f ?? for F-valued, ?-integrable functions /
defined ?-a.e. on S, If g; S -> F is a function such that / = g, ?-a.e,, and g e ?[·(?).
then we define J f ?? = f g??, and the integral f /?? is independent on g.
2.4. The spaces ^{?)
27. Definition, Let 1 < ? < oo. We denote by Lt(?) the set of ?-measurable functions
f:S -> F such that |/K> e L'(m), i.e.. such that /\/\??? < oo. For / e ?//-(?), we
denote
11/11/, = (j\f\pd^
28. ?^(?) is a vector space and the mapping / н» \\f\\P is a seminorm on it. The
topology defined by this seminorm is called the topology of convergence in the mean of
order p.

Vector integration in Banach spaces and application to stochastic integration 355
29. LPF (?) is complete and Sy(Ef) is dense in Lpt^),
30. Definition. We denote by ?^(?) the set of ?-measurable functions such that
II/IIoc = inf{a: 0 ^ a ^ +oo, | f(s)\ < a, ?-a.e,} < oo,
31. L^?(?) is a vector space and \\f\\oc is a seminorm on it, for which it is complete.
We have /„ ->¦ / in ?^(?). iff there is a ?-negligible set N С S such that /„ -> /
uniformly on S\N.
32. If F = R. the set S(Z) of real valued. ?-step functions is dense in Loc(?) :=
If F is an infinite-dimensional space, the set Sy(E) is no longer dense is L^(?).
The following property asserts the uniform ?-additivity and the uniform absolute
continuity of the indefinite integrals of functions in a Cauchy sequence.
33. THEOREM. Let (/„) be a Cauchy sequence in V\ (?) with \ ?. ? <oo. Then:
(a) The measures ?„ : ?(?) —» R defined by
f \fn
J A
?«(?)= / \/?\???. forAeE(u),
J A
are uniformly ? -additive, for n e N.
(b) lim,J(A)^(,/A ???'^? = 0. uniformly for ? e N.
(c) For every ? > 0, there is a set Sf e ? with MEf) < oo, such that
/ \&\???<?, for all ? eN.
Js\sf
Using Theorem 33, one can prove the Vitali theorem.
34. THEOREM (Vitali). Let (/„) be a sequence from ?^-(?) with 1 ^ ? < oo and
f :S -> F a function such that f, -» f, ?-a.e. Then f e ?,?-(?) аиЛ /„ -»· / /n ?^-(?) /#
the following conditions are satisfied:
(b) Нт^д^о/д \fn\p ?? = 0. uniformly for ? e N.
(c) For every ? > 0, i/itre /s a set SF ? ? with ?(??) < oo, such that
? |/„|/??<?, for all ? eN.
Js\sf
If ?E) < oo, condition (c) is automatically satisfied with Sf = S. From the Vitali
theorem, we can deduce the Lebesgue theorem in Lp (?).
35. THEOREM (Lebesgue). Let 1 ^ ? < oo, (/„) a sequence from Lp(?), f: S -> Fa
function and g e Lp^), such that

356
N. Dinculeanu
(a) fn -> /, ?-a.e,,
(b) l/«l^, ?-a.e., for each ?.
Then f e L'^(m) and fn -> / /и ?/^(?). //> = 1, ?/геи / /„ ??? -> f f ??.
Finally, we state Holder's inequality.
36. Let 1 < p, q ^ +oo be conjugate numbers, i.e.. ^ + ^ = 1. Let E, F, G be Banach
spaces such that, ? ci(F,G).
(a) If / e ЬрЕ(ц) and g e ?*-(?), then /g e ?^(?) and we have
I/
/g^M
Here, the product function /g: S" —» G is defined by
(fg)(s) = f(s)g(s), iorseS.
(b) Let TZ be a ring generating the <5-ring ?/. Then for every / e ?-^(?) we have
l/ll
/> = sup / \fg\d? = sup \f\\g\dp
the supremum being taken for g e Sf (TZ) with ||g||9 ^ 1.
(c) If G = R and / e ??-(?), then
l/ll/. = sup / />??
/
forge5/-(^)with||g||i/< 1.
3. Integration with respect to measures with finite variation
In this section we present the third stage in the development of the integral. We study
measures with finite variation and integration with respect to such measures. Integrability
with respect to a measure with finite variation reduces to Bochner integrability with
respect to the variation measure. We then study the indefinite integral and the Radon-
Nikodym theorem for vector measures. For a more detailed account, the reader is referred
to Dinculeanu A967) and B000).
3.1. Measures with finite variation
The framework for this section is a ring TZ of subsets of S, a Banach space ? and an
additive measure m : 7? —» E.

Vector integration in Banach spaces and application to stochastic integration 357
1. Definition. For every set Ac S, the variation of m on A is a number, finite or +00,
denote by var(w, A) or m(A) and defined by
var(w, A) = sup2_] \m(A-i)\
i€l
the supremum being taken for all finite families (A,);e/ of disjoint sets from 1Z contained
in A.
If A e 7Z|0C, we write |m|(A) = var(w. A). We say that m has finite (respectively
bounded) variation if var(w, A) < 00 for every A e TZ (respectively var(w, S) < 00). The
variation of m has the following properties:
2. \m(A)\ <; |w|(A), for AeTZ.
3. If А С S, then var(w, A) ^ var(w, ?).
4. var(w, A) = 0iff m(?) = 0 for every В е П with ? С A.
5. w is superadditive: for an arbitrary family (A,)(e/ of disjoint subsets of S we have
т(\^АЛ^^г}1{А,).
^i€l ' i€l
6. var(w -f ?, ?) ^ var(w. A) -f var(/z. A).
7. var(aw, A) = |or| var(w. A), for ? e R.
8. The variation \m\ :ll\oc -> [0, +00] is additive. If m is ?-additive on H, then |w| is
?-additive on Ti\ac.
9. Let ?: 72. -> R be a real-valued, additive measure. Then for every set А с S we have
sup{^(fi)|: ВеП, BcA}^m(A)^2sup{|m(B)|: ? e 7г. ficA}.
If ? is complex-valued, we replace 2 by 4.
10. A real valued, ?-additive measure ? : S -> R on a 5-ring ? has finite variation; if
? is a ?-ring, ? has bounded variation.
3.2. Integration with respect to a measure with finite variation
The framework for this section and for the following sections of this paragraph consists of
a 5-ring V of subsets of S, three Banach spaces ? с L(F, G) and a ?-additive measure
m:V —* ? with finite variation |w|.

358
N. Dinculeanu
Then I/я I is ? -additive on the ? -algebra P|OC in particular on the ? -algebra ? = ??(?>)
generated by P, possibly with infinite values.
Integrability with respect to w of vector-valued functions f:S—* F is reduced to
Bochner integrability of / with respect to the positive measure |w |.
11. DEnNlTlON. We say that a set А с S is w-negligible (respectively w-measurable) if
A is |w|-negligible (respectively |w|-measurable). We denote E(m) = E(\m\). We say a
property ? is true w-a.e. if it is true |w|-a.e.
We say that an F or R-valued function defined w-a.e. on S is w-negligible (respectively
w-measurable, w-integrable) if it has the same property with respect to |w |.
12. For 1 $5 ? ^ +oo we denote
LF(m) = LF{\m\)
and endow LF(m) with the seminormof Lft-(|m|):
\\?\\?=\\\?\\\?={\\?\? d\m\
for / e L^(w), if 1 ^ ? < oo and
IIу Hoc = II I/1 Hoc-
13. The vector space LF(m) is complete for the seminorm Ц/Ц,,.
14. If 1 ^ ? < oo, the set Sf(T>) of P-step functions f:S^> F is dense in LF(m).
However, if F is infinite dimensional, St (?) is no longer dense in Lf{m).
15. The Vitali and the Lebesgue convergence theorems are valid in L^(w).
16. The following assertions are equivalent for an w-measurable function /: S —> F:
(a) / is w-integrable;
(b) / is |w|-integrable;
(c) l/l is |w|-integrable.
17. We can now define the integral j f dm for functions / e LF(m).
For a P-step function / = ?''=| ?,??/ with A, e ? and х-, е F we defined the integral
/· "
/ fdm = Yum(A-)xibG.
J ?=?
>//'

Vector integration in Banach spaces and application to stochastic integration 359
If the sets A, are mutually disjoint, then
?
1/1 = ? ??, I*, I
i=\
and
I/
fdm
= Y^m(Ai).Xi k]T|w(A,)||.v,|
^Yi\m\(Ai)\xi\= { \f\d\m\ = \\fh.
It follows that the linear mapping L : Sy{V) -> G defined by
Ufi-ff*.. tofts*!».
is continuous for the seminorm ||/lh . Since Sy(T>) is dense in LJ. (m), the mapping L can
be extended to a continuous linear mapping L*: LF(m) —* G.
The value L*F(f) of the extension for a function / e L j. (w) is denoted by j f dm and
is called the integral of f with respect to m.
18. We still have
\J fdm U f\f\d\m\=\\fh, for feL\(m).
If fn -> / in L[-, then / /„ dm-> j fdm.
19. If / e L^(w) and Л e E(m), we have /хд е L\(m).
We denote, as usual,
/ fdm= / fxAdm.
20. If / e L^(w), the mapping Л ы- fA fdm from 2T(iw) into G is ?-additive and
lim [ \f\d\m\ = 0.
3.3. 77?e indefinite integral
Let w : V -+ ? С L(F, G) be a ?-additive measure with finite variation |w | on a 5-ring P.
and let ? be the ?-algebra generated by V.

360
N. Dinculeanu
21. ?????????. Let g e L\{m). We denote by gm : ? -> G the ?-additive measure
defined by
(gm)(A) — I gdm, for A e 2?.
and we call it the indefinite integral of g with respect to m, or the measure with density g
and base m, or the product of g and m.
22. If g e Lf(w), then |g| e L'(|w|) and we can consider the indefinite integral
Ы|ш|:Г^К+:
(Ы|ш|)(А) = f \g\d\m\, ??????.
The measure gm has finite variation \gm\ satisfying
\gm\ < |g||w|.
23. THEOREM. Let g e LF(m). If either g or m is real-valued, then we have equality
\gm\ = \g\\m\.
In particular, if ? is a positive measure and if g e L\ (?), then
I#mI = IglM-
For the proof, see Dinculeanu B000), Theorem 2.29.
24. Letg e L]F(m). ? \gm\ = |g||m| and if
J gdm = 0, for every A e ?,
then g = 0, w-a.e.
We now state some theorems concerning integrability with respect to the measure mg.
We consider first the case of real-valued measures.
25. THEOREM. Let ?:?> -> Ш be a ?-additive measure on a S-ring V with finite
variation |?|.
Let g e L]E^) and f: S —> Fa ? -measurable function. Then
(a) / is \g\\?\-integrable iff f is g\x-integrable.
(b) If f is g[L-integrable, then fg is ?-integrable and we have
J fd(gp) = f fgd/i

Vector integration in Banach spaces and application to stochastic integration 361
and the associativity formula
f(8H) = (/S)M-
(c) If g is real valued, then f is g\x-integrable iff fg is ?-integrable.
For the proof see Dinculeanu B000). Theorem 2.30.
For vector valued measures we have a similar theorem:
26. THEOREM. Let m :T> -> ? С L(F', G) be ? ? -additive measure with finite variation
\m\ on a S-ring V. Assume that either
(i) g e L' (m) is a real-valued function and f: S -> F is ?-measurable, or
(ii) g e L\{m) is a vector valued function and f:S -> R is a real-valued, ?-
measurable function.
Then:
(a) / is \g\\m\-integrable iff fg is m-integrable.
(b) If fg is m-integrable, then f is gm-integrable and we have
/ fd(gm)= / fgdm,
and the associativity formula
f(gm) = (fg)m.
(c) If g is real valued, then f is gm-integrable iff fg is m-integrable.
For the proof see Dinculeanu B000). Theorem 2.31.
3.4. The Radon-Nikodym theorem
In this section we state a Radon-Nikodym-type theorem for vector measures with finite
variation. This theorem is very useful in proving the equality \??\ = n ? for the measure
?? associated to a process X with integrable variation, in Section 8.
27. Let ?: Tl -> R+ be a positive, additive measure defined on a ring 7? and m :!?_--* ?
an additive measure. We say m is absolutely continuous with respect to ? or д-^Ьяокиегу
continuous and we write m <^?, if
lim m(A) = 0.
We state first the classical Radon-Nikodym theorem for real valued ? -additive measures
on a X'-algebra.

362
N. Dinculeanu
28. THEOREM (Radon-Nikodym). Let ?:? ^ Ш+ be a positive ?-finite, a-additive
measure and m : ? -> R a real-valued, ?-additive measure, such that m«/i.
Then there is ? ?-measurable function g e L' (?) such that m = g?. i.e.,
L
m(A)= I gdii, forAe?.
'a
Then
\m\(A)= J \g\dii. for ?&?.
J A
29. Remarks, (a) The theorem is valid under more general conditions, for measures
having the direct sum property, in particular for regular measures on a locally compact
space. (See Dinculeanu A967), Theorem 5. p. 182.)
(b) The theorem remains valid for vector-valued measures m: ? —> ? with finite
variation \m\, provided that the space ? has the Radon-Nikodym property (RNP).
A Banach space ? has the RNP. if for every finite measure space (S, ?, ?) and every ?-
additive measure m : .?7 —» ? with finite variation \m\ such that m«/i, there is a function
g e ??{?) such that m = g?, i.e.,
m(A)= I gd?, for A e ?.
J a
Then
-L
\m\(A)= / Igld/x. forAeT.
'a
Examples of Banach spaces with the RNP are reflexive spaces and separable duals of
Banach spaces.
For a detailed study of the Radon-Nikodym theorem for vector measures, the reader
is referred to Diestel and Uhl A977). We shall state now a more general and very useful
version of the Radon-Nikodym theorem, without imposing the Banach space ? to have
the RNP. But the density g is no longer measurable; it is only weakly measurable.
30. THEOREM (Radon-Nikodym). Let m: ? -> ? С L(F.G) be ? ?-additive measure
with finite variation \m\on ? ?-algebra ? and ?: ? -> R+ a positive, ?-finite, ?-additive
measure such that ш«д.
Let ? С G* be a space norming for G. Assume F and ? are separable. Then there is a
function Vm : S -> L(F,Z*) having the following properties:
(a) For every ? e F and ? e Z, the function (V,„.x, z) is ?-measurable, ?-integrable
and
[m(A)x.z)= I {?,„?,?)??, for A e ?.
J A

Vector integration in Banach .spaces and application to stochastic integration 363
(b) |V,„| is ?-measurable, ?-integrable and \m\ — \ ?,„\?. i-e.,
-L
\m\(A)= / \?„\??, forAeE.
'a
(c) ?/\??\ = ?, then\V„,\=\.
(d) For every function f e L[(m) andevery ;e ? the function (Vmf, z) is ?-integrable
and
(/'H=/
(??,/.?)??.
For the proof see Dinculeanu B000), Theorem 2.34. The theorem is valid without
assuming F and ? separable; but the proof involves the lifting theory (Dinculeanu A967,
Theorem 5, p. 269) and A. and С Ionescu Tulcea A969)).
The following theorem is in a certain sense the converse of the preceding one. It is used
to prove the existence of dual predictable projections of processes with integrable variation.
31. THEOREM. Let ?: ? -> R+ be a positive, ?-finite, ?-additive measure, ? С G*
a space norming for G and U: S -> ? С L(F', G) a function having the following
properties:
(i) The function (Ux.z) is ?-measurable for every ? & ? and ? e ?;
(ii) The function \U\ is ?-integrable.
Then there is a ?-additive measure m: ? -> L(F, Z*) with finite variation \m\ satisfying
the following conditions:
(a) For every A e ?. ? e F and ? e Z, the function (pA(Ux, z) is ?-integrable and
L·"--
[m(A)x,z)= / {??,?)??.
(b) \m\(A)^fA\U\dii,fi>rAtEJ.e., \m\^\U\ii.
(b') If F and ? are separable, then \m\ = |?/|?.
(c) For f e 0?(\?\?) and ;e Z, the function (Uf, z) is ?-integrable andwe have
{ifdm-?l
(?/.?)??.
(d) The measure m has values in L(F.G) if either G is separable, or G is the dual of a
Banach space ? and we take ? = H.
For the proof see Dinculeanu B000, Theorem 2.35) and Dinculeanu A967. Theorem 6,
p. 274).

364
N. Dmculeami
4. Semivariation of vector measures
In this section we study measures with finite semivariation, which will be used in the next
section to define the integral with respect to such measures. We also study extension of
measures from a given class to a larger one.
For the proof of the results in this section, the reader is referred to (Dinculeanu B000),
§4 and §7).
4.1. The semivariation
The framework for this section is a ring ?? of subsets of S, three Banach spaces E, F,G
such that ? с L(F, G) continuously and an additive measure m : Ш -> ? с L(F. G).
1. Definition. For every set А с S, the semivariation of m on A. relative to the
embedding ? с L(F, G) (or relative the pair (F, G)) is a number, finite or +oo, denoted
svar/-.c(w, Д) or /Mf.c(A) and defined by the equality
ту.с(А) = sup
]Pm(A,).v,
ie/
the supremum being taken for all finite families (A,),e/ of disjoint sets from TZ contained
in A and all families (*,¦),¦ e/ of elements of F\.
We say m has finite (respectively bounded) semivariation wu.g on 1Z of my.c(A) < oc
for every A e1Z (respectively my.ciS) < oo).
2. Equivalently, the semivariation m y.c can be defined by the following equality:
I ( I
mf.c,(A) = sup / sdm\
where the supremum is taken for all 7^-step functions s: S -> F with \s\ ^ <рл-
This alternative definition of the semivariation will be extended for any function /
instead of ?д. The semivariation has the following properties.
3. If А с S then/и/-.с(А)^ш/-.с(А).
4. w/..G(A) = 0iff w (?) = 0 for every set ? e TZ with fie A.
5. mr.G(A) ? var(w. A).
6. If the embedding ? с L(F, G) is an isometry, then
\m(A)\ ^fny.G(A), for A e 7г.

Vector integration in Banach spaces and application to stochastic integration 365
7. If m, ? : 1Z —> ? are additive measures and ael, then
(m+n)F.G(A) ? ту.с(А)+пу.с,(А).
(am)t G(A) ^ \a\mF.G(A).
8. The set function my.c is finitely subadditive on 1Z\0c- If m is ?-additive on 1Z, then
ffif.G is ?-subadditive on TZ\ac.
9. Assume ? с L(F,R) isometrically. Then
™fm(A) — var(w. A), for any А с S.
10. We have
If the embedding ? с L(F, G) is an isometry, then
mR.? ^mF.G-
4.2. Semivariation and norming spaces
We maintain as a framework an additive measure m : 7? —» ? с L(F,G) defined on a ring
7?, and a space ? с G* norming for G.
11. For each ? e ? we define the set function m:: 1Z -> F* by the equality
(jc, w-(A))= (m(A)jt, c), for.? e F and AeK.
Then w, is additive. If we consider the embedding F* = L(F, R). then by property 9 we
have (w;)f.K = inz.
The semivariation inF.G can be computed by means of the variations m-\
12. PROPOSITION. For any space ? С G* norming for G and for every- set А С S we
have
wf.c(A) = sup m-(z).
If F = R, the semivariation mm.? has special properties:

366
N. Dinculeanu
13. If m: 1Z -> ? — L(R, E) is an additive measure, then:
(a) for every set А с S we have
mK.?(A)^2sup{|m(B)|: Bell, В С A).
(b) m is locally bounded (respectively bounded) iff m^jr(A) < oo for every A elZ
(respectively wKi-( S) < oo).
To say that m is locally bounded on 7? means that for every set A e TZ, m is bounded on
the ring 72. П A.
For the proof, we use a corresponding property for the variations m- and then we apply
Proposition 12.
4.3. Semivariation of ?-additive measures
The semivariation of ?-additive measures has additional properties.
14. Let m:T> -> ? С L(F, G) be a ?-additive measure on a <5-ring P.
Then for any ? e G*, the measure /и- is ?-additive on ? and the variation \m:\ is ?-
additive on the ?-algebra ? — ??(?>) generated by V.
REMARK. It is possible that \m-\ is ?-additive for every с e G*, but the measure m is not
?-additive.
15. Let m-.Ti^f ? с L{F,G) be an additive measure on a ring TZ. Assume m can be
extended to an additive measure m : V —> ? с L(F,G) on the <5-ring V generated by 1Z.
If \m-\ is ?-additive for every ? in a space ? С G* norming for G, then
svarf.c(w', A) = svar/-.c('«. A), for A elZ.
The ?-additive measures on a <5-ring have always finite semivariation wjR.t.
16. THEOREM. Let m:T>^ ? be a ?-additive measure on a S-ring V. Then
(a) m has finite semivariation wj..?.
(b) /fS = (Jn S„ w/f/? S„ e P, then there is a positive, finite, ?-additive measure ? on
V such that wr.? <? ? locally and ? ^ /и?..?.
Remark. The local absolute continuity in ? ? <? ? means that for every set A e V we
have
lim wK.?(B) = 0.
?(?)->-0. BCA
If ? is a ?-algebra, then taking A = 5, we obtain the usual absolute continuity.

Vector integration in Banach spaces and application to stochastic integration 367
4.4. The family m f z of measures
7. Let m : TZ —>¦ ? с L(F, G) be an additive measure with finite semivariation my.G on
a ring TZ and ? с G* a space norming for G.
We denote by w/. z the set of positive, finite measures {\m-\: z& Z\] defined on TZ.
If m is ?-additive, then all measures \m:\ are ?-additive. We notice that the converse is
not true. However, the converse is true provided that m y.z consists of uniformly ?-additive
measures.
18. THEOREM. Let m:TZ—> R С L(F.G) be an additive measure with finite
semivariation in f.G on a ring TZ and ? С G* a space norming for G.
The following assertions are equivalent:
(a) m ? c* is uniformly ? -additive on TZ.
(b) »i f.z is uniformly ?-additive on TZ.
(c) mFG{An) -^ Oas A„ j 0 in TZ.
(d) fnFc(An) -> fflf.c(A) for any decreasing sequence (An) of sets from TZ with
intersection A = P| A„ eTZ.
(e) fflf,c(\) —>· 0 for any sequence (An) of disjoint sets from TZ with union A —
Each of these assertions imply that m is ?-additive on TZ.
The following three theorems give sufficient conditions for the uniform ?-additivity of
mFG.
19. THEOREM. Let m:V -> ? = L(R. E) be a ?-additive measure on a S-ring V. Then
the semivariation wr.? is finite and the set of measures mjj.p is uniformly ? -additive.
20. THEOREM. Let m : V —>¦ ? — L(F.G) be a ?-additive measure with finite
semivariation in f.g on a S-ring V. If со (?_ G, then in f.G is uniformly ? -additive.
21. THEOREM. Ifm : TZ -> ? С L(F, G) is a ?-additive measure with finite variation \m\
on a ring TZ, then m f.g* is uniformly ?-additive.
4.5. Extension of measures
In many cases, a measure is defined initially on a semiring and is first extended to the ring
generated by the semiring. This is the case, for example, of a function g: R -> ? and the
finitely additive measure mg defined for each interval of the form [a, b[ by
mg([a,b[) = g(b)-g(a).
The class of the intervals [a, b[ is a semiring V and we extend w? to an additive measure
on the ring TZ = r(V) generated by V- This extension is done by using the following
proposition:

368
N. Dinculeami
22. PROPOSITION. Let m: V -> ? or Ш+ be a finitely additive measure defined on_a
semiring V. Then m can be extended uniquely to an additive measure m': TZ -> ? or M.+
on the ring TZ = r{V) generated by V.
Ifm is ?-additive, then m' is also ? -additive.
If A e TZ, then A = (J?=| ?, with ?, e V mutually disjoint. The measure m' is defined
by
/1
m'(A) = ]Pm(B,).
i = l
We present now some theorems about the extension of a ?-additive measure from a ring
TZ to the <5-ring V = 8r(TZ) or to the ?-algebra ? = ??(??) generated by TZ.
4.6. Extension of positive measures
23. A positive, ?-additive measure ?:1?—> [0, + oo] on a ring TZ can be extended
by the Caratheodory procedure to a ?-additive measure ?': S -> [0, + oo] on the ?-ring
5 = ar(TZ) generated by TZ. If supAe-^(A) = c, then 5????5?'(?) = с.
If 5 is not a ?-algebra, we can perform a second ?-additive extension ?": ? —>
[0, +oo] on the ?-algebra ? = ??G?) generated by TZ:
?"(?) = $\??{?'(?): AeSDB). for В е Г.
If ? is finite on 7?, then ?" has the FMP.
If S = Uw=i S« with Si e 7?, then ? = S and the second extension is no longer
necessary.
If ? is ?-finite on TZ, then ?' is the only ?-additive extension of ? from TZ to 5.
However, ?" is not necessarily the only ?-additive extension of ? to ?.
Example. Let #[о.ц and #[о.ц be the class of Borel subsets of [0. 1[ and [0, 1],
respectively. Then ?[?. ? [ is a ?-ring in [0, 1 ] and the ?-algebra generated by it is B[o. ? ]. Let
? be a positive, finite, ?-additive measure on B\q. \ \. For any a ^ 0 we obtain a ?-additive
extension ? : B[o, ? ] -> ? of ?, be setting ?{ 1} = a.
To distinguish ?" from other extensions of ? to ?, we shall call ?" the canonical
extension of ? to ?. To unify the language, we shall call ?' also the canonical extension of
? to S. The canonical extension of ? to X1 is the smallest ? -additive, positive extension of
?' to X\ We shall extend below the notion of canonical extension to vector valued, additive
measures.
We shall continue to denote ?' and ?" by ?. If ? is finite on 7?, then ? is finite on the
5-ring ? generated by 7?; if ? is bounded on 7?, then ? is bounded on S and on ?, and
there is a set So e 5 such that ?(?) = 0 for every В e S with ? П 5? = 0 (Dinculeanu
B000), Theorem 3.1).
We mention the following property.

Vector integration in Banach spaces and application to stochastic integration 369
24. For every set A e ? with ?(?) < oo and for every ? > 0, there is a set В eTZ with
?(???) <?.
This property is no longer true for an extension different from the canonical one. This
property means that if ? is bounded, then TZ is dense in ? for the topology defined by the
semidistance p(A, ?) = ?(???), which allows extensions of vector valued, ?-additive
measure from TZ to ? (Theorems 25 and 26 below).
4.7. Extension of ?-additive measures
A ?-additive vector measure m:TZ-* ? does not necessarily have a ?-additive extension
on the <5-ring V = 8a (TZ). We present some theorems giving sufficient conditions for the
extension of m.
We consider first the extension of a measure that is absolutely continuous with respect
to a positive measure.
25. THEOREM. Let m:TZ-* ? be ? ? -additive measure on a ring TZ and assume there is
a positive, finite, ?-additive measure ? : 7U. —» Ш+ such that m <g ?. Then:
(a) m and ? can be extended uniquely to ?-additive measures m' and ?' respectively,
on the S-ring V — Sr(TZ) generated by TZ and we still have m' <K ?'.
(b) If ? is bounded on TZ, then m and ? can be extended uniquely to ? -additive
measures m' and ?' respectively, on the ? -algebra ? = ??(??) such that ?' is the
canonical extension of ? and we still have m' «?'.
If m has finite variation \m\, we can take ? — \m\ and obtain the following theorem:
26. THEOREM. Let m;lZ-* ? be a ?-additive measure with finite variation \m\. Then
(a) m can be extended uniquely to a ?-additive measure m :T> —> ? with finite
variation \m'\ on the S-ring V — Sr(TZ) and we have
\m'\(A)=\m\(A), for A eTZ.
(b) If m has bounded variation \m\ on TZ, then m can be extended to a ?-additive
measure m': ? -> ? with finite variation \m'\ on the ?-algebra ? = ??A?) such
that the variation \m'\ is the canonical extension of the variation \m\.
4.8. Canonical extensions
In order to shorten the language, we give the following definition.
27. Dehnition. Let m: TZ -> ? с L(E, F) be an additive measure and ? с G* a
norming space for G, such that for every ? e F and zeZ, the real-valued measure
(m(-)x, z) is ?-additive and has finite (respectively bounded) variation \(mx, z)|(-)-

370
N. Dinculeami
An additive extension m' :V -* L(F. Z*) (respectively m': ? -> L(F, Z*)) of m is
called the canonical extension of m, if for every ? e F and ? e Z, the real-valued measure
(m'(-)-*, z) is ?-additive and has finite (respectively bounded) variation \(m'x,z)\(-) which
is the canonical extension of this variation \(mx.z)\(-)-
One can prove easily that the canonical extension is unique, if it exists.
If m is ?-additive and has finite (respectively bounded) variation \m\ and if m has a
canonical extension m': V -» ? (respectively m': ? -> ?), then m' has finite (respectively
bounded) variation \m'\ which is the canonical extension of \m\.
4.9. Canonical extension of additive measures
Given an additive measure m:lZ -> ? С L(F,G) on a ring TZ, in order to apply the
integration theory of Section 5 below, it is enough to extend m to ? finitely additive (not
necessarily ?-additive) measure on the <5-ring V or on the ?-algebra ? generated by TZ.
We have the following extension theorem for additive measures. We defined a measure
m:TZ-* ? to be locally bounded, if for every set A eTZ, the restriction m : 1Z П A -> ? is
bounded.
28. THEOREM. Let m :H -> ? с L(F.G) be an additive measure and ? С G* anorming
space for G** (for example ? = G*).
(i) Aiiwwe ?/?a? w is locally bounded (respectively bounded) and that for every ? e F
and ? e Z, ?/ге real-valued measure (m(-)x. z) is ?-additive.
Then there is a unique locally bounded (respectively bounded) additive measure
m':T> -> L(F,Z*) (respectively m': ? -> L(F,Z*)) which is the canonical
extension ofm.
(ii) Aiiwwe f/га? w has finite (respectively bounded) semivariation m^.c a«^ that for
each z& Z, the measure m-\TZ^- F* is ?-additive.
Then m has a canonical extension m' :T> -> L(F, Z*) (respectively m': ? ->
L(F,Z*)) with finite (respectively bounded) semivariation m'FZ,, such that for
each ? e Z, ?/ге measure m'.:V -* F* (respectively m'.: ? -> ?*) /? ?-additive
and has finite (respectively bounded) variation \m'.\ and we have
m'FZ,(A) = in f.g(A), forAelZ.
If ? = G*, then m' takes on values in L(F, G**) and we have
mFG**(A) = ту.с(А), forAeTZ.
If G = D*, where D is a Banach space and if we take ? = D, then m' takes on
values in L(F, D*) and we have
™'f.d*(A) = ™f.d*(A). forAeTZ.

Vector integration in Banach spaces and application to stochastic integration 371
For the proof see Bongiorno and Dinculeanu B001), Theorems 3.6 and 3.7.
The following theorem gives conditions that ensure that the canonical extension is ?-
additive.
29. THEOREM. Let m :?? -> ? С L(F.G) be an additive measure and ? С G* a space
norming for G.
Assume cq <? E, that m is locally bounded (respectively bounded) and that for every ? e
F and ? e Z, the real-valued measure {m(-)x.z) is ?-additive. Then m is ?-additive and
has ? ? -additive canonical extension m :T> -> L(F,G) (respectively m : ? -> L(F.G)).
For the proof, see Bongiorno and DinculeanuB001). Corollary 3.9.
5. Integration with respect to a measure with finite semivariation
1. The framework for this section is an additive measure m.V -> ? С L(F,G) with
finite semivariation mF.G on a <5-ring V and a space ? с G* norming for G, such that
for each ? e Z, the measure m::V -> F* is ?-additive. ? = ??(?>) is the ?-algebra
generated by V. We do not assume that m is ?-additive.
In this section we present the fourth stage in the development of the integral J f dm, for
functions /: S -> F. For this purpose we define a seminorm my g (/) for such functions,
then the space TF G(m) of measurable functions / with w/-\g(/) < сю and then the
integral f fdm e Z* for functions / e Ty,c(m).
This is the most important part of the chapter. In fact, most interesting vector measures
do not have finite variation, but may have finite semivariation. This is the case, for example,
of the stochastic measure ?? associated to a summable process X, even if X is real-valued.
We shall apply the integration theory of this paragraph to obtain the stochastic integral.
Some of the results are valid under additional conditions such as:
(a) mF C(S) < oo, or
(b) S=\JS„, with S„ eP, or
(c) т is ?-additive.
Conditions (a) and (b) are satisfied if ? is a ?-algebra. But most results are valid without
imposing these restrictions and can be used in a wider range of applications, such as the
integral representation of Gaussian measures or the Riesz representation theorem.
If the reader is not interested in this generality, he or she can assume from the very
beginning that ? is a ?-algebra and in is ?-additive.
For the proof of the results stated in this section, the reader is referred to (Dinculeanu
B000), Section 5).
5.1. Measurability with respect to a vector measure
We define first negligible sets and functions with respect to m.

372
N. Dinciileanu
2. A set A e ? is said to be m-negligible if m(B) = 0 for every В e ? with В С A.
If follows that a set Л e ? is m-negligible iff |m|(A) = 0, iff w/. c(A) = 0, iff A is
m;-negligible for every jeZ.
3. A set А С S is said to be m-negligible if it is contained in an m-negligible set В е ?.
A countable union of m -negligible sets is again m -negligible.
If А С S is m-negligible, then A is m--negligible for every ? e Z. Conversely, if А С S
is mc-negligible for every ? e ? and if ? is separable or if the measures \mz\ with ? e Z|
are uniformly ?-additive, then A is m-negligible.
4. A property valid outside an m-negligible set is said to be valid m-almost everywhere
(m-a.e.).
5. A function /: S —>¦ D or R+ is said to be m-negligible if / = 0, m-a.e.
An m-negligible function / is m_--negligible for every ;eZ. Conversely, if / is m--
negligible for every ? e Z, then / is m-negligible if either / is ^-measurable, or ? is
separable or the measures \m-\ with ;eZ| are uniformly ?-additive.
We define now measurability with respect to m.
6. A function /: S —>¦ D or R+ is said to be m-measurable if it is equal m-a.e. to a ?-
measurable function, i.e., if there is a sequence (/„) of D or R+-valued, ?-step functions,
converging to /, m-a.e. Moreover, we can choose the functions /„ such that |/„| ^ |/|,
m-a.e., for each n.
If / is m-measurable, then it is m;-measurable for every ze Z. Conversely, if / is m--
measurable for every ? e Z, then / is m-measurable, provided that ? is separable or the
measures \mz\ with ? e Z\ are uniformly ?-additive (Dinculeanu B000), Proposition 5.5).
5.2. The seminorm m pc(f)
The alternative definition of the semivariation given in 2 of Section 4 is extended now for
functions.
7. DEnNlTlON. For every function f:S^> D or R we define
I f I
wF.c(/) = sup / sdm\
where the supremum is taken for all iT-step functions ?: S -> D or R with |i| ^ |/|.
Compare with Definition 8 of Section 2 in Chapter 2, of the integral of positive functions.
We can compute mFG(f)m terms of the measures \m-J. Compare with Proposition 12 of
Section 4.

Vector integration in Banach spaces and application to stochastic integration 373
8. If /: S —>¦ D or ? is w-measurable, then
wV.c(/) = sup{ j\f\d\mz\: zezA.
If the spaces F and G are understood, we shall write w instead of w f.g ¦
We have the following properties of wV.c(/) for w-measurable functions /:
9. (Kw(/)^oo.
10. w(/) = w(|/|).
11. w(/) = 0 iff / = 0, w-a.e.
12. If l/l = |g|, w-a.e., then w(/) = w(g).
13. If l/l ^ |g|, w-a.e., then w(/) ^ w(g).
14. m(f + g)^m(f) + m(g).
15. w(a/) = |a|w(/),fora eK.
16. MONOTONE CONVERGENCE THEOREM. // (/„) is an increasing sequence of
positive, m-measurable functions /„ : S —» E+, then
w(sup/„) = supw(/„).
17. For every sequence (/,) of positive, w-measurable functions /„ : S -> E+ we have
(ОС \ DC
n=l / n=\
18. FatOU'S LEMMA. It (/„) /? ? sequence of positive, m-measurable functions, then
wiliminf/,,) ^ liminfw(/„).
\ n^oo / л^зс
19. If / is w-measurable and с > 0, then
w({|/|>c})<~w(/).
20. If /: S -> E+ is w-measurable and w(/) < oo, then / < oo, w-a.e.
21. If /: S -> D or Ё is w-measurable, then the set {/ ? 0} is contained in the union
of a sequence (A„) from ? with w(A„) < oo.

374 N. Dinculeanu
5.3. The space ??(??у q)
We define now a space similar to the space L' of integrable functions.
22. Definition. We denote by Toinn.G)· the set of all m -measurable functions /: S ->
D with yhf c(f) < oo.
If the spaces F and G are understood, we shall write To(m) instead of fo(mf.c).
The functions / e ? Dim) are called w-integrable functions.
23. For each г eZ| we have /|/|d|m:| ^ mt.c(f)· hence То(т) С L[,(w:),
therefore
^о(»я)Ср|^{,Aя:.).
If ? is a closed subspace of G*. then we have the equality
TD{fu)=^L^D(m:).
;eZ
24. To(m) is a vector space and mt.G(f) is a seminorm on it. We shall consider on
To(m) the topology defined by this seminorm.
25. THEOREM. // (/„) is a Cauchy sequence in L}-,(/??). then there is a function f e
Td(™) and a subsequence (flk) such that:
(a) fn~* f in Fn(m).
(b) fk —> f m-a.e. and in To{m).
26. COROLLARY. The space То(т) is complete for the seminorm in.
27. Definition. Let /,,, /: S -> D be w-measurable functions, ? e N. We say that the
sequence (/„) converges in m-measure to /, if for every ? > 0 we have
nlim)i»i({|/B-/|>e})=0.
28. Proposition. Let f„,f e TD(m),n e N. Ifm(f„ - f) -> 0 then f„ -> / in m-
measure.
29. Remark. The set of ?-step functions of То(т) is not necessarily dense in JFD(m).
If the measures (m-) with ;eZ| are uniformly ?-additive, then the set of iT-step
functions of To{m) is dense.

Vector integration in Banach spaces and application to stochastic integration 375
30. We shall denote by Td(B, in) the closure of the set of bounded functions of Td(™)-
This set has properties similar to the usual L' -spaces, especially concerning the Lebesgue
theorem.
5.4. The integral
In the particular case D = F, we can define the integral j f dm for functions / e
^Fimr.G) and the integral belongs to Z*.
To simplify the notation we shall write Tf.c(m) or Ту.с(т) instead of ^"f(w/-.g)·
30. The construction of the integral / f dm is done in the following way:
Let / e TF.c(m)- Then / e L\(\m:\) for every ;eZ. Since m-.? -* F* has
finite variation |m-|, the integral f fdmr is defined in the sense of the third stage and
J f dm- e R. The mapping ? н> / / dm- is a continuous linear functional on Z:
\ifdm-\4
fdm-\<i I \f\d\m:]<:\z\m(f).
We denote the linear mapping / h-> f f dmz by / f dm and call it the integral of with
respect to m. We have j f dm e Z*.
(//<».*)-/.
fdm-
and
I/
/iiw
^mF,G(f).
From this last inequality it follows that the mapping / h->- f f dm from Ту,с(т) into Z*
is continuous for the topology of TF.dfn).
31. If we take ? = G*, we have / f dm e G** for / e Ту.о(т).
Let us denote, for the moment, by (Z) j f dm the integral corresponding to the norming
space ? с G*. If Z, Z' are two subspaces of G*, norming for G, and if ? с ?', then the
integral (Z) f f dm is the restriction to ? of the integral (Z') / dm. We have therefore
/(?) ? /Лш, z\ = /(?') ? /Лш, zl for z e ?.
In particular,
/(?) ? fdm,z\ = l(G*) f fdm,z\, for ? e ?.

376
N. Dinculeami
32. We are particularly interested in the case when j f dm e G. This is evidently the
case if G is reflexive and we take ? = G*. If the measures \m-\ with ? e Z\ are uniformly
?-additive, then f f dm e G for every bounded function / e Тг.в(т).
Assume m has finite variation \m\. Then m has also finite semivariation mr.G- We can
consider the space LF(m) = LF(\m\) of functions /:S -» F which are m-integrable, in
the sense of stage 3. For each ? e G* we have ]w:] ^ \m\, hence mF^{f) ^ / |/]d]m| =
11/111, the norm in LF(m). It follows that LF(m) с TF.c(m) and the embedding is
continuous.
If / e i^(m), then J f dm is the same, whether we consider / in LF(m) or in
•^f.g('w).
5.5. Convergence theorems
We state first the analog of Egorov's theorem.
33. THEOREM (Egorov). Assume that the measures \mz\ with ? e Z\ are uniformly ?-
additive. Let /„, /: S —» D, ? e N, be m-measurable functions such that /„ -> / m-a.e.
Then:
(a) For every ief A e ? w/7/г m(A) < oo and for every ? > 0, ?/геге ?'? a set В e ? with
В С A such that m(A \ ?) < ? and fn -> / uniformly on B.
(b) /n —> / /и m-measure.
Uniform convergence implies convergence in To(m).
34. THEOREM. Lei (/,) foe ? sequence from To{m) converging uniformly to a function
f-.S^D. Assume there is a set A e ? with m(A) < oo iwc/г г/шг all functions f„ vanish
outside A. Then f e TD(m) and m(f„ - f) -> 0. If D = F, ?/геи j fndm-> f f dm.
35. THEOREM (Vitali). Lei (/„) foe ? sequence from Td(™) and f;S-*Danm-
measurable function such that:
(a) /n —> / /и m-measure, or
(a') /„ -> / m-a.e. and the measure \m-\ with z& Z\ are uniformly ?-additive; and
(b) lirrim(A)^o rn(fn<PA) — 0, uniformly for ? e N.
77геи / e То(т) and m(fn — f) —»¦ 0.
//D = F, ?/геи f fndm-> f f dm.
For the Lebesgue's theorem, we have to restrict ourselves to the space Td(B, m).
36. THEOREM (Lebesgue). Assume S eT>. Let (/„) foe ? sequence from TD(B,m),
f e То(т) and g e Jk(i3, m) ? positive function. Assume that:
(a) /„ -> / /и m-measure, or
(a') /„ -> / m-a.e. and the measures \m-\ with ? e Z\ are uniformly ?-additive;
(b) |/n| ^ g, m-a.e., for each ? e N.
Г/геи / e .FD(fi, w) шЛ w(/„ - /) -> 0.
If ? = F, ?/геи / /„ dm -> / /dm.

Vector integration in Banacli spaces and application to stochastic integration 377
5.6. The indefinite integral of measures with finite semivariation
The definition of the indefinite integral is the same as for measures with finite variation.
37. Dehnition. Let g e TFG(m). We denote by gm: ? -> ?* the additive measure
defined by
(gm)(A) — I gdm, for A e ?,
and we call it the indefinite integral of g with respect torn, or the measure with density g
and base m.
The measure gm is not necessarily ?-additive; but for each ? e Z, the measure (gm)z is
?-additive and we have (gm)z = gmz.
The following proposition gives sufficient conditions for the ?-additivity of gm.
38. Proposition.
(a) Assume the measures \m:\ with ? e Z\ are uniformly ?-additive. Then f f dm e G,
for every f e Ty.ciB, m). In particular f f dm e ? for every f e T^,yiB,m).
(b) Assume со <t G. Then f fdm e G. for every f e Ty.Gi.rn). In particular, if со <? ?.
then f f dm e E, for every f e T^.yim).
(c) If g e Ty,cim) and fAgdm e G for every A e ?, then the measure gm is ?-
additive. In particular, if g e Jr.?(w) and fAgdme ? for every A e ?, then the
measure gm is ?-additive.
We state first the associativity formula in case the density is real-valued.
39. THEOREM. Assume m is ?-additive and S = (J?L| Sn with Sn eT>. Let ?: S -» К
be a real-valued, m-measurable function such that wr,?(<p) < сю and my ?(?) < oo and
assume that fA ? dm e ? for every A e ?. Then:
(a) The measure <pm is ? -additive and has finite semivariation i<yrn)y.G-
(b) If f ^ 0 is ?-measurable, then
i<pm)m.Eif) = m'm.Ei<pf)
and
i<Pm) y.Gif) = m F.Gi<Pf)·
(c) We have f e Ty.cifm) iff?? e Ty.cim) and m tms case we have
/ fd(<pm)= / ?????

378
N. Dinculeanu
and the associativity formula
f((pm) = (f<p)m.
Now we state the associativity formula for vector-valued densities.
40. THEOREM. Assume m is ?-additive and S = U«t ? S" with Sn eV.Let f e :Ff G (m)
and assume fA f dm e G for every A e ?. Then:
(a) The measure fm is ?-additive and has finite semivariation (frriMG on ?.
(b) If ? ^ 0 is ?-measurable, then
(c) If ? is real-valued and ?-measurable and if ?? e Tf.g(™), then ? e Jk.c(/W)
and we have
I tpd(fm) — I (pf dm
and the associativity formula
(p(fm) = (<pf)m.
5.7. Integral representation of linear operations on L'' -spaces
As a first application of the integration theory presented in the section, we state the
following theorem.
41. THEOREM. Let (S, ?, ?) be a measure space, ? t the S-ring of the sets A e ? with
?(?) < oo, F, G, Banach spaces, 1 ^ ? < oo, and U : ?^-(?) -> G a continuous linear
operation.
Then there is a ?-additive measure m : ?/ -> L(F, G) with finite semivariation mF.G
such that:
(a) mF.G(A) < ||t/||M(A)'/>\/0r A e ?,;
(b) fnFG(f) ^ \\U\\ \\f\\p ^ +00 for ?-measurable functions f: S -> F;
(c) LPF^)cTF.G(my,
(d) U(f) = ffdm,forfeLpFfa).
If U is an isometry, then:
(a') wf.G(A) = M(A)l//,,/orAeX'/;
(b') mfc(f) — WfWp ^ +oo,/or ?-measurable functions f: S -> F;
(c') LF^) = !FF.G(m), isometrically.
The measure w is defined for every A e ?/ by m(A)* = 1/(^дл), for * e F.

Vector integration in Banach spaces and application to stochastic integration 379
5.8. The Riesz representation theorem
The integration theory of this section can also be used to state and prove the Riesz
representation theorem:
42. THEOREM. Let К be a compact Hausdorff space, B(K) the ?-algebra of the Borel
subsets of К and Cf(K) the space of continuous functions f : К -> F. endowed with the
sup norm. Let U :Cf(K) -> G be a continuous linear operation. Then there is an additive
measure m :B(K) -> L(F, G**) such that
U(f)=ffdm, forfeCy(K) and \\U\\ = mFmC,.(K).
Moreover, for each ? e G*, the measure m- :B(K) -> F* is regular, ?-additive and with
finite variation.
For the proof see Foias. and Singer A960), Dinculeanu A967) and Diestel and Uhl
A977).
There are cases when the measure in in the above theorem is neither ?-additive, nor
regular, nor with values in L(F, G). It is an open problem to give a characterization of the
linear operations U for which the corresponding measure has one or more of the above-
mentioned properties. There are partial answers to this problem.
43. If U :Cu(K) -> G is weakly compact, then the corresponding measure m is ?-
additive, regular and has values in G.
For a detailed presentation of this case see Diestel and Uhl A977).
44. An operation U: Cf (К) -> G is said to be dominated, if there is a positive, regular
Borel measure ? on К such that
|?/(/)|< ?\?\??, for/e СИ*)·
If U :Cf(K) -> G is dominated, then the corresponding measure m is ?-additive, regular,
with values in L(F, G) and with finite variation.
For a complete presentation of this case see Dinculeanu A967).
45. A continuous linear functional U:Cy(K) -> ? is dominated, therefore, the
corresponding measure m : B(K) -> F* is ?-additive, regular and with finite variation.
6. The Stieltjes integral
In this section we study the functions with finite variation or semivariation on the real line
and define the Lebesgue-Stieltjes integral with respect to such functions, as an application
of the integration theory presented in Section 5.

380 N. Dinculeanu
6.1. The variation and the semivariation of a function
Let g :E -> ? с L(F, G) be a function.
1. Definition.
(a) For any interval /, the variation of g on / is a number denoted var(g, /) and defined
by
var(g, /) = sup]T|g(r, + |)-g(r,)|,
the supremum being taken for all finite divisions d: fy < f ? < · - · < f„ consisting of
points from /.
(b) For any interval /, the semivariation of g on /, relative to (F, G), is a number
denoted by svarf.c(g, /) and defined by
svarf.cC?. /) = sup
J>(',- + i) - *('¦¦)>¦¦
the supremum being taken for all finite divisions d: tQ < t\ < - - ¦ < t„ consisting of
points from / and all families (xq, x\ ?»-?) of elements from F|.
2. It follows that for any interval / we have
0 ^ svarf.G(g, /) ^ var(g, /) ^ +oo.
3. If / С J are intervals, then
var(g, /) ^ var(g, J) and svaiF.Gig-1) < svar/.-.c(g. ·/)·
Sometimes the variation and the semivariation are equal:
4. PROPOSITION. Assume ? С L(F, R) isometrically. Then, for every interval I we have
svarf.R(g,/) = var(g,/).
The relationship between the semivariations corresponding to different embeddings is
stated in the following proposition:
5. PROPOSITION. For any interval I we have
s\aiF.G(g, I) < svar?..p.(g, /) = var(g, /).
If the embedding ? С L(F, G) is an isometry, then for any interval I we have
svarR.?(g, /) ^ svarf.cC?. /)-

Vector integration in Banach spaces and application to stochastic integration 381
6.2. Semivariation and norming spaces
Let g: ? -> ? с L(F,G) be a function and ? С G* a norming space for G. For every
? e ? consider the function g;: ? —» F* defined by
[x,gz(t)) = [g(t)x.z), fori eEandjc e F.
Considering g-: ? -> F* — L(F,G), by Proposition 4. for any interval / we have
svarf.R(g-, /) = var(g;, /).
The semivariation of g can be computed by means of the variation of the functions g::
6. PROPOSITION. For any interval I we have
svarf.cC?, /) = sup var(g-, /).
In particular, for any t e ? we have
svarf.G(g, ]-oo, t]) = sup (gz, ] - oo, i]).
I--KI
7. It is convenient to denote
|g|@ = var(g,]-oo,r]) and g/.-.c@ = svarf.G(g,]-oo.r]).
|g| is called the variation function of g and g/.-.c is called the semivariation function of g,
relative to (F, G). If F and G are understood, we shall write g instead of gr.G-
We have the following properties:
8. |g| and g are increasing.
9. gf.c^lgl·
10. If ? CL(F,E) is an isometry, then
gFM=\g\-
11. If g is right continuous, then
\ar(g,[a,b]) = vai(g,(a,b]) and svar(g, [a, b]) = svar(g. (a. b]).
12. If a < b, then
\g\(b)=\g\(a)+v<iT{g,[a,b]).

382
N. Dinculeanu
13. If g has finite variation function |g| and if a < b, then
\g(b) - g(a)\ < \g\(b) - \g\(a) = var(?, [a, b]).
14. If g has finite semivariation function g and if a < b, then
g(b) - g(a) ^ svar(g, [a, b]).
15. If g: R —> R is increasing, then
|g|(fe)-|g|(fl) = g(fe)-g(fl).
An important property of g is the right or left continuity, which ensures that the
measure m4 (defined in the next section) is ?-additive. The right or left continuity of g
is inherited by its variation function |g|.
16. THEOREM. Assume g has finite variation function \g\. Then g is right (respectively
left) continuous iff\g\ is right (respectively left) continuous.
6.3. The measure associated to a function
We denote by TZ the ring generated by the intervals of the form ]a,b]. Each set A e TZ is a
finite union of disjoint intervals ]a,·, bj], 1 ^ i ^ n.
Letg:R^ ? С L(F, G) be a function.
17. We associate to g a finitely additive measure mK : H -+ E, defined first for intervals
]a, b] by
mg]a,b] = g(b)-g(a)
and then extended by additivity to the whole ring TZ.
We have the following properties:
18. For two functions g. g': R -> ? we have ms = m4· iff g — g' is a constant.
19. If g : R -> R is real-valued, then m K > 0 iff g is increasing.
20. For every ? e G we have
In particular, for every л* е ?* we have
i*m( = mx*K.

Vector integration in Banach spaces and application to stochastic integration 383
21. If g is right continuous, then for every interval / we have
var(w?, /) = var(g, /) and svar/.-.G(«i(., /) = svar/.-.c(g, /)·
22. If g has finite variation function \g\ and if g is right continuous, then
\mg\ = mlgl.
The following two theorems ensure the extension of the measure nig to the <5-ring V or
to the ?-algebra В{Ш). We consider first the case where g has finite variation function |g|.
23. THEOREM. Assume g:R -> ? is right continuous and has finite (respectively
bounded) variation function \g\.
Then ffij can be extended uniquely to ? ?-additive measure m.V —> ? on the S-ring
V of bounded Borel sets (respectively in :B(R) -> E), with finite variation \m\ and the
variation \m\ is the unique extension of the variation \m,,\ = m\g\.
For functions with finite semivariation we have a similar result.
24. THEOREM. Let g : R -> ? с L(R. G) be a function with finite (respectively bounded)
semivariation gu.c ???" ? С G* a norming space for G** (for example, ? = G*). Then
the measure m ч :??—* L(F, ?*) has finite (respectively bounded) semivariation my.G-
(i) Assume that for every ? e F and ceZ, the real-valued function (g(-)x, z) is right
continuous. Then:
(a) For every ? e F and ? e Z, the real-valued measure (m4(-)x, z) is ? -additive
and has finite (respectively bounded) variation.
(b) The measure m(, has a canonical finitely additive extension m:T> —> L(F,Z*)
(respectively m:B(R) -> L(F.Z*)) with finite (respectively bounded) semi-
variation m r.z-
(ii) Assume that for every ;eZ, the function g:: R -> F* w n'g/?? continuous. Then, for
every ? e Z, the measure m- :T> -> L(F. Z*) (respectively in: :B(R)-> L(F.Z*))
is ?-additive and has finite (respectively bounded) variation.
IfZ = G*, then m takes on values in L(F. G**). IfZ = G = D*, where D is a
Banach space, then m takes on values in L(F, D*).
Ifco<?E, then mg is ?-additive.
For the proof, see Bongiorno and Dinculeanu B001), Theorem 4.21.
6.4. The Stieltjes integral
Let g:R -+ ? с L(F,G) be a function and let mg:Tl -> ? be the finitely additive
measure associated to g.

384
N. Dinculeanu
25. We define first the Stieltjes integral f fdg in case g is right continuous and has
finite variation function |g|. In this case, by Theorem 23, the measure mx can be extended
to a ?-additive measure m:V -> ? with finite variation \m |. We shall denote m still by wg;
we have
We can consider the space LF(mg) = L\(\mg\), in the sense of stage 3 of the development
of the integral.
We shall denote LF(mg) by LF(g). For every / e LF(g) we define the Lebesgue-
Stieltjes integral f fdg by the equality
/ fdg= / /d»v
If feLF(mR), then |/| e L1 (wi|.?|) and we have
\f\dm\g\.
therefore
\J fdgU f\f\d\g\.
26. We consider now the case where g has finite semivariation function (g)-$..E and there
is a space ? с G* norming for G** (for example ? = G*), such that for each ? e Z, the
function gc: R -^· F* is right continuous. Then, by Theorem 24, ш, can be extended to an
additive measure mg :T> -> L(F, Z*) with finite semivariation (w?)/.-.z* such that for each
? e Z, the measure (m?): is ?-additive. We can consider then the space TF.z*(m$) defined
in Section 5. We denote J7F.z*(m<>) by Fy.z*(g) and for every function / e Fy.z'ig) we
define the LebesgueStieltjes integral f f dg by the equality
/ fdg= / /dwi?
,, eZ*.
We have then
fdgU(m!!)F.z.(f).
If ? = G*, then
I/
j fdg = J fdmxeG**.

Vector integration in Banach spaces and application to stochastic integration 385
If, in addition, cq <? G, then
J fdgeG.
27. Assume g is right continuous and has finite variation function |g|. Then g has also
finite semivariation function gr.G relative to any embedding ? с L(F, G). The measure
mg has finite variation |w?| = m^ and finite semivariation (in^r.G- We have
LxE(mg) CTV.c^i;)·
For / e LF(mg), the Stieltjes integral f f dg is the same, whether we consider / in
LF(mg) or in TF.c(mg).
28. We could consider the semiring V of the intervals of the form [a, b[ and the ring
Ti! generated by V, and define the measure т':Т1'—> Е by
m'g[a,b[ = g(b)-g(a).
Then m' is ?-additive and has finite variation (respectively finite semivariation) iff g is
/e/r continuous and has finite variation (respectively finite semivariation). If w'? has finite
variation, then w' can be extended to a ?-additive measure m' with finite variation on
the <5-ring V. If m has finite semivariation and if for every ? e Z, the function g,- is left
continuous, then w' can be extended to an additive measure m' with finite semivariation
on V, such that for every ? e Z, the measure m'_ is ?-additive.
If we start from a function g: R -> ? with finite variation but not necessarily left or right
continuous, it is more appropriate to define the measure m:TZ —>¦ ? by
m]a,b] = g(b+) - g(a+)
and the measure m': 1Z -> ? by
w'[a>[ = g(fo-)-g(a-).
Then both measures are ?-additive and can be extended to the same ?-additive measure
on V. In fact, the function g+(t) = g(t+) is right continuous and the function g~(t) =
g(t—) is left continuous and we have
m]a,b] = mg+]a,b] and m'[a,b[ = mg_[a,b).
Similar considerations can be made in case g has finite semivariation. But in this case.
g does not necessarily have lateral limits in G. However, there are elements G(t+) and
g(t—) in G** such that for every ? e F and ? e G* we have
lim(g(s)x,z)=(g(t+)x,z) and \imig(s)x,z}= (g(t-)x,z).
sit iff

386
N. Dinculeanu
7. The stochastic integral
The main application of the integration theory with respect to a measure with finite
semivariation is the stochastic integral.
7.1. Notations and definitions
The reader is supposed to be familiar with the general theory of stochastic processes, as
presented in any book, for example, in Dellacherie and Meyer A975-1980). We present
below a few definitions and notations that will be used in the sequel.
1. E, F, G are Banach spaces with ? с L(F, G).
2. (?,?, ?) is a probability space. The P-negligible (respectively P-integrable) sets
or functions are called, simply, negligible (respectively integrable). Instead of P-a.e. we
shall write a.s. (almost surely). The space LF(P) with 1 ^ ? ^ oo will be denoted by L!'F.
A set ? с R+ ? ? is called evanescent if it is contained in a set of the form R ? A with
А С ?, negligible. {Tt)t€r+ is a filtration, i.e., each Tt is a ?-algebra contained in ? and
? С Tt if s ^ t. We assume the filtration satisfies the usual conditions, i.e., Tt = C\s>t ^
for every t ^ 0 and each Tt contains all the negligible sets. _
A stopping time (or optional stopping time) is a function ? : ? -> E+ such that
{T ^ t) e T, for every t > 0. If 5 ^ ? are two stopping times, we define the stochastic
interval
]S,T] = {(t,w)eRx ?: S(w) < t ? T(w)\.
Other stochastic intervals, [S, T[, ]S, T], [S, T] are defined similarly. The graph of a
stopping time ? is [?] := [?, ?].
3. 1 ^ ? < oo. From ? с L(F, G) we deduce L? с L(F, Lpc).
4. 7? is the ring of subsets of R+ ? ? generated by the semiring of predictable
rectangles {0} ? A with A e Tq and (s,t] ? A with Л е Ts. The ?-algebra generated
by TZ is called the predictable ?-algebra and is denoted by V¦ The predictable ?-algebra
is also generated by the adapted, left continuous, real valued processes (adapted processes
are defined below).
A T-measurable process is called a predictable process.
A stopping time ? is said to be predictable if the stochastic interval [T, oo) is
predictable. The optional ?-algebra О is the ?-algebra generated by the adapted, right
continuous processes. It is also generated by the stochastic intervals ]S, T] with S ? ?
simple stopping times. An 0-measurable process is called an optional process.
Other ?-algebras of interest are Tt and Tt-, where ? is a stopping time. Tt is the
?-algebra of the sets A eT with А П [T ^ t] e T, for every t ^ 0. The ?-algebra Tt- is
generated by the sets of the form А П {t < ?} with t ^ 0 and A e Tt.

Vector integration in Banach spaces and application to stochastic integration 387
5. A function F:I+xi2-»fl, with values in a Banach space D is called a process,
or a stochastic process.
We say ? is adapted if for each t > 0, the random variable Y, is ^-measurable. ? is
cadlag if for each ? e ?, the path t -» Yt (?) is right continuous and has left limits.
In the sequel, X : E+ ? ? -> ? will denote a cadlag, adapted process with X, e i,? for
every t ^ 0. We consider X automatically extended ????? with X, = 0 for t < 0. Then
Xo- = 0. We extend also the filtration with Tt = JT() for t < 0.
Let ? : R+ ??-> D be a process and ? a stopping time. We denote by Hj the function
defined by ??{?) = #7-(?)(?) for ? e ? with ?(?) < ??, and by HtI\t<oc) the function
equal to Яг on {T < oo) and equal to 0 on {T = oo}.
The stopped process HT, obtained by stopping the process ? at the time ? is defined
by #,r = Н,лТ. The process #r~ obtained by stopping the process ? before ? is defined
by Hj" = H, if t < ? and Я,7^ = HT^ iit^T.
A process ? is said to be a modification of a process ?, if for every t > 0 we have
^ = ?,, a.s., the negligible set depending on t.
A process ? is said to be evanescent if the set {? ? 0} is evanescent.
Two processes ? and Z, with values in the same Banach space, are said to be
indistinguishable, if ? — ? is evanescent.
7.2. The measure ??. Summable processes
Let 1 ^ ? < oo. Let X :R+ ? ? -> ? с L(F. G) be an adapted, cadlag process with
X, e L? for every t ^ 0.
6. We associate to X the additive measure ??.??-* L!'E с L(F, LPG) defined, first, for
predictable rectangles by
/*({()} ? A) =lAXo, forAeJb
and
Ix{]sj]xA)=lA(Xt-XJ, foTAeT,,
and then extended by additivity to the whole ring 1Z.
We have
lx{[0,t] x A) = lAXt, for t ^ 0 and A e To-
In particular,
??([0.?]??) = ?,, forr>0.
If the process X is understood we shall write / instead of ? ?.

388
N. Dinculeanu
7. Since LPE с L(F, L^), we can consider the semivariation of ?? relative to the pair
(F, Lq). To simplify the notation, we shall write /or (Ix)f.g instead of (Ix)F Li>:
If.g(A) = sup
?/?(A,);
/e/
, for A e TZ,
where the supremum is taken for all finite families (Аг )i€/ of disjoint sets A, from ??
contained in A and for all families (jc/)ie/ of elements of F\.
8. Dehnition. We say that the process X is p-summable relative to the pair (F, G), if
?? has a ? -additive extension ? ? : V -> L^ with finite semivariation relative to (F, L^).
If ? = 1, we say that X is summable relative to (F, G).
The stochastic integral Я · X will be defined with respect to p-summable processes X.
Examples of p-summable processes are:
(a) Processes with integrable variation.
(b) Processes with integrable semivariation, provided that cq <?_ ? and cq <? G.
(c) Square integrable martingales X, in case ? and G are Hubert spaces.
We have the following useful criterion for the extension of the measure ? ? to a ? -additive
measure on V.
9. THEOREM. If cQ (? E, assertions (a)-(d) below are equivalent. If ? is any Banach
space, assertions (b), (c) and (d) are equivalent and (a) implies (b).
(a) ??.??—* LPE can be extended to ? ?-additive measure ?? : V -> L!E.
(b) ?? is bounded on TZ.
Let ? С L?t, ^ + ^ =\, be a closedsubspacenormingfor LE-
(c) For every g e Z, the real-valued measure (??, g) is bounded on 1Z.
(d) For every g e Z, the real-valued measure {Ix,g} is ?-additive and bounded on ??.
If ?? has ?-additive extension to V and has finite semivariation on 1Z relative to (F, G),
then the extension of ? ? has finite semivariation on V relative to (F,G) hence X is
P-summable relative to (F, G).
7.3. The stochastic integral
10. Let 1 ^ ? ^ oo and X: R+ ? ? -> ? с L(F, G) be a cadlag, adapted process.
Assume X is p-summable relative to (F, G). Consider the ?-additive measure
/x:-p-i.?ci.(F.i.?)
with finite semivariation /f G relative to (F, L?).
We can apply the integration theory presented in Section 5, replacing S, ?, m with
???,?, /x and E, F, G with LE. F, LPG. respectively.

Vector integration in Banach spaces and application to stochastic integration 389
LetZcZ^,, \ + \= 1 be a norming space for Lpc. For ? e Z, consider the measure
(/?): -V ^ F*, defined for A e ? and у е F by
(у,(/х);(А)) = (/х(АK\г)= ?(??(?)(?)?,?(?))??(?),
where the bracket in the integral represents the duality between G and G*. Then
(/x)f.z.? = sup{|(/x);|:zeZ, ||г||,<1}.
If ? and X are understood, we shall write I = ?? and /f .c = IF ^/. ¦ For a Banach space D
we denote TdUf.g) = FdUf Lr_), the space of predictable processes Я: R+ ? ? -> D.
such that
If.g(H) = sup) /"|tf|d|(/x),|: ||~||„ < 11 < oo.
Then TdUf.g) is a vector space and /f.c is a seminorm on TdUf.g), f°r which it is
complete.
The simple processes are not necessarily dense TdUf.g)·
11. If D = F we shall write J>G(X), TF,p(X), TfgUx) or TF ,?(??) instead
of Tf(Ax)f.g)- In this case we can define the stochastic integral ? ¦ X for processes
Я eJy.c(X) as follows.
Let ? e :Ff.G(X); then Я e L^((/.v);) for every с e Z, hence the integral j Hd(Ix)-
is defined and is a scalar. The mapping ? ь* j ??(??): is linear and continuous on Z. We
denote it by / ????. We have, therefore J ?dlx e Z*.
If Hdlx,z)= f Hd(Ix):, for:
eZ
and
I/
tfd/x
^If.g(H).
IfwetakeZ = (L?)*, then / ?dlx e (L?)**.
We are interested in those processes ? e Tf.g(X) for which the integral /[(O] Я^/?
belongs to Lg, for every t ^ 0. In this case we denote by the same symbol the equivalence
class f[0l]HdIx in Lpc, as well as any random variable belonging to this equivalence
class. We obtain in this way a process (L· . ?dIx),^o with values in G. This process is
always adapted; but it is not necessarily cadlag. This leads to the following definition:
12. Dehnition. We denote by L\- g(X) the set of processes ? e Ff.gW) satisfying
the following two conditions:

390
N. Dinculeanu
(a) /[0 t]Hd1x e LG, for each t ^ 0.
(b) The process (L· t, HdIx)t^o has a cadlag modification.
The processes ? e L]h G(X) are said to be integrable with respect to X.
If ? e LF G(X), any cadlag modification of the process (/](),| H ???),^? is called the
stochastic integral of ? with respect to X and is denoted by ? ¦ X or / ? dX:
(??),(?)=( ? HdXJ (?) = (? ? dlx\w), a.s.,
for each t ^ 0.
It follows that the stochastic integral ? ¦ X is defined up to an evanescent process, and
is a cadlag, adapted process.
The following theorem gives the jumps of the paths of a stochastic integral.
13. THEOREM. For any process ? e L\- g(X) we have
Л(ЯХ) = ЯЛХ, where &X, = X, -X,^.
The relationship between summability and stopping times is stated in the following
theorem:
14. THEOREM. Assume X is p-summable relative to (F, G). Let ? e L\ g(X) and ? a
stopping time. Then:
(a) XT is p-summable relative to (F. G).
(b) HaLFG{XT).
(С) 1|0.ПЙ?^С(Х).
(d) (HX)T = HXT = (\[0.T]H)X.
If ? is predictable, then:
(a') XT~ is p-summable relative to (F, G).
(b') HeL[.G(XT~).
(c') l[Q.T[H e L^iX).
(d') (?-?)?- = ?-??- = (???.?\?)-?.
7.4. Convergence theorems
Let 1 ^ ? < oo and X : R+ ? ? -> ? с L(F.G)a p-summable process relative to (F. G).
15. THEOREM. 77?е space L^ G(X) ;'i complete for the semivariation h,c-
16. THEOREM (Lebesgue). Lei (#") foe ? sequence from LFG(X), converging pointwise
to a process H. Assume there is a positive process ? e Tj.Uf.g ) such that
]#"] ^ ?, for each ?.

Vector integration in Banach spaces and application to stochastic integration 391
Assume, in addition, that ? can be approximated in Tr{1f.g) by bounded processes and
that the measures |(/?)-| are uniformly ?-additivefor ? e LG,. — + - = 1 ·
Then ? e LFG(X), H" -> ? in L\- g(X) and {Hn ¦ X), -> (H ¦ X), in LG for each
A Vitali-type theorem can be stated along similar lines.
7.5. Summability of the stochastic integral
If X is a p-summable process, the stochastic integral ? ¦ X is not necessarily p-summable.
We state below two theorems giving sufficient conditions for the summability of the
stochastic integral ? ¦ X.
We consider first the case when ? is real-valued. Let 1 ^ ? < oo and X : E+ ? ? —>
? с L(F, G) be a p-summable process.
17. THEOREM. Let ? e ?,? ?(?). Assume JAHdIx e L^ for each AeV. Then
(a) Я · X is p-summable relative to (F, G).
(b) К e LF G(H ¦ X) iff ? ? e L\g(X) and /? ??/'? cave we have the associativity
formula
K(HX) = (K- H)X.
Then we consider the case where ? is vector-valued.
18. THEOREM. Let ? e LF G(X). Assume fA Hdlx e LG for every AeV. Then
(a) Я · ? /? p-summable relative to (F, G).
(b) If К is a real-valued, predictable process and if KH e L]FG{X). then К е
Ljj G (H ¦ X) ?/iii we have
K(H ¦ X) = (KH)- X.
7.6. Local summability and local integrability
Let 1 ^ ? < oo and X :R+ ? ? -> ? с L(F. G) be a cadlag, adapted process with
X e LPE for each ? > 0.
We shall define now the local summability of X and the stochastic integral ? ¦ X.
19. DEHNITION.
(a) We say X is locally p-summable relative to (F. G), if there is an increasing
sequence G„) of stopping times with T„ \ эс. such that for each n. the stopped
process XT" is p-summable relative to (F. G).

392
N. Dinculeanu
(b) Assume X is locally p-summable relative to (F, G). A predictable process ?: E+ ?
? —>¦ F is said to be locally integrable with respect to X, if there is an increasing
sequence G„) of stopping times with T„ \ oo, such that, for each n, XT" is p-
summable relative to (F, G) and 1\q.t„]H is integrable with respect to XT".
Then, for each n, the stochastic integral (?^.?,,?^) ¦ XT" IS defined. The following
theorem states the existence of the pointwise limit of the above sequence of stochastic
integrals.
20. THEOREM. Assume X is locally p-summable relative to (F, G) and let //:К+хй->
F be a predictable process, locally integrable with respect to X. Let (T„) be an increasing
sequence of stopping times with T„ f oo. determining the local integrability of ? with
respect to X. Then, the limit
\im(\{0.T)l]H)XT"
exist pointwise, outside an evanescent set, is cadlag, adapted and independent of the
sequence (T„).
The limit in the above theorem is called the stochastic integral of ? with respect to X
and is denote by ? ¦ X or / ? dX. For each Tn we have
(H-X)T"=(l{0.Tn]H)-XT".
The stochastic integral with respect to locally p-summable processes has the main
properties of the stochastic integral with respect to summable processes.
8. Processes with integrable variation or integrable semivariation
In this section we state the summability of processes X with integrable variation or
integrable semivariation. For this purpose, we associate to X a stochastic ?-additive
measure ??. The stochastic integral can be computed pathwise as a Stieltjes integral.
We shall denote ? = B(R+) ? ?. We say that a process X with values in a Banach
space is measurable, if it is measurable with respect to M. Every process X : E+ ? ? —>¦ ?
is automatically extended with Xt (?) = 0 for t < 0 and ? e ?. A measurable process
which is not adapted is called a raw process.
8.1. Processes with finite variation or semivariation
Let X : R+ ? ? -+ ? с L(F, G) be a process.
1. Definition. We say that the process X has finite variation (respectively finite
semivariation relative to (F,G)), if for every ? e ?, the path t ь* ?((?) has finite

Vector integration in Banach spaces and application to stochastic integration 393
variation (respectively finite semivariation relative to (F, G)) on each interval [0, t], or,
equivalently, on each interval ]—oo, t].
We denote
|X|,M = var(X.(w),boo,r])
and
(Xf.gMw) = svarf.G(X.M, ] - oo. t]).
We say that X has p-integrable variation |X| (respectively p-integrable semivariation
Xf,c)if |Х|эо е L] (respectively (Xf.g)tc e L]).
2. Proposition.
(a) Assume X is right continuous and has finite variation |X|. If X is measurable
{respectively optional, predictable), then so is \X\.
(b) Assume X is right continuous and has finite semivariation X^.c- Assume also that
either F or G is separable. If X is measurable (respectively optional, predictable),
then so is Xf.G-
8.2. Optional and predictable stochastic measures
We state first the existence of optional and predictable projections of bounded processes.
3. THEOREM. Let ? : R+ ? ? -н>- R be a real-valued, bounded, measurable process. Then
there is a real-valued, optional process "? and a real-valued, predictable process !'?
satisfying:
Е{Фт1\т<ос}\Тт\) = "ФтЦт<ос). a.s.,
for every stopping time T, and
Е(.Фт1\т<эс\\Тт-) = рФтЦт<^\- a.s.,
for every predictable stopping time ?.
For the proof, see Dellacherie and Meyer A975-1980), VI 43.
The process "? is called the optional projection of ? and the process ГФ is called the
predictable projection of Ф.
Using the optional and predictable projections of processes, we can define the optional
measures and the predictable measures.
4. Dehnition. Let m : ? -> ? be a ?-additive measure.

394
N. Dinculeanu
(a) We say that m is a stochastic measure if it vanishes on evanescent sets.
(b) We say that m is optional (respectively predictable) if for every set ? e ? we
have
m{M)= J "(\M)dm (respectively \"(\м)ат).
In this definition, the characteristic function 1 м can be replaced by real-valued, bounded
processes Ф.
5. THEOREM. ? ?-additive measure ш : ?1 -> ? is optional (respectively predictable) iff
for every real-valued, bounded, measurable process ? we have
I 0dm = I "Фат (respectively I рФат).
The measure m has automatically finite semivariation in?..?-, therefore the above
integrals are defined.
8.3. The measure ? ?
To a process X with integrable variation or integrable semivariation we can associate a
stochastic measure ??, which is used to prove the summability of these processes, as
well as the summability of square integrable martingales. We consider first processes with
integrable variation.
6. THEOREM. Let X: R+ ? ? -» ? be a measurable, right continuous process with
integrable variation (i.e., \X\X e L1). Then there is a stochastic measure ??.?—> ?
with finite variation \??\ satisfying the following conditions:
(a) ??(?) = E(j \MdXs), for ? e M, and
\??\(?) = E(f \Md\X\s),for MeM. i.e.,
\??\ =?\?\-
(b) // ? С L(F; G) and ? :R+ ? ? ^ F is a process, then ? e ?,,(??) iff ? is ??-
measurable and E(J \Hs\d\X\s) < oo. In this case E(f HsdXs) is defined and we
have
f ???? = ?(? HsdX,\ and f ???,?\ = e( f Hs d\X\s
(c) The measure ?? is optional (respectivelypredictable) iffX is optional (respectively
predictable).
(d) If X is real-valued, then ?? ^ 0 iff X is increasing.

Vector integration in Banach spaces and application to stochastic integration 395
A similar theorem is valid for processes with integrable semivariation, but we have to
impose cq<? E.
7. THEOREM. Assume со <? E. Let X : R+ ? ? -> ? be a right continuous, measurable
process with integrable semivariation Xp_.? {i.e., (?^.?)? € ?-')¦ Then there is a
stochastic measure ??.?,^? ? with finite semivariation (??)?,.? satisfying the following
conditions:
(a) ??{?) = E{f\MdXs),for ? eM.
(b) if ? : R+ ? ? —» R is a measurable process with
?(?*.?(\?\)) ¦= Е((тХш)-.^(Ф((о)) < эо.
then ? e 3-?..?{??) and we have
[???? = ?([?!!???
eE.
(b') Assume, in a addition, that X has integrable semivariation X e.g and со <t G. Then
?? has finite semivariation {/1x)f.g- If ? :R+ ? ? -> F is a measurable process
with
E(XF.G(\H\)) := E{mXiU!))FG(H{w)) < oo.
then ? е^г.с(дх) and we have
J ???? = ?( ?HsdXA eG.
(c) The measure ?? is optional {respectively predictable) iffX is optional {respectively
predictable).
8.4. Summability of processes with integrable variation or integrable semivariation
The following theorem asserts the summability of processes with integrable variation or
semivariation. Moreover, in this case, the stochastic integral can be computed pathwise, as
a Stieltjes integral.
8. THEOREM. ^гХ:М+хй->?с L{F, G) be a cadlag, adapted process. Assume
that X has integrable variation \X\ {respectively со <? ?, со <? G and X has integrable
semivariations Xr.? and X f.g)- Then:
(a) The measure ? ? : 7? -> L^ can be extended to ? ?-additive measure ? ? :V ->
LfE with finite variation \??\ {respectively with finite semivariations {Ix)r.e and
Ux)f.g)-
(b) X is summable relative to (F, G).

396 N. Dinculeanu
(c) For every M eV we have
??(?) = ?(??(?)).
If X has integrable variation, then
\??\(?) = \??\(?) = ?]?](?) = ?(???(?)).
(d) If ? e LFG(X) and if f \Hs(w)\d\X\s(w)) < oo, a.s., (respectively the function
? —>¦ Xf.g(H(co)) is integrable), then
(??),(?)=\ Hs(w)dXs(w). a.s., fort^O.
(e) If E(f \Hs(co)\d\X\s(co)) < oo (respectively the function ? ь* XFG(H(a>)) is
integrable), then
j ???? = e( I Hdlx\ =?(? Hs(aj)dXs(oj)
9. Martingales
A martingale ? :R+ ? ? —>¦ ? с L(F. G) is not necessarily summable; but if it is, the
stochastic integral ? ¦ X is again a martingale.
If ? and G are Hilbert space and if ? is a square integrable martingale, then ? is
2-summable.
1. THEOREM. Let ? :Ш+ ? ? -> ? С L(F,G) be a martingale. Assume ? is p-
summable relative to (F, G) and let ? e Ту.с(М).
If L)t,H dlM e ^? /or every t ^ 0, ?/?en ? e L\ g(M) and the stochastic integral
? ¦ ? is a uniformly integrable martingale, bounded in L!'G.
IfLG is reflexive, then UF G(M) = TFG(M).
2. THEOREM. Assume ? and G are Hilbert spaces and let ? : Ш+ ? ? -> ? С L(F, G)
be a square integrable martingale. Then ? is 2-summable relative to (F,G). We have
L^Li(M) = FFLi
(M)
and for every ? e L1 , (?), the stochastic integral ? ¦ ? is a square integrable
h.L-G
martingale.
The proof of summability can be done either by using the summability criterion
(Theorem 9 of Section 7), or by using the Doleans measure ?^?) associated to the sharp

Vector integration in Banach spaces and application to stochastic integration
397
bracket {M) (defined below), which is an increasing process with integrable variation, and
proving that IM <<??<??>·
Let ? : R+ ? ? -> ? be a square integrable martingale. Assume ? is a Hubert space.
Then \M\2 is a submartingale of class (D) and has a Doob-Meyer decomposition
\M\2 = N + {M),
where N is a martingale of class (D) and (M) is a predictable, integrable, increasing
process, called the sharp bracket of ?.
Consider the Doleans measure ? ?? defined for every A e 1Z by
?(?)(?) = E(I{M)(A)).
3. PROPOSITION. The measure ???) can be extended to a ?-additive measure on the
predictable ?-algebra V and we have
?{?}(?)=\\??(?)\\~? forAeV.
4. COROLLARY. The process (M) is summable relative to ?
?{?)(?) = ?(?{?)(?)), forAeV.
and we have
5. THEOREM. Let ? :R+ ? ? -> ? С L(F,G) be a square integrable martingale.
Assume ? and G are Hilbert spaces. Then
(a) ?^(?(??)) С 0F G(M) and for ? e ?^-(?(??)) we have
I/
ЯЛ/М
^ ^(/м)^-ЧЯ)^||Я|Ц(;,л,
(b) For the particular embedding ? = L(R, E) we have
¦ 2
Lr(jhm)) = Lr_lif(M)
and for ? ? ?-flj(M(Ai)) we have
I/
HdIM
„=(/«)к.^(«) = 11«111:(м,и,
(с) // ?? /? real-valued and D is a Hilbert space and we consider the embedding
M.CL(D,D), then
L2D(M(M)) = ZA 2(M)
U.LD

398
N. Dinculeanu
and for H e L' ·, (M) we have
D.L-D
JHdlA =(iM)DLJ)(H) = \\H\\L])Ul„y
Remark. Here is the classical approach of the stochastic integral of real valued
semimartingales. A semimartingale is a process of the form X = ? + A, where ? is a
local martingale and ? is a process with finite variation. We define separately the stochastic
integrals ? ¦ ? and ? ¦ A and then define the stochastic integral HX = HM + HA.
The stochastic integral ? ¦ A is defined pathwise, as a Stieltjes integral:
(??),(?)=\ Hs{w)dAs{w), for t > 0 and ? e ?.
J\o.t]
The stochastic integral ? ¦ ? reduces to the case where ? is a square integrable
martingale. So, assume ? is a square integrable martingale. We identify the space M.-
of cadlag, square integrable martingales with the space L2(P), by identifying a martingale
? e M2 with Мое e L2(P) and endow M2 with the norm of L2(P). For a simple process
?
# = ??1? + ]?]?,1]??
we define the stochastic integral ? ¦ ? by
/1
(? ¦ ?), = ?0??0 + ]??,(????, - ???)?,).
;"=?
Then ? ¦ ? & ?2- If we consider the simple process Я as an element of ?,-(?(??)) =
L2{V, ?.(?)), then one can prove that the mapping ? ь* ? ¦ ? is an isometry:
||Я-А#||м2 = ||Я||/.2(/1м).
Since the simple process are dense in L2(?/л?)) one can extend the above isometry to an
isometry of the whole space ?,2(?(??)) into M2- The value of this extension for a process
? e ?,2(?(??)) is called the stochastic integral of ? with respect to M.
The isometry between the spaces M2 and ?,2(?(??)), which is the starting step for the
classical approach of the stochastic integral with respect to a square integrable martingale,
is also obtained in Theorem 5 above, using the measure-theoretic approach presented in
this chapter.
References
Bartle, R.G. A956), A general bilinear vector integral, Studia Math. 15, 337-352.

Vector integration in Banach spaces and application to stochastic integration 399
Bochner, S. A933), Integration von Funktionen. deren Werte die Elemente eines Vektorraumes sind, Fund. Math.
20, 262-276.
Bourbaki, N. A927-1959), Integration. Chapitres I-VI, Hermann, Paris.
Bongiorno, B. and Dinculeanu, N. B001), The Riesz representation theorem and extension of additive measures,
J. Math. Analysis and Appl. 261, 706-732.
Brooks, J.K. and Dinculeanu, N. A976), Lebesgue type spaces for vector integration, linear operations, weak
completeness and weak compactness, J. Math. Analysis Appl. 54. 348-389.
Brooks, J.K. and Dinculeanu, N. A991), Stochastic integration in Banach spaces. Seminar on Stoch. Proc.,
Birkhauser, 27-115.
Dellacherie, С and Meyer, P.-A. A975-1980), Probabilities et Potentiel, Hermann, Paris.
Diestel, J. and Uhl, J.J. Jr. A977), Vector Measures, AMS Math. Surveys, Vol. 15, Amer. Math. Soc.,
Providence, RI.
Dinculeanu, N. A967), Vector Measures, Pergamon Press, Oxford.
Dinculeanu, N. B000), Vector Integration and Stochastic Integration in Banach spaces, Wiley, New York.
Dobrakov, I. A970a), On integration in Banach spaces I, Czech. Math. J. 20(95), 511-536.
Dobrakov, I. A970b), On integration in Banach spaces 11, Czech. Math. J. 21 (96), 680-685.
Dunford, N. and Schwartz, J. A958), Linear Operators. Part I. General Theory, Wiley, New York.
Foias, С and Singer, I. A960), Some remarks on the representation of linear operators in the space of vector-
valued, continuous functions. Revue Roumaine Math. Pures Appl. 5, 729-752.
Gowurin, M. A936), Ober die Stieltjessche Integration abstrakter Funktionen, Fund. Math. 27, 255-268.
Halmos, PR. A950), Measure Theory; Van Nostrand. New York.
Ionescu Tulcea, A. and Ionescu Tulcea, С A969), Topics in the Theory of Lifting, Springer. New York.
Kluvanek, I. A961), On the theory of vector measures I, Mat. Fyz. Casopis 11, 173-191.
Kluvanek, I. A966), On the theory of vector measures II, Mat. Fyz. Casopis 16, 76-81.
Kussmaul, A.U. A977), Stochastic Integration and Generalized Martingales, Pitman, London.
Metivier, M. A982), Semimartingales, de Gruyter, Berlin.
Metivier, M. and Pellaumail, J. A980), Stochastic Integration, Academic Press, New York.
Pap, E. B002), Some elements of the classical measure theory. Handbook of Measure Theory, E. Pap, ed.,
Elsevier, Amsterdam, 27-82.
Pellaumail, J. A973), Sur I'integrale stochastique et la decomposition de Doob-Meyer, Asterisque, Vol. 9.
Rao, M.M. A979), Stochastic Processes and Integration, Sijthoff and Noordhoff, Alphenaan den Rijin.
Rao, M.M. A996), Stochastic Processes: General Theory, Kluwer Academic Publishers, Dordrecht.
Rudin, W. A973), Real and Complex Analysis, McGraw-Hill, New York.

CHAPTER 9
The Riesz Theorem
Joe Diestel*
Banach Center, Kent State University: Kent. OH 44242-0001, USA
E-mail: j_diestel @ hotmail. com
Johan Swart*
Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria,
Republic of South Africa
E-mail: jswart@math. up.ac.za
Contents
Introduction 403
1 .The classical Riesz Theorem 404
1.1. A brief historical discussion 404
1.2. The abstract characterization of C( A") as a Banach lattice 406
1.3. The abstract characterization of С (A") as a Banach algebra 406
1.4. The Josefson-Nissenzweig Theorem 408
2.The Riesz Theorem and the Hahn-Banach theorem 411
2.1. The dual of the injective tensor product of two Banach spaces 411
2.2. Choquet's representation theorem 413
2.3. The Stone-Weierstrass theorem 417
2.4. The Pietsch domination theorem 419
3.The Riesz Theorem for operators 421
3.1. Weakly compact operators on C(K) 421
3.2. The Dunford-Pettis property and property V 422
3.3. Absolutely summing, integral and nuclear operators on C(K) 424
4.The Riesz theorem for vector-valued continuous function spaces 430
*The work of this author on the report was done while visiting the Mathematics Department of the University of
Missouri at Columbia; sincere thanks must be extended to the members of the faculty and staff there for providing
such a good atmosphere in which to work. Also appreciative thanks must be extended to Kent State University's
Research and Sponsored office and the powers-that-be at KSU who provided support for a sabbatical year at
Missouri.
+The work of this author on the report was done while on sabbatical leave at Kent State University; thanks are
extended to the Mathematics Department there for providing a pleasant working environment. The author also
wants to thank the South African NRF as well as the Mellon Foundation for financial support.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
401

402 J. Diestel and J. Swart
5.Notes and remarks 435
5.1. Notes and remarks to Section 1 436
5.2. Notes and remarks to Section 2 437
5.3. Notes and remarks to Section 3 439
5.4. Notes and remarks to Section 4 440
5.5. Notes and remarks on tensor products 441
References 444

The Riesz Theorem
403
Introduction
The objective of this essay is a discussion of the following representation theorem,
henceforth referred to as 'The Riesz Theorem'.
THEOREM 0.1. The dual С (K)* of the space С (К) of continuous scalar-valued functions
defined on the compact Hausdorff space К (equipped with the supremum norm ||/|| =
sup{|/(&)|: к e K\) is the space M(K) of all regular Borel measures defined on K. That
is, there is a correspondence
x* eC(K)* <s> ???(?)
between members x* ofC(K)* and regular Borel measures ? on К given by
**(/)= ? f(k)dv(k);
Jk
this correspondence is a linear isometric isomorphism between C(K)* and M(K),
equippedwith the total variation norm \\?\\ \ = \?\(?), where
\?\{?) — sup] У^ |?(?,)|: (?,),€?; is a Borel partition of К \.
We open with a brief discussion of the history of this theorem tracing its development
from R Riesz's earliest papers through Kakutani's finishing touches. Along the way we
give several applications that are more or less classical.
Next, we investigate the Riesz Theorem when used in tandem with the Hahn-Banach
theorem. Surely this has been the most successful partnership in abstract analysis.
The representation theory of Banach space-valued linear operations on C(K) is our
next task. Here the wish to give The Riesz Theorem direct generalization meets with
considerable success and some elegant, far-reaching applications; such generalizations
have rather severe limitations though. Once these limitations are understood, the Riesz
Theorem - scalar-valued case - takes center stage once again and directs the action,
motivating the pertinent questions.
The Riesz Theorem for continuous Banach space-valued functions is investigated in our
fourth section. Again, despite some initial successes, the applicability to understanding
spaces of continuous vector-valued functions is shown to be fraught with difficulties.
The structure of spaces of Banach space-valued continuous functions on a compact
Hausdorff space is considerably more complicated than the scalar theory. We'll make some
observations about why this is so and discuss the situation vis-a-vis some of the most
important invariants arising in the study of C(KYs.
Finally, we make a few (very few) choice comments about bimeasures and polymea-
sures, as well as about variations on The Riesz Theorem for other tensor norms.
Alas many have contributed to our understanding of these topics. We must acknowledge
the following: Witold Bogdanowicz, Jean Bourgain. Pilar Cembranos, Bill Davis, Leonard

404
J. Diestel and J. Swart
Dor, Paddy Dowling, Barbara Faires, Tadek Figiel, Ben Garling, Bob Huff, Hans Jarchow,
Bill Johnson, Nigel Kalton, Chris Lennard, Dan Lewis, Paul Lewis, Victor Lomonosov,
Jose Mendoza, Olek Pelczynski, Albrecht Pietsch, Elias Saab, Paulette Saab, Chuck
Seifert, Anton Stroh, Andrew Tonge and last, but far from least, Jerry Uhl. Each of these
mathematicians explained things to us that added to our understanding and clarified much
of our thinking.
1. The classical Riesz Theorem
1.1. A brief historical discussion
The study of the dual of C(K) started, naturally enough, with the case of К = [a, b], a
closed bounded interval of the real line. The main player was F. Riesz who returned to the
problem on several occasions. By 1914 his increasing awareness of the measure-theoretic
underpinnings of the subject was plain.
J. Radon extended Riesz's work to describe the dual of C(K) when К is a compact
subset of E". It was Radon who first laid bare the measure theoretic character of C(K)*.
The first to successfully attack the subject of C(K)* for abstract K's was S. Banach,
in an appendix to S. Saks' classic monograph "Theory of the Integral". To be sure,
Banach's note relies on his earlier work in an annexe to Chapter XII of his famous
"Theorie des Operationes Lineaires". Saks was not one to let the grass grow underfoot
and he soon presented a more 'standard' proof constructing the representing measure via
Caratheodory's procedure.
The finishing touch was supplied by S. Kakutani who established the Riesz Theorem in
a form more-or-less as stated in Theorem 0.1.
It's important that we mention that these results did NOT happen in a mathematical
vacuum. Rather, they were part and parcel of the rapid development of functional analysis
before World War II and played a central role in that development.
We take a moment to discuss just a few of the ramifications of the study of С (К )-spaces
and their duals in the development of functional analysis.
One of the first applications of the Riesz Theorem is in characterizing weakly convergent
sequences in C(?)-spaces. Proved by Banach in the case of compact metric spaces К
(without use of the Riesz Theorem!), the following is an excellent example of how measure
theory can be used effectively in abstract analysis.
THEOREM 1.1. Let (/„) be a sequence in C(K). Then in order that (/„) be weakly
convergent to f e C(K) it is both necessary and sufficient that the sequence be uniformly
bounded and f(k) = limbec fn(k)for each к e К.
Of course, if (/„) converges weakly to / \x\C{K), then {/„: ? ^ 1}U{/} is bounded in
C(K) by the Banach-Steinhaus principle of uniform boundedness; further, if к е К, then
Sk(g) :=g(k), forg e C(K), is a member-in-good-standing of С (К)* and so

The Riesz Theorem
405
f{k) = Sk(f) = lim Sk(f,)= lim /„(*)
holds for each к е К.
On the other hand, if (/„) is a uniformly bounded sequence in C(K) and / e C(?)
satisfies
/(*) = Mm f„(k)
77^ ОС
for each к e K, then / is the weak limit of (/„). In fact, one needs to test the convergence
of (**(/„)) to **(/) for each x* e C(K)\
But the Riesz Theorem tells that any x* e C(K)* is represented by a regular Borel
measure ? on К with jr*'s action given by
**(*)= I g{k)dvL{k)
JK
for g eC(K). Hold on! (/„) is uniformly bounded on К and pointwise convergent to /;
Lebesgue's bounded convergence theorem tells us that
**(/)= [ f{k)dii(k)= [ lim fn{k)dii{k)
= lim [ ????&)= lim x*(fnl
And this shows that (/„) converges to / weakly in C(K)L
As mentioned above Banach proved Theorem 1.1 without recourse to the Riesz
Theorem, but deduced it nevertheless at least in the case of compact metric space domains.
Along the way, Banach also uncovered when uniformly bounded sequences of scalar-
valued functions converge weakly to zero on any set; his result, which seems to have been
forgotten by abstract analysts, follows from delicate manipulation with lim suprema and
lim infima and his own secret weapon: Banach limits. The result?
THEOREM 1.2 (Banach). Let (/„) be a uniformly bounded sequence of scalar-valued
functions defined on a set S. In order that (/„) converge weakly to zero in the space B(S)
of bounded functions on S (equipped with the supremum norm) it is both necessary and
sufficient that for any sequence (s^) of points in S we have
lim lim inf I/„ (ii) I =0.
77^00 k^-OC
This is still a marvelous theorem.
The appearance of С(?)-spaces m arguments of functional analysis became more
frequent. Often such appearances were a simple consequence of the Hahn-Banach
theorem. Here's how the reasoning went: take a Banach space X and look at the closed
unit ball ??* of the dual X* of X; when equipped with the so-called weak* topology, the

406
J. Dieslel and J. Swart
topology of pointwise convergence on ?, ??* is a compact Hausdorff space (it is compact
and metrizable precisely when X is separable). X always is isometrically isomorphic to a
closed linear subspace of the space C(Bx·. weak*) of continuous scalar-valued functions
defined on ? ?* with respect to this weak* topology. The map jc i—> x(-) of evaluation of
xeXat member x* of ? ?* affects an isometric isomorphism with a tip of the hat to the
Hahn-Banach theorem.
Banach and S. Mazur used this idea to show that every separable Banach space is
isometrically isomorphic to a closed linear subspace of C(/i), ? the Cantor discontinuum,
or of C[0, 1], results that've been employed by many in the more than half century since
their discovery.
In fact,the identification of C(?)-spaces among the Banach spaces was a cause celebre
in each of the real and complex cases.
1.2. The abstract characterization ofC(K) as a Banach lattice
To characterize the C(?)-spaces among the real Banach spaces, it is natural to call on the
notion of a Banach lattice. Recall that a real linear space X is called ordered if there's a
partial ordering "<" that's compatible with the linear structure: if л ^ у then .v + с ^ .v + с
for all ? and ax ^ ay for all scalar a > 0; an ordered linear space is a Riesz space if its
order is a lattice ordering. In a Riesz space, if ? is a given element, then ? is the difference
of two non negative disjoint elements
? = x+ — x~
where x+ = sup{.x,0} and x~ = sup{— .v.0); the absolute value of x is given by
|jc| =x+ +x~.
A real Banach space is a Banach lattice if it's a Riesz space for which ||.v|| ^ ||>?
whenever |jc| ^ \y\.
Here's how Kakutani characterized the C(?)-spaces among the (real) Banach lattices.
THEOREM 1.3 (Kakutani). A real Banach lattice X is isometrically isomorphic, as a
Banach lattice, to a space C(K) of all continuous real-valued functions on some compact
Hausdorff space К if and only if X is an "M-space with unit", that is, for x, у е X with
x,y^0,
\\sup[x,y}\\ = sup{M-хЦ. ||_v|l}
and there is a largest element (a unit) in the closed unit ball ofX.
1.3. The abstract characterization ofC(K) as a Banach algebra
The characterization of С (^)-spaces among the complex Banach spaces followed close on
the heels of Kakutani's gem and has became one of the fundamental results in all analysis.

The Riesz Theorem
407
It is due to I.M. Gelfand and M.A. Naimark. The structure that gives the suitable framework
within which to formulate the Gelfand-Naimark theorem is that of a Banach algebra.
A complex Banach space X that's also an algebra is a Banach algebra if \\x - >'|| ^ ||-v||||y||
whenever x, у е X; should the algebra have an identity we suppose that the identity has
norm = 1. An involution on the Banach algebra X is a map .т и .т* of the algebra onto
itself for which (x + y)* = x* + y*, (??)* = ?.?*. (? у)* = y*x*,x** = x and ||.v*|| = ||jc||;
here x's and y's range over X and ? is an arbitrary complex number. If in addition,
\\x*x\\ = \\x\\2 for each ? then X is called a C*-algebra.
Here's the remarkable theorem of Gelfand and Naimark.
THEOREM 1.4 (Gelfand-Naimark). Suppose X is commutative C*-algebra with identity.
Then X is isometrically isomorphic, as a C*-algebra, to the space C{K) of all continuous
complex-valued functions on some compact Hausdorff space К with the identity of X
corresponding to the constantly = 1 function on K.
In each case, be it Kakutani's characterization of real C(K)'s or the Gelfand-Naimark
characterization of complex C(K)'s, the volatile mix of algebraic and analytic structures is
key. In Kakutani's proof, the space К is the collection of norm one linear functional on X
that're lattice homomorphisms while Gelfand and Naimark use the norm one linear
functional that're multiplicative; in each case, the weak* topology serves to make К compact.
Unquestionably, the Gelfand-Naimark theorem quickly paid handsomer dividends and
continues to do so. Here's the basis of many of the theorem's most penetrating applications:
start with a normal (bounded linear) operator ? acting on a Hubert space H. Look at the
smallest closed *-subalgebraCr inside the algebra B(H) of all bounded linear operators on
? that contains ?, ?* and the identity; Ст is a commutative C*-algebra with identity and
so, thanks to Gelfand and Naimark, we know that Cr is isometrically isomorphic, as a C*-
algebra, to the space C{K) of all continuous complex-valued functions on some compact
Hausdorff space K. It's part of the magic of C*-algebras that one can show that the К
under question is a (non-empty) compact subset of C! In fact, ? = ?(?), the spectrum
of ? as an operator on //!! It's plain and easy to consider p(T) if ? is a polynomial but
now, thanks again to Gelfand and Naimark, we can consider f(T) where / e C{K)\ For
/ e C(K) we define f(T) as the operator in Ст that corresponds to / under the above
mentioned isometric isomorphism. For the polynomial p, this correspondent will just be
p(T), exactly the polynomial in T. After all the Gelfand-Naimark theorem guarantees
that the identity on ?(?) corresponds to ? itself. Here's what's so: (f\ + fi)(T) is just
/? (?) + f2(T), (/? · f2)(T) = /? (T) ¦ f2(T) - where /? (?) ¦ f2(T) is the product of two
operators - and f{T)* = f(T). In other words the interpretation of f(T) is faithful.
What's remarkable about the above interpretation of f(T) is that it leads to a reasonable
interpretation of ?(?) for bounded Borel functions <p defined on ? =?(?). Here's where
the Riesz Theorem enters the field of action. To find an operator in Ст that behaves like
?(?) is a bit too much to ask. But one knows that whatever the operator ?(?) is to be,
it's necessary to know how its values at x's act vis-a-vis y's, that is, we need to understand
what (?(?)?, у) looks like for any ?, у e ? where (, ) is H's inner product. This in mind,
let S range over members of Ст (which we know is C(a(T))) and for x, у e ? consider
the functional Sh> (Sx, y) onC(a(T)). Each such functional on C(a (T)) is continuous

408
J. Diestei and J. Swart
and linear, with norm ^ ||?||||}??. by the Cauchy-Schwartz inequality. It follows that each
pair x, у е И generates a regular Borel measure ??,? ???(?). Of course, /?{?)????{?)
makes sense for any bounded Borel function ? acting on ?(?). To make sense of ?(?)
is now a (non-trivial) matter of uncovering an operator - we'll call it ?{?) - on ? that
satisfies
(<p(T)x,y) = ?(?)????(?) foranyj, ve H.
Jo(T)
Just how this is done is well-documented in many texts in functional analysis starting
with L.H. Loomis' classic "Abstract Harmonic Analysis", where it seems this idea was
first shown the light of day, on up to W. Rudin's masterful "Functional Analysis". Among
the many byproducts of the above approach one funds the Spectral Theorem for normal
operators and generalizations thereof.
All this is classical.
1.4. The Josefson-Nissenzweig Theorem
An application of the Riesz Theorem of more recent vintage is due to B. Josefson
and A. Nissenzweig. To put their theorem into perspective recall that for any infinite
dimensional reflexive Banach space X there is always a sequence of norm-one functionals
(x*) in X* that is weakly null. Since the dual of a reflexive space is also reflexive, this
sequence (x*) is weak* null as well. What happens in general Banach spaces? This is
answered by the Josefson-Nissenzweig theorem; we indicate a proof of this below, a proof
shown to us by Leonard Dor many years ago. It emphasizes how regularity comes into
play and along the way makes use of the Lebesgue space L2.
THEOREM 1.5 (Josefson-Nissenzweig). If X is an infinite dimensional Banach space,
then there exists a sequence (x*) in X* of norm-one functionals that go to 0 in the weak*
topology.
Outline OF the proof. Nissenzweig used an observation of A. Pelczynski to construct
a measure-theoretic Cantor set. A bit of notation clears the way to describing the results of
Nissenzweig's combinatorial designs.
If<pe{0, 1}", и e N, then we'll denote by (?, ?) the member of {0, 1}"+I obtained from
? by tacking on ее {О, 1}.
Following Pelczynski's generous advice, Nissenzweig found an a > 0, a sequence (z,*)
of norm one elements in X*, a sequence (x„) of norm one elements in X and for each
? e {0, 1}" (any и) an infinite subset ?? of {?*,- ? e N} and a yv e X in such a way that
(i) for any ?, A№.0) and A№j > are pairwise disjoint infinite subsets of ??;
(ii) inf(A(V.o) - A№.|))(yv) > 0, for each ?;
(??) ?" ?<?€[?.?)"-' mf(^(<p.0)-*H ~ ?(?.\ )xn) ^ ?·
Once Nissenzweig had done this (admittedly highly non trivial) chore, he proceeded as
follows: for each ? let
?? ~ weak*closure A,

The Riesz Theorem
409
By (ii)
?(?.?) ? E(ipj) = 0.
It follows that if ?, ? e {0, 1}" and ???, then
?? П ?? = 0.
From this and (iii) we see that
— ]P ??{(?(?,0}?„ - ?(?. ? }?„) ^a.
<pe[0.1)"-'
For any ? e {0, 1}" choose и* е ?? and let
<pe[0.l)"
Each ?„ is a regular Borel probability measure on ? = (??*, weak*). The Banach-
Alaoglu theorem assures us of the existence of a weak* limit point ? e BCiK)* in the
closed unit ball of C(K)*. ? is a regular Borel probability on ?. It's more, much more. If
? ^ m and <p e {0, 1}'" then
1
?„(??) = —;
after all, there are exactly 2"~m ? 's in {0,1}" the first m of whose entries match ? exactly.
It follows that for any ? e {0, 1}'"
1
Indeed, if ?{??) < ^, then there'd be an open set G с К containing ?? so that
?(?) < 2^. It follows that there's a g eC(K), (Kg ^ 1, so that g(x*) = 1 for x* e ??
and g(x*) = 0 for ? <? G. Of course,
J 8??^?@)<^.
But / g ?? must be a limit point of the sequence (/ gd?,,) and so must be
^???^??,??^ —.

410
J. Diestel and J. Swart
Think about it:
1 С 1
— > M(G) > / %?? > limsupM„(?v) = — ·
*¦ 7 /1-»3C ^
OOPS!
We conclude that ?(??) > 4r for each ? e {0. 1}'". But there are precisely 2'" pairwise
disjoint E/s as ? ranges over members of {0. 1}'" so ?(??) = ^ is all that remains as a
conclusion - after all, ? is a probability.
Now define /„ e L2(K, ?) by
<pe|0.l]»-'
Then (/„)„epj is an orthonormal sequence in ?2(?,?). Let ;' :C(?) ^ ?-(?,?) be
the natural inclusion jx = jc(). For any ? e X
? I I-
|0'-*- /'?)?.-(???)?
I Г 2
? / ;дс(дс*)/й(дс*)^(дс*)
If Jt* e X* is given by
<M)= ? ??(?*)??(?*)??{?*),
JK
then we just saw that for any ? e ? ?,
?>,??|4?.
What about ||**||?
?? * и -^
X S>
<(*„) = ? ??*)???*)<??(?*)= ? ?*(?„)?,,(?*)<1?(?*)
Jk Jk
= ? ?? ?*(?„)??(?*)- ?
**(*„) ??(**)
<ре|0.1]
>
?
<ре[0.|)"-'
1
1 ¦ г * !
inf ? ?„ — — sup ? x„
2" л*€?,<.„,
2"
i*e?.
?.1)
= 2^ ?! ???(?[?,0)?„- ?{??)?„)
(pe|0.l)"-'
^? >0.

The Riesz Theorem
411
In tandem
||**|| ^ a > 0 and ]P |-v,?(*)|~ < oo.
?
for each ? e X and this finishes the proof of Theorem 1.5. ?
2. The Riesz Theorem and the Hahn-Banach theorem
The partnership of the Riesz Theorem with (various forms of) the Hahn-Banach theorem
is the most powerful in all abstract analysis. In this section we offer a glimpse of just how
this one-two punch can act.
We'll start with the most basic form of the Hahn-Banach theorem: extending bounded
linear functionals from a subspace of a Banach space to the whole space without changing
the norm. Our application will be the wonderful characterization by A. Grothendieck of
the dual of the injective tensor product of two Banach spaces.
Next, we apply Banach's version of the extension theorem wherein we extend a linear
functional that's dominated by a positively homogeneous, subadditive functional to the
whole space with domination remaining. Here the objective is the ever-fresh theorem of G.
Choquet to the effect that each point of a compact metrizable convex subset of a locally
convex space is the barycenter of a probability Borel measure supported by the set of
extreme points of that set.
We follow this with L. de Branges' famous proof of the Stone-Weierstrass theorem. We
use here the complex version of the Hahn-Banach theorem.
Finally, the separation theorem is applied to derive A. Pietsch's profound
characterization theorem for absolutely /^-summing operators.
2.1. The dual of the injective tensor product of two Banach spaces
If X and ? are Banach spaces (over the same scalar field), then their tensor product X <g> ?
can be normed via the so-called projective tensor norm in such a way that the dual of the
completion Х®У under this norm is (isometrically isomorphic to) the space B(X, Y) of
all continuous bilinear functionals on ? ? ?. The norm? If и e X <g> ?, then the projective
norm |и|л of и is
1л = 1пП]Г]||.х,|||Ы|·. и = ]Г\х, ®y,- |.
This was known to J. von Neumann and his co-workers F.J. Murray and R. Schatten. They
used this fact to considerable advantage when X and ? were both Hubert spaces. What
they could not do was identify the dual of the injective tensor product.
If и e X <g> Y, then the injective norm \u | v of и is given by
|M|v = sup{|u*®}>*)(M)|: x*eBx*.y*eBY*}.

412
J. Diestel and J. Swart
where
**® y*(j> ®y«) = J>*(*«)>'*0'«).
It's easy to see that | |v is a norm on X <g> ? which is a uniform cross-norm: \x <g> _y|v =
||*|| ||.y|| and if и ? : ? ? -> У ? and и:: ?: -> ^2 are bounded linear operators then и ? <g> i<2 is
a bounded linear operator from X| <g>X2 to Y\ ®?? with bound the product of the individual
norms when each is equipped with the injective norm. This norm is in fact the least such
norm and the completion of X <g> ? with this norm is called the injective tensor product of
X and ? and denoted by Х®У.
For any и & X <g> Y, it's easy to see that
|k|v ^ |к|л-
So each member of (Х®У)* is a member of the dual B(X, Y) of Х®У. The question is:
just which continuous bilinear functionals on ? ? У are continuous on Х<Й>У?
Here's what Grothendieck saw: given X and Y, the dual balls Bx> and By are each
compact in their respective weak* topologies and so the product space
К = (Bx*, weak*) ? (??>, weak*)
is compact, too; if и e X <g> Y, then и can be viewed as a member Л of C(?)-just define й
at a typical point (**, y*) of К by
й((х*.У*)) = (х*®У*)(")·
It's easy to see (especially once it's been told us) that the map
МИ Й
is well-defined linear isometry from (X <g> ?. \ |v) into C(K)l This map extends to an
isometric embedding of X<g>? intoC(^).
Okay, so take ? e (Х®У)*. Extend ? to a member of C(K)* with the extension having
the same norm as ?. The Hahn-Banach theorem will come in handy at this juncture. The
extension is naturally identifiable with a regular Borel measure ????; the Riesz Theorem
strikes! If we take ? e X and у e Y, then on realizing ? as a member of B(X, Y) we see
that
4>{x,y) = (p{x®y)
= ? ?®?(?*.?*)??(?*.?*)= f ?*(?)?*()')<1?(?*, y*).
Jk Jk
? is an integral bilinear form whose norm in (X®Y)* is just the norm \?\(?) of ? as a
member of C(K)*. To put it the way Grothendieck did, we have the following.

The Riesz Theorem
413
THEOREM 2.1 (A. Grothendieck). The dual (X®Y)* of the injective tensor product X®Y
of two Banach spaces can be identified with the space ??(?; ?) of integral bilinear
forms on ? ? Y; under this isometric identification a bilinear functional ? in (Х<8>У)*
corresponds to a measure ? on К (as above) via the formula
<p(x,y)= ? **0r)v*(v)d/i(**.v*).
Jk
2.2. Choquet 's representation theorem
Next we wish to talk about Choquet's representation theorem. We need some words of
introduction. Our setting is a locally convex (Hausdorff) linear topological space E; inside
of ? we have a (non-empty) compact convex set K. Each such set К has extreme points:
? e К is extreme if ?\{.x} is convex. In fact, it's a celebrated result of M. Krein and
D. Milman that any non-empty compact, convex subset К of a locally convex space ? has
extreme points and, in truth, К is the closed convex hull of its set of extreme points. If
К is a non-empty compact convex subset of a locally convex space ? and ? is a regular
Borel probability on К then ? has a barycenter ?(?) e К: .?(?) is the unique point of К
for which
?*(?(?)) = ( **(?)??(?)
Jk
for each x* e E*. Here's a stunning fact-of-life discovered first by G. Choquet; we follow
F. Bonsall's advice about the truth of Choquet's representation theorem.
THEOREM 2.2 (G. Choquet). Let К be a non-empty compact metrizable convex subset
of the (real) locally convex space E. Then each point of К is the barycenter of a regular
Borel probability measure ? on К which is supported by the set e\(K) of extreme points
ofK.
To be sure, when К is metrizable then the set cx(K) is a Borel set, a Ga-set, and so our
claim is that given jco e К there is a regular Borel probability ? on К whose barycenter is
xq for which ?(&?(?)) = 1.
Outline of Bonsall's proof. We start with the closed linear subspace A(K) of C(K)
consisting of affine functions; f eC(K) is affine if given ?. ? e К and 0 ^ ? ^ 1 we have
f(Xx + A - X)y) = ?/(.?) + A - ?)/(.?).
Of course we also consider convex and concave functions on К: /is convex if given
x,y e К and 0^? ^ 1 we have
f(kx + A - k)y) <: kf(x) + A - ?)/(?);

414
J. Diestel and J. Swart
while / is concave if — / is convex. To help us in our search for a suitable dominator, we
consider the concave envelope / of / e C(K):
J(x):=M{h(x): heA(K). /(v)O(y), foryeK].
It's not hard to verify that
• / is concave, bounded and Borel;
• / ^ / with / = / precisely when / is concave;
• f_+g^f + gand7f = tfift^0:
• i/-gmi/-giioc
• f + a = f+a ifaeA(K).
Take xq & K. Define ? onC(K) by
/>(/) =/(*>)¦
? is positively homogeneous and subadditive.
Since К is compact and metrizable, C{K) is separable and so is A(K). Hence there's a
sequence in the closed unit sphere Sa = {a e A: \\a\\x = 1} of A(K) that's dense therein;
if (a„) is the sequence let
/о е С (К) (by Weierstrass's M-test) and /o is strictly convex. Define F on A{K) + E/o
by
F(a + ?/?) = a(x[)) + a/0(.v0).
Notice that on A(K) + R/0, F is dominated by p! If ? ^ 0, then a + a/0 =_a + a/0; if
a < 0, then a + a/o is concave so that a + ?/? = a + ?/? which is ^ ? + ?/(). In either
case,
F(a+a/0) s?p(a+a/b).
We can extend F from A(K) + R/o to all C(K) in a linear fashion with its extension,
which we'll call ?, still dominated by p; that is, for any g e C(K)
4>{g) ^P(g) =?(¦*())¦
Of course, for any ? e A(?).
<p(a + ?/?) = ?(.??) + ?/(.го) = F(a + ?/0).
If g e C(?) is non-positive, then
0 > g(xo) = Pig) > <p(g)\

The Riesz Theorem
415
so ? is non-positive on non-positive members of C(K); <p is non-negative on non-negative
members of C(K). ? is non-negative member of C(K )*; there is a non-negative measure
?, a regular Borel non-negative measure on ?, so that
-/.¦
<P(g)= I gd^-
1 eA(K) and
\ = \(xQ) = F(\) = <p(\)= ? \?? = ?{?).
Jk
? is a probability.
A look at our strictly convex friend /o reveals that
/ /??? = <?(/0) = /0(?·0):
but /o ^ /0 so
/ /?^?^ / /?^?-
On the other hand, if a e A and ? > /?, then ? > /0, too; so
a(*0) = .F(a) = <p(a) = / ?<*?> / /0^? ^ / /?<*? = ?(/?) =/()Uo)·
A look at the definition of /o(*o) shows all above are equal! So
/ /0??= ?()??.
Jk Jk
Wow! _
/o is strictly convex so any point л e К where /o(.v) and /o(.v) are the same must be
extreme: if у ? ?, then
f(y + z\ /o(.v) , /o(c) . 7o(v) + 7o(c) . 7 ?? + ¦
Hence
{jteK: /o(*) = 7(>W}?ex(K).
However, /o ^ /(, and fK /0?? = fK ????.

416
J. Diestel and J. Swart
It follows that
1 = м([/о = 70]Нм(ех(К))«:1.
This proof is complete. ?
One remarkable consequence of Choquet's theorem is the following characterization of
weakly null sequences in Banach spaces
COROLLARY 2.3 (J. Rainwater). Let (x„) be a bounded sequence in the Banach space X.
Suppose that
lim;t*(jt„) = 0
?
for each extreme point x* of ? ?*. Then (xn) is weakly null.
PROOF. Suppose (x„) is a bounded sequence in X satisfying the extremal condition: for
each x* e cx(Bx*)
lim;t*(jt„) = 0.
?
If we could just suppose X were a separable Banach space, then ??* would be weak*
compact and weak* metrizable and so Choquet's theorem would be "on call". So what?
Well, if Xq e ??* is arbitrary then there must be a regular Borel probability measure ? on
(??*, weak*) such that ?(??(??*) = 1 and for each ? e X
x%(x)= ? ?*(?)??(?*)= ? ?*(?)??(?*).
J Bx* ??(??»)
Of course, the hypothesis that lim„ x*(x„) = 0 for each x* e ex(Sx>) just says that as
functions on ??*, (jc„) tends to zero ?-almost everywhere. But (x„) is also uniformly
bounded on ? ?*, so Lebesgue's still-wonderful bounded convergence theorem says
\imx*(x,,)= [ ?*(?„)??(?*)= ? ?„(?*)??(?*)=0.
" J вх* Jbx,
(?,,) is weakly null.
How we can deduce the general situation from the separable one? Well, the closed linear
span S of the sequence (x„) is separable and (x„) is weakly null in S precisely when
(x„) is weakly null in X. Does (x„) satisfy the hypotheses that lim„.s*(.x„) = 0 for each
s* e ex(Ss*)? If so, then all's right and the Corollary follows.
Here's the punchline: if s* is an extreme point of Bs*. then one ofs*'s Hahn-Banach
extensions to all X is an extreme point of ??*. This keen observation of Ivan Singer is
easy-to-see (once it's been pointed out!): if you consider the set ? of all Hahn-Banach
extensions of a given extreme point s* to all of X, then ? is a non-empty weak* compact

The RieSi Theorem
417
convex subset of Bx- arid so has at least one extreme point x*\ it's easy to see x* is also
an extreme point of ? ?*. ?
2.3. The Stone-Weierstrass theorem
Joined by the Krein-Milman theorem and playing under the brilliant direction of L. de
Branges, our dynamic duo soon produces a famous abstract proof of the Stone-Weierstrass
theorem in its complex setting. Recall the theorem's statement.
THEOREM 2,4 (Stone-Weierstrass). Let К be a compact Hausdorff space and X be an
algebra inside C(K), the space of all continuous complex-valued functions defined on K.
Suppose X is 'self-adjoint', that is, f e X /// e X. Assume that
A) ifkeK, then there's an f e X so f(k) ? 0;
B) if p,q e К and ? ? q, then there's an f e X so f(p) ? f{q).
Then X is dense in C(K).
Suppose that X is not norm dense in C{K).
The crux of the de Branges's argument is in the careful study of the family of measures
U{X) = {? e BC(K)*'- ? vanishes on X and is real}.
a weak* compact convex set which contains a non-zero member, thanks to the Hahn-
Banach and Riesz partnership. By all that's holy (that is, the Krein-Milman theorem),
U(X) has many non-zero extreme points. Let's take a look at such.
LEMMA 2,5. If ? e cx(U(X)) and ? ? 0 and if g: К -> С is a ?-essentially bounded
Borel function such that
f /?<?? = 0
for each f e X, then g is ?-essentially constant.
PROOF. The proof of this lemma involves masterful manipulation of measures and of
integrals having complex values. It's worthy of the closest attention.
It's plain that ||?|| = 1. Further,
jlgd? = 0
for each / e X; after all, X is 'self-adjoint'. But each ? e U{X) is real so for any ?-
integrable h,
I hd? = I hd?\

418
J. Diestel and J. Swart
from this we see that if / e X, then
/
0 = j fgdix = j fg-?? = j fgd/l
and so for each / e X
fgdv = 0.
Ah ha! If / e X, then
0 = J fgdix + j fjdix = j f(g + g)d^ = 2 j /Re^M.
We can assume that g is real valued.
We do so.
Adding real constants to g doesn't affect g's Borel measurability nor its ?-essential
boundedness nor its real-valued nature. So if we add С to g we have for any / e X that
by ?^ membership in U{X)
0= j fgd^ + C j /??= ? fig + ???.
In short, we can assume g is positive ?-almost everywhere. Normalize and we can assume
as well that
/
8?\?\=\.
?
?'
then |g| ^ 1 so g, being real-valued, satisfies
Suppose |g| ^ ?, ?-almost everywhere, where 0 < ? < ??. Could it be that ? > 1? If so,
J \\-8\?\?\ = j ??\?\- j 8?\?\=1-1=0
and 11 - g\ = 0, ?-almost everywhere! This says g is 1 ?-almost everywhere which we've
supposed not to be the case. So 0 < ? < 1.
The finale is in sight. Define ?? and ?? by
?,(?)= ? &??, ?:(?)= ? \—
J в J в ' —
— ??.
?
for ? a Borel subset of K. Both ??, ?: e U{X). Since g is not ?-essentially constant,
neither ?? nor ?? is ?. Yet
? = ??? + A -?)??,
contradicting ?'5 extremality. The lemma's proof is complete. ?

The Riesz Theorem
419
On to the proof of the Stone-Weierstrass theorem. Suppose X is a self-
adjoint subalgebra of C{K) satisfying B) yet failing to be norm-dense in C{K). Then
U{X) ? {0} and so U{X) contains a non-zero extreme point ?. Let g ? X, For each
/ e X, f · g e X, too. Hence
for each / e X, Since g e X. g is a bounded Borel measurable function and so our
Lemma 2.5 kicks in to tell us that g must be ?-essentially constant. But this says that
g is constant on ?'5 support; after all, g e C(K). Since \?\(?) = 1.
f gdM = g(k)
is the constancy achieved by g on ?'5 support, g does not change anywhere throughout
?^ support.
Look what we have done; any g e X is constant on ? 's support; B) says that support
must be a singleton {ко}.
There must be an a e C, |a| = 1 so that ? = aSin, where <5*„ is the 'point charge' or
'Dirac delta' at ко, ?*„(/) = f{h)) for / e C(K). It follows that for any g e X
0= / gd? = ag{k())
and so A) fails. This completes the proof, ?
2.4. The Pietsch domination theorem
In the late sixties, A. Pietsch initiated the study of absolutely p-summing operators
between Banach spaces. A bounded linear operator и : X -> ? between the Banach spaces
X and ? is absolutely p-summing A ^ ? < oo) if there is a constant ? > 0 so that for
any finite collection {jc ? xn} of vectors in X
???,??") <? sup ??>*?'') ; (О
if и is /^-summing then we define the /^-summing norm np(u) to be inf{Ai; A) holds}.
The importance of p-summing operators lies in large part in the way it measures m's
size. The comparison found in A) is not at all classical and it's a fact that being able to
compute the right-hand side often leads one to classical notions, be they from harmonic
analysis, operator theory or probability theory; as a result, p-summing operators are offen
unexpected players in a serious mathematical morality play.

420
J. Diestel and J Swart
While the definition of /^-summing says nothing about measures, be assured - measures
are always nearby, everready to make a difference. Pietsch made sure of that by proving
the following
THEOREM 2.6 (Pietsch domination). Let и: X -> ? be p-summing. Then there exists a
regular Borel probability measure ? defined on compact space (??*, weak*) such that for
any ? e X
\\ux\\ <л>(и)( ? \?%?)\???(?*)
V Jbx,
An aside; if we wish we can view each ? e X as a member of C{(B\*, weak*)); this
has been our schtick on several occasions already. The point of Pietsch's Theorem 2.6 is
that и being a p-summing operator allows и to be extended in a natural fashion from X
viewed as inside C((Bx*, weak*)) to be bounded linear operator (with bound ^ n,,(u)) on
the ??(?) closure of X.
PROOF, Let's look at just how this theorem is proved. Our proof follows Pietsch's spirit
but we let B. Maurey lead the way.
For simplicity we denote by К the compact space
? := (??*, weak*).
For ? ? ,x„ ? X define /v, ,„ e C(K) by
?-, ??**) = <(") ? ?**(*;)? - ? ""¦*; II"·
]?? j ??
Let C be
C=\fXl h,eC(K):xu,...xneX};
it's a pleasant surprise to find out that С is a convex cone. More, by the very definition of
np(u), each member of С is somewhere non-negative. Hence
СпЛ^=0, where Л/"= {g e C(K): g is everywhere negative}.
But ? has a non-empty interior so the separation theorem assures us of the existence of
a non-zero member ? e C(K)* such that
?(??(??(/), forgeWand/eC, B)
The left-hand side of B) ensures that ? is a non-negative measure; B) is impervious to
positive scaling, so we can assume the non-zero ? is, in fact, a probability. Finally, if ? e X
'//'

The Riesz Theorem
421
then
<??(/?)= ? (n^(u)\x*(x)\p -\\??\\?)??(?*)
J ? ?*
= ?';(??) ? \x*(x)\"dv(X*)-\\ux\\P.
Jbx,
since ? is a probability; A) follows from this. ?
3. The Riesz Theorem for operators
3.1. Weakly compact operators on C(K)
The first, and as yet the most successful, line of generalization of the Riesz Theorem, is to
operators as opposed to functionals.
In the main, here's what's involved.
Let и :C(K) -> X be a bounded linear operator from the Banach space C(K) to the
Banach space X, Then и** is bounded linear operator from C(K)** to X**; if В is a
Borel subset of K, then ?? acts continuously and linearly on C(K)* via the formula
??(?) = ?(?). An X**-valued measure is borne: F„(B) = и**(хв). The basic properties
ofF„?
• Fu is bounded and finitely additive into X**,
• Fu is weak*-countably additive.
• F„ is weak*-regular.
It's a wonderful discovery of R.G. Bartle, N. Dunford and J.T Schwartz that tells us
when Fu is norm-countably additive.
THEOREM 3,1 (Bartle-Dunford-Schwartz). Let u:C(K) -> X be a bounded linear
operator and Fu:Bq{K) -> X** be it's 'representing measure', as above. Then the
following are equivalent:
A) и is a weakly compact linear operator, that is, u(BciK)) 's relatively weakly compact
in X,
B) Fu takes all its values in X.
C) F„ is norm-countably additive.
D) Ftl is norm-regular.
The point of the correspondence is just this: for any operator и : С (К) -> X we have
that
x*uf= f f(k)dx*Fu(k)
Jk
holding for each / e C(K) and each x* e X*. If и is weakly compact, then the x*'s are

422
J. Diestel and J. Swart
extraneous, the integral exists in the norm topology and the integral
[ fdF
Jk
is a powerful ally in the study of и.
3.2. The Dunford-Pettis property and property V
To mention but one application of the Bartle-Dunford-Schwartz Theorem 3.1 we cite the
following basic result due to A. Grothendieck that's found in his fundamental paper about
weakly compact linear operators on C(K )'s.
THEOREM 3.2 (Grothendieck). Any weakly compact linear operator u:C(K) -> X
carries weakly convergent sequences to norm convergent sequences.
L[-spaces share with the C(?)-spaces the phenomena described above, a fact
discovered by N. Dunford and B.J. Pettis; in light of this, Grothendieck called such
spaces 'spaces with the Dunford-Pettis property'; after Grothendieck's discovery that
C(?)-spaces have the Dunford-Pettis property it was to be several decades before other
significant examples were uncovered and when they were it was with the aid of the Riesz
theorem in subtle, analytically-delicate situations.
A word or two about how the Bartle-Dunford-Schwartz description of weakly compact
operators provides an approach to Theorem 3,2. If (/„) is a weakly null sequence in C(K),
then the Banach-Steinhaus theorem tells us there's a constant ? > 0 so that \f,,(k)\ ^ ?
for all ? e N and all к e K; further, Banach tells us that (f„(k)) tends to 0 for each
к е К. But the point of a norm countably additive integral is that the Lebesgue bounded
convergence theorem holds, for much the same reason as for scalar integrals: Egoroff's
theorem. So (/ /„ dFu) tends to 0 in X's norm and with it lim„ \\uf„ || = 0, too.
Before you get carried away with the apparent elegance of this proof, we rush to
warn you that Grothendieck's proof, while never explicitly naming names or specifying a
representing vector measure, follows pretty much the same line of attack as that of Bartle-
Dunford-Schwartz. He does more. His deep analysis of weak compactness in C(K)*
permits him to prove the converse of Theorem 3.2. In fact he shows the following.
THEOREM 3.3 (Grothendieck). Let К be a compact Hausdorff space and X be a Banach
space. Suppose u:C(K) —> X is a bounded linear operator. Then the following are
equivalent.
A) и is weakly compact.
B) и is completely continuous, that is, и takes weakly convergent sequences to norm
convergent sequences, or, what's the same, и takes (relatively) weakly compact sets
to (relatively) compact sets.
C) и is weakly completely continuous, that is, и carries weakly Cauchy sequences onto
weakly convergent sequences.
D) и takes weakly Cauchy sequences into norm convergent sequences.

The Riesz Theorem
423
It is Grothendieck's supple handling of regularity that wins the day; paving the way is
his stunning improvement of a result of his mentor, J. Dieudonne. The result?
THEOREM 3.4 (Dieudonne-Grothendieck). For a bounded subset В of С (К)* to be
relatively weakly compact it is necessary and sufficient that given any sequence (G„) of
pairwise disjoint open subsets of К we have
lim sup |m(G„)| =0,
Grothendieck was not only one who had something to say about interesting variants of
weakly compact operators on C(K). Soon after Grothendieck, A. Petczynski introduced
the notion of an unconditionally converging operator, и : X —* ? is unconditionally
converging if whenever ?„ ?? is a series of terms in X for which ?„ ?-**(?'?)? < oo for
each x* e X*. then ?„ u(xn) is unconditionally convergent in Y. The celebrated theorem
of W. Orlicz and B.J. Pettis assures that weakly compact operators are unconditionally
converging regardless of their domain/codomain. Petczynski showed the converse for
operators acting on C(K)'s.
THEOREM 3.5 (Petczynski). A bounded operator и :C(K) -> X is weakly compact if and
only if и is unconditionally converging.
Again this special result about operators onC(^)'s leads to the isolation of an important
Banach space invariant. With Petczynski, we say that a Banach space X has property V if
any unconditionally converging operator и : X -> У is weakly compact.
It's an elegant piece of functional analysis that there is but one possible obstruction to
an operator being unconditionally converging: the classical Banach space cq of all null
sequences of scalars; indeed, as noted by Petczynski, a bounded linear operator и ; X -> ?
fails to be unconditionally converging if and only if there is a subspace Xo of X that's
isomorphic to со such that u's restriction to Xq is an isomorphism.
This leads to a fundamental consequence about the structure of Banach spaces of C(K)
ilk.
THEOREM 3.6 (Petczynski), If X is a complemented (closed linear) subspace ofC(K)
and X is infinite dimensional, then X contains an isomorphic copy o/cq.
Complementation means there is a bounded linear projection ? :C(K) -> C(K) whose
range is X. Were X to be without a subspace isomorphic to со, then by what we're
said above, ? is unconditionally converging, after all, P's range has no cq's in it. But
Theorem 3.5 tells us that ? is weakly compact. Theorem 3.3B) tells us that ? is also
completely continuous. Let's take stock: start with a bounded sequence (/„) in C(K);
apply ? and the resulting sequence (Pf,) has a weakly convergent subsequence (Pg„)\
apply ? again and the result (P2g„) = (P(Pg„)) is norm convergent. P2 (= P) takes
bounded sets to relatively compact sets; ? is a compact linear operator! The only closed

424
J. Diestel and J. Swart
linear subspaces that could possibly serve as the range of a compact linear operator are
finite-dimensional ones.
We rush to point out that Pelczynski called on the Bartle-Dunford-Schwartz Theorem
3.1 to prove a version of Theorem 3.5 that led him to Theorem 3.6; more precisely, he
showed that // X contains no copy of со, then even,· и : C(K) -> X is represented by an
X-valued Fu and so is a weakly compact operator.
As yet no one has proved Theorem 3.5 by purely vector measure theoretic techniques
using Theorem 3.1.
Before leaving this aspect of the Riesz theorem for operators we'd like to point out
that Bartle, Dunford and Schwartz based much of their analysis on a basic feature of
vector measures discovered during their work: If ? is ? ?-algebra, X is a Banach space
and F: ? —> X is norm-countably additive, then there is a countably additive scalar
valued ? on ? so that the family {x*F: \\.x*\\ ^ 1} is uniformly absolutely continuous
with respect to ?. Consequently, if и :C(K) —» X is a weakly compact linear operator
then there is a regular Borel probability ? on К so that u*X* с Ll(?); of course,
u* :X* -> L'(m) Я: C(K)* is still a weakly compact operator and u*B\* is bounded and
uniformly integrable. An old chestnut of de la Vallee Poussin now provides us with a
convex increasing ? : [0, oof-» [0, oof such that
Ф(х)>0 (ifjc>0),
г ?(?) ? г ?(?) ,
lim = 0, hm = +oo and
.v->-0 X л->-+зс ?
f *(|iiV(*)|)d/i(*Kl
JK
for each x* e Bx*\ u* is actually a bounded linear operator into the Orlicz space L0 (?),
A bit of tender love and care allows us to factor и through L* (?) where ? is the N-
function conjugate to ?, in fact through the 'absolutely continuous' part of L* (?). Much
can be derived from this; we mention but one analytic consequence: if и :C(K) -> X is
a weakly compact linear operator and (/„) is a bounded sequence in C(K), then (u(f,))
has a subsequence with norm convergent arithmetic means.
What of other classes of operators? Naturally, compact linear operators are weakly
compact; what of their representing measures? Bartle, Dunford and Schwartz were up
to the task: a bounded linear operator u;C(K) —> X is compact precisely when the
representing Borel measure Fu is X-valued and has a relatively compact range.
3.3. Absolutely summing, integral and nuclear operators on C(K)
More intriguing is the following
THEOREM 3.7. The operator и ,C(K) -> X is absolutely {-summing if and only if the
representing Borel measure Fu is of bounded variation; in this case, и is integral and

The Riesz Theorem
425
admits the following factorization
C(K) *¦ X
L\\FU\)
where a :C(K) -> Ll(\Fu\) is the natural inclusion and b ; L[ (|Fu\) -> X is given by
b(f)= f f(k)dF«(k).
We take note of the easily-verified fact that a bounded linear operator between Banach
spaces is absolutely summing precisely when its biadjoint is, with both sharing the same
sized absolutely summing norm; it's not hard from this to verify that the representing
measure Fu of an absolutely summing operator u:C(K) -> X is of bounded variation.
If F„ has bounded variation, then the factorization above is quickly verified and with it
the absolutely summing nature of и; after all, the natural inclusion of С (К) into L](\FU\)
is absolutely summing, thanks to Beppo Levi, and an operator with absolutely summing
factor is absolutely summing.
Such a result is absolutely pregnant with possibilities. Here's one: when is Fu Radon-
Nikodym differentiable with respect to | F„ | ? The answer is marked by its simplicity.
THEOREM 3.8. Let и :C{K) —> X be an absolutely summing operator with representing
Borel measure Fu. Then d Fu /d\ Fu \ exists as a Bochner integrable function precisely when
и is a nuclear operator, that is, и admits a representation in the form
"(/) = ][>„(/)*„. feC(K).
?
where (д„)С С'(К)*, U„) С X andJ2„ IImJKII < °?·
As an application of Theorem 3.8 we present a proof of a gorgeous theorem of Pietsch. It
may be viewed as a natural generalization of the fact that in Hilbert spaces, the composition
of two Hilbert-Schmidt operators is an operator of trace class.
Recall from Section 2 that an operator и : X -> У is absolutely /^-summing if there is a
? > 0 so that if ? \, xn e X then
(][>**||? <Msup{ (J] |.ir*(**)
inf{Ai > 0: above obtains} is the absolutely /^-summing norm of u, denoted by ??(?).
Here 1 < ? < oo.
ЧР
<1

426
J. Diestel and J- Swart
Now it is relatively painless exercise to see that и: X -> ? is absolutely /^-summing
precisely when и carries weakly p-summable sequences in X to absolutely p-summable
sequences in Y. The sequence (x„) is weakly p-summable if for each x* e X*
?^\?*{?„)\? <??;
?
this linear space of X-valued sequences is a Banach space (if X is) when equipped with
the Carnorm
|(^|^к=5ир{(]Г|.х*(;с,,)|Л :дг*еВх.|.
The sequence (y„) in ? is absolutely p-summable when
?>???"<??;
?
naturally, ||(jn)llt'' is just (?„ 113"cr 11я)'/ я - Again, €^1гопа(У) is a Banach space (if У is).
It's part and parcel of the theory of absolutely /^-summing operators that и : X -> ? is p-
summing when (u(x„)) e t'^mm„(Y) whenever (л„) е ^eak№ an(J, in fact, лр{и) is just
the operator norm of the induced operator from ^.eak(X) to iflronB(i'). The p-summing
operators are an Operator ideal' so the composition of two operators one of which is p-
summing is /^-summing with an estimate of the form np(uv) ^ тг/,(ы)||и|| or ?^(?)||?||
available depending on the /^-summing factor. Finally, if и is /^-summing and ? < r < oo,
then и is r-summing, too, with ?, (и) < ?,, (и). The /^-summing operator scale behaves
much like the L''(?)-scale, a fact that in light of Pietsch's Domination theorem is not
surprising. But surprises do exist. Here's one.
THEOREM 3.9 (Pietsch). If u:X -> ? and v.Y -> ? are absolutely 2-summing
operators, then ? о и is nuclear
First, we'll write и in factored form. By the Pietsch Theorem 2.6, there is a regular Borel
probability measure ? on К = (??*, weak*) so that for any ? e X
\\UX\\ ^n2(u)\\x(-)\\L2uty
If we temporarily denote by ?? the closure of {.v(-): ? e X} in ?2(?), then this inequality
says that и can be (linearly) extended to a bounded linear operator й: ?? -> Y\ we can even

The Riesz Theorem
427
assume ||?|| ^ Л2(и)- But Hubert spaces being Hubert spaces, й can be extended to all of
?2(?) in a linear, norm-preserving manner just by letting ? vanish on X^. The result:
-* ?
C(K) - L2(M)
where a : X -> C(?) is evaluation and /' :C(K) -> ?2(?) is the natural inclusion.
Our first task will be to show that и о и is absolutely 1 -summing and so we need a handle
as what и does to sequences (x„) that're in
CakW- Of course, if (*„) e ^,eak(X), then
(jc„) e ^weak^)'too; s0 ("¦*'<) 's absolutely 2-summable. More can be said.
Leta„ = (^ |jc„(^*)|iiM(.v*)I/2.Then
???"?2= ? ?\??(?*)\??(?*)= ? ^i^*(^)i^u*)^ii(^)ii(.
(?)"
So(a„)er and||(a„)||2iC||(.v„)
|1/2
*'....l(A·)'
Now we call on the duality implicit in our factorization of и through ?~(?): for
y*eY*, й*(у*)е?2(м)* = ?2(м);
|й*(У)| < 11й*И1Ь-*И = ???????.??? «? л-2(и)||у*Н-
Moreover, for ? e X and у* e ?*, й*у* е L2(m)* and х(-) e ?2(?) so
у*(и(дс)) = й*0'*)(дг(-)) = (*(-). "Y(-))t2lA)
= ( дс(дс*)й*у*(дг*)^(.г*).
./к
|у*(и(дс))|< ( |**(дО||й*у*(дг*)|<*д(дг*)
./к
= ( |.х*и)|1/2|.х*(.х)|'/2|й*у*(л-*)|^м(.х*)
./к
which by Cauchy-Schwartz is,
<|к(-)Г/2|2||дс(-)Г/2|й*у*(-)||2
= |U-(-)|!/2||.v(-)il/2|i7V(-)i|U.

428
J. Diestel and J. Swart
If we let y„ = и {x„ /?„) then
\У*(У„)\2 = \y*{u(x„/a„))\2 = -i-| y*M(.v„)|2
< ^||k,(oi1/2iuV(-)i|; = \\\х,л-)\1,1\й*y*l·)
\?„\?
= f \х„(х*)\\й*у4-х*)\2<1?(?*).
JK
It follows that for any NeN,
??,·?:«?/?-'*>??*?<·<·>|^·>
= /(EI*-^l)l--vu-)|2^)
<|(^)|tiiiik(x,ii5Viii2(/11:
from this we see that (y„) e ^^еак(У) and
l(}'")lt-;eUk(>')^ll(":")lli\i,ak^'7r2(")-
So? Well if (x„) e ^eak(-O then we can write u{x„) in the form
?(?„) = ?„}'„
where
(a„)ei2 and II(cr„)II ^ ||(-v„)II {"
and
(Уп) ? tleak With ||(y„)| 2 (n<|Ui)|ii2 ,v,T2(«).
"'weak1'' («еак(Л'
Now let's apply the absolutely 2-summing operator ? to (u(x,,)) = {u(a„y„))\ the result
is the sequence (a„v(y„))-a sequence which is plainly absolutely summable! To be sure,
(}'„) e ^Суеак(^) and и is absolutely 2-summing so (u(.v„)) is absolutely 2-summable, that
is, (||f(}'„)ll) e i2; it follows that (?„?(\„)) is in ?l with the i[ -norm gauged as follows:
им
MDI, = 11Aк^(л)||)||, = 1A^111^11I,
< ||K)|,|(ikv„ii)|, < IMI'i2 .x^i(v)\\(y„)\\t2 Y
(«еак,л' «сак1

The Riesz Theorem
429
This shows that not only is и о и absolutely summing but
? ? (v ou) ^ ?2(??)?2(?)
as well.
Now we look at the figure of и о и;
* ?
C(KY) —^ L2(My)
and concentrate on the following factor of ? о и:
C(KX) —^*- L2(MA·) —^ ? >¦ C(KY) —^- L2(My).
It's easy to see that ix and /y are absolutely 2-summing and so
iY o(Y^ C(KY)) ouoix:C(KX) -> L:(Mr)
is absolutely 1-summing. Theorem 3.8 says this operator is represented by an ?-(ду)-
valued measure that's countably additive and has finite variation; but L_^y)-valued
measures of finite variations always have Bochner integrable Radon-Nikodym derivatives
with respect to their variations (Hubert spaces have the "Radon-Nikodym property'); so
? о и has a nuclear factor and it's easy to see that this ensures us that и о и is itself nuclear.
It is relatively easy to show that between Hilbert spaces, the 2-summing operators and
Hilbert-Schmidt operators are precisely the same while the trace-class coincides with the
nuclear operators. So the above result is a generalization, a Banach space generalization,
of a familiar and fundamental feature of operators on Hilbert spaces.
No doubt the role of Theorem 3.8 in Banach spaces has been most profound when used
in tandem with the theory of differentiability of vector-valued measures. Without going
into details (they can be found in the monograph of J.J. Uhl and J. Diestel A978)) we
mention a couple of the most striking such applications
• If и : X -> ? is an integral linear operator (that is, the bilinear functional ?„ on
X x Y* given by
<Р(х,У*) = У*(и(х))
is integral in the sense as defined earlier) and v:Y->Zisa weakly compact linear
operator, then ? о и is nuclear.
• IfX* has the approximation property and the Radon-Nikodym property, then X* has
the metric approximation property.
C(KX) *· 1?(??)

430
J. Diestel and J. Swart
Recall that a Banach space У has the approximation property if regardless of the Banach
space X, the finite rank bounded linear operators from X to У are dense in the space of all
bounded linear operators from X to У relative to the topology of uniform convergence on
compact subsets of X. У has the metric approximation property if for any Banach space X
and any bounded linear operator и : X -> У. и is in the closure of all finite rank bounded
linear operators between X and У of norm ^ ||u||, again relative to the topology of uniform
convergence on compact subsets of X.
We hasten to observe that the Radon-Nikodym property's role in the proof may be
extraneous; as yet, it's unknown whether dual spaces with the approximation property
cannot be renormed to have the metric approximation property.
4. The Riesz theorem for vector-valued continuous function spaces
We turn now to the Riesz Theorem in a vector-valued setting; more specifically, we
describe the dual Cx(K)* of the space of continuous X-valued functions defined on the
compact Hausdorff space K, The basic result is due to I, Singer.
THEOREM 4.1, The dual, C\(K)*, of the space of all continuous X-valued functions
defined on the compact Hausdorff space К (and equipped with the supremum norm
11/11 = sup{||/(?)||: к e K}) is (isometrically isomorphic to) the space of all countably
additive, regular X*-valued Borel measures on К that have finite total variation where the
norm of such an F is given by \\F\\ = total variation of F over K.
Key to establishing this theorem is a result of J, Dieudonne which says that Cx(K) is
just the closed linear span of the 'elementary tensors'/(О* (= / <g> x), where / e C(K)
and ? e X. Using this by-now-classical result, we are able to recognize Cx(K) as the
injective tensor product С(КЩХ discussed in Section 2, Building on the discussion
of Section 2, most particularly, Grothendieck's description of the dual of any injective
product, we soon see that Cx(K)* = (C(K)<&>X)* is precisely the space of the integral
bilinear forms on C(K) ? ?. But each such bilinear form gives use to an integral linear
operator и : С (К) -> X* which through the good grace of Theorem 3.7 in Section 3 forces
a representing measure F„ on us. F„ = F is the culprit we're looking for and it fills the bill:
Theorem 3.7 of Section 3 tells us that F is an X*-valued countably additive, regular Borel
measure of finite total variation; what's more, the total variation of F, the integral norm
of u, the absolutely summing norm of и, the integral norm of the bilinear functional, the
norm of the same bilinear functional as a member of (C(K)®X)* - they're all the same.
Finally, if / e C(K) and ? e X, then
[ f(k)dF(k)(x)= f f®xdF
Jk Jk
defines the fundamental action of a member F of the space of measures at an elemental
f <8>x &Cx(K). The natural extension of this action to all of Cx(K) defines
f [f(k),dF(k))= f fdF.
Jk Jk

The Riesz Tlieorem
431
Theorem 4.1 provides some with the smug feeling that the structure of Cx(K) is much
like that of a convenient mix of C(K) and X, Woe befalls possessors of such naivete. The
structure of C\(K) is subtle and, while it a fortiori must reflect that of C(K) and X, it also
has more than a few hidden secrets.
Here's one small bit warning: in the case of C(K), once you know that К is an infinite
compact Hausdorff space, C(K) contains an isomorphic copy (an isometric copy even) of
со, Indeed, under such circumstances, you can build a sequence (/„) in C(K) of norm-
one elements so that /„, · /„ = 0 whenever тфп; it's easy to show that the closed linear
span [/„] in C(K) of the /,,'s is isometrically isomorphic to c(>. Sometimes, the result is
complemented in C(K) and sometimes it's not. In fact, it's so that for C(K) to contain
a complemented copy of с о it is necessary and sufficient that there exists a weak*-null
sequence (?„) in C(K)* that's not weakly null. This delicious byproduct of Theorem
3.4 is due to Pilar Cembranos. Rest assured that there are C(K)'s for which weak*-
null sequences in C(K)* are always weakly null: the first examples of this phenomenon
were uncovered by Grothendieck (and so Banach spaces sharing in this property are called
Grothendieck spaces) who showed that // К is extremally disconnected (the closure of any
open set in К is open), then weak*-null sequences in C(K)* are weakly null. Thanks to
Kakutani we know that the 2nd dual space C(K)** is a C(K**) for some (extremally
disconnected) compact Hausdorff space K** and so, in a sense, the C(K)'s that're
Grothendieck spaces are in the majority. It's still not known precisely which compact
Hausdorff spaces К produce C(K )'s that're Grothendieck spaces; be forewarned, there are
some connected spaces to be accounted for in any characterization. Take heart: even among
the O-dimensional compact Hausdorff spaces no usable characterization, be it topological
or Boolean, exists.
In case of vector-valued functions, not only can you find co's, under minimal conditions
there are complemented co's! Here's what's so:
THEOREM 4.2 (Cembranos-Freniche). If К is any infinite compact Hausdorff space and
X is any infinite dimensional Banach space, then Cx(K) contains a complemented copy
of c0.
(We follow F, Bombal in the proof of Theorem 4.2) How can such a thing be? Well it's
because of X's involvement in the make-up of Cx(K). X is infinite dimensional so the
Josefson-Nissenzweig Theorem 1,5 provides us with a sequence (x*) in Sx* (so all have
11-11 = 1) that's weak*-null. For each x* there's an x„ e X so that x* (x„) = 1 and ||jr„|| ^ ?,
say. If we pick the sequence (/„) in C(K) to be disjointly supported and each of norm-one,
then (f„(-)x„)„ spans а со in Cx(K); if k„ e К satisfies \\fn(k„)x„\\ = \\f„(-)xn\\cX{.K),
then the linear operator ? ;Cx(K)^> Cx(K) given by
/>(/) = ?>,;(/(M)/„(-).v„
?
is a bounded linear projection, whose range is the closed linear span in Cx(K) of
(fn(-)xn)n, an isomorph of со.

432
J. Dieslel and J. Swart
Remark 4.3. It is important to understand that in the above proof the fact that for any
/ e C\(K), the sequence (f(k„)) is relatively compact ensured that (x*,(f(k„))) is in со
so ?„ ?*?/^?))/,?-)?? makes sense.
Since weak* null sequences in the dual Z* determine operators from ? to cq while
weakly null sequences in Z* determine weakly compact operators from ? to c<), if weak*-
and weakly null sequences in Z* are the same, then every operator from ? to cq is weakly
compact. Whatever else is so, a weakly compact linear operator between Banach spaces
cannot have cq as its range - Banach's open mapping theorem ensures the reflexivity of
any Banach space that's the image of a weakly compact linear operator acting on a Banach
space. So, the Cembranos-Freniche Theorem 4.2 has the following consequence.
COROLLARY 4.4. if К is an infinite compact Hausdorff space and X is an infinite
dimensional Banach space, then C\(K) is NOT a Grothendieck space.
After all, Cx(K) always contains a complemented subspace isomorphic to c(> and such
containment is forbidden to Grothendieck spaces.
What about operators from C\(K) to any other Banach space Y? Formally, the basic
representation of operators from Cx(K) to ? in terms of measures is much like that of
Section 3. If we start with a Borel subset В of K, an ? e X and a v* e ?*, then define
FU(B) to be the member of ?(X; Y**) given by
Fu(B)(x)(y*)=u*y*(B)(x)
- keeping in mind the fact that u*y* takes its values in C\ (K )* a space of X* -valued Borel
measures - then it's plain that we have the right object to represent u.
THEOREM 4.5. Bounded linear operators и : Cx(K) -> ? correspondto bounded, finitely
additive measures Fu defined on the Borel subsets of К with values in ?(X; Y**). For
any y* e Y*, Fu(B)(x)(y*) is countably additive in B, X*-valued with finite variation,
regular; for any x, Fu(B)(x)(y*) is a weak*-countably additive, weak*-regular Borel
measure having values in Y**. For any f e Cx(K), we have
Ы,у*)= f (/(*), <*fi, (*)(/))¦
J к
An excellent place to read of this theorem is in it's discoverer N. Dinculeanu's book
"Vector Measures" [34]
Дп important a.lly in the study of operators и on С ? (?) is the 'Operator semi-variation',
Fu, of Fu. Foj a 8o/e.l set В с К,
FAB) = ||F„||0p(S) - supj ?? Fu(E)(Xi)
where the sypiemum is tafen ?.??? all finite Bo/el partitions я ©f В and. ail members ц of
the close4 unit ball ?,? of X.

The Riesz Theorem
433
Here the theory diverges drastically from that of Section 3. In fact, it's still unknown
when Fu takes all its values in C(X; Y) - this is a long-standing question of N. Dinculeanu.
In a similar vein, a usable characterization of the weak compactness of и in terms of
properties of Fu remains open.
There have been several serious assaults on Dinculeanu's question. No doubt the starting
point of the ultimately successful solution will be the following artful variation of the
theorem of Bartle-Dunford-Schwartz discussed in Section 3.
THEOREM 4.6 (J.K. Brooks and P.W. Lewis). Let u:Cx(K) -> ? be a bounded linear
operator and Fu be the C(X; Y**)-valued representing Bore I measure. Then the following
are equivalent statements.
A) Fu takes all its values in C{X; Y).
B) For each ? e X, Fu(-)(x) is norm-countably additive.
C) Each operator ux:C{K) -> Y, ux(f) = u(f()x) is a weakly compact linear
operator.
In their extensive study of operators и : С ? (?) -> ?, Brooks and Lewis also isolated
important special cases in which properties of Fu itself ensure that it takes all its values
in C(X; Y). The most interesting of these is the continuity of Fu at 0; // FU(B„) -» 0
whenever (B„) is a decreasing sequence of Borel sets with f\ B„ = 0, then all of Fu 's
values lie in C(X; Y). While sufficient, this condition is not necessary.
A modified example of I. Dobrakov due to P.W. Lewis and J.P Ochoa serves as a warning
to any 'quick-fix artist'! Let К be the one-point compactification of N, X = i] and ? = cq.
Set Fu([oo}) = 0. If ? e Iх and В is a Borel subset of K, put F„(B)(k) = ?„???""«.
where u\ = e\, иг = щ = ег/2, щ — и$ — иь = еу/Ъ Fu is ?(€'; co)-valued and
countably additive but
Fu{{\\) = Fu({2, 3}) = F„({4, 5, 6}) = · · · = 1
so Fu is not continuous at 0 (to be continuous at 0 it is necessary and sufficient that
Fu(Bn) -> 0 whenever (B„) is a sequence of pairwise disjoint Borel sets). This example is
covered by a treasured result of Lewis and Ochoa.
THEOREM 4.7 (Lewis and Ochoa). Suppose X is a Banach space with a Schauder basis,
and that u:Cx(K)->Ya bounded linear operator with representing measure Fu. F„
takes all its values in C(X; Y) if and only if there is a sequence (u>„) of weakly compact
linear operators w„ :Cx(K) -> ? such that
(i) u(f) — \\mn w„(f), for each f e Cx(K) and
(ii) // Fw„ represents w„ then for each x e X and Borel set В <ZK,
F«(B)(x)=limFWn(BXx)-
?
This brings us to the issue of weakly compact operators и: Cx(K) -> У-how to tell
when и is weakly compact (if possible) in terms of its representing measure Fu. Here
one comes face^to-.face with the problem of what do the weakly compact subsets of

434
J. Diestel and J. Swart
Cx(K)* look like? This problem is open. In fact, it's not that long ago that the question
of weak compactness in the considerably less-intricate Lebesgue-Bochner space ??(?)
was resolved by A. Ulger with finishing touches provided by J. Diestel, W. Ruess and
W. Schachermayer. As of yet this result has not lead to any new insight into C\{K)*, even
when X* has the Radon-Nikodym property - new insights that might help resolve the
question of what weakly compact operators on С ? (К) look like.
While the general study of weakly compact operators on С ? (?) remains in its infancy,
there are results in case X is a special space and the objective is to deduce something about
weakly compact operators. For instance, there has been considerable work of substance
about when Cx(K) has the Dunford-Pettis property. The following result is by no means
all that's known, just the sharpest.
Theorem 4.8.
(P) J. Bourgain: if X = ?'(?) or X = CL\Ul)(J) or X = L[ ] (J)(v) or ..., then
Cx (K) and all its duals have the Dunford-Pettis property.
D. Contreras/S. Diaz, N. Randrianantoanina: ifX is the disc algebra A, then Cx(K)*
(and so Cx(K) ) has the Dunford-Pettis property:
(?) ?. Talagrand: There is a Banach space X so that in X* weakly convergent
sequences are norm convergent yet Cx([0, 1]) does not have the Dunford-Pettis
property.
Talagrand's example is particularly noteworthy for the warning it contains: even when
the weakly compact sets in X* are easily understood, those in Cx(K)* need not be.
In Section 3 we told of Pelczynski's theorem to the effect that unconditionally
converging operators on C(^)-spaces are weakly compact; stated formally, 'C(^)-spaces
have property V. In light of the Cembranos-Freniche theorem, it is natural to ask whether
or not Cx(K) has property V if X does. Natural to ask but difficult to know. Nevertheless
the question has a pleasing answer.
THEOREM 4.9 (Randrianantoanina). If X is a separable Banach space with property V,
then Cx(K) has property V, too.
The proof is a deliciously delicate mix of Banach space ideas, adroit handling of
liftings and weak*-measurable maps with a pinch or two of descriptive set theory.
Randrianantoanina gets around the question of weakly compact sets in С ? (? )* by calling
on M. Talagrand's proof that Cx(K)* is weakly sequentially complete if X* is - a highly
non-trivial result of its own. It comes into play because X* is weakly sequentially complete
whenever X has property V. Now it's not enough to just call on Talagrand: there's a need
to know that in weakly sequentially complete Banach spaces bounded sets are relatively
weakly compact precisely when they contain no sequences that act like U 's unit vector
basis (a classical result of H.R Rosenthal), and what this means in the case of Cx(K)*;
here a deep understanding of Talagrand's work is called on. But Randrianantoanina has
such understanding and more, much more.
Whether or not the separability assumption on X is needed is not known. For instance,
the conclusion of Theorem 4.9 holds if X is any reflexive Banach space - something

The Riesz Theorem
435
first observed by Pelczynski, himself. In truth though even this fact can be deduced
from Theorem 4.9 thanks to the property of reflexive spaces that separable subspaces are
contained in separable complemented subspaces.
Curiously, while weak compactness of operators on Cx(K) are a quagmire not easily
escaped, the situation with absolutely summing and integral operators is clear.
THEOREM 4.10 (C. Swartz). The operator и :Cx(K)-> ? is absolutely summing if and
only if F„ takes its values in ?\ (?; ?), the space of absolutely summing operators from
X to Y, in a countably additive fashion with finite variation (each with respect to the
absolutely summing norm).
THEOREM 4.11 (P. Saab). The operator u:Cx(K)-> ? is integral if and only if Fu takes
its values in I(X; Y), the space of integral operators from X to Y, in a countably additive
fashion with finite variation (each with respect the integral norm).
While in the case of scalar valued functions, absolutely summing operators and integral
operators coincide, such is not the case of С ? (?). Indeed, it's a penetrating theorem of JR.
Retherford and С Stegall that says that absolutely summing operators on a Banach space
? are integral precisely when ? is a Cx -space, that is, Z's finite dimensional structure is
much like that of a C(?)-space; it follows that for most X's there are absolutely summing
m's on Cx(K) that are not integral.
To continue the path set in Section 3, we mention what the case is with nuclear operators;
again, things are unclear: Here's the best that's been said so far.
Theorem 4.12 (P. Saab and B. Smith).
(I) If u:Cx(K) -> ? is nuclear, then the representing measure F„ takes all of its
values in the space N(X\Y) of nuclear operators from X to Y, is countably
additive and has finite variation with respect to the nuclear norm.
(II) и : Cx(K) -> ? is nuclear, whenever the operator u# : C(K) -> N(X\ Y) given by
u*(f)(x) = u(f®x)
is nuclear (into the space N(X; Y)).
(Ill) In order that regardless of К and Y, u's nucleariry implies that ofu# it is necessary
and sufficient that X* have the Radon-Nikodym property.
5. Notes and remarks
This last section is a sort of catch-all. We provide bibliographical data. We discuss results
related to those of the previous sections. We broach several topics that were left unattended
earlier.
To be sure, the Riesz Theorem is too broad a subject to be given complete coverage,
even in the limited compact Hausdorff context. Our choices of topics were driven by our
interests and tempered by our deep, unabiding ignorance.

436
J. Diestel and J. Swart
5.1. Notes and remarks to Section I
To say that F. Riesz started the ball rolling with regards to the 'Riesz Theorem' is an
understatement of vast propositions. After discovering the proper description of C[a, b]*
in [84], he set upon applying the result in [85] as well as detailing its proof. Riesz was
to return to the result on several occasions; of particular interest to any serious student of
the evolution of ideas is his elegant proof presented in [86] - no one reading this classic
piece of mathematics can doubt that the time for a measure-theoretic treatment was rapidly
approaching.
Soon this promise came to fruition in J. Radon's ground breaking work [79].
As mentioned in Section 1, S. Banach [4] was to provide the first abstract version of
the Riesz Theorem. As if to cement the importance of this result, Banach [5] then derived
the existence of Haar measure on locally compact separable metric groups; Banach's proof
was to serve as a model for the most general results on the subject of invariant measures
on homogeneous spaces as witnessed for example in R. Steinlage's [95] characterizations
of existence and uniqueness of measures invariant under group actions and С Bandt's [7]
work on metric invariance.
S. Saks [91] must have felt obliged to answer Banach's functionally derived
characterization of C(K)*, for К a compact metric space, with a measure-theoretic approach; in this
he was eminently successful.
In the early forties, S. Kakutani [59,60] embarked on a program of characterizing C(K)-
and L1-spaces among Banach lattices. It was out of this efforts that he was inspired
(compelled?) to obtain a usable description of C(K)* for any compact Hausdorff space
K. Our formulation of the Riesz Theorem is but one consequence of Kakutani's efforts.
The paper of S. Banach and S. Mazur [6] highlights the central role that C(?)-spaces
are bound to play in the general theory of Banach spaces. Though Banach and Mazur
restricted their attention to separable Banach spaces, their result on universality is a model
for general spaces; indeed, it's this fact that forces C(?)-spaces (and with them, the Riesz
Theorem) into our analytic consciousness.
After all, if X is any Banach space then the closed unit ball ??* is weak* compact
and norming; hence evaluation of x e X on Bx» is a linear isometry of X into
C((BX*, weak*)). Hence, with the Hahn-Banach theorem in hand, analysis of functionals
on X can often be turned to analysis of members of a C(K)*. that is, a 'help!' call to the
Riesz Theorem.
This being said, we must to add that the Banach-Mazur theorem is special and much of
its value lies in its special character. That every separable Banach space is linearly isometric
to a closed linear subspace of C(A), where ? is the Cantor set, is important not least of
all because C(A) has a Schauder basis, for instance. This feature of the Banach-Mazur
universality theorem has been used by many abstract analysts.
The Josefson-Nissenzweig theorem is due, independently, to B. Josefson [57] and
A. Nissenzweig [69], who gave different proofs that were hauntingly similar in spirit, if not
in execution. It was after H.R Rosenthal's [87] profound I' -theorem that the similarity was
explicated; we owe a great debt to J. Hagler and W.B. Johnson [55] for such an explanation.
The part played by regularity in the proof Josefson-Nissenzweig seems to be indicative
of a general theme: the closer one is to a direct application of the Riesz Theorem the more

The Riesz Theorem
437
likely that the regularity of the measure will be a real player in the proof. Sometimes its
principal use is to ensure that a space of continuous functions injects continuously onto
a dense linear subspace of an L''-space and sometimes, as in Section 1, it is used more
bluntly.
5.2. Notes and remarks to Section 2
The subject of tensor products of Banach spaces is widely shunned yet possesses a certain
elegance once the arid preparations have been overcome. The earliest work by J. von
Neumann, F.J. Murray and R. Schatten was inspired by the Hilbert space situation and so
doomed to limited success. To be sure, A. Grothendieck was the first to penetrate the finer
structure of tensor products, first in his 'Memoir' [54] and later in much greater depth in his
'Resume' [53]. The integral operators of Section 2, be they linear or bilinear, were to form
the cornerstone to Grothendieck's theory of Banach spaces. The proof of Theorem 2.1 can
be found in his Memoir, wherein several fundamental consequences are drawn; however,
it wasn't until his Resume that the pivotal part played by the integral operators in his view
of Banach spaces became clear.
For those adventuresome souls who'd like to learn more about tensor products, we
recommend R. Ryan's forthcoming Cambridge introduction [88]; a detailed expose of the
Resume by the authors and J. Fourier [29] might be helpful for a fuller understanding of
Grothendieck's game plan for the study of Banach spaces and their structure. Last but far
from least, the monograph [26] of A. Defant and K. Floret gives a clear overview of tensor
product techniques vis-a-vis operator ideal notions.
Choquet's theorem made its debut in [23]. It soon found applications in diverse areas
of mathematical inquiry such as C* -algebras, ergodic theory, non commutative harmonic
analysis and Banach space geometry. As befitting such a gorgeous theorem, immensely
successful efforts were made to simplify its proof, generalize and expose.
F Bonsall's proof [12] probably provided the last word in elegance with regards to
Choquet's theorem in the metrizable case.
With regards to generalization, probably the most well-known (and as of yet the most
often used) is due to E. Bishop and K. de Leeuw [11].
THEOREM 5.1 (Bishop-de Leeuw). If К is a non-empty compact convex subset of a
locally convex Hausdorff linear topological space E, then each point ? of К is the
barycenter of a regular Borel probability measure ? on К that's supported by the extreme
points of К 's in the sense that ? (В) = Ofor any Baire subset В of X that contains none of
К 's extreme points.
G. Edgar [43] surprised the mathematical world when he provided a generalization that
relaxed the compactness condition.
THEOREM 5.2 (Edgar). Let К be a non-empty closed bounded convex subset of the
Banach space X. Suppose К is separable and has the Radon-Nikodym property. Then
each point is the barycenter of a regular Borel probability measure supported by the set of
К 's extreme points.

438
J. Diestel and J Swart
Separability plays the role of providing access to the methods and results of descriptive
set theory, most particularly, the Kuratowski/Ryll-Nardzewski selection theorem; the
Radon-Nikodym property enters the foray through ingenious use of the Martingale
Convergence Theorem. The monograph of R. Bourgin [15] should be consulted here.
Another fascinating effort at generalization (albeit in a different direction) was expanded
by S. Khurana [61,62] whose aim was a finitely additive representation theory. As yet, this
promising approach has been just that: promising.
The Choquet theorem has been the inspiration of several brilliant treatises. First,
R.R. Phelps' still-wonderful pocket-book [73] is a must; it's too bad it's out-of-print.
Next, E.M. Alfsen's monograph [ 1 ] presents the subject in a clear and compelling manner.
Finally, the master himself puts his theorem (and, oh. so much more) into perspective in
his treasured three volumes [20-22].
Rainwater's theorem is due, of course, to the ubiquitous John Rainwater [80].
It has descendants of outstanding pedigree, to be sure. John Elton [44] showed the
following.
THEOREM 5.3 (Elton). In order that every series ???" m a ^anac^ space X for
which ?„\?*(
xn)\ < oo for each x* e ??(??*) be unconditionally convergent it is both
necessary and sufficient that X contains no isomorphic copy o/cq.
Be forewarned: if you recall the classic result of С Bessaga and A. Pelczynski
[9] which asserts that it's precisely in spaces X without copies of cq that ???" 's
unconditionally convergent when J2„ \?*(??)\ < °° f°r eac'1 -v* e %*· men be aware that
Elton's Theorem 5.3 is considerably more subtle, relying on ideas related to the Radon-
Nikodym property. In particular, adroit use is made of another classical result of Bessaga
and Petczynski [10] this time to the effect that in separable duals every non void closed
bounded convex set has extreme points and is the closed convex hull of the set of such.
J. Bourgain and M. Talagrand [14] also entertained questions involving extremal
convergence.
The applications of the Riesz Theorem in tandem with the Hahn-Banach theorem
often involve extreme point considerations. Nowhere is this more powerfully demonstrated
than in de Branges's proof of the Stone-Weierstrass theorem. This paper [25] was
somewhat typical of many beautiful proofs from the 1960's involving function algebras.
The monographs of ? Gamelin [47] and L. Stout [96] provide clear exposition of many of
the best results of the times.
De Branges's proof still has a lot of kick in it. Most recently, Victor Lomonosov [66]
used ideas implicit in de Branges's proof to show that there exists a uniform algebra A
and a non-void closed bounded convex subset К of A such that the set of members of A*
that attain their maximum modulus on К is not norm dense in A*. Thus one of the most
beloved consequences of the Bishop-Phelps theorem is not so in complex Banach spaces.
Now Lomonosov is just the kind of mathematician who knows a good thing when he
sees it and soon he turned his counter-example into a positive result [67] showing that all
dual uniform algebras in which the full Bishop-Phelps result holds are self-adjoint. This
is startling mix of algebraic and geometric thinking, and the Riesz Theorem is again at the
center of the action.

The Riesz Theorem
439
A. Pietsch's stunning Domination Theorem appeared in the paper [74] in which he
first introduced the class ?? of absolutely /^-summing operators. Our proof is a small
modification and special case of an argument of Bernard Maurey [68].
Pietsch's isolation of the absolutely /^-summing operators was a singular event in
abstract analysis in general and Banach space theory in particular. It's true that in one
guise or another Grothendieck recognized ?\ and ?? but in the latter case he did not have
the same norm as Pietsch did. This is important, as witnessed by the remarkable result
discovered by D.J.H. Garling and Y. Gordon [48], and, independently, ?.?. Kadec' and
M.G. Snobar [58]: // ? is a finite dimensional Banach space, then 7n(id?) = vdimf.
Precise computations like this are frequently encountered in the literature on absolutely
summing operators, particularly as they relate to other aspects of mathematical enquiry. We
cannot overstate our respect for Pietsch's accomplishment. Our attitude is buttressed by the
many monographs that center on the Pietsch theorem and its ramifications. To be precise
let us mention: [72] (where connections are made with spaces of analytic functions), [77]
(which shows the role played by factorization schemes in harmonic analysis and operator
theory), [63,75] (each of which draws deep conclusions about eigenvalue distributions
for classical operators from the theory of /^-summing operators and their relatives), [99]
(where Minkowski spaces are the object of attention from a ??-perspective), [78] (which
studies volume ratios and the geometry of convex bodies with a ??,,-helper) and [76] (where
classical studies on orthonormal systems are given a refreshing new look). In addition the
Cambridge books of [30] and [102] give ample overview to how summing operators 'fit'
within the fabric of abstract analysis.
5.3. Notes and remarks to Section S
The original proofs of Theorems 3.1 through 3.6 in Section 3 are to be found in the classics
of A. Grothendieck [52], R.G. Bartle, N. Dunford and J.T Schwartz [8] and A. Pelczynski
[70,71 ]. We can think of no better advice to the students of the Riesz Theorem than to study
these papers closely.
The observation about weakly compact operators and arithmetic means was made by
J. Diestel and C.J. Seifert [32]. R. Anantharaman and J. Diestel [2] were able to improve
this result by dealing directly with the range of a countably additive vector measure.
Tracking the origins of Theorems 3.7 and 3.8 is a bit touchy, in large part because we've
taken a vector measure theoretic stance. However, the results are pretty enough that there's
plenty of credit to share so we mention L. Schwartz [92], J. Gil de Lamadrid [49,50],
A. Tong [100] and J. Diestel [28]; the book [33] gives these results careful discussion.
Theorem 3.9 appears already in Pietsch's original paper. It is a good example of measure
theory in action. Thanks to Pietsch's Domination Theorem, tools like Cauchy-Schwartz
and the Holder inequalities are made available for direct use in operator theory. It is so (and
somewhat more complicated to show) that if и : X -> ? is /^-summing and ? : ? -> ? is
<7-summing where ? and q are conjugate indices between 1 and oo then vu is 1 -summing.
Our parting comments about the composition of weakly compact and integral operators
and their relevance to approximation properties goes back to A. Grothendieck [54]. For a
modern read on these issues the paper [65] is highly recommended.

440
J. Diestel and J. Swart
5.4. Notes and remarks to Section 4
Theorem 4.1 is due to Ivan Singer [93]. [56] should also be consulted.
Our comment following Theorem 4.1 regarding C{K)'s that contain complemented
copies of со is due to Pilar Cembranos [17] who also noted that Cx(K) cannot be a
Grothendieck space if К is infinite and X is an infinite dimensional reflexive Banach space.
The stunning Theorem 4.2 was discovered independently by Cembranos [18] and
Francisco Freniche [46]. Their result soon lead to a spate of papers with interesting
variations, twists and turns on the same basic theme. Cembranos, Nigel Kalton and the
Saabs [19] showed that the injective tensor product of any two infinite dimensional Banach
spaces contains a complemented copy of со once one coordinate space contains a copy
of со. G. Emmannuelle [45] looked at the Lebesgue-Bochner spaces, Paddy Dowling [42]
considered the Hardy-Bochner spaces and vector-valued spaces of harmonic functions and
Diaz, Fernandez, Florencio and Paul [27] looked at spaces of Pettis integrable functions.
In each case that a complemented copy of со emerges in the function space/tensor
product after just appearing in one or another 'coordinate', it comes about as a result of
the following phenomenon: anytime a Banach space contains an unlimited copy of the unit
coordinate vector basis of со, it contains a complemented copy of со. Unlimited? Well,
a set К in a Banach space X is limited if any weak* null sequence in X* goes to zero
uniformly on К; К is unlimited if К is not limited. If (x„) is an unlimited unit vector basis
of со in X then there's a weak* null sequence (x*) in X*. an ?? > 0 and a subsequence
(*„.„) of (x„) so that
xm (x'ht< ) > ?
for all m; if we define и: X -> c(> by u(x) = (x*n(x)) then the series ?,? ?">„ satisfies
?», \x*(x>h„)\ < oo for each x* e X* (?,,,.?,,,,, is a 'wuC') yet since \\ux„m\\ >
I*m (¦*"»,) I > ?< Hmux"m ls not unconditionally convergent. We can apply the Bessaga
and Petczynski [9] result alluded to above to conclude that there is a subsequence (vk) of
(x,i,„) such that (f/t) and (u(vk)) are each equivalent to the unit vector basis of со and m's
behavior on the closed linear span [и* ] of the и* 's is that of an isomorphism. If we look at
? = u([vk]) we (should) see an isomorph of со inside со; but со is 'separably injective' and
so there is a bounded linear projection ?: со -> со with Pco = ?- The operator Q : X —> X
defined by
is a bounded linear projection on X with QX = [v„], an isomorph of со.
A space related to Cx(K) that's found application in the study of the invariant subspace
problem is the space WCx(K) of weakly continuous functions from К to X equipped
with the supremum norm. Though there is no known case in which WCx(K)* has been
described (unless К is finite and dimX is, too) it has been shown [3] that Cx(K)* is
complemented in WCx(K)*. This follows from the somewhat surprising fact that each
member ofWCx(K) is integrable with respect to each member ofCx(K)*.
Theorem 4.5 is due to N. Dinculeanu [34].

The Riesz Theorem
441
Theorem 4.6 comes from the paper of Jim Brooks and Paul Lewis [16] which contains
much of interest with regards to to the Dinculeanu problem. The paper of Lewis and Ochoa
[64] has a refreshing new look - hypotheses on a Banach space lead to conclusions of a
special and gratifying nature. Go figure.
The papers of Ulger [101] and Diestel, Ruess and Schachermayer [31] 'settle' the
problem of relative weak compactness in the Lebesgue-Bochner spaces Vx for 1 ^ ? <
oo. Regarding weak compactness in C\{K)*, only the paper of N. Randrianantoanina and
E. Saab [83] contains any real progress. This being so, [83] is a paper that should be closely
studied.
None of the papers that gave birth to Theorem 4.8 are 'easy reading'. Bourgain [13]
uses some techniques rooted in Riesz products; Contreras and Diaz [24] and Narcisse
Randrianantoanina [82] each need random versions of a nearest point technique that're
non-trivial and Talagrand's [98] is typical Talagrand - interesting but hard.
Theorem 4.9 is from [81].
Theorem 4.10 is found in [97], Theorem 4.11 in [89] and Theorem 4.12 in [90]. The
paper of Retherford and Stegall [94] contains a number of characterizations of ?? -spaces;
even when applied to C(K )-spaces there's much to be learned and enjoyed from Retherford
and Stegall.
5.5. Notes and remarks on tensor products
The Riesz Theorem takes on different guises when describing the dual (C(K) <8>„ X)* for
tensor norms a that're stronger than the injective norm.
The most important and best known of these tensor norms is the projective tensor norm
'л' (often called '?'). Here if и is a typical member of C(K) <g> X then
l"L = infj]P||/,'||ocll-*/l|: и = ]Г/, ®Xi
and C(K)®X is the completion of (C(tf)® X, I U). It is known that (C(K)®X)* is the
space B(C(K), X) of continuous bilinear functional on C(K) ? ?; in linear forms,
(C(K)®X)* = C(C(K);X*)
and so can rely on the Bartle-Dunford-Schwartzframework to deal with (C(K)®X)*. The
result is that members of (C(K)®X)* are bounded, additive, weak* countably additive
measures from the Borel ?-algebra of К to X*. Should X* contain no copy of со, then
each such measure is, in fact, countably additive and regular in the norm topology of X*.
So life isn't so bad.
Now a most interesting case is when X is itself a C(?)-space and we're looking at
(C(K\)®C(K2))*. In this case, C(K2)* is known to be weakly sequentially complete -
it's, of course, the space of regular Borel measures on ??-??? so does not contain an
isomorphic copy of cq. It follows that any и* е (С(К\)®С(.Кг))* corresponds to a (weakly

442
J. Diestel and J. Snarl
compact) linear operator и : C(K\) —» C(/o)*. which in turn corresponds to a countably
additive regular Borel measure Fu acting on the Borel subsets of ? \, and with values in
C(K2)* ¦ If we let B\ and Bi be Borel sets in K\ and Kz, respectively, then
F.,(Bi)(xb2)
makes sense: a mapping from the Cartesian product of the collection Bo(K \) and Bo(Ki)
of Borel subsets of ? ? and K2 is borne. This mapping, call it ?, acts on
B0(K\) * B0(K2)
via
M(S,,S2)=F„(S|)(xb:).
and for fixed B\ is countably additive and regular in Bi and vice-versa. It is a bimeasure.
Bimeasures are objects worthy of close scrutiny and close scrutiny has been paid them.
In a series of papers, I. Dobrakov [35-41] and his collaborators detailed the theory of
integration with respect to bimeasures and multimeasures, even if they're vector-valued.
Much remains to do and this is in large pact due to the role spaces like C(?|)<g)C(/o)
(and, more generally, C(K\ )<g> · · · ®C(Kn)) play in affairs of harmonic analysis or rather
harmonic synthesis.
A word or two about this last comment. Part of the charm of the projective tensor
product ('charm' being a word rarely appearing near the phrase 'tensor product'!) is the
fact that the projective tensor product of Banach algebras is a Banach algebra; in case
we're dealing with C(^i)<g)C(^2). the Banach algebra is commutative, has an identity
and, remarkably enough, a recognizable state space: ? ? ? ?2. This last fact was uncovered
by N. Varopoulos who laid bare the deep connections between these 'tensor algebras' and
questions in harmonic analysis. Let us whet the appetite of more ambitions readers. Let G
be a compact Abelian topological group and ? be its (discrete) dual group.
A(G) is the linear space of all / e C(G) for which there's a member F e Ь](Г),
the convolution group algebra (= ?' (?) in our present situation), so / = F, the Fourier
transform of F. If we norm A(G) by
II/IIa@ = ????.?(?) = ll^ll(i(D
then A(G) is a Banach space.
THEOREM 5.4 (N. Varopoulos). Define the operators ?, ? as follows
M:C(G)-> C(G ? G), P:C(G ? G) -+ C(G),
(Mf)(x,y):=f(x+y) and (Pf)(z):= f f(z-x.x)dx,
Jg
where dx is the normalized Haar measure on G.

The Riesz Theorem
443
A) ? is an isometric embedding of A(G) into C(G)®C(G).
B) ? takes C(G)®C(G) into A(G) with norm ?. 1 action.
C) PoM = idA{G).
D) ? ? P:C(G)®C(G) -> C(G)<g>C(G) ? ? norw о«е linear projection of
C(G)®C(G) onto A(G).
Some words-to-the-wise are in order. First, using the fact that C(K) has the
approximation property of Grothendieck, the inclusion
C(G)<g>C(G) <-+ C(G)®C(G)
is one to one; as a consequence members of C(G)®C(G) may be viewed as members of
C(G)<g)C(G), which we're already seen is CciG)(G) or C(G ? G). This might clarify
some of the above theorem's claims and bona-fides.
Next, for members of C(G)®C(G) we can compute the norm as follows: if / e
C(G)<g)C(G), then
' II '
where ?„ II/и II ос II gn Hoc < oo, and и = ?? f» ® gn- This is a famous characterization of
the norm in the completed projective tensor norm.
The proof of this theorem is a sequence of direct computations.
Remark 5.5. If the statement of the above theorem warms the cockles of your
mathematical soul, then you can't go wrong reading the essays of the stimulating
monograph of С.С Graham and O. McGehee [51], wherein the proof of the above theorem
and much more is given complete and clear exposition.
Sadly we are neither here to discuss the many wonders of spaces like C(K\)®C(Ki)
(which are called Varopoulos algebras and usually denoted by V(?|, ?2)) nor are we
qualified to do so. Rather, it's to point out the relationship with the Riesz Theorem through
bimeasures and to note that there are a fine gradation of other related spaces of 'measures'
between CX(K)* and (C(K)®X)*.
In his Resume, Grothendieck isolated a dozen other 'natural' tensor norms ? which, with
their associated continuous linear/bilinear forms, fit neatly in the general theory of Banach
spaces. In each case, the dual of C(K) <g>„ X is some sort of space of measures; in many
cases the precise determination of just which measures arise is not clearly understood. For
instance, the precise nature of which measures represent operators (all of which are weakly
compact) from a C{K) to an abstract L-space is unknown. Similarly, the determination of
which operators from C{K) to X have absolutely summing adjoints is open.

444
J. Diestel and J. Swart
References
[1] E.M. Alfsen. Compact Convex Sets and Boundary Integrals. Ergebn. Math. Grenzgeb.. B. 57. Springer.
New York A971).
[2] R. Anantharaman and J. Diestel. Sequences in the range of a vector measure. Comment. Math. Prace Mat.
30B) A991) 221-235.
[3] J. Arias de Reyna, J. Diestel. V. Lomonosov and L. Rodriguez-Piazza. Some obsenations about the space
of weakly continuous functions from a compact space into a Banach space, Quaestiones Math. 15 D)
A992L15^425.
[4] S. Banach. Theorie des Operations Lineaires. Monografje Matematyczne. Vol. 1 A932).
[5] S. Banach, The Lebesgue integral in abstract spaces. Note II in S. Saks' "Theory of the Integral", Dover
Publications A964).
[6] S. Banach and S. Mazur, Zur Theorie der linearen Dimension. Studia Math. 4 A933) 100-112.
[7] C. Bandt, Metric invariance ofHaar measure, Proc. Amer. Math. Soc. 87 A) A983) 65-69.
[8] R.G. Bartle, N. Dunford and J. Schwartz. Weak compactness and vector measures, Canad. J. Math. 7
A955J89-305.
[9] C. Bessaga and A. Petczynski, On bases and unconditional convergence of series in Banach spaces, Studia
Math. 17A958) 151-164.
[10] С Bessaga and A. Petczynski, On extreme points in separable conjugate spaces, Israel J. Math. 4 A966)
262-264.
[11] E. Bishop and K. de Leeuw, The representations of linear functionals by measures on sets of extreme
points, Ann. Inst. Fourier Grenoble 9 A959) 305-331.
[12] FF. Bonsall, On the representation of points of a convex set. J. London Math. Soc. 38 A963) 332-334.
[13] J. Bourgain, On the Dunford-Pettis property. Proc. Amer Math. Soc. 81 B) A981) 265-272.
[14] J. Bourgain and M. Taiagrand, Compacite extremale. Proc. Amer Math. Soc. 80 A) A980) 68-70.
[15] R.D. Bourgin, Geometric Aspects of Convex Sets with the Radon-Nikodym Property. Springer, Berlin
A983).
[16] J.K. Brooks and P.W. Lewis, Linear operators and vector measures. II. Math. Z. 144 A) A975) 45-53.
[17] P. Cembranos, Una caracterizacion de cuando C(K, E) es un espacio de Grothendieck para espacios ?
reflexives, Actas de las VIII Jomades Luso Espanholas Mathematica, Univ de Coimbra A979-1982).
[18] P. Cembranos, C(K, E) contains a complemented copy ofcu, Proc. Amer. Math. Soc. 91 D) A984) 556-
558.
[19] P. Cembranos, N.J. Kalton, E. Saab and P. Saab, Pelczynskis property (V) on C(i2, E) spaces. Math.
Ann. 271A) A985) 91-97.
[20] G. Choquet, Lectures on Analysis. Vol. I: Integration and Topological Vector Spaces, W. A. Benjamin, New
York A969).
[21] G. Choquet, Lectures on Analysis. Vol. II: Representation Theory. W.A. Benjamin, New York A969).
[22] G. Choquet, Lectures on Analysis. Vol. Ill: Infinite Dimensional Measures and Problem Solutions, W.A.
Benjamin, New York A969).
[23] G. Choquet and P.-A. Meyer, Existence et unicite des representations inte'grales dans les convexes
compacts quelconques, Ann. Inst. Fourier Grenoble 13 A963) 139-154.
[24] M.D. Contreras and S. Diaz, C(K. A) and C(K. H^) have the Dimford-Pettis property. Proc. Amer.
Math. Soc. 124 A1) A996) 3413-3416.
[25] L. de Branges, The Stone-Weierstrass theorem. Proc. Amer. Math. Soc. 10 A959) 822-824.
[26] A. Defant and K. Floret, Tensor Norms and Operator Ideals. North-Holland, Amsterdam A993).
[27] S. Diaz, A. Fernandez, M. Florencio and P.J. Paul, Complemented copies of cq in the space of Pettis
integrable functions, Quaestiones Math. 16 A) A993) 61-66.
[28] J. Diestel, The Radon-Nikodym property and the coincidence of integral and nuclear operators. Rev.
Roumaine Math. Pures Appl. 17 A972) 1611-1620.
[29] J. Diestel, J.H. Fourie and J. Swart, The metric theory of tensor products, Grothendieck s Resume revisited
B001), in preparation.
[30] J. Diestel, H. Jarchow and A. Tonge. Absolutely Summing Operators, Cambridge University Press.
Cambridge A995).

The Riesz Theorem
445
[31] J. Diestel, W.M. Ruess and W. Schachermayer, On weak compactness in ?.'(?, X). Proc. Amer. Math.
Soc. 118 B) A993) 447^453.
[32] J. Diestel and C.J. Seifert, The Banach-Saks ideal. I Operators acting on C(Q), Comment. Math. 1 A978)
109-118. (Special issue dedicated to Wtadystaw Orlicz on the occasion of his seventy-fifth birthday.)
[33] J. Diestel and J.J. Uhl, Jr., Vector Measures, Math. Surveys. No. 15, Amer. Math. Soc., Providence, RI
A977). (With a foreword by B.J. Pettis.)
[34] N. Dinculeanu, Vector Measures, Pergamon Press. Oxford A967).
[35] I. Dobrakov, On integration in Banach spaces. VIII. Polymeasures, Czechoslovak Math. J. 37A12) C)
A987L87-506.
[36] I. Dobrakov, On extension of vector polymeasures. Czechoslovak Math. J. 38A13) A) A988) 88-94.
[37] I. Dobrakov, On integration in Banach spaces. IX, X. Integration with respect to polymeasures,
Czechoslovak Math. J. 38A13) D) A988) 589-601. 713-725.
[38] I. Dobrakov, On integration in Banach spaces. XI. Integration with respect to polymeasures, Czechoslovak
Math. J. 40A15) A) A990) 8-24.
[39] I. Dobrakov, On integration in Banach spaces. XII. Integration with respect to polymeasures,
Czechoslovak Math. J. 40A15) C) A990) 424-440.
[40] I. Dobrakov. On integration in Banach spaces. XIII. Integration with respect to polymeasures,
Czechoslovak Math. J. 40A15) D) A990) 566-582.
[41] I. Dobrakov, On extension of vector polymeasures. 11. Math. Slovaca 45 D) A995) 377-380.
[42] P.N. Dowling, Complemented copies ofc() in vector valued Hardy spaces, Proc. Amer. Math. Soc. 107 A)
A989J51-254.
[43] G.A. Edgar, A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 A975) 354-358.
[44] J. Elton, Extremely weakly unconditionally convergent series. Israel J. Math. 40 C^4) A982) 255-258.
[45] G. Emmanuele. On complemented copies o/c(> in U'x. 1 < ? < ос. Proc. Amer. Math. Soc. 104 C) A988)
785-786.
[46] EJ. Freniche, Barrelledness of the space of vector valued and simple functions. Math. Ann. 267 D) A984)
479^486.
[47] T.W. Gamelin. Uniform Algebras. Prentice-Hall. Englewood Cliffs. NJ A969).
[48] D.J.H. Garling and Y. Gordon, Relations between some constants associated with finite dimensional
Banach spaces. Israel J. Math. 9 A971) 346-361.
[49] J. Gil de Lamadrid, Measures and tensors. Trans. Amer. Math. Soc. 114 A965) 98-121.
[50] J. Gil de Lamadrid, Measures and tensorv II. Canad. J. Math. 18 A966) 762-793.
[51] C.C. Graham and O.C. McGehee. Essays in Commutative Harmonic Analysis. Springer, New York A979).
[52] A. Grothendieck, Siirles applications lineaires faiblement compactes d'espaces du type C(K). Canadian
J. Math. 5A953) 129-173.
[53] A. Grothendieck, Resume de la theorie metrique desproduits tensoriels topologiqnes, Bol. Soc. Mat. Sao
Paulo 8 A953/1956) 1-79.
[54] A. Grothendieck, Produits tensoriels topologiqnes et espaces nucle'aires. Mem. Amer. Math. Soc.. No. 16
A955) 140.
[55] J. Hagler and W.B. Johnson, On Banach spaces whose dual balls are not weak* sequentially compact,
Israel J. Math. 28 D) A977) 325-330.
[56] W. Hensgen, A simple proof of Singer's representation theorem. Proc. Amer. Math. Soc. 124 A0) A996)
3211-3212.
[57] B. Josefson, Weak sequential convergence in the dual of a Banach space does not imply norm convergence,
Ark. Mat. 13A975O9-89.
[58] M.I. Kadec' and M.G. Snobar, Certain functional s on the Minkowski compactum. Mat. Zametki 10A971)
453^457.
[59] S. Kakutani, Concrete representation of abstract [D-spaces and the mean ergodic theorem. Ann. ot Math.
BL2A941M23-537.
[60] S. Kakutani, Concrete representation of abstract (M)-spaces. (A characterization of the space of
continuous functions). Ann. of Math. B) 42 A941) 994-1024.
[61] S.S. Khurana. Barycenters. extreme points, and strongly extreme points. Math. Ann. 198 A972) 81-84.
[62] S.S. Khurana. Barycenters, pinnacle points, and denting points. Trans. Amer. Math. Soc. 180 A973) 497-
503.

446
J. Diestel and J. Swart
[63] H. Konig, Eigenvalue Distribution of Compact Operators. Birkhauser, Basel A986).
[64] P.W. Lewis and J.P. Ochoa. The range of a representing measure. Math. Proc. Cambridge Philos. Soc. 124
B) A998) 365-369.
[65] A. Lima, O. Nygaard and E. Oja. Isometric factorization of weakly compact operators and the
approximation property. Israel J. Math. 119 B000) 325-348.
[66] V. Lomonosov, A counterexample to the Bishop-Phelps theorem in complex spaces. Israel J. Math. 115
B000) 25-28.
[67] V. Lomonosov, The Bishop-Phelps theorem fails for uniform non-selfadjoint dual operator algebras, to
appear.
[68] B. Maurey, The'oremes de factorisation pour les ope'rateurs line'aires a valeurs dans les espaces L1'.
Asterisque, Vol. 11. Societe Mathematique de France, Paris A974). (With an English summary.)
[69] A. Nissenzweig, W* sequential convergence, Israel J. Math. 22 C^4) A975) 266-272.
[70] A. Petczynski, Projections in certain Banach spates. Studia Math. 19 A960) 209-228.
[71] A. Petczynski, Banach spaces on which every unconditionally converging operator is weakly compact.
Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 10 A962) 641-648.
[72] A. Petczynski. Banach Spaces of Analytic Functions and Absolutely Summing Operators. CBMS Regional
Conf. Series in Math.. Vol. 30, Amer. Math. Soc, Providence, RI A977).
[73] R.R. Phelps, Lectures on Choquet's Theorem. D. Van Nostrand, Princeton A966).
[74] A. Pietsch. Absolut p-summierende AbbiUhmgen in normierten Raumen. Studia Math. 28 A966/1967)
333-353.
[75] A. Pietsch, Eigenvalues and s-Numbers. Cambridge University Press. Cambridge A987).
[76] A. Pietsch and J. Wenzel, Orthonormal Systems and Banach Space Geometry. Cambridge University Press,
Cambridge A998).
[77] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces. Conference Board of the
Mathematical Sciences. Washington, DC A986).
[78] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press,
Cambridge A989).
[79] J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen. Sitzungsber. Akad. Wiss.
Wien 122A913) 1295-1438.
[80] J. Rainwater. Weak convergence of bounded sequences. Proc. Amer. Math. Soc. 14 A963) 999.
[81] N. Randrianantoanina. Pelczynski's property (V) on spaces of vector-valued functions. Colloq. Math. 71
A) A996) 63-78.
[82] N. Randrianantoanina, Some remarks on the Dunford-Pettis property. Rocky Mountain J. Math. 27 D)
A997I199-1213.
[83] N. Randrianantoanina and E. Saab, Weak compactness in the space of vector-valued measures of bounded
variation. Rocky Mountain J. Math. 24 B) A994) 681-688.
[84] F. Riesz, Sur les operationes fontionelles lineaires, C. R. Acad. Sci. Paris 149A909)974-977.
[85] F Riesz. Sur certains systemes singuliers d'e'quations integrates. Ann. Ecole Norm. Sup. C) 28 A911)
33-62.
[86] F. Riesz, Demonstration nouvelle dun theoreme concernment les operationes fonctionelles lineares. Ann.
Ecole Norm. Sup. C) 31 A914).
[87] H.P. Rosenthal. A characterization of Banach spaces containing /'. Proc. Nat. Acad. Sci. USA 71 A974)
2411-2413.
[88] R. Ryan. An introduction to tensor products of Banach spaces, to appear.
[89] P. Saab. Integral operators on spaces of continuous vector-valued functions. Proc. Amer. Math. Soc. Ill
D) A991) 1003-1013.
[90] P. Saab and B. Smith, Nuclear operators on spaces of continuous vector-valued functions. Glasgow Math.
J. 33B)A991J23-230.
[91] S. Saks. Integration in abstract metric spaces. Duke Math. J. 4 A938) 408^411.
[92] L. Schwartz, Produits tensoriels topologiques. d'espaces vectoriels topologiques. Espaces vectoriels
topologiques nucleaires. Applications. Seminaire Schwartz 1953-54. Faculte des Sciences de Paris A954).
[93] I. Singer. Linear functionals on the space of continuous mappings of a compact Hausdorff space into a
Banach space. Rev. Math. Pures Appl. 2 A957) 301-315.

The Riesz Theorem
447
[94] C.P. Stegall and J. R. Retherford. Fully nuclear and completely nuclear operators with applications to L\-
and Loc-spaces. Trans. Amer. Math. Soc. 163 A972) 457^492.
[95] R.C. Steinlage, On Haar measure in locally compact ?? spaces, Amer. J. Math. 97 A975) 291-307.
[96] E.L. Stout, The Theory of Uniform Algebras, Bogden & Quigley, Inc.. Tarrytown-on-Hudson, NY A971).
[97] С Swartz. Absolutely summing and dominated operators on spaces of vector-valued continuous functions.
Trans. Amer. Math. Soc. 179 A973) 123-131.
[98] M. Talagrand, La propriete de Dunford-Pettis dans C(K. E) et Ll(E), Israel J. Math. 44 D) A983)
317-321.
[99] N. Tomczak-Jaegenmann, Banach-Maiur Distances and Finite-Dimensional Operator Ideals. Longman
Scientific & Technical, Harlow A989).
[100] A.E. Jong, Nuclear mappings onC(X). Math. Ann. 194A971) 213-224.
[101] A.Ulger, Weak compactness in Li(?,X), Proc. Amer. Math. Soc. 113A)A991) 143-149.
[102] P. Wojtaszczyk, Banach Spaces for Analysts. Cambridge University Press. Cambridge A991).

CHAPTER 10
Stochastic Processes and Stochastic Integration
in Banach Spaces
James K. Brooks
University of Florida. Gainesville. FL 32611-2082. USA
Contents
Introduction , 451
1. Stochastic integration in Banach spaces , 451
Introduction 451
1.1. Notation and definitions , . . . , , 452
1.2. The stochastic measure , 453
1.3. The summability question , 454
1.4. The stochastic integral 458
1.5. Closures of subsets in 7fC(J[) 461
1.6. Completeness of the space L^ c(x) 4f>2
1.7. Stopping the stochastic integral 464
1.8. Convergence theorems in L^- G(X) 466
1.9. Weak completeness and compactness of L^ C(B. X) 467
1.10. Local summability and local integrability 468
I.I I. Ito's formula 469
1.12. Further extensions of the stochastic integral 477
2. Regularity and the Doob-Meyer decomposition of abstract quasimartingales 478
Introduction 478
2.1. Notation 479
2.2. The Doleans function 480
2.3. The mean variation of X and quasimartingales 480
2.4. Regularity of quasimartingales 481
2.5. Cadlag modification without RNP 493
2.6. The Doob-Meyer decomposition of quasimartingales 496
Appendix A 497
A.l. The Doleans function 497
A.2. Mean variation 498
A.3. Quasimartingales 499
A.4. Right continuous quasimartingales 500
References 501
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
449

This page intentionally left blank

Stochastic processes and stochastic integration in Banach spaces
451
Introduction
In this chapter, the following two topics are presented;
A) stochastic integration in Banach spaces;
B) regularity of Banach valued processes and the Doob-Meyer decomposition of these
processes.
In Section 1, the stochastic integral in Banach spaces is developed, with the aid of a
vector bilinear integral. Convergence theorems are given and applied to establish Ito's
formula - the essential tool used in stochastic calculus. Section 2 deals with regularity
problems. It is of the upmost importance to establish when a process (Yf)t^o is right
continuous. For example, if this regularity condition is absent, we have no guarantee that
Yj is measurable, when ? is a stopping time - a disastrous state of affairs! Moreover, as we
shall see, regularity is equivalent to the Doob-Meyer decomposition of the process, a very
useful property. We show that essentially a process is regular if and only if it is weakly
regular. To establish these results, quasimartingale theory (in Banach spaces) is needed;
some of the background is given in the Appendix of Section 2, The material in this chapter
is based on a series of papers by Brooks and Dinculeanu (see references).
If the reader is familiar with classical stochastic analysis, this chapter is a quick
tour of the machinery used in the infinite-dimensional case. For the reader lacking this
background, the presentation can serve as a (very!) brief introduction to some aspects of
stochastic analysis - just assume that all Banach spaces are the scalar field, but even in
this case the stochastic measure is infinite dimensional, so a (hopefully enjoyable) detour
through vector bilinear integration theory is necessary. We recommend, for the first account
of Banach space processes. Metivier's impressive book A982), but be warned that at that
time even the regularity of Banach valued martingales was not known. (It was proved in
Brooks and Dinculeanu A987).) In Dellacherie and Meyer A978, p. 65), the existence
theorems for regular versions of real-valued supermartingales are regarded as "possibly
the most important single result in the whole theory of continuous parameter stochastic
processes".
1. Stochastic integration in Banach spaces
Introduction
Pellaumail A973) had the wonderful idea of representing a stochastic integral /HdX
as genuine integral J ? dm, where m is a "stochastic measure" on the ?-algebra of
predictable subsets of R+ ? ??, where (?. ?. ?) is the underlying probability space.
The measure m is L' -valued, that is. it is an infinite-dimensional vector measure. Due
to the lack of a satisfactory integration theory, his goal could not be completed - even
the existence of a cadlag modification of the process (j'0 ? dX), was not established.
Kussmaul A977), using Pellaumail's approach, was able to define a measure-theoretic
stochastic integral for real valued processes. Various definitions of a stochastic integral
have been given (Yor, 1973. 1974; Kussmaul. 1978; Gravereaux and Pellaumail, 1974;
Metivier, 1982; Metivier and Pellaumail, 1980; Pratelli, 1988), however either the Banach

452
J. K. Brooks
spaces were too restrictive, or the construction did not yield the convergence theorems
necessary for a full development of stochastic integration theory.
A general theory of the stochastic integral in Banach spaces was developed in Brooks
and Dinculeanu A991), using the bilinear vector integration (Brooks and Dinculeanu,
1976) with respect to a Banach valued measure with finite semivariation. This theory seems
to be tailor-made for applications to the stochastic integral. Let W:R+xi2-> F be a
predictable process, and assume Х:Е+хй-> ? с L(F. G) is a cadlag, adapted process,
with X, e LE, where F, G and ? are Banach spaces. The stochastic integral fQ ?dX
is defined to be j \[Q.t\Hdlx, where ?? is a stochastic measure on V, the predictable
?-algebra, with values in L[E с L(F. LlG). We call X summable if ??, defined on the
ring ??, generated by the predictable rectangles, can be extended to a ?-additive measure
on V, and the extension has finite semivariation.
The space of ? ? -integrable processes is a Lebesgue-type space with a seminorm, and
the space is complete. Unlike classical Lebesgue spaces, the simple processes are not
necessarily dense; this causes considerable difficulty since most properties of the integral
are usually proved first for simple functions and then extended by continuity. To overcome
this difficulty, a Lebesgue-type theorem is proved, which ensures the convergence of the
integrals (rather then convergence in the Lebesgue space itself). The Vitali and Lebesgue
convergence theorems are established; weak compactness criteria and weak convergence
theorems for stochastic integrals are presented, and are new even for the scalar case.
The reader is referred to Brooks and Dinculeanu A991) for a quick treatment of the
bilinear integration theory. The book Dinculeanu B000) is for the most part based on the
works of Brooks and Dinculeanu, and covers among other topics the bilinear integration
theory, stochastic integration in Banach spaces, regularity of processes, strong additivity,
weak compactness, Ito's formula, etc. - see the references to the joint work listed in this
chapter.
1.1. Notation and definitions
The spaces E, F, G will denote Banach spaces. The dual space of ? is denoted by ?". We
write ? с L(F, G) to mean that ? is isometrically embedded in L(F, G). the bounded
linear operators from F to G.
Let (?, ?, P) be a probability space. LPE = L''(i2. ?. P), 1 < ? < oo, is the space of
.^-measurable, ?-valued functions such that / \f\p dP < oo. If ? = oo, L^ denotes the
space of ?-valued essentially bounded .^-measurable functions.
We say that (Et)t-^Q, an increasing family of sub-?-algebras of ?, is a filtration. Set
?,+ to be f][Es: s > t}; the family is right continuous if ?,+ = ?, for each t. A process
O't)t^o is adapted to (?,) if Y, is XV-measurable for each t. We say that the filtration
satisfies the usual conditions if it is complete, that is (?. ?. ?) is complete and all the
P-negligible sets belong to ??, and if the filtration is right continuous. We shall always
assume that the filtration satisfies the usual conditions. A process (Yt) is said to be cadlag
if it is right continuous on [0, oof with finite left hand limits on ]0, oof. Let (Xt) and
(Yt) be two processes with values in the same space. We say that (Yt) is a modification
of (Xt) if Xt = Yt a.s. for each t ^ 0. We say (Xt) and (Yt) are indistinguishable if

Stochastic processes and stochastic integration in Banach spaces
453
for almost all ? e ?, Xt(o>) = Yt(a>). If А С ?* ? ? (R+ = [0, oof) and the process
Aд)ь t ^ 0 is indistinguishable from the zero process 0, we call A evanescent - that
means that the projection of A on ? is contained in a P-negligible set. Thus (Xt) and
(Yt) are indistinguishable if and only if the set {(t. ?): Xt(co) ? Yt{u>)) is evanescent.
We shall always assume in Section 1 that (X,) is a process; X :R+ ? ? —* ? с L(F, G)
and X is cadlag, adapted, with X, e i,fp for each t, where 1 < ? < oo. The terminology of
Dellacherie and Meyer A978) will be used.
We shall denote by ?? the ring of subsets of R+ ? ? generated by the predictable
rectangles [Од], with A e Eq and ]s, t] ? A, with 0 < s < t < oo and A e Х\. The
? -algebra V of predictable sets is generated by TZ.
1.2. The stochastic measure
Now we shall define the crucial concept used in the construction of the stochastic integral.
The stochastic measure Ix:Tl-> LPE. where ? С L(F.G) and X : E+ ? ? -> ? is first
defined on the predictable rectangles by
Ix([0A]) =lAX0 and /x(]i.r]xA) = lA(X,-XJ.
and then we extend it in an additive fashion to ??. Frequently we write / in place of ? ?. We
consider LPE с L(F, L?) and we define the semivariation of / with respect to (F, L?),
denoted by If.g, on R by
If.g(A) = sup\^2 I (Ai)xj
for A e R.
where the supremum is extended over all finite families of .v, e F\ and disjoint sets A, from
?? contained in A. If / can be extended to V, extend the semivariation to V in an analogous
fashion. We say that / has finite semivariation on TZ if it is finite on each set AelZ.
Definition 1.1. We say that X is /?-summable relative to (F. G) if ?? has a ?-additive
L^-valued extension, which will be unique, to V, and ?? has finite semivariation on V
relative to (F, Lp). If ? = 1, simply say X is summable relative to (F.G). If we regard
? = L(R, E), and X is p-summable relative to (E. E). we say that X is p-summable.
Observe that X is p-summable if and only if ?? has a ?-additive extension to V,
since in this case ? ? in bounded in L? on V, which implies bounded semivariation
relative to (R, L?). If 1 </>'</>< oc, and if X is /?-summable relative to (F. G),
then it is p'-summable relative to (F.G). In particular, p-summable relative to (F, G)
implies summable relative to (F. G), hence most results proved for summable processes
remain valid for p-summable processes. If X is p-summable relative to (F,G), then
X is summable relative to (R, E). If X is a process with integrable variation, then X
is p-summable relative to any pair (F,G) such that ? с L(F,G). If ? and G are

454
J. K. Brooks
Hubert spaces, then any square integrable martingale X with values in ? с L(F.G) is
2-summable relative to (F, G).
Note that X is p-summable relative to (F. G) if and only if ? ? has a ? -additive extension
to V and /? has bounded semivariation on H. rather than on V. Hence the problem of
summability reduces to a great extent to that of obtaining a ? -additive extension of ? ? from
?? to 73. Once the summability of X is assured, we can apply the theory of vector bilinear
integration theory (Brooks and Dinculeanu, 1976) and define the stochastic integral as the
bilinear integral with respect to vector measure ? ?.
Suppose now ? ? has been extended to V. It is convenient to extend it further to sets of
the form {00} ? A where A e ??. by setting /({00} ? A) = 0. Then V U ({00} ? ?^)
is the ?-algebra ^[0. 00] of predictable subsets ofR+ ?? on which / is still ?-additive,
and I(]S, T]) has the same value whether (S. T] is regarded as a subset of R+ ? ? or
R+ ? ?. Similar considerations hold for other types of predictable stochastic intervals,
and in particular for I{[T}) if ? is a predictable stopping time. The following theorem
extends the computation of / from predictable rectangles to stochastic intervals.
THEOREM 1.2. Assume that X is p-summable relative to (F.G) and regard Ix as the
unique extension of ? ? to V. Then
(a) There is a random variable, denoted by Xoc· belonging to Lh-, such that
???,-эсХ, = Хэс in LP. and Ix(]t.oo[ ? A) = l.A(X^ - X,) for A e ?,. If X
has a pointwise left limit X~^ then X-*.- = X^ a.s.
Consider now X extended at 00, by a representative ofX-^, and define X^- to be ??.
(b) For any stopping time T, we have ?? e Lp and ??([0. ?]) = Хт-
(c) If T is a predictable stopping time, then Xj- e Lpy and ??([0. ?)) = ??- and
??([?]) = ???.
(d) If S < 7 are stopping times, then Ix((S. T]) = Xj - X$. If S is predictable, then
IX([S, T]) = XT- Xs-· If both are predictable then IX([S. T) = XT- - X.s-
1.3. The summability question
Obviously the main question is given X :R+ ? ? -> ?, when is X /?-summable? The
answer to this question in the case ? ^> cq is unexpected: the mere boundedness of ??
on 1Z implies that X is p-summable relative to (R. E) and bounded semivariation on Tl,
relative to (F, L^) implies that X is /?-summable relative to (F. G). We state this main
result below.
THEOREM 1.3. Assume that ? ^> c(>. If ?? is bounded on ??, then X is p-summable
relative to (R, E). If Ix has bounded semivariation on ??, relative to (F, L?), then X
is p-summable relative to (F,G).
COROLLARY 1.4. Assume X is real valued and regardR С L(F. F). Then X is summable
relative to (F, F) if and only if ?? has bounded semivariation on 1Z relative to (F. LP).
What are the steps leading to a ?-additive extension of ? ? by just requiring boundedness.
if ? 2$ со? The first step is to establish a new extension theorem for vector measures

Stochastic processes and stochastic integration in Banach spaces
455
m : ?? -> V requiring a very weak hypothesis, that is, an extension theorem requiring only
?-additivity of zm, for ? belonging to a norming subset of V'. where V is Banach space.
This is necessary since we do not have a representation of (L^)'. but we can work with
the norming set of simple G'-valued functions. The extension is the following theorem
(Brooks and Dinculeanu, 1991).
THEOREM 1.5. Let m:TZ—*V be finitely additive, where V is a Banach space. Suppose
that ? С V is norming for V. Then conditions (a), (b) and (c) are equivalent. ?? is a ring
of subsets.
(a) m is strongly additive on ?? and for each ? e Z. zm is ?-additive on 1Z.
(b) m is strongly additive and ?-additive on TZ.
(c) m can be extended to ? ?-additive measure on ?(??) which we denote by ?.
(d) Assume V ~fi cq. Ifm is bounded on ?? and if zm is ?-additive on IZfor each ;e Z,
then m can be extended uniquely to ? ?-additive V-valued measure on ?.
(e) Assume V 2$ cq. Let (?, ?. ?) be a measure space and let m :TZ —*¦ ??(?) be
finitely additive and bounded. For each ? e Z, define the measure zm : K-> ?/(?)
by (zm)(A) = (m(A), z),for AelZ. If zm is ?-additive on TZfor each ? ? Z, then
m can be uniquely extended to a ?-additive measure m \ : ? —> Lv (?).
The next step is the following fundamental summability extension theorem (Brooks and
Dinculeanu, 1991).
THEOREM 1.6 (Summability extension theorem). If ? ^> со. then thefollowtng assertions
(l)-F) are equivalent. If ? is any general Banach space, then assertions B)-F) are
equivalent and A) implies B).
A) ?? can be extended to a ?-additive measure on V.
B) ?? is bounded on ??, the ring generated by the predictable rectangles in R+ ? ?.
Let ? С LqE, be any closed norming sub space for LPE, where l/p + l/q = I. For
g e Z, let G denote the process defined b\ G, = E(g/?,). XG is the real valued process
«X,,G,)),^>.
C) For each g e Z, the real measure (??, g) is bounded on TL.
D) For each g e Z, XG is a quasimartingale on ]0, oof.
E) For each g e Z, XG is a quasimartingale on ]0, oo].
F) For each g e Z, the measure (??. g) is ? additive and bounded on 1Z.
PROOF (Sketch). Some of the quasimartingale theory used here is given in Section 2
infra. We shall proceed in the following order: A) =>· B) 4» C) O· D) О E) О
F)=>A).
The implication A) =>· B) is evident since ?-additivity on a ?-algebra implies
boundedness. Also B) =>· C) is evident. To prove C) =>¦ B), note that for each AeK,
the linear functional g -» (IX(A), g) on ? is continuous. Since ? is norming for LPE, we
can embed LE с ?' isometrically. If we assume C), then
sup{|(/x(A),g}|: AeTZ) < oo for each g e Z.

456
J. K. Brooks
Then by the Banach-Steinhaus theorem we deduce that
sup{\lx(A)\p. А еП} < oo.
which is B).
Now prove C) o· D). Let g e L\, and consider the real measure {Ix,g} on ?? defined
by
f{lx(A),
(Ix,g)(A)= I (lx(A),g)dP forAeft.
We shall show (Ix,g) is bounded on 7? if and only if XG is a quasimartingale on ]0, oof.
To prove this, we first show that
(??^)(?) = ???(?), for A eft,
where ??0 is the Doleans function of the process XG. In fact, for В е ?? we have
{lx,g)([0B]) = f 1b(Xq, g)dP=f UXoGodP = ???([0?]).
For ]s, t] ? ?, where В е Es, we have
(Ix,g)((s,t]xB)={lB{(Xt-X>),g)dP
= f (X,,Gt)dP- [\xs.Gs)dP
Jb Jb
= ^xc{\s.t] ? ?).
Hence, (Ix, g) is bounded on A]0, oof, the ring of all finite unions of predictable rectangles
]s, t] ? A, where 0 < s < t < oo and A e Es, if and only if ??? is bounded on A]0, oof,
which is true if and only if XG is a quasimartingale on @, oo). It follows that 3 o· 4
since TZ=A[0]UA]0,oo[,(ix,g}= ??? on A]0, oof and /? is bounded on A[0], where
Д[0] = {0} ? ??.
We now show D) <?> E). Obviously E) =>· D). If D) holds, then from B) <?> C) <?> D)
proved above, we deduce that ?? is bounded on ??. Thus for g e ? we have
|X,G,|, = J \X,G,\dP < |g|,|X,|„ = |g|,|/x([0,r] ? ?)\?,
hence
sup|X,G,|i <|g|,sup{|/x(A)| : A e ft} < oo.
t '
Thus XG is a quasimartingale on [0, oo], which is E).

Stochastic processes and stochastic integration in Banach spaces
457
Next we show E) о F). The implication 6 =>· 5 is evident. Assume E) and let g e Z.
Then XG is a quasimartingale on ]0, oo] where (XG)^ = 0 by definition. For each n,
define the stopping time T„ = inffr: \X,\ > n). Then T„ / oo and \Xt\ < ? on [0. 7),]. At
this stage we do not know if Xjn belongs to l?, but since XG is a quasimartingale on
@, oo], we know Xt„Gt„ e L1 and
I^GrKnlCH^rn+I^GrJli,^)·
Since G is a uniformly integrable martingale, it follows that XT" GT" is a quasimartingale
of class (D) on ]0, oo], hence the corresponding measure д(ХС|7„ is ?-additive with
bounded variation on A]0, oo] (where A[0, oo] allows ]s, t] ? A, where t < oo), therefore
it can be extended to a ?-additive measure with bounded variation on V]0, oo]. Now for
each predictable rectangle ]s, t ] ? A, s < t < oo, A e Es, we have
^(Л-СO" (К ']хЛ)= ???(]?, П х А П [0. 7,,]).
therefore
и(ВIХС)т„ = ???{? П [0. ?,,]) for S e 7>]0. ??].
It follows that ?(?? is ?-additive on the ?-ring V]0, oo] П [0, 7),]; consequently д^с
is ?-additive on the ring В = Ui<„oc 7-?0· °°] n №, 7",,]. On the other hand, ??? is
bounded on A]0, oof since XG is a quasimartingale on ]0, oo], hence ??? has bounded
variation on A]0, oo]. It follows that ??? is ?-additive and has bounded variation on the
ring В П A]0, oo], which generates V]0, oo); hence ??? can be extended to a ?-additive
measure with bounded variation on V]0, oo]. Since (??. g) = ??? on A]0. oof, it follows
that (Ix,g) is bounded on A]0, oo]. Since (Ix.g) is bounded and ?-additive on A[0], it
follows that {Ix,g) is bounded and ?-additive on 7? = Д]0. oo]; hence F) holds.
To prove F) =>· A), assume ? ^ cq. If we assume F), then (Ix,g) is bounded and
?-additive on ?? for g e Z. By Theorem 1.5, /? can be extended to a ?-additive measure
on ? = ?(??), and this is A). D
Since Hilbert-space valued stochastic processes are so important, let us turn our attention
to this setting and see how the summability theorem can be applied to these processes.
When one examines the Hilbert space case it is easy to understand how important the
summability theorem is. Without the geometry of Hilbert space, the above theorem allows
us to replace the condition that we have a square integrable martingale at our disposal.
The summable processes in this theory play the role of the square integrable martingales
in the classical theory. It turns out that every Hilbert valued square integrable martingale is
summable. One can show by an example that for any infinite-dimensional Banach space E,
there is an ? valued summable process which is not even a semimartingale.
Suppose ? and G are Hilbert spaces and F is a Banach space such that ? с L(F,G).
For example, ? = L(R, ?), ? = L(E. R) or ? с L(G. ? §>ws G), where HS indicates
that the Hilbert-Schmidt norm is used on ? <g> G. The main result in this setting is that

458
J.K. Brooks
any ?-valued, square integrable martingale ? is 2-summable relative to any embedding
ECL(F,G).
We say that ? is a square integrable martingale if ? : R+ ? ? —*¦ ?. ?, e LE for each t
and sup, \M,\i < oo; in other words, there exists a random variable ?? e LE such that
M, = E(Mx\E,),t^0.
Our task is to show that Im -V -> ^^- is the ?-additive extension of Im on 7?. having
finite semivariationrelative to (F, LG).
To show this, we begin with properties of Im .
(A) If ? and N are square integrable martingales taking values in E, then if A
and В are disjoint sets from 4 and if ?, ? e F. then /m(A)±//v(S) in L| and
Im(A)x 1 1ы(В)у mL2G.
(B) If ? is a square integrable martingale, then Im is bounded on ??.
Indication of proof. Let A e Tl and let A be a disjoint union of predictable rectangles
{0} ? A0, A0 e Tq and ]s,, t, ] ? A,, for A, e ?,. , 1 < / < и. Let 7 be the max of the i,
and set В = [0, 7] ? ?. Since /M(A)±/M(S - A) by (A), we have
\Im(B)\\ = \Im(B-A)\; + \Im(A)\].
which implies
\Im(A)\~2 < \Mt\j <sup|M,|5 < oo.
t
(C) ? ? can be extended to a ? -additive measure on V by the Summability Theorem,
since M, being a Hubert space, does not contain a copy of со·
(D) ? ? has finite semivariation relative to (F. LG) for any embedding ? с L(F,G).
a
1.4. 77?e stochastic integral
Now that we have our vector measure ?? in hand, we can apply the vector bilinear
integration of Brooks and Dinculeanu A976) (see also Diculeanu B000) and Dinculeanu
B002), Chapter 8 in this Handbook, for details regarding this theory). We shall freely use
this theory in the sequel.
We shall assume in this section that X:R+ ? ? ^ E. where ? с L(F, G) is a cadlag,
adapted p-summable relative to (F,G). We consider ?? as extended to V[0, oo], with
/x({oo} x A)=OforAer:=V^o^·
Our vector measure space is (R+ ? ?. V. ??) where ?? : V -> LE С L(F. LG). If
l/p + l/q = 1, and the space ? с LG, which contains the simple functions is norming
for L?.
For convenience, we let m denote ??. Define the measure m- = (??):: V[0, oo] —> F'
for ? e ? by
(y, m-(A)) = (m(A)y, z)= I [lx(A)(co)y. ?(?))??(?).

Stochastic processes and stochastic integration in Banach spaces
459
Then, by the definition of the semivariation m = Ix, we have
(I'^F.Lp=ffiF.L''. :=sup{|iw:|: ;eZ. |c|i;<l}.
where \m-\ denotes the total variation of m -. If ? is fixed, write If.,? to be the of all positive
?-additive measures |w-|, ? e ? and |;|(/ < 1.
For any Banach space D, we denote by Td(If.g) or To(IF Lr) the space of all
predictable processes ? :R+ ? ? ->¦ D such that
< 1 \ < oo.
iF.G(H)=mh. L^H) = supl j\H\d\m:\: \z\4 < 1
even if Я is defined on R+ ? ?, since the value is the same. We remind the reader
that TdUf.g) is a complete vector space with seminorm Iy.c an(J is contained in the
intersection of all the spaces 0D(\m:\), for ; e Z. For any set С С Tody.с), we denote
the closure of С in this space by Td(C. If.g)-
Obviously, the most important case is when D = F, then we can define j1\о.цН ??? :=
fQ ?dX e LPC for all t ^ 0, and we also demand that the process (L· ,, ?dI),^o has
a cadlag modification, for ? belonging to a subspace of Ту {Aу,о)У. this will be our
Lebesgue space of functions denoted by Ц, G(X). the space in which our stochastic
integration theory lies. But, a lot of work must be done before we reach this stage.
Now we begin work on constructing the stochastic integral. Let ? с (L^)' be norming
for Lpc and let ? e TF.C{X) := Ff(Ux)f.g)· Then since ? e L\-(\m:\) for each ; e Z,
we have that the mapping
/"¦
m ? dm- A)
is a continuous linear functional on Z.
Definition 1.7. Denote by f Hdlx the element of Z' defined by A).
Note f Hdlxe (LPG)" if ? = L4C..
Also note, if ? с Z|, then / ? dlx is a restriction to ? of the element defined by Z|.
We now present a useful type of Lebesgue Theorem for pointwise convergence H" -> ?
of elements in Tf g(^) relating to the convergence of J H"dlx -> f Hdlx, when
fH"dIxeL'Gc(L'G)".
THEOREM 1.8. Let (H")n^0 be a sequence of elements from Tf.g(x) s"ch that \H"\ <
\H°\ for each ? and H" -> ? pointwise. Assume that f H" dlx e L!G for each ? ^ 1,
and f H" d]x —* j ? dlx converges pointwise on ?, weakly in G.
Then f Hdlx eLG and f H"dlx -> f Hdlx in the a(L!G.LqG.) topology of L'G, as
well as pointwise, weakly in G. If (f H" dlx)n converges pointwise strongly in G, then
f H" dlx -> / ? d1x strongly in LG.

460
? ?. Brooks
Indication of proof. Since |//K \H°\, we deduce that ? e Ту.с(Х). Let ? e i/JL.
Apply Lebesgue's theorem in space Lj. (\m-\), and deduce that H" -> Я in L^(|m.-|), and
thus J H" dm- ->¦ f ? dm-, that is
/»¦
'rf/? K-),c(-))J -> ( / Hdlx,z
? h e LX(P), then hz e L^,, so we obtain
E\h()((j H,dIxX),z{)^^n Hd1x,hz\
Thus the sequence (((/ Hn dlx)(),z{))) is weakly Cauchy in Ll(P), and consequently
the indefinite integrals of the above sequence are uniformly absolutely continuous with
respect to P. Let ?(?) be the weak limit in G of (J Hndlx){w). The Vitali
Convergence theorem implies that (?(-),?(-)) e Ll(P) and ((/ H" dlx)(-),z(-)) converges
to (Ф(),г(·)) in L'(P), hence the expectations ?(((/H"dlx)(-),z(-))) converge to
?({?,?)). Since ? e ?-g. is arbitrary, we see that ? e LG. Consequently (Ф,г) =
?((*(¦), г(-))) = (f Hdlx.z), hence /ffrf/x = ? e LG and jH"d1x -»· (fHdIx)
pointwise, weakly in G. From the above, it follows that f H"dlx -*¦ f Hdlx in the
a(Lg, L^,) topology of Lji; in particular in the ?(?^, L^,) topology of L^. This in turn
implies that the indefinite integrals are uniformly ?-additive on ?. If (/ H"dlx)(w) ->
?(?) strongly, the Vitali convergence theorem for LG implies f H''dlx -> J Hdlx
mOG. ' D
Now we are in a position to describe our space of integrable functions ? for which we
can define the stochastic integral. First of all, it can be shown that if X is p-summable
relative to (F, G) and ? e Ty.G(X), then if / \iQj]HdIx belongs to Ll'E for each t > 0,
the process
t н»
il[0.,]HdIx
is adapted, that is f 1[0j\Hdlxis ?1,-measurable, for each t. But this process may not
be cadlag, which is necessary for any viable stochastic theory. So we make the following
definition.
Definition 1.9. We denote by L\ G(X) the set of processes ? e JFf.cih) satisfying
the following two conditions:
A) j\[0j]Hd1x eL?, for each? > 0;
B) the process (f lw.t]H dlx),^t) has a cadlag modification.

Stochastic processes and stochastic integration in Banach spaces
461
We say the processes ? e LlF G(X) are integrable with respect to X, and any cadlag
modification of the process in B) is called the stochastic integral of ? with respect to
X, denoted by ? ? X or J HdX:
(? ??),(?)=( f HdXj (?)= ( I 1\(,.,\H dlx )(?), a.s.
If ? = oo, set (? ? X)x = f Hdlx.
We note that the stochastic integral is defined up to an evanescent process. If ? is an
7^-step process taking values in F, then
(? ? ?),(?) = / Hs(w)dXs(w),
that is ? e LFG(X) and its stochastic integral is a Stieltjes integral. Also if X has
finite variation |X| and if X is p-summable relative to (F, G), and if ? belongs to
Tf.g(X) and satisfies f[Q r]\Hs(w)\d\X\s(w) < oo for each t and ?, then Я е Lf.G(X)
and the stochastic integral (? ? ^),(?) equals the pointwise integral /[() ,| Hs(tu)dXs{w).
Furthermore, if ? e LF G, then for every t e [0, oo], we have
(//xX),_eL? and (HxX),-= Hd1x.
•/I0.il
In particular, (Я ? ?)-?- = (? ? ?)^ = f ?dlx and the mapping f -> (Я ? ?), is
cadlag in L^,.
1.5. Closures of subsets in T(-\g(X)
As before, if С С TF.G{X), then Ty.c(C, X) denotes the closure of С in Т/.,С(Х). If
С consists of processes ? such that f Hdlx e LG, then by continuity of the integral,
we have f Hdlx e L'G for all ? e TF.G(C, X)- It turns out that if С С L\G{X), then
Ff.g{C,X)cLfg{X).
In this case, we write LF G(C,X) = Tf-.c(C, X).
Here are some spaces С of particular interest:
A) The space BF of bounded predictable processes with values in F;
B) The space SFD) (respectively SF(V)) of simple, F-valued processes over К
(respectively V);
C) The space ? of predictable, elementary, F-valued processes of the form
// = //0l|o)+ ? ?*\?,.?,-^ B)

462
J. K. Brooks
where (?,) ? ^,^„+i is an increasing family of stopping times with 7b = 0 and //, is
bounded and ??; measurable, for each /'.
One can show that SfCR) and ?f are contained in LFG(X). More generally, if the
family of measures If.g is uniformly ?-additive then
Тг.сE{П),Х)=Ту.с(Ву,Х) = 1\,с{Вь,Х),
and this is always true if F is replaced by ? and G is replaced by E. If X has integrable
variation, or if X is a square integrable martingale in a Hubert space E, then
J>.c(S(ft), *) = ?'-.c<X)·
The need for the family of measures /f.c to be uniformly ?-additive arises from the fact
that Egorov's theorem is not valid, in general, without this condition - a condition that
markedly differs from the classical setting. It turns out that if G ^> cq or if G is weakly
sequentially complete, then /f.c is in fact uniformly ?-additive.
We note that it can be shown that if ? is an elementary process of the form listed in B),
then ? e LFG(X) and (? ? ?), can be computed pathwise:
(H xX), = HQXQ+ ? //.-(??-,.,?,-??-?,).
1.6. Completeness of the space L h G (X)
It is of the upmost importance that our Lebesgue space L\. c(X) of functions ? integrable
with respect to X is complete. It is the cadlag aspect which is the difficult point, and this is
handled in the next theorem, which will imply that LF G(X) is in fact complete.
THEOREM 1.10. Let (//") be a sequence from L\- g(X) and assume that H" -> ? in
Ff.g(X)- Then we have ? e LF G and for every t we have (//" ? ?), -> (? ? X), in
LG and there exists a subsequence (nr) such that (H"r ? X), -> (? ? X), a.s. as г -> oo,
uniformly on every bounded time interval.
Indication of proof. Note, to get ? e L]F G(X). we use the uniformly convergent
subsequence to ensure that we have an adapted cadlag modification of (/ l[o.,|# dlx),.
(//") is a Cauchy sequence in LF G(X) converging in Ту.в(Х) to H. By passing to
a subsequence, if necessary, we can assume that /V.c(#" - Hn+[) ^ 4~" for each n; let
Z" = //" ? X, and for ?? > 0, define the stopping time
t/„=inf{r: \Z* -?;,+1|>2-"}??().

Stochastic processes and stochastic integration in Banach spaces
463
Let G„ —{Un< ?()}¦ For each stopping time v, we have Z" = / l[o.v\H"dIx, hence
E(\Znv-Z4, + l\) = E(\ f (H"-H"+])dIx)
\\J[0.v\ /
(H"-Hn+l)dIx
I/
./[Од·]
\f (//" - H"A
\JU).v\
l)dlx
L'c,
<;/>.с(я"-я"+')^4-".
/[Од
In particular, for ? = Un, we have
E(\Z"u„-zauV\)^-n-
We assert
P(G„)^2nE(\ZnUn-Z»+l\)^2-n.
To see this, note if ? e G„, then [/„(?) < ?<>; take a sequence ?* \ ?/„(?), with ^ < ??
such that |??(?) - ?"+1(?)| > 2"", for each it. Then, by the right continuity of the Z',
conclude that |?^(?) - Z?+' (?)| > 2"", and thus
E(\Z4jn-Z'^\)^2-"P(Gn),
and the desired inequality follows.
NowletGo = limsupG,,.ThenP(Go) = 0. ????? Go, there is a ? such that if и ^ ?,
we have ? ? G„, hence [/„(?) = ??· Thus
sup|z;'M-z;,+1|s;2-",
'<?
and we have the desired uniform Cauchy condition of (Z"(a>)) for t < to-
Define ?,(?) = lim„ ?"(?), ? ? Go. This process is adapted and cadlag and
?/ — Zf ? ? ^2
-n + l
It follows that for t < t0 we have Z, e L? and Z" -> Z, in L?. On the other hand,
1[0.,)Я" -»¦ 1[0.,)Я in TF.G{X), hence Z'' -»· f[0j]HdIx in (?.«,)'. Thus
/ ¦
Hd1x = Z, eL1'
Since ? is cadlag, we deduce that Я e L\ g(X) where we extend Z, consistently, for
r e [0, oof, and we also have (? ? ?), = ?, for each t. Thus LlFG(X) is complete, and

464
J. K. Brooks
since ?? was arbitrary, we have the desired uniform convergence for a suitable subsequence
on each bounded time interval, ?
Corollary 1,11, LF G(X) is complete,
COROLLARY 1.12, If IF,G is uniformly ? -additive (for example, if G ^> со or G is weakly-
complete), then LFG(X) contains all the F-valued, bounded, predictable processes (in
particular, if F = Й).
In fact, in this case Zf, the space of elementary processes is dense in Ff.g(Bf, X)·
Since EF с LrG(X), we have Ty.c(Br\ X) С L\G(X), One can also show that if X
has integrable variation or if ? and G are Hubert spaces, then LF G(X) = Рг'.вФ, X) =
Ff.g(X).
1.7. Stopping the stochastic integral
We present in this section a few results to show that the usual rules to stopping apply to
LlFG(X),
THEOREM 1,13, Let ? e L[FG(X) and let ? be a stopping time. Then 1[о.т\Н е
LlFG(X) and
A[0tj#) xX = (H xX)T,
If ? is predictable, then 1[о.;г[# е LF G(X) and
(ll0JlH)xX = (H xX)T~,
COROLLARY 1,14, For every stopping time T, we have l[o.T] x X = X ¦ If ? is
predictable, then l[o.r[ x X = XT~.
THEOREM 1.15. Let S ^T be stopping times and assume that either
(a) h : ? -> ? is bounded, ? ^-measurable and ? e LF G (X) or
(b) h; ? —> F is bounded, ?'^-measurable and ? e L^ E(X)·
Then
(i) 1M 7-)Я and hl]s.T]H are integrable with respect to X and
(hl]S.T)H) ? X = h[(l]S.TiH) ? X],
(ii) If S is predictable and h is ?<;--measurable, then l[s 7-)// and h\[s.T\H are
integrable with respect to X and
(hl[S.T]H) ? X = h[(l[S.T]H) ? X].

Stochastic processes and stochastic integration in Banach spaces 465
The next theorem gives a more complete description of the properties of X for a
p-summable process X,
THEOREM 1,16, Let ? be a stopping time,
(a) XT is p-summable relative to (F,G) and
/хг(А) = /х([0,Г]ПА), /orAe-PfO, oo],
(a') If ? is a predictable, then XT~ is p-summable relative to (F, G) and we have
???- (A) = Ix ([0, T] П A), for A e V[0, oo].
(b) For every predictable, F-valued processes H, we have
s\arF L^IXT(H) = s\aiF LrIx(l[0.T]H),
(b') If ? is predictable, then
s\aiF L?lXT-(H)=s\aiF L?lx(llo.T]H).
(c) We have ? e TF.G(XT) if and only if\[i).T]H e TF.G(X) and in this case
? Hdlxr= j ll0,T]HdIx.
(c') If ? is predictable, then ? e Ту,с{Хт~) if and only ifl[o.T[H e Tf.g(X) and in
this case
f HdIXT=fl[0.T[HdIx.
(d) We have ? e LF G(XT) if and only ifl{o.T]H e LF G(X), and in this case
? ? XT = (ll0.T]H) xX.
IfHe LFG(X), then ? e LFG{XT), 1?0.7?^ e lf.gW> and
(? ? X)T = ? ? XT = (Iio.t-)//) ? X.
(d') Assume ? is predictable. We have ? e L\ g(Xt~) if and only if l[o.T[H e
LF G(X), and in this case
? xXT~ = (l[l).T)H) xX.

466
J. K. Brooks
(e) If the set of measures (??) F L·/, is uniformly ? -additive, then so is (Ixi) FL4,
(e') If ? is predictable and if (? ?) h- L4 is uniformly ?-additive, then so is Aхт-)у L4 ,
Recall, that if (Yt) is a process, AY, = Y, — Y,-, assuming У,_ exists,
THEOREM 1,17, For any process ? e L\ - g(X), we have
A(H ? X) = HAX.
1.8. Convergence theorems in L^ G(X)
To be a genuine Lebesgue space, the Vitali theorem and the Lebesgue convergence
theorems should hold. These theorems are now stated for LF G(X).
THEOREM 1.18 (Vitali). Let (//") be a sequence from J7f.g(X) and let ? be an F-valued
predictable process. Assume that
(i) 1F.G(//" 1A) -> 0 as IF.G(A) -> 0, uniformly in ?
and that any one of the conditions (ii) or (iii) below is true;
(ii) H" -> ? in Ifg-measure;
(iii) H" -> ? pointwise and I/. ,G is uniformly ? -additive (for example, if F = R or G
is a Hilbert space).
Then
(a) ? e TF.G(X) and H" -* ? in FF.G(X),
Conversely if H'\ ? belong to FF.G(BF, X) and H" -+ ? in TF.G(X), then (i) and (ii)
hold.
Under the hypotheses (i) and (ii) or (iii), assume, in addition, that H" e LhG(X) for
each n. Then
(b) ? e LF C(X) and H" -> ? in L[- g(X)\
(c) For each t e [0, oo], we have (//" ? X)l -> (? ? ?), in L'G;
(d) There is a subsequence («,·) such that (?"' ? X), -> (? ? X), a,s., uniformly on
bounded time intervals.
We remark that if H" -> ? pointwise uniformly, we have a similar result,
THEOREM 1.19 (Lebesgue). Let (//") be a sequence from TF.G(X) and let ? be an
F-valued predictable process. Assume that
(i) There is a process ? e Fr(Bvx, X) such that
|Я"КФ for each n\
and that either (ii) or (iii) hold;
(ii) //" —> ? in IFG-measure;
(iii) H" -> ? pointwise and IF G is uniformly ? -additive (for example, if F = RorG
is a Hilbert space).

Stochastic processes and stochastic integration in Banach spaces
467
Then
(a) ? e TF.G(B, X) and H" -> ? in Ty.ciX).
Assume in addition that H" ei,|- G{X) for each n. Then
(b) ? eL]F G(X) and Hv -> ? in L\- g(X);
(c) For each t e [0, oo], we have (//" ? X), -> (Я ? ?), /? L?;
(d) 7/геге is a subsequence (nr) such that (//"' ? X), -> (Я х X), a.s., uniformly on
bounded time intervals.
1.9. WeaA; completeness and compactness of LF G(B, X)
The advantage and purpose of establishing a Lebesgue space for the bilinear vector
integral is the possibility to examine weak completeness and weak compactness - a major
consideration for any theory. In this section we address this topic. We write В for By.
THEOREM 1.20. Assume that F is reflexive and G ~J> cq. Then L\ g(B, X) is weakly
sequentially complete.
Now we shall apply the general theory of weak compactness in the bilinear integration
theory to LF G(B, X). Recall that a subset К in a Banach space is said to be conditionally
weakly compact if every sequence of elements from К contains a subsequence which is
weakly Cauchy.
THEOREM 1.21. Let X be p-summable relative to (F, G). Assume F is reflexive and If.· G
is uniformly ? -additive. Let К С L\g (?, ?) be a set satisfying the following conditions:
A) К is bounded in LFG(B,X)\
B) H\An -> 0 in LFG(B, X) uniformly for ? eK, whenever A„ e V and A„ \ 0.
Then К is conditionally weakly compact in Ly G(B, X). If in addition, G ^> cq, then К
is relatively weakly compact in LF G(B, X).
In the last case, for every sequence (H")from K, there exists a subsequence (H'h) such
that (f H"rdX), converges weakly in LG,for each t.
THEOREM 1.22. Let X be ?-valued and p-summable relative to (?, ?). Let К С
Ljjj E(B, X) be a set satisfying the following conditions:
A) К is bounded in L^ E(B, X);
B) fA ? dlx -> 0 in LPE uniformly for ? e K, whenever A„ e V and A„ \ 0.
Then К is conditionally weakly compact in L~ E(B, X). If in addition, ? ^ cq, then К
is relatively weakly compact in L^ E(B, X).
In the last case, for every sequence (H")from K, there exists a subsequence (//"') such
that (//"' ? X), converges weakly in LE, for each t.

468
J. K. Brooks
THEOREM 1.23. Let X be an ?-valued process, p-summable relative to (?. ?). Let
(//")„^0 be a sequence of scalar processes from L~ E(B.X). Suppose that ? ^> cq.
IfflAH"dX -> flAH°dX, for every AeV, then
H" -> H° weakly in LlEE(B. X),
hence
( I H" dx\ -> ( I H°dx\ weakly in LPE, for each t > 0.
1.10. Local summability and local integrability
As usual, X : R+ ? ? -> ? с L(F, G) is a cadlag, adapted process with X, e L[ for each
f >0.
Definition 1.24. (a) We say X is locally p-summable relative to (F, G), if there exists
an increasing sequence (T„) of stopping times, with T„ / oo, such that for each n, XT" is
p-summable relative to (F,G).
(b) Assume X is locally p-summable with relative to (F,G). A predictable process
? : R+ ? ? -> F is said to be locally integrable with respect to X if there is an increasing
sequence (T„) of stopping times with T„ / oo, such that, for each n, XT" is p-summable
relative to (F, G) and l[o.r„)" is integrable with respect to X1".
Note that in the above situation, the stochastic integral A|о.7"„]н) х XT" is defined.
The following theorem states the existence of the pointwise limit of the above sequence
of stochastic integrals.
THEOREM 1,25. Assume X is locally p-summable relative to (F,G) and let ? be an
F-valuedpredictable process, locally integrable with respect to X, Let (T„) determine the
local integrability of ? with respect to X (as above). Then the limit
lim(l[o.:r„itf) x X
?„
exists pointwise outside an advanced set. This limit process is cadlag, adapted and
independent of the sequence (T„).
The limit in the above theorem is called the stochastic integral of ? with respect to X
and is denoted by
? xX or / HdX
/»¦
We have that (? ? X)T" = A[0???]?) ? ??" for each n. This stochastic integral has the
main properties of the stochastic integral with the respect to a summable process.

Stochastic processes and stochastic integration in Banach spaces
469
1.11. Ito's formula
In order to compute the Riemann integral, a calculus had to be developed; the stochastic
counterpart of evaluating stochastic integrals is the Ito stochastic calculus, founded on
the famous Ito formula or change-of-variable formula. Ito established this formula for
continuous semimartingales, which was later extended by Kunita and Watanabe. In this
section we shall present the full extension in the Banach case setting to processes X which
are semi-locally summable (defined below). This was developed in Brooks and Dinculeanu
A990),
In order to motivate Ito's formula, consider the very concrete, but important, case when
(X,) is a Brownian motion. Assume С and g are suitable real-valued functions on R~ and
that f(x, t) is a continuous function on R ? [0, oof together with its derivatives /,·, /,,, /,,
Let (Y,) be a processes such that for any 0 ^ t\ <fiwe have:
Yn-Yt]=i C(t)dt+[ g(t)dXt.
J[t]-4] J\']-':\
Set dY, = C(t)dt +g(t)dX,. Then the differential form of Ito's formula is
df(Y,,t) = [?,(?,,?) + fAY,,t)C(t) +{f,AY1J)gZ(t)]dt
+ fAY,,t)g(t)dXt.
Note that if X, were continuously differentiable in ?, then by the standard calculus formula
for total derivatives, the term 1 /2fxxg- dt would be absent. It is this term that distinguishes
the Ito calculus from the classical case. Brownian motion, being a square integrable
martingale on [0, t], with continuous paths, yields this "nice" version of Ito's formula.
The general case requires quite some preparation.
In the general Banach space setting, dealing with processes of a much more complicated
nature, we must study jumps (AXSJ and processes [[X]], the quadratic variation, and
impose the condition that the quadratic variation is finite. Now we shall state Ito's formula
in its general form; all terms will be defined later, and some theorems involving the
relevant concepts will be presented in order to contribute to the reader understanding of
the framework of the theory.
THEOREM 1.26 (Ito's formula). Let X: R+ ? ? —> ? be semi-locally summable with
respect to the bilinear mappings B\ (x. y) = л <g> у of ? ? ? into ? <g)T ? and Bj(x, у) =
y(x) of ? ? L(E, G) into G (that is, we regard X, in this instance, as taking values in
L(L(E,G),G)).
Assume that X has finite quadratic variation [X]. Let f : ? —*¦ G be a function of class
C~ such that f"': ? -> L(E ®? ?, G) is uniformly continuous and bounded on bounded
subsets of E. Then for every t > 0, we have a.s.:
f(X,)=f(Xo)+ f f'(Xs~)dXs + \S f"(Xs-)d[[X]]s
J\Q.i\ l J]0.t]

470
J. K. Brooks
= /(^?) + ? f'(X,-)dXs + \ ? f"(X*-)d[[X]]\
J]0.i] J\0.i\
+ ? [уда-я*.-)-/'<?.-)??,].
0<.?<?
1.11.1. Preliminary results. Throughout this section, assume ? С L(F. G) and X : E+ ?
? -> ? is a cadlag, adapted process with X, e LJ.¦(/>), for ? > 0.
We shall reserve the notation (и(нД))^о, н = 1.2 , for a family of stopping
conditions, which we indicate by (*):
(i) for each n, v(n, 0) = 0 and v(n, k) / oo, as к —> oo;
(ii) Нт„5ирА.(и(иД + 1) - u(/;, A:)) =0;
(iii) there is a sequence a„ \ 0 such that for t e [>(//, k), v(n. к + 1)], we have
\%t - Xv(„.k)\ ^ an-
Suppose a„ \0. An example of such a family (v(n,k)) satisfying (*) is
u(n, 0)=0 and
v(n,k + 1) = inf{/ > v(u,k): \X, - XHn.k)\ > a,,} A(v(n,k) +a„).
For each n, we define
X" = / ,-Уг(п.;-I1г(п.;-).1'(п.;-+1I-
*^o
From condition (ii) we deduce that X" -> X_ pointwise, and uniformly on М+хй. Define
Уо- = 0, for any process Y.
The next theorem is important in proving Ito's formula.
THEOREM 1.27. Assume that F = L(E,G)\ hence ? С L(F,G). Assume X is locally
summable relative to (F,G). Let f: ? -> G be a function of class C1. Then /':?->
L(E, G) = F is continuous and f'(X-) is locally integrable with respect to X.
Assume further that either the set of measures (I)r.G " uniformly ?-additive (for
example, ifF = RorG^> c0) or the second derivative f":E-> L (? ®T E.G) is bounded
on bounded subsets of E.
Then for every t > 0, we have
f'(X,-)dXs =lim}f'(Xl(„.k)At)(^Hn.k+\)A! - Хцп.к)лО-
к>{)
/
Ло

Stochastic processes and stochastic integration in Banach spaces
471
1.11.2. Vector quadratic variation. In this section ? and D are Banach spaces and
?. ? ? ? -> D is a continuous bilinear mapping, denoted by ?(.?, y) = лгу, such that
1-х:| = sup{|*y|: |y| ^ l},for? e E.
Denote by ?': ? ? ? -> ?) the bilinear mapping defined by B'(.x, y) = ?( ?, л). Write
2
?:? = ? ¦ ?.
Important examples of such bilinear mappings are:
A) The tensor product B(x, y) = ? <g> у from ? ? ? into ? <g>.T ?. We write д·®- = ? <8>?.
B) The inner product S(jc, у) = {?, у), if ? is a Hilbert space.
We define a process Z, taking its values in ?, to be semi-locally p-summable relative to
the pair (?, G), if it is of the form ? = X + ?, where X is locally p-summable relative to
(F, G) and ? is an ?-valued cadlag, adapted process with finite variation.
If5:?x?->Disa bilinear mapping as above, we can embed isometrically
? с L(E, D). We say a process X : R+ ? ? -> ? is locally summable or semi-locally
summable relative to the bilinear mapping ?, if we regard X as taking values in L(E, D)
and X is locally summable or semi-locally summable relative to (?, D).
If X is semi-locally summable relative to both В and B\ then we can integrate ?-valued
processes H,and f ? dX and / dX ¦ ? have values in D. In particular, if X is semi-locally
summable relative to В and B', then X_ is locally integrable with respect to X for both В
and S', and the stochastic integrals
/ Xs-dXs and / dX,-Xs-
J[0.t] J\Q.i\
are defined and have values in D. Note that if В is the tensor product or the inner product,
then local summability relative to В implies local summability relative to ?'.
Definition 1.28. Assume Х:1+хй-> ? is semi-locally summable relative to В and
?'. The D-valued, cadlag, adapted process [[?]]? defined by
[[X]]f = Xj- [ (*,_¦</*,+</*.,-Хд._)
J [O.i]
is called the vector quadratic variation (or vector square bracket) of the process X with
respect to B. Note that [[X]]B = XBr
If B(x, y)=x®y and D = ?®? ?, we denote [[X]] = [[?]]?, and call [[X]] the tensor
quadratic variation of X. Hence, if X is semi-locally summable relative to (x, у) н> ? <g> у,
then
[[X]], = Xf2- f (XA_®i/X(+i/Xv®X.,_).
JI0.t \
If ? is a Hilbert space and X is semi-locally summable relative to the inner product ?, we
denote [X] = [[?]]?, and call [X] the scalar quadratic variation of X. In this case we have
[X], = |X,|2-2 [ Xs-dX„.
J[0.t\

472
J. K- Brooks
Shortly we shall present an extension of [X], when ? is a general Banach space.
Note that if ? is a Hubert space and X is a semimartingale, then both [[X]] and [X]
can be defined since in this case X is semi-locally summable relative to the corresponding
bilinear maps.
The following theorem gives us some insight into the nature of [[?]]?.
THEOREM 1.29. Assume X:R+xl2->?is semi-locally summable relative to В and
B'. For each t ^ 0, we have
[[X]]f = X5+limprob^(Xl.(„.jt+i)A, - Хш.к)л,J-
k^O
If X is locally summable then there is a subsequence such that the limit is a.s. uniform on
bounded time intervals.
Remark 1.30. If ? is a Hubert space and X is semi-locally summable relative to the
inner product, then [X] is an increasing, positive process.
1.11.3. Quadratic variation. In order to ensure that [[X]] has finite variation paths, we
need to examine the quadratic variation [X].
Definition 1.31. We say that X has finite quadratic variation if there exists a family of
stopping times (v(n,k)) satisfying condition (*) such that
[X], := |Xol" + hm / \Xvt„.k+\)Ai ~ Ху{п.к)л,\~
? *¦—'
k^O
exists and is finite a.s. for every t ^ 0. The process ([X]t) is called the quadratic variation
ofX.
Note that [X] is positive, increasing and adapted. If ? is a Hubert space and X is a
semimartingale. then X has finite variation [[X]] which is equal to the vector quadratic variation
[[?]]? relative to the inner product B.
An important fact is that if X is semi-locally summable relative to В and B' and if X
has finite quadratic variation, then [[?]]? has finite variation and
\[[X]]f-[[X]]f\^[X],-[Xh, for s^r.
When [[?]]? has finite variation and takes values in Z), one can integrate pathwise, with
respect to [[X]]B, process ? with values in F, if F с L(D, G). Although [[X]]f has an
integral representation, we cannot use this formula in evaluating / Hsd[[X]]f, since we
do not know if the processes j Xs- ¦ dXs and fdXs ¦ Xs_ have local finite variation or
if they are locally summable relative to the bilinear mapping D ? F -> G. The following
theorem studies this problem.

Stochastic processes and stochastic integration in Banach spaces
473
THEOREM 1.32. Assume that X is semi-locally summable relative to the bilinear maps В
and B' and suppose X has finite quadratic variation [X].
Let F = L(D, G) and let ? be an F-valued cadlag, adapted process. Then [[X]]B
has finite variation, hence the Stieltjes integral L· t.Hs-d[[X]]f is defined pathwise.
Let (v(n,k)) define the quadratic variation [X]. Assume that for each ? e ? there is
a sequence b„(w) 4- 0 such that for t e [v(n. k), v(n,k + 1)[ we have
\?,(?) - Я,,,,,.*,(?) | ?,,(?).
Then there is a subsequence {m} of{n] such that a.s.
Hs-d[[X]] = lim> Hl.lm_ii)(Xv{m^^.\)Al — Xi-im.k)M)'·
m *—'
The above theorem has a corollary which is used in the proof of Ito's formula.
COROLLARY 1.33. Assume that X is semi-locally summable relative to the tensor product
and has finite quadratic variation [X], and let f: ? -> G be a function of class C~,
such that f" is uniformly continuous on bounded subsets of E. Then the Stieltjes integral
f\p ,|/"("К.«-)^[["К]]? " defined pathwise and there is a family of stopping times (which
can be used in the definition of[X]) such that a.s.
f"(Xs-)d[[X]]s = \imyif"(Xrl„.k))(Xl.i„.k+i^l - Х|.<п.*>л,)®2-
Jt^O
1.11.4. The process of jumps. In this section, we shall state some results about the jumps
of X, which will enable us to prove Ito's formula.
LEMMA 1.34. Let X:R+ ? ? -> ? be cadlag and adapted. Let (v(n,k)) satisfy
condition (*). Then for every ? and ?, we have
{s: \AXA >2an} c{v(n,k): k^O}.
THEOREM 1.35. Assume that X is ?-valued, adapted, and has finite quadratic
variation [X], and let (v(n, k)) be a family of stopping times defining [X]. Then
A) thefamily {\??^\-\ s ^t] is summable for each t;
B) the process of jumps defined for any t ^ 0 by
J, :=J2\A*s\2
is increasing, cadlag, adapted and satisfies J, ^ [X]t\
C) J, = lim„ Хл>о |A-Kr<n.it)l2l|r(H.Jt)^f) uniformly for t on any bounded time
interval;
/
/
J]0

474
J. K. Brooks
D) let V," = J2k^o \Xvin.k+\)/\t - Хш.к)лА2 and assume that for every ? there exists a
subsequence («,-), which may depend on ?, such that V'h (?) converges uniformly to
[?],(?) — \Xq(w)\2 on bounded time inten'als. Then [X] is cadlag and the process
[?]?;·.= [?], -?|дх.,|2
is continuous.
Note that the assumption D) is satisfied if ? is a Hilbert space and X is locally summable
relative to the inner product.
Now we examine the regularity of [[?]]?.
THEOREM 1.36. Assume X is ?-valued, cadlag, adapted and has finite quadratic
variation [X]. Let (v(n,k)) be a family of stopping times defining [X]. Then
A) the family {(Xs)-: s ^t] is summable for each t;
B) the process of vector valued jumps
? J, := ?(???J
is cadlag, adapted and has finite variation \vJ\ which satisfies
\vJ\, <?\??,\2\
C) we have
^2(AXsJ = \im^2(AX,llKk)Jl[,ilukKl]
uniformly for t in any bounded time interval;
D) assume that X is semi-locally summable relative to В and B'. Then the process
[[x]f; ¦= [[xnf - ?(???J
s<i
is continuous;
E) let ? :R+ ? ? -> L(D, G) be a cadlag, adapted process. The family {//,_(??,J:
s ^ t} is summable and we have
J, ¦= ? Hs-(AX.sJ = lim^2Hv{,Kk)-(AXm.k)Jl[H„.k)^t}
0<s%i k^O
uniformly on bounded time intervals. The process J,(H, X) is adapted and right
continuous.

Stochastic processes and stochastic integration in Banach spaces
475
The following corollary is used in establishing Ito's formula.
COROLLARY 1.37. Assume X is ?-valued, cadlag, adapted and has finite quadratic
variation [X]. Let (v(n, k)) be a family of stopping times satisfying (*) defining [X], Let
f ; ? -> G be a function of class C2\ hence /":?-> L(E <§>.T E, G). Then the family
{f"(Xs-_)(???)-: s ^ t] is summable and
V /"(X,-)(AX,)®2=limV/"(X1.,n.A)-)®2l|,.,n.*)<0
1—1 n 1—/
0<.5<i i>0
uniformly on bounded time intervals.
1.11.5. Comments on the proof ofltd's formula. Now that we have the machinery in
place, we shall make some comments on the outline of the proof of Ito's formula, stated in
Theorem 1.26.
If / is the function in Corollary 1.37, Taylor's formula holds, that is, there exists
a function RE ? ? -> L(E ®? ?,G), with R(x,x) = 0, for ? e E, such that
limv_>..v R(y, x) = 0 uniformly with respect to .v belonging to any given bounded set and
such that for ?, у e ? we have
f(y) = /(*) + /'(*)<>· - -v) + fv/V)(y - -v)®2 + Жу, *)(y - ·*)®2-
Returning to Ito's formula, by the above, we have
f{X,) - f(Xo) = ? (/(^i-c.t+DAi) - /<X.-(,uw))
Jt^O
= /__,/ (-^?·(??.I·)??)(-??·(??.I + ?)?? — -^i'(ii.i)Ai) C)
jt>0
+ - 2__,/ '(-??·<?1.*)??)(-??·(?!.* + ?)?? — -??·(?!.*)??) " D)
)®2, E)
where we set R„.k := /?(?,4„.*+?,?? - Л:г|„.*)л,).
By Theorem 1.27, taking a subsequence if necessary, we see that the sum in C)
converges pointwise to L· t] f'(Xs-) dX,. By Corollary 1.33, again by taking a subsequence
if necessary, we deduce that the sum D) converges pointwise to , f,Qj, f"(Xs-) d[[X]]x.
The remaining task - an arduous one - is to prove that
lim > R,i.k(Xv{n.k + \)At — -Kr(njt)Ai)
/1 * '
k>0

476 J. K. Brooks
is equal pointwise to the sum
? (/<*¦«) " ¦№-) " /'(^.»-)??? - ?/(?,-)(??.?®2)-
0<?<?
To do this, write
Rn.k(Xi4n.k + \)At - Xr{n.k)At) ' = vn.k + Wij.k +Уп.к,
where
Vn.k = f(Xv(n.k + \)At) - f(Xi-(n.k+\)At-) — f (Xnii.k))&Xvln.k-<-UAt
- ? /"("^ !¦("¦* ))(?-^ !'('!.*+? ??/)8"·
W,uk = f(XvOi.k+\)/\t-) - f(Xv<n.k)) - f'(Xi4Kk))(Xvin.k + [)Ai- ~ Xv(n.k))
~ j/ (Xv{n.k))(Xvin.k + \)At- — Xvln.k)) '·
Уп.к = 2~/ (Xvln.k))((&XH,uk+\)At) ® (Xv(n.k+l)At ~ Xf{n.k))
+ (Xrln.k+\)At- — Xt-(n.k)) ® ??,.(„.?+|)??)-
First prove
lim^|iun.jt|l|l.(„.jt)<,)=0 F)
uniformly on bounded time intervals.
Then prove, if Aluklv{n,k]<t = {???,,,,.^?? > b„}, where b„ I 0. a„ ^ b\ and
2a„ ^ b„, then
Hm ^lji?.A|lA„.Al|f(n.*)<:i) = 0 G)
k^O
uniformly on bounded time intervals.
Next prove that for t ^ 0 we have
lim^ Я(Хн„д+|)л,, XV{n.k)Ai)(Xi-Ouk + \)Ai - ??(??)??) ~^A'nX =0 (^)
k^O
and
lim J^ /?(X,(„.jt+i), Xv(„.k))(Xvt„.k^\ ? - Хг(«.*))®21д;а ¦ 1[?·(«.*?? = °- (9)
k^o
Finally, the last two steps are to show
Ит?и„.аА„.;-1|,,,,Д)<,)= ? ?(??,^-)(??,)®2 A0)
k>0 0<s<i

Stochastic processes and stochastic integration in Banach spaces
477
and
Нгп?]и,,.*1и,,.*><,)= ? tf(Xs.Xs-)(AXs)®2. (И)
k^O 0<.s<i
All of the above are for the case X is bounded. For the unbounded case, obtain a sequence
Tj ? oo such that XT'~ is bounded. Note that XT,~ still has finite quadratic variation and
[XTi~], = [X],'~; therefore, for each XT<~ we have the formula. For a given ? and t > 0,
т. _
choose /' such that ?,(?) > t. Then for every 0 < s ^ t, we have Xs' = X.s, and the
equality holds for ?, (?). This completes the outline of the proof.
1.12. Further extensions of the stochastic integral
In this section we shall briefly sketch some extensions of the theory which was presented
in the previous sections.
If ?: R+ ? ? -> ? and X: R+ ? ? -> ? are processes, ? is a Hubert space, X is
a square integrable martingale and ? is assumed to be measurable (i.e., measurable with
respect to B(R+) ? ?), a stochastic integral ? ? ? was constructed in Brooks and Neal
A998), and properties of this integral were presented. In the case ? is an optional process
(i.e., it is measurable with respect to the optional ?-algebra - the ?-algebra generated by
right-continuous, adapted processes), ? ? ? agrees with the compensated (or optional)
stochastic integral developed by Yor (see Dellacherie and Meyer A978)). Moreover, when
? is bounded and measurable, then ? ? ? = °H с X. where °H is the optional projection
of Я. When ? is predictable, then ? ? ? agrees with the Hilbert-space stochastic integral
defined by Kunita A970). In Brooks and Neal A998) a very general stochastic integral
is developed when one of ? and X is real valued and the other is Hilbert-space valued,
assuming that ? is just measurable; the construction uses H1' and BMOp spaces of Hilbert-
space valued processes. A weak Pettis-type stochastic integral is presented in Brooks and
CandeloroB000).
Next we turn to the nuclear space setting. The space ? will denote a reflexive, complete,
bornological nuclear space, whose dual ?' under the strong topology is nuclear. Since
? and ?' are projective limits of separable Hubert spaces, we can choose appropriate
neighborhood bases of zero, U(E) and U(E'), respectively, and work with quotient
spaces ?([/), E'(U'), U e U(E), U' e U(E'), that are Hubert spaces induced by the
norms corresponding to U and U'; we have canonical mappings k(U):E—* E(U) and
k(U, V):E(V) -> ?([/), if V с U (also for the E'(U') system). Ustunel A982) defined
the concept of a projective system of stochastic processes in ?' as follows. Call the set
X = {Xu'\ U' e U(E')} a projective system of martingales if X1' is an ?'([/')-valued
martingale and if V" с U', then k(U', V')XV and Xй are indistinguishable. We say such
an X has a limit in ?' if there exists a weakly adapted mapping X' :E+ ? ? —* ?' such
that k(U')X' is a modification of XL', for each U'. Such a limit does exist if the projective
system consists of square integrable martingales, and X' will be strongly cadlag in ?'.
The predictable case is treated in Ustunel A982) and the general case in Brooks and
Neal A998b). If ? is bounded and weakly optional, a stochastic integral ? ? ? is defined

478
J.K. Brooks
in terms of ? о X u , where U' absorbs the range of ? in the sense that ? takes its values in
the Hubert space E[(U')°]. It can be shown that if {//"} is a sequence of weakly optional
E-valued functions on R+ ? ?, which converges pointwise in ? to Я and if each H"
takes its values in a fixed E[(U')°], then {#" ? ?] converges in the Hubert space of
square integrable real valued martingales, if X is a projective system of square integrable
martingales.
The general case X:R+ ? ? —> ? с L(F,G). where the spaces are nuclear are
treated in Brooks A999), where a stochastic measure ? ? on V takes its values in h\: and
a bilinear integration theory is constructed for certain processes X. We have the following
summability result in this setting: X is summable if ? ? is bounded on 1Z, the ring generated
by predictable rectangles.
Finally, we construct a Pettis (weak) stochastic integral for predictable processes
? : R+ ??^?, where ? is a Banach space; X is scalar valued, adapted and X, e L', for
each t. Call ? weakly integrable if there exists an E- valued process ? such that ? is cadlag
and Z, = (((*', //)) ¦ X), for each t. This integral is developed in Brooks and Candeloro
B000). The theory yields a norm for such H, and the space of weakly integrable processes
strictly contains Ll(X), if ? is infinite dimensional.
2. Regularity and the Doob-Meyer decomposition of abstract quasimartingales
introduction
In this section we shall examine the structure of quasimartingales taking their values in
a Banach space E. We are especially interested in the existence of cadlag modifications
and in the Doob-Meyer decomposition of such quasimartingales. The motivation of this
inquiry was twofold. First, the problems concerning regularity and the Doob-Meyer
decomposition of quasimartingales arose naturally in the attempt to develop a stochastic
integration theory for Banach space valued processes. Secondly, we wished to improve
on the existing results on the regularity and structure of quasimartingales, as presented by
Pellaumail A973), Metivier and Pellaumail A975) and Kussmaul A977,1978). The results
in this section were developed by J.K. Brooks and N. Dinculeanu A988).
Here is a sketch of the results. Let ? :?+ ? ? -> ? be a quasimartingale (not
necessarily cadlag).
A) Assume ? has the Radon-Nikodym Property (RNP), that is any ?-valued measure,
with finite variation and absolutely continuous with respect to a probability measure P,
has an ?-valued Radon Nikodym derivative with respect to P.
It turns out, surprisingly, that X is strongly cadlag if and only if X is weakly cadlag
B.12). Furthermore, X is cadlag if and only if X has a Doob-Meyer decomposition
X = ? + A, where ? is a cadlag local martingale process and A is a predictable, cadlag
process with finite variation and A() = 0. This decomposition is unique up to an evanescent
set.

Stochastic processes and stochastic integration in Banach spaces
479
B) Assume that ? has the RNP and X is a quasimartingale. Then X has a cadlag
modification if and only if X satisfies the following regularity condition:
(R) lim?(lFX,) = ?(lFX() fori < oo and F e ?„\
il.v
or even the following weak regularity condition:
(WR) \im(E(lf.-X,),z) = {E(lf.-X<,z) for s < oo and F e ?,,
il.v
г in a subset ? of ?", which is norming for E.
This is proved in Theorem 2.1 which gives 13 equivalent conditions for the existence of
a cadlag modification; these conditions are new even in the scalar case. As a corollary
(Theorem 2.13) we give a characterization of the ?-valued measure ??, which is
associated with a quasimartingale X.
C) Without any assumption on the Banach space E, we present results concerning the
existence of a cadlag modification of X in terms of its right limit X+ along the rationals
(Theorems 2.15-2.18). The fact that ?-valued martingales have cadlag modification was
established by J.K. Brooks and N. Dinculeanu A987).
D) Assume that ? has the RNR Then X is a quasimartingale of class (D) and satisfies
condition (R) if and only if X has a Doob-Meyer decomposition X = ? + A, with ? a
martingale of class (D) and A a predictable, cadlag process with integrable variation and
A0 = 0 (Theorems 2.20-2.23).
Conditions (D) and (R) together are also equivalent to the associated measure ?? being
? -additive with bounded variation.
Various concepts used will be stated in Appendix A and sections in this appendix will
be referred to as Sections A. 1, A.2, etc.
2.1. Notation
In this section, we shall present notation and definitions which will be used to prove the
main results in the next sections.
Throughout this part, (?, ?, ?) is a probability space, (?,I€-^. is a filtration satisfying
the usual conditions, ? is a Banach space with norm || ¦ ||, and dual space ?", and (Xt),€?__
is an ?-valued, adapted process with X, e L[(P), foreach t ^ 0. We shall always consider
X to be extended to oo, with X^ = 0. If X has a limit at do, it will be denoted by X^_.
For every a, with 0 < a ^ do, we shall denote by 1Z@. a], the ring generated by the
predictable rectangles of the form ]s, t] ? F, with 0 ^ s < t ^ a and F e ??, and by
7?]0, <*[, the union of the rings 7?]0, ?], with ? < a. In particular, 7?]0, do[ consists of all
finite, disjoint unions of bounded predictable rectangles ]s, t] ? F with t < oo, and this
ring generates the ?-algebra V of predictable subsets of ]0, oof ? ?. Note that 7?]0, a]
contains the stochastic intervals ]S, T], where S and ? are simple stopping times bounded
by a. We notice that the above rings do not contain subsets of {0} ? ?. In fact, these sets
have no role in problems of regularity.

480
J.K. Brooks
2.2. The Doleans function
For each predictable rectangle ]s, t] ? F, we define
fix(]s,t] ? F) = E(lr(X, - Xs)).
In particular, ?-xQs, oo] ? F) = — E(lf.-Xs).
Note that ?? is finitely additive on the semiring of predictable rectangles, and thus it
can be extended uniquely to an ?-valued, finitely additive measure on the algebra 7?]0, oo];
this extension will still be denoted by ??. It is called the Doleans function of the process
X. This function is invaluable in the study of quasimartingales. An immediate consequence
of the definition of ?? is the equivalence of the following two conditions:
YunE{\FXl) = E(\FX<), A2)
lis
ШпдлЧКг] x F)=0. A3)
? is
As we mentioned previously, we refer to the regularity condition above as condition (R).
Other properties of ?? will be stated in Section A.l.
2.3. The mean variation of X and quasimartingales
For any a with 0 < a ^ oo, we define the mean variation of X on ]0, a] to be the number
Varx]0,a] = sup53|?((X,,.1 " ?0\?',)\? < *>·
i
where the supremum is taken over all finite partitions 0 ^ ?i ^ · ¦ · ^ t„ = a. The mean
variation VarxJO, a[ of X on ]0, a[ is defined similarly by taking t„ <a.
It is important to be able to compute Vaix]0, a] by taking the supremum over different
sets, for example, random partitions. Results concerning different representations of
VarxJO, a] are presented in Section A.2, along with the relationship between Varx]0, a]
and ????. the variation of ??.
We say that X is a quasimartingale on ]0.a], or on ]0.a[, if VarxJO.a] < oo or
Varx]0,a[< oo, respectively. We shall be interested in three types of quasimartingales.
namely, those on ]0, oo], on ]0, oof, and those on every bounded interval ]0. a]. Elementary
facts concerning quasimartingales. along with some convergence properties are given in
Sections A.3 and A.4. In particular, X is a quasimartingale on ]0, a] or ]0, a[, with a ^ oo,
if and only if the Doleans function ?? has bounded variation \??\ on TZ]0,a] or 7?]0, <*[,
respectively.
We say that X is of class (D) if the family {X/: ? simple stopping time) is uniformly
integrable. We say X is of class (LD) if for each fixed a < oo, the above family, with
? ^ ?, is uniformly integrable.

Stochastic processes and stochastic integration in Banach spaces
481
2.4. Regularity of quasimartingales
In this section we shall present one of the main theorems concerning the existence ?? cadlag
modifications of a quasimartingale.
Recall that a family ? с ?" is called total (for E) if ? e ? and (x.z)=0 for alU e Z
implies that ? = 0. A family ? с ?" is called norming (for E) if for each xe?we have
Evidently, any norming set is total.
We remark that the RNP of ? in the next theorem is used only in steps 7 and 8 of the
proof, and is not needed in the hypotheses of the lemmas established in the course of the
proof, except for Lemmas 2.3(b) and 2.11.
THEOREM 2.1. Assume that ? has the RNP and that X is a quasimartingale on each
bounded interval and has a separable range. Let ? С ?' be any set which is norming for
the range of X. The following assertions are equivalent:
A) X has a cadlag modification;
B) X has a right continuous modification;
C) X is right continuous in the mean, that is, for the strong topology of LE;
D) X is right continuous for the weak topology of LE;
E) X is right continuous in probability;
F) lim,j.j E(lFXt) = E(\FXs)fors < oo and F e Es;
G) X = ? + A, where ? is a (not necessarily right continuous) local martingale and
A is a predictable, cadlag process with finite variation and Aq = 0;
A') (X, z) has a cadlag modification for each ? e Z;
B') (X, z) has a right continuous modification for each ;eZ;
C') {X, z) is right continuous in the mean, that is, in the strong topology of L', for each
zeZ;
D') (X, z) is right continuous in the weak topology ofO, for each zeZ;
E') (X, z) is right continuous in probability for each ? e Z;
F') \imlU{E(lrXt),z) = (E(lFX,),z) for s < oo, F e ?, andze Z;
(T) (X,z) — M(z) + A(z), for each ? e Z, where M(z) is a (not necessarily right
continuous) local martingale and A(z) is a predictable, cadlag process of finite
variation with Aq(z) = 0.
The decomposition in G) or (T) is unique up to an evanescent set.
PROOF. The proof will proceed along the following implications: 7=>1=>2=>3·?>5=>
6 => 6', 7 => 7' => 1' => 2' => З'о 5' => 6' => 7, 3 => 4 => 6' and 3' => 4' => 6'. The only
implications that require the space ? to have RNP are 6 => 7 and 6' => 7. All the other
implications mentioned above are valid for any Banach space E.
The implications 7 => 1 and 7' => ? follow from the fact that any local martingale
has a cadlag modification (see Theorem 2.15 below). The implications 2 => 3 o· 5 and
2' => 3' о 5' are stated in Proposition A.4.1. We shall prove below the implication 6' => 7.

482
J.K. Brooks
All the other implications are evident. The proof will be done in several steps. In each step
it will be assumed that condition 6' is satisfied. We remark that the RNP is not needed in
Steps 1-7. In the process of proving some of the steps, we shall establish properties which
are valid in more general situations and these will be stated as independent lemmas. In the
statements of these lemmas, no conditions will be assumed, unless otherwise specified.
We shall now prove the implication 6' => 7 assuming first that X is a quasimartingale
on ]0, oo]; therefore ?? has bounded variation on 7?]0, oo]. Without loss of generality we
can also assume that ? is separable and that ? is norming for ? and countable.
STEP 1. The measure ?? satisfies the following conditions:
lim|MX|(]i.f] ? ?)=0 fors < ос A4)
and
limlux(]s,(]xF) = 0 fors < oo and F e 27,. A5)
In fact, hypothesis F') is equivalent to
Нт(дх(]5,г] xf),;) = 0 fors < oc, F e ?, and ;eZ.
Then Step 1 will be a consequence of the following lemma.
LEMMA 2.2. Let ? be any Banach space, let ? С ?' be a total set, and let ?: TZ]0, oo] —>
? be an additive measure with bounded variation \?\. Thenfor every s < oo, the following
assertions are equivalent:
A) Нт1Ь|д|(]л,г]хЯ)=0;
B) lim, is ?(]?, t] ? F) = 0, for every F e Es;
C) lim, u (?(]?, t] ? F), z) = 0, for F e ?, and ? e Z.
PROOF. Obviously A) => B) => C). Assume now C) and prove B). Since ? has bounded
variation |?|, it is strongly additive on 7?]0. oc], that is ?(?„) -> 0 for every sequence (?„)
of disjoint sets from 7?]0. oo] (see Brooks and Dinculeanu A974) for results concerning
strong additivity). As a result, for every decreasing sequence A„ 4- 0 from 7?]0. oo], the
sequence (?(?„)) is convergent in ?. In particular, for A„ = ]s,t„] ? F, with tn i s and
F e 2T.V, the limit ? = lim„ ?(]?,?„] ? F) exists in E. By assumption C), we conclude
that (x, z) = 0 for every ;eZ, which proves B). Next assume B) and prove A) (cf. also
Metivier A974). Let t„ I s and let ? > 0. Let (?;) be a finite partition of ]s,t\] ? ?
consisting of disjoint predictable rectangles such that
|?|(]?.?|]??)<53|?(?|+?.
к

Stochastic processes and stochastic integration in Banach spaces
483
For any set A e 7?]0, t\ ] contained in ]?,?|]?? we have
? \\?(? П Rk)\\ 2 ? ?^?*)|| " ? У(ЛСП Rk)\\
к к к
? |?|(]0, ?,] ? ?) - ? - |м|(Л') = \?\(?) - ?.
If we apply the above to A = ]s. t„] ? ?, we obtain
\?\ (]s, t„] ??)^? \\?{?'.*?] x ? П ??) ? + ?.
Now let ? -> ?? and obtain lim„ |?|(].?,?„] ? ?) ^ ?, which proves A). ?
Step 2.
(a) For each z&Z, the quasimartingale {X, z) has a cadlag modification.
(b) There exists a cadlag negative submartingale ? such that \??\ = ?? on 7?]0, oo].
Assertion (a) follows immediately from Lemma 2.5 below, since {X,z) satisfies
condition F') which is equivalent to condition A) of this lemma, for (X,z). Assertion (b)
will be a corollary to the following two lemmas. In fact, from Lemma 2.3, we deduce first
that \?? | satisfies condition (i) in Lemma 2.3, and then deduce that there exists a negative
submartingale ? such that \??\ = ?? оп7?]0,оо]. By Step 1, \??\ satisfies condition (ii)
of Lemma 2.5, which insures that ? can be chosen to be cadlag.
Lemma 2.3.
(a) Let ? be any Banach space and let ? be an adapted ?-valued process with
Y, e L[E(P)forall t ^ 0. The ? is a quasimartingale on ]0, do] if and only if its Doleans
measure ? = ?? satisfies the following condition:
(i) For every t < oo, there is a function f, ^ 0 in L[(P) such that |?|(]?, эо] ? F) ^
E(\Ff)JorFeEt.
(b) Assume that ? has RNP and let д:7?]0,эо] —> ? be an additive measure
with bounded variation \?\ satisfying condition (i) above. Then there is an ?-valued
quasimartingale ? such that ? = ??.
PROOF, (a) Assume first that У is a quasimartingale on ]0,oo]. For any predictable
rectangle ]u, ?] ? F, set IY(]u, ?] ? F) = lF(Yv - Yu). Then ? ? is an L}(P)-valued,
finitely additive set function defined on the semiring of predictable rectangles; it can be
extended to an additive measure on 7?]0, do]. For every A e TZ]0, oc], we have ?? (A) =
E(IY(A)). Let t < oo and let ? be the set of all functions in L[(P) of the form
?\\?{??(?^\?,)\\

484
J. K. Brooks
where (A,-) is a finite family of disjoint predictable rectangles contained in ]t,oo] ? ?.
Note that for each such family, we have
J]||/y(A,)|1^|MH(]0,oo]xi2)<oo,
i
hence the set ? is bounded in L' (?). We next show that W is directed for the usual order in
L\P). If (A,) and (Bj) are two finite families of disjoint predictable rectangles contained
in ]?, oo] ? ?, and if (Bj) is a refinement of (A,), then using the additivity of ?? we obtain
^|?(/y(A,-)|r,)|iC^|?(/y(S,)|r,)[[.
i i
This result then implies that ? is directed in Lx (?). ?. being bounded and directed in the
space Ll(P) of equivalence classes of integrable functions, it has a supremum in L](P),
and we choose a representative f, in this equivalence class. For any set F e ?, and any
finite family (A,) of disjoint predictable rectangles contained in ]t,oo] ? F, we have
J2\\E{lY(Ai)\E,)\\ iClf/, a.s..
hence
Х)|ду(А/)|^?(Ь-/,);
consequently
lMH(]r,oo]xF)iC?(b-/,).
Conversely, if (i) is satisfied for t = 0, then ?? has bounded variation on ]0, oo] ? ?,
hence У is a quasimartingale on ]0, oo].
(b) Let ? be an F-valued additive measure on 7?.]0,oo] with bounded variation \?\
satisfying (i). For fixed t < oo, the mapping F н> ?(]?, oo] ? F) is a countably additive
measure on ?, with bounded variation and is absolutely continuous with respect to P.
Since ? has the RNP, there is a function Y, e L\. (P) such that
?(]?,??] ? F) = -E(\yY,) forFeXV
It follows that ? = ?? and since ? has bounded variation on 7?]0,oo], У is a
quasimartingale on ]0, oo]. ?
Remark 2.4. In the scalar case, a property similar to (i) in Lemma 2.3 was proved in
Kussmaul A977), in a different setting.

Stochastic processes and stochastic integration in Banach spaces
485
In the scalar case, the following lemma characterizes the right continuity of ? in terms
of the corresponding measure ??. The vector case will be considered later (Theorem 2.13).
LEMMA 2.5. Let ? be a real valued, additive measure on 1Z]0, oo] with bounded variation
\?\ satisfying condition (i) in Lemma 2.3 and let ? be a real valued quasimartingale on
]0, oo] with ? = ??. Then ? has a cadlag modification if and only if
lim^(]j,(]xF)=0 fors < oo and F e ?,. A6)
r is
If, moreover, ? ^ 0 on 1Z]0, oo], then ? is a negative submartingale; ? has a cadlag
modification if and only if
????(]5,?] ??)=0 fors<oo. A7)
tls
PROOF. If ? has a cadlag modification, then, by Proposition A.4.1, ? is right continuous
in the mean, hence lim,j,A Е{\уФ,) = Е(\уФн) for s < oo and F e Es, and this condition
is equivalent to A6); if ? ^ 0, this condition is equivalent to A7).
To prove the converse implication, assume first that ? ^ 0on7?]0, oo] and satisfies A7).
Then ? is a negative submartingale and A7) is equivalent to
Нт?(Ф,) = ?(ФЛ) fors<oo.
lis
By Brooks and Dinculeanu A987), ? has a cadlag modification. If ? is real valued and
satisfies condition A6), then by Lemma 2.2, its variation |?| satisfies A7). By the first part
of the proof, there are two cadlag negative submartingales ?+ and ?~ satisfying ?+ =
??+ and ?~ = ??-. Then ?? = ?,?+_?-), hence ?+ - ?~ is a cadlag modification
of ?. D
Remark 2.6. A similar result was proved in Kussmaul A977) for measures satisfying
a certain "condition (S)".
The above lemma will be extended to quasimartingales with values in a Banach space
(Theorem 2.13).
For every simple stopping time ?, ?? is an ? ? -measurable, integrable random variable.
We shall extend, in Step 3 below, the definition of ?? by means of a random variable ??,
where ? is an arbitrary stopping time T. To do this we need the following lemma.
LEMMA 2.7. If ? is an ?-valued quasimartingale on ]0, oo], then for every stopping
time ? and for every decreasing sequence (T„) of simple stopping times with T„ I T, the
sequence (?(???\??)) is Cauchy in L\-.
PROOF. We remark first that if ? ^ U ^ V, where U and V are simple stopping times,
then
E((YV -??)\??I<:\??\(]?.?]).

486
J. K. Brooks
In fact.
? \\E((YV - ??)\??)\\?? = sup J2 ? \\?((??-??)\??)??\\
= ™?? l· \\(Yv-Yv)dP\\
= sup?|/i>-(]t/. V]nR+ xFj)\\
i
?\??\(]?.?]).
where the supremum is taken over all finite families (F,) of disjoint sets from ?? ¦ Then,
taking U = Tn+\ and V = T„, we obtain
? \\?{(??„ ~ ??„+?)\??I iC ^|му|(]Г/| + 1. Г,,]) iC |My|(E+ ??)<??.
/1 /1
It follows that the series ?? ?((??„ - ??;;+] )|?Y) is convergent in L^. For any к we have
? ?((??? ~ ??„+] )\??) = ?(?? \??) - ?{??\??).
П<к
hence ?(??? \ ??) converges in L[-, and this proves Lemma 2.7. ?
Remark 2,8. This lemma was proved in Kussmaul A978) under the assumption that ?
is right continuous in the mean.
STEP 3. For any stopping time ? we can chose ? ??-measurable random variable
Хт е LF satisfying the following conditions: ___
(a) If(Tn) is a decreasing sequence of stopping times with T„ 4- T, then ?(??„ ? ??) ->
Хт in the mean,
(b) If ? is a simple stopping time, then Хт = Хт a.s.
(c) If X is right continuous, then Хт = Хт a.s.
(d) If X' is a modification of X, then X'T = Хт ci.s.
(e) If ? eZ andY = {X, z), then ?? = {XT.j) a.s.
(f) IfT„ I oo, then for any t < oo we have ??,,^? -*¦ %t я-*-
We first take a decreasing sequence (T„) of simple stopping times Tn with T„ 4. T.
By Lemma 2.7, the sequence (?(????\??)) is Cauchy in L\:, and has a limit и е L[.
This limit is independent of the sequence (T„). In fact, let (S„) be another decreasing
sequence of simple stopping times converging to T, and let ? = lim„ ?(?$„ \??) in LE.
Let ; e Z. By Step 2, the real quasimartingale ? = {?,?} has a cadlag modification
W. By Proposition A.4.2, we have WT e L1 and WT„ -> WT and W.Sn -> W/ in L1;

Stochastic processes and stochastic integration in Banach spaces
487
consequently both sequences (?(???\??),?) and (/?(?^,|.??), z) converge to Wt in L1.
Thus {u,z) = Wt = (v, z) a.s. Since ? is countable, we have и = ? a.s.
We denote by XT a representative of the element lim E(XtJEt) in L[. Then Xr is
?? -measurable.
For ? e ? and У = (X, с), we then have
(Хг,г} = (Нт?(Х7-„|Гг).;} = Нт?((Х7;,.;)|Гг) = ?г.
and this proves (e).
If X' is a modification of X, then X'T = ??„ a.s., for simple stopping times ?„ and (d)
follows. Also (b) follows in a similar fashion.
If X is right continuous, then (c) follows by Proposition A.4.2.
Property (a) holds by definition if (Г„) is a sequence of simple stopping times. Now
let (Г„) be an arbitrary decreasing sequence of stopping times converging to Г, and
prove that E{Xj„ \??) -> Xr in L\:. For each « choose a simple stopping time Sn with
?„ ^ S„ ^ ?„ + I In and
?^?,,,?^-??,,?^?/".
Hence
llECXsJ-^-^xYj^r)!, <!/"¦
We can choose the S1,, decreasing to ?, hence ?(X.y„ ?-??7) -»¦ %r and then (a) follows.
To prove (f), let ? e ? and let W be a cadlag modification of ? = (?, ?), which exists
by Step 2, Then a.s., (??„?,, с) = ?7„л, and W, = (?,,?). Let N С ? be a ?-negligible
set such that for all ? e /V, the above equalities are valid for a fixed f, for all integers
? and all г in a countable norming set Z. For ? e /V, choose « such that ?„(?) > ?.
Then (??„?,(?), г) = WTnAI(w) = W,(co) = (?,(?), ?), hence ?7„?,(?) = ?,(?), which
proves (f).
Step 4.
(a) For any stopping time ? we have \\??\\ ^ —?? a.s.
(b) For any stopping time S ^ ? iff have \\E(Xr — Xs)\\ ^ ?(?? — ?s)¦
For ? & ? and Л е 7Z]0,00] we have
\?{?.:)(?)\ = \(??(?),?)\ iC ||;||??(?).
hence | < X,, с > | ^ —Ф, a.s. for each t ^ 0. Let (Г„) be a sequence of simple stopping
times, T„ 4. Г. Then | {X7;,, c) | ^ -??„ a.s. for each «, therefore
?((Х7-„,г)|Г7-)^-?(Ф7„|Г/) a.s.

488
J.K. Brooks
Let ? —» oo and obtain
\{??,?)\^-?? as.
Since ? is countable and norming, we have \\?? II ^ — ?? a.s.
For 5 ^ ? simple stopping times we have
\\?(??-?3)\\ = \\??(]?,?])\\^\??\(]?,?]) = ??{]?,?])
= ?(?? - <?s)-
Then (b) follows by approximating general stopping times by simple ones. ^
For any stopping time ?, we define now the "stopped" process XT as follows: (XT), =
????, fori ^ 0.
STEP 5. For each stopping time T, the stopped process XT is a quasimartingale on ]0, oo]
satisfying the regularity condition
\imE(lFXf) = E(lFxl) for s < oc, F e ?,.
If 5? ^ Si ^ ¦ · ¦ < S„ is a finite family of simple stopping times, then by Step 4(b),
?№+, -xTs,)\\ = EHW, -xt.Si)\\
i i
^??(??^-< -*7-AS,)^Var*]0,oo].
The last inequality is valid by Proposition A.4.3 since ? is a cadlag quasimartingale.
Taking the supremum for all families E,) as above, and using Proposition A.2.3, we obtain
VarjrJO, oo] ^ Var<2>]0, do] < oo,
hence XT is a quasimartingale on ]0, oo].
Since ? is cadlag, it is right continuous in the mean by Proposition A.4.1. Then by
Proposition A.4.2, we have
]im Е(ФТмг -Фт^ч)=0,
r Is
therefore
\imE(lFXj) = E(lFXj).
and this proves Step 5.

Stochastic processes and stochastic integration in Banach spaces 489
STEP 6. If S and ? are stopping times with S ^ ?, then (XT)S is a modification of Xs.
We remark first that, by Step 5, we can apply Step 3 to XT and define for every t ^ 0,
(XT)sAt; for notational convenience, we shall abuse notation and we shall denote the latter
simply by (??)$??- Hence the process (XT)S is define by Step 5 and is a quasimartingale
satisfying the regularity condition (R). Let S„ I S, S„ simple stopping times, and let t ^ 0;
then Sn At 4- S At and Sn л ? л t I S a t, hence, since the Sn are simple.
= Xsai = X, in LE.
It follows that (XT)f = X? a.s., hence (XT)S is a modification of Xs.
Next, for each ? we define the stopping time
r„ = inf{i: -?,^?).
Sine ? is cadlag, we have Tn f oo. Set X"" = X7".
STEP 7. X(,,) is a quasimartingale of class (D) on ]0,oo] a/id satisfies the regularity
condition
lim?(lfX,(")) = ?(l/-X!")) forsoo, FeE,.
In fact, for simple stopping time ? we have by Step 4(a):
\\Х?\\^-Фтлтп ^п+ФТ11еЬ1.
since ? is cadlag (see Proposition A.4.2). This shows that X' is of class (D). The other
assertion was proved in Step 5.
STEP 8. X(,,) can be decomposed uniquely (up to an evanescent set) in the form X"" =
Min) + A"", where M(n) is an ?-valued martingale of class (D) and A"" is an E-valued,
predictable, cadlag process with integrable variation with Aj" = 0.
This is a direct consequence of Step 6 and Lemma 2.11 below. In the proof of
Lemma 2.11 we shall use the following lemma.
LEMMA 2.9. Let ? be an ?-valued, adapted process such that Y, e L\(P) for each t.
The ?? is countably additive and of bounded variation on 7?]0, do] if and only if ? is a
quasimartingale of class (D) on TZ]0, do] and satisfies
lim?(l/.y,) = ?(l/.ys) for s < oo and F e Es. A8)
lis

490
J.K. Brooks
PROOF. Assume first that У is a quasimartingale of class (D) satisfying condition A8).
This condition is equivalent to
Нтму(]5,г] ? F)=0 for s <oo and F e ?\. A9)
? l.s v '
We shall prove that ?? also satisfies the following condition: For any decreasing sequence
//„ from ?, with H„ I 0 we have
limsup{|My(]5,r])|}=0, B0)
" л
where ?„ is the family of stochastic intervals ]S, T], such that S and ? are stopping times,
S^T and] 5, T] C]0,oo] ? ?„.
In fact, for (#„) and S, ? as above, we have
\\??(]?, T])\\ = \\E(lHn(YT - Ys))\\ iC E(lHn(\\YT\\ + \\YS\\)).
and this last term tends to zero uniformly relative to ]S, T]. We now use the criteria in
Metivier A982, Lemma 2, p. 83) and deduce that conditions A9) and B0) imply that ??
is countably additive on 7?]0, oo].
Conversely, assume that ? у is countably additive and of bounded variation on 7?]0, oo].
Then it is immediate that У is a quasimartingale on ]0, oo] satisfying A8). Since \??\ is
also countably additive, we have
Ит|ду|(]5, t] ? ?) = 0 for.v<oo.
By Lemma 2.5, there is a cadlag, negative submartingale ? such that \??\ = ??. Then
||F(l/ry,)| <; -ЕAуФ,) fori ooandFeT,.
From this, we deduce that || Y, \\ ^ -Ф, a.s. As a result, || Yj || ^ —Фт a.s. for every simple
stopping time T. By Metivier A982, Theorem 13.3) ? is of class (D), hence ? is also of
class (D).
Remark 2.10. Metivier A982) proved the above lemma under the additional assumption
that ? is right continuous. His result follows from the above lemma since right continuity
implies right continuity in the mean (see Proposition A.4.1), which, in turn, implies A8).
LEMMA 2.11. Assume ? has the RNP. The following two assertions are equivalent:
(a) Y is a quasimartingale of class (D) on ]0, do] and satisfies
limF(l/-T,) = E(lFY,) for s < oo and F e ?,.
lis

Stochastic processes and stochastic integration in Banach spaces
491
(b) ? = ? + A, where ? is an ?-valued (not necessarily cadlag) martingale of
class (D) and A is an ?-valued cadlag process with integrable variation and with
The decomposition in (b) is unique (up to an evanescent set).
PROOF. Assume first (a). By Lemma 2.9, ?? is ?-additive and has bounded variation on
1Z]0, oo]; therefore it can be extended to a ?-additive measure, still denoted by ? у on the
?-algebraof all predictable subsets of ]0, do] ? ?. Consider the measure у defined on the
?-algebra ? of all predictable subsets of ]0, oo[ ? ?, equal to ?? on the predictable subsets
of ]0, ??[??, and equal to 0 on the predictable subsets of {0} ? ?. Then у is ?-additive
and has finite variation on V. Since ? has the RNP, there is a predictable ?-valued process
such that ||g|| = 1 and
v(B)= I gd\v\ for В e V.
f gd\v
Jb
Since ?? and у vanish on evanescent sets, there is a unique (up to an evanescent set),
predictable, cadlag, increasing, integrable process V associated to \v\ (Dellacherie and
Meyer, 1978,VI.65),by
\v\(B) = e( J lBdVj forBeP.
Since |y|({0} ? ?) = Owe have V0 = 0. Then
v(B) = e( (\Bgsdv\ for Sep.
The process A, = L· t,gsdVx is cadlag, predictable, with A(> = 0 and
v(B) = e( \\BdA \=??(?) for Be P.
The variation \A\ of ? is integrable, since it is predictable and satisfies \A\ ^ V. For
В е 7?]0, oof we have
??(?) = ?(?) = ??(?).
If we set ? = ? — A, then ?? = 0 on 7?]0, oof, hence ? is martingale. Moreover, ? is
of class (D) since both ? and A are of class (D), and this proves (b).
The uniqueness of the decomposition follows from the fact that, (as in the scalar case), an
?-valued, predictable, cadlag martingale with finite variation and vanishing at 0 is equal 0
everywhere (except on an evanescent set).
Conversely, if we assume (b), then ? is of class (D) and ?? = ? a on 7?]0, oof and
?? = ?? on {oo} ? ?. Since ?« and ?.\ are ?-additive and of bounded variation on

492
J.K. Brooks
{00} ? ? and 7?]0, ??[, respectively, it follows that ?>- is ?-additive and with bounded
variation on 7?]0, 00]; hence, by Lemma 2.9. У satisfies condition (a). D
STEP 9. We have X — ? + A as in assertion G) of the statement of the theorem.
By Step 8, for each ? we have a unique decomposition X"" = M"" + A"", where
M1 is a martingale of class (D) and A"" is a cadlag, predictable process with integrable
variation and Aq" = 0. Let N' be a cadlag modification of M(n) (see (Brooks and
Dinculeanu, 1987)). Then У("> = N(n) + A"" is cadlag modification of X("K By Step 6,
(X("+1))r" is a modification of X""; (X<"+1>O'« therefore (y<"+1>O'» is a modification
of У""; since they are cadlag, we have (У1"*"O" = y1 (outside an evanescent set). It
follows that (N("+U)T" = N{,,) and (A<"+"O« = A<">. Then we can define the cadlag
processes У, N, A such that these processes, stopped at T„ are equal, respectively, with
Y{"\ N[n), A"". It follows that N is a cadlag local martingale and A is a cadlag,
predictable process with finite variation and with A() = 0, and ? = N + A. We deduce,
using Step 3(f), that У is a modification of X. Hence the process ? = X - A is a
modification of the local martingale N, and for each « and ? we have
(MT»), = X\n) - A™ = N™ a.s„
hence M7" is a martingale of class (D). It follows that ? is local martingale and this proves
the implication 6' => 7 in case X is a quasimartingale on ]0, do].
STEP 10. Assume now that X is a quasimartingale on each bounded interval. Then for
each a < 00, the stopped process X" is a quasimartingale on ]0, do] satisfying assertion
F') of the theorem. By Step 9, we have X" = Ma + A" with M" and A" as in assertion
G) of the theorem. By the uniqueness of this decomposition, taking a = 1,2,..., we
have (X"+1)" = X" hence (M"+1)" = M" and (A"*1)" = A"; hence there is a local
martingale ? and a predictable cadlag process A with finite variation and Aq = 0 such
that ? and A stopped at ? are equal respectively with N" and A"; hence X = ? + A, and
the theorem is completely proved.
D
The next theorem gives equivalent conditions for X to be actually cadlag (rather than
having a cadlag modification). In particular, it states that X is a strongly cadlag if and only
if it is weakly cadlag.
THEOREM 2.12. Assume that ? has the RNP and that X has separable range and is
a quasimartingale on each bounded interval. Let ? С ?" be a set which is normingfor the
range ofX, The following assertions are equivalent (up to evanescent sets):
(a) X is cadlag;
(b) X is right continuous;
(c) X is weakly right continuous;

Stochastic processes and stochastic integration in Banach spaces
493
(d) (X, z) is right continuous for every ;eZ;
(e) X = ? + A, where ? is a cadlag local martingale and A is a cadlag predictable
process of finite variation with Aq = 0.
PROOF. The implications (e) => (a) => (b) => (c) => (d) are obvious and valid in any
Banach space (not necessarily having the RNP). Now assume (d) and show (e) holds.
We can assume ? is countable, since the range of X is separable. By Proposition A.4.1,
{X, z) is right continuous in the mean for each ; e Z, hence it satisfies condition F') of
Theorem 2.1:
\imE((lFX,,z)) = E({lFX,,z)) fori <эо, F e ?,, ze Z.
Then by Theorem 2.1, X = N + A where N is a local martingale and Л is a cadlag
predictable process with finite variation and A(> = 0. Let ? be a cadlag modification of N
(see Theorem 2.15). Then ? = ? + A is a cadlag modification of X. This implies that
(Y, z) = (X, z) up to an evanescent set, for each fixed ; e Z. Since ? is countable, X = ?
up to an evanescent set and the theorem follows. ?
Lemma 2.5 can now be completed as follows.
THEOREM 2.13. Assume that ? has the RNP. Let ?:??]0,??] -> ? be an additive
measure with bounded variation |?|. Then there exists a cadlag quasimartingale ? on
]0, oo] such that ? = ?? if and only if the following two conditions hold:
For every t ^ 0, there exists a function f, e L' (P) such that
|M|(]f,oo]xF)iC?(lf/,) forFeE, B1)
and
??????,?] ? F) =0 for s < oo and F e 2ГД. B2)
? is
PROOF. Since ? has the RNP, by Lemma 2.3, ? satisfies B1) if and only if there exists
an ?-valued quasimartingale W such that ? = ?\?. Then by Theorem 2.1, W has a cadlag
modification У if and only if condition F) of Theorem 2.1 holds, which is equivalent to
condition B2) above. ?
2.5. Cadlag modification without RNP
In this section we shall investigate the existence of right continuous or cadlag modifications
of X without assuming that ? has the RNP. The characterization of cadlag modifications
of X is given in terms of its right limits X+ along the rationals (any countable set dense in
E+ can be used). The following results are proved:
A) If X is a local martingale then X+ exists (outside an evanescent set) and is a cadlag
modification of X.

494
J, K. Brooks
B) If X is a quasimartingale and if X+ exists (outside an evanescent set), then X+
is a modification of X if and only if the regularity condition (R) holds. If X is a
real-valued quasimartingale, then X+ and X- exist (outside an evanescent set).
C) If X is a process with integrable variation, then X+ (which exists everywhere) is a
modification of X if and only if condition (R) holds.
2.5.1. Right and left limits. For ? e ? and t e R+, we denote (in this section only)
?,+(?) = lim ?,-(?), ?,-(?) = lim ?,-(?),
r U с ?'
for r rational, whenever these limits exist. We denote by X+ and X- the functions
(t, ?) н> ?,+(?) and (t, ?) н> ?,_(?), respectively.
We notice that if X+ is defined on a set ? = Ш+ ? F with F С ?, then X+ is right
continuous on M. Moreover, X+ is cadlag if and only if X- also exists on ?.
We know that if X is a real-valued supermartingale, then X+ and X_ exist a.s., i.e.,
outside an evanescent set, see Dellacherie and Meyer A978, VI.3.). It follows that if X
is a real-valued quasimartingale, then X+ and X- exist a.s. But if X is an ?-valued
quasimartingale, we do not know whether X+ and X exist. We list first some general
properties of X+.
Lemma 2.14.
(a) IfX has a right continuous (respectively cadlag) modification, then X+ (respectively
X+ and X-) exist a.s. and X+ is a right continuous (respectively cadlag)
modification of X.
(b) Let (T„) be an increasing sequence of stopping times Tn f oo, such that XT"
has a right continuous (respectively cadlag) modification for each n. Then X+
(respectively X+ and X-) exists a.s. and X+ is a modification of X.
(c) Let ? С ?" be a countably set, norming for the range of X. If X+ exists a.s. and
if (X,z) has a right continuous modification for each ? e Z, then X+ is a right
continuous modification of X.
PROOF, (a) If У is a modification of X, the assertion follows from the equality Xr = Yr
a.s. for each r rational.
(b) For each и, (Хт")+ exists, by (a), and is a modification of XT". Then X+ exists a.s.
and (XT,<),+ = XTt" a.s. for each t; letting ? -+ oo we get Xt+ = X, a.s. for each t.
(c) For each ? e Z, (X, z)+ exists a.s. and is a modification of {X, z). Then
(Xl+,z) = (Xl,z)+ = (X„z) a.s.
for each ? e ? and t ^ 0. Since ? is countable, X,+ = X, a.s. for each t. О
2.5.2. Local martingales. We call X a local martingale if there is an increasing
sequence (Г„) of stopping times with Г„ t oo a.s., such thatforeach n, XT" is a martingale
(not necessarily right continuous). This means that for each ? and t, X," is integrable;

Stochastic processes and stochastic integration in Banach spaces
495
but Xj is not necessarily integrable if ? is an arbitrary stopping time. We note that a
modification of a local martingale is not necessarily a local martingale.
THEOREM 2.15. If X is a local martingale then X+ and X- exist a.s.\ X+ is a local
martingale and a cadlag modification of X.
PROOF. Let ?„ f oo such that XT" is a martingale for each n. By Brooks and Dinculeanu
A987), each XT" has a cadlag modification. Then we apply Lemma 2.14, (a) and (b), to
deduce that (XT")+ is a modification of XT" and X+ is a modification of X. Moreover,
X+ is a local martingale, since for each ? we have (X+)T" = (XT")+ and (XT")+ is a
martingale. ?
2.5.3. Quasimartingales We first state a new theorem on real-valued quasimartingales.
In the sequel, condition (R) will be referred to as condition B3).
THEOREM 2.16. If X is a real valued quasimartingale on bounded intervals, then X+
and X- exist a.s. and are quasimartingales. X+ is a cadlag modification of X if and only
if condition B3) holds:
\imE(lFX,)=E(lFX,) fors < oo and F еГ,. B3)
PROOF. Without loss of generality we can assume that X is a quasimartingale on ]0, oo].
Then ?? has bounded variation \??\ on Щ0, oo] and ? + and ?? are positive additive
measures on 7?]0, oo]. Let ? and ? be negative submartingales such that ? J = ?<*> and
?? = ??. Then ?+, ?+, ?+ and ?- exist a.s. and are submartingales (Dellacherie and
Meyer, 1978, VI.3). From the equality ?? = ??-? on TZ]0, oo], we deduce that ? - ?
is a modification of X. It follows that X+ and X- exist a.s. and are modifications of
the quasimartingales ?+ — ?+ and ? — ?-, respectively; hence X+ and X- are also
quasimartingales.
If X+ is a right continuous modification of X, then X is right continuous in the mean
(Proposition A.4.1) and condition B3) follows.
Conversely, if condition B3) holds, then, by Lemma 2.5, X has a cadlag modification;
by Lemma 2.14(a), X+ is a cadlag modification of X. ?
THEOREM 2.17. Assume that X is a quasimartingale on bounded sets, has separable
range and X+ exists a.s. Then X+ is a right continuous modification of X if and only if
condition B3) holds.
PROOF. Assume first that X satisfies condition B3) and let ? с ?" be a countable set,
norming for the range of X. Then for each ? e Z, (X, z) is a real-valued quasimartingale
satisfying condition B3). By Theorem 2.16, (X, z)+ exists a.s. and is a cadlag modification
of (X, z). Then, by Lemma 2.14(a), X+ is a right continuous modification of X.
Conversely, if X+ is a right continuous modification of X, then X is right continuous in
the mean (Proposition A.4.1) and condition B3) follows. ?

496
J.K. Brooks
2.5.4. Processes with finite variation. The following theorem is new even in the scalar
case. We remark that a modification of a process with finite variation does not necessarily
have finite variation.
THEOREM 2.18. Assume X has finite variation \X\ and that \X\, is integrable for
every t. Then X+ {which exists everywhere) has integrable variation on bounded intervals.
Moreover, X+ is a modification of X if and only if condition B3) holds.
PROOF. The existence of X+ and X_ follows from the inequality \\X,-XS\\ ^ |X|,-|X|A
fori < t. From this inequality we deduce \\Xt+ — Xs + W ^ \X\t+ — I-KL + fori < t. Since
\X\, is increasing, it follows that X+ has finite variation |X+|, and that \X+\ ^ \X\ + . X+
is right continuous and adapted, hence its variation |X+| is right continuous and adapted.
From \X+\, ^ \X\,+ ^ |X|,+i e L1 we deduce that \X+\, e L1 for every t ^ 0. Now,
since X has integrable variation on bounded intervals, it is a quasimartingale on bounded
intervals. By Theorem 2.17, X+ is a cadlag modification of X if and only if condition B3)
holds, and the theorem is proved. ?
Remark 2.19. If the variation \X\, of X on [0. t] is integrable, then it is ^-measurable.
In fact, the set ? of functions of the form ]Г, !№, + , - %t, II with 0 = ?? < ¦ ¦ ¦ < ?„ = t is
directed and bounded in L\Et), hence sup<? exists in L[(Et). Since \X\, = sup<? a.s. it
follows that \X\, is ?1,-measurable.
2.6. The Doob-Meyer decomposition of quasimartingales
Theorems 2.1 and 2.12 can be viewed as giving equivalent conditions for the Doob-Meyer
decomposition X = ? + A of a quasimartingale, in terms of right continuity. Lemma 2.11
gives conditions for the Doob-Meyer decomposition of quasimartingales of class (D). The
following theorem is an extension of both Lemmas 2.9 and 2.11 for quasimartingales of
class (LD).
THEOREM 2.20. The following two assertions are equivalent:
(a) X is a quasimartingale on bounded intervals, is of class (LD) and satisfies B3);
(b) ?? has bounded variation on each ring TZ]0, a] with a < oo, and is ?-additive on
TZ]0,oo[.
If ? has the RNP, the above assertions are equivalent to
(c) X = ? + A, where ? is a (not necessarily right continuous or bounded) martingale
and A is a cadlag, predictable process with Aq = О and with integrable variation on
each bounded interval. The decomposition is unique up to evanescent set.
For the correspondence X ++ ?? ++ ? +? established above we can state the following
completions.
THEOREM 2.21. X is a quasimartingale on ]0, oo[ of class (LD) and satisfies condition
B3) if and only if ?? is ?-additive and of bounded variation and on Щ0, oo[ if and only
if ? is a martingale and A has integrable variation.

Stochastic processes and stochastic integration in Banach spaces
497
THEOREM 2.22. X is a quasimartingale on ]0, oo] of class (LD) and satisfies condition
B3) if and only if ?? has bounded variation on 1Z]0, oo] if and only if ? is a martingale
bounded in L]E and A has integrable variation.
THEOREM 2.23. X is a quasimartingale on ]0, oo] of class (D) and satisfies condition
B3) if and only if ?? has bounded variation and is ?-additive on 1Z]0, oo] if and only if
? is a martingale of class (D) and A has integrable variation.
PROOF. Theorem 2.23 is proven in Lemmas 2.9 and 2.11. To prove Theorem 2.20, we
apply Theorem 2.23 for each ? e N to the stopped quasimartingale X". The implication
(a) =>· (b) follows then by noticing that if ?? is ?-additive on each ring 7Z]0, и],
then it is ?-additive on Щ0, oof. The implication (a) => (c) is proved by decomposing
X" = M" + A" where M" is a martingale of class (D) and A" is a predictable, cadlag
process with integrable variation and ? ? = 0. From the uniqueness of this decomposition
we deduce the existence of two processes ? and A which stopped at ? are equal to M"
and A", respectively. Finally, we remark that ? is itself a martingale. The implications
(b) =>· (a) and (c) =>· (a) are evident. Theorems 2.21 and 2.22 are immediate. D
Appendix A
In this appendix we shall collect some properties of the concepts involved in this chapter.
АЛ. The Doleans function
The following properties hold for the Doleans function ? ? introduced in Section 2.
АЛЛ. X is a martingale if and only if ? ? = 0 on 7Z]0, oof.
АЛ.2. X is a submartingale if and only if ?? ^ 0 on 7Z]0, oof. X is a negative
submartingale if and only if ?? ^ 0 on7Z]0, oo].
АЛ.З. If ? is another process like X, then ?? = ?? on 7Z]0, oof if and only if X - ? is
a martingale. We have ?? = ?? on Щ0, oo] if and only if X is a modification of ?.
АЛА If У is a negative submartingale and if \??(?)\ ^ ??(?), for A e 7Z]0, oo], then
\X,\ ^ —Y, a.s. for each t.
АЛ.5. For any stochastic interval ]S, T], with S and ? simple stopping times, we have
??]8,?]=?{??-?8).
АЛ.6. Assume X is a quasimartingale on ]0, or], for each ? < oo and let ? be a total
subset of ?". The following conditions are equivalent for any s < oo:

498
J. K. Brooks
(a) ????.?|??|(]?,?] ??) = 0;
(b) lim(iJMX(]j,f]x F) = Oforany F e ?,-
(c) ????,?^???,?] ? F),z) = 0, for any Fel, and с e Z;
(d) \imasE(lFX,)=E(lFX,), for F e ?„\
(e) lim,is(?(lFA:,), г) = (?A/- Xs), c), for F e 2Tt and ? e Z.
The equivalence of (a), (b), (c) was proved in Lemma 2.2. The equivalences (b) о (d) and
(c) o· (e) are evident and are valid even if X is not a quasimartingale.
A.2. Mean variation
The mean variation was introduced in Section 3.
A.2.1. PROPOSITION. For any ? ? oo, we have
Varx]0,a] = sup5]||?(lfI(X,i-X.fl)|2:(,)||1,
where the supremum is taken over all finite families (]s,. ?,] ? F,) of disjoint predictable
rectangles contained in ]0, ?] ? ?.
To prove the above, arrange the endpoints л, and ?, in an increasing order, 0 ^ и \ ^
¦ ¦ ¦ ^ u„ = a. Then
]j/,f/] ? F, = [J ]и;, иу_ц] ? A/y,
where A,, = 0 if ]s,-, Г,] П ]иу, uJ + \] = 0 and Ai; = F, e 2Г„; if s, ^ uj < t,. For fixed /,
the sets A,y and A^ are disjoint if ?'? к. Then
Yd\E{\F,(X,i-Xs,)^,)\x^Yi\E{(Xlli^-Xl,l)\EU])\x^yax]0,a],
and this inequality is preserved if we take the supremum in the left hand side. The converse
inequality is evident.
A.2.2. PROPOSITION. For any a ^ oo, we have
|??|(]0.?]??) = ?8??]0.?] and |?? |(]0.?[??) = Varx]0,a[.
For the proof, use Proposition A.2.1 above and
|F(X,,+, -?,,|?,,)||,=5??|?((?(?,, + , -?,,\?,??))\
the sup being over the ?1,,-measurable step functions / with ||/||эс ^ 1- See Pellaumail
A973,2D1) for the real case and Metivier A982, ex. E11.2).

Stochastic processes and stochastic integration in Banach spaces
499
A.2.3. PROPOSITION. For any a < oo, we have
\arx]0,a] = sup^2\\E<,Xs,.l - XS,)\\ = sup J] \\E(Xs,.t
i i
the supremum being taken over all finite increasing sequences 5? ^
stopping times Si, with S„ ? a.
The proof is similar to that of the scalar case (Kussmaul, 1977, Theorem 9.2). If X is a
right continuous quasimartingale, the above equalities remain valid for arbitrary stopping
times Si (see Proposition A.4.3).
A.3. Quasimartingales
For the definition of quasimartingales see Section 3.
A.3.1. X is a quasimartingale on ]0, a[ or on ]0, a] if and only if ?? has bounded
variation on 7?]0, а[ or 7?]0, a], respectively.
A.3.2. Any martingale is a quasimartingale on ]0. oof; it is a quasimartingale on ]0, oo]
if and only if sup, \\X,\\i < oo; in this case Var;e]0, oo] = sup, HXJi.
A.3.3. Any negative submartingale and any positive supermartingale is a quasimartingale
on]0, oo].
A.3.4. A process X with integrable variation (that is, E(\X\X-) < oo) is a
quasimartingale on ]0, oo].
A.3.5. X is a quasimartingale on ]0, oo] if and only if X is a quasimartingale on ]0, oof
and sup, \\Xi\\\ < oo.
A.3.6. X is a quasimartingale on ]0,a] if and only if the stopped process Xa is a
quasimartingale on ]0, oo].
A.3.7. IfX is a quasimartingale on ]0, a], then ||X|| is a quasimartingale on ]0, a].
A.3.8. PROPOSITION. X is a quasimartingale on ]0, oof (respectively on ]0, oo]) if and
only if the following condition is satisfied:
For every t < oo, there is a positive function /, e L' (P) such that for every F e ?, we
have
№x\(]t,oc[xF)^E(lFf,) (respectively \??\(]?,??] ? F)^E(lyf)).
This is Lemma 2.3 for quasimartingales on ]0, oo].
-Х5,-1ЗД|Р
¦ ¦ ¦ ^ S„ of simple

500
J. K. Brooks
A.4. Right continuous quasimartingales
A.4.1. PROPOSITION. Assume X is a quasimartingale on each bounded interval. X is
right continuous in the mean if and only if it is right continuous in probability. If X has a
right continuous modification, then it is right continuous in the mean.
The above proposition is due to Orey A965-66) for the scalar case. The proof for the
vector case is similar. See also Kussmaul A977, Theorem 9.3).
A.4.2. PROPOSITION. Assume X is a right continuous quasimartingale on ]0, oo]. Then
(a) Xt is integrable for every stopping time T;
(b) IfT„ is a decreasing sequence of stopping times converging to T, then Xjn —> Xt
in the mean.
This is proved by Orey A965-66) in the scalar case. See also Kussmaul A977,
Theorem 9.4).
A.4.3. PROPOSITION. Assume X is a right continuous quasimartingale. Then the
equalities in Proposition A.2.3 remain valid for arbitrary stopping times Si, with S1, < <*.
This follows from Propositions A.2.3 and A.4.2. See also Kussmaul A977, Theorem 9.2)
for the scalar case.
A.4.4. PROPOSITION. Assume X is a quasimartingale on each bounded interval, and let
? С ?' be a total set for E. We have B3) in case any one of the following conditions is
satisfied for every ? e Z:
A) (X, z) (in particular X) has a right continuous modification;
B) (X, z) (in particular X) is right continuous in the mean;
C) (X, z) (in particular X) is right continuous in probability;
D) (??, ?) (inparticular ??) is countably additive.
Indication of PROOF. By Proposition A.4.1, A) => B) and B) <s> C). Then B)
implies, for every ? e Z, s < oo and F e ?^:
]imE(lF(X,,z)) = E(lF{X„.z)),
which is equivalent to
lim^x(]s,t]xF),z) = 0.
lis
Then apply Lemma 2.2 to deduce B3). Finally, D) => B3) is evident. D
The author is pleased to acknowledge the permission of Springer-Verlag to use portions
of the material in references Brooks and Dinculeanu A988, 1990, 1991) for this chapter.

Stochastic processes and stochastic integration in Banach spaces
501
References
Brooks, J.K. A976), Stochastic processes in Banach spaces. Quasimartingales and stochastic integrals. Rend.
1st. Mat. Univ. Trieste 26 A994) 6-33.
Brooks, J.K. A999), Stochastic integration in nuclear spaces. University of Florida. Preprint.
Brooks, J.K. and Candeloro, D. B000), The foundation of weak stochastic integration theory in Banach spaces.
Tech. Report N. 15, Perugia, 1^4.
Brooks, J.K. and Dinculeanu, N. A974), Strong additivirv. absolute continuity and compactness in spaces of
measures, J. Math. Anal. Appl. 45, 156-175.
Brooks, J.K. and Dinculeanu, N. A976), Lebesgue-rxpe spaces for vector integration, linear operations, weak
completeness and weak compactness, J. Math. Anal. Appl. 54. 348-389.
Brooks, J.K. and Dinculeanu, N. A987), Projections and regularity of abstract processes. Stochastic Anal. Appl.
5, 17-25.
Brooks, J.K. and Dinculeanu, N. A988), Regularity and the Doob-Mexer decomposition of abstract
quasimartingales. Seminar on Stochastic Processes, Birkhauser. Basel, 21-63.
Brooks, J.K. and Dinculeanu, N. A990), ltd's formula for stochastic processes in Banach spaces. Conference on
Diffusion Processes, Birkhauser, Basel. 349-397.
Brooks, J.K. and Dinculeanu, N. A991), Stochastic integration in Banach spaces. Seminar on Stochastic
Processes, Birkhauser, Basel, 27-115.
Brooks, J.K. and Neal, D. A988), The optional stochastic integral. Seminar on Stochastic Processes, Birkhauser.
Basel, 45-54.
Brooks, J.K. and Neal, D. A991), A characterization ofHilbert stochastic integrals. Atti. Sem. Mat. Fis. Modena
39, 113-129.
Brooks, J.K. and Neal, D. A992), A vector measure approach to the optional stochastic integral. Ulam Quart. 1,
17-25.
Brooks, J.K. and Neal. D. A998), Generalized stochastic integral for real and Hilbert processes, Atti. Sem. Mat.
Fis. Modena 46, 189-247.
Brooks, J.K. and Neal, D. A998b), Generalized stochastic integral on nuclear spaces, Atti. Sem. Mat. Fis.
Modena 46, 83-98.
Dellacherie, С and Meyer, P.-A. A978). Probabilities and Potential, North-Holland. Amsterdam.
Dinculeanu, N. B000), Vector Integration and Stochastic Integration in Banach spaces. Wiley-Interscience. New
York.
Dinculeanu, N. B002), Vector integration in Banach spaces and application to stochastic integration. Handbook
of Measure Theory, E. Pap, ed., Elsevier, Amsterdam, 345-399.
Fisk, D.L. A965), Quasimartingales, Trans. Amer. Math. Soc. 120. 359-388.
Folmer, H. A972), The exit measure of a supermartingale. Z. Wahrscheinlichkeitsth. verw. Geb. 21, 154-166.
Gravereaux, B. and Pellaumail, J. A974). Fonnule de ltd pour des processes a valeurs dans des espaces de
Banach, Ann. Inst. H. Poincare 10, 399^422.
Kunita, H. A970), Stochastic integrals based on martingales taking their values in Hilbert spaces. Nagoya Math.
J. 38, 41-52.
Kussmaul, A.U. A977), Stochastic Integration and Generalized Martingales. Pitman, London.
Kussmaul, A.U. A978), Regularitat und stochastische Integration von Semimartingalen mit Werten in einein
Banachraum, Dissertation, Stuttgart.
Metivier, M. A974), The stochastic integral with respect to processes with values in a reflexive Banach space.
Theory Prob. Appl. 14, 758-787.
Metivier, M. A982), Semimartingales, deGruyter, Berlin.
Metivier, M. and Pellaumail, J. A975), On Doleans-Folmer's measure for quasimartingales. Illinois J. Math. 77,
491-504.
Metivier, M. and Pellaumail, J. A980), Stochastic Integration. Academic Press, New York.
Orey, S. A965-66), F-processes, Proc. 5th Berkley Symposium, Vol. 2, 301-313.
Pellaumail, J. A973), Sur I'integrale stochastique et la decomposition de Doob-Meyer, Asterisque, Vol. 9, Soc.
Math. France.

502
J. K. Brooks
Pratelli, ?. A988), Integration stochastic et geometric des espaces de Banach, Seminaire de Probabilities,
Lecture Notes in Math., Springer-Verlag. New York.
Protter, P, A990), Stochastic Integration and Differential Equations, Springer-Verlag, New York.
Rao, K.M. A969), Quasimartingales, Math. Scand. 24. 79-92.
Strieker, C. A977), Quasimartingales, martingales locales, semimartingales et filtration naturelles. Z. Wahr-
schenlichkeitsth. verw. Geb. 39. 55-64.
Ustunel, S. A982), Stochastic Integration on Nuclear Spaces and Its Applications. Ann. Inst. H. Poincare 18 B).
165-200.
Yor, M. A973), Sur les imegrales stochastiques a valeiirs dans mi espace de Banach. С R. Acad. Sci. Paris 277.
467^469.
Yor, M. A974), Surles imegrales stochastiques ? valeiirs dans un espace de Banach, Ann. Inst. H. Poincare 10,
31-36.

CHAPTER 11
Daniell Integral and Related Topics
M. Diaz Carrillo
Departamento de Analisis Matematuo. Universidad de Granada, 18071 Granada, Spain
E-mail: madiai@ugr.es
Contents
Introduction 505
1. Daniell integral extension 505
1.1. Daniell systems of functions and their first extensions 505
1.2. Daniell upper and lower integrals. Summable functions 506
1.3. Characterisations and extensions of integrable functions 508
1.4. Generalisations and related topics 511
2. Integral representations for linear functionals 514
2.1. Measures induced by integrals. Daniell-Stone theorem 514
2.2. Related results and applications 516
2.3. The interplay between measure and topology. Riesz representation theorem 519
3. The abstract Fubini theorem 521
4. The Radon-Nikodym theorem 522
5. Integral norms. Local integral metrics and Daniell-Loomis integrals 524
Acknowledgements 526
References 526
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
503

This page intentionally left blank

Daniell integral and related topics
505
Introduction
The Riemann integral extension and its limit theorems have a grave drawback, all
connected with the fact that it does not produce enough integrable functions. Reformulating
Lebesgue's ideas, Daniell extension shows how to improve on this constructing the upper
and lower integrals, analogs of Lebesgue's outer and inner measures.
The present article is devoted to a description of the different formulations of the
general Daniell scheme of extending certain linear functionals on a vector space, that is not
based on a preliminary construction of a theory of measure, this appears as consequence
of the theory of the integral. We offer the appropriate abstract forms of classic results
as the Radon-Nikodym theorem and Fubini theorem. Moreover, we point out when the
"functional approaches" are effective and we note the way to other generalisations. In
particular, looking at the integral as an extension by continuity under a suitable seminorm
or integral metric, matters simplify considerably.
We shall use the following terminology and notational conventions: Ш denotes the
extended reals, with the usual conventions concerning the arithmetic and order structure.
An assertion ? about extended real numbers is said to hold, if ? is true provided that
every sum appearing in ? is defined. When a set Л с К is not bounded from above
or below in R, we write sup Л = oo or inf A = —oo, respectively; and if A = 0, then
inf A = — sup A = oo. For a nonempty set X we let Rx consist of all functions /: X -> R.
The algebraic operations, partial ordering and limits on Rx are defined pointwise. For a
subset AcMxwe write +A = {/ e A: / ^ 0}. If a sequence (/„) С Шх converges to /
we denote it by /«->/, and if it is increasing or decreasing, we write /„ f / or /„ 4- /,
respectively.
Given a real vector space В с Rx, we say that ? is a ring of functions if it is closed
under taking products, ? is a vector lattice or a Riesz space of functions if it is closed
under taking pointwise maxima and minima: f,g&B=$-fvgeB and / л g e В (or
equivalently, if / e В also |/| e B, with |/|(.r) := |/(.r)|). В is Stonian if it contains
/ л 1 for any / e В (Stone's condition).
1. Daniell integral extension
This section gives a brief discussion of the Daniell extension procedure. The extended
function class L\ (I) of summable functions is defined via three classic methods: equality
of the upper and lower integrals, difference of monotone limits with boundedness in
integral / and closure of the class of elementary functions with respect to an L \ (I) type
seminorm.
We provide a simplified treatment of the former method and compare it with the main
steps of the others. Well-known text books on this field are Pfeffer A977), Royden A968)
and Weir A974), among others.
1.1. Daniell systems of functions and their first extensions
Let X be an arbitrary set and 0 ? В С ?? a vector lattice (elementary functions). A linear
functional /: ? -> ? is said to be nonnegative if 1(h) ^ 0 for every h e +B\ then it is

506
?. Diaz Carrillo
monotone increasing. In particular, \I(h)\ ^ /(|?|) f°r апУ h e В. A. nonnegative linear
functional / on В is called a Daniell integral whenever (D): (/?„) С В, h„ I 0 implies
/(/?„) —> 0. The triple (X. ?, /) is called a Daniell system. There are several conditions
equivalent to (D) used by different authors (Daniell, Riesz, Loomis, Stone) that are basic
for the construction of the integral starting from nonnegative linear functionals (e.g., see
Bogdanowicz A967, Proposition 2)).
Denote by ?? the class of all E-valued functions / on X for which there is a sequence
(h„) С В such that h„ \ f. Let /+(/) = lim /(/?„). Since (D) is equivalent to (D'): if (/?„)
and (g„) are increasing sequences in ?, and if lim/?,, < limg,,, then lim /(/?„) ^ lim/(g„),
we see that /+(/) is well defined.
We list the more relevant properties: В с В *, ?? forms a positive cone and is closed
under taking limits of increasing sequences, finite infima and countable suprema. If
/ e B\ then /+(/) = sup{/(g): g e B, g ^ /} > —oo; /+ is additive, positively
homogeneous and increasing on B'. If (/„) с ?? and f„ \ /, then /+(/) = lim I+(f„),
that is, /+ is continuous along increasing sequences of ??. As a consequence, the fact
that /+ commutes with sup (and series) when it is applied to an increasing sequence
(respectively nonnegative functions) from ??, is also especially important.
One introduces ?^ and /~ in complete analogy with ?? and /+. The properties of
(?^, /~) derive from those of (??. /+) by means of the following elementary assertion:
/ eBl if and only if -/e fit, and in this case /"(/) = -/ + (-/). Denote ? = Bi ???.
Order-continuous systems. ? \ f means that ? is a directed upwards system of
functions on X such that / = supjg: g e ?] (with dual definition for ? I /).
A nonnegative linear functional / on ? is called a Bourbaki integral on В whenever (B):
? С ?, ? ? 0 implies inf{/(/): / e ?} = 0. ?^ denotes the class of all functions on X
for which there is a system ? с В such that ? f /. Put /г(/) = Нт/(Ф). ? (?*, /?)
extension is constructed in the dual manner.
Recall that B^ с B^ and every Bourbaki integral on В is also a Daniell integral on B,
and that has a slightly superior integration theory. Bourbaki integral extensions can be
found, e.g., in Pfeffer A977), as an easy kind of dualization.
For an arbitrary linear functional /, we can reduce it to the nonnegative case by means
of the functional Jordan decomposition (see, for instance, Papangelou A966), Gunzler
A973, Lemma 1, p. 169)). Under some additional assumptions Adamski A983, 1986)
gives several characterisations of Daniell lattices and strong Daniell lattices (i.e., vector
lattices В с Rx, where every nonnegative linear functional ? : В —> ? is ?-smooth or net-
continuous). Although, in contrast to Gould A964) and Jacobs A978), Adamski does not
use any compactification in the latter case.
1.2. Daniell upper and lower integrals. Summable functions
Section 1.1 provides a rich class of functions that will be used to play the role that
open sets play in the construction of Lebesgue's measure. Now, the Riemann upper and
lower integrals are replaced by slightly different ones, called the Daniell upper and lower
integrals.

Daniell integral and related topics
507
For an arbitrary function /eRx one defines the Daniell upper integral 1(f) =
inf{/+(g): / < g e ??}, and the Daniell lower integral /(/) := -l(-f). 7 is positively
homogeneous, increasing and subadditive for /, g for which / +g is defined everywhere;
/^Tonl^andif /e ?? nBi, then7(/) =/(/). Next, the following result, which is
usually referred to as the "monotone convergence theorem", is fundamental to the whole
theory of integration.
THEOREM 1.1. Let(f„)<zRx, f, t f andl(f) > -co, then 7(/„) 11(f)-
Now, using Theorem 1.1 we obtain the classic Fatous lemma:
PROPOSITION 1. Let (/„) cR* such that l(inff„)> -oo, thenl(inff„) ^ 7(liminf/„) ^
liminf/(/„)·
Theorem 1.1 and finite subadditivity of 7 imply its countable subadditivity, which is one
of those features of 7 that the Riemann upper integral does not share.
From now on we will use L\A) to denote the class of those f еШх for which
1(f) = /(/) e Ш (I-summablefunctions), and we define the common value of 1(f) and
1(f) as the integral of / e L\(I) and denote it by /(/).
PROPOSITION 2. / e fif andl(f) < oo implies f e L\. L\ is "almost" a vector lattice of
functions containing B, and I is a nonnegative linear functional on L\. The class L \ П R
(which forms a genuine vector space on Ш), is not "much smaller" than L\, in fact: if
f & L\ and g e Rx, g(x) = f(x) for each ? e X for which f(x) is finite, then g e L\ and
I(g) = I(f).
Observe that L \ contains functions with values oo and -oo, so L \ is not a vector space
(one cannot define a "+" on it such that one gets an Abelian group), this is an error that
appears already in Daniell A917/18, p. 288), and also in several books. Of course one
can circumvent this difficulty, in Pfeffer A977, p. 61), Floret A981, pp. 81-82), Stroock
A994, p. 132), the situation is handled correctly. Also, since the infinity values are attained
only on null sets, one can form equivalence classes of functions which differ only on null
sets. Another approach which avoids these difficulties is given by an der Heiden A970),
where the Daniell method is modified in such a way that automatically a vector space
arises.
Now, an application of Proposition 1 and its dual, gives the Lebesgue dominated
convergence theorem.
Theorem 1.2. Let (/„) с L\, g,h e L\, and g ^ /„ ^ h, ? e N. /// = lim/„, then
/eL, a^/(/) = lim/(/„).
Then, with the above results we can establish that the functional / ???,? nRx is a
Daniell integral that extends / on B, and L ? has the closure properties desired.
A detailed presentation of the complete results as stated above can be found in Pfeffer
A977, Chapters 4 and 5). Also along the same lines see also Royden A968), Taylor

508
?. Dial Carrillo
A965) and Stroock A994). Pfeffer's book contains an excellent collection of examples
and counterexamples in measure and integration.
In the remainder of this section we group some rich classes of well-known examples of
Daniell systems.
PROPOSITION 3 (Radon measures). В = Coo(X) (continuous functions with compact
support on a locally compact Hausdorff space ?). ? is a Stonian lattice ring of bounded
functions. Any nonnegative linear functional on Cqo(X) (it is customary to term such a
functional a positive Radon measure on X) is a Bourbaki integral on Q)q(X). The first
Daniell and Bourbaki extensions of Q)q(X) coincide if and only if every open subset of X
is ?-compact (Pfeffer, 1977, p. 37).
PROPOSITION 4 (Linear extensions of set functions). Let ? be an ?-additive measure
on a ring R of subsets, Br denotes the lattice ring of step functions over R. The map
f ??? = ?,.6??? ([h = r]), h e ?/?, is the unique linear extension of ? to ?/?. This
extension preserves ?-additivity, at least for nonnegative measures.
Riemann-Stieltjes integral. Let X = [a, b], В = C(X) (continuous functions on X), and
let g be an increasing real-valued function on X. For /efi, let /(/) = fj f(x)dg(x),
then / is a Bourbaki integral on В.
1.3. Characterisations and extensions of integrable functions
The Daniell mean. A variant functional approach to integration of interest considers
the integral as an extension by continuity under a suitable seminorm, as introduced by
Stone A948/49) and Aumann A952). For a more general exposition of this subject see
Section 5 in this chapter. A recent book by Bichteler A998) presents a complete treatment
of integration using the Daniell mean || · || associated with the Daniell upper integral /,
defined by ||/|| = 7(|/|) e +1 for all / e Rx. Bichteler point out that "already in the
case of the Riemann integral, a simple absolute-value under the upper integral, simplifies
matters considerably"; also, since the most frequently encountered class of functions (in
the context of integration) are lattice rings, he starts with an elementary integral system
(X, ?, /), where / is a Daniell integral on a lattice ring ? of bounded functions 0? a
set X, and studies the structure of the closure of ? under || · ||. A sequence (/„) с ??
is said to be converging in || · \\-mean to / if ||/„ - /|| -> 0. The functional || · || is
absolute-homogeneous, solid (|/| ^ \g\ => ||/|| ^ \\g\\), countable subadditive and finite
on ? (elementary integrands). We remark that the utility of || · || lies in that it is upper bound
for the elementary integral, \I(h)\ ?. \\h\\ for all /iei.
A function / is called || · \\-negligible if its mean ||/|| is zero. A subset A of X is || · ||-
negligible if its characteristic function ?? is negligible; and the notion || · \\-a.e. arises as
usual.
L* denotes the class of || · \\-a.e. defined functions with finite mean. Now, an a.e. defined
function / e Rx is called || · \\-integrable if it belongs to the mean-closure off in L*. The
class of integrable functions is denoted by L|(?, /). For / e L\ define 1(f) = lim/(/?„),

Daniell integral and related topics
509
where(/?„) с ? converges in mean to /. L\ is complete in mean; moreover, if (/„) C+L|
satisfies supA. ||?„_| f„\\ < oo then ||/„|| -> 0, and the properties concerning limits
(monotone and dominated convergence theorems) are formulated in terms of mean. For
these results, and for more in the same context (on integrable sets, and their connection
with integrable functions, measurable functions, L''-spaces, etc.) see Bichteler A998). In
this way it is easy to prove the following characterisation:
Theorem 1.3. A function f is || · \\-integrable if and only if 1(f) = /(/) e R.
We remark that the above construction can easily be extended to functions with values
in a Banach space (?, || · || ?¦). In fact, the class L\(E) (Bochner integrable functions) will
be the || · ||-closure of ? <g> ? in L*t-. where L*. denotes the class of all ?-valued defined
functions on which the mean ||/|| = l(\\f\\t) is finite (Schafke, 1970,1972; Hoffmann and
Schafke, 1992). Finally, if signed elementary integrals / are considered, the finite variation
/(/) = sup{|/(/?)|: h e ?, |?| <c /} e R for all / e +?, replaces the positivity. Here,
L,(/) = L|(|/|), since |/(/)| <c |/|(|/|) <c Daniell mean for |/| (see Bichteler A998,
p. 140), GunzlerA973,p. 169)).
The closure of a Daniell system. Another functional approach, based essentially on
the order structure of the Daniell system (X, B, I), is contained in Constantinescu and
Weber A985). The notion of B-exceptional sets associated with the Riesz space В in
Шх is introduced (i.e., А с ?, ???? e ?), which have the characteristics appropriate to
exceptional sets in a sense of "almost every where", in terms of Riesz spaces, and the
notion of B-a.e. also appears as usual. For an increasing functional / on a subset F с Шх,
the triple (X, F, /) is said to be closed if, for every monotone sequence (/„) С F with
(/(/„)) bounded in R, the function lim/„ e F and /(lim/„) = lim/(/„). For a closed
Daniell system, the ?-exceptional sets are exactly those subsets of X whose characteristic
function have integral zero. The explicit process for extending a given Daniell system to a
closed Daniell system allows several ways of characterising L\ (/), constructed as above,
with the notation S+ = {Д|1б1,: /„: (/„) с В. f„ ;}.
THEOREM 1.4. Let (?. ?, /) be a Daniell system. Then the following assertions are
equivalent, for every f e Шх, a e R
(i) For every ? > 0, there exists an ?-bracket of f relative to I (i.e.. a pair (h. g) e
?/ ? ?? such that h <c / <c g. /-(/;). I+(g) e R and I'(g) - r(h) < ?), and
1(f) = L(f) = a.
(ii) For every ? > 0, there exists an ?-bracket of f relative to I (h. g) ? Bi ? ?? such
that ?(h) iCa iC I+(g).
The equivalences are all consequences of the definitions of the space L\(I) and of the
functionals/+ and /~ (see Constantinescu and Weber A985, p. 214)).
The triple (X. L\(I), /) is the smallest closed Daniell system extending every Daniell
system (X, B. I). In addition, the properties of the given Daniell system are characterised
(Induction Principle, Constantinescu and Weber A985, p. 313)). We note that the closed
Daniell systems constructed here are not yet "integrals" in the sense of the above authors;

510
?. Diaz, Carrillo
an additional step is required adjoining the Daniell system in an obvious way, and
then taking the Daniell system closure. This method uses the notion of an "admissible
extension" of a Daniell system (X. B, /), that is to say, a closed Daniell system (X, B, I)
extending (X, ?, /) such that for every / e В there is a function g e L \ with f = g ?-a.e.
In a more recent book by Constantinescu, Filter and Weber A998), the above point of view
of integration theory via Daniell methods, is also treated.
Difference of monotone limits. For a Daniell system (X, B, I) another standard functional
approach to obtain the class of all /-summable functions (originally given by Daniell
A917/18)), can be found in classic texts such as Weir A974) or Shilov and Gurevich
A966).
The main steps are: А с X is said to be null if there is an increasing sequence (/?„) с В
for which /(/?„) is bounded and h„ (x) diverges for every ? e A. The corresponding notion
of a.e. follows as usual, and if h e Rv, h = 0 a.e. (null function), then h e L\{I) and
1(h) = 0. Let the positive cone Lmc = {/ e ??: 3(?„) С В, h„ \ f a.e. and /(/?„)
bounded}. For / e LC, define /(/) = lim/(/j„). One has that any function in L"K is
equal a.e. to a function of Вт П L\(I); thus, if / e Rx such that f=h-g,h,ge L"K,
then / e L|(/). On the other hand, a similar argument (?-bracket) used to prove ii) in
Theorem 1.4, and with the monotone convergence theorem (in the L"u'-process), proves
that every /-summable function can be expressed as the above difference h — g. A detailed
explanation of the results as stated above can be found in Weir A974, p. 11). Note that the
corresponding formulation of the monotone convergence theorem in terms of absolutely
convergent series (Weir A974, p. 15)), allows the whole theory of Daniell integrals to be
worked out in terms of absolute convergence; this method is considered by Asplund and
Bungart A966). Finally, observe that if f e L\(I), there is a function h e B* with h ^ /
and /(/) finite. The above definition of null set implies that h, and hence /, can only take
the value ooona null set А с X (analogy, with the value — oo), and 1(хл) = 0 (so, it is a
negligible set). This point is essential for the usual convention about "linear combination"
in ? and "admissible" extension mentioned above.
I-integrable functions. A final extension appears in Pfeffer A977, Chapter 6), to extend
В U L\ (I) adding to it a large class of functions whose integrals are infinite. L^ denotes
the class of all functions / e Rx such that / л h e L\ for all h e В (equivalently,
for all h e L\)\ and L~ :— —L^. A function / on X is called I-integrable whenever
/eli:=i.f Ui.7.
One has ?? с L+ and L\ = LJ П LJ. It is consistent to define /(/) = /(/) if
/ e L^, and the extension process is closed. Note that an / e +LJ is sometimes called
nonnegative /-measurable function (see, e.g., Taylor A965, p. 246), Royden A968, p. 295),
Volcic A980)). This final extension has useful properties for studying extended real valued
measures (respectively integrals) induced by integrals (respectively measures), as treated
in Section 2.
Note that if / e L^ and /(/) < 00, then / e L\\ and if 1 e L\, /-integrable sets
(Xa & L+) coincide with /-summable sets. We mention that if / is a Daniell integral with
corresponding space L\ (I) of summable functions, abstract LP{I) spaces containing all

Daniel! integral and related topics
511
the measurable functions (in the sense of Stone) for which \f\p e L\{I), ? > 1, can be
studied (see, e.g.. Weir A974, p. 206)).
1.4. Generalisations and related topics
The systematic search for more general basic systems (?,?,/), or certain particular
systems, to obtain new applications, as well as to unify methods, has given special
functional approaches to integration.
Non-lattice integration. Daniell-Stone integration also makes sense if the underlying
function vector space E, on which the nonnegative linear / is defined, is not a lattice.
The first author to have this idea seems to have been Leinert A982); his aim was to work
in abstract harmonic analysis (see Leinert A984) where he gives a proof of Plancherel's
theorem). The space L\ is constructed as the closure of ?" in an L \ -type seminorm.
Namely, for /eP,
7(/):=inf(?/(/„): f„e+E, ?/„>/);
11/11 = /(|/|) for every / e R*,and L\ is the norm closure of ?" = {/ e ?; ||/|| < oo} in
the class of all extended real-valued functions on X with finite norm. Then, the Beppo Levi
theorem is proven, and with some weak additional assumptions, the classic convergence
theorems follow, which are proved in the usual fashion. These results seem to have
been done essentially already by Aumann A952) with the same integral norm. He also
proves a Beppo Levi theorem. It could be checked whether the lattice structure is here
essential. A similar generalisation is investigated by Konig A982, 1992) with applications
to complex algebras of continuous functions closed with respect to uniform convergence
and to Hardy algebras. Starting with the sequential continuity condition (?): for each
increasing sequence (h„) с ? with lim/?,, > 0 we have lim/(/j„) ^ 0, then L]n(I) is
defined as the closure of ? in Rx with respect to the sequential upper integral I„(\ ¦ |)
(or Stone's integral norm, see Aumann A952)), where In(f) = infjlim/f/j,^: (/;„) С Е
increasing with / ^ lim/?,,}. Equivalently, h\ consists of all functions / e Шх such that
h(f) + In(—f) = 0, i.e., it is the linearity space of the sublinear functional l„. To
obtain convergence theorems / needs to fulfill a Beppo Levi condition; similar situation is
treated with "summenbeschrankt" by Aumann A952), "starke integralnorm" by Schafke
A977), and "upper S-norm" by Bichteler A973). Furthermore, in order to obtain the full
results, one has to assume the fortified continuity condition (?_): In(f) = I~{f)ior all
/ ^ 0, where /+ (satellite functional associated with I„), is defined for all / e Rx, as
above with (h„) с +Е. Now, the monotone representation theorem is proven: for / e L\
there are u, ? monotone limits from E, such that / = и — ? a.e.; which, combined with
the previous description of Lj,, show that the three afore-mentioned classic extension
methods to construct the spaces On continue to produce identical results under the present
assumptions. Integration in the sense of Bourbaki (which is not covered by Leinert), is
considered here as a parallel dualization.
Another generalisation along these lines was investigated by Leinenkugel A992), who
uses non-lattice integration procedures to specialize Denjoy and Perron-Li spaces. This

512
?. Diaz Carhllo
proves useful in some cases that could not be treated with the usual Daniell-Stone
integration theory.
Order theoretic, non-additive integration. The idea of an order theoretic presentation
of integration was most extensively dealt with Alfsen A958, 1963) (see also McShane
A953)). The complete "absence of linearity" requires modifications in the definitions and
proofs of standard methods of integration. This question has been reviewed by Riecan
A979), who gives a unified and simplified study, with the help of an ordering, of the
analogy between the measure theory and the integration theory. He starts with the notion
of full integral introduced by Alfsen: Let J: A —>¦ R be a monotone map defined on a
sublattice A of a lattice H, satisfying the conditions:
(i) J(x) + J(y) = J(x ? v) + J(x л v), for all jr. у e A (J is a valuation);
(ii) (x„) с ?, ? & A, x„ ? ? implies J(x„) —>¦ J(x) (x„ \ ? means x„ ^ x„+\, ? e N
and ? = \Jn x„)\ and the following Beppo Levi requirement·.
(iii) (xn) с A, x e #, x„ | ? and (J(x,,)) bounded implies ? e A.
If we assume that ? is a relatively ? -complete sublattice of an ? -continuous lattice H,
then (A, p) is a complete pseudometric space, where p(x. y) = J(x ? у) — J (? л у) for
all x,y e A (Alfsen A963, Theorem 2)). In particular, if A is the class of all ?-integrable
functions, J(/) = j f ?? and p(fg) = /(/-#)??, one has the Riesz-Fischer theorem.
Construction of a full integral can be solved in various ways. For instance, the first
extension of the Daniell system given in Section 1.1, and the well-known Caratheodory
method can be adapted (for results in this direction see Riecan A964, 1975), Futas
A971), Brehmer A974)). Various generalizations are also discussed by Riecan with a
general algebraic system (quasilinear structures and subadditive measures); in this setting,
an interesting decomposition theorem is presented including the Jordan decomposition
for Daniell integrals (Riecan A979, p. 227)). Also an der Heiden A970) considers
Daniell integrals with values in abstract boundedly complete partially ordered vector
space. A further generalized Daniell extension (using partially ordered groups, weakly
?-complete lattice-ordered groups) can be found in Wilhelm A988), where some theorems
are adaptations to this more abstract setting of the results known for function spaces (see
Schafke A971)), and in the more recent book by Riecan A997). For related results, using
natural generalisations of a nonnegative linear functional ("prelinear", "strong sublinear",
"monotone" functions) or of a nonnegative additive measure (subadditive measures,
capacity functions) see also Sipos A979a, 1979b). Greco A977). Schmeidler A986),
Denneberg A994) and Pap A995).
Non-additive measures and integrals might be of interest to pursue with respect to future
applications in problems that cannot be treated with additivity: problems in economic,
fuzzy measures, artificial intelligence, Bayesian decision theory, subjective probability,
among others. For an exposition of non-additive measure see Part 9 in this Handbook.
Essential integration. A unified functional approach to integration based on the notion of
the upper functional (an abstract version of the upper integral) is given by Anger and Porte-
nier A992). The procedure is different from the more abstract one discussed by Aumann
A952), Hoffmann and S_chafkeJ_1992), Schafke A977) or Bichteler A973) (see also our
Section 5). A given v:Rx -> R is called an upper functional if ? = {/ e R: v(f) <

Daniell integral and related topics
513
00} is a lattice cone on which ? is positively-homogeneous with v(f +g)^v(f Ag)
+ v(f v g) ^ v(_f) + v(g) (strongly sublinear on 7"), satisfying v(f) = inf{u(g): / ^
g e T] for / e Wx (determined by 7), with R=]-oc, 00]. Then, the class of all v-
integrable functions is defined J(v) = {/ e Rx: v(f) = u*(/) := -v(-f) e R}. i(u)
is closed in Rx with respect to v(\ ¦ |). The special case where a function cone 6сК"
that is closed under minima and ? : & —>¦ R is linear and regular (? = ?* on ?) gives
?*(/) = inf^(g): f ^ g e &} which is an upper functional and J (?*) is the abstract
proper Riemann integral (see the particular situation treated by Bobillo and Diaz Carrillo
A987)). Next, essential integration is investigated, which comprises abstract improper
Riemann integrals as well as essential Radon integrals. It is based on the essential upper
functional v*(f) = inf/,ey $upkeJ_ (v(f л(-к)) ? /г), where J- = {/ e i(u), / ^ 0}. v*
corresponds to Schafke's local integral norm (Schafke A977)). J(v') defines the class of
all essentially v-integrable functions. If / is a nonnegative linear functional on a vector
lattice S, with ? = /*, Diaz Carrillo and Mufioz A990) have introduced J(v').
An upper functional ? is called an upper integral if it satisfies Daniell's property. For
any increasing sequence (/„) с Rx, f(sup„ /„) = sup,, v(f„). Then, for upper integrals
the usual convergence theorems are obtained; furthermore, measurability in the sense of
Stone is considered. For related results (with the corresponding strong integral norm) see
also Bichteler A973). The usual Daniell integration theory is a special case of the above
general theory. In fact, if ? is a monotone linear functional and regular (always true for
vector spaces) on a function cone ? closed under minima, then the usual extension ?"
to monotone limits from ? gives upper functionals ? = ?&·. ?' satisfying Daniell's
condition; and the Daniell-?-summable functions are obtained by the above closure J(v) for
the cone of monotone limits from ? (see Theorem 9.10 in Bichteler A973)). Fubini
theorems are not investigated for essential integration, which are treated with integral
metrics (see Section 5). For the case of finitely additive set function essential integration has
been introduced by Gunzler A973), extending the Dunford-Schwartz integral (Dunford
and Schwartz A958)).
For alternative points of view, we note that techniques of categories and functors are used
to introduce in a new "axiomatic" way the notion of upper and lower integrals of "Daniell
measures" and to prove extension theorems and generalised monotone convergence
theorems (see Cicogna A976) and Bogdan A975)).
Further generalizations and related results have been considered to Daniell-type
integration theory: integrals having values in a ordered semigroup by Thiam A975);
integrals having values in a Dedekind-complete vector lattice by Kluvanek A965) (see
also Rudin A975), Muni A974) and an der Heiden A970)), with values in a Banach space
by De Lia and Mikusinski A995); integrals defined on a vector lattice of functions having
values in a quasi- complete locally convex Hausdorff space by Okada A984); and integrals
defined in locally convex lattices by Bogdanowicz A971), among others.
Non-continuity integration. Analogues to the Daniell extension process, with or
without weaker continuity assumptions on the elementary integral, have been treated by Au-
mann A952), Loomis A954) and Gould A966). More recently, Bobillo and Diaz Carrillo
A987, 1989) discussed an integral extension of Lebesgue power, starting in this case with
a vector lattice В с R* and a nonnegative linear functional / on B. The triple (X. B, I) is

514
?. Diaz Carrillo
called a Loomis system, for which we define a first extension BT = sup{M: 0 ? ? С В)
and / + (/) = sup{/(g): / > g e ?} for all / e R*. Since /+ is not additive on ??,
we introduce the class BT = {/ e ??: / + (/ + g) = /+(/) + /+(g) for all g e ??}·
Now, using the classes ?? and — ??. for each / e Rx, the upper and lower
integrals /(/)_and /_(/) are defined as usual (see Sections 1.1 and 1.2). The elements of
? = {/ e Rx: 1(f) = 1(f) e R) are called /summable (with respect to a Loomis
system). This extension permits us to obtain results similar to those for the Daniell-Bourbaki
case; for instance, convergence theorems are proven by Gunzler A991). We recall that
"no continuity conditions" on the starting elementary integral allows us to subsume
situations of integration with respect to finitely additive measures ?. Diaz Carrillo and Gunzler
A997) generalise the above process by "localization", using an appropriate local
convergence in measure, which is very useful for obtaining convergence theorems analogously
to the classic ones (some of which are not true for В ). Namely, the set L = L(B. I) of
/integrable functions is defined as the set of those / e Rx for which there exists an
7-Cauchy net (h„) с В with h„ -> /G), which means that 7(|/„ - /| л h) -> 0 for each
fixed h e +B. L coincides with the Daniell L\ in the classic case. A similar R\-theory
(abstract Riemann integration) replacing / by I~(f) := —/_( —/). was treated by Diaz
Carrillo and Munoz A993), which subsumes the space of abstract Riemann-?-integrable
functions R ? (?, R), defined as above, but with h„ -> / "?-locally" (Gunzler A985, p. 199,
70)). Then, given the Loomis system (X. Bq. /;,) induced by a finitely additive measure
?:?^- [0, oof, where ? is a semiring of sets in X, one has R^ (?. ?) (proper Riemann
?-integrable functions) с DS(X, ?. ?. R) (Dunford-Schwartz integral A958, p. 112))
С R\^,W) с L(Bn. //;), with coinciding integrals, and all С are in general strict. For
general Loomis systems the above extension classes are specialized to abstract Riemann,
Loomis, Daniell and Bourbaki integrals (see Diaz Carrillo and Gunzler A997, p. 1084)).
2. Integral representations for linear functionals
For integral representation in the scalar case, one usually has the following general
situation: Let us take an abstract set X, a vector lattice i,cMx, / : L —>¦ R linear and
continuous in some sense, and a class ? of subsets of X in some way associated with
(X, L, I). Then, one may ask if there is a set function ? on ?, possibly unique, such that /
is representable by ?, i.e., all functions / e L are ?-integrable and satisfy /(/) = f f ??-
The question is answered, in a very special and simple case, by the original
representation theorem of F Riesz, which states that any nonnegative linear functional / on С ([a. b])
can be given by means of a Lebesgue-Stieltjes integral. This result has been generalised
in several significant ways. For an overview of this subject see Batt A973), Weir A974,
Chapter 10), an der Heiden A978) and Glicksberg A951).
First we focus our attention on Daniell systems (X. B. I). Various generalisations and
special cases will be considered later.
2.1. Measures induced by integrals. Daiiiell-Stone theorem
Given a Daniell system, the main purpose of this section is to derive conditions for the
existence of a measure ? with the property stated above. (?. ?. ?) denotes a measure

Daniel! integral and related topics
515
space, whenever ? is a ? -algebra in X and ? an extended nonnegative measure on ?.
The elements of ? are called ?-measurable sets. The following theorem summarizes the
well-known facts.
THEOREM 2.1. Let (X, B, I) be a Daniell system, and suppose that L\(I) is Stonian. Let
A consist of all I -integrable sets А С X, and let v(A) = ?(?,\) for each A e A. Then
(X, A, v) is a measure space (induced by ?), ? is complete and saturated (every locally
?-measurable set is A-measurable).
The proof essentially uses the fact that +LJ is closed under taking limits and the
countable additivity of 7 on+L+ (see PfefferA977, pp. 122, 126)).
PROPOSITION 5. If M(A± denotes the family ofall ?-measurable functions in X (i.e.,
/"' (B(R)) с Л with В(Ш) Borel ? -algebra in I), one has for any f eRx: f e ? (A)
if and only if /^ e L+, and in particular, f e L\ if and only if f e ? (A) and
I(f+) — l(f~) has meaning.
Note that many of the results in Section 1.2 can be strengthened using the notion of
Ao-a.e., with the null ideal An = {A e A: v(A) = 0}.
Integrals induced by measures. Given a measure space (?.?,?) we shall define a
Stonian Daniell system Bq on X and a Daniell integral ?? on Bq that we extend to L \ (??)
(/?-integrable functions) (see Proposition 4 in Section 1.2). Namely, the Stonian vector
lattice Bq consists of all i2-simple functions on X (i.e., immeasurable functions / that
assumes only finite number of values and with ?([f ? 0]) < oo). Now, given the usual
canonical form of / e Bq, the map defined by /„(/) = ?/'=? c4fJ-(Aj) gives a Daniell
integral on Bq .
Next, using the notion of /-integrable function given in Section 1.3, we can extend the
integral ?? to L \ (??) and according to Theorem 2.1, define a new measure space (X. A, v)
induced by ??. The relationship between the measures ? and и (induced by ? via /;<), is
given for the following assertions·.
PROPOSITION 6. The measure ? is a complete and saturated extension of ?, i.e., ? С А
and v(A) = ?(?) for each A e ?. If, in addition, ? is complete and saturated, then
(?,?,?) = (?,?,?) (see Pfeffer A977'. p. 152)).
Now, applying the process of inducing measures by integrals and vice versa, the
following theorem is obtained.
THEOREM 2.2 (Daniell-Stone). Let (X.B.I) be a Daniell system, and suppose that
L\(I) is Stonian. Let (?,?,?) be the measure space induced by I (Theoren^lA), and
let (X, /?_д, Iv) be the Stonian Daniell system induced by v. Then L\(I) = L\(IV) and
Hf) = l,-(f) for all /el|(/).

516
?. Diaz Carrillo
The proof uses the fact that ? is complete and saturated; so, with Proposition 6, (?, ?, v)
is the measure space induced by / and /,., and the classic approximation of nonnegative
Д-measurable functions for nonnegative Д-simple functions and Proposition 7 (see Pfeffer
A977, Theorem 12.18)).
Abstract Lebesgue integral. Given a space of measure (?, ?, ?), for any nonnegative
simple function /, f f ?? is defined as above. Now, for any nonnegative immeasurable
function / on X, let f /?? = sup{f g??: g e ??, 0 ^ g ^ /}. An arbitrary
immeasurable function / is called ?-integrable(o? abstract Lebesgue ?-integrable) whenever
J f+ ??- f f~ ??:= f f ?? has meaning; and we let L\ (?) = {f ? L\ (?): f /????.)
where L \ (?) denotes the class of all / ?-measurables for which J f ?? exists (see Pfeffer
A977), Cohn A980, Chapter 2)).
The relationship between the integral /;< and the abstract Lebesgue ?-integral can be
obtained as follows:
Proposition 7. ?,(?) = ?,(/;1) ? ? (?) and j f ?? = l „(f) for all f e ?\(?). If ?
is complete ana saturatea, L\ (?) = L\ (/;1), so that, the abstract Lebesgue integral is a
restriction of a certain Daniell integral.
As a consequence, the following theorem tells us that the Daniell integral / on L \ (I) is
equivalent to the abstract Lebesgue integral with respect to the measure ? induced by /.
THEOREM 2.3. Let (X, B, /) be a Daniell (or Bourbaki) system. Assume that the class
of I-summable functions is Stonian. Let (?, ?. v) be the measure space inaucea by I
(Theorem 2.1). Then, L\(v) consists of all I -integrable functions ana f f dv = 1(f) for
all feL^v).
The proof follows directly from Theorems 2.2, 2.1 and Proposition 7 (see Pfeffer A977,
p. 155)). Alternative expositions of Daniell-Stone integral representation can be found in
general textbooks such as Royden A968, p. 297), Segal and Kunze A968, p. 57), Stroock
A994, p. 144), Konig A997, Appendix V) and also Gunzler A975, p. 143). It is well
known that one cannot dispense with "Stone's condition" (compare Fremlin A974) for
a counterexample), but other substitute conditions can be considered (see also Kindler
A983)). The study reported here has the advantage that it is an immediate consequence
of the previous treatment of measures (integrals) induced by integrals (measures). The
most frequently used general method of Caratheodory extension gives the same measures
as the Daniell construction. Uniqueness and approximation theorems can be found, for
example, in Weir A974, Chapter 13). For recent versions of the "traditional" Daniell-Stone
theorem in terms of the abstract Bourbaki integral, we refer the reader to Leinert A995,
Chapter 14) and Konig A992, Appendix). An interesting reference on these objects is the
survey textbook by Konig A997), with bibliography notes and comparative descriptions of
some previous works.
2.2. Relatea results ana applications
We use the notation ?(?) to denote the smallest ?-algebra over X with respect to which
all functions in the vector lattice ? are measurable, a(C) as usual denotes the smallest

Daniell integral and related topics
517
?-algebraover С С P(X). Let (X, B, I) be a Daniell system, and assume that 1 e B. Note
Lo = L\ f)Rx, then ?(?,?) = {А С X: хд е Lq}, which is the completion of ?(?) with
respect to the finite measure ? defined as usual on ?(?0), and the corresponding integral
representation holds.
As a consequence, one has the following well known Caratheodory extension: Let ?
be an algebra of subsets of X and suppose that ? : ? —>¦ [0, oc[ is a finitely additive
measure such that ?(?„) 4- 0 whenever (А„)сй decreases to 0. Then there is a unique
finite measure, ? on ?(?) with the property that ? coincides with ? on ?. Here one
has ?(?) = ?(??), /? on Bq is a Daniell integral, and Theorem 2.3 gives the desired
existence and uniqueness (see Stroock A994, p. 146), where an important application
to the "probability product" is also proven). We can still establish the uniqueness of ?
with a weaker assumption, e.g., there is an everywhere positive function / e L\ (Royden
A968, p. 295)). Moreover, by an obvious localization procedure, one can reduce problems
involving ?-finite situations to finite ones.
We also recall that the extensions of Daniell integral serve as background for the study
of probability. In fact, given a Daniell system (?, ?, /), we assume that 1 e В and
/A) = 1, let ? = {А с ?: ?? e В*} and define ?(?) = ?(??) for all A e ?. Then
?*(?) = ???(?@): А С D e ?] is a probability measure on the ?-algebra A = [H С Х\
?*(?)+?*(?€) = l},and ?* = ? on ?. Now, since for / e B\ [f > r] e ?, r e E,
thena(B) = a(Si) = a(i?).Now,forany А С ?, ?*(?) = inf{/(/): f e Br, f > ??},
and ?* is a measure on ?(?2). Then, integral representations follow: There is a unique
probability measure ? on ?(?) such that each / e В is P-summable and /(/) = f fdP-
The results are identical under the hypothesis that / is a Bourbaki integral, with ??
replaced by Bl and sequences by nets (see Ash A972, p. 175)). For applications to
probability theory see also Simonnet A996) and Neveu A964).
It is interesting to note that the capacity approach can be used to directly deduce the
Daniell method of integration and the main Daniell representation theorem. For instance.
Chapter 7 in Rao A987) is devoted to a comparative study of an outer measure and an
analogous capacity function.
Some fundamental properties of measures of fitzzy sets are discussed by Qu A983),
using the Daniell representation theorem; in this paper a fuzzy set is considered as a point
of a segment in a Riesz space.
A recent systematic treatment of measure and integration based on the notion of
regularity is investigated in a survey by Konig A997). This development started in the
work of Tops0e A976) and others, and the "main instruments are certain new (and
unconventional) envelope formations which resemble the traditional Caratheodory outer
measure". For a lattice cone ? с +ШХ let_r(?)= {Ac ?: ?.\ e ?}andL(?) = {[/ ^ r\.
f e ? and r > 0}. If ?-.ЦЕ) —>¦ +K is a monotone set function with <p@) = 0,
the "horizontal integral" is defined for all / e ? by f fd<p = J^ ?([/ > r])dr e +Ш
(an integral of Choquet type). Let /: ? -> +R be a nonnegative linear and increasing
functional (elementary integral); then, we define ? to be a representation (a source)
of / if /(/) = § f ?? for all / e ?. Next, we define the crude outer envelope
/*:+E* _> +K and the crude inner envelope h ¦ +ШХ -+ +K of / to be /*(/) =
inf{/(g): / ^ g e ?} and /*(/) = sup{/(g): / ^ g e ?}, which induce the two set
functions ?, ?:?,(?)-> +E defined by A(A) = ?*(??) and v(^) = /*(Xa) for all

518
?. Diaz Carhllo
A e L(E). In this context, ? с Шх is said to be Stonian if / e ? => / л r, (f - r)+ e ?
for all r > 0. The following representation theorem is due in essence to Greco A977) and
see also Denneberg A994).
THEOREM 2.4. Let /:?—>¦ +R be an elementary integral on the Stonian lattice cone
? С +RX. Then, I admits "sources" if and only if it has the truncation properties: Co (/
continuous at 0): /(/лг) J, Ofor r 4- 0, a/id Cx (/ continuous at oc): /(/ Л г) | /(/)
У»г r I oo, /or a// / e ?.
The sources of / have a natural and simple characterisation; in fact, if / fulfills Co +CX,
then a monotone set function ?: L(E) —>¦ +E is a source of / if and only if ? ^ ? ^ ?-
If, additionally, ? is downward ?-continuous, then ? = ?.
The above general statements can be seen in Koning A997, p. 149). Their proof use
a known method of "lower-upper aproximant" for the characteristic function of certain
"parameter spectral sets" of /, and the definition of Riemann integral. These results
are already contained in Gunzler A973, Theorem 2), even for / nonnegative, there the
conditions Co and Cx have been introduced.
Results in this direction had been obtained by an der Heiden A978), investigating the
representation of linear functionals by finitely additive measures. Integral representations
with prescribed lattice В by some finitely or ?-additive measure, were also discussed
by Gunzler A974, 1975) (e.g., if X is any topological space, В contains all continuous
functions, or if В contains all bounded functions); in addition, these results can be
generalised to abstract В with certain completeness conditions (see Gunzler A974) and
Gould A966)).
Moreover, generalizing the vector lattice ? to a "subtractive" subsemigroup of Kx, and
I: В —> ? additive, К and ? Abelian partially ordered groups, sufficient conditions are
given by Gunzler A975, p. 119), for the existence of an unique measure representing a
nonnegative /.
Still further main generalisations are discussed by Konig A997); we formulate a
particular situation: A measure ? : ? —>¦ +R on an ?-algebra ? is called an
?-representation of an elementary integral / if, for all / e E, f is immeasurable and /(/) = j f ??.
Then Konig A997, p. 159), gives the following substitute for the traditional Daniell-Stone
theorem:
THEOREM 2.5. Let I: ? —>¦ +R be an elementary integral on a Stonian lattice cone
? С +ШХ. Then the following are equivalent: (i) / admits ?-representations, (ii) I(v) =
I(u) + ??(? - и) for all и ^ и in E. In this case. /„(/) = / / d? for all f e E„. where ?
is the maximal inner ?-extension of A.
Here, for all /e +RX, I„(f) = sup{lim /(//„): (h„) С ? with/j„ \, some function ^ /}
(inner ?-envelope). This result includes the more classic theorems about representing a
given linear functional / as an integral with respect to a measure. See also the earlier
paper by Pollard and Tops0e A975), where / is defined on a cone E С +??, with some
variant assumptions on the Daniell condition, and (/ - g)+ e ? for all /, g e E. This last
condition is avoided by Tops0e A976), but similar results are obtained.

Darnell integral and related topics
519
Finally, since the Daniell extension scheme works for some non-linear functionals (see
our comments in Section 1.4), integral representations are investigated in this case by Sipos
A979c).
2.3. The interplay between measure and topology. Riesz representation theorem
It is suitable to use the Daniell approach to obtain the integral representation of an
important class of linear functionals. The main results are based on the notion of
"regularity", which, for a set function, means to determine its values from a particular set
system by approximation from above or below. In traditional measure theory this notion
is linked to topology. We assume that X is a locally compact Hausdorff space. Denote
by J and ? the classes of all compact and open subsets of X. respectively. Let / be a
Bourbaki integral on Cqo(X) and let (?.?, v) be the measure space induced by / (see
Theorem 2.1); we know that ? is complete and saturated. Furthermore, an extremely useful
property is shown: ? is regular, i.e.,
(i) О С A (so, A contains the Borel ?-algebra B(X))\
(ii) К eJ=>v(K) <oc;
(iii) О е 0 => v(O) = sup[v(K): К ej, К с О), and
(iv) A e A =>v(A) =inf{u(C>): OeO. А с О} (outer regular v).
Now, by Theorem 2.3 one obtains the following important result referred to in the
literature as the "Riesz representation theorem".
THEOREM 2.6 (Riesz). Let J be a nonnegative linear functional on Cqq(X). Then, there
is a unique measure space (X, A, v) satisfying the following conditions:
(i) ? is a regular complete and saturated measure, and
(ii) J(f) = ffdvforallfeCw(X).
The proof is essentially contained in the above explanation, only remains to be proven
the uniqueness. To do so is essential for ? to be regular, together with Propositions 7 and 6.
Here the fact that / is a Bourbaki integral is essential (see, in this sense, the proof of ?
regular by Pfeffer A977, p. 164)). For instance, in Ash A972), the different results under
hypothesis (D) and (B) in Section 1.1 are stated separately.
A nonnegative regular Borel measure ? on X (defined on the Borel ?-algebra) is referred
to in some literature as "Radon measure". Therefore, Theorem 2.4 establishes a one-to-one
correspondence between nonnegative linear functionals on Cqo(X) and Radon measures
in X. The reader should be warned that not everyone uses the same definitions for regular,
Borel (and Baire) measures. We therefore explain what it means to us. A Borel measure ?
is called inner regular if
v(A) = sup{v(K): KeJ. К с A)
for every Borel set A. Assume the hypothesis of Theorem 2.4, and suppose that X is ?-
compact, then there is a unique Borel inner and outer regular measure ? on X, which
represents to / on Cqo(X). We note that if О с Кп (countable union of compact sets)
and и is a nonnegative Borel measure such that v(K) < oc for all KeJ, then ? is inner

520
?. Diaz Carrillo
and outer regular. As a consequence, if / is a nonnegative linear functional on Coo(X),
there is a unique Borel measure ? such that Coo(X) С L\ (v) and /(/) = f fdv for all
/eCoo(X).
PROPOSITION 8. //X /5 compact Hausdorff and M(X) is the collection of all the
outer and inner regular finite signed measures on B(X), the map I н> ? is a isometric
isomorphism of the conjugate space ofC(X) (real valued continuous functions on X) and
M(X), where the norm of ? e M(X) is taken as \v\(X)(= \\I\\).
For X a metric space, not necessarily compact, and nonnegative measures. Ash
A972) investigates a weak*-convergence type of sequence (v„) С M(X) to a measure
? : / fdv,, —>¦ / fdv for all / e C(X). The results form the starting point for the study of
the central limit theorem of probability.
The classic paper by Glicksberg A951) merits special mention, as it shows that a
nonnegative functional / on Ci,(X) (bounded, real-valued continuous functions on a
topological space X) is representable if and only if Lebesgue's monotone convergence
theorem, restricted to elements of Ci,(X), holds for / (Dini's theorem works here). If X is
a completely regular topological space, Ascoli's theorem gives such a representation.
There are proofs of Proposition 8 using compactification theory (see Varadajan A958),
and the complete survey by Flachsmeyer and Terpe A977)). For the historical development
that leads to the traditional Riesz representation theorem, we refer the reader to the survey
article of Batt A973), as well as Anger and Portenier A992), Dunford and Schwartz A958)
and Weir A974). A treatment of the extensions of measures in topological spaces, and
related results in this line, can be seen in the classic books of Halmos A950), Berberian
A965), Royden A968), Taylor A965), Valdivia A979) and Weir A974). For adiscussion of
the different definitions of "Radon measures" (on arbitrary Hausdorff spaces), see Schwartz
A973) and Berg, Christensen and Ressel A984), notes and remarks in Chapter 2.
For an abstract Riemann-type approach given by Anger and Portenier A992) (see
Section 1.4 on essential integration), if &(X) denotes the lattice cone of all lower semi-
continuous R-valued functions on X, which are nonnegative outside some compact subset
of an arbitrary Hausdorff space X, and if ? is a regular linear functional on &(X) ("Radon
integral on X"), a related refinement on representation by Radon integrals is contained in
the following theorem by Anger and Portenier A992, p. 131).
THEOREM 2.7. Let ? be a lattice cone of lower semi-continuous functions such that
&-(X) С Тф (upper envelopes of upward-filtering families in T). If ? is a regular linear
functional on T, which is &(X)-tight (i.e., ?*(/) = inf{r*(/ vs): se ?_}), then ? is a
Bourbaki integral and the restriction of f dr (= ??') to &(X) is the only Radon integral
representing ?.
We note that, according to Konig A997, p. 168), the corresponding extended Riesz
representation theorem is obtained as a specialization of the above Theorem 2.5, in the
order continuous version. Finally, some historical notes and relations to previous works,
concerning to Riesz representation theorem, have been considered by Anger and Portenier
A992, Appendix §16). See the related Chapter 9 in this Handbook by Diestel and Swart
B002).

Daniell integral and related topics
521
3. The abstract Fubini theorem
The Daniell method developed in the previous sections allows us to establish a general form
of the powerful result that evaluates "multiple integrals by iterated integration". This was
originally proven by G. Fubini in 1907 for R"+"' and some improvements were provided
by M. Stone in the late 1940s.
Throughout this section we assume that for / = 1,2, (X, B,, Ij) are Daniell (or Bourbaki)
systems, and L\ denotes the class of all /j-summable functions on Xt. Let X = X\ ? X:;
for any function / on X we consider the section functions as usual and, if fx e Lj, we
define (h_f)(x) = h(f.x) for each ? e X |. Thus, we have defined functions hf and /| /
on ? ? and Xi, respectively. A vector lattice В on X is called a product system (with respect
to (S|, /|) and (Z?2, /2)) whenever, for each / e В the following conditions are satisfied:
(i) fx e B2 for each ? e X \ and /, e B\ for each у е Xi\
(ii) hf e S| and /| / e Вт, and
(iii) /|(/2/) = /:(/i/).
Next, let /: В -> R be defined by /(/) = /| (hf) for each / e B. One has that / is a
Daniell and Bourbaki integral (product). L\ denotes the class of all /-summable functions
on X. The (iterated) upper and lower integrals are defined as above; then, one has the
following main preliminary result:
Proposition 9. /// e L,, then /2/,72/ e L\. /,/,7|/ e L] and 1(f) = l\(hf) =
/|(/2/) = /2(/|/) = /2G|/).
The key for the proof is the following relations: For any / e Rx. 7(/) ^ max{7| G:/),
72G|/)}, and its dual.
Let (Xi,Ai,Vj) be the measure space induced by /,-,/ = 1,2 (Theorem 2.1). The
following well-known theorem holds.
Theorem 3.1 (Fubini). Let f e L\, then /v e L.} for v\-a.e. ? e X\, and /, e L\ for
vi-a.e. у e Xi. Moreover, if F and G are functions on X\ and X?, respectively, such that
F(x) = fr(fx) and G(y) = /|(/,), whenever the integrals on the right side exist, then
F e L\, G e L] and 1(f) = /, (/2/) = h(I\ /)¦
The proof follows the standard steps, using the function h = 11 / - I_, / on X2- which
belongs to +Lj and h = 0 u2-a.e., since 72(/;) = 0. Then, reiterated applications of
Proposition 9 yield the result (see Pfeffer A977. p. 186)).
We recall that the "tensor product" of the vector spaces B\ and ?2, ? ? <g> ?2, such that
I /I e ?| <g> ?2 whenever / e B\ <g> ?2, satisfies that B\ <g> ?2 С В (ample product system),
and it is a product system (see Pfeffer A977) and Floret A981, §13)). Starting with two
complete measures ?? and ?2 on X\ and X2, respectively, the above results are used to
construct a product measure in X\ <g> X2. and to obtain the corresponding Fubini (and
Tonelli) theorem, via the induced integrals. In this context, a new product measure in a
topological space, which is regular whenever the factor measures also (topological product
or measure topologically induced by the product measure) is also treated by Pfeffer A977,
Chapters 16 and 17).

522
?. Dial Carrillo
In Bichteler A998), products of elementary integrals are studied in the following way:
For/' = 1,2, (Xj, ?¦, /,) elementary integrals (see Section 1.3, Daniell mean; remember, ?,
are rings of bounded functions on X,), the elementary integrals onX| xXiare the finite
sums of the form h(x, y) —^2':i hJi(x)hJ2(\). hj e ?,; such a collection is denoted by ?3
or ?| <g> ?2 (ring of bounded functions on X \ xXi. but it is not, in general, a vector lattice;
this fact justifies the integration theory "of means" for rings). The elementary integral of
h e?, is well defined by /(/?) = (/| ? /2)(?) = ?/=? 1\(h\) ¦ h(h2). Products of additive
set functions and "Radon measures" are specialized. Let /''(/) = I \ (hf) (iterated upper
integral) and ||/f = 7,G2/), then || · ||ft is an upper bound for I = I\ ? h and that agrees
with / on +?, so we close ? with respect to || · ||л to obtain L\ (|| · ||);') and we extend the
elementary integral by continuity to obtain the extension / on L\(|| · \\h). Then, one can
evaluate the integral by iterated integration, but this uses a mean that refers to the order
of integration, which leads to choosing a new mean || · ||*: the largest mean that agrees
with / on +?. Let L\(I\ ? I2) be the closure of ? in T( || · ||*) (integrable functions for the
product), and extend /| ? I2 to L1 by continuity. In this framework, the classic Fubini and
Fubini-Tonelli theorems (with measurable functions with ?-finite carrier) are established
(Bichteler A998, pp. 130-131)). Here, some applications to convolution and interpolation
are studied. Order continuous situations can also be treated.
The Daniell extension is used by Kelly and Srinivasan A976) to give a unified proof of
the Fubini theorem for "Baire and Borel measures" on locally compact Hausdorff spaces.
Konig A997) investigates the product formations based on inner regularity, which can be
specialized to "Radon measures" (in the sense given here in 2.3). He made an extensive
study of the "sectional representation" and "monotone approximation of functions" to
present several versions of Fubini-Tonelli theorem; his bibliography notes (p. 229) are
worthy of consideration. See the related Chapter 17 in this Handbook by Grekas B002).
Fubini theorems are investigated by de Amo and Diaz Carrillo A995) for proper and
abstract Riemann integration with "non-continuity integration" discussed in Section 1.4.
The results obtained can be developed with Proposition 9 and they contain those of the
Riemann^-abstract integral due to Eisner A975).
Iterated (local) integral norms and corresponding abstract Fubini theorems are studied
by Hoffmann A977a, 1977b) and by Hoffmann and Schafke A992, Chapter 3).
4. The Radon-Nikodym theorem
Throughout this section we shall assume a Daniell system (X.B.I) which has been
extended to (X, L1 (/),/). A function h e +RX is said to be locally integrable with respect
to / if fh e L1 (/) for all / e В (note that if ? is the Lebesgue measure on the line, then /
is locally ?-integrable if and only if it is ?-integrable on every bounded interval). Then, the
functional defined by J (/) = I(fh). for all / e B. is a Daniell integral (sometimes called
the "measure with base / and density /7". written J = hi). Consider two assumptions about
the vector lattice В: В is Stonian and X is ?-finite with respect to B. These require the
existence of an increasing sequence (X„) of elementary sets X„ = {.v e X: h„(x) ^ 1}.
h„ e B, whose union is X, and ensures the ?-finiteness of the measure derived from any

Daniell integral and related topics
523
elementary integral on В (see Zaanen A958), where the above assumptions are discussed
in some detail).
THEOREM 4.1 (Radon-Nikodym). Let (X.B.I) be a Stonian Daniell system, and
suppose that X is ?-finite with respect to B. Let J be a Daniell integral on B, and
(i): let h e +RX such that J(g) = I(gh), for all g e B. Then, f e L\(J) if and only if
fh e L|(/), and in this case (ii): J(f) = Hfh).
Condition (i) is equivalent to (i'): every /-null subset of X is У-null (write J <Cl)Af h,
к both satisfy (i) then h — к is /-null. To prove that (i') => (i), one uses that for a bounded
linear functional J on L2(/), there is h e L2(I) such that J(f) = I(fh) for all / e L2(I)
(Riesz's theorem).
The proof can be found in Weir A974) and is based on four assertions:
(a) A 7-null => ? ? [h ? 0] is /-null;
(b) if/eL,G), then//? e L\(I) and (ii) holds;
(c) A is /-null => A is У-null; and
(d) if fh e L\ (/), then feL\(J) and (ii) holds.
The classic measure-theoretical Radon-Nikodym theorem (see, for example, Weir
A974, p. 216)) is a special case of Theorem 4.1. Also in the same context, an excellent
outline of the complete results (?-finite and general case), as well as the different methods
of proving it and the usual topics on duality of L''-spaces and conditional expectation, can
be found in Rao A987).
In the sense of extension of the integral, via the "Daniell mean" by Bichteler A998),
the measures with densities and the Radon-Nikodym theorem are also treated. In fact, the
?-additive elementary integrals of finite variation form a band M*(?) in the class of the
elementary integrals of finite variation on ?. If / is ?-finite (a similar condition to the
above one), every J belonging to the band generated by / can be obtained by the formula
J = hi. As application conditional expectation is also treated (p. 155).
An interesting generalization to the differentiation theory of Daniell integrals (in the
sense of Radon-Nikodym derivatives) has been considered by Volcic A980), without any
use of measure theoretic methods; hence, it does not suppose that Stone's condition holds.
Volcic proves a Hahn-type decomposition theorem and a Radon-Nikodym theorem for
Daniell integrals, using the notion of localizability of the basic integral /. We say that /
is localizable if the lattice M^ is complete, where M^ is the lattice of all extended real-
valued functions к for which kg is Stone-measurable, for any Stone-measurable g. For a
signed integral У, a set H+ locally-J-integrable is said to be positive (respectively H~
negative) if J(fxH+) > 0, for each / e +L\ (J) (respectively ^ 0 for H~)\ the disjoint
sets H+ and H~ with union X are said to form a Hahn decomposition of X with respect
to J. In this context (compare Zaanen A961). Floret A981) and Bichteler A973)), the
following main results are proven;
THEOREM 4.2.
A) I is localizable if and only if each J that is absolutely continuous with respect to I
(J <<C /), has a Hahn-decomposition.

524
?. Diaz Corrilh
B) /// is localizable and J «; /, then there exists g e M^(I) such that J(f) = I(fg)
forallfeLi(J).
The function g is given by max{/? e M^_: I(fh) < J(f)}. Assertion A) is a Kelley-
type characterization of localizability. For the measure case see Kelley A966). See also the
related Chapter 6 in this Handbook by Volcic and Candeloro B002).
Using similar methods, Volcic A981) proves that the lifting theory can be extended
to Daniell integrals. Essentially the same general setting is considered to investigate
the "localizable upper gauge" by Bichteler A973), where Radon-Nykodym theorems,
for scalar and Banach-valued "measures" are given. The existence of liftings is also
investigated.
Finally, we note that in the context of non-continuity integration, an approximate
functional Radon-Nikodym theorem is proven by de Amo, Chictescu and Diaz Carrillo
A999). In fact, for two nonnegative linear functionals J and / on a lattice algebra В сШх.
such that J is "absolutely continuous" with respect to /, one can express J as follows:
J(f) = lim/(/u„) for all /efl, where (i'„) is a fixed sequence in B. This result is the
"functional" similar to a previous fundamental result by Fefferman A967). In the same
spirit, the remaining question is whether we can find an exact Radon-Nikodym derivative.
We know some partial results using a notion of "exhaustion" of / on elements of a function
algebra lattice (for the finitely additive case, see the classic results by Maynard A979) and
Hagood A986)). See de Amo, Chictescu and Diaz Carrillo B001).
5. Integral norms. Local integral metrics and Daniell-Loomis integrals
The classic book by Bichteler A973) gives an excellent and extensive study of integration
for Banach space-valued functions and measures with upper integral norms and their
generalizations to upper gauges, "motivated by the desire to carry M.H. Stone's unified
treatment of set functions and Radon measures as far as possible". These questions were
first raised by the pioneering work of Aumann A952), some more general results and
variants were also treated by Schafke A970, 1972). In Bichteler, local integral norms
are not investigated, only "Daniell-continuous" integral norms (?-subadditive), and only
with the Stone condition assumed. The finitely additive case is not considered. The basic
assumptions and definitions in Bichteler are as follows: The pair (R.m) is called an
?-valued elementary integral on X, whenever R is a Stonian vector lattice (integration
lattice), and m (integral or measure) is only a linear map from R to some Banach space E.
The linear operator U : R® ? —>¦ F (Banach space) and its semivariation \\U\\ are defined
as usual; U has finite variation if there is a seminorm/;: +RX -> +RX such that ||?/(j)|| ^
n(\g\), for all g e R ® ? (see Dinculeanu A966)). An upper norm ? on X (M : +RX ->
+RX increasing, subadditive and positively homogeneous) is said to be an upper-S-norm
on X if additionally satisfies: (/,) с +ШХ, f„i and supM(/,) = M(sup/„) (equivalent
to Aumann's "Summenbeschrankt" A952). Next, the space L]f(R. M) is defined as the
M-closure of R ® ? in FE(M) := [k e Ex: M(\k\) < oo} ((R. M)-integrable functions
or "Bochner integrable" functions is also used). The scalar case is obtained similarly. Then,
the following main result is proven (Bichteler A973. Theorem 7.12, p. 79)):

Darnell integral and related topics
525
THEOREM 5.1. Let R be an integration lattice and ? an upper-S-norm on X, that is
finite on +R. Then LE(R, M) is an M-complete seminormed vector space and for any
M-Cauchy sequence (f,) С LE, there is a subsequence (f,k) that converges M-a.e. to
lim/,.
In particular, with a Jordan upper norm mJ(/) = {m(h): \f\ ^ h e R\ for m : R —>¦ К
nonnegative and linear, the Jordan-w-integrable functions space, L E (R. m J) is defined and
it agrees with /?'(/?, m), ? = R. but the abstract Riemann-integrable functions space
R\ (R,mJ) is not treated (see Diaz Carrillo and Mufioz A993), Diaz Carrillo and Giinzler
A993, p. 421)).
With the notion of weak upper gauge: ? an upper-S1-norm finite on R, such that
+R э f, <: f„+i ? /? e LlE(R, M) implies /,->¦/ e LE(R. M) and M(\f, - f\) -> 0
(converges in mean), weak convergence theorems are proven (pp. 83-84). Moreover, with
the additional condition that: +R э f„ ^ /,+i, supM(f,) < oo implies (f,) M-Cauchy
(upper gauge), monotone convergence theorem and Lebesgue's dominated convergence
theorem (with only \f„ | ^ к е +ШХ, M(k) < oo) are obtained (pp. 88-89). Next, it is
shown how the upper norms can be used to extend an elementary "measure": Given
U . R <g> ? —> F linear, ? upper norm on X with the condition that it majorizes U i.e.,
||?/(й)|| ^ M(\h\) for all h e R <g> E. then there is a unique M-continuous extension
of U to LE(R.M) (this corresponds with Aumann's extension A952) or Schafke's
extension A977)); but the above condition is stronger than Aumann's assumption). Given
a "measure" M, a necessary condition for the existence of a majorizing upper gauge is that
it be 5"-continuous.
Standard upper integrals are constructed by Bichteler A973. Section 9); in particular,
the Daniell mean has been already considered in our Section 1. Also, projective limits,
martingales (from the projective limit point of view), lifting theorems, integration of
measures and Radon-Nikodym theorems are discussed by this author. See also Bichteler
A975), where "measures" with values in non-locally convex spaces are considered and
interesting applications are discussed. Note that Schafke's approach A970. 1971, 1972) is
more general, but one comparison of conditions on integral norms, results and possible
applications should be an interesting task for the future. The book by Hoffmann and
Schafke A992) could give a good introduction in this field.
With the aid of Schafke's local integral metrics (Schafke A977)), the "non-continuity
integration process", given in Section 1.4, is generalized by Diaz Carrillo and Giinzler
A993). One has the usual convergence theorems using a suitable local mean convergence,
which can be tracked back to Loomis A954). More precisely, the functional q : +ШХ —>
+ШХ is called an integral metric on X if q@) = 0 and q(f) ^ q(g) +q(k) if / ^ g +k,
fg,k? +ШХ. If В С Шх, we define the class of all q-integrable functions Bq as the
closure of В in Rx with respect to q; if additionally an / : В —>¦ ? is given, which is
uniformly continuous on В with respect to q, the unique ^-continuous extension of /
to Bq will be denoted by 1q. In all of the following (X, B. I) will be a Loomis system.
For any integral metric q, the corresponding local integral metric of Schafke is defined
ЬУ ?/(/) = sup{<7(/ Ah): he +B) for all / e +RX; q\ is again an integral metric
with q\ -?, q, and one has В С Bq С Bqi. Г1 = Iе" on Bq. Moreover, we introduce q-
measurability, defined by the property that truncation by integrable functions leads to

526
?. Diaz Carrillo
integrable functions. The relevant convergence properties with respect to q or qp (where
?/>(/) = [<7(//')]l//', / e +??, ? > 1) are developed. This allows us to introduce the
abstract Daniell-Loomis spaces Rp as usual, and to prove that Rp = B'1'' (see Diaz Carrillo
and Gunzler A996)). In this framework. Giinzler A996) discusses Bq-improper integrable
functions < = (/ e Rx: /П/ie Bq for all h e+B and lim+? /(/ ? h) exists e E},
where / П h = (/ л h) ? (-/г). If / and q are equivalent on B, such improper integration
turns out to coincide with Schafke's local integral extension.
The results hold also in the finitely additive case, and it is shown that as special cases
proper- and abstract Riemann-, Loomis-, Lebesgue-, Daniell- and Bourbaki-integrals are
subsumed. In fact, for a general Loomis system (X, B, I) and q(f) = /"(/) :=inf{/(/?);
/ ^ h e +B) for all / e +RX, one has Bq = /?prop(S, /) = two-sided completion of
Loomis A954, p. 170) с S'" = R\(B.I) (abstract Riemann-/-integrable functions of
Diaz Carrillo and Mufioz A993)), containing the one-side completion of Loomis A954,
p. 178). Moreover, it is also true that with q(f) = 1(f) := inf{/+(g); / ^ g e ??] (as in
Section 1.4), Bq = ~B (/-summable function by Bobillo and Diaz Carrillo A987)) and
Bq< — L(I,B) (local extension by Diaz Carrillo and Gunzler A997)) = improper B-
integrable functions. Given the Daniell condition (D), then one can use Aumann's /? as_<7,
where /?(/) =???(?? I(h„): f ij Y^ h„,h„ e +B), then B'< = Daniell's L\(I,R)
and Bqi = L\ + {local L?-malfunctions}. Finally, in the Bourbaki situation, where / is
net-continuous, let q = 1+ with /+(/) := -/"(-/), then LT = Bq = L\.
For a related result concerning essential integration (see Section 1.4 and Anger and
Portenier A992)), one_has essential-u-integral functions J'(v) = ?1/, = ?'11, with q an
upper functional ? on +RX; which give new characterisations of R\ (B. I) and L(I, B), for
instance.
Acknowledgements
The author would like to express his gratitude to Prof. Hans Gunzler for his valuable
comments and suggestions.
References
Adamski, W. A983), Some characterizations of Daniell lattices. Arch. Math. 40 D). 339-345.
Adamski, W. A986), ?-smooth linear junctionals on vector lattices of real-valued functions, Studia Math. 83 B).
105-117.
Alfsen, E.M. A958), On a general theory of integration based on order. Math. Scand. 6 (S). 67-79.
Alfsen, E.M. A963), Oder theoretic foundations of integration Math. Ann. 149. 419^461.
an der Heiden, U. A970), Integration in Vektonerbanden nach Daniell, Diplomarbeit Univ. Gottingen.
an der Heiden, U. A978). On the representation of linear functionals by finitely additive set functions. Arch.
Math. 30, 210-214.
Anger, B. and Portenier, С A992), Radon Integrals. Progress in Math.. Vol. 103, Birkhauser, Basel.
Ash, R.B. A972), Measure, Integration and Functional Analysis. Academic Press. New York.
Asplund, E. and Bungart, L. A966), A First Course in Integration. Holt. Reinhart and Winston. New York.
Aumann, G. A952), Integralerweiterungen mittels Normen. Arch. Math. 3. 441^450.
Batt, J. A973), Die Verallgemeinerungen des Darstellungssatzes von F Riesz und ihre Anwendungen, Jahresber.
Deutsche. Math.-Verein. 74, 147-181.

Daniell integral and related topics
527
Berberian, S.K. A965). Measure and Integration. Macmillan. New York.
Berg, C, Chistensen. JR.P. and Ressel. P. A984). Harmonic Analysis on Semigroups. Graduate Texts in Math..
Vol. 100. Springer. Berlin.
Bichteler, K. A973), Integration Theory (with Special Attention to Vector Measures). Lecture Notes in Math..
Vol. 315, Springer, Berlin.
Bichteler, K. A975), Measures with values in non-locally comex spaces. Lecture Notes in Math.. Vol. 541.
Springer, Berlin, 277-285.
Bichteler, K. A998), Integration. A Functional Approach. Birkhauser. Basel.
Bogdanowicz, W.M. A967), An approach to the theory of integration generated by daniell fimctionals and
representations of linear continuous functionals. Math. Ann. 173. 34-52.
Bogdanowicz, W.M. A971), Minimal extension of Daniell functional.·, to Lebesgue and Daniell-Stone integrals.
Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 19. 1093-1100.
Bobillo Guerrero, P. and Diaz Carrillo. M. A987). Siimmable and integrable functions with respect to any Loomis
system, Arch. Math. 49, 245-256.
Bobillo Guerrero, P. and Diaz Carrillo. M. A989). On the summability of certain ?-integrable functions. Arch.
Math. 52, 258-264.
Bogdan, V A975). A new approach to the theory of probability via algebraic categories. Lectures Notes in Math..
Vol. 541, Springer, Berlin. 345-367.
Bourbaki. N. A952), Integration. Elements de Mathematique III. Livre VI. Hermann. Paris.
Brehmer, S. A974). Verbandtheoretische Charakterisierung des Mass und lntergralbegriffs von Caratheodory.
Potsdam Forsch. В C).
Candeloro, D. and Volcic, A. B002). Radon-Nikodym theorems. Handbook of Measure Theory. E. Pap. ed..
Elsevier, Amsterdam, 249-294.
Choquet. G. A953/54), Theory of capacities. Ann. Inst. Fourier (Grenoble) 5. 131-295.
Cicogna, G. A976), Un nuovo approcio alia teoria dell'integrazione reale. Bull. Un. Mat. Ital. A E) 13 A).
142-151.
Cohn. D.L. A980), Measure Theory, Birkhauser. Boston.
Constantinescu, С and Weber, K. A985). Integration Theory. Vol. I: Measure and Integral. Wiley-Interscience.
New York.
Constantinescu, C, Filter, W. and Weber. K. A99%). Advanced Integration Theory. Kluwer Academic. Dordrecht.
Daniell, P.J. A917/18), A general form of integral. Ann. of Math. B) 19. 279-294.
Daniell, P.J. A919/20), Furthers properties of the integral. Ann. of Math. 21, 203-220.
de Amo, E. and Diaz Carrillo, M. A995), On abstract Fubini theorems for finitely additive integration. Proc.
Amer. Math. Soc. 123 (9), 2739-2744.
de Amo, E. and Diaz Carrillo, M. A997), Fubini-integrals metrics. Rend. Circ. Mat. Palermo B) 44. 161-174.
de Amo, E., Chictescu, I. and Diaz Carrillo, M. A999). An approximate functional Radon-Nikodym theorem.
Rend. Circ. Mat. Palermo B) 48. 443^450.
de Amo, E., Chictescu. I. and Diaz Carrillo. M. B001), An exact functional Radon-Nikodym theorem for Daniell
integrals, Studia Math. 148 B), 97-110.
De Lia, A. and Mikusinski, P. A995), A Daniell-type integral with values in a Banach Space, Arch. Math. 65 E),
417-423.
Diestel. J. and Swart, J. B002). The Riesi theorem. Handbook of Measure Theory. E. Pap, ed.. Elsevier,
Amsterdam, 401^*47.
Denneberg, D. A994), Non-Additive Measure and Integral, Kluwer Academic. Dordrecht.
Diaz Carrillo, M. and Munoz Rivas, P. A990), Integral extension with locally-integral seminorm, Acta Univ.
Carolin. Math. Phys. 31, 25-27.
Diaz Carrillo, M. and Munoz Rivas. P. A993), Finitely additive integration: Integral extension with local-
convergence, Ann. Sci. Math. Quebec 17, 145-154.
Diaz Carrillo, M. and Gunzler, H. A993), Local integral metrics and Daniell-Loomis integrals. Bull. Austral.
Math. Soc. 48, 411^426.
Diaz Carrillo, M. and Gunzler, H. A996). Abstract Daniel l-Loomis spaces. Bull. Austral. Math. Soc. 53. 135-
142.
Diaz Carrillo, M. and Gunzler, H. A997). Daniell-Loomis integrals. Rocky Mountain J. Math. 24 D). 1075-1087.
Dinculeanu, N. A966), Vector Measures, VEB, Berlin.

528
?. Dial Carrillo
Dunford, N. and Schwartz, J.T. A958), Linear Operators. Port I: General Theory. Interscience. New York.
Eisner, J. A975), Zum "Sati von Fubini" fiir ein abstraktes Riemann-integral. Math. Z.. 265-278.
Fefferman, C.H. A967). A Radon-Nikodvm theorem for finitely additive set functions. Pacific J. Math. 23 B).
35^45.
Flachsmeyer, J. and Terpe. F A977), Some applications of the theory of compactifications of topological spaces
and measure theory, Russian Math. Surv. 32 E). 133-171.
Floret, K. A981), ???- und Integrationstheorie. Teubner. Stuttgart.
Fremlin, D.H. A974), Topological Riesi Spaces and Measure Theory. Cambridge University Press. Cambridge.
Futas, E. A971), Extensions of continuous functionals. Math. Cas. 21. 191-198.
Glicksberg, I. A951), The representation of junctionals by integrals, Duke Math. J. 19. 253-261.
Gould, G.G. A964), A Stone-Cech-Ale.xandroff-type compactification and its application to measure theory,
Proc. London Math. Soc. C) 14, 221-224.
Gould, G.G. A966), The Daniell-Bourbaki integral for finitely additive measures. Proc. London Math. Soc. C)
16, 297-320.
Greco, G.H. A977), Integrate monotono. Rend. Sem. Mat. Univ. Padova 57. 150-166.
Grekas, S. B002), On products of topological measure spaces. Handbook of Measure Theory, E. Pap, ed.,
Elsevier, Amsterdam, 745-764.
Giinzler, H. A973), Linear functionals which are integrals. Rend. Sem. Mat. Fis. Milano 43, 167-176.
Gunzler, H. A974), Stonian lattices, measures and completeness. Linear Operatoren und Approximation II,
Herausg. Butzer und Sz. Nagy, Intern. Ser Numer. Math., Vol. 25. Birkhauser, Basel. 113-126.
Gunzler, H. A975), Integral representations with prescribed lattice. Rend. Sem. Mat. Fis. Milano 45, 107-168.
Gunzler. H. A985), Integration, Bibliogr. Instit., Mannheim.
Gunzler, H. A991), Convergence theorems for a Daniell-Loomis integral. Math. Pannon. 2, 77-94.
Gunzler, H. A996), Improper integrals and local integrals. Benefit 96-14, Mathematischen Seminars Kiel. Revue
Rumaine.
Hagood, J.W. A986), A Radon-Nikodym theorem and Lp completeness for finitely additive vector measures,
J. Math. Appl. 113. 266-279.
Halmos, PR. A950). Measure Theory. Van Nostrand. New York.
Hoffmann, D. A977a), Zum "Satz von Fubini". J. Reine Angew. Math. 298. 138-145.
Hoffmann, D. A977b), Integralerweiterung durch Intergralnonn mil Werten in priigeordneten Halbgruppen,
J. Reine Angew. Math. 295, 187-201.
Hoffmann, D. and Schafke, FW. A992), Integrate. Bibliogr. Instit.. Mannheim.
Jacobs, K. A978), Measure and Integral. Academic Press, New York.
Kelley, J.L. A966), Decomposition and representation theorems in measure theory. Math. Ann. 163, 89-94.
Kelley, J.L. and Srinivasan, T.P A976), A unified proof of Fubini theorems for Baire and Borel measures. Lectures
Notes in Math., Vol. 541. Springer, Berlin. 25-29.
Kindler, J. A983), A simple proof of the Daniell-Stone representation theorem. Amer. Math. Monthly 90 F).
396-397.
Kluvanek, I. A965). Integrate vectorielle de Daniel/. Mat. Fyz. Casopis Sloven. Akad. 15. 146-161.
Konig, H. A982). Integraltheorie ohne Verbandspostulat, Math. Ann. 258. 447^458.
Konig, H. A992), Daniell-Stone integration without the lattice condition and its application to uniform algebras,
Ann. Univ. Sarav. Ser Math. 4. 1-91.
K6nig,H. A997), Measure and Integration. An Advanced Course in Basic Prodecures and Applications, Springer,
Berlin.
Leinert, M. A982), Daniell-Stone integration without the lattice condition. Arch. Math. 38, 258-265.
Leinert, M. A984), Plancherel's Theorem and integration without the lattice condition, Arch. Math. 42, 67-73.
Leinert, M. A995), Integration und ???, Vieweg. Weisbaden.
Leinenkugel, С A992), Daniell-Stone approach to the general Denjoy integral, Proc. Amer. Math. Soc. 114 A),
39-52.
Loomis, L.H. A954), Linear functionals and content, Amer. J. Math. 76, 168-182.
Maynard, H.B. A979), A Radon-Nikodym theorem for finitely additive bounded measures. Pacific J. Math. 83
B), 401^413.
McShane, E.J. A953). Order-Preserving Maps and Integration Processes, Ann. of Math. Stud.. Vol. 31, Princeton
University Press, Princeton, NJ.

Daniell integral and related topics
529
Muni, G. A974), Intorno adalcuni aspetti della teoria dell'integrate xectoriale. Rend. Mat. F) 7, 239-268.
Neveu, J. A964), Bases mathematique du calcul des probabilites, Paris.
Okada, S. A984), Vector Daniell integrals. Math. Slovaca 34 A), 35-65.
Pap, E. A995), Null-Additive Set Functions. Kluwer Academic, Dordrecht.
Papangelou, F. A966), Note on the functional analogue of the Jordan decomposition of measures. Arch. Math. 17
C), 253-255.
Pfeffer, W.F A977), Integrals and Measures, Marcel Dekker, New York.
Pollard D. and Tops0e. F. A975), A unified approach to Riesz type representation theorems. Studia Math. 54 B),
173-190.
Qu, Y.S. A983), Measures of fuzzy sets. Fuzzy Sets and Systems 9 C), 219-227.
Rao, M.M. A987), Measure Theory and Integration, Wiley-Interscience, New York.
Riecan, B. A964), Sur une extension continue des certaines fonctions monotones, Spisy Prer. Fak. Univ.
J.E. Purkyne 457, 481^483.
Riecan, B. A975), An extension of the Daniell integration scheme. Mat. Cas. 25 C), 211-219.
Riecan, B. A977), On the Caratheodory method of the extension of measures and integrals. Math. Slovaca 27
D), 365-374.
Riecan, B. A979), On the unified measure and integration theory. Acta. Fac. Res. Nat. Univ. Camen. Mathem.
27, 217-237.
Riecan, B. A983), On measures and integrals with values in ordered groups. Math. Slovaca 33 B), 153-163.
Riecan, B. A997), Integral, Measures and Ordering. Math. Appl., Vol. 411, Kluwer Academic, Dordrecht.
Royden, H.L. A968). Real Analysis, 2nd edn.. MacMillan, New York.
Rudin, V.N. A975), On the abstract theory of the Daniell integral. Math. Questions C). Trudy Tomsk. Gos. Univ.
220,57-61.
Schafke, FW. A970), Integrationstheorie I, J. Reine Angew. Math. 244, 154-176.
Schafke, FW. A971), Integrationstheorie II, J. Reine Angew. Math. 248, 147-171.
Schafke, FW. A972), Integrationstheorie und quasinormiert Gruppen, J. Reine Angew. Math. 253, 117-137.
Schafke, FW A977), Lokale Integralnormen und verallgemeinerte uneigentliche Riemann-Stieltjes Integrale,
J. Reine Angew. Math. 289, 118-134.
Schmeidler, D. A986), Integral representation without additivity. Proc. Amer. Math. Soc. 97 B), 255-261.
Schwartz, L. A973), Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford
University Press, Oxford.
Segal, I.E. and Kunze, M.A. A968), Integrals and Operators, McGraw-Hill, New York.
Shilov, G.E. and Gurevich, B.L. A966), Integral, Measure and Derivative: A Unified Approach, Prentice-Hall,
Englewood Cliffs, NJ.
Simonnet, R. A996), Measures and Probabilities, Springer, Berlin.
Sipos, J. A979a), Integral with respect to apre-measure. Math. Slovaca 29 B), 141-155.
Sipos, J. A979b), Nonlinear integrals. Math. Slovaca 29 C), 257-271.
Sipos, J. A979c), Integral representation of nonlinear fimctionals, Math. Slovaca 29 D), 333-345.
Stone, M.H. A948/49), Notes on integration I-IV, Proc. Nat. Acad. Sci. USA 34 A948), 336-342, 447^455,
483^490; 35 A949), 50-58.
Stroock, D.W. A994), A Concise Introduction to the Theon of Integration, Birkhauser, Basel.
Taylor, A.E. A965), General Theory of Functions and Integration, Blaisdell. New York.
Thiam, D.S. A975), Integrale de Daniell a valeurs dans un semi-groupe ordoenne, С R. Acad. Sci. Paris Ser.
A-B 281 E-8), Aii, A 215-A 218.
Tops0e, F. A970), Topology and Measure, Lecture Notes in Math., Vol. 133, Springer, Berlin.
Tops0e, F. A976), Further results on integral representations, Studia Math. 55 C), 239-245.
Valdivia, M. A979), Analisis Matemdtico V, Unidades Didact'icas. Vols. 4, 5. 6, UNED, Madrid.
Varadajan, VS. A958), On a theorem of F Riesz concerning the form of linear fimctionals. Fund. Math. 19,
209-220.
Volcic, A. A975), An extension of the Daniell integral scheme, Mat. Cas. 25 C), 211-219.
Volcic, A. A977), Sulla differenziazione degli integrali di Daniell-Stone, Rend. Sem. Mat. Univ. Padova 61,
251-258.
Volcic, A. A980), Differentiation of Daniell integrals, Proceedings Measure Theory. Oberwolfach 1979, Lectures
Notes in Math., Vol. 794, Springer, Berlin, 284-294.

530 ?. Diaz Carrillo
Volcic, A, A981), Liftings and Daniell integrals. Measure Theory. Oberwolfach 1981. Lectures Notes in Math..
Vol. 945, Springer, Berlin. 180-186.
Weir, A.J. A974). General Integration and Measure, II. Cambridge University Press. Cambridge.
Wilhelm, M. A988). Integral extension procedure in weaklv ?-complete lattice-ordered groups II. Studia Math.
89C), 231-239.
Zaanen, A.C. A958), An Introduction to the Theory- of Integration, Interscience. New York.
Zaanen, A.C. A961), The Radon-Nikodym theorem, I, II. Indag. Math. 23. 157-187.

CHAPTER 12
Pettis Integral
Kazimierz Musial
Institute of Mathematics, Wroclaw University, 50-384 Wroclaw, PI. Grunwaldzki 2/4. Poland
E-mail: imisial@math.imi.wroc.pl
Contents
Introduction 533
1. Preliminaries 534
2. Measurable functions 535
3. Scalar integrals, basic properties 538
4. Pettis integral 541
5. Limit theorems 550
6. The range of the Pettis integral 553
7. Universal integrability 557
8. Pettis integral property 558
9. Weak Radon-Nikody'm property and related properties 560
10. Conditional expectation 568
11. Differentiation 570
12. Fubini theorem 572
13. Spaces of Pettis integrable functions 573
13.1. Space of all Pettis integrable functions 573
13.2. Functions satisfying the strong law of large numbers 575
13.3. Functions with integrals of bounded variation 577
13.4. 11.?(?, X) equipped with the variation norm of integrals 578
13.5. Bounded Pettis integrable functions 579
References 580
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
531

This page intentionally left blank

Pettis integral
533
Introduction
Until 1968, there has been no significant progress in the theory of the Pettis integral since
Pettis's A938) original paper. The Pettis integral was known to be countably additive and
absolutely continuous. Due to Pettis A938) it was also known that the space of Pettis
integrable functions defined on the unit interval, integrated with respect to the Lebesgue
measure may be non-complete in the semivariation norm. There were also known some
facts concerning Pettis integrability of strongly measurable functions and a few examples
of non-strongly measurable but Pettis integrable functions but almost nothing more. Then,
Rybakov A968) proved that the Pettis integral is of ?-finite variation. In 1975, Thomas
presented some negative aspects of Pettis integrability. He proved that the space of Pettis
integrable functions with respect to a nonatomic measure in never complete if the range space
is infinite dimensional and that for each infinite dimensional Banach space X there exists
an X-valued function on [0, l]2 such that not almost all its section are Pettis integrable on
[0, 1]. Thomas also suggested that probably one could find a Pettis integrable function /
on [0, 1] such that lim/,^o II7^ fs+' f(t)dt\\ = 00 on a set of positive Lebesgue measure.
Then started a series of papers developing the theory of Pettis integration.
Edgar A979) investigated Banach spaces with the property that each scalarly integrable
function with values in the space is Pettis integrable (so-called PIP) and proved several
non-trivial facts about these spaces. He was also the first to consider the Lindolof property
(so-called (C) property) for closed convex sets in the context of Pettis integration.
Fremlin and Talagrand A979) constructed an example of a Pettis integrable function
with non relatively norm compact range of its integral, answering a question of
Grothendieck formulated in the language of operators. The same paper contains a proof
of Stegall theorem saying that on perfect measure spaces the range of a Pettis integral is
always norm relatively compact.
Musial A979) (preprint version was published in 1976) called attention to the problem
of the existence of Pettis integrable Radon-Nikodym derivatives of Banach space valued
measures proving that there is a strong connection between conjugate spaces with all
measures of finite variation possessing Pettis integrable densities and Banach spaces not
containing any isomorphic copy of l\.
I have tried to include all papers concerning Pettis integrability, but the text presents
my own point of view on the subject. As a result some papers are only mentioned without
details. There are some topics completely absent in this survey. In particular I consider
only Banach spaces, in spite of several papers dealing with locally convex spaces, where
the notion of the Pettis integral is quite natural. I also left out of account the approach via
Stone transform due to D. Sentilles, for the benefit of liftings, which seem to be more
measure theoretical. That approach however was fully exploited by Talagrand A984).
I skipped also some important geometric properties of Banach spaces possessing WRNP
type properties (containing trees, extreme points), mainly because of limited amount
of space. Totally overlooked is the theory of strongly measurable functions, which is
discussed in several other papers and books (cf. Diestel and Uhl A977). I also decided do
not write anything about Fremlin's generalization of McShane integral (Fremlin A995)),
which is intermediate between Bochner and Pettis integral, in spite of thinking that the
integral of Fremlin may turn out to be more suitable for several applications than the Pettis

534
A". Musial
integral. Also the approaches of Phillips A940), Birkhoff A935), Dobrakov A970a) and
Erben and Grimeisen A990) are beyond this paper. Most of the problems posed in the
survey paper Musial A991) are still open, so I do not formulate new ones.
1. Preliminaries
Throughout (?,?,?) is ? finite complete measure space with ? ? 0. If not stated
differently, the conjectural measure space will always be denoted by (?.?.?). ?+
is the collection of all measurable sets of positive measure, ??(?) is the ?-ideal of
?-null sets and ?* is the outer measure generated by ?. ? will denote a lifting on
(?, ?,?) (for the detailed information concerning liftings see Strauss, Macheras and
Musial B002) in this Handbook). ?o(M) is the space of all real-valued measurable
functions on (?,?,?), functions which coincide ?-a.e. are not identified. ?? is the
topology of pointwise convergence in the space of all real-valued functions. C\ (?) is the
space of all real-valued ?-integrable functions. A set ? с С ? (?) is uniformly integrable if
limM(A)^0supye? jA \f\d? = 0. С is the ?-algebra of Lebesgue measurable sets on the
unit interval [0, 1] or on the real line and ? is the Lebesgue measure on C. V(S) denotes
the collection of all subsets of S. If Q is a non-empty set with a topology T. then the
?-algebra of Borel sets on Q is denoted by Bo(Q. T). Ba(Q. T) is the ?-algebra of Baire
sets. Compact spaces are always assumed to be non-empty and Hausdorff. X is always an
infinite dimensional Banach space (unless otherwise stated), B(X) is its closed unit ball
and X* is the conjugate of X. If ? с X then ?? denotes the annihilator of ? in X*.
If L с X then L** is the ?(?**. X*)-closure of L in X** and linL is the linear space
generated by L in X, If /;? —>¦ X, then the composition of / with a functional x* is
denoted as x* f or (x*, f).
We say that X has Mazur's property if each sequentially weak*-continuous functional
on X* is in X. X has the property (C) or X is a Corson space if each family of closed
convex subsets of X with the countable intersection property has a nonempty intersection
(see Corson A961), Pol A980) and Drewnowski A986)). X has the property (K) if
each sequence of points x„ e X converging to zero contains a subsequence (x,n) with
convergent series Y^L\ х,ц.
By BC(X**) we denote the set of all x** e X** which are weak*-cluster points of
countable subsets of B(X).
If у : ? —>· X is a measure, then we say that у is ?-continuous if 1нп//(д)^о ?|?>(?)|| = 0.
у is dominated by ? if there exists ? > 0 such that ||?>(?)|| ^ ??{?) for all ? e ?.
As a rule we will assume in the proofs that ? = 1. |?| denotes the variation of у and
cabv^, X) is the Banach space of all ?-continuous X-valued measures of finite variation
endowed with the variation norm. са(ц. X) is the Banach space of all ?-continuous X-
valued measures equipped with the semivariation norm.
If у : ? —> X is a ?-continuous measure, then for every ? e ?+ the set
? v(F) ]
A.(E) := I -J—-: F e ?? and F с ?
is said to be the average range of у on E.

Pettis integral
535
AXIOM L (cf. Fremlin and Talagrand A979)). [0. 1] cannot be covered by less then the
continuum closed sets of the Lebesgue measure zero.
AXIOM К (cf. Fremlin and Talagrand A979)). There is a cardinal к possessing the
following properties:
(a) there is a set U с [0, 1] of cardinality к such that k*(U) = 1;
(b) [0, 1] is not a union of к Lebesgue negligible sets.
It is known (cf, Fremlin and Talagrand A979)) that Axiom L is a consequence of
Martin's Axiom and Axiom К follows from the existence of real measurable cardinals,
2. Measurable functions
Definition 2.1, Let Г be a linear total subset of X*. A function f :? -+ X is said
to be ?-scalarly ?-measurable, if x* f is ?-measurable for each x* e ?. If ? = X*,
then / is called scalarly ?-measurable. If X = Y* and ? = ? then / is called weak*
scalarly ?-measurable. f is strongly measurable if there is a sequence of X-valued simple
functions /„ = 53/J'i xihXE,„ with ?,„ e ?, which is ?-a.e. converging to /. If ? is
fixed the reference to it will be suppressed. This will concern also all further
definitions.
The following theorem, due to Pettis A938), explains the relationship between the strong
and weak measurability.
Theorem 2.2 (Pettis's measurability theorem). A function f-.?^? is strongly
?-measurable if and only if
(i) / is scalarly ?-measurable, and
(ii) / is ?-essentially separably valued, i.e., there exists ? e ??(?) such that f(? \ ?)
is a separable subset of X.
DEnNlTlON 2.3. We say that two ?-scalarly ?-measurable functions /. g : ? -> X are
?-scalarly ?-equivalent if x* f = x*g ?-a.e. for each x* e ?. If ? = X*, we say about
scalarly ?-equivalent functions and, if X = Y* and ? = ? then / and g are said to be
weak* scalarly ?-equivalent. Two strongly measurable functions / and g are ?-equivalent
if f = g M-a.e.
Below are presented some classical examples of ?-measurable functions.
EXAMPLE 2.4. A scalarly measurable function that is not strongly measurable but is
scalarly equivalent to a strongly measurable function. Let a : [0, 1 ] —> [0, oof be a function
such that the set ? := {t e [0. 1]: a(t) > 0} is of positive Lebesgue outer measure and let
[et: t e [0, 1]} be the canonical basis for the nonseparable Hubert space Ь([0. 1]). Define
fa : [0, 1] -> /t([0, 1]) by fa(t) = a(t)e,. It is a consequence of the Riesz Representation
Theorem that x* f = 0 ?-a.e. for each x* e Ь([0, 1])* (i.e., / is scalarly ?-equivalent to

536
A". Musial
the zero function). On the other hand, if ? с Я is of positive outer measure then fa(E)
is nonseparable. In virtue of Theorem 2.2, fa is not strongly ?-measurable. If a is non-
measurable then also the function ||/„||: t —>¦ ||/a@|| is not measurable.
As a specific example one can take a = ?\>, where V is the Vitali set.
EXAMPLE 2.5 (Ryll-Nardzewski). A weak* scalarly measurable function that is not
scalarly measurable and not weak* scalarly equivalent to any scalarly measurable
function. Define /: [0, 1] -> C*[0.1] by f(s) = <5S. / is obviously weak*-X-measurable,
since for у e C[0, 1], we have (y, f(s)) = y(s). To see that / is not scalarly measurable
denote by ?? the atomic part of ? e C*[0. 1], and let V be a ???-?-measurable subset of
[0, 1]. Define x** e C**[0, 1] by ?**(?) = ?,,(?). Since x**(f) = ?? the function / is
not scalarly measurable with respect to ?.
Since C[0, 1] is separable, / is not weak* scalarly equivalent to any scalarly (hence also
strongly) measurable function.
It is worth to notice that the norm of / is a measurable function.
EXAMPLE 2.6 (Hagler). A scalarly measurable function that is not scalarly equivalent
to a strongly measurable one. Let (A„) be a sequence of nonempty subintervals of [0.1],
such that:
(i) A, =[0,1],
(ii) A„ = Ain U ??„+? for each ? e N,
(iii) А; П Aj = 0 if /'? j and 2" ^ij< 2"+1.
(iv) ???„?(?„) = 0.
Define /: [0, 1] —>· ?? by /(f) = (Хд„@) fori e [0, 1]. Then / is scalarly measurable (cf.
Diestel and Uhl A977)).
To prove that / is not scalarly equivalent to a strongly measurable function it is enough
to show that / itself is not strongly measurable (because l\ is separable). This immediately
follows from Pettis's Measurability Theorem. Indeed, if ?(?) > 0 and t. s are two distinct
points of ? then there is ? such that t e A„ buts ^ A„. Hence ||/(f) — f(s)\\ = 1.
One can ask when a scalarly measurable function is scalarly equivalent to a strongly
measurable one. The first global non-trivial result of this type is due to D.R. Lewis (see
Stegall A975/76a, 1975/76b)), who proved that scalarly measurable functions taking their
values in a WCG space are scalarly equivalent to strongly measurable functions. Edgar
A977) has undertaken an effort to characterize the Banach spaces with the property that
each X-valued scalarly measurable function is scalarly equivalent to a strongly measurable
function. In case of a single function we have
THEOREM 2.7 (Edgar A977)). Let f: ? -> X be a scalarly measurable function. Then,
f is scalarly equivalent to a strongly measurable function if and only if the image measure
/(?): Ba(X,weak) —>· R is tight in the weak topology (i.e., for each ? > 0 there is a weakly
compact W С X with ?*(\?) > ?(?) — ?).
As a direct consequence of the above theorem, one obtains:

Pettis integral
537
THEOREM 2.8 (Edgar A977)). Let X be a Banach space. Given any (?, ?,?), each
scalarly measurable f:? —> X is scalarly equivalent to a strongly measurable function if
and only if (X,weak) is measure compact (i.e., each measure on (X,weak) is ?-additive,
what in the case of the weak topology means the tightness of each measure, that is given
? > 0 there is a norm compact К С X with ?*(?) > ?(?) — ?).
????????? 2.9. A function f-.? -> ? is ?-scalarly ?-bounded provided there is
? ^ 0 such that for each x* e Г the inequality \x*f\ ? М||л:*|| holds ?-a.e. If ? = X*
then we say about scalarly ?-bounded function, and in the case of X = Y* and ? = Y-
about weak* scalarly ?- bounded function.
An easy calculation proves that if /: ? —>¦ X is strongly measurable and scalarly
bounded, then it is bounded (i.e., there is ? > 0 such that sup{||/(a>)||: ? e ?} ^ ?
?-а.е.).
The following fact (usually presented in the context of a family of measurable scalar
functions) permits often to reduce the general situation to the case of scalarly bounded
functions.
PROPOSITION 2.10 (cf. Musial A979)). // f-.? -> X is ?-measurable then there exists
a non-negative measurable function ??, with the following properties:
(i) For each x* e ? we have \(x*. /(?))| ;??^(?)\\?*\\ ?-a.e.,
(ii) ??(?) ? \\f(co)\\r (=sup{|<**, f(co))\: x* e Г П B(X*)}) ?-a.e.,
(iii) If ф : ? —> [0, oo) is a measurable function satisfying (i) and (ii) (with ?^. replaced
by ф), then ??, ?. ф ?-a.e.
PROOF. Consider the set ? ? R endowed with the ?-algebra ?(? ? ?), and the
product measure ? ? ?, where к is any probability measure on С such that ??(?) =
??(?). Let S(x*) = {(?,?): \(x*,f(w))\ > л||.г*||} for x* e Г, and let a = sup{(M ?
КЮТ=\ S(x*)l х*еГП B(X*), ?? e ?}. Since a < oo there are x*, х*_,...еГП В(Х*)
such that a = (? ? k)[U,?Li s(x*)]. Now, it is enough to put <p^ = sup,, \x*,f\, where the
supremum is taken pointwise. ?
COROLLARY 2.11. /// : ? —>¦ X is ?-scalarly measurable, then there exists a sequence
ofpairwise disjoint sets E„ e ? covering ? and such that for each ? the function f is
?-scalarly bounded on E„.
It is easy to give an example of a scalarly bounded function that is not bounded
(e.g., setting a(t) = t in Example 2.4). It turns out however that the things can be more
complicated. Edgar (see Talagrand A984)) proved that there exists a scalarly bounded
function which is even not scalarly equivalent to any bounded function. Weak*-scalarly
bounded functions behave much better. To formulate the result we will introduce a notion
that will be in constant use further.

538
К. Musial
Let / be an X*-valued function which is weak* scalarly measurable and weak*-scalarly
bounded. If ? is a lifting on L,c(M) then we denote by po(f) the unique X*-valued
function defined by
(x,p0(f)(w)): = p(xf)(w),
where ? and ? run across X and ?, respectively.
It is known (see A. and С Ionescu Tulcea A969)) that po(f) is weak*-Borel measurable
(i.e., /o(/)_l (?) e ? for all weak*-Borel В с X*) and the measure ?? : = ????/)-1 is a
Radon measure on the completion ?A of the ?-algebra of weak*-Borel subsets of X* (see
von Weizsacker A978) and Edgar A978)). It is however totally not obvious whether po(f)
is in general scalarly measurable.
PROPOSITION 2.12. If f-.? -> X* is weak*-scalarly bounded and weak*-scalarly
measurable, then for an arbitrary lifting ? the function f is weak*-scalarly equivalent
to the bounded and weak* Borel measurable function po(f).
3. Scalar integrals, basic properties
Definition 3.1. Let ? be a linear total subspace of X*. A function /: ? -> X is ?-
scalarly ?-integrable if x*f e L ? (?) for each.v* e T.If ? = X*, then / is called scalarly
?-integrable, and in the case of / : ? -> X* and ? = X с X**, the function / is said to
be weak*-scalarly ?-integrable.
DEFINITION 3.2. A ?-scalarly ?-integrable f-.? -> X is ?-?-integrable if for each
? e ? there exists v/(?) e X such that
?*??(?) = ??*/??
for each jc* e ?. The set function v/¦: ? —>¦ X is called the indefinite Г-integral of /
with respect to ?, and ? f(E) is called the Г-integral of / over ?el with respect to
?. An X*-integrable function is called Pettis ?-integrable and an X-integrable function
(if / : ? -> X* and ? = X) is called vveaA:* ?-integrable (or Gelfand ?-integrable). The
Gelfand integral of / will be denoted by *v/. If f-.? -> X is considered as an X**-
valued function then its weak* integral in X** is called the Dunford integral and it is
denoted by v**. It is clear that each Г-integral is uniquely determined and it is an additive
set function (provided it exists). The Г-integral is also a ?-measure, i.e., x*v( is ?-ad-
ditive for each x* e ?. Sometimes, we shall use the following notations: ? - fF f ??,
weak* - fE f ?? and D- fEf ??.
If f-.? -> X is scalarly ?-integrable, then an operator Tf.X* -> ?.|(?) associated
with / is defined by Tf{x*) = x* f.
From the integral point of view, the functions with the same indefinite Pettis integrals
are non-distinguishable, they are scalarly equivalent. We shall denote by ?(?, ?) (or

Pettis integral
539
by ?((?, ?,?),?) if necessary) the space of classes of scalarly ?-equivalent Pettis
?-integrable X-valued functions. ?(?. ?) is a linear space with ordinary algebraic
operations. One defines a norm on ?(?. ?) by
|/| =sup\ f \(x*. /)\?·?: л* e B(X*)\ =sup{\X*vt\(Q): .x* e B(X*)\.
It is known that
ill f 1
sup I / / ?? : ? e ? J
\\\Je I
defines an equivalent norm on ?(?, ?).
We will investigate also the space of all Pettis integrable functions with norm relatively
compact range of their integrals: ?, (?, X) : = {/ e ?(?, ?): ?/(?) is norm relatively
compact}. Besides ?(?, ?) we will be often considering its subsets ?(?, ?) consisting of
functions taking their values in a set К с ?.
It is one of the main problems in the theory of vector integration to find conditions
guaranteeing the existence of the Pettis integral.We shall start with two classical results.
PROPOSITION 3.3 (Gelfand A936)). Each weak* scalarly ?-integrable /:? -> X* is
weak* ?-integrable.
As an immediate corollary we get the following fact
PROPOSITION 3.4 (Dunford A937)). Each scalarly ?-integrable function f :? -> X is
Dunford ?-integrable.
If X is reflexive then the Dunford and Pettis integrals coincide. When X is not reflexive,
this may not be the case.
EXAMPLE 3.5 (cf. Diestel and Uhl A977)). A Dunford integrable function that is not
Pettis integrable. Define / : @, 1] —> c\) by
/@ = BХC-1.||@,22ХB-:.2-.|(г),..-.2',хB-я.2-^1|(г),.-.)-
If jc* = (??,«2,...) e /| = c*, then
x*f = ?^??2"?0~„ 2_„*i| and / \x*f\dX ? ]? |?„| < oo.
,7=1 ^0 /7=1
It follows from Proposition 3.4 that / is Dunford ?-integrable. On the other hand, it is
easily seen that for each ? e С
D- [ fdk= {2k(E П B-', 1]) 2"?(? П B"", 2-"+l]),. -.}.

540
A". Musiat
In particular
D- f /<Я = A,1,1,...,1.-..)^С(,
??.?]
and so / is not ?-Pettis integrable.
At this place one should recall that such a phenomenon cannot happen if X is a separable
Banach space not containing any isomorphic copy of cq. Unfortunatelly, the result does not
extend in this form to non-separable Banach spaces (cf. Edgar A979)).
THEOREM 3.6 (Dimitrov A971), Diestel A973)). If X is a separable Banach space
without an isomorphic copy o/co, then each X-valued Dunford integrable function is Pettis
integrable.
The function fa considered in Example 2.4 is Pettis integrable with vfu =0.
But there are also non-trivial examples of Pettis integrable functions.
Example 3.7. Let / be the function considered in Example 2.6. Since ||/(r)ll ^ 1
everywhere, / is Dunford integrable and for each ? e С and ? = ?\ + ?? e l^c = ? ®co>
we have
DC
j' ???? = j m(f)dk+ { rtf)dk = j ???„?\{{?})??
n= I
DC
= ]Г щ ({?})?(? ПА„) = (???, (?(? П ?,,))) = (?. (?(? П ?,,))).
The last equality follows from the fact that lim„ ?( A„) =0 and so v(E) = {?(??\ A,,)) e C(>.
But /ji considered as a functional on ?? belongs to c(| and so ???(?) = 0 for each ? e ?.
It follows that / is Pettis ?-integrable.
In case of a quite arbitrary ? nobody has been too much interested in describing the
properties of the ?-integral. The most interesting case is when ? = X*.
THEOREM 3.8. /// is Pettis ?-integrable, then v/ is a ?-continuous measure ofa-finite
variation. Moreover, \vf\(E) = fF<pfdp for each ? e ? (we put here ? / instead of ?*
for simplicity). In particular, the collection [x*f: ||.v*|| ^ 1} is uniformly ?-integrable.
The ?-additivity and continuity of v/ is due to Pettis A938). The ?-finiteness of the
variation was proved by Rybakov only in 1968.
Remark 3.9. A result similar to that in Theorem 3.8 for an arbitrary total Г с X* is
false. If ? is norming (i.e., for each ? e X the equality ||jc|| = sup{|(jc*,.r)|: ||л-*|| ^ 1,

Pettis integral
541
x* e ?} holds) and / is ?-integrable, then |v/1 is a ?-finite measure and \vf\(E) =
§???(?? for all ? e ? (see Musiat A979)). In particular, if f-.? -> X* is weak*-
bounded and satisfies for a lifting ? the equations xf = p(xf) for all ? e X, then
| v/ |(E) = /? 11/11 ?? for all ? e ?. This clearly generalizes the well known equality for
the Bochner integral. It may however happen that u/ is not countably additive in the norm
topology of X, and it is not ?-continuous.
It is so for the function presented in Example 3.5, when considered as an /^-valued
function. Vf : С —>¦ lx is weak*-countably additive but not countably additive in the norm
topology and not ?-continuous.
4. Pettis integral
We shall start describing conditions equivalent to the Pettis integrability with a classical
result that is always a starting point, when one wants to find conditions guaranteeing the
Pettis integrability of a single function.
THEOREM 4.1. Let f : ? —>¦ X be scalarly integrable. Then f is Pettis ?-integrable if and
only ifT/ : X* —>¦ L\ (?) is weak*-weakly continuous ifandonly ifTf : B(X*) —>¦ L\ (?) is
weak*-weakly continuous.
The following corollary is a simple consequence of Theorem 4.1.
COROLLARY 4.2. A scalarly integrable f :? —>¦ X is Pettis integrable if and only if the
set {x* e X*: x* f = 0 ?-a.e.} is weak* closed.
COROLLARY 4.3. // f-.? -> X is Pettis ?-integrable, then Tf.X* -> ? ? (?) is weakly-
compact and the set Zf := {x* f: \\x* || ^ 1} is weakly closed in L \ (?).
To formulate the next characterization of Pettis integrable functions we are going to use
an idea which is a generalization of Huff's A986) conception of separable-like functions.
Dehnition 4.4. A scalarly measurable function f-.? -> X is determined by a space
? с X if for every x* e Y^ the equality x* f = 0 holds ?-a.e.
The characterization presented below has been discovered by Drewnowski A986)
(without (iii)). Stefannson A992) proved then the equivalence of (i) and (iii).
THEOREM 4.5. ///: ? —>· X is scalarly integrable and the induced operator Tj : B(X*)
—> L\ (?) is weakly compact, then the following conditions are equivalent:
(i) /eP(M,X);
(ii) / is determined by a weakly compactly generated space ? ^ X;
(iii) / is determined by a space ? С X possessing Mazur's property;
(iv) / is determined by a Corson space ? С Х.

542
A". Musial
PROOF (Sketch). If / e ?(?, X), then one can take as ? the space generated by ?/(?).
Since each WCG space has Mazur's property (cf. Diestel A975, p. 148)) and is a Corson
space, we have (ii) =>· (iii) and (ii) =>· (iv). To prove (iii) =>· (i) we are going to use a
simple idea of Stefannson. Let /: ? —>¦ X be scalarly measurable and determined by a
weakly compactly generated ? с X. First denote by Ту the operator Ту : Y* —>¦ Ь\(ц.)
defined by 7>(y*) := Tt(y*x]), where y*xl is an arbitrary extension of y* to the whole X.
In a standard way one proves that Ту is weak*-weakly continuous if and only if 7/ is
weak*-weakly continuous if and only if Ту is ?(?*. y)-weakly continuous. Since 7/ is
weakly compact and / is determined by Y, one can prove, applying Mazur's theorem on
the equality of weak and norm closures of convex sets that 7/ is sequentially ?(?*, Y)-
weakly continuous. Hence Ту is sequentially ?(?*. y)-weakly continuous. Now, since
? has Mazur's property and Ту is sequentially ?(?*. y)-weakly continuous, we get the
weak*-weak continuity of Ту and hence the Pettis integrability of /. For the proof of
(iv) =>· (i) we refer to Drewnowski A986). ?
Notice that each of the conditions (ii)-(iv) of Theorem 4,5 can be equivalently written as
??(?) с ?**, where ? is respectively WCG. Mazur or Corson. Example 3.5 shows that
the weak compactness of Tf in the above theorem is necessary.
To formulate more general result, we need a notion introduced by Drewnowski A986).
Definition 4.6. If ?? is the collection of all Corson subspaces of X then the Corson
envelope of X is defined by
X:= (J Г*.
YeCx
Drewnowski A986) strengthened the implication (iv) =>· (i) of the above theorem to the
following form:
THEOREM 4.7. Iff-.?^? is scalarly integrable, T,¦ : X* -> L\ (?) is weakly compact
and ?**(?)(??, then /еР(д.Х).
Geitz A981) introduced a notion related to Rieffel's essential range of strongly
measurable function (Rieffel A968)).
Definition 4.8. Let /: ? -> X be a function and let ? e ? be an arbitrary set. The core
of / over E, denoted by cor/¦(?), is the set defined by
cor/(?):=P|{conv/(?\yV): ????(?)}.
Geitz A981) also noticed that if/??(?,?). then for each ? e ?+
сот/(Е) = convA ; (E).
The following lemma is quite useful:

Pettis integral
543
LEMMA 4.9. /// is weakly measurable and cor /(?) ?? for all ? e ?+, then x* f = 0
?-a.e. if and only ifx* = 0 on cor/ (?).
Then Andrews A985) introduced the weak*-core of a weak* scalarly measurable
function /; ? —>¦ X*\
cor*f(E) := P|{conv*/(? \ ?): ? ???(?)}.
Drewnowski A986) introduced the weak**-core (he called it the weak*-core) setting
cor**(?) = Q{conv**/(?\yV): ? ???(?)}
and proved that if / e ?(?, ?), then
cor}*(?) = cbrTTEj" = Л~{Ё~)**.
THEOREM 4.10 (Talagrand A984)). Let f :? -> X be scalarly integrable. Then f is
Pettis integrable if and only if ?t :X* —>· ?,|(?) is weakly compact and cor / (E) ?^/ifor
each ? e ?+
PROOF. Assume that Tf :X* —> ?.|(?) is weakly compact, cor/ (?) ? 0 for each ? e 2?+
and / ? ?(?, ?). Then, according to Lemma 5-1-2 of Talagrand A984) (which remains
true for functions with weakly compact Tf), there exist functionals y*. .v* e B(X*) such
that ?{? e ?: ?*/(?) ??*/(?)} > 0 and у*/ is in the pointwise closure of the set
??/:={?*/???:?*/ = ?*/?-?*.}.
Let {x*} с W/ be pointwise convergent to y*f. Without loss of generality, we may
assume that x* —>¦ y* in ?(?*. X). Since for every a we have x*f = z* f ?-a.e., we
have .х*|сог;Ш) = r*|Cor,№)· Hence y*|a)r,(i2) = c*L-or,<i?) and consequently y*f - z*f
?-a.e.
This however contradicts our assumption, and so / e ?(?, ?). ?
Another proofs of the above result were presented by Edgar A982) and Huff A986).
Certainly the above theorem has been inspired by earlier characterizations of Pettis
integrability obtained by Geitz A982) in case of a perfect measure (that was a little bit
more technical) and by Sentilles's A981) characterization (also for perfect measures only)
of Pettis integrability in terms of the Stonian transform.
Talagrand A984) formulated also an equivalent condition guaranteeing Pettis
integrability of /, which has been then reformulated and reproved by Drewnowski A986) in the
following form:
THEOREM 4.11 (Drewnowski A986)). Let f: ? -> X be scalarly integrable. If
сот*,*(Е) П ? ? Я for each ? e 2?+ and Tf is weakly compact, then f is Pettis integrable.

544
A". Musial
????????? 4.12 (Riddle, Saab and Uhl A983)). A family ? of real-valued functions on
(?, ?, ?) has the Bourgain property if for each ? e 2?+ and each ? > 0 there is a finite
collection fcpffjn ?+ such that for each function h eH one can find Fef with
suph(F) - infh(F) < ?.
In case of a uniformly bounded ? an equivalent definition is as follows: for each
? & ?+ and each pair a < b of reals, there is a finite collection ? с V(E) П .?7+
such that for each function h eH one can find F e ? with inf{/?(o>): ? e F) ^ a or
sup{/?(o>): ? e F} < b.
It is easy to see that the Bourgain property of ? yields the same property of the pointwise
closure of ?. Moreover the Bourgain property of a function constrains its measurability.
PROPOSITION 4.13 (Riddle, Saab and Uhl A983)). If ? satisfies the Bourgain property,
then each function in ? is measurable and each function in the pointwise closure of ? is
the almost everywhere pointwise limit of a sequence from ?.
PROOF. In order to prove the proposition take / e HT<' and an ultrafilter WonH which
has / as a cluster point. Then, put for ? e 2?+ and ? > 0
?(?,?)= {heH:suph(E)-mfh(E) <?).
Each ? e ?+ contains F e ?+ with H(F. ?) e U. Using the Zorn-Kuratowski lemma,
one can find for each positive ? a maximal family Ve of pairwise disjoint sets in 2?+ such
that H(F, ?)eW for each F eVF. It is obvious that ?(? \ \JFeVf F) = 0. Moreover, if
??? is the family of all finite subcollections of Ve, then
/e f) f) ??(?,?)'' eU.
KeK, EeK
Now let for each m e N the sequence {A,„.„: ? e N} be an enumeration of V\/„, and let
?,„„ e AmM be arbitrary. Taking for each meNa function
in m
/mep|p|W(At.(I;l/fc)
* = l n=l
such that
|/m(Wjt.„)- /(Wjt.„)| < 1/*
for each 1 ^ k, ? ^ m, we get a sequence (/,„) с ? that is ?-a.e. convergent to /, D
Dehnition 4.14. Let S be a topological space and let ? be a positive finite measure
defined on a ?-algebra В containing all Borel subsets of S. ? is said to be selfsupporting,
if for each В е В+ there exists A eV(B)C) B^ such that ?\ A is strictly positive (i.e., for
each open U, we have t/nA = 0ort/nAe #+).

Pettis integral
545
It is easily seen that if ? is selfsupporting, then for each В e #+ there exists A e V (?) ?
?+ such that ?\? is strictly positive and ?(?) = ?(?). We shall write A = supp(M|S).
It is well known that Radon measures are selfsupporting. Also if ? is a lifting on ?, Tp
is the topology defined by taking as its basis, the family {A e ?: Ac p(A)} (cf. A. and
С Ionescu Tulcea A969, p. 59)), then ? is selfsupporting with respect to Tp.
PROPOSITION 4.15. Let S be a topological space and let ? be a uniformly bounded
family of real-valued continuous functions on S. Moreover, let ? be a selfsupporting
measure on ? ?-algebra В 5 Bo(S). Then, if ? does not contain any sequence equivalent
to the standard unit vector basis of I \ (in the sup norm), then ? has the Bourgain property.
PROOF. Suppose that ? does not have the Bourgain property. Then, there exists ? e BjL
such that ? — supp^|7) and there exist a < b such that for each finite collection
П с V(T) П B+ there is / e ? with inf f(R)<a and sup f{R) > b, for every R e K.
We shall now construct inductively a collection {A„.,„: m = 1 2"; ? e N} of sets
from ?+ and a sequence (/,,) с ? satisfying the following properties:
A„+\.2m-\ UA„ + |1,„ С Anj„.
fn+\(s)<a ifi e A„+i.2„,_|.
fn+\(s)>b ifs e ?,,+ ?.:,,,.
Assume, we have already constructed {/„,: m = \,...,k) and {A„.,„: m = 1 ,2";
? = 1,... ,k]. By the assumption, we can find for each m e {1 2k] a set
Tk.meB+nV(Ak.,„)
such that Tk ,„ = 5ирр(д|7А.,„) (for the first inductive step set 7bi = 7o? = T). Moreover,
there is fk + \ &H. with
inf fk+\ (TkJn) < a and sup fk+\(Tk.m) > b
for every m e {1,..., 2k}.
Put now
Ak+\.2m-\ = [s eTk,,„: fk+\(s) <a),
Ak+\.bn = {s e Tkjn: fk+\(s) > b).
It follows that Ak+\.m e B+ for every m e {1 2*+'}. Rosenthal's argument A974)
shows that the sequence (/„) is equivalent to the standard basis of l\ in the sup norm. D
Dehnition 4.16. Let К с X* be a nonvoid set. A set W с X is weakly precompact
with respect to К (or weakly ?-precompact) if each bounded sequence in (x„) с W has
a subsequence (x,n) such that for each x* e К the sequence (x*,x„k) is convergent. If
К = B(X*), then W is called weakly precompact (equivalently, no bounded sequence
in W is equivalent, in the norm topology, to the unit vector basis of/|).

546
A". Musial
As particular instances of Proposition 4.15 we get the following results:
COROLLARY 4.17. Let ? be a lifting on ?^(?, ?, ?) and let ? be a uniformly bounded
family of real-valued functions defined on ? and such that ?. is weakly precompact in
^cc(M)· Then p{H) has the Bourgain property.
Corollary 4.18 (Riddle and Saab A985)). If{xf: \\x || ^ 1} has the Bourgainproperty,
then f e ?(?, X*) and the range ofvf is norm relatively compact.
PROOF. Fix ? > 0, ? e ? and ?** e B(X**). Then put
H=[(x,f):xe B(X). \(x** - .v. *vf(E))\ <?].
Clearly ?. has the Bourgain property.
It follows from the Goldstine theorem that (x**, /) is in the pointwise closure of ?.
and so (Proposition 4.13) there are x„ e ?, ? e N, such that {xn, /)eW for each ? and
lim„U„, /) = (***, /) ?-a.e.
Hence, we get
<2?
by the Lebesgue Convergence Theorem, and this proves that / e ?(?, ?*). The norm
relative compactness of ?/ (?) — *v/ (?) follows from the properties of Tj: if {x„) is
weakly convergent, then (Tf(x„)) is a.e. convergent (cf. Riddle, Saab and Uhl A983)). D
COROLLARY 4.19 (Haydon A976)). Let X be a Banach space containing no isomorphic
copy ofl\. If ? is a complete finite Radon measure on B(X*) equipped with the weak*
topology, then the identity function on B{X*) is ?-Pettis integrable.
PROOF. In view of Proposition 4.15 the family ? = B(X) has the Bourgain property.
Hence Corollary 4.18 yields the Pettis integrability of the identity function. D
Among several questions concerning Pettis integrability the following seems to be quite
interesting: assume that / e ?(?. ?*) is weak*-scalarly bounded and ? is a lifting on
^(?). When is the function po(f): ? —>¦ X* Pettis integrable?
The next result is due to Talagrand A984) but his proof is different from that presented
here.
PROPOSITION 4.20. Let f :? —>· X* be a weak*-scalarly bounded and weak*-scalarly
measurable function. Assume that for each S > 0 there is ? e ? with ?(? \ ?) <
? and such that the set {(/. jr)/?-: ||л|| < 1} is weakly precompact in ?,^(?). Then
Po(/) e ?(?, ?*). In particular, if X does not contain any isomorphic copy of l\, then
й(/)еПд,П
PROOF. The assertion is an immediate consequence of Corollaries 4.17 and 4.18. D

Pettis integral
547
As a consequence we get a result due to Riddle, Saab and Uhl A983) and deduced by
them from Theorem 9.10 via Theorem 9.7.
COROLLARY 4.21. Let X be a separable Banach space and let f: ? -> X* be a weak*-
scalarly bounded and weak*-scalarly measurable function. Assume that for each S > 0
there is ? e ? with ?(? \ ?) < S and such that the set {(/, x)xe- \\x\\ < ? " weakly
precompact in ?,^??). Then f e ?(?, ?*).
THEOREM 4.22 (Macheras and Musiat B000)). Let f :? -> X* be a scalarly bounded
Pettis integrable function. If each x** e X** is ^-measurable, then A)(/) e ?(?. ?*)-
PROOF. According to Theorem 4.5 and the weak*-Borel measurability of A)(/) we have
only to show that for an arbitrary с e ?/-B?)? the equality (z, A)(/)) = 0 holds true ?-
a.e. Suppose that there exists с e ?/ (?)^ such that ?{? e ?: (?, ?)(/)(?)) > 0} > 0 and
||z|| = 1 and let K°f be the the weak*-closureof ft)(f)№) in X*. Then, since ?0 is aRadon
measure and ??(?°.) = ?(?), there exist a weak*-compact set L с {.?* e AT^-. (z.x*) > 0}
and a positive real number a such that
?o(^)>0, г is continuous on L and (c, .x*} > a for each x* e L.
Take now an arbitrary net (ха)аел in B(X) that is ?(?**, ?*)-convergent to с Then, the
convergence to ? is uniform on ?/ (?). To avoid unnecessary complications, we assume
at once that the initial net (ха)аел is Mackey convergent to с Then, for each ? e N there
exists an e A such that
\{xa,vf(E))\ < l//j for all ? e ? and all a ^a„.
If A„ := {a e A: a ^ a,,}, then for each collection of points x* .v* e i, there is an
index or.-* . * e A„ such that
\(?,?*)-(???, ,...v,*}| < 1/" foreach/'^/j.
Equivalently,
L"ci{J{x*:\(z.X*)-(xa.x*)\<l/n}".
aeA„
Now, as a consequence of the compactness of L and the continuity of z\L, there exists a
finite set B„ с А„ such that the inclusion
L"c (J{**: \{z,x*)-{xa.x*)\<l/n}n
aeB„
holds true. It follows that z\L is a pointwise cluster point of the countable set {.va|L: xa e
U^i S,,}. Consequently, there exists x** e X** that is a weak*-cluster point of the

548
К. Musial
set {xa: a e Ц^, ?,,} and x*f\L = z\L. It follows from the construction of x™ that
Xq* e Vf(EI and so
f (x?*,po(f))d?>a?{po(f)-\L))>O = {x**.v/{p0(f)-](L))).
J^frHL)
On the other hand, one can easily show that if /: ? —>¦ X* is a scalarly bounded Pettis
integrable function, then
[x**
,?/(?)) = ?(?**,??{/))??
for each x** e BC(X**) and each ?el
This gives the required contradiction. ?
The next corollary is part of Proposition 4.20 but can be also derived from Theorem 4.22
if one takes into account Haydon A976) who proved that if l\ <? X isomorphically,
then each functional x* e X* is universally measurable with respect to the weak* Borel
structure of X* (this is in fact contained in the assertion of Corollary 4.19).
COROLLARY 4.23. If l\ <? X isomorphically and f-.?^-?* is weak* scalarly
measurable and weak* scalarly bounded, then for each lifting ? the function po(f) is
Pettis integrable.
Trying to overcome difficulties met while characterizing Pettis integrable functions
Fremlin imposed an additional requirement besides the scalar integrability. The property
was called by him "proper relative compactness" (Fremlin A982)) and then "stability" by
Talagrand A984).
Definition 4.24. Let ? be a collection of real valued functions defined on ?. ? is said
to be stable if for each A e 2T| and arbitrary reals a < ? there exist k, I e N satisfying the
inequality
l4+i ( U (/ < «}* x (/ > ?)' n ?"+?) < H(A)k~
where ?*+/ is the direct product of к +1 copies of ?.
One may assume in the above definition that к = I. Moreover (Talagrand A984)), if ?
is stable and pointwise bounded then it is pointwise relatively compact in ??(?) and its
pointwise closure is also stable.
Definition 4.25 (Talagrand A984)). A function / : ? -> X is properly measurable if
the set
Zf:=[x*f: |И|<1}
is stable.

Pettis integral
549
THEOREM 4.26 (Talagrand A984)). // f :? -+ X is properly measurable and Zf is
uniformly integrable, then f e ?(?, ?) and the range of ? f is norm relatively compact.
If ? is perfect, Axiom L holds and B(X*) is weak*-separable, then each scalarly
measurable f: ? —> X is properly measurable (and hence it is Pettis integrable if Zf
is uniformly integrable).
PROOF (Sketch). We have to prove only that the map x* -> x*f from the unit ball of X*
into L\(?) is weak*-norm continuous. Let (x*f)ses be a net that is pointwise convergent
to x*f and ||jc*|| ^ 1 for all s e S. According to Talagrand A984, Theorem 9-5-2), the
net (x*f)ses is convergent to x^f in ?-measure. Because of the boundedness of the net
in ?,^?) the Lebesgue dominated convergence theorem may be applied, yielding the
convergence in the norm of L \ (?).
The second assertion is a direct consequence of Axiom L (Talagrand A984, Theorem
9-3-3)). D
In case of a weak*-scalarly bounded and weak*-scalarly measurable function
/: ? —> X* we have the following two results:
THEOREM 4.27. Let ? Cl X be a bounded norming subset of X and let f : ? -> X* be a
function. If each countable subset of the set {xf: ? e ?} is stable then for each lifting ? on
Loc(?) the function po(f) is an element of ?(?. ?*) and the range of its Pettis integral is
norm relatively compact. If ? is consistent, then po(f) is properly measurable.
PROOF. If ? is a lifting, then each countable subfamily of {xpo(f): ? e ?} is stable. If
moreover ? is consistent, then the whole collection {xpo(f): ? e A] is stable (see Musial
B000)). The Pettis integration of po(f) and the norm relative compactness of its Pettis
integral are now consequences of Theorem 4.26.
Let now ? be an arbitrary lifting on Loc(?). Then again each countable subfamily
of {xao(f): ? e ?} is stable. Due to the countability of the set we may apply Lemma
9-3-2 of Talagrand A984) to get the stability of each countable subset of ? with
respect to ???(/)~' defined on the completion of weak*-Borel subsets of X* with
respect to ??o(f)~]. But as all elements of ? are weak*-continuous, we may apply
Theorem 9-4-2 of Talagrand A984) obtaining the ???(/)_? -measurability of all elements
of X** (notice that in the proof of the implication (c) =>· (a) of 9-4-2 Axiom F is not
applied). Since ??(/) is weak*-scalarly measurable, we get the weak scalar measurability
of ??(/). The Pettis integrability of ??(/) follows now from Theorem 4.22. Since the
Pettis integral of ??(/) coincides with that of po(f), this completes the whole proof. D
THEOREM 4.28 (Talagrand A984)). (Axiom L) Let (?, ?, ?) be perfect, f : ? -+ X*
be a function and let ? с X be a bounded norming set. Moreover, let A a be the union of
the weak*-closures in X** of countable subsets of ?. Then the following conditions are
equivalent:
(a) For each x** e A a the function л**/ is measurable;
(b) If ? is a consistent lifting on Lr^(?), then po(f) is properly measurable;

550
К. Musial
(c) If p is a lifting on ?,^??), then po(f)e ?(?. ?*) and the range of its Pettis integral
is norm relatively compact:
(d) If ? is a lifting on Loc(?), then po(f) is scalarly measurable.
PROOF. If ? is perfect, Axiom L holds true and (a) is satisfied, then each countable subset
of {xf: ? e ?] is stable by Talagrand A984, Theorem 9-3-3). ?
5. Limit theorems
It is the purpose of this section to prove the convergence theorems of Vitali and Lebesgue
type for the Pettis integral. The theorems had been proved first by Geitz A981) under
the assumption of perfectness of the basic measure space and then they were proved in
full generality by Musial A985). Geitz's proofs were based on a James's characterization
of weakly compact subsets of a Banach space (see James A964)) and on Fremlin's
subsequence theorem (see Fremlin A975). Applying the classical Mazur theorem on the
equality of weak and norm closures of convex sets instead of Fremlin's theorem I was able
to get rid of perfectness of the measure space (Musial A985)). Below I am presenting still
another proof based on a Grothendieck characterization of weakly compact sets that seems
to be more elementary than James's characterization (cf. Holmes A975, p. 157 for a simple
proof)). In fact one can obtain the result directly from Theorem 4.5 but the proof presented
here seems to be more exciting.
PROPOSITION 5.1 (Grothendieck A952)). A bounded nonvoid set W С X is relatively
weakly compact if and only if for every two sequences (v„) С W and (x*) С В(Х*) the
equality
lim lim(x* ,?,? — Hmlim(r* . .v„)
П 111 ' 111 ?7 '
holds true, provided all the limits exist.
The theorem we are going to present is an analogue of Vitali's convergence theorem.
Conditions (a) and (b) of this theorem guarantee that for each x* e X* and ? e ? the
sequence {$??*^??: ? e ?} is convergent to fEx*fdp , and that the set {x*f: x* e
B(X*)} is weakly relatively compact in ?.|(?). They may be replaced by any others
guaranteeing the above weak compactness and the convergence of the appropriate sequence
of scalar integrals.
THEOREM 5.2 (Geitz, Musial). Let f :? -> X be a function. If there exists a sequence
of Pettis integrable functions /„ : i? —>¦ X such that:
(a) The set {x* f,: \\x*\\ ^ 1, ? e N} is uniformly ?-integrable,
(b) lim„ x* fn =x*f in ?-measure, for each x* e X*,
then f is Pettis ?-integrable and,
lim / ?„??= \ f ??
weakly in X,for each ? e ?.

Pettis integral
551
PROOF. Fix ? e ?. Since the classical Vitali convergence theorem yields the convergence
lim„ fEx*f„dp = fEx* f ??, for each ? e X*. we see that the sequence (fE f, ??) is
weakly Cauchy.
In order to prove our assertion, it is sufficient to show that given (/„,„)», and {x\)k С
B(X*), we have
a := limlim/^*, / /„,„ ??) = ? := limlim/.v^, / /„,„ ???
provided all the limits exist.
It follows from (a) and (b) that the sequence (.v^, /)* is uniformly integrable and
bounded in L\ (?). Hence it is weakly relatively compact. This yields the existence of a
subsequence (z*k)k of (xf)k and a function h ??-?(?) such that z*kf ^ h weakly in ?,|(?).
Mazur's theorem yields now the existence of functionals wf e conv{~*: /' ^ k} such that
lim / \w*kf - ??\?? = 0 and limu^/ = h ?-a.e.
If Wq is a weak*-cluster point of (wf)k, then ? = u>(*/ ?-a.e. Consequently,
<* = wofdV-
On the other hand, we have
limU*. J . /«„ ??\ = lim/;*. ? /„,„ ??) = lim/n*. j /„„, ??
= (»'()- ( /«„ ^М) = J >'?· ??,„)??.
It follows from the classical Vitali theorem that
? = \??[(??*0,?„,)?·?= /"?,/^?.
'" Je Je
This proves the required equality a = ?. ?
As an immediate consequence of Theorem 5.2, we get the following generalization of
the classical Lebesgue Dominated Convergence Theorem.
THEOREM 5.3 (Geitz, Musial). Let f : ? -> X be a function satisfying the following two
conditions:
(a) There exists a sequence of Pettis ?-integrable functions /,,:i2 —>¦ X such that
lim„ x*fn = x* f in ?-measure, for each x* е X*,
(?) There exists h e ?.|(?) such that for each x* e B(X*) and each ? e N, the
inequality \x*f,\ < h holds ?-a.e. (the exceptional sets dependon x*).

552
A". Musial
Then f e ?(?, ?) and
lim / fndix= \ ???.
" Je Je
weakly in X, for all ? e ?.
PROPOSITION 5.4. Assume that the assumptions of Theorem 5.2 or 5.3 are satisfied.
Assume moreover that ? is perfect and the functions f, are scalarly bounded. Then
Hindoo /,, = / weakly in ?(?, ?).
PROOF. According to Collins and Ruess A983) if/? л:*/<*M^ fE x* f ?? for all ? e ?
and all x* being extreme points of B(X*), then the sequence (/„) is weakly convergent
??/???(?, ?). ?
It would be interesting to know whether the unit ball of X* in the assumptions of
Theorems 5.2 or 5.3 may be replaced by its extreme points.
It is natural to ask when the condition (a) of Theorem 5.3 is sufficient for the Pettis
integrability of a function. If X contains an isomorphic copy of cq, then (a) is too weak to
guarantee the integrability. Indeed, let / be the function considered in Example 3.5. With
the same notation, we have
ОС
? f = 2_ian2 "XB-.-2-"-
/1=1
If
/,(?) = B?B-,ц(г),..., 2"??-„_2-„~?,@,0,0,0,...)
then clearly
lim / ?*^?? = ? ?*f ??
" Je Je
foralljt*eX*and Ее ?, but / i ?(?. ?)
It turns out however that со is the only exceptional Banach space.
THEOREM 5.5. Let X be without any isomorphic copy of cQ. If f: ? ^ ? is scalarly
?-integrable and there are functions f, e ?(?, ?) such that
lim / ?*^??= / ?*f ??
" Je Je
for all ? ?? and ?* e X*, then f e ?(?. ?) ???
lim / ???? e / f ??
weakly in X for all ? e ?.

Pettis integral
553
PROOF. In virtue of Corollary 2.10 the set ? can be decomposed into pairwise disjoint
set ?„ e ?, such that for each ? e N and x* e X* the inequality |.?*/??„ I ^ «II-"*"* II holds
?-a.e. It follows from the comments preceding Theorem 5,2 that / is Pettis integrable on
each ?„. Now it is sufficient to apply the series characterization of a Banach space not
containing cq (Bessaga and Pelczynski A958)). ?
Notice that if X is separable, then the assumption of scalar integrability itself yields the
required assertion (Theorem 3,6),
6. The range of the Pettis integral
For several years it has been an open question of Pettis whether for each Pettis integrable
function the range of v/ is always norm relatively compact,
Fremlin and Talagrand A979) gave a negative answer to that question, Before providing
their example let us first present a positive result of Stegall (see Fremlin and Talagrand
A979)),
THEOREM 6,1 (Stegall), If ? is perfect then for each scalarly integrable f : ? -> X such
that Tf is weakly compact, the range ?**(?) of the Dunford integral of f in X** is norm
relatively compact.
PROOF. We may assume that / is scalarly bounded. It is obvious that for each ? e ?
the equality ??-?? = v*f*(E) holds, so in order to prove the norm relative compactness of
vj( ?) it is sufficient to show the compactness of Tt. To do it, choose any sequence (x„) с
B(X*). Since / is scalarly measurable, (x*f) has a subsequence (x*Iltf) that converges
a.e. Otherwise, we could apply Fremlin's subsequence theorem (Fremlin A975)), to get a
subsequence without measurable cluster points, in the space of all real-valued functions
endowed with the topology of pointwise convergence.
If x* is a weak* cluster point of {(x* )), then x*nf -> x*f pointwise, and hence in
L ? (?), because of the Lebesgue theorem.
Thus, Tf is compact, and the assertion is proved. ?
Since perfectness of the basic measure space in not necessary for a Pettis integral to have
norm relatively compact range there is an obvious question when it can happen. It turns
out that the answer depends on approximation of Pettis integrable functions by simple
functions.
THEOREM 6.2 (Musial A985)). If f e ?(?, ?), then vt (?) is norm relatively compact
if and only if f is a limit of a sequence of X-valued simple functions, in the norm topology
?/?(?, ?).
We say that X has the ?-PCP (Pettis compactness property) if for each / e ?(?, ?)
the set ?/(?) is norm relatively compact. If this property is satisfied for an arbitrary
(?, ?, ?), then X has the PCP No general description of PCP is known but there are
interesting partial solutions. In particular we have the following result of Talagrand:

554
К. Musiai
THEOREM 6.3 (Talagrand A980)). (MA) If l^ is not a quotient of X, then X has the
PCP.
Talagrand A980) presents also an example of X possessing PCP and having lx among
its quotients.
More can be said about the following stronger property.
DEHNITION 6.4. A set 0 ? К с X has the ?-Compact Range Property (CRP) if every
X-valued ?-continuous measure of finite variation with its average range contained in К
has norm relatively compact range.
The global property in this context (i.e., CRP of X or, equivalently, of B(X)) has been
studied by Musiat A979). Then it was localized by Riddle, Saab and Uhl A983). They
called К to be a set of complete continuity.
Definition 6.5 (Riddle, Saab and Uhl A983)). A subset ? ? 0 of a Banach space X
is called a ?-weak Radon-Nikodym set (respectively, a ?-weak**-RN set) if for every ?-
continuous measure of finite variation у : ? —>¦ X satisfying for all sets ? e ? the inclusion
Av(E) с К there exists a ?-valued (respectively, a AT**-valued) Pettis integrable density
of у with respect to ?. In the above definitions the ?-continuity may be replaced by ?-
domination. If AT is a ?-weak Radon-Nikodym set for all finite complete ?, then it is
called a weak Radon-Nikodym set. Similarly for ?-weak**-RN set. Sometimes we will
say also that К has the weak (or weak**) Radon-Nikodym property. If К = B(X) then we
say about the weak RNP (or weak** RNP) of X.
WRNP was introduced in Musiai A979) and W**RNP by Janicka (see Musiai A980)). It
is an immediate consequence of the complementability of X* in X*** that X* has WRNP
if and only if it has W**RNP
Remark 6.6. L? [0, 1] is an example of a Banach space without the W**RNP Indeed,
L,[0, 1] is complementable in L**[0, 1], so the W**RNP of L,[0, 1] would imply the
RNP of the space, and it is well known that L | [0, 1 ] does not enjoy the last property.
Since each measure space can be embedded as a thick subset of a perfect measure space
(cf. Musiai A979)), we get the following conclusion from Theorem 6.1:
THEOREM 6.7 (Musiai A979) in case of К = X). If К has the W**RNP (in particular
WRNP), then К has also the CRP.
In case of the separable range of Vf the complete characterization is given by the
following theorem that has been obtained independently by Talagrand A984) and Musiai
A985):
THEOREM 6.8. /// e ?(?, ?), then the following are equivalent:
(i) {x*f: x* e B(X*)} is a separable subset of ^?(?);

Pettis integral
555
(ii) There exists ? ?-algebra ГС1 such that (?, ?,?\?) is separable, and f is
scalarly measurable with respect to ?;
(iii) There exists a sequence (/„) of X-valued simple functions, such that {x*f,'- " ? ?,
||**|| ^ 1} is uniformly integrable and for each x* e X* the sequence (x*f,)„eU is
?-a.e. convergent to x*f;
(iv) There exists a sequence (/„) of X-valued simple functions, such that for each
x* e X* the sequence (x*f,}ne'::. is convergent to x*f weakly in L\ (?);
(?) у / (?) is a separable subset ofX.
PROOF, (i) =>· (ii) Assume that the set {x* f\ x* e B(X*)} is separable. Then there exists
a sequence (x*) in B(X*), such that {x*f: ? € ?} is dense in {**/: x* e B(X*)}. If ? is
the ?-algebragenerated by all x*f and by ??(?) then, clearly ?\? is separable and each
x* f is ?1-measurable.
(ii) =>· (iii) Assume that / is scalarly measurable with respect to a separable measure
space (?, ??,?\??) and let ? = ?{(?„: ? e ?}) с Д, be a countably generated ?-
algebra that is ?\ ??-dense in ?0. Moreover, let ?„ be the partition of ? generated by the
sets E\,..., E„.
Put for each ?
_ ^ Vf(E)
}"-^ ?{?)??
Еел„
with the convention 0/0 = 0.
Since {x*f: \\x*\\ ? 1} is uniformly integrable, this yields the uniform integrability of
{x*f„: ? e N, ||jr*|| ^ 1}. As by the assumption ? is dense in ??, we have E(x*f\E) =
x* f ?-a.e., and so lim„ x* f, =x*f ?-a.e..
(iv) =>· (v) The condition (iv) means that for each ? e ? the sequence (\>jn(E)) is
weakly convergent to vj(E). Hence, y/ (?) is contained in the weak closure of the
set U,^| ?f„(?) and the last set is separable, since the ranges of all vj„-s are finite
dimensional.
(v) =>· (i) Suppose that [x*f: x* e B(X*) is non-separable and take an arbitrary
jc* e S(X*) and h\ e ?.^(?) such that {h\,x*f) = 1. Assume then that we have
already constructed for an ordinal ? < ?\ a family {(x*. ha): a < ?] with the following
properties:
(a) x*eS(x*),
(?) ha e ?-??(?),
(у) **/elin{**/: ? < у} for each ? < ?,
«>«.,«>=(: ¦;::;;;*:
Since {x*f: x* e S(X*)} is non-separable, one can find Xp e S(X*) such that .v^/ ?
linjjc*/: ? < /3}. Then, applying the Hahn-Banach theorem we get hp e ?.^(?) such that
(hft,x*pf) = \ ma (hp,x*f) = \ ??? -?\ ? < ?.
Consequently, we get a net {(**, /?„): a < ?\} satisfying (?)-(?) for all ?, ?, ? less
than ? ?.

556
К. Musiat
It follows that
\T}(hp)-T}(ha)\>\
whenever ? < ?, and so the set 7",*(?-^(?)) is non-separable in X**. But 1??{??: ? & ?)
is norm dense in /^(?) and so linv(.?) is norm dense in ?*(?^(?)). It follows that
?(?) is non-separable. ?
Stefansson A992) noticed that one can add an additional equivalent condition to
Theorem 6.8:
/ is determined by a separable space.
Combining Theorem 5.2 with Theorem 6.8 we get the following characterization of
Pettis integrability:
THEOREM 6.9 (Geitz, Musiat). Let f: ? ^ ? be a function. Then, f e ?(?, ?) and
?/(?) is a separable set if and only if there exists a sequence (/„) of X-valued simple
functions, such that:
(j) The family {x*f,: ? e N, x* e B(X*)} is uniformly integrable,
(jj) For each x* e X* we have lim„ x* f„ = x* f ?-a.e.
If f is scalarly bounded, then (/„) can be taken to be bounded (i.e., sup,, ||/,?(?)|| ^ ?
?-a.e.).
The first result of the above type was obtained by Geitz A981) who proved it in case of
a perfect measure ?. It was then generalized by Musial A985).
Plebanek A993) introduced the following notion: a weak Baire measure ? on X (i.e.,
a measure on the ?-algebra of Baire sets in the weak topology of a Banach space X) is
scalarly concentrated on a subspace ? of X if x*\Y = 0 implies x* = 0 ?-a.e., for all
functionals x* e X*. Then he proved the following characterization of Pettis integrable
functions with separable range of their integrals:
PROPOSITION 6.10. /// : ? —>· X is scalarly bounded and scalarly measurable then f e
?(?, X) and ? [(?) is separable if and only if the measure /(?) is scalarly concentrated
on a separable subspace of X.
In the particular case of spaces of continuous functions on compact spaces a few further
results are known. Rosenthal A970, Theorem 4.5) proved that if (and only if) a compact
space К satisfies the countable chain condition (CCC), then all weakly compact subsets of
C(K) are separable. Hence all C(?)-valued Pettis integrals have then norm separable
ranges. In particular it is so in case of К carrying a strictly positive Radon measure.
Another type of a sufficient condition was formulated by Plebanek A993).
PROPOSITION 6.11. If every sequentially continuous function f : К —> R is continuous,
then every Pettis integral of а С (К)-valued function has a separable range.

Pettis integral
557
EXAMPLE 6.12 (Fremlin and Talagrand A979)). Let / be an arbitrary nonvoid set and
let ? be the product measure on ? ? := {О, 1}' of the measure 1/25[0| + l/25[ ц. If ? is
the completion of the product ?-algebra, then let ? be the ?-algebra on ? ? defined in the
following way: ? e ? if and only if there exists a free non-measurable filter W on I and a
set F e ? such that EDW = FCiW.lt can be proved then that there is a unique extension
of ? to a complete measure Jl on ? (cf. Talagrand A984, Theorem 13-2-1)). Let ? be the
canonical injection of ? ? into loc(I)- Then, ? is scalarly /Z-integrable but it is not /Z-Pettis
integrable.
The detailed proof of these facts can be found in Fremlin and Talagrand A979) or in
Talagrand A984, Chapter 13).
EXAMPLE 6.13 (Fremlin and Talagrand A979)). Let ? be the measure described in
Example 6.12 and let к = Jl <§> ? be its complete product on ? ? ? ??. If / :?/ ? ?/ —>
/??(/) is defined by
f(x,y) = (p(x)-(p(y)
then / e P(/c,/oc(/)), but vt (? ® ?) is not relatively compact if / is infinite, and non-
separable if / is uncountable.
7. Universal integrability
Definition 7.1. Let ? ? 0 be a compact space. A function h:K -> R is said to
be universally measurable if it is measurable with respect to the completion of each
Radon measure defined on K. f.K —> X is ?-universally scalarly measurable if**/
is universally measurable for every x* e Г с Х*-
THEOREM 7.2 (Riddle, Saab and Uhl A983)). Let ? be a Radon measure on a compact
space К and let f : К —> X* be a scalarly bounded and scalarly universally measurable
function. If X is weakly compactly generated and f takes its values in a weak*-separable
subspace ofX*, then f e ?(?, ?*).
PROOF. Assume first the separability of X and let ? > 0 be arbitrary. Since X is separable,
there exists a compact set L с К such that ?(? \L) < ? and (/, x) is continuous on L for
each ? e X. Let
A=[(f,x)\L: II*K1}
and Mr (L) be the set of all real-valued universally measurable functions on L equipped
with the pointwise convergence topology. As / is universally measurable, the set A
is relatively compact in Mr(L). According to Theorem 2F of Bourgain, Fremlin and
Talagrand A978), every sequence in A has a pointwise convergent subsequence and so,
it is weakly precompact in C(L). A direct application of Rosenthal's theorem (Rosenthal
A974)) says that A contains no copy of the standard unit vector basis of /|. Since,

558
К. Musial
the canonical embedding of C(L) into ?,^?^.?) is a contraction, the set {(/,x)xl'-
||jr|| ^ 1} contains no copy of the /1-basis in the L^(^^)-norm either. Thus, it is weakly
precompact and Lemma 4.20 completes the proof. The non-separable case is a consequence
of separable complementability of X. ?
Assuming the continuum hypothesis, Plebanek A998) proved that the weak*
separability of the range cannot be omitted. He proved namely that if ? is the Haar
measure on 2W>, then L\ (?) is a non-separable WCG space and there is a bounded function
/:[0, 1] -> Loc(?) such that (x*, f) is Borel measurable for every x* e L~c(?)* and
/^?(?,^(?)).
Applying some consequences of Proposition 4.20 Bator A988a) and Stefansson A995)
have got the following decomposition properties of universally measurable functions:
PROPOSITION 7.3. Let К ^ 0 be a compact space and let f : К -> X* be bounded and
universally scalarly measurable. Then:
(a) (Bator) for each Radon measure ? on К there exist functions g and h such that g
has the Bourgain property for ?, h is weak* scalarly ?-null and f = g + h;
(b) (Stefansson) if X is a WCG space, then for each Radon measure ? on К there exist
functions /| and fi such that f\ is universally Pettis integrable, /i is weak* scalarly
equivalent to zero and f = f\ + fi-
As an immediate consequence of Proposition 7.3 and Corollary 4.17 we get the
following result:
COROLLARY 7.4. Let ? be a Radon measure on a compact space К and let f .K —>¦ X*
be bounded and universally scalarly measurable. Then for each lifting ? on Loc(?) the
function po(f) is Pettis ?-integrable.
The following result of E. Saag generalizes Corollary 4.19 of Haydon A976).
PROPOSITION 7.5. Let fi ? К С X* be a convex weak*-compact set equipped with the
weak* topology. If B(X) is weakly precompact with respect to K, then the identity function
on К is universally Pettis integrable and the integrals take their values in K.
PROOF. In view of Proposition 4.15 if ? is a Radon measure on K, then the family
? = B(X) has the Bourgain property. Hence Corollary 4.18 yields the Pettis integrability
of the identity function. The Hahn-Banach theorem applied to the weak* topology of X*
proves that the integral takes its values in К. ?
8. Pettis integral property
Dehnition 8.1. X has the ?-Pettis integral property if each X-valued, scalarly bounded
and scalarly ?-measurable function is ?-Pettis integrable. If such a property holds true
for all complete measures, then X is said to have the Pettis Integral Property (PIP). If ?

Pettis integral
559
is the Lebesgue measure on the unit interval, then we say about the Lebesgue PIP. If X
has the ?-??? with respect to all Radon measures, then it is said to have the universal PIP.
A measure space (?, ?, ?) has PIP if for every Banach space X each bounded scalarly
?-measurable X-valued function is Pettis integrable.
X has the universal PIP if for every compact К each bounded universally scalarly
measurable function /: К —>¦ X is universally Pettis integrable.
The first example of a Banach space without the Lebesgue PIP was given by Pettis
A938). He constructed an example of a bounded scalarly Borel measurable /: [0, 1] ->
/oc[0, 1] that is not ?-Pettis integrable. Then, Edgar A979), assuming (CH), published an
example of C[0, ? ? ]-valued function, which is also not ?-Pettis integrable.
No complete characterization of Banach spaces possessing PIP is known, but there are
several partial answers.
THEOREM 8.2 (Fremlin and Talagrand A979)). If Axiom К is true, then every Banach
space has the Lebesgue PIP. If either Axiom К or Axiom L is true, then l-^ has the ?-???
for every perfect ?.
As observed by Plebanek A993) the original proof of the first part of the above result
gives in fact more. To formulate it let us assume that ? is non-atomic and denote by
???(?/"?) the minimal cardinality of a set ? с ? which is not in ???. Then, let cov(Aflt) be
the minimal cardinality of a subfamily of ??? covering ?. Moreover, let ?? be the standard
product measure on 2K.
THEOREM 8.3. //поп(Л/1,) < co\(M\k )¦ then the measure ?? has PIP.
The main positive result is an immediate consequence of Theorem 4.10.
THEOREM 8.4 (Talagrand A984)). If X is Corson, then X has PIP.
As a direct consequence of Theorem 2.8 we get PIP of measure compact spaces (Edgar
A979)). Since each weakly countably determined space X (see Vasak A981)) (in particular
every K-analytic or WCG) is Lindelof in the weak topology of X (Vasak A981)), and
every Lindelof space is measure compact (see Varadarajan A961)), all such spaces have
PIP (Edgar A979)). If card ? = ? ? then, /| (?) is measure compact, hence it has PIP. On
the other hand, /|(K|) is not Corson (see Edgar A979), where several other important
observations concerning PIP can be found).
It turns out also that there is a strong connection between PIP and real-valued measurable
cardinals.
THEOREM 8.5 (Edgar A979)). l\ (?) has PIP if and only //card ? is not a real-valued
measurable cardinal.
THEOREM 8.6 (Andrews A985)). If the least real-measurable cardinal is not less than
the continuum, then 1\(?) has UPIPfor all ?.

560
К. Musial
Andrews A985) presented also a few other conditions guaranteeing UPIP of conjugate
Banach spaces with metrizable weak* compact and separable subsets.
Edgar A979) observed also that Mazur's property implies PIP, and since each X with
angelic (X*,weak*) is Mazur, also such spaces possesses PIP. As noticed by Plebanek
A993), if К is a first countable compact space, then C(K) is Mazur. Consequently, we get
the following result.
THEOREM 8.7 (Plebanek A993)). If К is a first countable compact space, then C{K)
has PIP and every C(K)-valued Pettis integral has a separable range.
In case of UPIP certainly the most important result has been obtained by Riddle, Saab
and Uhl and it is simply a reformulation of the main part of Theorem 7.2.
THEOREM 8.8 (Riddle, Saab and Uhl A983)). IfX is separable, then X* has UPIP.
No general characterization of UPIP even for conjugate spaces is known.
9. Weak Radon-Nikodym property and related properties
The following two results due, independently, to Dinculeanu A967) in case of Banach
space valued measures of finite variation and Rybakov A968) in case of ?-finite variation
are the starting points of all non-separable Radon-Nikodym type theorems.
THEOREM 9.1. Let ? : ? —>¦ X* be a weak* measure. If\v\ is a ?-finite measure, such
that ??(?) ^??(\?\), then there exists a weak* scalarly integrable function f : ? —> X*
such that
/(i2)Cconv*A,(i2) and (x, v(E)) = ? (?, ?)??
for each ? e X and each ? e ?.
If ? dominates ? and ? is a lifting on ??(?) then f can be chosen to satisfy also the
equality (x, f) = p({x, f)) for all ? e X.
PROOF. Assume first that и satisfies the inequality |u|(?) ^ Мд(?Кога11 ? e ?. Denote
for ? e X by fx the Radon-Nikodym derivative of the measure (x,v) with respect to ?.
Clearly, \fK\^M ?-a.e., and so |p(/v)| ^ ? everywhere.
Defining / : ? -+ X* by
{x,f(w)) = p(f,)(co)
for each ? e ? and ? e X we get the equality
J (x, f)dp={ ?(/?)?? = J ? ?? = {?, ?(?))
for each ? e ? and each ? e X.

Pettis integral
561
Consider now the family ? of all finite partitions ? of ? such that if ? = {? \,..., E„}
then p(Et) = Ei for all i <^n. We say that ? \ < ?? if each element of ? ? is a union of
elements of 7?2. Then, let
fc€7T
Applying Lemma VI.8.3 of Dunford and Schwartz A958), we see that xf(w) =
lim^/y xfn(co) for all ? e ? and all ? e X (cf. Kupka A972)).
The general case follows by decomposing of ? into pairwise disjoint sequence of sets
?„ e ?, such that \v\(E ? ?„) ?. ??(? ? ?„), for all ? e ? and ? e N. D
The property saying that for each X*-valued measure ? of ?-finite variation that is ?-
continuous there is a weak* measurable / : ? —>¦ X* such that
{x,v(E)) = ?{?,/)?? A)
for each ? e ? and ? e X, may be called the ?-weak* Radon-Nikodym Property
(?-W*RNP). Without loss of generality one may assume that и is ? dominated. So it is
a consequence of Theorem 9.1 that each conjugate Banach space has the W*RNP (please
notice that we use this name in a different meaning than it is used in Talagrand A984)).
/ will be called a weak* density (or a weak* Radon-Nikodym derivative) of ? with respect
to ?.
More generally, a set 0 ? К с X* has weak* RNP if for each (?, ?, ?) and each
?: ? -> X* satisfying v(E) e ?(?) ¦ К for every ? e ? there is /: ? -> К such that A)
is satisfied. Then weak* compact convex subsets of X* have W*RNP.
As a consequence of Theorem 9.1 we obtain the following result:
THEOREM 9.2. Let ? : ? —> X foe ? ?-continuous measure of ?-finite variation. Then,
there exists a weak* measurable f : ? —> X** such that
f^)^coiw**AvW) and {?*,?(?))= ? [?*, ???
for each x* e X* and ? e ?. If ? dominates ? and ? is a lifting on ?.^(?) then f can be
chosen to satisfy also the equality {x*. f) = p((x*, f)) for all x* e X*.
It is interesting and useful to know that the W**RNP and the WRNP are determined by
a single measure space.
THEOREM 9.3 (MusialA982)if К = B(X)). Let К =convK фЪ be a bounded subset
of X. If К has the X-W**RNP (respectively ?-WRNP), then it has also the W**RNP
(respectively WRNP).

562
К. Musial
PROOF. Let (?, ?, ?) be an arbitrary complete probability measure space and let и : ? —>
X be a nonatomic measure satisfying for every ? e ? the relation v(E) e ?(?) ¦ K.
(A) Assume first that ? is the completion of a countably generated ?-algebra ГСГ
with respect to ?\?.
Let (?„) с ? be a sequence generating ? and let ? : ? —>¦ [0, 1] be its Marczewski
function:
?(?) = 2]?3-"??-„(?).
It can be easily checked that ?-1 :? ? ?(?) —>¦ ?(?) is a Boolean ?-isomorphism of
??[0.i] П ?(?) onto Г. Let ?:? -> [0, 1] be the image of ? under ? and, let ? :[0, 1]^·
[0, 1] be the function defined by ?([0,9(t)]) = ?([0, t]). If ? = ? ? ?, then for each ? e ?,
we have ?[?~' (?)] = ?(?) and the measure algebras of ? and ? are isomorphic.
Let now ? : ? —>· X be given by
?(?) = ?[?'(?)].
We have ||?(?)|| ^?(?) for each В e ?. Hence, by the assumption, there is / ??(?,?**)
(respectively ?(?, ?)), such that ?(?) = Д- f ??.
It follows that for each ? e ?
-L
v(E)=P- ? /????.
Moreover, in view of Theorem 6.7, the set ?(?) is norm relatively compact.
(B) Assume now that ? is arbitrary and notice that ?(?) is a norm relatively compact
subset of X** (respectively X). Denote by ? the collection of all complete measure spaces
(?, ?, ?\?) with 4СГ being the completion of a countably generated ?-algebra ? ?
with respect to ?| ? ? and order ? upwards by inclusion.
In view of (A), for each ? there is fA e ?(?\?, ?**) (respectively ?(?\?, ?)), such
that
L
v(E)=P- J /???
for each ? e ?.
We shall prove that the net (fA) is Cauchy in the norm of ?(?, ?**) (respectively
P(m.X)).
To prove it, fix ? > 0 and take a simple function hF ·. ? —>¦ X , such that
sup
ЕеГ
v(E)- ? ????
<?
(cf. Musial A980)).

Pettis integral
563
Now fix ? e ? such that he is /i-measurable. Then, for each ? ^ ?
\?? ~he\ < 4 sup
???
It follows that for ? \, /}2 ^ /i the inequality |/j, - /j,| < 2? holds, and so the net is
Cauchy, as required.
But ? is countably directed, so there exists /}(> such that for each ? ^ Д> we have
|/л-/л„|=о.
It follows that each such fA is scalarly ?-equivalent to fAn and so for each ? e ?, we
get the equality
v(E)=P- j /??)?·?.
This completes the proof. ?
Definition 9.4. Given a directed set (Я, <), a family of ?-algebras ?? с Г, and
functions /? e ?((?, ??,?\??); X) with ? e ?, the system {/т,Гт;л· е Я) is a
martingale if ? ^ ? yields Гт С Г,, and ?(/?\??) = /?. The martingale is bounded
if there is ? > 0 such, that for each x* e X* and each ? e Я the inequality | (**, /T)| <
Af ||.x*|| holds ?-a.e. The martingale is convergent in ?(?, X) if there is / еР(д, X) such
that liin^ \f„ - /| = 0. The collection of all finite ^-partitions of ? into sets of positive
measure is denoted by Яг- We order it in the following way: ?\ < ?? if each element
of ? ? is, except for a null set, a union of element of ?2 -
The following theorem is a martingale characterization of the WRNP and the W**RNP.
THEOREM 9.5 (Musial A980) for К = S(X)). For a bounded set К = conv К ^0С X
the following conditions are equivalent:
(i) К has the WRNP (respectively W**RNP),
(ii) Given any (?, ?, ?) and any bounded martingale {/„, ?„; ? e ?) of К-valued
Pettis ?-integrable (simple) functions, then {/„,.?„; ? e N) is convergent in
?(?,?) (respectively ?(?, ?**)).
PROOF. Assume (i) is satisfied and take a bounded martingale {/„,.?„; ? e N) in
?(?,?). Assume, that ? > 0 is such that |(.v*,/„)| ^ M||.v*|| ?-a.e. (the exceptional
sets depend on**).
Let ?? = U^L ? ?» and 'et ? ·" Д) —>¦ -^ be given by
u(?) = lim / /,,??.
" Je
v(E)
-I.
hf ??
< ?.
for each ? e ??.

564
A". Musiat
We have ||?(?)|| < ? ?(?) and ? (?) e ?(?) ¦ ? for all ? e ?. The set function ?
extends uniquely to a measure v\ : ? = ?(?^) —>¦ X satisfying the similar conditions for
all ? e ?. Setting for each ? e ?
v(E) = J E{XE\S)dv]
(where the integral is in the sense of Bartle, Dunford and Schwartz A955)) we get an
extension of t>| to the whole ? satisfying for all ? e ? the relations
\\v(E)\\ ^ ??(?) and u(E) e ?(?) - ??.
Since К has the WRNP (or W**RNP), we get / e ?(?, ?) (respectively ?(?, ?**))
being the density of ? with respect to ?.
Since К has the W**RNP, it has the CRP (by Theorem 6.7). Thus, in a similar way, as
it has been done in the proof of Theorem 9.3, one can show that {(/„, ?„); ? e ?) is a
Cauchy martingale in ?(?, ?**).
Since (?\?„)(?) = fEf„dp for each /; e N, we have for each x* e X* (x*. f„) =
E((x*,f)\E„) and this gives
limi \E(x*f\E)-x*f„\dii = Q.
Together with the Cauchy condition, this yields
lim|/„-/| =0.
Assume now that (ii) is satisfied and take a measure ? : ? —> X satisfying for each ? e ?
the relations
|?(?)|^?(?) and u(E) e ?(?) ¦ AT.
Define for each ? e ? ? the function /T by
f - ? OIEI/F
Jn — /^ c XL·
and let E„ =?(?). {(/?, ??); ? e ? ?) is a bounded martingale in P( ?, ?)???\ ^ ?? <
• - then, by the assumption, {(/T„, 2TT„); // e N) is convergent in ?(?, ?) (respectively
?(?,?**)).
Let / : ? —> X** be a weak* density of ? with respect to ?:
foreach;c*eX*, ? eE.

Pettis integral
565
One can easily see that there exists in ? ? a sequence ?\ < ?? < ¦ ¦ ¦ such that
limsupl jfj(**, /- /*,)\??: ?* e B(X*)\ = 0
and so, in particular
]???[(?*,/?„)<?? = (?*,?(?))
" J ?
for each x* e X* and ? e ?.
Hence, if g = lim„ /^ e ?(?, ?) (respectively ?(?, ?**)), then for each ? e ?, we
get the required equality
u(E)= / #??. D
PROPOSITION 9.6 (C.Ryll-Nardzewski, see Musi at A979)). lx does not have the WRNP.
PROOF. Let ?„ be the dyadic partition of [0, 1 ] into 2" intervals and, let ?„ be the
collection of all possible unions of elements taken from ?„. If (A„) is an enumeration
of Ui^li ^n, then clearly lim„ ?(?„) = 0. Define a measure ?: С -+ со с /эс by setting
?(?) = (?(???„))?:.
Then, f(?) is a norm relatively compact subset of со, ||v(?)|| ^ ?(?) for each ? e ?and ?
is without Pettis ?-integrable derivative in ??. Indeed, let /: [0, 1] -> lx = I* be a weak*
density of и with respect to ?. It means in particular that if (en) is the standard basis in /?,
then
к(ЕПА„) = {е„,у(Е))= ? (?„,/)??
for each ? e N.
But the sequence (хд„) is pointwise dense in {0, 1}1°". Thus, if хд is a ???-?-
measurable cluster point of (хд„), then хд is ?-a.e. equal to a pointwise cluster point of
((e„, /)). Such a point is of the form (x*,f) for a functional x* e lx. This means that / is
not scalarly measurable and hence it cannot be a Pettis integrable density of ? with respect
to ?. ?
The next theorem has been first proved by Musiai A976) (see also A979)) for separably
complementable X. Then, in full generality the necessity has been proved by Musiai
and Ryll-Nardzewski A978) and the sufficiency by Janicka A979), Musiai A979) and
J. Bourgain.
THEOREM 9.7. X* has the weak Radon-Nikodym property if and only if X contains no
isomorphic copy ofl\.

566
A". Musiat
Assume that X contains no copy of l\. If ?: ? —>· X is a ?-dominated measure and ?
is a lifting ?? Loc (?), then by the weak* Radon-Nikodym property of X*, there exists a
weak* density /:?—>- X* of и with respect to ? such that
p((*,/)) = (¦*,/) for each .veX.
Since X contains no isomorphic copy of/|, it follows from Corollaries 4.17 and 4.18 that
[x**ME)) = f[x*\f)dvi
by the Lebesgue Convergence Theorem, and this proves the WRNP of X*.
Assume now that X contains a subspace that is isomorphic to l\ and let ? :l\ —>¦ X be an
isomorphic embedding. Then T* : X* —>· /^ is a surjection. If ? :C —>¦ lx is the measure
constructed in the proof of Proposition 9.6, then according to Musial and Ryll-Nardzewski
A978), there is a ?-dominated measure к : С -> X* such that ?*? = v. Since ? is not Pettis
differentiable, also к cannot have a Pettis integrable density.
At this place it is also worth to recall a characterization of WRNP of X* in terms of
functions. The result is a direct consequence of the W*RNP of conjugate spaces.
COROLLARY 9.8 (Musial A979)). X* has WRNP if and only if for every complete
(?, ?, ?) and each weak*-bounded and weak*-scalarly measurable f: ? —>¦ X* there
exists g e ?(?, ?*) and a weak*-scalarly null h-.?^-?* with f = g + h.
More or less at the same time Rybakov A977) presented another characterization of
Banach spaces not containing l\ (see also Musial A979)).
THEOREM 9.9. X does not contain any isomorphic copy ofl\ if and only if X* has CRP.
The next result is a generalization of Theorems 9.7 and 9.9. It was also an essential step
towards localizing of the weak RNR
THEOREM 9.10 (Riddle, Saab and Uhl A983)). For a given operator ?: X -> ? the
following conditions are equivalent:
(i) The set T(B(X)) is weakly precompact;
(ii) ? factors through a Banach space containing no isomorphic copy ofl\;
(iii) The set T*(B(Y*)) has WRNP;
(iv) The set T*(B(Y*)) has CRP;
(?) ? * factors through a Banach space with the WRNP.
The most general description of sets possessing WRNP will be given in Theorem 9.15.
In order to present an idea of its proof we need however yet a few additional facts.
LEMMA 9.11 (Rosenthal A974)). Let (x„) be a pointwise bounded sequence of real-valued
functions defined on a set S and having no pointwise convergent subsequence. Then there

Pettis integral
567
exists a subsequence (x„k) of (x„), a real number r and ? > 0 such that for every infinite
subset Mof{nk: k^ 1}, there is a point s e S with
Xm (s) > r + ? for infinitely many m e ?
and
x,„ (s) < r for infinitely many in e ?.
THEOREM 9.12 (Rosenthal A978)). Let Q be an uncountable Polish space and let
(A„, S„),i> ? be a sequence of pairs of closed and disjoint subsets of Z. Assume that the
sequence (A„, B„ )„ ^ ? has no convergent subsequence. Then, there exists a compact subset
L of Q, a homeomorphism h from L onto the Cantor set ? = {0, 1}'", and an increasing
sequence (nk) such that
A„KC\L = h~\vk) and B,„ ? L = /?"' (V[)
for all к (here Vk = {t = (t„) e ?: tk = 0}).
The next lemma is taken from Matsuda A985).
LEMMA 9.13. Let X be separable and let Q ? 0 be a weak*-compact convex subset
of X*. If B(X) is not Q-weakly precompact, then there is a Radon measure on (Q, weak*)
and a measure ? : ? —>¦ X* such that
(a) v(E) e ?.(?) ¦ Q for each ? e ?;
(b) ?(?) is not relatively compact in the norm topology.
PROOF (Sketch). Let (x„) be a sequence in B(X) without subsequence pointwise
convergent on Q. Without loss of generality we may assume that (x„), r and ? > 0 satisfy
the conclusion of Lemma 9.11. Setting A„ := (x* e Q: (x*,x„) ^ r +?] and B„ := [x* e
Q: (jc*, jc„) < r] we get a sequence (A„, S„)„^i that has no convergent subsequences. In
virtue of Theorem 9.12 there exists a compact subset L of Q, a homeomorphism h from L
onto the Cantor set ? = {0, 1 }?, and an increasing sequence (nk) such that
A„,nL = A"l(l/i) and ?„1??, = ??(??')
for all к (here Vk = [t = (t„) e ?: tk = 0}).
Let ? be the normalized Haar measure on ? and let ? be the Radon measure on L such
that h(rj) = ?. ? ? is the extension of ? to the whole Q.Then
?(?„?) = ?^](?,)) = ?(??<)=\/2 = ?(??4)
and
?(?„????4) = ?(?,??<) = 1/4

568
A". Munal
whenever ?'? j. Let ? be the weak*-integral of the identity function on Q with respect
to ?. ? is a vector measure on the ? -algebra of weak*-Borel subsets of Q.
The conclusion (a) of the lemma is an easy consequence of the separation theorem. To
see that (b) is also fulfilled notice that if /' < / then
|u(A„,)- u(AHi)| >(u(A„,)-u(A„,),.r„,}
> (r + ?)?(?„, П ВП]) - ??(?„? П ?,,,) = ?/4. ?
Definition 9.14 (Talagrand A984)). A weak* compact subset ? of X* is a Pettis set
if the identity function is universally scalarly measurable (with respect to Radon measures
on (K, weak*)).
The next theorem summarizes most of the results presented earlier in this paper and
concerning WRNR
THEOREM 9.15. Let К фЧ)Ьеа weak*-compact convex subset ofX*. Then the following
conditions are equivalent:
(i) B(X) is weakly precompact with respect to К;
(ii) the identity function is universally Pettis integrable on K;
(iii) К is a Pettis set;
(iv) К has WRNP;
(?) ? has ?-WRNP;
(vi) К has CRP,
(vii) К has ?-CRP
PROOF (Sketch), (i) =>· (ii) follows from Proposition 7.5. (iii) =>¦ (i) is a consequence of
Bourgain, Fremlin and Talagrand A978). (i) =>· (iv) follows from Corollaries 4.17 and 4.18.
(iv) =>· (vi) follows from Theorem 6.7 and (vii) =>· (i) from Lemma 9.13. ?
Applying the result of Talagrand A984) which says that if К с X* is a Pettis set then
its weak* closure has WRNP, one gets further generalizations of Theorem 9.15.
10. Conditional expectation
Let ? С ? be a ?-algebra. If / e ?(?, ?), then a function ?(/\?):? -> X is a
conditional expectation of / with respect to ? if E(f\E) e ?(?, ?, ?) and
J ?{/\?)?? = ?' ??? B)
for all ? e ?.
The first example of a Pettis integrable function without conditional expectation was
published by Rybakov A971). It was an /4-valued function. Heinich A973) published
then an example of /1-valued Pettis integrable function on [0, 1]-, which does not admit

Pettis integral
569
the conditional expectation with respect to a sub-? -algebra of the ? -algebra of Lebesgue
measurable sets. These examples can be classed in the following pattern: If / e ?(?, ?)
has infinite variation \Vf\ and ?-algebra ? С ? is such that |u|<a| is not ?-finite then
the existence of E(f\S) would contradict Theorem 3.8. Since for each non-atomic ?
and infinite dimensional X there is / e ?(?, X) with infinite \vf\, there are a lot of such
examples.
The following global result is an obvious consequence of the above considerations:
PROPOSITION 10.1 (Musial, A976) for K = B(X)). Let К С X be a closed and convex
set with WRNP. Iff e ?(?, ?) and ? is ? ?\?-complete sub-?-algebra ???, then f has
the conditional expectation with respect to ? if and only if the measure ?/ \? is of ?-finite
variation.
First examples of scalarly bounded Pettis integrable functions without conditional
expectations were presented by Talagrand A984, 6-4). One of them is / e F(k,Ix(I))
defined in Example 6.13 (see also Musiat A985)). Indeed, if ? is the completion of the
Borel algebra of ?/ ? ?/ with respect to ? <g> ?, then ? is к -dense in ? ®?. If there
existed a Pettis integrable E(f\S), then the equality B) would be true for all к -measurable
sets. This is however impossible in case of infinite /, since according to Stegall's result
(Theorem 6.1) the range of vFAj^) is norm relatively compact.
Talagrand A984) obtained the following interesting result covering the particular case
of the conditional expectations:
THEOREM 10.2. (Axiom L) Let (?,?,?) be a complete probability space and let
? :?,|(?) —>¦ L\(v) be a bounded operator. Then, let f: ? —>¦ X* be a weak*-scalarly
bounded function such that each countable subset of {xf: \\x\\ < 1} is stable. Then there
exists a properly measurable function g :(9 —>¦ X* such that T(xf) = xg v-a.e. for all
? e X. /// e ?(?, ?*), then g = Tf e P(u, X*).
Riddle and Saab A985) proved another sufficient condition guaranteeing the existence
of conditional expectations.
THEOREM 10.3. // / ??(?, X*) is scalarly bounded and the set {xf: \\x\\ < 1} is weakly
precompact in ?,^??), then f has conditional expectation with respect to all sub-?-
algebras of ?.
PROOF. Assume that / e ?(?, X*) satisfies the above assumptions, let ? be a sub-?-
algebra of ? and let ? bea lifting on L~x_ (?). Define g-.? -> X* by setting xg =
p(Es(xf)), for each ? e X. Since {.xf: \\x\\ ^ 1} is weakly precompact in /.^(?) and
the conditional expectation operator ?? :?^(?) —>¦ /^(??-?) is a contraction, the set
{xg: ||jr || $5 1} is weakly precompact in L^c (?\?). In view of Corollaries 4.17 and 4.18
we have g e ?(?\?, X*). Now, since the equality fExgd? = jE xf ?? for all ? e X and
the functions / and g are Pettis integrable, we get the same equality for all ? e X**. D

570
К. Musial
11. Differentiation
Already Pettis A938) noticed that if / e ?(?, ?) then for each x* e X* there exists a set
A(x*) e С of measure one, such that the equality
h~* ?1+\?\??))??=?*?(?)
lim
/i — 0
holds true for all t e A(x*). Pettis asked then whether in case of a strongly measurable
/ e ?(?, ?) the sets A(x*) could be replaced by a single set of full measure, i.e., whether
the Pettis integral of / is a.e. weakly differentiable.
The answer is negative in case X = h (Phillips A940)) and X = C[0, 1] (Munroe
A946a)). Thomas A976) conjectured that such a counterexample could be constructed
in every infinite dimensional X.
In A994) Kadets proved that for every infinite dimensional Banach space X there is
a strongly measurable and Pettis integrable function /:[0, 1] —>¦ X for which g(t) :=
/o f(s)ds is non-differentiable on a set of positive measure.
Then Dilworth and Girardi A995) generalized the above result proving that always there
exist functions that have nowhere weakly differentiable Pettis integrals. To formulate the
main result more precisely let ? be the collection of all increasing functions ? : [0, oo) —>
[0, oo) satisfying the growth condition
DC
]??B"/'"-,)?/2^<^· C)
n=\
for some increasing sequence (p„);^l() of integers and such that ?@) = 0.
THEOREM 11.1 (Dilworth and Girardi A995)). For each X and ? e ? there exists a
strongly measurable f e ?(?, ?) such that
II/fdk
WJi
?(?(?))
for every non-degenerated interval I с [0, 1 ].
If moreover
ОС
n=\
thenf?L\(k,X),
PROOF (Sketch). Let {/J': ? = 0, 1,2,...; к = 1,2, ...,2") be the dyadic intervals on
[0,1], i.e.,
4
'k-\ к
2" ' 2"

Pettis integral
571
Define inductively a collection {A'].: /7 = 0, 1,2 ; к = 1,2 2") of pairwise disjoint
sets of positive measure such that А" с /".
Fix К > 1. By a theorem of Mazur there is a basic sequence {.v„} in X with basis constant
at most К. Take a blocking {F„} of {x„} with each subspace F„ of large enough dimension
to find (using the finite-dimensional version of Dvoretzky's Theorem (Dvoretzky, 1961))
a 2"-dimensional subspace E„ of F„ such that the Banach-Mazur distance between E„
and /|" is less than 2. Let T„ :l;" —>· E„ be for each ? an operator such that ||7"„|| ^ 2
and ||?„"?| = 1. If {u'j.: к = 1,2 2") are the standard unit vectors in /;", then let
<!:=Г„<:.
Let (p„) be an increasing sequence of integers, with po = 0 satisfying
DC
??D- 2~p"~t)-j2^ < oo.
»=?
Define/:[0, !]->¦ X by
/(?) = 2*]?>D.2-''»-0]??<,,??") -',f.
»=l jt=l
One can easily check that for each ? e С
ос 21'"
f /^ = 2^^?D-2-/'-)??«")"' / XA["dkepk\
J E - ,? J ?
Since (E„) form a finite-dimensional decomposition there is a projection of ?„ ?„ onto
EPn of norm not exceeding 2K.
Now, if / is an arbitrary interval in [0, 1] one can find first /"' С / such that 4?(/"') ^
?(/) and then ? satisfying p„-\ ^m < pn.
It follows that
\l
fdk
> VD-2~/'""')
_A = I |J/
\P-
and so since A^" с lj!" С /'" С / we have
I/
/??
> VD-2-',"~').
But ? is increasing and 4 ¦ 2 /,n~' ^4-2 '" > ?(/) and so
II/
fdk\
?{?(?)).

572
A". Musiat
If ?~ ? tfrB-"«-'J"" = oo, then ??=? *<4 ¦ 2-"-')?*=? ?'?? = oo and so / ?
L ? (?, X) (according to Diestel and Uhl A977, p. 55)). D
As a corollary we get an answer to the original Pettis's question.
COROLLARY 11.2 (Dilworth and Girardi A995)). There exists a strongly measurable
f e ?(?, ?) that has nowhere weakly differentiable integral.
PROOF. Taking in Theorem 11.1 ?(?) = f^4 we get a function / satisfying for each
t e [0, 1 ] the equality
lim
?—?
/'
1 /"+"
- / ?(?)??
¦ oo.
If the Pettis integral of / were weakly differentiable at some t, then the above limit would
be finite. D
As noticed in Dilworth and Girardi A995)) it follows from Theorem 11.1 the existence
of a Pettis integrable function such that its integral is of infinite variation on every
subinterval of [0, 1]. This generalizes earlier result of Janicka and Kalton A977) where
the Banach space valued measure possessing that property could not be represented as a
Pettis integral (A good reference to the circle of problems concerning vector measures of
infinite variation on all sets of positive measure is the work of Drewnowski and Lipecki
A995)).
With some further effort one can get other results connecting the differentiability
problem with absolutely summing operators and cotype of X (see Dilworth and Girardi
A995)).
12. Fubini theorem
The classical Fubini theorem holds true not only for real functions but also for Bochner
integrable functions on an arbitrary measure space. Thomas A976) observed that for
every infinite dimensional Banach space X does exist a strongly measurable and ?--Pettis
integrable function /: [0, l]2 —>¦ X with the property that for every jte[0,l] the function
у —>¦ f(x,y) is not ?-Pettis integrable. Recently Michalak B000) attempted to resolve the
problem in case of arbitrary bounded Pettis integrable functions.
THEOREM 12.1 (Michalak B000)). Let (?, ?, ?) and (<9, ?,?) be complete probability
spaces and let X be a WCG space not containing any isomorphic copy of l\. If f e
?(?<§?, X*) is such that the set {\xf\: ? e B(X)} is order bounded in ?,|(?®?), then
there exists a function j:flx0->X* which is scalarly equivalent to f and
(i) the function g(-,0) ??(?, X*) for v-a.e. ? ев;
(ii) the function g(w, ¦) eP(u, X*)for?-a.e.? e ?;

Pettis integral
573
(iii) fA ?? %?? ® ? = fA (fB g(co, ?)??(?)) ??(?) = jB(jA g(co, ?) ??(?)) dv(9) for
all Ae ? and В еТ.
The above theorem cannot however be extended to all Banach spaces. If ??(?, ?) is
the space of all functions f :? —>¦ X such that ??€? ??/(?)||2 < oo, then one gets the
following result:
THEOREM 12.2 (Michalak B000)). Let (?, ?, ?) and (<9, ?, v) be complete probability
spaces and let X be a Banach space. Assume that ? vanishes on points and there
is a nonmeasurable subset of ?. Then for every bounded Pettis integrable function
f:? ? ? -> 12(?, X) there exists a bounded function g-.? ? ? -> ??(?, ?) which is
scalarly equivalent to f and
?({? e ?: g(a>, ¦) is not scalarly ?-measurable}) ^ 1/2.
13. Spaces of Pettis integrable functions
13.1. Space of all Pettis integrable functions
Pettis A938) noticed that ?(?, ?-?(?)) is non-complete. Rybakov A970) proved that
?(?, со) is non-complete. Then Thomas A976) proved that ?(?, ?) is non-complete in
case of an arbitrary not purely atomic ? and infinite dimensional X. Janicka and Kalton
A977) proved the same in case of the Lebesgue measure on [0, 1].
Modifying the example of Pettis, Drewnowski, Florencio and Paul A992) showed that
?(?, ?-2(?)) does not have property (K). Then, Drewnowski and Lipecki A995) proved
that ?(?, ?) is never ultrabarrelled if ? is non-atomic, and so it is neither Baire nor has
property (K). However, the following holds true:
THEOREM 13.1 (Drewnowski, Florencio and Paul A992)), ?(?, ?) is always barrelled.
PROOF (Sketch). Let us say that a locally convex space ? admits an (?, ?, ?)-???1?3?
algebra of projections (see Drewnowski et al. A992) for details) if there exists a set
{Рд: A e ?] of linear projections in ? such that:
A) ?? is the identity on Z, PAnB = Pa ¦ Рв for all ?,???, and РАив = Рл + Рв
for all disjoint А, В е ?;
B) Pa is continuous for every A e ?;
C) for every ? e Z, the vector measure Fx: ? —>¦ ? defined by F,(A) := Pa(x) is
?-continuous (that is Рд(.х) = 0 if ?(?) = 0).
Now we need the following
PROPOSITION. Let ? be a metrizable locally convex space admitting an (?,?,?)-
Boolean algebra of projections {Рл: A e ?]. Assume also that the projections satisfy
the following condition:

574
A". Musial
D) If(A„) is a sequence of pairwise disjoint elements of ? and (x„) is a null sequence
in ? such that Pa„(x„) =xn. ? = 1,2 then there exists a sequence (и*·) in N
such that the series ^ д-,ч is convergent.
IfW is a closed subspace of ? such that Pa(W) с W for all A e ? and (?, ?, ?) is
atomless, then W is barrelled.
Now we are ready to present the proof. Assume first that (?, ?, ?) is atomless. For
A & ? and / e ?(?, ?), define P,\(f) := хд · /. One can easily check that the conditions
(l)-C) are fulfilled. We have to check yet the condition D) of the just formulated
proposition. So let {A,,} be a pairwise disjoint sequence in ? and let (/„) be a null sequence
in ?(?, X). Without loss of generality, we may assume that ? := ?„ |/»| < °°· Let
/(?) := ]ГН /н (?) be defined for all we ?. Then, for every x* e X* we have
f \{?*,??))\??= ? ?\(?*,/,??))\????\\?*\\
J ? Jn „
and so / is scalarly integrable. Since for every A e ?, we have ]T„ || fA f, ??\\ < oo, the
series ]TH fA f, ?? is convergent to an element v(A) e X. One easily see now that
(?*,?(?))=?? [?*,?)??=?[?*.?)??.
„ Jada,, Ja
This completes the proof in case of an atomless ?.
If ? is atomic, then ?(?, ?) is a Banach space. In the general case one decomposes
the measure space into atomless and atomic parts obtaining ?(?, ?) as a direct sum of a
barrelled space and a Banach space, which is again barrelled. ?
The above result has been then generalized by Diaz et al. A995) to the following form:
THEOREM 13.2. ?(?, X) is always ultrabornological.
PROOF (Sketch). The proof is based on the following fact (see Diaz et al. A995) for
details).
PROPOSITION. Let [PA: A e ?] be an (?, ?,?)-??????? algebra of projections in a
metrizable locally convex space Z. Assume that (Рл: А е ?} is an equicontinuous family
of operators and satisfies the following condition:
E) // (?„) is a decreasing sequence in ? with ?(?\?„) = 0, (л„) is a bounded
sequence in ? such that ??„(.?„) = x„ for all n, and (a„) e /|, then the series
^„а,д„ is convergent in Z.
If Pa(Z) is ultrabornological for each ?-atom A, then ? is also ultrabornological.
The rest of the proof is similar to the previous one. Рд(/) := Хд · / with / e ?(?, ?)
and A e ?. This is an equicontinuous family of projections forming an (?. ?, ?)-???1?3?

Pettis integral
575
algebra of projections in ?(?, ?). One can easily check the validity of the condition E).
If A is an atom then ??(?(?, ?)) is isomorphic to X. Hence the assumptions of the
proposition are satisfied and so ?(?, ?) is ultrabornological. ?
There was also an attempt to introduce a complete metric on ?(?, X) (necessarily not
equivalent to the original one). Heilio A988) defined such a topology composing the
original semivariation norm with the convergence in measure. Setting for an arbitrary X-
valued function /
\/\?:=?{?: ?*{???: |/(?)| ? a } ? a}.
Heilio set then for each scalarly integrable /
111/111··= |/| +l/l„
and proved that ?(?, ?, ||| · |||) is complete. One of the consequences of this fact is another
proof of the incompleteness of ?(?, ?).
There is a wide class of problems concerning the possibility of embedding of a
space ? into ?(?, X) provided ? is an isomorphic subspace of X. Diestel proved in 1988
(unpublished) that if the range of ? is infinite and X contains an isomorphic copy of со,
then the completion of ?, (?, ?) contains a complemented copy of со. Emmanuele A992)
generalized it to ?(?, X). Diaz et al. A993) proved that if X contains an isomorphic copy
of со and ? is non-atomic and perfect, then ?(?, ?) contains a complemented copy of cq.
This has been then generalized by Freniche.
THEOREM 13.3 (Freniche A998)). If the range of ? is infinite, then the following are
equivalent:
(i) X contains a copy o/co;
(ii) ?(?, ?) contains a copy of cq;
(iii) ?(?, ?) contains a complemented copy of cq.
The next result concerns ?(?, X) as a subspace of the space ca(C, ?. ?) of all X-valued
?-continuous measures equipped with the semivariation norm.
THEOREM 13.4 (Drewnowski and Lipecki A995)). If X is separable then ?(?, ?) is an
T„a but not T„ subset ofca(C,k, X).
13.2. Functions satisfying the strong law of large numbers
Hoffmann-J0rgensen A985) and Talagrand A987) introduced the space ???(?, ?) of X-
valued functions satisfying the law of large numbers.
X: 3a, eX lim
1
af - -V/??,)
? L—'
i = l
:0 fo^^-a.e. (?;) &?~
where ?°° is the countable direct product ??????^- the countable product of ?.

576
A". Musiat
Talagrand A987) defined the Glivenko-Cantelli seminorm on LLN(?, ?) setting for an
arbitrary function / : ? —>¦ X
||/1|cc =Hm sup / ?,,??*,
where
??)= sup -]?|.?*(/(?,))|-
[|л*|[<1 « /<л
It has been proved by Beck A963) that Bochner integrable functions satisfy the strong
law of large numbers. If X is separable and a function /: ? —>¦ X satisfies the strong law
of large numbers, then / e L \ (?. ?), i.e., / is Bochner integrable (Hoffmann-J0rgensen
A985)).
Talagrand A987) described completely the class of functions satisfying the law of large
numbers. His proof is however too long to be presented here.
THEOREM 13.5 (Talagrand A987)). For a function f-.?^? andw = (?„) e ?? set
Sn(w) = (\/n) 53"_| /(?,). Then the following conditions are equivalent:
(a) / satisfies the strong law of large numbers;
(b) / is properly measurable and f? ||/|| ?? < oo;
(c) For almost allwe ?^, the sequence (\/n)S„(co) converges in norm;
(d) For each ? > 0 there is a simple function g : ? —>¦ X such that \\f — g|| gc ^ ?·
It easily follows from the above theorem that the Glivenko-Cantelli norm coincides on
???(?, ?) with the ordinary | -1 norm and each function satisfying the law of large
numbers is Pettis integrable. Functions in ???(?, ?) that are scalarly equivalent are not
distinguishable by the GC-norm. This permits us to identify scalarly equivalent elements
of LШ(?, X) and investigate the quotient space (denoted also by LLN (?. ?)).
In the context of LLN(?, X) Talagrand A987) successfully applied the concept of stable
sets to description of the so called Glivenko-Cantelli classes of functions.
Dobric A990) posed a question about completeness of LLN(?, X). It turned out that the
space is almost never complete and in general it is even not barrelled.
THEOREM 1 3.6 (Musial B001a)). Assume that ? is not purely atomic. Then the space
LLN (?, X) is non-complete. If moreover LLN (?. ?, var) is complete, then LLN(^, X) is
even not barrelled.
PROOF. ??·(?, ?) endowed with the Pettis norm is non-complete (see Thomas A976)).
Let (/„) be a Cauchy sequence in ?((?, X) that is not convergent in ?,(?, ?). It
follows from Musial A979, Proposition 3) that for each ? e N there exists a simple
function /2„·.? —>· X with |/„ - h„\ < l/n. Since simple functions are properly
measurable and LLN(^, X) с P( (?, X), the sequence (hn) is also Cauchy in LLN(?, X).

Pettis integral
577
Clearly, the sequence (h„) is divergent in ???(?, ?). This completes the proof of the
noncompleteness.
If ???(?, ?) were barrelled then applying the closed graph theorem to the identity map
from ???(?, ?) to LLN(^, X, var) one would get its continuity. Hence the two norms
would be equivalent. This is however impossible if X is infinite dimensional and ? is not
purely atomic. ?
If each X-valued Pettis integrable function is weakly equivalent to a strongly measurable
function (it is in case of X possessing RNP or in case of a measure compact X), then
???^?, X, var) = L ? (?, ?) and so ???(?, ?) is not barrelled.
13.3. Functions with integrals of bounded variation
We denote by PV(m, X) the set of those / e ?(?, ?) for which \vf\ is finite. ??(?, ?)
can be considered as a subspace of саЫ(ц., X), endowed with the variation norm. We
define a ||-1| у norm on ??(?, ?) by setting
||/||?:=|?>/|(?).
Clearly |/| <: \\f\\v for each / e ?(?, X).
Exactly as in the proof of Theorem 13.1 one can obtain
THEOREM 13.7 (Musial B001a)). PV(M, X) is barrelled.
PROPOSITION 13.8 (Musial B001a)). // X has the weak Radon-Nikodym property, then
PV(M, X) is a Banach space. In particular if ? is a Banach space not containing any
isomorphic copy ofl\, then ??(?, ?*) is complete.
It is not known whether the space ??(?, ?*) is always complete. Assuming the validity
of Axiom L, one can prove however the completeness of ? V (?. ?*) for an arbitrary perfect
measure ?.
THEOREM 13.9 (Musial B001a)). (Axiom L) If ? is perfect then FV (?, ?*) is complete.
Proof (Sketch). Let (/„) be a Cauchy sequence in ??(?, ?*)) and let ? be the limit of
(vfn) in cabv(\i, X*). The classical Radon-Nikodym theorem guarantees the existence for
each ? e N of a function h„ e L\ (?) such that
\\>-\>fn\{E) = f hndn. D)
for each ? e ?. Moreover, since X* has the W*RNP, there exists a weak* measurable
function /: ? —> X* such that for each .x e X and each ? e ?
xv(?) = / xfd\x.

578
К. Miisial
The theorem will be proved if we show that there exists a Pettis integrable function / that
is weak*-equivalent to /. To do it let us notice first that for each ? from the closed unit
ball of X and for each ? e N the equality D) yields the relation
\(x,f-fn)\^h„ ?-a.e. E)
It follows that for each /; e N and each x** e BC(X**) we have
\(x**,f-f„)\^h„ ?-a.e.
Since the sequence (h„) is convergent to zero in the norm of ?-?(?), we obtain the
measurability of all functions x**f, with x** e BC(X**). If / is weak*-scalarly bounded,
then the Pettis integrability of po(f) follows from Theorem 4.28 and we may put / =
/}()(/)- If / is arbitrary, we apply the decomposition Corollary 2.11. This completes the
proof. ?
The next result shows that sometimes the assumptions concerning Axiom L and the
measure space are superfluous. We omit a direct and easy proof.
PROPOSITION 13.10 (Musial B001a)). Lei X be a separable Banach space and let ? be
a closed linear subspace ofX*. Then ??(?, ?) is complete for an arbitrary ?.
It is not known whether the space ??(?, ?*) is always complete. There is however an
example due to D. Fremlin of a non-conjugate X, such that PV(A, X) is non-complete.
Since ? is perfect, it follows that in general the structure of the Banach space X is more
important, than the properties of ?.
EXAMPLE 13.11 (Fremlin, see Musial B001a)). For each t e ]0, 1] let e, e /,J0, 1] be the
unit vector at t: e,(s) = 0 if ? ? t and e,(s) — 1 if s = t. Consider ??).,] as an element of
Loc[0, 1] and set for each t e]0, 1]
x(t) = Xio.i\ + ei/r.
Then, let X be the closed linear subspace of L-J0, 1] ? /^[0, 1] generated by all x(t),
te]0, 1] and all (??,?) with ?? e C[0, 1].
THEOREM 13.12 (Musial B001a)). The space FV(X,X) does not have the (K) property.
13.4. LLN^, X) equipped with the variation norm of integrals
It has been proved in Talagrand A987) that Pettis integrals of functions from LLN(^, X)
are measures of finite variation. Thus, it makes sense to equip the space ???/(?, ?) with
the variation norm of the integrals. It will be denoted by ???(?, X,var). Contrary to
??(?, ?*), the space LLN(^, X* ,var) behaves quite well.

Pettis integral
579
THEOREM 13.13 (Musial B001a)). ??(?, X*,var) is a Banach space.
PROOF {Sketch). Let ? be a consistent lifting on (?, ?, ?) and, let (/,,) be a Cauchy
sequence in LLN{p, X*, var). Moreover, let ? be the limit of (t>/„) in cabv(^, X*) and, let
f-.?^ X* be a weak*-density of ? with respect to ?. We shall split the proof into two
parts. Assume first that / is weak*-scalarly bounded. Without loss of generality, we may
assume then that / = po(f). Moreover, let for each ? a function ?„ e L \ (?) be the RN-
density of the measure \vtn\ and let ??„, := ?({? e ?: ? „(?) ^ m}). Since ?„ e L\(p),
there exists for each ? e N a number m„ such that ?(?'? ,„ ) < \/n.
Since f„xn„„, is scalarly bounded and properly measurable, it follows from the
consistency of ? that g„ := A)(/hX#„,„„) is bounded and properly measurable, for each
? e N. Hence, g„ e ???(?, X*, var) and since f - gn= po(f - g„), we see that for each
? e N and some ? > 0
II/-««lice <|?-??|(?) + ???(??_ )-
That is / is approximated in the Glivenko-Cantelli norm by elements of ???/(?, ?*). In
virtue of Theorem 13.5, / satisfies the law of large numbers.
The general case follows by an appropriate decomposition of ?. ?
13.5. Bounded Pettis integrable functions
We denote by ??(?, X) the linear space
(/??(?,?): ||/||^:= sup ||.v*/lk<^|.
1 iiv-iki '
where ||.х*/11зс is the /^(?^???? of x*f. One can easily check that || · \\p^ is a norm.
Then, let ?^(?, ?) := {/ e ?^(?. ?): vt(E) is norm relatively compact).
Identifying weakly equivalent functions - we denote by LZJVoc(M. X) the linear space
\feLLNte,X):\\f\\P^:= sup \\x*f\\x < oo|.
1 ??.???? '
PROPOSITION 13.14. IfX has the WRNP then Лс (?, ?) is complete.
THEOREM 13.15 (Musiat B001b)). (Axiom L) If ? is perfect, then ?^(?,?*) is
complete.
PROOF. Let (f„)%\ be a Cauchy sequence in ?^(?, ?*). Then for each m,n eN
SUp \\x**f„ - X**f„, | = SUp ||-v/„ - Xf„, \\oc
ll.v"Kl ' 11-vlKl

580
К. Musiat
and
sup ||*/„ - xf,„\\x = sup! ??/,,)(?) - ?)(/»;)(?)||, F)
where ? is a lifting on Lx(^). Consequently, the sequence (po(f„)) is uniformly
convergent to a function h-.? ^> X* such that /г = aj(O- Since for each x** e SC(X**)
the functions A) (/«) are measurable, according to Talagrand A984, Theorem 6-2-1)(where
the Axiom L is used), the functions A)(/i) are m ??(?·%*) and so ^ is scalarly
measurable. Then, it is a consequence of Theorem 5.2 that h e ??(?, ?*).
Thus, using F), with h rather then /,„, we get
lim||/„ - h\\P =lim sup ||jv/„ — дсй||эс
" ?????
= limsup|A)(/i)(w)-?)(?)(?)| =0.
This proves the completeness of ??(?, ?*). ?
The above proof makes it obvious that in fact the following more general result holds
true:
THEOREM 13,16 (Musiat B001b)). Let ? and X be arbitrary: If for each countable family
? С ??(?, X*) there exists a lifting ? such that Art/) is ?-Pettis-integrable for each
f e T, then ??(?, ?*) and ??(?, ?*) are complete.
COROLLARY 13.17. // X is separable, then for each ? the spaces ?^(?,?*) and
??(?, ?*) are complete.
Analysis of the proof of Theorem 13.15 when ? is consistent shows the validity of the
next result.
THEOREM 13.18 (Musiat B001b)). The space ????(?, ?*) is complete.
Considering each X-valued function as an X**-valued function we get the following
result in case of an arbitrary Banach space X:
THEOREM 13.19 (Musiat B001b)). The completion of the space ???(?, ?) is a
subspace ofLLN'?(?, ?**). If Axiom L is satisfied and ? is perfect then the completion of
??(?-, ?) is a subspace of ??(?, ?**).
References
Andrews, K.T. A985), Universal Pettis integrability, Canad. J. Math. 37, 141-159.
Battle, R.G., Dunford, N. and Schwartz, J. A955), Weak compactness and vector measures, Canad. J. Math. 7.
289-305.
Bator, E.M. A985), Pettis integrability and the equality of the norms of the weak*-integral and the Dunford
integral, Proc. Amer. Math. Soc. 95, 265-270.

Pettis integral
581
Bator, E.M. A988a), A decomposition of bounded scalarh- measurable functions taking their values in dual
Banach spaces, Proc. Amer. Math. Soc. 102, 850-854.
Bator, E.M. A988b), Pettis decomposition for universally scalarh measurable functions, Proc. Amer. Math. Soc.
104, 795-800.
Bator, E.M. and Lewis, P. A987), Some connections between Pettis integration and operator theory. Rocky
Mountain J. Math. 17, 683-695.
Bator, E.M. and Lewis, P. A989a), Weak precompactness and the weak RNP, Bull. Polish Acad. Sci. 37, 7-12.
Bator, E.M. and Lewis, P. A989b), Remarks on completely continuous operators. Bull. Polish Acad. Sci. 37,
409^413.
Beck, A. A963), On the strong law of large numbers, Ergodic Theory, Proc. Intemat. Sympos., New Orleans,
1961, Academic Press, New York.
Bessaga, С and Petczynski, A. A958), On bases and unconditional convergence of series in Banach spaces,
Studia Math. 17, 151-174.
Bilyeu, R.G. A978), A metric inequality characterizing bancentres and other Pettis integrals, Proc. Amer. Math.
Soc. 68, 323-326.
Bilyeu, R.G. and Lewis, P.W. A985), Differentiability of convex functions, Pettis integrability, and Rybakov's
theorem, Bull. Polish Acad. Sci. Math. 33, 581-585.
Birkhoff, G. A935), Integration of functions with values in a Banach space, Trans. Amer. Math. Soc. 38, 357-378.
Bongiomo, В., Di Piazza, L. and Musiat, K. B000). An alternate approach to the McShane integral. Real Anal.
Exchange 25, 829-848.
Bourgain, J., Fremlin, D.H. and Talagrand, M. A978), Pointwise compact sets of Baire-measurable functions,
Amer. J. Math. 100, 845-886.
Brooks, J.K. A969a), Representations of weak and strong integrals in Banach spaces, Proc. Nat. Acad. Sci. USA
2, 266-270.
Brooks, J.K. A969b), On the Gelfand-Pettis integral and unconditionally convergent series. Bull. Acad. Polon.
Sci. Ser. Sci. Math. Astronom. Phys. 17, 809-813.
Brooks, J.K. A972). Sur la representation de I'espace dual de I'espace des functions integrables an sens de
Pettis, C. R. Acad. Sci. Paris Ser A-B, A1099-A1101.
Brooks, J.K. and Dinculeanu, N. A980), On weak and strong compactness in the space of Pettis integrable
functions. Proc. Conf. on Integration, Topology and Geometry in Linear Spaces (Univ. North Carolina. Chapel
Hill, NC, 1979). Contemp. Math., Vol. 2. 161-187.
Brooks. J.K. and Dinculeanu. N. A982). On weak compactness in the space of Pettis integrable functions. Adv.
in Math. 45, 255-258.
Chakraborty. N.D. and Jaker, A.Sk. A993), On strongly Pettis integrable functions in locally convex spaces. Rev.
Mat. Univ. Complut. Madrid 6. 241-262.
Chatterji, S.D. A974), Sur Vintegrabilite de Pettis, Math. Z. 136, 53-58.
Chictescu, I. A990), A parametrical example of Dunford, Pettis and Bochner integration. Stud. Cere. Mat. 42,
405^41 8.
Collins, H.S. and Ruess, W. A983), Weak conipactness in spaces of compact operators and of vector-valued
functions. Pacific J. Math. 106. 45-71.
Corson, H.H. A961), The weak topology of a Banach space. Trans. Amer. Math. Soc. 101. 1-15.
Diaz, S., Fernandez. ?.. Florencio, M. and Paul. P.J. A993). Complemented copies of cq in the space of Pettis
integrable functions. Quaestiones Math. 16. 61-66.
Diaz, S.. Fernandez. ?., Florencio. M. and Paul. P.J. A995). A wide class of uhrabomological spaces of
measurable functions, J. Math. Anal. Appl. 190. 697-713.
Diestel, J. A973), Applications of weak compactness and bases to vector measures and vectorial integration. Rev.
Roum. Math. Pures Appl. 28, 211-224.
Diestel, J. A975), Geometry of Banach Spaces - Selected Topics. Lecture Notes in Math.. Vol. 485, Springer,
Berlin.
Diestel, J. A984), Sequences and Series in Banach Spaces. Graduate Texts in Math., Vol. 92. Springer, Berlin.
Diestel, J. and Faires. B. A974). On vector measures. Trans. Amer. Math. Soc. 198. 253-271.
Diestel, J. and Uhl, J.J. A977). Vector Measures. Math. Surveys. Vol. 15, Amer. Math. Soc., Providence. RI.
Diestel, J. and Uhl, J.J. A983), Progress in vector measures 1977-83. Proc. Measure Theory and Its Appl.
(Sherbrooke, 1982). Lecture Notes in Math.. Vol. 1033. Springer. Berlin, 144-192.

582
A". Musial
Dilworth, S.J. and Girardi. M. A993). Bochner vs. Pettis nann: Examples and results. Banach Spaces (Merida.
1992), Contemp. Math.. Vol. 144. Amer. Math. Soc., Providence. RI. 69-80.
Dilworth, S.J. and Girardi, M. A995). Nowhere weak differentiability of the Pettis integral, Quaestiones Math.
18, 365-380.
Dimitrov. D.B. A971). One remark concerning Gelfand integral. Functional Anal. Appl. 5. 84-85 (in Russian).
Dinculeanu, N. A967), Vector Measures, Pergamon Press.
Dinculeanu, N. A981). Uniform a-additivirs- and uniform comergence of conditional expectations in spaces of
Bochner or Pettis integrable functions. General Topology and Modem Analysis (Proc. Conf. Univ. California.
Riverside. 1980), Academic Press, New York. 391-397.
Dinculeanu, N. A983). Uniform с-additivitx in spaces of Bochner or Pettis integrable functions over locally
compact group. Proc. Amer. Math. Soc. 87. 627-633.
Dinculeanu, N. and Uhl, J.J. A973), A unifying Radon-Nikodym theorem for vector measures. J. Multivariate
Anal. 3, 184-203.
Dobrakov, I. A970a). On integration in Banach spaces I. Czechoslovak Math. J. 20. 51 1-536.
Dobrakov, I. A970b), On integration in Banach spaces II. Czechoslovak Math. J. 20. 680-695.
Dobric, V. A990), The decomposition theorem for functions satisfying the law of large numbers. J. Theoretical
Probab. 3, 489^496.
Drewnowski, L. A986), On the Dunford and Pettis integral. Probability and Banach Spaces (Zaragoza. 1985).
Lecture Notes in Math.. Vol. 1221. Springer. Berlin. 1-15.
Drewnowski, L., Florencio. M. and Paul. P.J. A992). The space of Pettis integrable functions is barreled. Proc.
Amer. Math. Soc. 114. 687-694.
Drewnowski, L. and Lipecki. Z. A995). On vector measures which have everywhere infinite variation or non-
compact range. Dissertationes Math. 339. 3-39.
van Dulst, D. A980), Characterization of Banach Spaces not Containing /'. CWI Tract. Vol. 59.
Dunford. N. A937), Integration of vector-valued functions. Bull. Amer. Math. Soc. 43. 24.
Dunford. N. and Schwartz. J. A958). Linear Operators I. Pure and Appl. Math.. Interscience. New York.
Dvoretzky, A. A961). Some results on convex bodies and Banach spaces. Proc. Intemat. Sympos. on Linear
Spaces, Jerusalem A961). 123-160.
Edgar, G. A977). Measurability in a Banach space I. Indiana Math. J. 26. 663-667.
Edgar, G. A978), On the Radon-Nikodym property and martingale convergence, vector space measures and
applications II, Proc. Dublin A977). Lecture Notes in Math.. Vol. 645. Springer. Berlin. 62-76.
Edgar, G. A979). Measurability in a Banach space II. Indiana Math. J. 28. 559-579.
Edgar, G. A982), On pointwise-compact sets of measurable functions. Measure Theory (Oberwolfach. 1981).
Lecture Notes in Math., Vol. 945. Springer. Berlin. 24-28.
Egghe, L. A982-83), Extensions of the martingale comergence theory in Banach spaces.
Egghe, L. A984), Convergence of adapted sequences of Pettis integrable functions. Pacific J. Math. 114. 345-366.
Emmanuele, G. A992), On complemented copies ofc^ in spaces of operators, Comm. Math. 32, 29-32.
Emmanuele, G. A993), Precompactness in the space of Pettis integrable functions. Acta. Math. Hungar. 62,
333-335.
Emmanuele, G. and Musial, K. A990), Weak precompactness in the space of Pettis integrable functions, J. Math.
Anal. Appl. 148. 245-250.
Erben, W. and Grimeisen, G. A990), Integration by means of Riemanii sums in Banach spaces I, Z. Anal.
Anwend. 9. 481-501.
Erben, W. and Grimeisen. G. A991), Integration by means of Riemann sums in Banach spaces II, Z. Anal.
Anwend. 10, 11 -26.
Farmaki, V A995), A geometric characterization of the weak Radon-Nikodym property in dual Banach spaces.
Rocky Mountain J. Math. 25, 611-617.
Fink, J.P. A979), The Pettis integration of a perturbed wave equation. Math. Proc. Cambridge Philos. Soc. 86,
145-159.
Fremlin, D.H. A975), Pointwise compact sets of measurable functions, Manuscripta Math. 15, 219-242.
Fremlin, D.H. A982), Properly relatively compact sets of measurable functions. Note of 1 July 1982.
Fremlin, D.H. A994), The Henstock and McShane integrals of vector-valued functions, Illinois J. Math. 38,
471^479.
Fremlin, D.H. A995). The generalized McShane integral, Illinois J. Math. 39. 39-67.

Pettis integral
583
Fremlin, D.H. and Mendoza, J. A994), On the integration of vector-valued functions. Illinois J. Math. 38. 127—
147.
Fremlin, D.H. and Shelah, S. A993). Pointwise compact and stable sets of measurable functions. J. Symbolic
Logic 58, 435^455.
Fremlin, D. and Talagrand, M. A979). A decomposition theorem for additive set functions and applications to
Pettis integral and ergodic means. Math. Z. 168. 117-142.
Freniche, F.J. A998), Embedding cq in the space of Pettis integrable functions. Quaestiones Math. 21. 261-267.
Geitz, R. A981), Pettis integration, Proc. Amer. Math. Soc. 82. 81-86.
Geitz, R. A982), Geometry and the Pettis integral. Trans. Amer. Math. Soc. 269, 535-548.
Gelfand, I.M. A936), Sur un lemme de le theorie des espaces lineaires. Comm. Inst. Math. Mec. Univ. de
Kharkoff et Soc. Math. Kharkoff 13. 35^40.
Ghoussoub, N. and Saab, E. A981). On the weak Radon-Nikodvm property. Proc. Amer. Math. Soc. 81, 81-84.
Gordon, R. A990). The McShane integral of Banach-valued functions. Illinois J. Math. 34. 557-567.
Gordon, R. A991), Riemann integration in Banach spaces. Rocky Mountain J. Math. 21, 923-949.
Graves, W.H. and Ruess. W. A984), Compactness and weak compactness in spaces of compact-range vector
measures. Canad. J. Math. 36. 1000-1020.
Grothendieck, A. A952), Criteres de compacite' ge'neraux dans les espaces fonctionnels. Amer. J. Math., 168-
186.
Gulisashvili, A. A983), Estimates for the Pettis integral in interpolation spaces with some applications. Banach
Space Theory and Its Applications (Bucharest. 1981), Lecture Notes in Math., Vol. 991, Springer. Berlin,
55-78.
Hashimoto, K. A983), On the sequential approximation of scalarly measurable functions by simple functions,
Tokyo J. Math. 6, 153-166.
Hashimoto, K. and Oharu, S. A983), Gelfand integrals and generalized derivatives of vector measures. Hiroshima
Math. J. 13, 301-326.
Haydon, R. A976), Some more characterizations of Banach spaces containing l\. Math. Proc. Cambridge Philos.
Soc. 80, 269-276.
Heilio, M. A988), Weakly summable measures in Banach spaces, Ann. Acad. Sci. Fennicae. Ser. A, I. Math.,
Dissertationes, Vol. 66.
Heinich, M.H. A973). Esperance conditionelle pour les functions vectorielles. C. R. Acad. Sci. Paris Ser. A 276.
935-938.
Helmer, D. A994). On the Pettis measurability theorem. Bull. Austral. Math. Soc. 50. 109-116.
Hoffmann-j0rgensen, J. A971), Vector measures. Math. Scand. 28. 5-32.
Hoffmann-j0rgensen, J. A985). The law of laige numbers for non-measurable and non-separable random
elements. Coll. en l'Honneur Laurent Schwartz. Vol. 1. Asterisque 131. 299-356.
Holmes, R.B. A975). Geometric Functional Analysis anil Its Applications. Graduate Texts in Math., Vol. 24.
Springer. Berlin.
Huff, R. A986). Remarks on Pettis integrability. Proc. Amer. Math. Soc. 96. 402^404.
Ionescu Tulcea, A. and Ionescu Tulcea, С A969), Topics in the Theory of Lifting, Ergeb. Math. Grenzgeb. B. 48.
Springer. Berlin.
Jaker. A.Sk. and Chakraborty. N.D. A997), Pettis integration in locally coiwex spaces. Anal. Math. 23. 241-257.
James, R.C. A964). Weak compactness and refiexivitv, Israel J. Math. 2, 101-119.
Janicka. L. A979), Some measure theoretical characterizations of Banach spaces not containing l\. Bull. Acad.
Polon. Sci. 27. 561-565.
Janicka, L. and Kalton. N.J. A977), Vector measures of infinite variation. Bull. Acad. Polon. Sci. 25, 239-241.
Kadets. V.M. A994), Non-differentiable indefinite Pettis integral, Quaestiones Math. 17, 137-139.
Kluvanek. I. and Knowles, G. A975). Vector Measures and Control Systems. North-Holland/American Elsevier.
Komlos, J. A967). A generalization of a problem ofSteinhaus, Acta Math. Acad. Sci. Hungar. 18. 217-229.
de Korvin, A. and Roberts. Ch.E. A979). Interchange of vector-valued integrals when the measures are Bochner
or Pettis indefinite integrals. Bull. Austral. Math. Soc. 20, 199-206.
Kupka. J. A972). Radon-Nikodym theorems for vector valued measures. Trans. Amer. Math. Soc. 169. 197-217.
Luu. D.Q. A997), Convergence of Banach-space-valued martingale-like sequences of Pettis-integrable functions.
Bull. Polish Acad. Sci. Math. 45. 233-245.
Macheras. N.D. and Musial. K. B000). Liftings of Pettis integrable functions. Hiroschima J. Math. 30, 215-219.

584
К. Musial
Matsuda, M. A983), On Sierpinski's function and its applications to vector measures. Math. Japon. 28, 549-560.
Matsuda, M. A985), On Saab's characterizations of weak Radon-Nikodym sets, Publ. Res. Inst. Math. Sci.,
Kyoto Univ. 21, 921-941.
Matsuda, M. A986), A characterization of weak Radon-Nikodym sets in dual Banach spaces, Publ. Res. Inst.
Math. Sci., Kyoto Univ. 22, 551-559.
Matsuda, M. A991), A characterization of Pettis sets in dual Banach spaces. Publ. Res. Inst. Math. Sci., Kyoto
Univ. 27, 827-836.
Matsuda, M. A993), A characterization of'поп-Pettis sets in terms of martingales. Math. Japon. 1. 177-183.
Matsuda, M. A994), Remarks on Pettis sets. Rep. Fac. Sci. Shizuoka Univ. 28. 17-23.
Matsuda, M. A995), A characterization of Pettis sets in terms of the Bourgain property; Math. Japon. 41, 433-
439.
Matsuda. M. A996), On localized weak precompactness in Banach spaces. Publ. Res. Inst. Math. Sci., Kyoto
Univ. 32, 473^491.
Matsuda. M. A998), On localized weak precompactness in Banach spaces II. Hiroshima Mat. J. 28. 399-418.
Matsuda, M. A999a), ? generalization of the Radon-Nikodvm property in dual Banach spaces, fragmentedness,
and differentiability of convex functions. Publ. Res. Inst. Math. Sci.. Kyoto Univ. 35. 921-933.
Matsuda. M. A999b). A remark on localized weak precompactness in Banach spaces. Comment. Math. Univ.
Carolin. 40. 271-276.
Michalak, A. A996). Translations of functions in vector Hardy classes on the unit disk. Dissertationes Math..
Vol. 359.
Michalak, A. B000). On Pettis integrabilin of translations of functions in L^. Bull. Polish Acad. Sci. Math. 48.
113-119.
Michalak, A. B001), On the Fubini theorem for the Pettis integral for bounded functions. Bull. Polish Acad. Sci.
Math. 49, 1-14.
Moedomo, S. and Uhl, J.J. A971), Radon-Nikodvm theorems for the Bochner and Pettis integrals. Pacific J. Math.
38,531-536.
Munroe, M.E. A946a), A note on weak differentiability of Pettis integrals. Bull. Amer. Math. Soc. 52, 167-174.
Munroe, M.E. A946b), A second note on weak differentiability of Pettis integrals. Bull. Amer. Math. Soc. 52,
668-670.
Musial, K. A976), The weak Radon-Nikodvm propertv in Banach spaces, Aarhus University. Preprint Series
1976/77, No. 7.
Musial, K. A979), The weak Radon-Nikodym property in Banach spaces, Studia Math. 64, 151-174.
Musial, K. A980), Martingales of Pettis integrable functions. Measure Theory (Oberwolfach, 1979), Lecture
Notes in Math., Vol. 794, Springer, Berlin. 324-339.
Musial, K. A982), A characterization of the weak Radon-Nikodvm property in terms of the Lebesgue measure,
Proc. Conf. Topology and Measure III. Vitte/Hiddensee. 1980, Wiss. Bait. Emst-Moritz-Amdt-Univ. Greifs-
wald, 187-191.
Musial, K. A984), The weak Radon-Nikodym Property in conjugate Banach spaces. Proc. Conf. Topology and
Measure IV, Trassenheide, 1983, Wiss. Bait. Ernst-Moritz-Amdt-Univ. Greifswald, 163-168.
Musial, K. A985), Pettis integration. Proc. 13th Winter School on Abstract Analysis (Smi, 1985), Rend. Circ.
Mat. Palermo, Ser II Suppl. 10, 133-142.
Musial, K. A987), Vitali and Lebesgue convergence theorems for Pettis integral in locally convex spaces, Atti
Sem. Mat. Fis. Univ. Modena 35. 159-166.
Musial, K. A990), Pettis integration of dominated functions. Atti Sem. Mat. Fis. Univ. Modena 38. 261-265.
Musial, K. A991), Topics in the theory of Pettis integration. School on Measure Theory and Real Analysis (Grado,
1991), Rend. Istit. Mat. Univ. Trieste 23, 177-262.
Musial, K. A994), A few remarks concerning the strong law of large numbers for non-separable Banach space
valued functions. Workshop di Teoria della Misura e Analisi reale (Grado, 1993), Rend. Istit. Mat. Univ. Trieste
Suppl. 26, 221-242.
Musial, K. B000), Lifting and some of its applications to the theory of Pettis integral. Workshop di Teoria della
Misura e Analisi reale (Grado, 1997). Rend. Istit. Mat. Univ. Trieste Suppl. 31 A). 107-141.
Musial, K. B001a), The completeness problem in the space of Pettis integrable functions. Quaestiones Math. 24
441^452.

Pettis integral
585
Musiat, K. B001b), The completeness in spaces of bounded Pettis integrable functions and in spaces of bounded
functions satisfying the law of large numbers. Demonstratio Math. 30, 329-334.
Musial, K. and Plebanek, G. A989). Pettis integrabilirx and the equalin of the weak* -integral and the Dunford
integral. Hiroshima Math. J. 19, 329-332.
Musial, K. and Popa, D. A997), The weak Radon-Nikodvm property in spaces of nuclear operators. Quaestiones
Math. 20, 677-683.
Musial, K. and Ryll-Nardzewski, С A978). Liftings of vector measures and their applications to RNP and WRNP,
Vector Space Measures and Applications II. Proc. Dublin A977). Lecture Notes in Math.. Vol. 645. Springer.
Berlin, 162-171.
vanNeerven, J.M.A.M. A994). Remark on a theorem of Riddle, Saab, and Uhl. Quaestiones Math. 17. 183-187.
Odell, E. and Rosenthal. H.P. A975). A double-dual characterization of separable Banach spaces containing l\.
Israel J. Math. 20, 375-384.
Pallares, A.J. and Vera. G. A985). Pettis integrability of weakly continuous functions and Baire measures.
J. London Math. Soc. 32. 479^487.
Pettis, B.J. A938), On integration in vector spaces. Trans. Amer. Math. Soc. 44. 277-304.
Pettis, B.J. A939), Differentiation in Banach spaces. Duke Math. J. 5. 254-269.
Phillips, R.S. A940). Integration in a convex linear topological space. Trans. Amer. Math. Soc. 47. 114-1456.
Plebanek, G. A993), On Pettis integrals with separable range. Colloq. Math. 64. 71-78.
Plebanek. G. A998). On Pettis integral and Radon measures. Fund. Math. 156. 183-195.
Pol, R. A980), On a question ofH.H. Corson and some related problems. Fund. Math. 109. 143-154.
Riddle, L.H. A982), The geometry of weak Radon-Nikodvm sets in dual Banach spaces. Proc. Amer. Math. Soc.
86, 433^438.
Riddle, L.H. A984), Dunford-Pettis operators and weak Radon-Nikodym sets. Proc. Amer. Math. Soc. 91. 254-
256.
Riddle, L.H. and Saab. E. A985). On functions that are universally Pettis integrable. Illinois J. Math. 29. 509-531.
Riddle, L.H., Saab, E. and Uhl, J.J. A983). Sets with the weak Radon-Nikodym property in dual Banach spaces.
Indiana Univ. Math. J. 32, 527-541.
Rieffel, M.A. A968). The Radon-Nikod\m theorem for the Baclmer integral. Trans. Amer. Math. Soc. 131. 466-
487.
Rosenthal, H.P. A970), On injective Banach spaces and the spaces L^- (?) for finite measures ?. Acta Math.
124, 205-248.
Rosenthal, H.P. A974), A characterization of Banach spaces containing l\. Proc. Nat. Acad. Sci. USA 71. 2411-
2413.
Rosenthal. H.P. A978), Some recent discoveries in the isomorphic theory of Banach spaces. Bull. Amer. Math.
Soc. 84, 803-831.
Rybakov, VI. A968). On vector measures. Izv. Vyssh. Uchebn. Zaved. Mat. 79. 92-101 (in Russian).
Rybakov. VI. A970), On the completeness of the space of Pettis integrable functions. Uchebn. Zap. Moskov. Gos.
Ped. Inst. 277, 58-64 (in Russian).
Rybakov, VI. A971), Conditional mathematical expectation for functions that are integrable in the sense of Pettis.
Mat. Zametki 10, 565-570 (in Russian).
Rybakov, VI. A975), A generalization of the Boihner integral to locally convex spaces. Mat. Zametki 18, 577-
588 (in Russian).
Rybakov, VI. A977), Some properties of measures defined on a normed space possessing RNP. Mat. Zametki 21,
81-92 (in Russian).
Saab, E. A982a), On Dunford-Pettis operators that are Pettis representable. Proc. Amer. Math. Soc. 85. 363-365.
Saab, E. A982b), Some characterizations of weak Radon-Nikodym sets. Proc. Amer. Math. Soc. 86. 307-311.
Saab, E. A988), On the weak*-Radon Nikodym property. Bull. Austral. Math. Soc. 37. 323-332.
Saab, E. and Saab, P. A983), A dual geometric characterization of Banach spaces not containing l\. Pacific J.
Math. 105,415^425.
Schachermayer, W., Sersouri, A. and Werner. E. A989). Moduli of non-dentability and the Radon-Nikodym
property in Banach spaces, Israel J. Math. 65. 225-257.
Sentilles, D. A981), Stonian integration of vector functions. Proc. Conf. on Measure Theory and Appl.. Northern
Illinois Univ., 123-135.
Sentilles. D. A983), Decomposition ofweaklx measurable functions. Indiana Univ. Math. J. 32, 425^437.

586
A". Musial
Sentilles, D. (unpublished). Differentiation of measures on boolean algebras: Lifling and Radon-Nikodym,
Preprint.
Sentilles, D. A984), Some measure theoretic implications for the Pettis integral. Measure Theory (Oberwolfach,
1983), Lecture Notes in Math.. Vol. 1089. Springer. Berlin. 165-170.
Sentilles, D. and Wheeler, R.F. A983), Pettis integration vie the Stonian transform. Pacific J. Math. 107. 473^496.
Sierpinski, W. A934), Hypothese du continu. Monogratie Matematyczne.
Sikorski, R. A964), Boolean Algebras, Ergeb. Math. Grenzgeb. B. 25, Springer, Berlin.
Stefansson, G.F. A992), Pettis integrability. Trans. Amer. Math. Soc. 330, 401^418.
Stefansson, G.F. A995), Universal Pettis integrability property. Proc. Amer. Math. Soc. 123, 1431-1435.
Stegall, Ch. A975/76a). A result ofHaydon and its applications. Seminaire Maurey-Schwartz, Expose No. II.
Stegall, Ch. A975/76b), Errata. Seminaire Maurey-Schwartz.
Strauss, W.. Macheras, N.D. and Musial, K. B002). Liftings. Handbook on Measure Theory, E. Pap, ed., Elsevier,
Amsterdam, 1131-1184.
Talagrand, M. A980), Sur les mesures vectorielles definies par une application Pettis integrable. Bull. Soc. Math.
France 108, 475^483.
Talagrand, M. A984), Pettis Integral and Measure Theorv, Mem. Amer. Math. Soc., No. 307, Amer. Math. Soc.,
Providence, RI.
Talagrand, M. A987), The Glivenko-Cantelli pmblem, Ann. Probab. 15. 837-870.
Talagrand. M. A996), The Glivenko-Cantelli pmblem, ten years later. J. Probab Theoret. Probab. 9, 371-384.
Thomas, E. A974), The Lebesgue-Nikodym Theorem for Vector Valued Radon Measures. Mem. Amer. Math.
Soc, No. 139, Amer. Math. Soc, Providence. RI.
Thomas, ?. (?976), Totally summable functions with values in locally convex spaces. Measure Theory
(Oberwolfach, 1975), Proceedings, Lecture Notes in Math., Vol. 541, Springer, Berlin, 117-131.
Tortrat, A. A976), ?-regularite des lois, separation an sens <le A. Tulcea et pmpriete de Radon-Nikodym. Ann.
Inst. H. Poincare 12, 131-150.
Uhl, J.J. A972), ? characterization of stronglx measurable Pettis integrable functions, Proc. Amer. Math. Soc.
34, 425^427.
Uhl, J.J. A972a), Martingales of stronglx measurable Pettis integrable functions. Trans. Amer. Math. Soc. 167,
531-536.
Uhl, J.J. A973), ? sunvy of mean convergence of martingales of Pettis integrable functions. Vector and Operator
Valued Measures and Applications (Proc Symp., Alta. Utah, 1972). 379-385.
Uhl, J.J. A973a), Erramm to "Martingales of strongly measurable Pettis integrable functions", Trans. Amer.
Math. Soc. 181, 507.
Uhl, J.J. A976/77), Pettis mean convergence of vector-valued asxmptotic martingales. Z. Wahrscheinlichkeitsth.
verw. Geb. 37, 291-295.
Uhl, J.J. A980), Pettis's measurability theorem. Proc. Conf. on Integration, Topology and Geometry in Linear
Spaces (Univ. North Carolina. Chapel Hill, NC, 1979). Contemp. Math.. Vol. 2, Amer. Math. Soc. Providence,
RI, 135-144.
Vasuk, L. A981), On one generalization of weakly compactly generated Banach spaces, Studia Math. 70, 11-19.
Varadarajan. VS. A961), Measures on topological spaces. Mat. Sb. 55. 35-100. (English translation: Amer. Math.
Soc. Transl., Vol. 48 A965), 161-228.)
von Weizsacker, H. A978), Strong measurability, liftings and the Choquet-Edgar theorem. Vector Space
Measures and Applications II, Proc. Dublin A977), Lecture Notes in Math., Vol. 645, Springer, Berlin, 209-
218.

CHAPTER 13
The Henstock-Kurzweil Integral
Benedetto Bongiomo
Dipartimento di Matematica ed Applkcrjoni, Via Archirafi 34, 90123 Palermo, Italy
E-mail: bongiom @ unipa. it
Contents
Introduction 589
1. Partitions and Riemann sums 589
2. The Henstock-Kurzweil integral on the real line 590
2.1. The primitives 592
2.2. Convergence theorems 593
3. Further Riemann-type integrals on the real line 596
3.1. The McShane integral 596
3.2. The C-integral 596
3.3. Integrals induced by differentiation bases 597
4. Multidimensional Riemann-type integrals 599
4.1. The Henstock-Kurzweil integral 600
4.2. The ?-regular Henstock-Kurzweil integral 601
4.3. The regular Henstock-Kurzweil integral 602
4.4. The p-integral 602
4.5. The divergence theorem 603
4.6. Fubini's theorem 608
5. The Henstock-Kurzweil integral of vector valued functions 608
6. The Henstock-Kurzweil integral on general spaces 609
References 610
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
587

This page intentionally left blank

The Henstock-Kurzweil integral 589
Introduction
The problem of recovering a function from its derivative is called the problem of primitives.
In his thesis Lebesgue A902) remarked that his integral does not integrate all unbounded
derivatives and left open the problem of defining a more general integral which solves
the problem of primitives. This was done independently by Denjoy A912a. 1912b) and
Perron A914). However, the advantages of Lebesgue's theory were so evident that only
few mathematicians were interested into the Denjoy-Perron integral.
In spite of the general conviction that no modification of Riemann's method could
possibly give such powerful results as that of the Lebesgue's integral, independently
Kurzweil A957) and Henstock A963) introduced a Riemann-type integral that is
equivalent to the Denjoy-Perron integral.
The fact that the Riemann integral has an intuitive appeal and it is well known to
each mathematician, and the fact that a Riemann-type integral was as good as Lebesgue's
attracted the interest of many scientists. So the theory of nonabsolutely convergent integrals
has advanced considerably in the last 30 years.
In this chapter we give an overview of this theory. It is far from complete, as research
is still being carried out in the area; however we hope that this note gives resonable
informations on the current status of the subject.
1. Partitions and Riemann sums
The method of Kurzweil and Henstock is based on the notions of gauge and partition.
Let [a,b] be a subinterval of the real line E. A gauge on [a,b] is, by definition,
any positive function <5 defined on [a,b]. A partition of [a, b] is, by definition, any
collection ? = {(A,·,*i)}f=1 of pairwise disjoint intervals A, and points .v, e A, such that
[a,b] = U, Ai- If Ц Ai is strictly contained in [a, b]. then ? is called a partial partition
of [a,b]. ? is said to be
• S-fine whenever А, С ]л",- - <5,, xt¦ + S, [, /' = 1, 2,..., p,
• anchored on a set ? whenever .x, e E, i = 1, 2 p.
COUSIN'S LEMMA. For any gauge 8 on [a, b] there exists a S-fine partition of [a, b].
Given a function /: [a, b] -> R and a partition ? = {(A,, .v,)}f=1 of [a, b] the Riemann
sum of / with respect to ? is defined as follows
?
]?/ = ]?/(*,)?(?,),
? i=l
where ?( ?,) stands for the Lebesgue measure of A,.

590
В. Bongiorno
2. The Henstock-Kurzweil integral on the real line
Dehnition 2.1. It is said that / is Henstock-Kurzweil integrable on [a, b] whenever
there is a real number / satisfying the following condition: for each ? > 0 there exists a
gauge S such that
?/-'
< ?,
for each <5-fine partition ? of [a. b].
Remark that Cousin's lemma implies that the number / from Definition 2.1 is
determined uniquely by the integrable function /. In fact, assume that J ? I also satisfies
Definition 2.1 and take ? = \I - У1/2. Then we can find two gauges <5| and <5i on [a.b] so
that | J2p / - /| < ? for each <5|-fine partition ? of [a, b], and | ?? f - J\ < ? for each
<52-fine partition ? of [a. b]. Now let S — min{<5|, <5i), and use Cousin's lemma to find a
S-fine partition ? of [a, b]. Then
I/-./K
?/ + ?/
<2e = \I-J\
which is a contradiction.
The number / is called the Henstock-Kurzweil integral of / on [a.b] and we set
/ = fa f. It is said that / is Henstock-Kurzweil integrable on a set ? с [a. b] if /??
is Henstock-Kurzweil integrable on [a.b]. Here ?^ denotes the characteristic function
of E.
If / is Henstock-Kurzweil integrable on [a.b], then it is Henstock-Kurzweil integrable
on each subinterval of [a.b] and the function F(x) = f* f is called the indefinite
Henstock-Kurzweil integral or the Henstock-Kurzweil primitive of f.
Fundamental theorem OF calculus. // F is a differentiable function on [a,b],
then its derivative F' is Henstock-Kurzweil integrable on [a, b] and
f
F' = F(b)-F(a).
We remark that in the definition of the Henstock-Kurzweil integral the gauge S can
be chosen to be Lebesgue measurable (see Liu A987-88), Pfeffer A988), and Buczolich
A997-88)).Moreover if there existsaBaire 1 function^ suchthatg > |/| (and only in this
case) the gauge 8 can be chosen to be nearly upper semicontinuous (i.e., almost everywhere
equal to some upper semicontinuous function) (see Buczolich A995)).
A frequently used lemma in the theory of Henstock-Kurzweil integral is the following
lemma.

The Henstock-Kurzweil integral
591
HENSTOCK'S LEMMA. // / is Henstock-Kurzweil integrable on [a,b], then for each
? > 0 there exists a gauge 8 such that
?/(*,-)(&¦-?,-)- ? ' f
b,
"' < ?.
?'=?
for each S-fine partition {]a,, fr, [,-*<)}> ' = ', 2,.. ·, p, of [a. b].
A proof can be found, for example, in McLeod A980) and in Gordon A994). The
Henstock's lemma is used, in particular, to prove that:
The indefinite Henstock-Kurzweil integral F(x) = /(|* / is a continuous
function, differentiable almost everywhere on [a, b] with F' = f almost everywhere
on [a, b].
Notice also that
(a) each Henstock-Kurzweil integrable function on [a, b] is Lebesgue measurable;
(b) each nonnegative Henstock-Kurzweil integrable function on [a, b] is Lebesgue
integrable and the two integrals coincide;
(c) // a function is Henstock-Kurzweil integrable on every measurable subset ? of
[a, b], then it is Lebesgue integrable on [a, b] and the two integrals coincide.
From (b) it easily follows the monotone convergence theorem and the dominated
convergence theorem.
Monotone convergence theorem. If fi ^ fi ^ ¦ ¦ ¦ fn ^ ¦ ¦ ¦ is a sequence of
Henstock-Kurzweil integrable on [a. b] functions such that lim„ fj f„ exists finite, then
f = lim fn is Henstock-Kurzweil integrable on [a, b] and fa f = lim,, f(l /„.
DOMINATED CONVERGENCE THEOREM. If {fn} " a sequence of Henstock-Kurzweil
integrable on [a, b] functions such that
(i) inf„ /„ and sup„ f„ are Henstock-Kurzweil integrable on [a, b];
(ii) lim„^oo fn = f almost everywhere on [a. b];
then f is Henstock-Kurzweil integrable on [a, b] and fa f = lim,, fa fn.
The fact that the Henstock-Kurzweil integral includes the Lebesgue integral was proved
by Henstock A968), McShane A969). Davies and Schuss A970).
It is however interesting to note that each Henstock-Kurzweil integrable function /
on [a,b] is Lebesgue integrable on some subinterval of [a,b] (see Buczolich A991a)),
and that it is Lebesgue integrable on a sequence {X„} of measurable sets such that
[a, b] = U„ Xn and /x / -»· /j" / (see Lu and Lee A990-91)). Liu A992-93) proved
that each X„ can be chosen closed, and Gordon A991-92) proved that each measurable set
X„ can be chosen such that fx f = f f.
If / is a Lebesgue measurable function, then it is Henstock-Kurzweil integrable if and
only if

592
В. Bongiorno
(i) it has locally small Riemann sums; i.e., for each ? > 0 there exists a gauge <5 such
that for each ? e [a, b], for each subinterval [c,d] of ]x — ?(?), ? + S(x)[ and for
each <5-fine partition ? of [c, d] we have | ?? f\<?-
or, if and only if
(ii) it has functionally small Riemann sums; i.e., for each ? > 0 there exist a nonnegative
Lebesgue integrable function g and a gauge <5 such that | ??. f\<?> f°r every
S-fine partition ? (where PK denotes the subpartition of P) consisting of all
(Ai.Xi) e ? such that |/(.r,-)| > g(xt).
Condition (i) was proved by Schurle A986a. 1986b), Bullen and Vyborny A991), and
condition (ii) by Lu and Lee A990-91).
2.1. The primitives
Lusin A912) introduced the following notion of generalized variation to give a descriptive
definition of the Denjoy-Perron integral.
Definition 2.2. A function F is said to be AC* on a set ? wheneverfor each ? > 0 there
exists ? > 0 such that ?? co(F, ]a,-. ?>,·[) < ? (where ? denotes the oscillation of F over
]a,-, bl¦[), for every sequence of non-overlapping intervals {]a,·. i>,-[} with ]T- \bi — ?,-1 < ?
and a,·, fr, e E. Further, F is said to beACG* on [a, b] whenever it is continuous and there
exists a sequence of closed sets {?„} such that [a,b] = U„ E„ and F is AC* on each En.
Henstock A963), Kurzweil A978), Kubota( 1980), Lee and WittayaNaak-in A982), and
Gordon A986-87) showed that the Denjoy-Perron integral is equivalent to the Henstock-
Kurzweil integral by proving the following characterization of the Henstock-Kurzweil
primitives.
THEOREM 2.3. The family of all Henstock-Kurzweil primitives coincides with the class
of all ACG ^-functions.
A slight modification of the notion of ACG„-function, due to Henstock A963) (see also
Gordon A989-90a) and Lee A989-90)) gives a different descriptive characterization of
the Henstock-Kurzweil primitives.
DEnNlTlON 2.4. A function F is said to be ACs on a set ? whenever for each ? > 0
there exist a positive number ? and a gauge ? such that | ]?;. F(A,-)| < ? for each у-fine
partial partition {(Aj. jc,)} anchored in ? and such that ]?( | A, | < ?.
F is said to be ACGs on [a, b] whenever there exists a sequence {?„} of sets such that
[a,b] = (J,, F„ and F is АСц on each En.
It is clear that an ACGs function is ACs on each set of Lebesgue measure zero.
Bongiorno et al. A995-96) proved that the converse is also true. It is then possible to
formu late a third characterization of Henstock-Kurzweil primitives in terms of the absolute
continuity of an useful extension of the classical notion of variation of a function. Let

The Henstock-Kurzweil integral
593
F: [a, b] -> R, ? с [?, fr] and let 5 be a gauge on [a, b]. The 5-variation of F on ? is
defined as follows
Var(F, S, E) = sup ]T| F(AA)|,
л
where the supremum is taken over all <5-fine partial partitions {(A;. *<)}f=| anchored on E.
The infimum over all gauges S of the 5-variations Var(F, S. ¦) is a metric measure called
f/?e variational measure associated with F and denoted by ?/-; i.e.,
Mf(F) = infVar(F,<5, F).
Remark that ?/? coincides with the classical variation of F if and only if F is absolutely
continuous (i.e., a Lebesgue primitive).
THEOREM 2.5 (Bongiorno et al. A995-96)), The family of all Henstock-Kurzweil
primitives coincides with the class of all functions F whose associate variational measure
?/r is absolutely continuous with respect to the Lebesgue measure.
Lee A989-90), and Kurzweil and Jarnik A991, 1992a), gave a characterization of
Henstock-Kurzweil primitives equivalent to the previous one, but with the additional
hypothesis that F is differentiable almost everywhere in [a,b]. Moreover the condition
"?/? is absolutely continuous with respect to the Lebesgue measure" is displayed in terms
of ? and <5 and it is called the strong Lusin condition of F in Lee (see also Lee and
Vyborny A993)). The same property is called F is well behaved on sets of measure zero in
Kurzweil and Jarnik A991), and F is variational^' normal in Kurzweil and Jarnik A992a).
2.2. Convergence theorems
Let F(x) = jc2sin;c~2 when ? ? 0 and F@) — 0. The sequence /„ = F'x|„-i_|j, ? =
1,2,..., converges to / = F' everywhere on [0. 1] as ? -> oo, but it is not monotone and
it is not dominated by any Henstock-Kurzweil integrable function. Yet each fn and / are
Henstock-Kurzweil integrable on [0, 1] and we have
[ f= lim f fn.
Jo "^°°Jo
A convergence theorem that includes the monotone convergence theorem, the dominated
convergence theorem and covers the above case, is an extension of Vitali's theorem, called
controlled convergence theorem.
CONTROLLED CONVERGENCE THEOREM. Let{f,,} be a sequence of Henstock-Kurzweil
integrable functions on [a, b] such that
(i) /,,(jc) -> f(x) almost everywhere in [a, b]\

594
В. Bongiorno
(ii) the primitives F„ of /„ are ACG* uniformly in n; i.e., [a.b] is the union of
a sequence of closed sets {?<.} satisfying the following condition: for each к
and for each ? > 0 there exists ? > 0 such that sup„ J2i co(F„, ]a,, fo, [) < ? for
every sequence of non-overlapping inten>als {]a,, b,[] with J2i \bi ~ ai\ < Л ап<^
ai, bi e Ek;
(iii) the primitives F„ converge pointwisely to a continuous function F:
then f is Henstock-Kurzweil integrable on [a. b] and F is its primitive.
This theorem (under the additional hypothesis of uniform convergence of the primitives),
was proved by Chelidze and Djvarsheishvili A978), Lee and Chew A985), Lee and Chew
A987) and Gordon A990-9la). In the present version it was proved by Chew A980),
Liao A987) and Fleischer A996-97). A weaker version was given by Liao A989).
If we assume that {F,,} is uniformly ACG& (instead of uniformly ACG*), then
condition (iii) can be skipped (see Bongiorno and Di Piazza A991-92)). Some other
suitable notions of uniform generalized absolute continuity has been used to skip
condition (iii) in the controlled convergence theorem (see Lee A990) and Pujie A993—
94)). Remark that although the notions of function ACG* and ACGs are equivalent, a
sequence {F,,} is uniformly ACG s if and only if it is uniformly ACG* and equicontinuous
(see Bongiorno et al. A993-94, Theorem 3)). If a sequence {/„} satisfies conditions (i),
(ii), (iii), then it is said that {/„} is control convergent to /. It is interesting to notice that
for every Henstock-Kurzweil integrable function / on [a, b] there exists a sequence of
step functions {sn} which is control convergent to / (see Chew and Lee A990)).
A sequence {/„} of Henstock-Kurzweil integrable functions on [a, b] is said to be equi-
convergent to / whenever f„{x) -» f(x) for each ? e [a, b], and for each ? > 0 there
exists a gauge S such that
sup
n
? Jit
< ?.
for each <5-fine partition ? of [a, b],
THEOREM 2.6 (Kurzweil A980)). Let {/„}, be a sequence of Henstock-Kurzweil
integrable functions on [a,b]. It {/„} is equi-convergent to f, then f is Henstock-Kurzweil
integrable on [a, b] and fj f = lim„ f'f„.
Gordon A992-93) proved that the notions of equi-convergent sequence and control
convergent sequence are equivalent (for everywhere convergent sequences). Further
conditions equivalent to the equi-convergence have been given by Bongiorno et al.
A993-94), Leader A993-94) and Wang A993-94). See also Leader A992-93) for more
convergence theorems.
Finally we give three necessary and sufficient conditions for the convergence of a
sequence of Henstock-Kurzweil integrals.

The Henstock-Kurzweil integral
595
THEOREM 2.7 (Bartle A994-95)), Let {/„} be a sequence of Henstock-Kurzweil
integrable functions on [a,b] such that /„(л) -> f(.x), for each ? e [a,b]. Then f is
Henstock-Kurzweil integrable on [a, b] and
rb rb
/ /= lim / /„
if and only if for each ? > 0 there exists a positive integer NF such that for each ? ^ NF
there exists a gauge S„ with \ ? ? fn ~ ? ? /? < ? for aH S„-fine partition ? of [a, b].
THEOREM 2.8 (Gordon A995-96)). Let {/„} be a sequence of Henstock-Kurzweil
integrable functions on [a,b] such that f„(x) -> f(x), for each ? e [a,b]. Then f is
Henstock-Kurzweil integrable on [a, b] and
pb pb
/ /= lim / /„
if and only if for each ? > 0 there exists a gauge 8 such that for each S-fine partition ? on
[a, b] there exists a positive integer ? ? with \ ?? fn — fj fn\ < ? for all ? ^??.
A sequence {F,,} is called asymptotically-ACG* convergent if there is a sequence of
closed sets {X*} such that [a, b] = {jk Xt, and for each ? > 0 and each к there exists
щ > 0 such that
limy^a>(F„; Ah) < ?,
? '—?
h
for each collection {A/,} of nonoverlapping intervals with end points on Xk and with
?????? < ik.
A non-negative function ? is said to be an essential gauge on [a, b] if the set {x e
[a, b\. ?(?) = 0} is a Lebesgue null set. Let F: [a, b] -> ? and let ? с [a, b].
The essential variation of F on ? is defined as follows
Vess(F,?) = infsupV|F(A„)|.
? ? ?
where the supremum is taken over all y-fine partitions {(A,..v,)}-'=1 anchored on ? and
the infimum over all essential gauges ?,
THEOREM 2.9 (Bongiorno et al. A995)). // {/„} is a sequence of Henstock-Kurzweil
integrable functions on [a, b] such that
(!) /h ->¦ / a.e. in [a,b],
B) the primitives Fn of'/„ converge pointwise to a continuous function F(x) everywhere
on [a, b], then f is Henstock-Kurzweil integrable on [a, b] with primitive F if and only if.
(i) {Fn} is asymptotically ACG* on [a, b],
(ii) there exists a sequence of measurable sets {Xk} such that [a,b] = {J^Xk and
Wfi, - F, Xk) -> O.for each к е N.

596
В. Bongiorno
3. Further Riemann-type integrals on the real line
The method of Henstock and Kurzweil was adapted in many ways to different situations.
The first and most striking modification was done by McShane A969). It concerns the
notion of partition. A collection ? = {(A,, .v,)}^, of pairwise disjoint intervals A, and
points Xi e [a, b] is said to be a McShane partition if [a, b] = Ц A,. Remark that each
partition in the sense of Section 1 is a McShane partition, but not vice versa.
3.1. The McShane integral
Dehnition 3.1. It is said that / : [a, b] -> ? is McShane integrable on [a, b] whenever
there is a real number / satisfying the following condition: for each ? > 0 there exists a
gauge S such that
?/-'
< ?.
?
for each 5-fine McShane partition ? of [a, b].
The fact that in a McShane partition {(?,, ?,)} the point x, may be chosen outside the
interval A, increases the number of <5-fine partitions, so it is more difficult for a function to
be integrable. In fact,
THEOREM 3.2 (McShane A969)). A function f is McShane integrable on [a, b] if and
only if it is Lebesgue integrable on [a. b].
3.2. The C-integral
Bruckner et al. A986) remarked that the solution of the problem of primitives provided by
Denjoy, Perron, Kurzweil and Henstock possesses a generality which is not needed for this
purpose. In fact the function
F(x)=\XS'm^-- °<^'-
lo, x = 0.
is a primitive for the Denjoy-Perron-Kurzweil-Henstock integral, but it is neither a
Lebesgue primitive, nor a differentiable function, or a sum of a Lebesgue primitive and
a differentiable function.
A Riemann-type definition for the minimal integral which includes Lebesgue integrable
functions and derivatives was given by Bongiorno A996) and Bongiorno et al. B000),
using the following small modification of the McShane integral.

The Henstock-Kurzweil integral 597
DEHNITION 3.3. A function / : [a, b] -> ? it is said to be C-integrable on [a, b] if there
exists a constant / such that for each ? > 0 there is a gauge <5 so that
]?/(*,)?(?,)-/
/ = l
< ?.
for each <5-fine McShane partition {(A\,x\), (??,??)} satisfying the condition
Ef=1distU,, A,)< 1/?.
The mentioned minimality of this integral is an immediate consequence of the next
theorem.
THEOREM 3.4 (Bongiorno et al, B000)), A function f:[a.b}^> ? is C-integrable if
and only if there exist a derivative f\ and a Lebesgue integrable function /2 such that
/ = /1 + /2.
3.3. Integrals induced by differentiation bases
We recall that a differentiation basis on [a, b] is a collection ? of pairs (A, x), where A is
a subinterval of [a, b] and ? is a point of A, such that inf{X(A): (А. л) e ?} = 0 for each
? & [a, b]. A partition ? = {(Aj, xi)}l.= [ is called a /i-partition whenever (A,-,*;) e /i for
J = 1, 2,...,p.
Let ? be a family of differentiation bases {?(?): a el} satisfying the partitioning
property; i.e., a /i(a)-partition of [a, b] exists for each or e I.
A function f:[a,b]-> ? is said to be B-integrable, with integral /, on [a, b] if for
every ? > 0 there exists a differentiation basis ?(?) e В such that | ]T/> / — /| < ?, for
each ? (a)-partition ? of [a, b]. The number / is called the ?-integral of / on [a, b].
Examples of such integrals are the dyadic integral, the approximate integral, the
symmetric integral and the approximate symmetric integral.
3.3.1. The dyadic integral. Let a and b be dyadic numbers and let S be a gauge
on [a,b]. We denote by /id(<5) the class of all pairs (A,x) such that ? e А с [a,b],
A = [;/2*, (; + \)/2k], for some naturals ; and k, and Ac].t- &{x),x + S(x)[. Let
??d = {Ad(S)}, where 5 runs on the family of all gauges on [a,b]. The Sj-integral is
called the dyadic integral. It was studied in many papers, in particular in connection
with some problem in Dyadic Harmonic Analysis (see Skvortsov A969, 1988, 1992-93),
Kahane A988) and Gordon A990-9lb)).
The primitives have been characterized by Bongiorno et al. B000) by an extension of
Theorem 2.5 in which the variational measure is constructed using Bd partitions.
3.3.2. The approximate integral. Let {Sx: ? e [a. b]} be a family of measurable subsets
of [a,b] such that ? e Sx and ? is a point of density of ??¦. The class of all pairs ([c,d], x)
such that c,d e Sx and ? e [c.d] is, by definition, an approximate differentiation basis.

598
В, Bongiorno
Let ??ap be the family of all approximate differentiation bases. The ??ap-integral is called
the approximate integral.
A Perron equivalent definition of the approximate integral was defined by Burkill A931),
The approximate integral was studied by Henstock A991), Carrington and Pacquement
A972), Bullen A983), Liao A987), Lee A989), Gordon A990-9 la), Skvortsov A992-93),
Fu A993-94), Liao and Chew A993-94), Liao A993-94), Gordon A994), Lu A996-97),
Bongiorno et al. B000).
The problem of recovering an approximately continuous function from its approximate
derivative is solved by the approximate integral (see Gordon A994)), while the Denjoy-
Khintchine integral recovers the continuous primitives only (see Saks A964) and
KhintchineA927)),
A function F is the primitives of / with respect the approximate integral if and only
if it is approximately differentiable on [a, b] except for a countable set with approximate
derivative F' — f almost everywhere on [a, b]. A different characterization in terms of
generalized absolutely continuous function was given by Gordon A994—95):
THEOREM 3.5. A function F:[a,b]-+ Ш is a primitive of the approximate integral if
and only if it is ACG&; i.e., there is a sequence of measurable sets {E„} such that
[a, b] = Un E„, and for each ? > 0 and each ? e N there exist ?„ > 0 and an approximate
differentiable base ?„ such that
J2(F(di)-F(Cl))
v,e?„
< ?
for each ?,,-partition {(]c/,d,-[, xj)\ i — 1. 2...., p\ with ?,,.^,,^/ — c<) < '!¦
Convergence theorems are due to Darmawijaya and Lee A988).
3.3.3. The symmetric integral. Given a gauge S on [a. b], we denote by /isym
(<5) the
collection of all pairs (I,x) such that / = [x -h,x +h], withO < h < S(x) and ? <x-h,
x + h <Ь,от I = [a,a + h] with 0 < h < S(x), or / = [b - h, b] with 0 < h < S(x). Let
fisym = {Asym(S)}, where <5 runs on the family of all gauges on [a, b]. The ??sym-integral is
usually called the symmetric integral.
The symmetric integral was studied by Henstock A963), Kurzweil and Jarnik A987,
1990a) and Thomson A994). In fact the integral considered by Thomson (called the (R[s )-
integral) is somewhat more general than the /isvm-integral. It is defined by special
partitions called symmetric partitions. For each gauge S, a /iSymE)-partition {(A/, Xj)} is said
to be a symmetric partition if the conditions a e A-,,b e Aj imply ?(?,) = X(Aj).
Remark that 4sym- (respectively G?s)-) integrability on [a.b] does not imply ?
sym"
(respectively (R[s)-) integrability on each subinterval of [a, b] and that /}Sym- (respectively
(/?])-) integrability on contiguous intervals does not imply /iSym- (respectively (/?])-)
integrability on their union.
The primitives of the (tf')-integral are characterized by symmetrically ACG*
functions defined on all [a, b] except possibly some countable set (see Thomson A994,
Theorem 9.34)), or, in some particular cases, by an extension of Theorem 2.5 in which the

The Henstock—Kurzweil integral
599
variational measure is constructed using Bsym partitions (see Bongiorno et al. B000,
Corollary 5)).
Remark also that in some very simple situations the symmetric integral Asym
(respectively (/?])) may be used to solve the coefficient problem for trigonometric series; i.e.,
if
ОС
f(x) = — + /J(fljt cosk.x + bk sinkx). for each ? e R.
*=o
how may the coefficients of the series be determined?
Of course if / is Lebesgue integrable then the coefficients are determined by the usual
Fourier formulas using Lebesgue integrals. But in general / is not Lebesgue integrable,
neither Henstock-Kurzweil integrable. This is for example the case of each function
f(x) = ?„ b„ ???? with b„ > 0 and ]Tn b„/n = +oo. Nevertheless, even in the most
general case b„ \ 0, / is symmetrically integrable (in both senses) on any interval of
length 2? and the coefficients are determined by the usual Fourier formulas using the
symmetric integrals.
3.3.4. The approximate symmetric integral. A complete solution for the coefficient
problem is given by the approximate symmetric integral, introduced by Preiss and
Thomson A989).
A differentiation basis ? is said to be a measurable approximate symmetric
differentiation basis if there is a measurable set Г с R ? ]0, oo[ such that
(i) ([x - t, ? +1], x) e ? whenever (x, t) e ?,
(ii) limsup,lV) \{t e @, h); (?, t)<?T}\/h = 0, for each ? e R.
Let ??as be the family of all measurable approximate symmetric differentiation bases. As
before, in the definition of the ??as-integral the partitions are "symmetric partitions". The
resulting integral is the approximate symmetric integral.
Solutions of the coefficients problem have been also done by Denjoy A941^49),
Marcinkiewicz and Zygmund A936), James A950) and Burkill A951). They involve the
inversion of a second order symmetric derivative:
en f/ ч r F(x+h) + F(x-h)-2F(x)
SD2F(x)— lim .
ft^o+ h2
A Riemann-type integral which gives F in function of SDtF was defined by Freiling
et al. A997). The definition of this integral involves several elaborate partitioning
arguments for rectangles in the plane.
4. Multidimensional Riemann-type integrals
A subset A of R" is said to be an interval if it is the Cartesian product of ? no degenerate
intervals of R. Given a gauge S: A -> R+ and a partition ? we say that ? is 5-fine if
d(Aj) < S(xj), i = 1,2,..., p, where "d" stands for "diameter".

600
В. Bongiorno
Given a function /: A -> Ш and a partition ? = {(A,, jt,)}''=1 of A, we set ? ? f =
??=? /(¦*/')?"(?,-), where ?"(?,) stands for the и-dimensional Lebesgue measure of A,.
4.1. 77?e Henstock-Kurzweil integral
DEHNITION 4.1. Let A be an interval of E" and let /: A -> R. It is said that / is
Henstock-Kurzweil integrable on A whenever there is a real number / satisfying the
following condition: for each ? > 0 there exists a gauge S on A such that ? ?/> / — /| < ?,
for each <5-fine partition ? of A.
Many of the properties of the one dimensional Henstock-Kurzweil integral extends to
the и-dimensional one (see Henstock A963. 1988)), in particular
- the Henstock-Kurzweil integral includes the Lebesgue integral;
- each nonnegative Henstock-Kurzweil integrable function is Lebesgue integrable;
- monotone and dominated convergence theorems hold;
- the Henstock-Kurzweil integrability on A implies the Henstock-Kurzweil integrabil-
ity on each subinterval of A;
- the Henstock-Kurzweil integrability on A implies the Lebesgue integrability on some
subinterval of A (Buczolich A991a));
- the Henstock-Kurzweil integrability on A implies the existence of a sequence of
closed sets [X„] such that A = U„ X» and /x / -> /j' / (see Lee et al. A996-97)),
The indefinite и-dimensional Henstock-Kurzweil integral F(B) = fB f is an additive
continuous interval functions, but in general the set Of of all points ? e A such that F
is not differentiable at ? has positive Lebesgue measure; i,e„ for each ? e Of. there exits
к &N such that for each m e N the condition
F{B) ft ч
- fix)
?» (S)
>k
holds for some interval В with ? e В and d{B) ^ \/m.
For each teNwe set
rk(F)= \(B,x): xe Of and —!—- - f(x) ^k\.
? ?»(?) )
THEOREM 4.2 (Lu and Lee A999)). An interval function F defined on the subintervals of
A is the primitive of a Henstock-Kurzweil integrable function f on A if and only if for any
к e N and ? > 0 there is a gauge S such that
]??"(?)<? and Y^\F(B)\<e,
for each S-fine partial partition ? = {(A,. jc,-)}/Li with iA>-xi) e A-(F) and*; e 0F,for
i = 1, 2,..., p.

The Henstock-Kurzweil integral
601
Other characterizations of the primitives of the «-dimensional Henstock-Kurzweil
integral has been given by Lee and Ng A996-97), and Kurzweil A998).
Integrals of Riemann-type which primitives are differentiable almost everywhere and
absolutely continuous in some generalized sense are the ?-regular Henstock-Kurzweil
integral, the regular Henstock-Kurzweil integral, the p-integral.
4.2. The ?-regular Henstock-Kurzweil integral
The regularity of an interval В is the ratio of its shortest and longest side; it is denoted
by r(B). Let 0 < ? ^ 1. A partition ? = {(A,,jc,)}''=1 of A is said to be ?-regular if
r(Aj) ^ ? for each /'.
Dehnition 4.3. A function /: A -* R is said to be ?-regularly Henstock-Kurzweil
integrable on A whenever there is a real number / satisfying the following condition:
for each ? > 0 there exists a gauge S such that | J2P f - I\ < ?. for each <5-fine ?-regular
partition ? of A,
Note that for each 0 < ? < 1 there exists a function / which is ? ?-regularly Henstock-
Kurzweil integrable for each a\ e (a. 1), and is not ??-regularly Henstock-Kurzweil
integrable for each aj e @, ?),
An interval function F is said to be a-differentiable at a point ? e A with ?-derivative
F^{x) if for every ? > 0 there exists <5(;t) > 0 such that
F(B) F>, ч
?» (S)
< ?
for each interval В satisfying the conditions ? e B, d(B) ^ <5(;t) and r{B) ^ a.
Remark that if 0 < a < ? ? 1 and F is ?-differentiable to / at x, then F is ?-
differentiable to / at ? (see Kurzweil and Jarnik A992a)).
Let ? С [a, b) and let S be a gauge on E. The ?-regular S-variation of F on ? is defined
as follows
V„(F,a,E) = sup?|F(Aft)|,
li
where the supremum is taken over all ?-regular S-fine partial partitions {(Ал ,*/,)} anchored
on E. The ?-regular variation of F on ? is defined as follows
V„(F. E) = mfVa(F,8,E),
where the infimum is taken over all gauges S.
V„(F, ¦) is a metric measure. If F is additive and V„(F. ¦) is absolutely continuous with
respect to the Lebesgue measure on A then F is ?-differentiable almost everywhere on A
(see Di Piazza B001)).

602
В, Bongiorno
THEOREM 4.4 (Kurzweil and Jarnik A992a)). An interval function F defined on the
subintervals of A is the primitive of an a-regularly Henstock-Kurzweil integrable function
f on A if and only if
(i) F is additive;
(ii) the measure Va(F. ¦) is absolutely continuous with respect to the Lebesgue
measure;
(iii) F^{x) = f{x) at almost every ? e A.
4.3. The regular Henstock-Kurzweil integral
Given a function ? ; A -> @. 1), a partial partition ? = {(Ai.Xi)}j'_] of A is said to be
?-regular if r(A;) >0 (*,·).
An interval function F defined on the subintervals of A is said to be regularly
differentiable at a point ? & A with derivative F'(x) if for every ? > 0, there exist S(x) > 0
and 0 < ?(?) ? 1 such that
F (x)
< ?
?» (S)
for each interval В satisfying the conditions ? e B, d(B) ? S(x) and r(B) ^ ?(?).
DEnNlTlON 4.5. A function f:A -> R is said to be regularly Henstock-Kurzweil
integrable on A whenever there is a real number / satisfying the following condition: for
each ? > 0 there exist a gauge <5 and a function ? : A —* @. 1) such that \?? f ~ I\ < ?>
for each <5-fine /3-regular partition ? of A.
Let ? с A. An interval function F is said to be AC* on ? if for every ? > 0
there exist a gauge 8, a function ?;? -* @, 1) and a positive constant ? > 0 such that
53,- |F(A,)| < ? for each <5-fine /S-regular partial partition {(A/,-*,)}f=1 anchored on ?
such that ?], ?"(A,) < /j.
A continuous function F is said to be ACG* on A if A is the union of a sequence of sets
{?,} such that F is AC* on each ?,-.
THEOREM 4.6 (Chew A989-90)). Аи interval function F definedon the subintervals of A
is the primitive of a regularly Henstock-Kurzweil integrable function f on A if and only if
F is ACG* on A and F'(x) = f(x) almost everywhere in A.
4.4. The p-integral
Given a function p:AxR+->@,l) and a partial partition ? = {(A,, -*/)}''=| of A, it is
said that ? is yo-regular whenever r( A,) > ? (.*,-, d( A,·)) for each /'. If p(x,d) < 1 for each
jc e A and ii e E+, then given a gauge S on A there exists a <5-fine p-regular partition of A
(see Jarnik and Kurzweil A995)).

The Henstock-Kurzweil integral
603
The /O-regular variation VP(F, E) of F on a set ? с A is defined as the ?-regular
variation by using yo-regular partitions. If F is additive and VP(F,-) is absolutely
continuous with respect to the Lebesgue measure on A, then F is regularly differentiable
almost everywhere on A (see Di Piazza B001)).
Dehnition 4.7. Let ?: ? ? R+ -> @. 1) with p(x.d) < 1 for each .v e A and d e К"
and let / : A —* R. It is said that / is yo-integrable on A whenever there is a real number
/ satisfying the following condition: for each ? > 0 there exists a gauge S such that
? ?? / — /| < ?, for each <5-fine yo-regular partition ? of A.
THEOREM 4.8 (Jarnik and Kurzweil A995)). Let ?: ? ? E+ -> ]0, 1[ be a function
satisfying the conditions:
A) p(x,d) < 1 for each ? e A a/idd e R+:
B) 0 < liminfi/_0+ p(x,d) ^ limsup(/^(r p(x.d) < \ for each ? e A.
Then an interval function F defined on the subintervals of A is the primitive of a p-
integrable function f on A if and only if
(i) F is additive;
(ii) the measure Vp(F, ¦) is absolutely continuous with respect to the Lebesgue
measure;
(iii) F'(x) = f(x) at almost every ? e A.
Each /O-integrable function is the limit of an equi-convergent sequence of step-functions
(see Jarnik and Kurzweil A997)).
4.5. The divergence theorem
The main force of the one dimensional Henstock-Kurzweil integral is the integrability of
each derivative. A corresponding result in higher dimensions should be the integrability of
the divergence of any differentiable vector field. Unfortunately this does not seem to be the
case using the и-dimensional Henstock-Kurzweil integral. Modifications of the Henstock-
Kurzweil method producing integrals for which the divergence of any differentiable
vector field is integrable, are due to Mawhin A981a, 1981b), Jarnik et al. A983), Jarnik
and Kurzweil A984, 1985, 1988), Pfeffer A986 1993), Jurkat A993), Jurkat and
Nonnenmacher A994a, 1994b, 1994c), Nonnenmacher A994, 1995, 1995-96).
4.5.1. The GP-integral.
DEHNITION 4.9. It is said that / : A -> R is GP-integrable on A whenever there is a real
number / satisfying the following condition: for each ? > 0 and 0 < a ^ 1 there exists a
gauge S on A such that I ?/> / — /| < ?. for each partition ? of A which is 5-fine and
?-regular.
The GP-integral was introduced by Mawhin A98 lb). The divergence of a differentiable
vector field is GP-integrable and the Gauss-Green formula holds. However, there are

604
В. Bongiorno
examples when A and В are non-overlapping intervals, the GP-integrals fA f and fB f
exist, but fAljB f does not (see Jarnik et al. A983)). Pfeffer A986) suggested to solve this
problem by using special r-regular partitions. A partition ? = {(А,, .г,·), / = 1,2, — p} is
called special if jc, is a vertex of the interval A, for each /'. Then the problem arises whether
for any gauge S and for any 0 < ? ? 1, there exists a <5-fine ?-regular special partition of A.
Buczolich A991a) answered positively this problem in R2 if ? — 1/1000, and Fejzic et al.
A993-94) if ? < 1/-У2. In dimensions higher than two the problem about the existence of
special partitions is still unsolved.
4.5.2. The M\-integral. Jarnik et al. A983) modified the notion of ?-regular partition,
obtaining an additive integral which preserves the good properties of the GP-integral.
Given a partition ? = {(A/, -*,¦)},'!= ? · we set
/'
2T(P) = ??*„_, OA;)d(A;)
; = i
where ЭА; denotes the boundary of A, and HH~\ (ЭА,) its (n — 1 )-dimensional Hausdorff
measure.
Definition 4.10. It is said that /: A -> R is M\ -integrable on A whenever there is a
real number / satisfying the following condition: for each positive ? and С there exists a
gauge <5 on A such that | ? ? / — /| < ?, for each <5-fine partition ? of A with ?(?) ^ C.
4.5.3. The Nonnenmacher's integral. Nonnenmacher A994) extended the M\ integral
by considering only those <5-fine partitions for which the sum of the (n — 1 )-dimensional
Hausdorff measures of the boundaries of the non-regular intervals occurring in the partition
is bounded.
DEnNlTlON 4.11. A function / : A -> R is said to be Nonnenmacher's integrable on A
whenever there is a real number / satisfying the following condition: for each ? > 0, r > 0,
К > 0 there exists a gauge S on A such that | ?? f — Ц < ?, for each <5-fine partition
P = {(Ai,Xi)}?=i of A such that Eru,)<rHn-№Aj)) ??.
The divergence theorem holds with respect to each vector field which may not be
differentiable on countably many hyperplanes and its continuity may fail at contably many
points where it is only assumed the vector field to be of Lipschitz type with a non-positive
exponent ? > 1 — п.
4.5.4. The Pfeffer's integral. An extension of the Nonnenmacher's integral was obtained
by Pfeffer A986) by using a relative regularity condition with respect to a finite family of
planes parallel to the coordinate axes.
Let 0 ^ к ? ? — 1. A k-plane is a A:-dimensional linear submanifold of R" which
is parallel to к distinct coordinate axes. The regularity of an interval В relative to a
family V of ^-dimensional planes is the number r(B.V) defined as follows: if V = 0,

The Henslock-Kurzweil integral 605
then ?(?,-?) = k"(B)/(d(B))u; if V consists of a single ?-plane H, then r(B.H) =
?*(? ? H)/{d{B))k (here ?*(·) denote the A:-dimensional Lebesgue measure) whenever
? П ? ? 0 and r(fi. ?) = k"(B)/(d(B))n otherwise; finally, if V ? 0 is arbitrary, then
r(fi,-p) = sup{r(fi.{#}): ? eV].
Dehnition 4.12. A function / : A -> ? is said to be Pfeffer's integrable on A whenever
there is a real number / satisfying the following condition: for each 0 < ? < 1/2 and each
finite family of A:-planes V, with 0 ^ к ^ ? — 1. there exists a gauge <5 on A such that
| ?? / — /| < ?, for each 5-fine partition ? = {(А/. л";)}''=| of A such that r(A(, V) ^ ?,
i = 1,2,..., p.
Using this integral it is possible to formulate a divergence theorem in which the vector
field may not be differentiable on countably many (n — l)-planes and its continuity may
fail at contably many (n — 2)-planes.
Moreover the usual transformation formula holds for a continuous map ? such that A is
the union of nonoverlapping intervals A,, /' = 1.2. p, and ? is affine on each A;.
4.5.5. The PV'-integral. A better transformation formula holds for the Pt/-integral. It
was introduced by Jarnik and Kurzweil A985, 1988) with the use of С '-partitions of unity
(br. Pt/-partitions) instead of the usual partitions. Given an interval А с ?", a Pi/-partition
of A is a collection {(9j,Xi)}f=i of C1-functions ?, with compact supports and points
?? e A such that 0 ^ ?f=, ?, (t) ^ 1 for each t e E". and ?f=, ?, (t) = 1 on A.
If S is a gauge on A, then a Pi/-partition {(#,, .v/M'Li is said t0 be <5-fine if ft vanishes
outside the ball of center x-, and radius ?(?,). for each /'.
Dehnition 4.13. A function /: A -> ? is said to be Pt/-integrable on A whenever there
is a real number / satisfying the following condition: for each ? > 0 there exist a gauge S
on A and two constants a > 0 and ? > 1 such that
? fix,) / Wdt-I
< ?.
for each 5-fine Pt/-partition {(ft..v,)}''=| of A such that 9,(t) < A +orH;(.x;) for each
t e E", and
/ |D0/(O||dr < — / 0/(r)dr.
Jr" Pi Jr»
where || · || denotes the euclidean norm of E", D9, denotes the gradient of ?,, and
Pi — sup{ || t - ?/1|: t e supp ft}.
The usual transformation formula holds for С' -diffeomorphisms; therefore it is possible
to extend the Pt/-integration to differentiable manifolds. The divergence theorem holds
with respect to each continuous vector field which may not be differentiable on a
hyperplane.

606
В. Btmgionio
4.5.6. The g-integral. The g-integral, introduced by Pfeffer A993), is an integral which
uses a more traditional concept of interval partition, but a larger class of gauges. A function
S : A -> R+ U {0} is said to be a weak gauge on A whenever its null set has ?-finite (n — 1 )-
dimensional Hausdorff measure.
Dehnition 4.14. Let A be an interval. A function / ; A -> R is said to be g-integrable
on A whenever there is a real number / satisfying the following condition: for each ? > 0
there exist a weak gauge <5 and a sequence of positive real numbers {?;}, j = 1, 2 , so
that | ?? f - I\ < ? for each 5-fine ?-regular partial partition ? = {(?,, .?,)}''=| such that
A \ U(;L| A-j is the union of disjoint intervals B\. Zb Bk with 7i„-\ (SBj) < l/? and
??{?])<?]???] = 1,2,.... ?.
The divergence theorem holds with respect to each continuous vector field which may
not be differentiable on a set of ?-finite (n — 1 )-dimensional Hausdorff measure. However
the g-integral is not rotation invariant (Buczolich A992-93)). This deficiency is removed
by the ^"-integral.
4.5.7. The ?-integral. The :F-integral. introduced by Pfeffer A993). uses partitions with
figures (finite union of intervals).
Definition 4.15. Let A be a figure. A function /: A -> R is said to be JF-integrable
on A whenever there is a real number / satisfying the following condition: for each ? > 0
there exist an essential gauge <5 and a sequence of positive real numbers {/jy}, j = 1.2
so that | J2p f - l\ <? for each 5-fine partial partition ? = {(A,. .v,)}''=1 such that
A) A, isafigurewithX"(A,)/(i/(A,)W„-i(A,)) > ? for/= 1.2 p:
B) A \ (Jf=| A, is the union of disjoint figures B\, B-> Bk with H„-\(dBj) < 1/?
and ?"(?j) < ?, for j = 1.2 k.
The J"-integral satisfies the change of variable formula for each injective Lipschitz map
? : A -> R" such that ?~] is also Lipschitz (lipeomorphism) and ?(?) is a figure.
However, there is some lipeomorphism ?:?-> R" such that ? (A) is not a figure. This
lack is removed by the ??V-integral.
4.5.8. The BV-integral. The SV-integral. introduced by Pfeffer A99la, 1993), is based
on partitions which use BV-sets instead of figures.
Given a set E, the essential closure of ? is defined as the set of all ? e R" such that ?
is not a dispersion point for E. It is denoted by cL?. The set deE =cl,.? nd,.(R" \ E) is
called the essential boundary of ?.
A BV-set is a bounded subset ? of R" such that H„-\ (de(E)) < +oo. The intervals
and the figures are particular В V -sets.
Dehnition 4.16. Let A be a BV set. A function /:cl,.A -> R is said to be BV-
integrable on A whenever there is a real number / satisfying the following condition:
for each ? > 0 there exist an essential gauge S and a sequence of positive real

The Henslock-Kurzweil integral
607
numbers {/jy}, j = 1, 2 , so that | ]Tp / — /| < ? for each <5-fine partial partition
{{A\,x\), (A2,X2), ¦¦¦, (Ap,xp)} such that
(?) A, isaBVsetwithXtAO/O^A/iK-i^A,)) > ? for / = 1, 2,.... p;
B') A \ Uf=| A, is the union of disjoint BV sets ?|. B2 Bk with H„-1 (9^бу) <
?/? and ?"(?,) < /jy for ; = 1. 2 к.
The SV-integral is invariant for general lipeomorphisms. The divergence theorem holds
with respect to each continuous vector field which may not be differentiable on a set with
?-finite (n — 1 )-dimensional Hausdorff measure.
4.5.9. The v-integrai The v-integral was introduced and studied by Jurkat A993).
Jurkat and Nonnenmacher A994a, 1994b. 1994c). Nonnenmacher A995, 1995-96). It is
a coordinate free integral for which the divergence theorem holds with respect to each
unbounded vector field with singularities of order greater than 1 — /;.
Dehnition 4.17. Let A be a compact BV set. A function / : A -* Ш" is said to be v-
integrable on A whenever there are a real number /, a decomposition A = Eq U (U/? ? ?/')
of A and a function ?: N -* [0, ?] ? {1,2} such that
A) the sets ??> E\,... are disjoint;
B) EQ is a subset of the interior of A, and ?" (? \ ?(>) = 0;
C) if 00) = (?, 1) with a < n, then Ha(Ej) < do;
D) if 0(/) = (n, 1), then ?"(?;) = 0;
E) if 0(/) = (a. 2), then W„(?,) = 0;
and such that, for each positive ?,?,?\ there exist a gauge S on A and a sequence of
positive real numbers ?\,?? so that | ?]/>/ — /| < ? for each <5-fine partial partition
? = {(Ak, **)}?=| U {(AJr ^',)}^=| of A satisfying the following conditions:
A) the {Ak} and the {A'h) are compact BV subsets of A;
B) if jc* e ?0, then Л(А*)" <; K\Ak\„ and H„-1 (ЭЛА) ? Kd{Akf-];
C) for / = 1,2, the collection ?, of all A* such that xk e E-, satisfies one of the
following conditions:
(a) if 0@ = (?, 1) (respectively 00) = (?. 2)) with 0 ^ a < ? - 1, then ?,
is a subcollection of a finite family {Cj]"-=i of compact SV sets such that
7in-\(B(x,r) П ЭСУ) ?? АГ,г"~' for all .r e R" and r > 0. Ey^C/)" ij AT,
(respectively Zjd(Cj)a < 6»,). and ?y. Xc,(-v) < */;
(b) if 0(/) = (n - 1. 1) (respectively 00) = (?/ - 1.2)). then П„-\(дАк) ? AT,,
(respectively H„- \ (ЭС,¦) ^ 6>,). for each Aa e ?,;
(c) if 0(z') = (a, 1) (respectively 00) = (a. 2)) with ? - 1 < a < л, thenrf(A^)" ^
AT, ?" (Л*), Нп-\(ЕдАк) ^ AT/, (respectively |Э(Су)„_| ^ 6),·) for each A* e
r,;
(d) if 00) = (и, 1) or 00) = 0/.2), then d(Ak)" ? K,\Ak\„ and H„-\(dAk) ?
KidiAk)"-* for each Ак еГ,;
D) ?Л W„_ ? (Э A;,) ^ AT, and for /; = 1. 2 <7 either x'h e ?0 either there exist / e N
and a e ]n - 1, и] such that 0@ = (<*- ;')¦ j =1.2, and .v^( e ?,-.

608
В. Bongiorno
4.6. Fubini 's theorem
One of the main tool of the multidimensional Lebesgue integral is the Fubini's theorem.
This is not the case for many of the integrals considered in this section. In fact
Fubini's theorem is incompatible with the divergence theorem, in the sense that for each
multidimensional integral that satisfies Fubini's theorem and in dimension one coincides
with the Henstock-Kurzweil integral, it is possible to construct a differentiable vector field
that is not integrable (see Pfeffer A993)).
Fubini's theorem holds for the и-dimensional Henstock-Kurzweil integral (see Kurzweil
A973) and Henstock A988)) but it does not hold for the regular integrals (see Ostaszewski
A986)).
5. The Henstock-Kurzweil integral of vector valued functions
The Henstock-Kurzweil method is readily adaptable to provide a theory of integration for
vector-valued functions. However in case the range space is an infinite dimensional Banach
space the corresponding theory differs from the real-valued case in some important points:
A) the Henstock's lemma does not hold (see Skvortsov and Solodov A998-99), and
CaoA992));
B) there exist integrable functions which are not measurable (for example / : [0, 1 ] ->
/oo[0, 1] denned by f(t) = ?[0,?]);
C) there exists integrable functions which primitives are not ACG* neither ACGs (same
example).
But if the range space is a vector space endowed with nuclearity, then the Henstock's
lemma holds (see Nakanishi A994)); therefore in particular Henstock's lemma holds if the
range space is one of the spaces S, S', V and V occurring in the Schwartz's theory of
distributions.
As mentioned in Section 2.4 a slight modification of Henstock-Kurzweil method
produces the McShane integral which is equivalent (for real valued function) to the
Lebesgue integral.
We recall that if X is an infinite dimensional Banach space the Bochner integral and
the Pettis integral are the most known generalizations of the Lebesgue integral. A function
/: [a, b] -> X is said to be Bochner integrable on [a, b] if there exists a sequence of
simple functions {s,,} such that lim„ fh \\f — s„ \\ = 0. where || · || denotes the norm in X.
A function /: [a, b] -> X is said to be Pettis integrable on [a, b] if the scalar function
x* о f is Lebesgue integrable on [a. b] for each x* e X*, the dual of X, and there exists
we X such that fix* of =x*(w) for each x* e X*.
The class of all McShane integrable functions is strictly contained between the class of
all Bochner integrable functions (see Gordon A990)) and the class of all Pettis integrable
functions (see Fremlin and Mendoza A994)).
A Pettis integrable on [a, b] function / is McShane integrable in the following cases:
A) /is strongly measurable (see Gordon A990));
B) / is Henstock-Kurzweil integrable on [a. b] (see Fremlin A994)).

The Henstock-Kurzweil integral
609
Therefore it follows that even in the Banach-valued case a function / is McShane
integrable on [a, b] if and only if //? is Henstock integrable for every measurable
Ec[a,b].
Nakanishi A994) showed that the McShane and the Bochner integrals are equivalent for
functions with range in a vector space endowed with nuclearity.
6. The Henstock-Kurzweil integral on general spaces
The Henstock-Kurzweil method was developed in a very general situation by Henstock in
many papers, in particular in A991). The basic elements of the theory are a generic fixed
set X and a generic fixed family ? of non-empty subsets of X, called intervals. Every finite
disjoint union of intervals is called an elementary set. A partition of an elementary set ?
is as usual any finite collection ? = {(A,·,*,-)} of disjoint intervals A, and points ?? e A,
such that [Jj A, = E. In this general situation there is no meaning to define a gauge, then
Henstock defines the "<5-fine partitions" in the following axiomatic way.
A collection S of pairs (A,x) with A e ? and ? e A is called a dividing collection. It is
said that S divides an elementary set ? whenever there exists an <S-partition of E. A family
? = {Sa: a e T] of dividing collections is fixed.
This family is the vital component of the theory, differences between the families being
reflected in the differences between the corresponding integrals.
To assure the unicity and the additivity of the resulting integral it is assumed that ?
satisfies the following conditions:
A) for each elementary set ? there exists a e ? such that Sa divides E;
B) if a and ? are elements of ? such that Sa and Sp divide a given elementary set E.
then there exists ? e ? such that SY divides ? and SY С Sa ? Sp;
C) if E\ and Ej are elementary sets and Sa divides E\ U E2, then there exists a \, ai e ?
such that Sa] ={(A,x)eSa: А с E\] and ?„, = {(A.jc) e <S„: Ac E2):
D) if?| and Ej are disjoint elementary sets, <Sff| divides ? \, Sai divides Ei and А с Е,
for each (A, x) e Sa,, / = 1. 2. then there exists ? e ? such that 5y divides ?|U?i
and <Sy С <Sa, U <Sa2.
Let h(A,x) be a real-valued function defined on the family of all pairs (A. x) e «S„, for
each or e Г. It is said that h is T-integrable on an elementary set ? with integral fEdh if
for each ? > 0 there is a e ? such that
y^h(Ai,Xi) - dh\<s
Je I
for each Sa -partition of E.
Notice that even in this general setting monotone and dominated convergence theorems
hold, and that the corresponding integral on product spaces satisfies Fubini-type and
Tonelli-type theorems.
It is interesting that this integral applies to the unification of Wiener and Feynmar
integration in a single mathematical framework, and to the availability of new and powerful
techniques for handling some long-standing mathematical difficulties in formulating a

610
В. Bongiomo
definition and calculus of the Feynman integral (see Muldowney A987) and Chapter 6
ofHenstockA991)).
References
Alexiewicz, A. A948), Linear junctionals on Denjoy integrable functions. Coll. Math. 1. 289-293.
Bartle. R.G. A994-95), A convergence theorem for generalized Riemann integrals. Real Anal. Exchange 20.
119-124.
Bongiomo, B. A994), Relatively weakly compact sets in the Denjoy space, J. Math. Study 27 A). 37^44.
Bongiomo, B. A996), Un nuovo integrate per it problema delle primitive. Le Matematiche 51B). 299-313.
Bongiomo, B. and Di Piazza, L. A991-92). Convergence theorems jor generalized Riemann-Stieltjes integrals.
Real Anal. Exchange 17, 339-361.
Bongiomo, В., Di Piazza, L. and Pfeffer, W,F. A993-94), A full descriptive definition of the BV-integral.
Comment. Math. Univ. Carolin. 36. 461^469.
Bongiomo, В.. Di Piazza. L. and Preiss. D. B000), A constructive minimal integral which includes Lebesgue
integrable functions and derivatives. J. London Math. Soc. 62 A), 117-126.
Bongiomo, В.. Di Piazza, L. and Skvortsov, V.A. A993-94). Uniform generalized absolute continuity and some
related problems in H-integrability, Real Anal. Exchange 19 A), 290-301.
Bongiomo, В., Di Piazza, L. and Skvortsov, V.A. A995). The essential variation of a function and some
convergence theorems. Anal. Math. 22. 3-12.
Bongiomo, В.. Di Piazza, L. and Skvortsov. V.A. A995-96). A new full descriptive characterization of the
Denjoy-Perron integral. Real Anal. Exchange 21 B). 656-663.
Bongiomo, В., Di Piazza, L. and Skvortsov. V.A. B000). On variational measures related to some bases, J. Math.
Anal. Appl.. to appear.
Bongiomo, В., Pfeffer, W.F. and Thomson. B.S. A996). A full descriptive definition oj the gage integral, Canad.
Math. Bull. 39D), 390-401.
Bruckner, A.M., Fleissner, R.J. and Foran. J., A986). The minimal integral which includes Lebesgue integrable
junctions and derivatives, Colloq. Math. 50 B). 289-293.
Buczolich, Z. A987-88), Nearly upper semicontinuous gauge junctions in K'", Real Anal. Exchange 13 B).
436-440.
Buczolich, Z. A991a), A general Riemann complete integral in the plane. Acta Math. Hungar. 57C^4). 315-323.
Buczolich, Z. A991b), Henstock integrable junctions are Lebesgue integrable on a portion, Proc. Amer Math.
Soc. 111A), 127-129.
Buczolich, Z. A992-93), The g-integral is not rotation invariant. Real Anal. Exchange 18. 437^447.
Buczolich, Z. A995), Characterization oj upper semicontinuoiisly integrable junctions,]. Austral. Math. Soc. 59.
244-254.
Buczolich. Z. and Pfeffer. W.F. A997). On absolute continuity, Czechoslovak Math. J. 47. 525-555.
Buczolich. Z. and Pfeffer. W.F. A998). On absolute continuity. J. Math. Anal. Appl. 222 A). 64-78.
Bullen. PS. A983). The Burkill approximately continuous integral, J. Austral. Math. Soc. Ser. A 35. 236-253.
Bullen. PS. and Vybomy, R. A991). Some applications of theorem of Marcinkiewicz. Canad. Math. Bull. 34 B),
165-174.
Burkill, J.C. A931), The approximately continuous Perron integral. Math. Z. 34. 270-278.
Burkill. J.C. A951), Integrals and trigonometric series, Proc. London Math. Soc. 1. 46-57.
Cao. S.S. A992), The Henstock integral for Banach-valued functions. Southeast Asian Bull. Math. 16. 35^40.
Carrington. D.C. and Pacquement. A. A972), Sur line extension duproce'ded'integration de MR. Henstock,C. R.
Acad. Sc. Paris Ser. A 274. 1901-1904.
Chelidze. V.G. and Djvarsheishvili. A.G. A978). Theon of the Denjoy Integral and Some of Its Applications,
Tbilisi (in Russian).
Chew, T.S. A980). On the generalized dominated convergence theorem. Bull. Austral. Math. Soc. 37. 165-171.
Chew. T.S. A989-90), On the equivalence of Henstock-Kurzweil and restricted Denjoy integrals in K". Real
Anal. Exchange 15. 259-268.

The Henstock-Kurzweil integral
611
Chew. T.S. and Lee. P.Y. A990). The topology of the space of Denjoy integrable functions. Bull. Austral. Math.
Soc. 42C). 517-524.
Darmawijaya. S. and Lee, P.Y. A988). The controlled convergence theorem for the approximately continuous
integral of Burkill. Proc. of the Anal. Conf.. Singapore. 1986. North-Holland Math. Stud.. Vol. 150. North-
Holland, Amsterdam.
Davies. R.O. and Schuss, Z. A970). Aproof that Henstock's integral includes Lebesgue's, J. London Math. Soc.
2B). 561-562.
Denjoy, A. A912a), Une extension de I'integrale de M. Lebesgue, C. R. Acad. Sci. Paris 154. 859-862.
Denjoy, A. A912b). Calcul de la primitive de la fonction derivee la plus generate. С R. Acad. Sci. Paris 154,
1075-1078.
Denjoy, A. A941^49), Legns sur le calcul des coefficients d'une serie trigonome'trique. Hermann, Paris.
Di Piazza. L. A994). Weak convergence of Henstock integrable sequences. J. Math. Study 27 A), 148-153.
Di Piazza. L. B001). Variational measures in the theory of the integration in K'". Czechoslovak Math. J..
to appear.
Fleischer. I. A996-97). A Vitali-like convergence theorem for the Henstock integral. Real Anal. Exchange 22 A),
174-176.
Fejzic. H.. Freiling. С and Rinne. D. A993-94). Two-dimensional partitions. Real Anal. Exchange 19 B). 540-
546.
Freiling. C. Rinne. D. and Thomson. B.S. A997). A Riemann-type integral based on the second symmetric
derivative. J. London Math. Soc. 56. 539-556.
Fremlin. D.H. A994). The Henstock and McShane integrals of vector-valued functions. Illinois J. Math. 38 C).
471^479.
Fremlin, D.H. A995). The generalized McShane integral. Illinois J. Math. 39. 39-67.
Fremlin. D.H. and Mendoza. J. A994). On the integration of vector-valued functions. Illinois J. Math. 38 A).
127-147.
Fu. S. A993-94). The S-Henstock integration and the approximately strong Lusin condition. Real Anal. Exchange
19A). 312-316.
Gordon. R.A. A986-87). Equivalence of the generalized Riemann and restricted Denjoy integrals. Real Anal.
Exchange 12 B). 551-574.
Gordon. R.A. A989-90a). A descriptive characterization of the generalized Riemann integral. Real Anal.
Exchange 15. 397^*00.
Gordon. R.A. A989-90b). Another look at the convergence theorem for the Henstock integral. Real Anal.
Exchange 15 B). 724-728.
Gordon. R.A. A990). The McShane integral of Banach-valuedfunctions. Illinois J. Math. 34. 557-567.
Gordon. R.A. A990-91a). The inversion of approximate and dyadic derivatives using an extension of the
Henstock integral. Real Anal. Exchange 16 A). 154-168.
Gordon, R.A. A990-91b). Another approach to the controlled convergence theorem. Real Anal. Exchange 16.
306-310.
Gordon, R.A. A991), A general convergence theorem for nonabsolute integrals. J. London Math. Soc. 44. 301 —
309.
Gordon. R.A. A991-92). Riemann tails and the Lebesgue and Henstock integrals. Real Anal. Exchange 17. 789-
795.
Gordon, R.A. A992-93). On the equivalence of two convergence theorems for the Henstock integrals. Real Anal.
Exchange 18A). 261-266.
Gordon. R.A. A994). The Integrals of Lebesgue. Denjo\. Perron, and Henstock. Graduate Stud, in Math.. Vol. 4.
Amer. Math. Soc. Providence. RI.
Gordon. R.A. A994-95). Some comments on an approximately continuous Khintchine integral. Real Anal.
Exchange 20 B). 831-841.
Gordon, R.A. A995-96), An iterated limits theorem applied to the Henstock integrals. Real Anal. Exchange 21
B), 774-781.
Gordon. R.A. A997-98). Some comments on the McShane and Henstock integrals. Real Anal. Exchange 23 A).
329-342.
Henstock. R. A961). Definition of Riemann type of the variational integral. Proc. London Math. Soc. 11 C).
402^418.

612
В. Bongiorno
Henstock, R. A963). Theory of Integration. Butterworths. London.
Henstock, R. A968), A Riemann-tspe integral ofLebesgue power, Canad. J. Math. 20. 79-87.
Henstock. R. A968). Linear Analysis, Vol. 16. Butterworths. London.
Henstock. R. A969). Generalized integrals of vector-valued functions. Proc. London Math. Soc. 19 C). 509-536.
Henstock, R. A976). Integration in product spaces, including Wiener and Feynman integration. Proc. London
Math. Soc. 27. 317-344.
Henstock, R. A988). Lectures on the Theory of Integration. World Scientific. Singapore.
Henstock, R. A991). The general theory of integration. Oxford Mathematical Monographs. Clarendon Press.
Oxford.
James. R.D. A950), A generalized integral (II). Canad. J. Math. 2. 297-306.
Jarnik, J. and Kurzweil. J. A984). A nonabsolutelv convergent integral which admits С' -transformations, Casopis
Pest. Mat. 109B). 157-167.
Jarnik, J. and Kurzweil. J. A985). A nonabsolutelv convergent integral which admits transformation and can be
used for integration on manifolds, Czechoslovak Math. J. 35. 116-139.
Jarnik, J. and Kurzweil. J. A988). A new and more powerful concept of the PU-integral. Czechoslovak Math. J.
38. 8-48.
Jarnik, J. and Kurzweil. J. A995). Perron type integration on ?-dimensional intervals and its properties.
Czechoslovak Math, J. 45. 79-106.
Jarnik. J. and Kurzweil. J. A997), Another Perron type integration in ? dimensions as an extension of integration
of stepfunctions, Czechoslovak Math. J. 47 C). 557-575.
Jarnik. J.. Kurzweil, J. and Schwabik, S. A983). On Mawhin's approach to multiple nonabsolutely convergent
integrals, Casopis Pest. Mat. 108. 356-380.
Jurkat. W.B. A993). The divergence theorem and Perron integration with exceptional sets. Czechoslovak Math. J.
43. 27-45.
Jurkat. W.B. and Nonnenmacher. D.J.F. A994a). An axiomatic theory of non-absolutely convergent integrals
in R". Fund. Math. 145 C). 221-242.
Jurkat. W.B. and Nonnenmacher. D.J.F. A994b). A generalized ?-dimensional Riemann integral and the
divergence theorem with singularities. Acta Sci. Math. 59. 243-258.
Jurkat. W.B. and Nonnenmacher. D.J.F. A994c). A theory of non-absolutely convergent integrals in K" with
singularities on a regular boundary. Fund. Math. 146 A). 69-84.
Jurkat. W.B. and Nonnenmacher. D.J.F A995). The fundamental theorem for the v^-integral on more general
sets and a corresponding divergence theorem with singularities. Czechoslovak Math. J. 45 A). 69-77.
Jurkat. W.B. and Nonnenmacher. D.J.F. A995-96). A constructive definition of the ?-dimensional v(S)-integral
in terms of Riemann sums. Real Anal. Exchange 21 A). 216-235.
Kahane. J.-R A988). Une theory de Denjoy des martingales dyadiques. Enseign. Math. B) 34 C—4). 255-268.
Khintchine. A. A927). Recherches sur la structure des-functions mesurables. Fund. Math. 9. 212-279.
Kubota. J. A980). An elementary theory of the special Denjoy integral. Math. Japon. 24 E). 507-520.
Kurzweil. J. A957), Generalized ordinary differential equations and continuous dependence on a parameter.
Czechoslovak Math. J. 7. 418—446.
Kurzweil. J. A973). On Fubini theorem for generalized Perron integral. Czechoslovak Math. J. 23 (98). 286-297.
Kurzweil. J. A978). The Perron-Ward integral and related concepts. Appendix A in: Jacobs. K.. Measure and
Integral, Academic Press. New York.
Kurzweil. J. A980). Nichtalsolut konvergente Integrate, Teubner. Leipzig.
Kurzweil, J. A998). The set of primitives of the Henstock-Kurzweil integrable functions on a convergence vector
space, Tatra Mt. Math. Publ. 14. 29-37.
Kurzweil, J. and Jamfk. J. A987). On some extensions of the Perron integral on one-dimensional intervals. An
approach by integral sums fulfilling a svmmetrv condition, Funct. Approx. Comment. Math. 17. 49-55.
Kurzweil, J. and Jarnik. J. A990a), On a generalization of the Perron integral on one-dimensional inteivals, Ann.
Polon. Math. 51. 205-218.
Kurzweil. J. and Jarnik. J. A990b). The PU-integral: its definition and some basic properties. New Integrals.
Lecture Notes in Math., Vol. 1419. Springer-Verlag. Berlin.
Kurzweil. J. and Jarnik. J. A991). Equi-integrabilitv and controlled convergence of Perron-type integrable
functions. Real Anal. Exchange 17 A). 110-139.

The Henstock-Kurzweil integral
613
Kurzweil. J. and Jarnik. J. A992a). Differentiability and integrability in ? dimensions with respect to a-regular
intervals. Results Math. 21 A-2). 138-151.
Kurzweil. J. and Jarnik. J. A992b). Equivalent definitions of regular generalized Perron integral. Czechoslovak
Math. J. 42 B). 365-378.
Kurzweil. J. and Jarnik, J. A993), Generalized multidimensional Perron integral involving a new regularity
condition. Results Math. 23 C^4). 363-373.
Kurzweil. J.. Mawhin. J. and Pfeffer. W.F. A991). An integral defined by approximating BV partitions of unity,
Czechoslovak Math. J. 41. 695-712.
Kubota. Y.. A direct proof that the RC-integral is equivalent to the D*-integral, Proc. Amer. Math. Soc. 80.
293-296.
Leader. S. A992-93). Basic convergence principles for the Knrzweil-Henstock integral. Real Anal. Exchange 18
A). 95-114.
Leader. S. A993-94). Uniform Kurzweil-Henstock integrability. Real Anal. Exchange 19 A). 173-193.
Lebesgue. H. A902). Integrate, longueur, aire, Ann. Mat. Рига Appl. 7. 231-359.
Lee. P.Y. A989). Lanzhou Lectures on Henstock Integration, Series in Real Analysis. Vol. 2. World Scientific.
Singapore.
Lee. P.Y. A989-90). On ACG* functions. Real Anal. Exchange 15. 754-759.
Lee. P.Y. A990). Generalized convergence theorems for Denjov-Perron integrals. New Integrals. Lecture Notes
in Math.. Vol. 1419. Springer-Verlag. Berlin.
Lee. P.Y and Chew. T.S. A985). A better convergence theorem for Henstock integrals. Bull. London Math. Soc.
17. 557-564.
Lee. P.Y. and Chew. T.S. A987). Л short proof of the controlled convergence theorem for Henstock integrals. Bull.
London Math. Soc. 19. 60-62.
Lee. P.Y. and Ng. W.L. A996-97). The Radon-Nikodvm theorem for the Henstock integral in Euclidean spaces.
Real Anal. Exchange 22. 677-687.
Lee. P.Y. and Vybomy. R. A993). Kurzweil-Henstock integration and the strong Lusin condition, Boll. Univ.
Mat. Ital. 7D). 761-773.
Lee. P.Y. and Wittaya Naak-in A982). A direct proof that Henstock and Denjoy integrals are equivalent. Bull.
Malaysian Math. Soc. B) 5, 43^47.
Lee. T.Y. Chew. T.S. and Lee. P.Y. A996-97). On Henstock integrability in Euclidean spaces. Real Anal.
Exchange 22A). 382-389.
Liao. K.C. A987). A refinement of the controlled convergence theorem for Henstock integrals, SEA Bull. Math.
11.49-51.
Liao. K.C. A989). The weak controlled convergence theorem for Henstock integrals, SEA Bull. Math. 13 A).
15-18.
Liao. K.C. A992-93). On the descriptive definition of the Burkill approximately continuous integral. Real Anal.
Exchange 18. 253-260.
Liao. K.C. A993-94). Two types of absolute continuity used in Henstock and A P integration are equivalent. Real
Anal. Exchange 19A). 212-217.
Liao. K.C. and Chew. T.S. A993-94), The descriptive definitions and properties of the AP integral and their
application to the problem of controlled convergence. Real Anal. Exchange 19 A). 81-97.
Liu. G.Q. A987-88). The measurability of ? in Henstock integration. Real Anal. Exchange 13 B). 446-450.
Liu. G.Q. A992-93). On necessary conditions for Henstock integrabilin. Real Anal. Exchange B). 522-531.
Lu. S.R A996-97). Notes on the approximately continuous Henstock integrals. Real Anal. Exchange 22 A). 1-5.
Lu. S.P and Lee. P.Y. A990-91). Globally small Riemann sums and the Henstock integral. Real Anal. Exchange
16 B). 537-545.
Lu, J. and Lee. P.Y. A999). The primitives of Henstock integrable functions in Euclidean space. Bull. London
Math. Soc. 31. 173-180.
Lusin. N.N. A912). Sur lesproprietes de I'integral de M. Denjoy, C. R. Acad. Sci. Paris 155. 1475-1478.
Marcinkiewicz. J. and Zygmund, A. A936). On the differentiability of functions and the summability of
trigonometrical series. Fund. Math. 26. 1^4.4.
Mawhin. J. A981a). Generalized Riemann integrals and the divergence theorem for differentiable vector fields,
E.B. Christoffel. Birkhauser, Basel. 704-714.

614
В. Bongiomo
Mawhin, J. A981b), Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable
vector fields, Czechoslovak Math. J. 31, 614-632.
McLeod, R.M. A980), The Generalized Riemann Integral. Carus Mathematical Monograph. Vol. 20.
Mathematical Association of America.
McShane, E.J. A969), A Riemann-Tvpe Integral that Includes Lebesgue-Stieltjes, Bochner and Stochastic
Integrals. Amer. Math. Soc. Memoirs, No. 88. Amer. Math. Soc.. Providence. RI.
McShane, E.J. A973), A unified theory of integration. Amer. Math. Monthly 80. .349-359.
McShane, E.J. A983), Unified Integration. Academic Press. New York.
Muldowney, P. A987). A General Theorv of Integration in Function Spaces. Pitman Res. Notes in Math. Series.
Vol. 153, Longman Scientific and Technical, Harlow.
Nakanishi, S. A994), The Henstock integral for functions with values in nuclear spaces. Math. Japon. 39 B).
309-335.
Nonnenmacher. D.J.F. A994). A descriptive, additive modification of Mawhin's integral and the divergence
theorem with singularities. Ann. Polonici Math. 59 A). 85-98.
Nonnenmacher, D.J.F. A995). Sets of finite perimeter and the Gauss-Green theorem with singularities. J. London
Math. Soc. 52 B), 335-344.
Nonnenmacher, D.J.F. A995-96), A constructive definition of the ?-dimensional v(S)-integral in terms of
Riemann sums. Real Anal. Exchange 21 A). 216-235.
Ostaszewski, K.M. A986), Henstock Integration in the Plane. Mem. Amer. Math. Soc., No. 35.4, Amer. Math.
Soc. Providence, RI.
Perron, O. A914), tlber den Integralbegriff. Sitzber. Heidelberg Akad. Wi.ss.. Math. Naturw. Klasse Abt. A 16.
1-16.
Pfeffer, W.F. A986), The divergence theorem. Trans. Amer Math. Soc. 295 B). 665-685.
Pfeffer. W.F. A987), The multidimensional fundamental theorem of calculus. J. Australian Math. Soc. 43. 143-
170.
Pfeffer, W.F. A988). A note on the generalized Riemann integral. Proc. Amer. Math. Soc. 103 D), 1161-1166.
Pfeffer. W.F. A991a), A descriptive definition of a variational integral and applications. Indiana Univ. Math. J.
40, 259-270.
Pfeffer, W.F. A991b), The Gauss-Green theorem. Adv. Math. 87 A). 93-147.
Pfeffer, W.F. A992), A Riemann type definition of a variational integral. Proc. Amer. Math. Soc. 114. 99-106.
Pfeffer, W.F. A993), The Riemann Approach to Integration. Cambridge University Press, Cambridge.
Preiss, D. and Thomson, B.S. A989), The approximate symmetric integral. Canad. J. Math. 41 C). 508-555.
Pujie, W. A993-94), Equi-integrabilin and controlled convergence for the Henstock integral. Real Anal.
Exchange 19 A), 236-241.
Saks, S. A964), Theory of the Integral. Dover, New York.
Schurle. A.W. A986a). A function is Perron integrable if it has locally small Riemann sums, J. Austral. Math.
Soc. A 36, 220-232.
Schurle, A.W. A986b), A new property equivalent toLebesgue integrability. Proc. Amer. Math. Soc. 96. 103-106.
Skvortsov, V.A. A969). A certain generalization of Perron's integral. Vestnik Moskov. Univ. Ser. 1 Mat. Mekh.
(English translation: Moscow Univ. Math. Bull. 24. 48-51).
Skvortsov, V.A. A988). Some properties ofdvadic primitives. New Integrals. Lectures Notes in Math.. Vol. 1419.
Spnnger-Verlag, Berlin. 167-179.
Skvortsov, V.A. A992-93). On some questions of R. Gordon related to approximate and dyadic Henstock
integrals. Real Anal. Exchange 18 A). 267-269.
Skvortsov, V.A. A997). A variational measure and sufficient conditions for the differentiability of an additive
interval function. Vestnik Moskov. Univ. Ser. 1 Mat. Mekh. B). 55-57 (English translation: Moscow Univ.
Math. Bull. 52 B), 38-39).
Skvortsov, V.A. A998), Variation and variational measures in integration theory and some applications. J. Math.
Sci. (New York) 91 E). 3293-3322.
Skvortsov, V.A. and Solodov. A.P A998-99). A variational integral for Banctch-valued functions. Real Anal.
Exchange 24 B). 799-806.
Skvortsov. V.A. and Thomson. B.S. A995-96). Symmetric integrals do not have the Marcinkiewicz property. Real
Anal. Exchange 21 B), 510-520.

The Henstock-Kurzweil integral
615
Thomson. B.S. A971). Covering systems and derivatives in Henstock division spaces. J. London Math. Soc. B)
4. 103-108.
Thomson, B.S. A985). Real Functions. Lecture Notes in Mathematics. Vol. 1170. Springer-Verlag. Berlin.
Thomson. B.S. A994). Symmetric Properties of Real Functions. Monographs and Textbook in Pure and Appl.
Math.. Vol. 183. Marcel Dekker. New York.
Thomson, B.S. A998-99). Some properties of variational measures. Real Anal. Exchange 24 B). 845-854.
Wang. P.J. A993-94). Equi-integrability and controlled convergence for the Henstock integral. Real Anal.
Exchange 19 A). 236-241.

CHAPTER 14
Set-Valued Integration and Set-Valued Probability
Theory: An Overview
Christian Hess
Universite Paris Dauphine, Viabilite, Jeux, Contrdle, 75775 Paris Cedex, France
E-mail: hess@viab.ufniid.daupluiie.fr
Contents
1. Introduction 619
2. Notations and preliminaries 620
3. Integration of strongly measurable multifunctions 622
4. Weakly measurable multifunctions and graph-measurable multifunctions 629
5. The Aumann integral 634
6. The set-valued conditional expectation of closed valued multifunctions 639
7. Set-valued measures 643
8. The probability distribution of a measurable multifunction 647
9. Set-valued strong laws of large numbers 652
9.1. Convergence in the Hausdorff metric topology 652
9.2. Convergence in the sense of Painleve-Kuratowski 654
10. Set-valued martingales 658
11. Gaussian multifunctions. The set-valued Central Limit Theorem 660
12. Set-valued versions of the Fatou Lemma 662
13. Epigraphical convergence 663
14. Concluding remarks 665
References 666
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
617

This page intentionally left blank

Set-valued integration and .set-valued probability theory: An overi'iew
619
1. Introduction
The mathematical modelization based on multifunctions (alias set-valued maps,
correspondences, etc.) has shown its great adaptability and relevance for a long time. It allows one
to take into account the multiplicity of possible choices, the lack of information and/or
the uncertainty in a lot of situations ranging from Optimal Control to Economic Theory.
For example, this can be seen in the monographs by Hildenbrand [118], and Aubin and
Frankowska [16]. In particular, measurable multifunctions (alias set-valued random
variables, random sets, etc.) are of interest in Probability Theory and in Statistics, not only
from the theoretical point of view, but also for applications. Here, our aim is to present the
main results from the theory of measurable multifunctions. We will look more closely at
integration, conditional expectation and convergence theorems such as strong laws of large
numbers and martingales convergence theorems.
The notion of random set is intuitive and can be considered from different points of
view. It is possible to go back as far as the classical Buffon's needle problem (see, e.g.,
[125]) that implicitly involves this notion. In the twentieth century, the first papers dealing
with random sets seem to be those of Robbins [172,173] in the middle forties. Robbins
proved a celebrated formula related to the measure of a random set in an Euclidean space.
Later, Kudo [135] and Richter [171], motivated by Mathematical Statistics, examined
measurability and integration problems. The literature of the sixties and the early seventies
was more abundant. At this time, a great deal of works were inspired by problems
arising from Control Theory and Mathematical Economics. We can cite the papers by
Aumann [17], Debreu [65], Castaing [33], Castaing and Valadier [40], Van Cutsem [203-
205] and Valadier [199,200]. During the seventies and the next decades, the number of
works on the field increased rapidly. Let us mention, for example, the monograph by
Castaing and Valadier [41], and the paper by Hia'i and Umegaki [117]; both were published
in the late seventies and were highly influential. A lot of further references will be given in
the sequel, but one can also cite general monographs on multifunctions (not especially
devoted to integration theory), for example that by Klein and Thompson [130] in the
eighties, and that by Aubin and Frankowska [16] in the early nineties.
This paper is organized as follows. In Section 2, we collect some elementary facts
and introduce our notations. Section 3 deals with the integration of strongly measurable
multifunctions and, more particularly, with those that can be approximated by simple
measurable multifunctions in the sense of the Hausdorff distance. Furthermore, these
multifunctions are assumed to be integrably bounded, whence almost surely bounded
valued. In Section 4, we present weakly measurable multifunctions in connection with
graph-measurability, and we recall two versions of the Measurable Selection Theorem.
Section 5 is devoted to the Aumann integral, whose construction is based on integrable
selections and which has become the most popular set-valued integral. The Hia'i-Umegaki
set-valued conditional expectation is examined in Section 6. Set-valued measures (alias
multimeasures) are considered in Section 7; as we shall see, three distinct definitions can
be given. A few basic results on the probability distribution of measurable multifunctions
are presented in Section 8. Section 9 is devoted to set-valued strong laws of large
numbers; there, two convergence concepts are considered on the space of subsets,
namely the Hausdorff metric convergence and the Painleve-Kuratowski convergence. In

620
С. Hess
Section 10, we introduce the definition and the main properties of set-valued martingales.
In Section 11, we recall some basic facts about Gaussian multifunctions and the set-
valued Central Limit Theorem. Section 12 is devoted to set-valued versions of Fatou's
Lemma. Last but not least, we give a short introduction to epigraphical convergence, which
derives from set convergence and has appeared to be an interesting functional convergence.
There, we shall give applications in a stochastic context. Finally, a few concluding remarks
indicate other topics of interest in the field. They are followed by a substantial reference
list which shows how wealthy this domain is.
2. Notations and preliminaries
We consider a separable Banach space X. whose norm is denoted by || · || and Borel ?-field
by B(X). We define the following families of subsets of X
• 2X: the family of all subsets of X.
• Q: the family of open subsets of X,
• B{X): the Borel ?-field (or Borel ?-algebra) of X, generated by Q,
• C(X): the set of all closed subsets of X,
• CC(X): the set of all closed convex subsets of X.
• Сь(Х): the set of all closed bounded subsets of X.
• /C(X): the family of all nonempty compact subsets of X
• /CC(X)'· the family of nonempty compact convex subsets of X,
• /CW(X): the family of all nonempty weakly compact subsets of X.
The meaning of Ct>c(X) and ?*-с(Х) is clear. On 2X we consider the Minkowski addition,
denoted by "+" and the scalar multiplication, respectively defined by
C + C' = {x+x': ? eC. v'eC'}. aC = {ax: ? eC],
where С, С е 2X and a e R. These operations satisfy the following properties
aC + aC'=a(C + C), (a + /3)C с cC + /3C. B.1)
Easy examples show that the inclusion in B.1) is strict when С is not convex. When С is
convex, inclusion B.1) becomes an equality, namely
(a+fi)C = aC + pC. ?, ? e R.
Given С е 2х, the distance function d(-,C) and the support function s(-,C) of С are
respectively defined by
d(x,C) = mf{\\x-y\\: veC}. ? e X.
i(;y,C) = sup{ (у.*): лес}, ? e X*.
where (y,x) stands for the duality pairing and X* for the dual space of X. We denote
by clC (respectively, coC. coC) the closure (respectively, the convex hull, the closed

Set-valued integration and set-valued probability theory: An oveniew
621
convex hull) of C. Recall that the distance function (respectively, the support function)
characterizes a closed set (respectively, a closed convex set). Indeed, for every closed set С
(respectively, every closed convex set C), we have
C={x eX: d(x,C) = 0},
and
C= f]{xeX: (y.x)^siy.C)}. B.2)
vefl*
where B* denote the closed unit ball of X*. Equality B.2) is a consequence of the Hahn-
Banach Theorem.
The support function satisfies the following simple properties that are stated for easy
references.
s(yX) = s(y,coC), Ce2x.
s(y, C + C')= s(y, C) + s(y, C'). ? e X, С С е 2х \ {0}.
It is positively homogeneous and subadditive.
s(ay,C) = as(y,C), a > 0. ? e X, С е 2х. B.3)
s{y + y',C)iis(y.C) + s{y'.C), y.y'eX. Ce2x. B.4)
In particular, taking у' = —у in B.4) we obtain
s(y,C) + s(-y,C)^s@.C) = 0, Ce2x\{0). B.5)
For every pair (С, С) е Съ, we also define the Hausdorff distance between С and С by
/j(C, C')=inf{a >0: CCC'+afiandC'cC + fffi),
where ? denotes the closed unit ball of X. Equivalently, if the excess of С over C' is
defined by
e(C, C')=sup{d(x,C): .teCJ,
we have
AfC,C')=max{«(C,C').e(C',C)).
These formulas also makes sense when С and C' are closed sets, possibly unbounded,
but in this case, h is only a pseudo-distance in that it can take on the value +oo. The

622
С. Hess
topology generated by h is called the Hausdorff metric topology and is denoted by гн. For
every С е 2х \ {0} we set
||C|| = sup{||jc||: ? e C),
which is the Hausdorff distance between {0} and С Some other properties of the Hausdorff
distance are listed below for easy references. The proof can be found in [65, p. 362].
PROPOSITION 2.1. For every С, С', К, К' e Съ(Х) and a > 0 the following properties
hold
(a) h(c\(C + C), c\(K + K')) <; h(C, K) + h(C, K');
(b) h(aC,aK)=ah(C,K);
(c) h(coC,coK)<:h(C,K).
In particular, the maps (C. K) -^ cl(C + K), (a, C) -> <*C and С -> соС are
continuous in the Hausdorff metric topology on Сь(Х).
It can be shown that the space С endowed with the pseudo-distance h is complete, and
that Сь, Cc, & are closed subspacesofC (see, e.g., [41. Chapter II]). Using a lemma due to
Grothendieck (see, e.g., Lemma V.6 in [41]), the same property can be proved for /Cw.
Finally, we recall the Hormander formula (see, e.g.. Theorem 11.18 of [41]) which
expresses the Hausdorff distance of bounded convex sets in terms of support functions.
For every C,C e Ct>c one has
h(C,C')= sup \s(y, C)-s(y,C')|. B.6)
vefl*
The above equality admits several variants or special cases that will be useful. Because of
B.3), the supremum can also be restricted to
S={yeX*: \\y\\ = l].
the unit sphere of X*. In the special case where C' = {0}, it is readily seen by using B.5)
and B.6) that
||C|| =sups(yX).
yeB
3. Integration of strongly measurable multifunctions
Let (?, ?, ?) be a probability1 space. A multifunction is a map, defined on ?, whose
values are subsets of some given set. In this section, we shall restrict our attention to
the space Съ(Х) (or Сь) of closed bounded subsets of X, endowed with the topology гн
'We have choose this setting forsake of simplicity. In fact, a lot of results of this Handbook remain valid when
? is a finite or ? -finite positive measure.

Set-valued integration and set-valued probability theory: An oven'iew
623
generated by the Hausdorff distance h. Further, we consider the Borel ?-field В(Съ(Х), гц)
generated by the гн-ореп subsets of Съ(Х). A multifunction F-.? -> Сь(Х) is said to
be stmngly ?-measurable (or simply, strongly measurable) if, for every member W of
В(Съ, тц), one has F~l(W) e A. A more precise criterion is given in Proposition 3.3
below for the case where F takes on its values in a гн-separable subspace of Сь(Х).
Multifunctions that enjoy some measurability property are also called "random sets". The
proofs of the results of this section can be found in [110].
We begin by defining the set-valued integral or expectation of a simple multifunction,
i.e., of a multifunction F assuming only a finite number of values, More precisely, let
{? ?,,.,, Ak} be an Д-measurable partition of ? and let F: ? -> F (?) be the measurable
multifunction taking on the value К, е Съ for any ? e A, (/' = 1,..,,k). This can be
written
к
F = J24Ki·
i = \
F is called a simple multifunction. Clearly, F is strongly measurable. In this equality, 1д,
denotes the indicator function of A/, namely
? 1, if we Aj,
In this case, the expectation (or integral) of F is the member of Съ defined by
E(F) = cl
?>(Ai)*,-|, C.1)
where the sum refer to the Minkowski addition. When X is finite dimensional, the closure
operation is not necessary. The following lemma which is valid for a general metric
space allows for identifying the class of strongly measurable multifunctions that can be
approximated by a sequence of strongly measurable simple multifunctions.
LEMMA 3.1. // F-.? -> С is a closed valued multifunction, then the following two
statements (a) and (b) are equivalent.
(a) F is the pointwise limit on ? of a sequence of strongly measurable simple
multifunctions.
(b) F(i2) is a тц-separable subspace of С and, for every К e ?(?), the map ? ->
h(K, F(<w)) is measurable.
Recall that /C and /Cc are гн-separable subspaces of Сь- However, when X is infinite
dimensional, neither Сь nor Сьс is гн-separable. For example, consider the space X = I
of sequences (?„)„^ ? such that ??>\ ??" I converges and, for any subset / of N, the set of
positive integers, the subset
С ? =co{e„: ? e /}.

624
С. Hess
Recall that e„ denotes the sequence whose all terms are zero excepted the /zth equal to 1. It
is readily seen that if / and J are two distinct subsets of N, one has h (C/, Cj) > 1 · Since
the family
is uncountable, the nonseparability of Сы. follows. In particular, there exists strongly
measurable multifunctions that cannot be approximated by a sequence of strongly
measurable simple multifunctions.
Given a subspace С of Сы it is useful to introduce the following three classes (A), (B)
and (C) of multifunctions:
(A) ?' (С, Л) — the class of strongly Д-measurable multifunctions with values in С
such that
E||F|| < +oo,
where "E" stands for the expectation. Such multifunctions are said to be integrably
bounded;
(B) S(C, A) = the subclass of ?' (С. A) whose members are strongly Д-measurable
simple multifunctions;
(C) С1(С, A) = the subclass of ?' (С. Л) of those F that can be approximated by
simple multifunctions, i.e., such that one can find a sequence (F„)n^\ in S(C, Л)
verifying
lim h(F(cu),Fn(cu))=0 a.s.
Further, given F,G e ?' (С. Л) we set
/}(F,G) = E/?(F,G).
Similarly to the case of random vectors, it is easy to show that ? is a metric on
С1(Съ,Л), modulo the ?-almost sure equality. In other words, ? satisfies the following
three properties, for every F, G, ? in ?' (Съ- Л)
(i) /i(F, G) = 0 if and only if F(w) = G(w) a.s.
(ii) /i(F,G) = /i(G.F);
(iii) /i(F, G) ^ /i(F, ?) + ?(?. G).
In addition, if we denote by F' and G' two other multifunctions in С1(Съ-А), the
following three properties can be easily deduced from Proposition 2.1.
(iv) /i(cl(F + F'), cl(G + G')) < /i(F. G) + /i(F', G');
(v) 4(aF,aG)=a4(F,G),a>0;
(vi) /i(coF,coG)^/i(F, G).
Further, it can be proven that С1 (Съ- Л) endowed with ? is complete. More precisely, if
(Fn)n~z\ is a Cauchy sequence in (С\Съ-А). ?), i.e., if it satisfies
lim /}(F,„,F„) = 0.
/n.n—>зо

Set-valued integration and set-valued probability theory: An overview
625
then there exists a strongly measurable multifunction F such that
lim A(F,Fn)=0.
n—>oo
Moreover, F is unique up to a modification on a null set. and there exists a subsequence
(Fn(k))k>\ of (F„) such that
lim h(F, F„(jt)) =0 a.s.
The following lemma, inspired from Lemma V.2.4 of [157], shows that for every
гн-separable subspace С of Съ, every member of C](C ,A) can be approximated by a
sequence of strongly Д-measurable simple multifunctions. The limit exists almost surely
(in fact for every ? e ?) in the sense of the quasi-metric ?.
LEMMA 3.2. Let C' be a ??-separable subset ofCb(X). Then, for every F e ?' (C, A),
there exists at least a sequence (F„) of simple multifunctions in С (С, A), satisfying the
following three properties
(a) lim^oc h(F(w), F„(w)) = 0,Vwei2:
(b) ||F„(w)|| <; ||F(w)||, 4?&?:
(c) \imn^ooA(F,F„) = 0:
(d) Moreover, when the values of F are convex (respectively, compact, weakly
compact), the F„ can be chosen so as to be convex-valued (respectively, compact-
valued, weakly compact-valued).
It is interesting to mention the following characterization of С)}(Съ, A) in the space
Cl(Cb,A).
PROPOSITION 3.3. For every F e С](Съ-А), the following two statements are
equivalent:
(i) F is a member of C)} (Съ, A);
(ii) There exists a x\\-separable subspace C' ofCb such that F(a>) e С a.s.
Now, let us explain how to construct the integral of a strongly measurable multifunction
F :? -> С', where С is a rH-separable subspace of Cb- For the sake of simplicity, we
limit ourselves to the case of a convex valued multifunction, i.e., when С с Ссъ- We also
assume that С is гн-closed, and stable under the Minkowski addition and multiplication
by positive scalar. This is not restrictive, because the closed convex cone generated by a
??-separable subset of Сьс is still гн-separable. The difficulties encountered in the case of
a multifunction whose values are not convex are briefly discussed in Remark 3.8.
So, given a multifunction F: ? -» С', we define the map ? : S(C, A) -> С by
0(F) =E(F), C.2)
where E(F) is defined by C.1). The following result displays the main properties of the
map ? on S(C, A).

626
С. Hess
PROPOSITION 3.4. Let F and G be two strongly measurable simple multifunction with
values in C, i.e., members ofS(C, A), and let a ^ 0. Then, the map ? defined by C.2)
satisfies the following properties
(a) 0(F + G) = c\[0(F) + 0(G)},
(b) <P(aF) = a<P(F),
(c) h@(F), 0(G)) s: Eh(F, G), and, in particular,
|?(F)|^?(||F||),
(d) for every у e X, one has
i(y,E(F))=ET(v.F).
(e) if F с G a.s., then E(F) с E(G) (monotonicity property).
The next result shows that the integral (or expectation) can be defined for every F e
?,',(?', A), and that the properties listed in Proposition 3.4 still hold in this more general
framework.
THEOREM 3.5. The map ? : S(C, A) -> С defined by C.2) can be extended to a map ?
from ?' (C\ A) into C. It will be also denoted by ${F) = E(F), for F e ?' (C, A). This
map still enjoys properties (a) to (e) of Proposition 3.4. More precisely we have
(a) $(F + G) = c\[$(F) + $(G)},
(b) $(aF)=a$(F),
(c) h($(F), $(G)) < E/?(F, G), and, w particular,
|E(F)|^F||F||.
Moreover, the map ? still enjoys properties (d) and (e) of Proposition 3.4.
Remark 3.6. The above approach for defining the set-valued integral is explicit, in
that it starts with simple multifunctions and uses only elementary operations such as the
Minkowski addition and the scalar multiplication. It is also intrinsic in that it does not
involve any auxiliary space.
In spite of these nice features, it is interesting to sketch briefly an alternative approach
for constructing the set-valued conditional expectation for a compact convex multifunction.
This approach was adopted by Debreu [65]. Although of a more implicit and abstract
nature, it is shorter. Indeed, it is based on the possibility of embedding isometrically the
metric space (Сьо Ю. endowed with the Minkowski addition and the scalar multiplication,
in a Banach space. More precisely, let C(B*) denote the set of real-valued continuous
functions defined on ?*. Further, it is assumed that C(B*) is endowed with the uniform
norm || · ||u, i.e.,
IN|u = sup{|u;(;y)|: yefl*), weC(B*).

Set-valued integration and set-valued probability theory: An oven-iew
627
As already mentioned, we assume that Сы,- is endowed with the Hausdorff distance h. Then,
the map ? : Сы: -> C(B*) defined by
^@ = ?(·,0, СеСьс.
is an isometry from (Cbc, h) onto (<^(Сьс)· II ¦ llu)· This is an immediate consequence of the
Hormander formula. Moreover, ?@^) is a convex cone, and ? is an isomorphism with
respect to the convex cone structure. In other words, it satisfies
*(с(С) = а*(С), С.С еСьс- а>0.
? is called the Radstrom embedding. In addition, we have the implication
Ccc4W)^(C). C.C'eCbc·
If ? is restricted to a гн-separable subspace С of Cbc then ?@'), as well as c\(V(C) -
?(?')), the Banach subspace of C(B*) generated by V(C), are separable. Thus, it is
possible to define the integral of a random vector /: ? -> C(Z?*). This is similar to the
construction of the Bochner integral for random vectors. Let us recall that this integral
is appropriate when one has to deal with functions taking on their values in an infinite
dimensions Banach space (see, e.g., [68, p. 44]). Then, given an integrably bounded
multifunction F :? -> С', the integral of F. can be defined by simply setting
E(F) = «^-'(E(^(F))). C.3)
In Debreu's approach, the properties of the set-valued integral are inherited from those of
the integral of C(Z?*)-valued random vectors and from the properties of the map ?. Thus,
our approach is a little more general than that of Debreu, because Equation C.3) is not
required to define E(F), but is only a consequence of the definition.
Remark 3.7. When the dimension of X is greater than one, the convex cone ?(€^)
is infinite dimensional. Indeed, it contains an infinite sequence of linearly independent
vectors. For example, assume that X = R2 and consider the sequence (C„ )„ ^ ? of compact
convex subsets defined by
C„ = the polygon whose vertices are the points
of coordinates (cos2kn/n. пп2кл/n),
where к = 0,1,... ,n. It is easily checked that, for every ? > 1, the support functions
s(-, Ci), ·. -,s(-, C„), regarded as vectors in C(B*). are linearly independent.
Remark 3.8. Let us say some words about a possible definition of the integral of a
multifunction F: ? -> Съ(Х), whose values are not necessarily convex. For this purpose,

628
С. Hess
we recall that a member A of Л is called an atom of (? .?. ?) if ? (A) > 0 and if for every
В e A such that В с A, one has P(B) = 0 or P(A \ B) = 0. If ? ? and A2 are two distinct
atoms, then P(A\ П A3) = 0. It follows that the collection of atoms of a probability space
is finite or countable. Consequently, given a probability space (?, ?, ?), the set ? can be
split into the purely atomic part i?pa and the nonatomic part i?na (see, e.g., [118, p. 45]).
The purely atomic part ??? expresses as a finite or countable union of atoms A*, namely
If the multifunction F: ??? -> Сь(Х) is defined by
where Q e Сь (^ ^ 1), and if F is integrably bounded, i.e.,
||F||d/i = J]/i(A*)||C*|| < +oo,
then it is natural to define the integral of F over i2pa by
ЕA^) = ]ГМ(А,)Сч-. C.4)
where the series converges in the Hausdorff metric. It was shown in [110] that, for any
F e ?J,(Cb, A), the integral can be defined by
E(F) = d{E(lnp.F)+E(lnoacoFdM)},
where E(l^paF) is defined by C.4) and E(l^nacoF) is defined as in Theorem 3.5.
When X is finite dimensional, the closure operation is no longer necessary.
Remark 3.9. (i) Using the approach displayed in this section, several other properties
of the set-valued integral could be proved. For example, it would not be difficult to prove
set-valued analogs of the Dominated Convergence Theorem. On the other hand, the same
construction also allows for defining the conditional expectation of strongly measurable
multifunctions (see [110]). However, a other more general approach will be given in
Section 6.
(ii) Other approaches based on the Riemann integral were adopted by Hukuhara [122,
123], by McShane [149], and by Artstein and Burnes [8]. In Section 5, we shall present
the Aumann integral, whose definition involves selections and which even makes sense for
non integrably bounded multifunctions.
/,

Set-valued integration and set-valued probability theory: An overview
629
4. Weakly measurable multifunctions and graph-measurable multifunctions
This measurability concept is the most largely used, because it is not too strong, whence
easy to check. It allows for a lot of stability properties and for general results of existence
of measurable selections, especially in connection with the graph measurability that we
shall also present.
On 2X, we consider the Effros ?-field ? generated by the subsets U~ defined by
U~ = {Ce2x: СГШ^0},
where U ranges over Q, the set of open subsets of X. Clearly, the restriction of ? to X,
considered as a subset of 2X, coincides with B(X). The restriction of the Effros ?-field to
some subspace S of Iх will be denoted by ? («S).
We continue to denote by (?, A) an abstract measurable space. As already mentioned,
a map F from ? into Iх is also called a multifunction. The domain and the graph of F
are, respectively, defined by
dom(F) = {we ?: F(w) ^ 0} and Gr(F) = {(?. .?) e ? ? ?: ? e F(w)\.
F is said to be Effros measurable or simply measurable, if for every W in ?, F~](W) is
a member of A ("weakly measurable" in the terminology of Himmelberg [ 120]). From the
definition of the Effros ?-field, it follows that F is measurable if and only if, for any open
subset U of X, the subset
Г'(Г)=Ц«: F(w)ni/^0}.
is a member of A- Thesub-a-field F~'(?) generated by F is denoted by Ar-
Clearly, a multifunction F: ? -> 2х is measurable if and only if the multifunction cl(F)
is measurable. Further, if F is measurable then its domain is a member of A- This is a
consequence of the equality
dom(F) = F-'(X-).
A strongly measurable multifunction is measurable, which justifies the terminology. This
follows from the relationship
(F-1(i/-))c = F-'({CeC: Cct/e})
valid for all open sets U and from the гн-closedness of the set [С еС: С с t/c}, which
shows that F~'(i/~) is a member of В(Съ, тн) (in the above equality, the superscript
"c" stands for the complement operation). Conversely, if F is a weakly measurable
multifunction taking on its values in a rn-separable subspace С of Сь(Х), then it is
also B(C, TH)-measurable (see Remark 4.9(H)). On the other hand, there are measurable
multifunctions which are not strongly measurable (see, e.g., Example 3.4 in [ 117] or [23]).

630
С. Hess
The set of all measurable multifunctions with values in X is denoted by M(A, 2X)
or simply M{2X). The set of closed-valued measurable multifunctions is denoted by
M{C{X)).
A multifunction F: ? —» 2х is said to be graph-measurable if Gr(F) is a member of
the product ?-field A <g> B{X). The connections between Effros measurability and graph
measurability are given by Theorem 4.1 below. The proofs can be found in [41]. First, the
following notion from measure theory is needed. The ?-field of universally measurable
subsets, associated with this measurable space, is denoted by A and is defined as the
intersection of all ?-fields ?? (the ?-completion of A), where ? ranges over the set of all
positive bounded (or probability) measures on (?. A). Consequently, one has the inclusion
А с A and, when the ?-field A is ?-complete with respect to some fixed positive bounded
measure ?, one has the equality
For example, analytic subsets and coanalytic subsets of a complete separable metric space
are universally measurable (see, e.g., [67]).
Theorem 4.1.
(a) If a closed valued multifunction F is Effros measurable, then it is graph-measurable.
(b) Conversely, every graph-measurable multifunction F : ? —> 2х is Effros
measurable with respect to A, the ?-field of universally measurable subsets of(?, A).
(c) Consequently, if F is defined on a complete probability space (?. ?. ?), a closed
valued multifunction F is Effros measurable if and only if it is graph-measurable.
PROOF. Statement (a) is easily proved by relying on the equality
Gr(F)= {(?,?)?? ? X: d(x, F(w)) = 0}.
and by noting that the function (?. л) -> d(x.F(co)) is A ® ??(X)-measurable. The
proof of statement (b) is a consequence of the Projection Theorem (see the proof of
Theorem 4.6). ?
From Theorem 4.1(c), one can deduce a lot of stability properties for measurable
multifunctions. For example, we mention the following result concerning countable unions
and intersections.
THEOREM 4.2. Assume that (?. A. P) is complete. If (F„),i^i is a sequence of
measurable multifunction, then the multifunctions F and G defined by
F=\jF„ and G=f]F„.
are measurable.

Set-valued integration and set-valued probability theory: An overview
631
A crucial notion for working with measurable multifunctions is that of selection and of
Castaing representation.
Dehnition 4.3. Let F: ? -> 2х be a multifunction.
(i) A selection of F is a map / from ? into X such that /(?) e F(a>) for every
? e dom(F).
(ii) A Castaing representation is a sequence (/„) of measurable selections, such that
for every ? e dom(F), F(oj) is equal to the closure of the countable subset
{/„(?): ? 1} (see, e.g., [41. Chapter III] or [1741).
The following theorem provides two existence results of a measurable selection. It does
not involve the linear structure of X. The proof of part (a) can be found in, e.g., [41,124,
136,174,208,209]; that of part (b) is contained in [41, Chapter III].
Theorem 4.4.
(a) IfF e M(A, C(X)) and ifdom(F) is nonempty: then F admits at least one A-meas-
urable selection.
(b) If F is a graph-measurable multifunction with values in Iх and if Gr(F) is
nonempty, then F admits at least one ?-measurable selection.
In the above result, it can be noted that, when F is closed valued, it has at least one
Д-measurable selection. When no particular hypothesis is made on the values of F, one
can only assert the existence of an Д-measurable selection. As already mentioned, when
(?, A) is endowed with a positive finite measure ?, this entails the existence of one
?? -measurable selection.
As the next result shows, the measurability of a closed-valued multifunction can be
expressed in terms of distance functions and is connected with the notion of Castaing
representation.
THEOREM 4.5. If F is a closed valued multifunction, the following three statements are
equivalent
(a) F is Effros measurable, i.e., F_l (U~) e Afor all open subsets U of X.
(b) For every ? e X, the function d(.x. F(·)) is A-measurable.
(c) F admits a Castaing representation.
Equivalence (a) o· (b) is valid in a more general setting. Indeed, a multifunction F
whose values lie in a separable metric space is weakly measurable if and only if, for
every ? e X, the positive function d(x. F(·)) is measurable. This is a consequence of the
following equality
F-'(?(*,(*)) = {?&?: d(x.F(co)) <a\.
valid for all ? e X and a > 0 (where B(x, a) denotes the open ball of radius a centered
atjc).

632
С. Hess
THEOREM 4.6. If F is a closed valued multifunction defined on a complete probability
space (?,?,?), then each of the following statements is equivalent to anyone of
Theorem 4.5.
(d) F is graph measurable.
(e) F'BeAforall BeB(X).
(f) F~C e Afar all С eC(X).
PROOF. We only sketch the main implications. Implication (d) => (e) is an immediate
consequence of the Projection Theorem (Theorem 111.23 in [41]). Indeed, for every В е
B(X), the subset
F~ В = ???)? (Gr( F)A(fix8)),
is a member of A = A- As to the implication (b) => (d), simply observe that, for any
countable dense subset D of X, the following equality holds
Gr(F)= []{???: d(.x. F(a>)) = 0}.
Implication (f) => (a) follows from the fact that, in a metric space, every open subset U is
the countable union of closed sets, namely
u=\Jc„,
which yields
u- = {J c-. ?
Remark 4.7. Consider the property (g) below.
(g) F~K eA for all К е /C(X).
When X is a metric space the implications (f) => (a) => (g) hold. The three properties
are equivalent when X is ?-compact (i.e., a countable union of compact sets), especially
when X is a finite dimensional Banach space (see, e.g., [120] or [41]). Further results on
the measurability of multifunctions and counter-examples can be found in [121].
One of the advantages of considering the Effros ?-field in order to define the
measurability of a multifunction is that this ?-field is equal to the Borel ?-field of
a separable metrizable topology, namely the Wijsman topology. This topology was
introduced by Wijsman [213,214] on the space of closed convex subsets in an Euclidean
space, but it also makes sense in a general metric space (X, d). In this setting, the Wijsman
topology is defined on C(X) as the weakest topology rWu/) determined by the family
{d(x, ·): ? & X). Equivalently, rw(f/) is the topology of pointwise convergence of distance
functions on X.

Set-valued integration and set-valued probability theory: An overview
633
For every С е C(X), the distance function d(-,C) is Lipschitz continuous (with
Lipschitz constant one). It follows that, when the metric space (X, d) is separable, the
Wijsman topology is metrizable and separable (see [101]). Further, Beer [26] proved that
when (X, d) is Polish2 (C(X), rW(j,) is Polish too. The following result displays the Borel
structure of the Effros ?-field. The proof can be found in [101, Proposition 3.1.1] or [27,
Theorem 6.5.14]).
THEOREM 4.8. If(X, d) is a separable metric space, then, one has
?(C(X))=B(C(X)
Remark 4.9.
(i) The equality of Theorem 4.8 is no longer true when the Wijsman topology is
replaced with the Hausdorff metric topology гн- In general, the Effros ?-field is
strictly included in B(C(X), ??) (see [23]).
(ii) However, when we restrict to the space IC(X) of compact subsets of X, we have the
equality
?(?(*)) =S(/C(X),th).
The proof can be found, e.g., in [27,41,177]. More generally, according to a remark in
the beginning of the present section, if С is a тн-separable subspace of C(X), then one has
E(C')=B(C',tv/{j))=B(C',th).
This follows from the second countability of the restriction of гн to С and from the fact
that the /г-closed balls are members of the Effros ? -field. In turn, the latter is a consequence
of the equality
h(C,C') = sup\d(x,C)-d(x,C')\,
valid for every pair (C, C') e C(XJ, where D denotes a countable dense subset of X.
There are a lot of specific situations where one wants to find a measurable selection. For
example, a result which can be useful in many situations, for example in Control Theory,
is the so-called Measurable Implicit Function Theorem (Theorem III.38 in [41]).
THEOREM 4.10. We consider another measurable space (T.T), a Suslin topological
space ? (i.e., the continuous image of a Polish space), a multifunction F: ? -> 2Y such
that Gt(F) eA<S>B(Y),a multifunction G : ? -> 2"'" such that Gr(G) e ? ? ? and a map
g-.???^? which is measurable with respect to ? ? ?(?) and T. If it is assumed that
g(w, F(w))nG(w)^0, 4???,
-A topological space is said to be Polish if it is separable, metrizable and complete for a suitable metric.

634
С. Hess
then there exists a map f: ? —> ?, whose graph is A <8> ? (?) -measurable, which is a
selection of F and which satisfies
g(w,f(w))eG(w). VweC.
Remark 4.11. (i) Another useful notion for closed convex valued multifunctions is that
of scalar measurability. A multifunction F : ? -> CL- is said to be scalarly measurable if for
every у e X*, the function s(y, F(·)) is measurable. It is not hard to show that a measurable
multifunction is scalarly measurable. Indeed, for every у е X* and every real a, one has
{C eCc: s(y,C)>a} = W~.
where W = {x e X: (y, x) > a). The converse implication does not hold in general, but
it holds for the class of countably supported multifunctions (see Definition 5.5 below)
satisfying a suitable analyticity property. The reader may consult [24] for further details
and [23] for the connections with strong measurability.
(ii) Let us also mention the works of Leese [141,142] who presented another approach
of measurable multifunction, involving the Suslin operation and Suslin spaces. The case
of multifunctions whose values lie in a nonseparable Banach space was considered by
Barcenas and Urbina [25].
5. The Aumann integral
Consider a probability space (?. ?, ?) and a sub-a-field В of A. By ?\?, ?, ?; X) we
denote the space of all (classes of) measurable functions from (?. B) into (X.B(X)). For
every F e M{C(X)), we also define
S(F, B)={fe L°(i2. ?, ?: ?): /(?) e F(co).
for ?-almost every ? e dom(F)}.
It is known that L°(i2, ?, ?; ?) (= L°(X)) endowed with the topology of convergence in
probability is a metrizable topological vector space, provided one identify two functions
that coincide ?-almost surely. Since a sequence converging in probability admits an almost
surely converging subsequence, it is clear that, for any sub-a-field В of Л, the set S(F, B)
is closed in L°(i2, ?, ?; X). We denote by ?(?,?,?.?) (or L](X) for short) the
subspace of L°(X), whose members are Bochner integrable. Given a sub-?-field В of A
and a multifunction F, we define the following L1 (X)-closed subset of ^(?, ?, ?; ?)
S](F,B)= {/??\?,?,?:?): f(w)e F(w),
for ?-almost every ? e dom(F)}.
We begin by recalling a result on the measurability of infimums (respectively, supremums).

Set-valued integration and set-valued probability theory: An overview
635
PROPOSITION 5.1. Let F eM(C(X)) and ?: ? ? ? -> Ш be an A® B(X)-measurable
function. If for every ? e ?, ?(?. ·) is upper semicontinuous, then the function тф defined
by
? -» ???(?) = \??{?(?, ?): ? e F(a>)}.
is measurable. On the other hand, if?(?.¦) is lower semicontinuous, >Пф is A-measurable.
A similar result can be stated for the supremum by replacing ? with —?.
If ?: ? ? ? -> R is A® ??(X)-measurable then we can define the integral functional
^опЬ\Х)Ъу
Mf)= f ?{?,/(?))??, feC](X).
The following result allows for interchanging the infimum and the integration operations,
and plays a crucial role in set-valued integration. Its proof can be found in [117,
Theorem 2.2].
PROPOSITION 5.2. Let ?: ? ? ? -> R be as in Proposition 5.1. Then one has
inf{70(/): /eS'(F)}= [ тф{Ш)а^х.
JQ
Applying Theorem 5.2 to the function (?, ?) -» ? (?, ?) = \\?\\, it is not hard to see that
when the following inclusions hold
AF с В с А,
the set S] (F, ?) is nonempty if and only if the positive function d@, F(·)) is integrable.
In such a situation, we say that the multifunction F is integrable. Observe that when F is
integrable, one has
M(dom(F)) = 1.
On the other hand, F is said to be integrably bounded if the function
? -> /?(F(o>)) = sup{ll-x||: xeF(a))} (also denoted by ||F(?)||)
is integrable. In this case, every measurable selection of F is integrable, so that one has
S](F,A) = S(F,A).
Further, the values of F are almost surely bounded.

636
С. Hess
The following notion of integral for multifunctions was introduced by Aumann [17].
For any measurable multifunction F and any sub-?-field В of A, the set-valued (Aumann)
integral of F over ?, with respect to B, is denoted by I(F, B) and is defined by
I(F,B) = U /??: feSl(F.B)\.
I(F,A) is simply denoted by 1(F); it is nonempty if and only if F is integrable. The
integral of F over A e A is denoted by I a (F).
On the other hand, F is said to be Aumann-Pettis integrable if it admits at least a Pettis
integrable selection. For the definition and elementary properties of the Pettis integral, the
reader may consult [68, p. 52] and [155,156]. Let us only recall that in finite dimensional
spaces the Pettis integral coincide with the usual integral, whereas in infinite dimensions
there exist Pettis integrable functions that are not Bochner integrable.
Example 5.3. An integrably bounded multifunction is integrable, whence Aumann-
Pettis integrable, but the converse implication is false. Indeed, consider the measurable
multifunction F defined by
F(w) = ?@, r(o))) = the closed ball of radius r(w) centered at the origin,
where r : ? -> @, +oo) is a given nonintegrable measurable function. In this case, the
Aumann integral (see below) of F over ? is equal to X.
Here are some properties of the Aumann integral, whose proofs can be found in [ 17].
THEOREM 5.4. Let F,G e M(C(X)). If we assume in addition that F and G are
integrable, then the following properties hold.
(a) I(c\(F + G)) = d(I(F)+I(G)).
(b) cl/(coF) = co/(F).
(c) ?/(?,?, ?) is nonatomic, then c\I(F) is convex and one has
c\ I (co F)=c\ 1(F).
(d) //(?, ?, ?) is nonatomic and if X is finite dimensional, then 1(F) is convex and
I(F) = I(coF).
Moreover, if F is integrably bounded, 1(F) is compact.
(e) For all у е X* one has
s(y,I(F))= [ s(y,F)d?.
JQ

Set-valued integration and set-valued probability theory: An over\iew
637
(f) If F and G are integrably bounded then the following inequality holds
fc(cl/(F),cl/(G))< f h(F,G)dp.
(g) If F is integrably bounded and takes on its values in a тц-separable subspace of
Със(Х), then one has
E(F) = cl/(F)
where E(F) denotes the integral as defined in Section 3 (Theorem 3.5).
The first part of property (c) is a consequence of the Lyapunov Convexity Theorem
(see [117, Theorem 4.2]). The first part of statement (d) was proved by Richter [171]. The
proof of statement (g) can be found in [ 117 ]. The others properties are due to Aumann [17].
Other properties related to statement (b) will be given in Theorem 5.9.
Theorem 5.6 below concerns the class of countably supported multifunctions, whose
definition is recalled hereafter.
DEFINITION 5.5. A multifunction F-.? -> CC(X) is said to be countably supported if
one can find a countable subset D of B* such that
F(w) = P| {x e X: (y,x) < s(y, F(co))}, for all ?? ?.
If we want to emphasize the role of D, we say that F is D-countably supported. For
example, if ? is a topology on X, compatible with the duality pairing, every multifunction
whose values are weakly compact and convex is D-countably supported for every ?-
dense subset of B*. However, the class of countably supported multifunctions is much
wider (see [24]). In particular, a countably supported multifunction may have unbounded
values. On the other hand, every member F of ?/'(Cc. Л) is countably supported with
respect to a countable subset of X*. generally depending on F (Theorem 3.6 in [23] and
Proposition 2.4(g) in [24]).
The first part of the following theorem characterizes integrable selections of a countably
supported multifunction, the second part provides a criterion for two such multifunctions
to be almost surely equal. Theorem 5.6 remains valid for Aumann-Pettis integrable
multifunctions (see [78]).
THEOREM 5.6. Let F be a measurable, integrable, closed convex valued multifunction
and f eL'(X).
(a) If F is countably supported then feS](F)ifandonly if
fd^?c\IA(F). for all A e A-
I

638
С. Hess
(b) Let G be another integrable, closed convex valued multifunction. If G is countably
supported then F(co) = G (?) for almost every we ? if and only if
clIA(F) = c\IA(G). for all A e A.
The set-valued Aumann integral is not always closed. Thus, it is interesting to indicate
situations where it is. This problem has been considered by several authors, such as
Castaing [33,37], Hia'i and Umegaki [117], Byrne [31] and KM [128] among others.
THEOREM 5.7. If F e M(C(X)) is integrably bounded and convex weakly compact
valued then 1(F) is weakly compact.
PROOF. Clearly, the set 51 (F) of integrable selections of F is L1 (X)-closed and convex.
By Theorem 3.6(ii) of [129]. S'tF) is also weakly relatively compact, whence weakly
compact. Further, since the map ?-.f —* §Qfdpi from L](X) into X is linear and
strongly continuous, it is also continuous with respect to the weak topologies. The desired
conclusion is obtained by noting that 1(F) = ^(S'(F)). ?
Now, we turn to an interesting question: given a subset 5 of L' (X), is it possible to find a
necessary and sufficient condition for the existence of an integrable multifunction F such
that S = S*(F)? The answer is provided by Theorem 5.8 below, which is Theorem 3.1
of [117]. Recall that a subset S с L](X) is said to be decomposable (with respect to A), if
for every A e A and every /. g e S. t.\f + 1дс? is a member of S.
THEOREM 5.8. Let S be a nonempty closed subset of ZJ(X). Then, there exists
an integrable multifunction F e M(C(X)) such that S = S](F) if and only if S is
decomposable.
The following result addresses the problem of interchanging the set-valued integration
and the convex hull in finite dimensions.
THEOREM 5.9. Assume that X = W1 and consider a multifunction F, whose values lies
in R'l, the positive cone of X. IfGr(F) e A<S>B(X), then, one has
со / Fd\x= I coFi^.
Remark 5.10. (i) The equality of Theorem 5.9 was proved by Aumann [17] and was
extended later by Wagner [210]. Another proof based on the theory of Young measures
was given by Balder [20]. On the other hand, the case of multifunctions with a countable
range was examined by Khan and Sun [126].
(ii) It is interesting to mention another area of research concerning the integration
of both, single and set-valued functions, namely integration of random variables taking
on their values in a metric space with no linear structure. This can be traced back
to Frechet [84] and was taken further by S. Doss [72.73], Benes [28]. Herer [90-93],

Set-valued integration and set-valued probability theory: An oven'iew
639
H. Doss [71], Raynaud de Fitte [170], Es-Sahib and H. Heinich [79], and Hess [1121.
In some of these works, one can find extensions of the strong law of large numbers and of
martingales convergence theorems.
(iii) Let us mention a result that has applications in Control Theory and Optimization.
Given a measurable multifunction G defined on a product space ? ? ?. the integral over ?
is a multifunction depending on the second variable, namely
?-.t -> / ??,?)?(??).
In this situation, one may look for a measurable selection ? of the multifunction ?, which
has the following form
?-.t -> / ?(?,?)?(??).
where g is a suitable measurable selection of G. This question was raised and solved
by Artstein [6] when X is finite dimensional. An infinite dimensional extension was
established recently by Saint-Pierre and Sajid [178].
(iv) A notion of variance for random sets was introduced by Kruse [ 134] by means of
selections.
6. The set-valued conditional expectation of closed valued multifunctions
The construction of the set-valued expectation is a natural extension of the multivalued
integral, but it is more delicate. It was examined by several authors. Let us mention Van
Cutsem [203-205], Neveu [158], Bismut [30], Dynkin and Evstigneev [77], Castaing and
Valadier [41], Hia'i and Umegaki [117]. Hia'i [114] and Papageorgiou [163]. Given a sub-
?-field В of A, and a integrable Д-measurable multifunction F. Hia'i and Umegaki [117]
showed the existence of a ?-measurable integrable1 multifunction G such that
S\G,B) = c\{E(f\B): /eX'(FJ)},
the closure being taken in L1 (?; X). The multifunction G is the (set-valued) conditional
expectation of F relatively to В and is denoted by E(F\B). Clearly, it is defined up to a
?-null set. The existence of G is an immediate consequence of Theorem 5.8. Indeed, it is
easy to check that the set
(?(/|?): feS\F,A)}.
is decomposable with respect to the sub-a-field В and that the L1 (X)-closure of a
decomposable set is still decomposable.
¦^In fact, Hia'i and Umegaki assumed that F is integrably bounded, but their proofs easily extend to the integrable
case (see [114]).

640
С. Hess
One may observe that the above definition is global and only defines E( F\B) by mean of
the set of its ?-measurable integrable selections. Thus, it is desirable to provide examples
where a 'pointwise' characterization is available, i.e.. a formula allowing one to calculate
E(F\B)(w) for almost every ? e ?. Several examples of that type are presented hereafter.
The more general ones are examples (e) and (f)-
Example 6.1.
(a) If the multifunction F reduces to a single-valued function /:? -> X, the
conditional expectation of F clearly coincides with the ordinary conditional
expectation for vector-valued random variables.
(b) If ? = {?, 0}, then for any integrable multifunction F we have
E(F\B)(co) = c\I(F) a.s.
(c) Let r be a real positive integrable function and /: ? -> X be a vector-valued
integrable function. If the multifunction F is defined by
F(w) = B(f(co),r(co)).
then its conditional expectation with respect to В is given by
E(F\B)(w) = ?(?(/|?)(?), ?(?·|?)(?)).
This can be seen by using Theorem 6.2(c) below.
(d) Let и be a real-valued integrable function and у е X. If we define the multifunction
Fby
F(co) = {xeX: (y,x)=v(w)},
then
E(F\B)(w) = {x e X: (y, ?) = ?(?|?)(?)}.
(e) Assume that X* is strongly separable and that F is an Д-measurable integrable
multifunction whose values lie in Съ^Х)- Then, the following equality holds
(see [117, Theorem 5.5])
E(F\B)(w)= f]{.xeX: (y.x) < E(j(v. F)\B)(w)}.
A similar equality holds when F takes on its values in /CWC(X) and when D is
a countable Mackey dense4 subset D of X*. It is proved by replacing the strong
topology on X* by the Mackey topology.
4Such a countable subset exists in X* as soon as X is separable. For the definition and the basic properties of
the Mackey topology, see, e.g.. [184. p. 1311.

Set-valued integration and set-valued probabilin theory: An overview
641
(f) Assume now that there exists a regular conditional probability distribution of ?
given B, i.e., a map Q: ? ? ? -> [0, 1] satisfying
(i) for every В e B, the map ? -> <2 (?|?) is Д-measurable,
(ii) for every we ?, the map В -> Q (?\?) is a probability measure on (?. B),
(iii) VAe AVBefi, ?(???) = /4 <2(?|?)?(??).
In this case, it was shown by Valadier [201, Theoreme 1] that the set-valued
conditional expectation is given by
E(F|B)(fl>) = cl{J f($)Q(d$\w): feS\F,A)\ a.s.
We end this section by recalling some basic properties of the set-valued conditional
expectation, which are natural extensions of those of real-valued or vector-valued random
variables. The proofs can be found in [117] for the integrably bounded case and in [114]
for the integrable case. As the reader will see, the classical properties of the conditional
expectation are recovered only for convex valued multifunctions.
THEOREM 6.2. If F and G are two integrable multifunctions with closed values in X,
and В is a sub-?-field of A, then the following properties holds.
(a) E(cl(F + G)\B) = cl(E(F|B) + E(G|S)) a.s.
(b) Ifr is a real В-measurable function such that rF is integrable, then
E(rF\B) = rE(F\B) a.s.
(c) If g is a bounded scalarly B-measurable5 function from ? into X* then
s(g,E(F\B)) = E(s(g,F)\B) a.s.
where s(-, C) denotes the support function of the subset С (see Section 2).
(d) E(coF\B)=coE(F\B)a.s.
(e) If F and G are integrably bounded, then
A(E(F\B),E(G\B))^A(F.G).
In other words, the map F —> E(F|??) is nonexpansive from С (Съ, A) onto
C\Cb,B).
(f) If F c, G a.s., then
E(F|#) с E(G\B) a.s. (monotoniciry property).
In the following theorem, the values of the multifunction F are required to be convex.
Thai is, for every ? e X, the function ? —> (g(a>).x) is ?-measurable.

642
С. Hess
THEOREM 6.3. Let F be an ?-measurable integrable multifunction, with values in Cc.
(a) IfB\ ? ? ? A, thenE(E(F\B)\B\) = E(F\B\).
(b) If F is B-measurable and if r is an ?-measurable positive bounded function such
that rF is integrable, then
E(rF\B) = E(r\B)F a.s.
In particular, E(F\B) = F.
(c) If F is B-measurable and if В] СбсД then the conditional expectation of F
relative to B\ taken on the base space (?.?.?) is equal to the conditional
expectation of F relative to B\ taken on (?. ?. ?).
Remark 6.4. (i) The assumption of r being positive cannot be removed in
Theorem 6.3(b) as the following example shows. Consider the measure space ([0. 1), ?, ?),
where ? is the Lebesgue measure, В the trivial ?-field [?, 0} and F the constant
multifunction defined by F(a>) =[-1,1] (here X = R). Also consider the function r such
that
r(w) = -l if0^w<l/2 and r(w)=l. if 1/2 ?? ? <1.
Then E(rF\B) = F, but E(r\B)F = {0}.
(ii) If F is not assumed to be convex valued, properties (b) and (c) of Theorem 6.3 are no
longer true. Indeed, assume that X = R and that (?.?. ?) is nonatomic. Let ? = {?, 0}
and F be the constant multifunction such that F(co) = {0. 1), ? e ?. Theorem 5.4(c)
shows that
E(F|?) = cl/(F) = [0. 1].
so that the equality E(F|??) = F does not hold.
In view of the next result, we recall that for every G e MBX) and every sub-?-field ?
of Л, we have set (see Section 5)
h(G,T)= J f fdix: /6S'(G.f) ¦ BeT.
THEOREM 6.5. Let F be an integrable multifunction with closed values.
(a) For every В е В one has
c\IB(E(F\B).B)=c\IB(F.A).
(b) If F is convex valued, then for every В е В one has
c\IB{E(F\B),A)=c\IB(F.A).

Set-valued integration and set-valued probability theory: An over\'iew
643
(c) If F is a member of C[(C^.A) (see Proposition 3.3), then E(F\A) is uniquely
determined as a member of L\ (С, B) satisfying the equality in (b)for every В eB.
Here is an extension of Theorem 5.5 of [ 117] and of our Example 6.1(e). The proof is
similar, but is based on our Theorem 5.4.
THEOREM 6.6. Let D be a countable subset of B* and F : ? -* CC(X) be a D-countably
supported, integrably bounded multifunction. Then one has, for almost all ? e ?,
E(F\B)(w) = p| {x e X: (y,x) <; E(i(y. F(c))\B)}.
veD
Remark 6.7. The multivalued conditional expectation is closely connected with the
conditional expectation of random variables depending on a parameter. This problem is
of great importance in many applications. See, e.g., [41,43,81,201,202,204].
7. Set-valued measures
The theory of set-valued measures (we shall rather use the synonym "multimeasures"),
has been developed by Vind [206], Schmeidler [186], Debreu and Schmeidler [66], Art-
stein [2], Godet-Thobie [86-88], Coste [48-50], Pallu de la Barriere [162], Drewnowski
[74], Coste and Pallu de la Barriere [52-55], Hiai [113,114], Thiam [193-196], Papageor-
giou [164] and Klei" [127] among others. One of the motivations was the applications to
Mathematical Economics or to Statistics.
Basically, multimeasures are measures whose values are subsets of some Banach space
(or, more generally, topological vector space). As we shall see, there are at least three
possible definitions depending on the series summability concept considered in the space
of closed sets.
We first recall that a sequence (x„ )„ ;> ? in X is said to be unconditionally convergent
if for every one to one map ? from N onto itself the series ?,^\ -*0<»> 's convergent.
An analogous definition can be given for a sequence (C„) in Със(Х) endowed with the
Hausdorff distance. Given a sequence (C„)„^i in Iх, the infinite sum ?,?>\ С» is defined
by
J]C„ = {* e ?: ? = ??"· x" e Ся. О 11. G.1)
where the convergence of the series ?]„> | ·*„ is unconditional.
Further, recall that a map ?: A -> Iх \ {0} is said to be additive if for every pair (?, ?)
of disjoint members of A one has
M(ADB)=c\(M(A) + M(B)).

644
С. Hess
where the "+" sign stands for the Minkowski addition. When the values of ? are weakly
compact, the closure operation is no longer necessary. This definition can be extended to a
pairwise disjoint family of members of A. The first definition of a multimeasure uses the
Hausdorff distance on the space of closed sets.
DEFINITION 7.1. An additive map ? : A -> C(X) is called an h-multimeasure if ?@) =
{0}, and if for every sequence (A„)„^ ? of pairwise disjoint sets in A, we have
Дт^ UMjA„.cl^M(A,) =0.
The second definition refers to the summability notion introduced by equality G.1).
DEFINITION 7.2. An additive map ?: A -> 2х \ {0} is called a strong multimeasure if
??@) = {0}, and for every sequence (A„)„^| in A of pairwise disjoint sets, we have
mI\JaA=c\J2m(A„).
The third definition involves support functions.
DEFINITION 7.3. An additive map ? : A -> 2х \ {0} is called a weak multimeasure if for
every у & X*, the map A -> s(y, M(A)) is an extended real-valued measure.
Clearly, if ?: A -» 2X is a weak multimeasure, then so are the maps M' =
c\M and M" = coM. The same property is valid for /г-multimeasures (for M", use
Proposition 2.1(c)).
The main connections between the above three definitions are provided in the next
proposition.
Proposition 7.4.
(a) If M.A-* Съ(Х) is an h-multimeasure, then it is also a strong multimeasure.
(b) IfM.A—* Съ(Х) is a strong multimeasure, then it is also a weak multimeasure.
(c) IfM.A—* /CWC(X) is a weak multimeasure. then it is also an h-multimeasure. Thus,
in this case the three notions of multimeasure coincide.
Proposition 7.4(a) follows easily from the following result which provides another
formulation of the countable summability for /г-multimeasures. It is due to Drewnowski [74].
PROPOSITION 7.5. //(C„)„^i is a sequence in Съ(Х), then the following two statements
are equivalent.
(i) The series (C„) is unconditionally convergent in Съ(Х) endowed with the Hausdorff
distance.

Set-valued integration and set-valued probability theory: An overview
645
(ii) For every sequence (x„)„^ \ in X such that x„ e C„ for all ? ^ 1, the series (*,,) is
unconditionally convergent.
Moreover, the following equality holds
cl ]T C„ = ell ??": x" e C» \
Let ? : A -> Iх \ {0} be a strong multimeasure. For every A e Л we define V^(A) by
VM(A) = sup
]Г||М(А,)||
I /=1
where the supremum is taken over all finite Д-measurable partitions {? ?,..., A„} of A.
From Proposition 1.1 of [113], it is known that Vm is a positive measure. If Vm(&) < +°o,
then ? is said to be of bounded variation.
Example 7.6. Let us give an example of strong multimeasure. For this purpose, consider
Q e Със(Х) and the collection S of vector measures m : A -> X such that for all m e S
and A e A, one has m(A) e ?(?)?). Now, define the map ?: A -> 2? \ {0} by
??(?) =
y^w,(A,): w, e5, A, e Л (pairwisedisjoint), ? ^ 1 >.
I i = l J
It is not difficult to check that ? is additive and that it is a strong multimeasure.
Example 7.7. A simple example of /г-multimeasure can be given in the following way.
Let F e М(Съ(Х)). Assume that F is integrably bounded and define ?: A -> Съ(Х) by
M(A) = c\IA(F). AeA.
where Ia(F) denotes the Aumann integral of F over A (see Section 5).
Example 7.8. If in Example 7.7 the multifunction F is no longer assumed to be
integrably bounded, but only Aumann-Pettis integrable (i.e., the set of its Pettis integrable
selections is nonempty), then the map ? is a weak multimeasure, but not necessarily
an /г-multimeasure. Indeed, as Example 5.3 shows, there exist Aumann-Pettis integrable
multifunctions that are not integrably bounded.
Further, ? is said to be ?-continuous if ?(?) = 0 implies ? (A) = {0}.
Given a weak multimeasure ?: A -> Iх \ {0}, a selection of ? is a vector measure
m : A -> X such that m{A) e ? (A) for every A e A. The set of all selections of ? is
denoted by <S,w· The multimeasure ? is said to be rich (of selections) if
M(A) = cl{w(A): m eSM\,

646
С. Hess
for all A e A. In this case, the multimeasure can be recovered from its selections. The
following result is due to Coste [48].
THEOREM 7.9. Let M:A-* Съ(Х) be a strong multimeasure.
(a) If A is countably generated and if ? is ?-continuous, then ? is rich.
(b) IfX is separable and if ? takes on its values in Cbc(X), then ? is rich.
(c) If ? takes on its values in /C« (X), then for every A e A, one has
M(A)= \m(A): m e<S,w}.
A member A of Л is called an atom of the multimeasure ? : A -> 2X \ {0} if ? (?) ?
{0} and if either ? (?) = {0} or M(A\B) = {0} holds for every ? с А, В е A. A measure
space having no atom is said to be nonatomic.
The Banach space X is said to have the Radon-Nikodym property (RNP) if for each
finite measure space (?, ?. ?) and each vector measure m : A -> X which is of bounded
variation and ?-continuous, there exists f e L] (?,?.?: X) such that
m(A) = I f ??.
for all A e A. For example, it is known that reflexive Banach spaces and separable dual
spaces have the RNP (see [68] or [69]). The following result concerns the convexity of the
values of a multimeasure. The first part is due to Hiai [113], while the second part can be
found in [2] or [49].
THEOREM 7.10. Assume that X has the RNP. Let ? : A -> 2?\{0) be a nonatomic strong
multimeasure of bounded variation. Then for every A e A, cl ? (A) is convex. Moreover,
the set
cl
\JM(AI
1ЛеЛ J
is convex.
Example 7.11. Let m : A -> X be a vector measure. Define ?: A -> 2х \ {0} by
? (A) = \m(B): В eA. B^A}. AeA.
Then ? is a strong multimeasure, and ? is nonatomic if and only if m is so. Further,
Vm(A) is equal to the variation of m and ?(?) is equal to the range of m.
Now, we turn to versions of the Radon-Nikodym theorem for multimeasures.

Set-valued integration and set-valued probability theory: An merview
647
Definition 7.12. Let ? : А -> 2х \ {0} be a multimeasure. F eMBx) is said to be a
set-valued Radon-Nikodym derivative of ? with respect to ? if
M(A)= I Fd\x, AeA.
JA
and a generalized Radon-Nikodym derivative of ?? with respect to ? if
clM(A) = cl / F<*m. AeA.
J A
Let us give a version of the Radon-Nikodym Theorem concerning multimeasures,
whose values are weakly compact. It is due to Kle'i [127,129]. It provides a necessary
and sufficient condition. Earlier versions of this result had been proved by several authors
(see, e.g., [2,50,54,66,86-88,164,206]).
THEOREM 7.13. Let X be a Banach space. Then, the following two statements are
equivalent.
(a) X has the Radon-Nikodym property:
(b) For every probability space (?. ?. ?), for every multimeasure ? :?—> ?,?,(?) of
bounded variation, there exists an integrably bounded Radon-Nikodym derivative
F of ? with weakly compact values.
Remark 7.14. Let us also mention the works of Thiam [193-196] who studied the
Daniell integral for functions with values in an ordered semi-group and gave applications
to the integration of real-valued functions with respect to a multimeasure. On the other
hand, Puri and Ralescu [166] proved a version of the strong law of large numbers for
vector-valued random variables with respect to a multimeasure.
8. The probability distribution of a measurable multifunction
This section is devoted to the presentation of fundamental properties of the distribution
of measurable multifunctions. This domain was explored by several authors for various
purposes (applications to mathematical economics, stochastic optimization, etc.). Let
us mention, for example, the works of Hart and Kohlberg [89], Hildenbrand [118],
Artstein [3], Gine et al. [85], Artstein and Hart [10], Hess [96,97,103,111], Salinetti and
Wets [182], Lavie [137,138,140] and Raynaud deFitte [169].
Here, we denote by M(X) the space of all probability measures on (X, B(X)). M(X)
is endowed with the weak topology, also called the topology of narrow convergence (see,
e.g., [29]). For each measurable function /: ? -> X, the distribution of / is denoted by
? f and defined on B(X) by
?/(?) = ?{/-'(?)}, BeB(X).

648
С. Hess
Let ft be a ?-field on the space Iх and F: ? -> 2? an Д/ft-measurable multifunction.
Like for real or vector valued random variables, the distribution of F is the measure ?^
on B*,ft) defined by
Mf(tf) = M(F-' (/?)), tfeft. (8.1)
Two measurable multifunctions F. G : ? -> 2? are said to be independent if for every
R, S eTZ one has
M(F-|(/?)nG-|E))=M(F-|(/?))M(G-|E)).
This amounts to the equality
M(F.G) =MF®MG,
on the product space (C(X) ? C(X), 7г ® ft)).
In the sequel, it will be sufficient to consider the case where ft is equal to the Effros
?-field ? (this will be explained in Section 9).
It is natural to consider measurable selections that are measurable with respect to the ?-
field Af = F~]{?) (i.e., the ?-field generated by the measurable multifunction F). This
raises the question of the relations between the sets of distributions of ?h -measurable
selections and those of Д-measurable selections. The answer is given in Theorem 8.3,
and in Corollary 8.5(a) for integrable selections. In view of application to the SLLN,
the multivalued integrals are considered in Corollary 8.5(b). On the other hand, we
shall provide results allowing one to extract equidistributed and independent selections
from sequences of measurable multifunctions having similar properties. For this purpose,
appropriate characterizations of equidistribution are proved at the end of this section
(Proposition 8.6), as well as a connection with the multivalued integral. First, we recall
a simple and useful criterion for two measurable multifunctions to have the same
distribution.
PROPOSITION 8.1. If F and G are members of M(C(X)), then the following three
statements are equivalent:
(a) F and G have the same distribution on (C(X), ?).
(b) For any open subset U of X, P{F~(U)) = P[G~(U)}.
(c) For any finite subset ? = {.x\, л>} of X (or of some countable dense subset),
the Rk-valued random vectors (d(x,F))x€y and (d(x,G))x€Y have the same
distribution.
PROOF. It follows from the definitions that statement (a) implies (b) and (c). To prove the
implication (a) => (b), we define the subset W+ of C(X) by
W+ = {CeC(X): CCIV).

Set-valued integration and set-valued probability theory: An oven'iew
649
for every subset W of X, and we observe that for each open subset U of X one has
(U-)c = (Uc)+ = {CeC(X): C^UL}.
where Uc denotes the complement of U. This class generates ? and is stable under
finite intersections, which by a classical result yields the desired implication. For proving
implication (a) => (c), remember that the Effros ?- field is also generated by the family of
distance functions
С ->d(x,C),
where ? ranges over X (or over a countable dense subset). Like in the first part of the proof,
it is enough to observe that the class, whose members are defined by
(C eC(X): d(x,,C)<a,, V/= 1 it}
(where к e N, x, e X and or,- e R), generates ? and is stable under finite intersection. D
For every sub-a-field ? of A, consider the space ?\?. ?, ?; X) of all (classes of)
measurable functions from (?,?) into (X.B(X)). Further, define the following subset
associated with the multifunction F
?(?.?)=|? = ?/??(?); feS(F.F)}.
where S(F,T) is the set of all J"-measurable selections of F (see the beginning of
Section 5).
So, M(F,T) the set of all probability measures ? on (X.B(X)) such that each
? e M(F,T) is the distribution of some ^"-measurable selection of F. Before stating the
results on the set of selections and its associated set of probability measures, an example is
useful.
Example 8.2. Consider the case where ? = [0, 1], ? = ?(?), ? = Lebesgue's
measure, X = ? and F = {0, 1} (i.e., F is a constant multifunction). Clearly we have
?? = [?,?), S(F,Af) = (f = Q,f=\) and M(F,AF) = [h.&\\.
On the other hand, one has
S(F,A) = (f = tA: AeA],
This shows that the inclusions
S(F,AF)?S(F.A) and M(F,AF)?M(F.A)
M(F.A) = {p8{) + (\- p)o\: pe[0.\]}.
may be strict.

650
С. Hess
The following theorem is the main result of the present section. It was already stated
in [96,97], but the proof contained a gap; the correct proof was given in [103,111]. It
provides a fundamental equality, which is the starting point for the study of the distribution
of multifunctions in connection with their measurable selections. Its consequences,
especially Corollary 8.5(b) concerning the multivalued integral, play an important role
in the proof of the multivalued SLLN for measurable multifunctions whose values may be
unbounded.
THEOREM 8.3. IfF is a measurable multifunction with closed values in X, then, in M(X)
endowed with the topology of weak convergence of measures, the following equality holds
true
coM(F,A)=coM(F,AF). (8.2)
where "со" denotes the closed convex hull operation.
We denote by ?' (X) the subset of ? (?) whose members ? satisfy
\?\\?? < +oo.
Given a sub-a-field ? of A and a measurable multifunction F, we define the following
subset of M(X)
M\F,T) = {^,: feS\F.T)}.
where 51 (F, T) was defined in Section 5. Now, consider an integrable multifunction F.
Obviously, the inclusion M] (F,T) с M(F. JF) holds for any sub-a-field ? of A. The
following simple lemma shows that the closure of both sides are equal in ? (?).
LEMMA 8.4. For any sub-? -field ? ofA and any integrable multifunction F whose values
are members ofC(X) the following equality holds true
c\M\F,F) = c\M(F.T).
the closure being taken in M(X) in the weak (narrow) topology.
From Theorem 8.3 and Lemma 8.4, we easily deduce the following corollary, whose first
part makes equality (8.2) more precise when F is integrable. Part (b) is an application of (a)
to the multivalued integral. It was given by Artstein and Hart [10, Theorem 2.2] when F
is an integrable measurable multifunction with closed values in a finite dimensional
space. Let us note that Artstein and Hart's proof relies on a previous result of Hart and
Kohlberg [89] valid for integrably bounded multifunctions. A self-contained proof was
given in [111].
/

Set-valued integration and set-valued probability theory: An overview·
651
Corollary 8.5.
(a) For every integrable multifunction F whose values are in C(X) the following
equality holds true
со M\F, А) = со M](F.Ah), (8.3)
the closure being taken in M(X) in the weak topology.
(b) For any integrable multifunction F whose values lie in C(X), one has
со I (F, A) = со I (F,Ar). (8.4)
The following result states that two measurable multifunctions F and G have the
same distribution if and only if the sets of distributions of selections that are measurable
relatively to the ?-fields Af and Ac respectively, are equal. As we shall see, this property
plays a crucial role in the proof of the multivalued SLLN in an infinite dimensional Banach
space. Considering the ?-fields generated by the multifunctions (instead of the ?-field A)
allows for more precise results. Indeed, unlike in [10]. the equalities of statements (b)
and (c) below do not involve any closure operation. This is important in view of the
extension of the SLLN in infinite dimensional spaces.
PROPOSITION 8.6. Let F and G be two measurable multifunctions with closed values
in X. Then, the two following statements (a) and (b) are equivalent:
(a) F and G have the same distribution on the measurable space (C(X), ?).
(b) In M(X), the following equalit}· holds true
M(F,Ar) = M(G,AG).
Moreover, if F and G are integrable then each of the above statements is equivalent
to
(c) In ?' (X), the following equalit}- holds true
M\F.Af) = M\G.Ag)-
Consequently, if F and G have the same distribution one has
I(F,AF) = I(G,AG).
Remark 8.7. (i) An alternate proof of the last statement of Proposition 2.6(c) was
given by Hia'i [116, Lemma 3.1B)] using the properties of the multivalued conditional
expectation, as defined in Section 6.
(ii) It is worthwhile to remark that Theorem 8.3 and Proposition 8.6 ((a) and (b)) remain
valid when X is only assumed to be a complete separable metric space.
(iii) Let us mention a result of Artstein [3], who considered the following problem:
given a probability distribution ? on (K,(X). ?), is it possible to find ? e M(X) which is

652
С. Hess
the distribution of some measurable selection / of a random set F : ? -» ?,(?) whose
distribution is equal to ??
(iv) The study of the distribution of measurable multifunctions was initiated by
Choquet [45]. Given F e M(C(X)), Choquet introduced the capacity functional
redefined on fC(X) by
??(?) = ?(?-?(?-))=?{???: ?(?)?\??9). KelC(X).
Choquet showed that, when X is finite dimensional, 7> characterizes the distribution of F.
He also showed that if ? is a capacity satisfying suitable additional requirements then there
exists a unique probability distribution ? on (C(X).E) such that
T(K) = v(K-) = v({CeC(X): СПК^0}). К е ?,(?).
Later, Matheron [148] gave a probabilistic proof of Choquet's result. See also
Norberg [160,161] and Salinetti and Wets [182] for related results. A more abstract
approach was considered by Ross [176] without topological assumptions. It was shown
recently by H.T. Nguyen and N.T. Nguyen [159] that the above result on the existence
of у may fail when X is not locally compact. Finally, let us note that the convergence
in distribution of multifunctions was studied by Norberg [160,161], and by Salinetti and
Wets [182]. The convergence in probability was examined by Salinetti and Wets [181], and
Salinetti et al. [179].
9. Set-valued strong laws of large numbers
It is remarkable that the classical Kolmogorov strong law of large numbers (SLLN)
makes sense in the set-valued case. Firstly, we shall present set-valued SLLN involving
the convergence in the Hausdorff metric topology гн on Сь(Х). Secondly, we shall
be interested in SLLN involving the Painleve-Kuratowski convergence, which allows
for multifunctions taking on unbounded values. In both cases, we shall observe a
convexification phenomenon. More precisely, in multivalued SLLN the limit is always
convex, even if the given multifunctions are not convex-valued. On the average, the Cesaro
sums tend to be convex.
9.1. Convergence in the Hausdorff metric topology
As to the measurability issues, we have already noted in the beginning of Section 8 that the
distribution and independence of multifunctions are relative to a given ?-field, say ??, on
C(X). In this subsection, it is natural to take the ?-field ?? equal to В(Съ, ти). However,
since we shall restrict to a гн-separable subspace, we can take TZ equal to the Effros ?-field
as well (see Remark 4.9(ii)).
THEOREM 9.1.1. Consider a strongly measurable multifunction F with values in a
тц-separable subspace С of Съ(Х) and a sequence (F„)„~^\ of pairwise independent
multifunctions having the same distribution as F. Then, the following two statements are
equivalent

Set-valued integration and set-valued probability theory: An overview
653
(a) F is integrably bounded.
(b) There exists С &Съ such that
lim h I C, - V Fi (?) ) = 0 a.s.
Moreover, when С exists it is equal to col(F) and the convergence also holds in the ?-
metric, i.e.,
lim /}(C, -Vf,) =0.
The proof of Theorem 9.1.1 can be done in two steps. When F is assumed to be convex
valued, it suffices to use the embedding ? from Сы.-(Х) into C(B*) (see Remark 3.6)
and to invoke the SLLN for vector-valued random variables. This is the easier step, but it
requires the гн-separability of С The second step consists of a so-called convexification
procedure, namely of a property of the following type, which is of a purely deterministic
nature.
The convexification property (#). Consider a pair (X,C), where X is a Banach
space and С asubspaceofCb(X). For every sequence (C„)„~^\ in С and С eC satisfying
(i) С is convex,
(ii) Нт„^эсй(С,^5:!'=|СоС/) = 0.
one has
lim hlc-f^d) =0.
Example 9.1.2. Let us mention three situations where property (#) holds.
(a) X is finite dimensional and С = ?,(?) (Artstein and Vitale [11]).
Using the Shapley-Folkmann inequality, it can be shown that for every sequence
C|,..., C„, in K(X) one has
hi yjcoC,, /Jc,- J ^ \idmax{r(Cj): i = 1 /?}.
\/=i ;=i /
where d is the dimension of X and r(C) denotes the radius of the subset C, i.e.,
r(C) — inf sup \\x — y\\.
¦te*veC
An alternative proof in special cases was given by Cressie [59]. A variant of this theorem
has been rediscovered recently by Uemura [198], and Taylor and Inoue [192].

654
С. Hess
(b) X is a separable Hubert space and С = /C« (X).
This was proved first by Cassels [32] using probabilistic arguments and later by
Hess [95,99] by a geometric approach in the framework of (countable) Hubert spaces.
This was extended later by Puri and Ralescu [166,167] when X is a Banach space of type
? e A, 2]. In all these papers, the proof is based on an inequality of the following form
]ГсоС,,]Гс,и ]Гг(С,У
( = 1 ( = 1 / I ( = 1
where к = 2 when X is Hubert and к = ? e A. 2] when X is of type p.
(c) X is a separable Banach space and С = /C(X), the set of strongly compact subsets
of X.
In this case, property (#) was proved independently by Artstein and Hansen [9] and by
Hia'i [115]. Further, it was shown by Hess [98,99] that it is not possible to replace /C(X) by
/CW(X).
It is interesting to mention a set-valued version of the Birkhoff ergodic theorem. Assume
that a measurable map ?: ? -> ? is given. ? is said to be measure preserving if for every
A e ? one has ?(?_? (?)) = ?(?). The ?-field I of invariant subsets of ? is the set of
those A e ? such that A and T~l (A) are equal up to a ?-null set. The following result is
due to Hess [94-96].
THEOREM 9.1.3. Let F be a strongly measurable and integrably bounded multifunction
with values in a z\\-separable subspace С ofCb(X)- Then, for almost all ? e ? one has
lim h ( coE(F|I), - Vfir?)) - 0.
n^oc \ ? *—i y ' J
As it is known for single-valued random variables, this result can be stated equivalently
in terms of stationary sequences (see [94—96]). Extensions and variants of this result
have been proved by Schurger [189] and, more recently, by Krupa [131]. On the other
hand, assuming conditions on the moments, set-valued SLLN have been proved for
nonidentically distributed, independent sequences of integrably bounded multifunctions.
See, e.g., Lyashenko [146] and Hia'i [115].
9.2. Convergence in the sense of Painleve-Kuratowski
Now, we consider the extension of the SLLN to the case of measurable multifunctions
whose values are unbounded. In this case, the Hausdorff distance is no longer appropriate
to formulate the convergence, because it is too strong. Instead, we consider the Painleve-
Kuratowski convergence, whose definition is recalled hereafter.
Let ? be any topology on X. If (Cn)„^\ is a sequence in 2X we set
?-LiC,, = ? ? e ?: ? — ?- lim x„. xn e C„, ? ^ 11,
I / Л

Set-valued integration and set-valued probability theory· An overview
655
and
r-LsC„ = \x e ?: ? — ?- lim лг*. xt e C„{k). к > 1 .
where (Cn^))k^\ is a subsequence of (C„). The set ?-LiC,, (respectively, ?-LsC,,) is
called the lower limit (respectively, the upper limit) of (C„) relatively to the topology ?.
These subsets are also called the inferior limit and the superior limit, respectively, which
explains the notation. When ? is metrizable, these subsets are both ?-closed. Further,
?-Li C„ and ?-Ls C„ are unchanged if each C„ is replaced with its ?-closure.
The sequence (C„) is said to converge to С in the sense of Painlevi-Kuratowski (PK),
relatively to ?, if one has
С = ?-Li C„ = ?-Ls C„
(see, for example, [ 14,27,45,180,190]). This is denoted by
C = PK(r)- lim C„,
n —>эс
or simply
С = PK- lim C„.
when X is finite dimensional and ? is the usual topology on X. A convergence criterion in
the general case was given by Couvreux and Hess (Corollary 1, p. 944, in [56]).
The convergence in the Hausdorff distance entails the PK-convergence, but the converse
implication is false, as the following simple example shows.
Example 9.2.1. Let X = R2 be endowed with any norm and consider the subsets
С = a straight line passing through the origin.
С,, = a straight line such that the angle between С and C„ is \/n.
It is readily seen that the sequence (C„)„j>i PK-converges to C, but does not гн-converges.
In the present section, the ?-field TZ considered in Section 8 is taken equal to E,
the Effros ?-field on C(X). Thus, the definition of the distribution and independence
of multifunctions are relative to ? and we consider an Effros-measurable
multifunction F: ? -> C(X). As to the integrability condition, integrable boundedness is no longer
appropriate because an integrably bounded multifunction is almost surely bounded valued.
Instead, we assume that F is integrable, namely that S] (F) is nonempty. The proof of the
SLLN for multifunctions whose values may be unbounded are mainly based on the results
presented in Section 8, especially on Corollary 8.5(b) and Proposition 8.6(c).

656
С. Hess
THEOREM 9.2.2. Let X be a separable Banach space, F e M(C(X)) be an integrable
multifunction and (F„)„^| be a sequence ofpairwise independent multifunctions having
the same distribution as F. Then, there e.xists a null set N such that
1 "
coI(F,A)=PK- lim - Vf,(w). ???\?.
?—>эс /; ?—1
? = ?
A converse to the above theorem can be given when the values of the multifunctions
do not contain any line. They may be unbounded, but may contain half-lines only. For
sake of simplicity, we only present the finite dimensional version of a result due to Hess
(Theorem 9 in [98,99]).
THEOREM 9.2.3. Consider a finite dimensional normed linear space X and a closed
subset L of X which contains no whole line. Let (Fn)n^\ be a pairwise independent,
equidistributed sequence in Л4(С(Х)). Moreover, assume that for almost all ? e ? the
following inclusion holds
Fd<o)QL. (9.1)
Under the above assumptions, if F\ is not integrable, i.e., ifS(F\) =0, then
1 "
PK- lim - Vf,(w)=0.
1 = 1
For example, if X = M.d, the subset L can be taken equal to the positive cone W+. When
the F„ are single-valued, condition (9.1) simply means that the X-valued random variables
F„ have positive values.
Remark 9.2.4. (i) Theorem 9.2.2 has been proved by a lot of authors. It was proved
first by Artstein and Hart [10] when X is finite dimensional, for mutually independent,
identically distributed sequences. In the early eighties, infinite dimensional versions
were proved independently by Hiai [116] and Hess [98,99], when the PK-convergence
is replaced by the Mosco convergence, an infinite dimensional extension of the PK-
convergence (see the next remark). Moreover, using a result of Etemadi [80], Hess [98,99]
relaxed the hypothesis of mutual independence, assuming only the pairwise independence.
Afterwards, the set-valued SLLN was proved with respect to other extensions of the PK-
convergence such as the Wijsman convergence [111] and the slice convergence [107,111],
that are of interest in infinite dimensional spaces.
(ii) The above mentioned Mosco convergence is one of the infinite dimensional
extensions of the PK-convergence. It was introduced by Mosco [153,154] and has
interesting variational properties that was exploited in the areas of calculus of variations
(see, e.g., [62,64]) and optimization [175]. If X is infinite dimensional and if the sequence
(C„) PK-converges to С relatively to both the strong topology and the weak topology of X
(denoted by s and w, respectively) then (C„) is said to be Mosco convergent to C. This

Set-valued integration and set-valued probability theory: An oven'iew
657
is denoted by С = M- lim„ C„. However, Mosco convergence has nice properties only in
reflexive spaces so that several other topologies have been introduced. For an extensive
study of convergences and topologies on spaces of subsets of a metric or a normed linear
space, the reader is referred to the monograph by Beer [27] and to the paper by Sonntag
and Zalinescu [190], who presented a classification of set-convergences.
(iii) In some extent, it is possible to deduce the set-valued SLLN for unbounded valued
multifunction from the integrably bounded case. This approach relies on a truncation
argument. More precisely, let F e M(C(X)) be an integrable multifunction. For every
integer к ^ 1, we define the multifunction Fk by
Fk(w) = c\(F(w)DBk(w)), ?&?.
where Bk(co) denotes the open ball of radius ?(?) = k(d@, F(a>)) + 1), centered at 0.
Now, if (F„)„^ | is a sequence of pairwise independent measurable multifunctions having
the same distribution as F, we set
Fk(w) = d(F„(<u)r)Bk(co)), ???,
where Bk{a>) is defined as above except that F(a>) is replaced by F„(a>). For every k, ? ^ 1,
Fk is Af„-measurable and integrably bounded (because d@, F„(-)) is integrable). The
Af„ -measurability follows from the Effros measurability of the map
С->с1(СПВ@,г)).
for every r > 0.
Thus, when X is finite dimensional, Fk takes on compact values, which allows us to
apply Theorem 9.1.1 to the sequence (Fk)n^\, for every к ^ 1. This yields
1 "
coE(F*) = lim -Vf*(w) a.s.
Then, using the simple equalities
k^\ i = l / = l jt^l /'=|
and Lemma 5.11 of [106], it is readily seen that this yields the PK-convergence as in
Theorem 9.2.2. This technique can be used for proving other strong limit theorem for
multifunctions.
(iv) There are other kinds of set-valued SLLN where the multifunctions are no longer
assumed to be identically distributed, but only independent (see, e.g., [115,116] and [169]).
Let us also mention set-valued versions of the Komlos theorem (see [22,39,133,216]).
Like in the SLLN, Komlos theorem involves Cesaro means, but is valid for L' -bounded
sequences. An interesting application to fuzzy random sets was obtained by Colubi et
al. [47].

658
С. Hess
10. Set-valued martingales
In this section, X generally stands for a finite dimensional Banach space. Infinite
dimensional extensions will be mentioned in Remark 10.4.
Dehnition 10.1. Let (?,,)??^? be a nondecreasing sequence of sub-a-fields of ?.
A sequence (F„)„^| of measurable multifunctions with values in CC(X) is said to be
adapted to (Bn) if, for any /; ^ 1, F„ is B„-measurable, i.e., F~](U~) e B„ for every
U e Q. An adapted sequence (F„) is said to be a set-valued martingale if the following
two conditions (a) and (b) hold, for every integer /; ^ 1:
(a) S] (F„,B„) is nonempty (i.e., F„ is integrable),
(b) F„ =E(Fn+i\Bn).
Further, if, instead of equality (b), one has the inclusion
F„ с E(Fn+i |?„) (respectively, F„ 5 E(F„+| \B„)),
we shall say that (F„) is a submartingale (respectively, supermartingale).
Example 10.2. In the following examples (G„) denotes a sequence of integrable CC(X)-
valued multifunctions and, for every integer// ^ \,B„=a(G\ G„) is the sub-a-field
generated by the multifunctions G \ G„.
(i) In this example, we assume that the G„ are centered in the sense that
0eE(G„+||i3„) a.s. for all О 1. (Ю.1)
If we define the sequence F„ by
F(I=cl j^G,¦[, (Ю.2)
then (F„) is a submartingale. Indeed, from A0.1) and A0.2), and from Theorems 6.2
and 6.3, we obtain, for every ? ^ 1,
E(F„+i|S„) = e(c1(^]g,+G„.
Bn
>i = l
= cl{F(I+E(G(I+i|B(I)}2Fn.
(ii) Now, we consider the sequence (G„) and (?„) as in (i). In addition, the G„ are
assumed to be integrably bounded, but condition A0.1) is no longer assumed to hold. If the
sequence (F„) defined by A0.2) is a supermartingale, then for all ? ^ 2 the multifunctions
G„ are single-valued. Thus we have
F, = G ? + g2 + h g„.

Set-valued integration and set-valued probability theory; An overview
659
where g-,: ? -> X are integrable random vectors with zero expectation. Only G\ can be
set-valued. Indeed, for every ? ^ 1, one has
E(F„+| \B„) = F„ + E(G„+| \B„) с F„.
which implies
E(G„+i|i3„) = {0}. for all « > 1.
In turn, this yields the existence of a Bn+\-measurable random vector gn+\ such that
G„+\ = [gn+\ }¦ This result is due to Ezzaki ([83, p. 17]).
(iii) If we consider (G„) and (B„) as at the beginning of the present example, then the
sequence (F„) defined by
F„=co(|Jg,Y for all О 1.
is a submartingale. Indeed, using Theorems 6.2 and 6.3 it is readily seen that for every
? ^ 1 and almost every we ?, one has
E(F„+||i3„) = E(F„UG„+i|i3„)
2 E(F„\B„) UE(G„+| \B„) = F„ UE(G„+| |?„) Э F„.
(iv) Let (G„) be a sequence of multifunctions as in example (iii). If we assume that
OeG„ a.s. forallO 1, A0-3)
then it is readily seen that the sequence (F„) defined by
F„ = f]Gi,
is a supermartingale. Indeed, for every ? ^ 1 one has almost surely
E(F„+||i3„)=E(F„nG„+i|i3„)
с E(F„\B,,) ПЕ(С,|Д,) - F„ ПЕ(С,|Д,) с F„.
Condition A0.3) prevents F„ from taking empty values.
(vi) Examples (iv) and (v) are special cases of more general situations. Indeed, consider
an adapted sequence (F„) of multifunctions. If (F„) is a nondecreasing (respectively,
nonincreasing) sequence of multifunctions, then it is a submartingale (respectively, a
supermartingale).
(vii) Finally, it is worthwhile to observe that the following property is an immediate
consequence of our definitions: if a set-valued submartingale or supermartingale is single-
valued (i.e., its values are points of the space X), then it is a martingale.

660
С. Hess
The following result presents a property of almost sure convergence of set-valued
martingales whose values may be unbounded. It can be seen as an extension of the classical
Doob's Convergence Theorem.
THEOREM 10.3. Let {B„)n^\ be a nondecreasing sequence of sub-?-fields of ? and
(F„)„j,| an adapted sequence of multifunctions with values in CC(X). If (F„) is a set-
valued supermartingale satisfying the condition
sup Ed@, Fn) < +oo.
then there exists an integrable multifunction F e M(C(X)) such that, for almost every
? e ?,
F(w) = PK- lim F„(w).
II —>OC
Remark 10.4. (i) Theorem 10.3 is a special case of Theorem 5.12 in [106], where X is
assumed to be an arbitrary separable Banach space (possibly infinite dimensional). There,
the following additional condition was required.
(WB) there exists a multifunction H, whose values are weak ball-compact, such that
Fn(w)?H(w), OUefi.
Recall that a subset С of CC(X) is said to be weak ball-compact if its intersection with
every closed ball is weakly compact. Condition (WB) is automatically satisfied when X
is reflexive or, more particularly, finite dimensional. The proof is based on a truncation
argument and involves Lemma 5.11 in [106], already described in Remark 9.2.4(iii). On
the other hand, existence results for martingale selections can be found in [105,106]. Here,
for the sake of simplicity, we have only presented convergence results when X is finite
dimensional.
(ii) There are other references where other set-valued versions of the Doob convergence
theorem are proved. For example, let us mention the works of Neveu [158], Daures [63],
Hiai and Umegaki [117], Coste [51], Luu [144,145], Bagchi [18], Choukairi [46], Wang
and Xue [211], Dong and Wang [70], Hess [109], Krupa [131,132] and Lavie [139].
A version of the optional sampling theorem can be found in [1]. An application to the
convergence of fuzzy sets was given by Li and Ogura [ 143 ]. A lot of further references can
be found in [131].
11. Gaussian multifunctions. The set-valued Central Limit Theorem
First recall that an X-valued random variable /: ? -> X is said to be Gaussian if for every
finite sequence (y\,..., yy) e (X*O, where j' > 1, the Ry-valued random vector
«3?,/>,-¦¦, <O./».

Set-valued integration and set-valued probability theory: An overview
661
is Gaussian. More generally, an integrably bounded multifunction F ; ? —* /C(X) is said
to be Gaussian if, for every finite sequence (vi, v/) e (X*)J, the R^-valued random
vector
(s(y],F),...,s(yj,F)),
is Gaussian. The following result characterizes Gaussian multifunctions with compact
values.
THEOREM 11.1. Let F-.? -> /CC(X) be an integrably bounded multifunction. The
following two statements are equivalent.
(a) F is Gaussian.
(b) F has the following form
F = K + f.
where К e K,C(X) is nonrandom and /:?-> ? is a Gaussian vector with zero
expectation.
Theorem 11.1 was proved by Lyashenko [147] when X is finite dimensional and by
Vitale [207]. It was extended later to the case where X is infinite dimensional by Puri and
Ralescu [168], and by Meaya [150].
On the other hand, there exists a version of the central limit theorem for multifunctions
whose values are compact. First, we need some notations.
Let Sk ~' be the unit sphere of X = Жк and С (Sk~') be the space of continuous functions
defined on S*_l, endowed with the uniform norm || · ||u. The dual space C(S*_I)* is the
space of Radon measures on Sk~]. The duality pairing is defined by
(k,u)= I udX, ue<C(Sk-^). XeCiS'*)*.
Jsk-X
Given a C(Sk~' )-valued random variable /, the expectation E(/) is defined as a Bochner
integral. This is possible because CE'A^1) is separable. Further the covariance function
rf is defined on the product space C^)* xCEi_l)* by
?/(??, ?2) = ?{(? ?, / - ?(/))(?2, / - ?(/))}
= ? (?,. /(?) - ?(/))(?2· /(?) - ?(/)) ?(??).
When F e _M(/C(X)) is an integrably bounded multifunction, the covariance function of F
is denoted by /> and is defined as the covariance function of the C(Sk~' )-valued random
variable
y-> s(y, F).

662
С. Hess
THEOREM 11.2. We assume that X = Rk. Let F e M(K(X)) be an integrably bounded
multifunction and a sequence (F„)„^| of independent multifunctions having the same
distribution as F. Then the sequence
Vnhl -]Tf,,EcoF J,
converges in distribution to \\g\\u, where g is a centered Gaussian C(Sk~])-random
variable with covariance function Гу.
The proof of Theorem 11.2 appeals to the central limit theorem for C(S*_l)-random
variables, which allows for proving the result in the case where F e М(К,^(Х)), and to the
Shapley-Folkmann inequality in the general case. Theorem 11.2 is due to W. Weil [212].
Previous versions had been settled by Cressie [60,61 ] in the case of multifunctions taking
on a finite number of values.
12. Set-valued versions of the Fatou Lemma
The classical Fatou Lemma of Integration Theory asserts that, given a sequence (/„)„^ ?
of positive measurable functions, the following inequality holds
/ liminf/^P^liminf / f„dP.
JR n^oc n^x J?
Extensions of this inequality to the case where the /„ take on their values in X = R'
(d ^ 2) or in infinite dimensional Banach spaces, were established in order to get the
existence of equilibria in Economic Theory. In these extensions, the inequality must be
changed into an inclusion. It is worthwhile to note that, because of the partial character of
the order relation in W1, this sort of result cannot be deduced directly from the classical
Fatou Lemma.
For the same modelization purpose, the case where the /„ are replaced with measurable
multifunctions F„ was examined by several authors (see, e.g., [118, Theorem 6, p. 68]). In
the set-valued case, there are in fact two versions of the Fatou Lemma for multifunctions:
one for the PK-lower limit and another for the PK-upper limit. Indeed, it is no longer
possible to deduce one version from the other by merely changing Fn into — F„. On the
other hand, it was shown by Balder and Hess [21 ] that there is a strong connection between
the set-valued version and the vector version of Fatou's Lemma. In fact, one can be deduced
from the other.
As mentioned above, several authors have studied this subject. Let us mention the
works by Aumann [17], Schmeidler [185], Hildenbrand and Mertens [119], Artstein [5],
Balder [19], Yannelis [215],Castaing and Clauzure [38], and Balder and Hess [21] among
others. We refer the reader to [21 ] for a more complete list of references and for a discussion
of the various existing results. Basically, the first version of the following result is due to
Aumann [17].

Set-valued integration and set-valued probability theory: An overview
663
THEOREM 12.1. Let X be a separable Banach space and (F„) be a sequence in
M(C(X)).
(a) If the sequence (d@, F„))n~^\ is uniformly integrable, then the following inclusion
holds
cl/(s-LiF„)CS-Li/(F„).
(b) If the sequence (|| F„ \\ )„^\ is uniformly integrable and if there exists a multifunction
G : ? -» /CW(X) such that
F„(a))CG(a)), шей. ii^l.
then one has
w-Ls / (F„) с cl /(со w-Ls F„).
Remark 12.2. (i) In statement (a) the multifunctions are only assumed to be integrable,
whence can have unbounded values. In statement (b), the F„ are assumed to be integrably
bounded, thus they are almost surely bounded valued. Versions of statement (b) for
integrable multifunctions have been given, but they are of a more delicate nature.
For example, they involves conditions on the asymptotic cones of the multifunctions
(see [21,104]). On the other hand, results involving the convergence in distribution in
connection with the approximation of the multivalued integral were given by Artstein and
Wets [12].
(ii) Similar properties have been proved for the set-valued conditional expectation
(see [102,104,116]).
13. Epigraphical convergence
The epigraphical convergence, "epiconvergence" for short, is closely related to PK-
convergence and its infinite dimensional extensions. Let и ·. X -> R be an extended real-
valued function. Its epigraph (or upper graph) is the subset of ? ? Ш denoted by epi(w)
and defined by
epi(n) = {(x,a)e XxR: u(x) iCa}.
The conjugate (or the polar) function of и is denoted by u* and defined on X* by
K*00 = sup{()>,;0 -u(x): ? e X}. ye X*.
The map и —» и* is often called the Young-Fenchel transform. Further, let (?,,),,^? be a
sequence of extended real-valued functions. Consider a topology ? on X. The sequence
(и„) is said to converge epigraphically (or "epiconverge" for short) to u, relatively to ?, if
ерЦк)=РК(г)- lim epi(M„). A3.1)
/?—?-ОС

664
С. Hess
When X is finite dimensional and ? is the usual topology, this is denoted by
и = lime и„,
n—*oc
where the subscript "e" stands for "epigraphical". Of course, a symmetric notion involving
hypographs (alias lower graph) can be introduced.
This type of convergence was considered by a lot of authors in order to study the
convergence and the approximation of optimization problems, especially in the calculus
of variations. Let us mention the works of De Giorgi [64], Mosco [153,154], Attouch [14],
Dal Maso [62], Rockafellar and Wets [175] and Artstein and Wets [13], where further
references can be found. Also, applications to the convergence of estimators, in Statistics,
were given by Hess [108] and Choirat et al. [43].
Without entering into details, let us mention the following facts (see, e.g., [14] and [62]).
If (V„) denotes the minimization problem
mm[un(x): ? e X), /ieNU{oo),
and if (и„) epiconverges to ux, then under additional compactness assumptions, it can be
shown that
infi/3C(;c)= lim inf m„(jc).
.t€X ?-*????
Moreover, the solutions of (Voc) can be approximated by solutions of (Vn) for ? large
enough.
The connection of epiconvergence with that of measurable multifunctions is clear.
Indeed, let
be an extended real-valued function. The multifunction F defined by
F(a>)=epi(<p(w. ¦)) = {(x,a)e XxR: ?(?.?) ^a\. A3.2)
It is called the epigraphical multifunction associated with ?. When ? is ? <8> B{X)-
measurable, it is called an integrand. It is not difficult to show the following equivalence
(see Lemma VII. 1 in [41])
Gr(F)e A®B(X) <s> ? is ? ® ?(X)-measurable.
Let us give an example involving epiconvergence in connection with the SLLN. First, we
need the following simple auxiliary result, which is of independent interest. The proof can
be found in [107].

Set-valued integration and set-valued probability theory: An oven-iew 665
Proposition 13.1.
(a) The multifunction F defined by A3.2) is integrable if and only if there exists
f e L1 (X) such that the function ? -> ?(?, /(?))+ is integrable.^
(b) In ? ? R, f/?e distance function from the origin to the epigraphical multifunction is
given by
w^d(@,0),F(w))=\nf{\\x\\ +?(?..?)+: xe X}.
Theorem 13.2.
(a) The stochastic discrete infimal convolution ?„ (?. ?) defined on ? ? X by
II " 1 " ]
-VWffl.i,): (.vi x„) e X", -У]*/ =x \
? *—' ? i—1
1=1 1=1 J
epiconverges to the convex lower semicontinuous regularization of the continuous
infimal convolution ? which is the deterministic function defined by
?(?)=?\{ ?(?,/(?))??: feL](X). J /(?)??=?\.
(b) For almost all ? e ?, the so-called mean functional defined by
?(?.?)??.
??
is the epi-limit of the sequence of the sample mean given by
1 "
(?,?) -> -?^?(?,?).
? ?—1
/=?
Statement (a) can be easily proved by reformulating Theorem 9.2.2 in terms of
epigraphs. The proof of statement (b) is based on the continuity of the Young-Fenchel
transform with respect to epiconvergence (see [14,62]).
Remark 13.3. There is another approach allowing one to prove more general versions
of Theorem 13.2(b). It is based on the formulation of the epigraphical limit in terms
of Lipschitz approximation. It has applications to Statistics [76,108], Optimization
Theory [13,57,58], Ergodic Theory and in Econometrics [43].
14. Concluding remarks
As illustrated above, the theory of measurable multifunctions raises a great variety of
problems involving probability theory and mathematical analysis arguments. It allows
For every real a. we set a+ = max@. a).
L

666
С. Hess
for new points of view and is often at origin of new approaches for classical questions.
Of course, we could have presented other topics of interest. Let us mention, for
example, the measurability issues concerning the extreme points of multifunctions (see,
e.g., [41, Chapter IV]). Further, measurable multifunctions depending measurably or
continuously on a second parameter play a important role in several applications. We refer
the reader to [34,35] and [165] for continuous dependence (Caratheodory multifunctions),
and to Godet-Thobie [87] and Artstein [7] for measurable dependence. Concerning
convergence problems, one can mention the weak convergence of multifunctions studied
by Artstein [4], using two different approaches: one based on support functions, the other
on the embedding technique. A result on large deviations for sums of random sets was
proved recently by Cerf [42].
The theory of random sets has also beautiful and rich geometric aspects, especially
when X is finite dimensional. Let us mention, for example, the books of Kendall and
Moran [125], Matheron [148], Santalo [183], and that of Stoyan et al. [191]. In this
context, a lot of interesting probabilistic and statistical applications have been given, for
example, to Sterology and to Image Processing. In fact, the Statistics of random sets,
which has connections with spatial stochastic processes, is a very lively domain. One can
find works dealing with theoretical aspects, and others with engineering and numerical
applications. Let us mention [ 148,191 ], Molchanov [151,152], Schmitt and Mattioli [ 187],
Aubin [ 15], and Choirat and Seri [44]. The interested reader may consult the proceedings of
the international symposium on "Advances in Theory and Applications of Random Sets",
edited by D. Jeulin (World Scientific, 1997), where a large variety of recent papers are
available along this line.
References
[1] R. Alo. A. de Korvin and R. Roberts. The optional sampling theorem for convex set-valued martingales.
J. Reine Angew. Math. 310 A979), 1-6.
[2] Z. Artstein, Set-valued measures. Trans. Amer, Math, Soc, 165 A972).
[3] Z, Artstein, Distribution of random sets and random selections. Israel J. Math, 46 D) A983).
[4] Z, Artstein, Weak convergence of set valued functions and control. SIAM J, Control Optim. 13 D) A975),
865-878.
[5] Z. Artstein, A note on Fatou's lemma in several dimensions. J, Math. Econom. 6 A979). 277-282.
[6] Z. Artstein, Parametrized integration of multifunctions with applications to control and optimization.
SIAM J. Control Optim. 27 F) A989), 1369-1380.
[7] Z. Artstein, Relaxed multifunctions and Young multimeasures. Set-Valued Anal. 6 C) A998). 237-255.
[8] Z, Artstein and J,A. Bumes. Integration of compact set-valued functions. Pacific J. Math, 58 B) A975).
297-307,
[9] Z. Artstein and J.C, Hansen, Conve.xification in limit laws of random sets in Banach spaces. Ann, Probab.
13A985), 307-309,
[10] Z. Artstein and S, Hart, Law of large numbers for random sets and allocation processes. Math, Oper, Res,
6A981). 482^492,
[11] Z, Artstein and R.A, Vitale. A strong law of large numbers for random compact sets. Ann. Probab. 3
A975), 879-882.
[12] Z, Artstein and R. Wets. Approximating the integral of a multifunction, J. Multivariate Anal. 24 A988),
285-308,
[13] Z, Artstein and R, Wets, Consistency of minimizers and the SLLNfor stochastic programs. J, Convex Anal.
2 A-2) A995), 1-17.

Set-valued integration and set-valued probability theory: An overview
667
[14] H. Attouch. Variational Convergence for Functions and Operators. Applicable Mathematics Series,
Pitman Advanced Publishing Program, New York A984).
[15] J.P. Aubin, Mutational and Morphological Analysis. Birkhauser. Boston A999).
[16] J.P. Aubin and H, Frankowska. Set-Valued Analysis. Birkhauser. Boston A990).
[17] R.J, Aumann. Integrals of set-valued functions. J. Math, Anal. Appl. 12 A965). 1-12,
[18] S, Bagchi. On a.s. convergence of classes of multivalued asymptotic martingales. Ann. Inst, H. Poincare.
Probab, Statist, 21 D) A985). 313-321.
[19] E, Balder, A unifying note on Fatou's Lemma in several dimensions. Math, Oper, Res, 9 A984), 267-275,
[20] E, Balder. A unified approach to several results involving integrals of multifunctions. Set-Valued Anal, 2
A994). 63-75.
[21] E, Balder and С Hess. On the unbounded multivalued version of Fatou's Lemma. Math, Oper Res. 20 A)
A995),
[22] E, Balder and С Hess, Two generalizations of Komlos' theorem with lower closure-type applications.
J, Convex Anal. 3 A996). 1-20.
[23] A, Barbati. G. Beer and С Hess, J. Convex Anal, 1 A) A994), 107-119,
[24] A, Barbati and С Hess. On the largest class of closed convex valued multifunctions for which Effros
measurability and scalar measurability coincide. Set-Valued Anal. 6 A998). 309-326,
[25] D, Barcenas and W. Urbina, Measurable multifunctions in nonseparable Banach spaces. SIAM, J. Math,
Anal, 28 E) A997). 1212-1226.
[26] G, Beer. A Polish topology for the closed subsets of a Polish space. Proc, Amer. Math, Soc. 113 A991).
1123-1133,
[27] G. Beer. Topologies on Closed and Closed Convex Sets. Mathematics and Its Applications, Kluwer Acad,
Publ., Dordrecht A993),
[28] V.E. Benes, Martingales on metric spaces. Theory Probab, Appl. 7 A962). 81-82.
[29] P, Billingsley, Convergence of Probability Measures. Wiley A968).
[30] J.M. Bismut, Integrates convexes et probabiite's. J. Math, Anal, Appl, 42 A973). 639-673.
[31] C.L. Byrne. Remarks on the set-valued integrals of Debreu and Aumann. J, Math. Anal. Appl. 62 A978),
243-246,
[32] J.W.S. Cassels. Measure of nonconvexity of sets and the Shapley-Folkman-Starr theorem. Math. Proc,
Cambridge Phil. Soc, 78 A975), 433^436,
[33] С Castaing. Sur les multi-applications mesurables. R.I.R.O. 1 A967). 91-126.
[34] C, Castaing, Sur /'existence des sections separement mesurables et separement continues d'une
multifunction. Sem. Anal, Convexe de l'Universite de Montpellier, No. 14 A975),
[35] C. Castaing. A propos de /'existence des sections separement mesurables et separement continuez d'une
multifunction separement mesurable et separement semi-continue infe'rieureinent. Sem. Anal, Convexe de
l'Universite de Montpellier, No, 6 A976),
[36] C, Castaing, Me'thode de compacite' et de decomposition, applications: minimisation, convergence des
martingales, lemme de Fatou multivoque. Ann. Mat. Рига Appl, D) 164 A993). 51-75.
[37] C, Castaing. Weak compactness and convergences in Bochner and Pettis integration, Vietnam J, Math, 24
C) A996),
[38] С Castaing and P, Clauzure. Lemme de Fatou multivoque. Atti, Sem, Mat, Fis, Univ. Modena 39 A991),
303-320.
[39] C, Castaing and F, Ezzaki. Convergence of convex weakly compact random sets in super-reflexive Banach
spaces. Preprint. Universite de Montpellier II A996).
[40] С Castaing and M, Valadier. Equations differentielles muttivoques dans les espaces vectoriels localement
convexes. R.I.R.O, 16 A969). 3-16,
[41] С Castaing and M. Valadier. Convex Analysis and Measurable Multifunctions. Lecture Notes in Math,,
Vol, 580. Springer, Berlin A977),
[42] R, Cerf, Large deviations for sums ofi.i.d. random compact sets. Proc. Amer, Math. Soc, A998).
[43] С Choirat. С Hess and R, Seri. A functional version of the Birkhoff ergodic theorem for a normal
integrand: A variational approach. Working paper No. 0101. Centre de Recherche Viabilite. Jeux.
Controle, Universite Paris Dauphine B000). to appear in Ann, Probab,
[44] С Choirat and R, Seri, Confidence sets for the integral of random closed sets, in: Proceedings of the 8th
International Conference IPMU. Madrid. July 3-7 B000). 509-514.

668
С. Hess
[45] G. Choquet, Theory of capacities. Ann. Inst. Founer 5 A953/1954). 131-295.
[46] A. Choukairi-Dini. M-convergence et regularite des martingales multivoques: epi-martingales. J.
Multivariate Anal. 33A) A990).
[47] A. Colubi. M. Lopez-Diaz. J.S. Dominguez-Menchero and M.A. Gil. A generalized strong law of large
numbers. Probab. Theory Related Fields 114 A999). 401^417.
[48] A. Coste. Contribution a la theorie de {'integration multivoque. These d'Etat. Universite Pierre et Marie
Curie, Paris A977).
[49] A. Coste, Sur les multimesures ? valeurs ferme'es bornees d'un espace de Banach. C. R. Acad. Sci. Paris
280A975). 567-570.
[50] A. Coste. La propriete de Radon-Nikodim en integration multivoque. С R. Acad. Sci. Paris 280 A975).
1515-1518.
[51] A. Coste, Sur les martingales multivoques. С R. Acad. Sci. Paris 290 A980). 953-956.
[52] A. Coste and R. Pallu de la Barriere. Un theoreme de Radon-Nikodym pour les multimesures a valeurs
fermees localement compactes sans droite. C. R. Acad. Sci. Paris 280 A975). 255-258.
[53] A. Coste and R. Pallu de la Barriere. Sur I'ensemhle des selections d'une multimesure a valeurs fermees,
C. R. Acad. Sci. Paris 282 A976). 829-832.
[54] A. Coste and R. Pallu de la Barriere. Radon-Nikodxm theorems for set-valued measures whose values are
convex and closed. Roczniki PTM Prace Mat. 20 A978). 283-309.
[55] A. Coste and R. Pallu de la Barriere. On set-valued measures with convex closed values and their set of
selections. Comment. Math. Spec. Issue 2 A979). 43-57. (Special issue dedicated to W. Orlicz.)
[56] J. Couvreux and С Hess. A Levy type martingale convergence theorem for random sets with unbounded
values. J. Theoret. Probab. 12 D) A999).
[57] J. Couvreux and C. Hess. Approximation an sens de Mosco d'integrandes et defonctionnelles integrates.
С R. Acad. Sci. Paris Ser. I 321 A995). 159-164.
[58] J. Couvreux and C. Hess. Mosco approximation of integrands and integral functionals. J. Optim. Theory
Appl. 90 B) A996), 335-356.
[59] N. Cressie. A strong limit theorem for random sets. Adv. in Appl. Probab. Suppl. 10 A978). 36-46.
[60] N. Cressie. Random set limit theorem. Adv. in Appl. Probab. Suppl. 11 A979). 281-282.
[61] N. Cressie. A central limit theorem for random sets. Z. Wahrscheinlichkeitsth. verw. Geb. 49 A978). 37-
47.
[62] G. Dal Maso. An Introduction to Г-Convergence. Birkhaiiser. Boston A993).
[63] J.P. Daures, Version multivoque du theoreme de Doob. Ann. Inst. H. Poincare 9 B) A973). 167-176.
[64] E. De Giorgi, Convergence problems for functions and operators: Proceedings of International Meeting
on Recent Methods in Nonlinear Analysis (Rene. 1978). E. De Giorgi. E. Magenes and U. Mosco, eds,
Pitagora. Bologna A979). 131-188.
[65] G. Debreu. Integration of correspondences. Proceedings of Fifth Berkeley Symposium on Mathematical
Statistics and Probability. Vol. II. Part I A966). 351-372.
[66] G. Debreu and D. Schmeidler. The Radon-Nikadym derivative of a correspondence. Proceedings of the
Sixth Berkeley Symposium on Mathematical Statistics and Probability A971).
[67] С Dellacherie and P.-A. Meyer. Probability et Potentiel. Chapitres I a IV. Hermann. Pans A975).
[68] J. Diestel and J.J. Uhl Jr.. Vector Measures. Math. Surveys. Vol. 15. Amer. Math. Soc.. Providence. RI
A977).
[69] N. Dinculeanu. Vector Measures. Pergamon Press. Oxford A967).
[70] W.L. Dong and Z.P Wang. On the representation and regularity of continuous parameter multivalued
martingales, Proc. Amer. Math. Soc. 126 F) A998). 1799-1810.
[71] H. Doss. Esperance mathematique et martingales dans un espace metrique. Laboratoire d'Analyse et
Modeles Stochastiques. annee 1988-99 A988). 23-27.
[72] S. Doss. Sur la moyenne d'un element aleatoire dans un espace distancie. Bull. Sci. Math. 73 A949).
1-26.
[73] S. Doss. Moyennes conditionnelles et martingales dans un espace metrique. С R. Acad. Sci. Paris Ser. A
254 B2) A962).
[74] L. Drewnowski. Additive and countably additive correspondences. Ann. Soc. Math. Polon. Comm. Math.
19A976). 25-54.

Set-valued integration and set-valued probability- theory: An overview
669
[75] N. Dunford and J.T. Schwartz, Linear Operators. Parti; General Theory. 3nd edn. Interscience Publishers
Inc. Wiley, New York A966).
[76] J. Dupacova and R. Wets. Asymptotic behaviour of statistical estimators and of optimal solutions of
stochastic optimization problems. Ann, Statist. 16 D) A988), 1517-1549.
[77] E.B. Dynkin and I.V. Evstigneev. Regular conditional expectations of correspondences. Theory Probab,
Appl. 21 B) A976). 325-338.
[78] K, El Amri and С Hess. On the Pettis integral of closed valued multifunctions. Set-Valued Anal. 8 B000).
329-360.
[79] A, Es-Sahib and H, Heinich. Barycentre canonique pour un espace metrique a courbure negative. Preprint.
UPRES-A 6085, Analyse et Modeles Stochastiques A998),
[80] N, Etemadi, An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitsth. verw. Geb,
55A981). 119-122,
[81] I.V, Evstigneev, Regular conditional expectations of random variables depending on parameters. Theory
Probab, Appl, 31 C) A986), 515-518.
[82] I.V, Evstigneev. Measurable selections theorems and probabilistic control models in general topological
spaces, Math, USSR Sb, 59 A) A988),
[83] F, Ezzaki. Convergence des esperances conditionnelles d'ensembles aleatoires et LFGN. These d'Etat,
Universite de Rabat, Morocco, June A996),
[84] M. Frechet, Les elements aleatoires de nature quelconque dans un espace distancie. Ann, Inst, H, Poincare
10D) A948). 215-310,
[85] E, Gine, M.G. Hahn and J. Zinn. Limit theorems for random sets; an application of probability in Banach
space results. Probability in Banach Spaces IV, A, Beck and K, Jacobs, eds. Lecture Notes in Math..
Vol, 990. Springer. Berlin A983), 112-135.
[86] С Godet-Thobie. Sur les multimesures de transition. C. R, Acad. Sci. Paris 278 A974). 1367-1369,
[87] C, Godet-Thobie, Selections de multimesures. Applications a un theoreme de Radon-Nikodym multivoque.
C. R, Acad, Sci, Paris 279 A974). 603-606.
[88] С Godet-Thobie. Some results about multimeasures and their selections. Measure Theory at Oberwolfach
1979, Lecture Notes in Math, Vol, 794. Springer, Berlin A980). 112-116.
[89] S. Hart and E, Kohlberg. Equally distributed correspondences. J, Math, Econom, 1 A976).
[90] W, Herer. Esperance mathematique au sens de Doss d'une variable aleatoire ? valeurs dans un espace
metrique. С R, Acad, Sci. Paris Ser. I 302 C) A986).
[91] W. Herer, Esperance mathematique au sens de Doss d'une variable aleatoire ? valeurs dans un espace
metrique. С R. Acad. Sci. Paris Ser. I 305 A987). 275-278.
[92] W. Herer, Esperance mathematique au sens de Doss d'une variable aleatoire a valeurs dans un espace
metrique. С R. Acad. Sci. Paris Ser. I 306 A988). 681-684.
[93] W. Herer, Martingales of random subsets of a metric space of negative cunature, Set-Valued Anal. 5
A997), 147-157.
[94] C. Hess, Theoreme ergodique et hi forte des grands nombres pour des ensembles aleatoires. C. R. Acad.
Sci. Paris Ser. A 288 A979). 519-522.
[95] C. Hess, Sur les multi-applications ? valeurs fermees dans un espace de Frechet. Partie 4: theoreme
ergodique et loi forte des grands nombres. Cahiers Math, de la Decision, No. 7932, Universite Paris
Dauphine A979), 1-27.
[96] C. Hess, Loi de probabilite des ensembles aleatoires a valeurs fermees dans un espace de Banach
separable, C. R. Acad. Sci. Paris 296 A983).
[97] C. Hess, Loi de probabilite et independance pour des ensembles aleatoires ? valeurs fermees dans un
espace de Banach, Sem. Anal. Convexe de l'Universite de Montpellier, No. 7 A983).
[98] C. Hess, Quelques theoremes limites pour des ensembles aleatoires home's он поп, Sem. Anal. Convexe
de l'Universite de Montpellier, No. 12 A984).
[99] C. Hess, Loi forte des grands nombres pour des ensembles aleatoires поп homes a valeurs dans un espace
de Banach separable. С R. Acad. Sci. Paris Ser I 300 A985), 177-180.
[100] C. Hess, Mesurabilite. convergence et approximation des multifonctions a valeurs dans un e.l.c.s.. Sem.
Anal. Convexe de l'Universite de Montpellier, No. 9 A985).
[101] C. Hess, Quelques re'sultats sur la mesurabilite' des multifonctions ? valeurs dans un espace metrique
separable, Sem. Anal. Convexe de l'Universite de Montpellier, No. 1 A986).

670
С. Hess
102] С. Hess, Theoreme de la convergence dominee pour Vintegrale et I'esperance conditionnelle des
ensembles aleatoires поп home's et des integrandes, C. R. Acad. Sci. Paris Ser. I 306 A988), 139-142.
103] С Hess, Sur la loi de probabilite d'un ensemble aleatoire a valeurs fermees dans tin espace metrique
separable. Sem. Anal. Convexe de I'Universite de Montpellier, No. 19 A991).
104] C. Hess, Com-ergence of conditional expectations for unbounded random sets, integrands and integral
junctionals. Math. Oper. Res. 16 C) A991), 627-649.
105] C. Hess, Sur I 'existence de selections martingales et la convergence des surmartingales multivoques. C. R.
Acad. Sci. Paris Ser. I 312 A991). 149-154.
106] C. Hess, On multivalued martingales whose values may be unbounded: martingale selectors and Mosco
convergence, J. Multivariate Anal. 39 A) A991).
107] С Hess, Multivalued strong laws of large numbers in the slice topology. Application to integrands. Set-
Valued Anal. 2 A994), 183-205.
108] C. Hess, Epi-convergence of sequences of normal integrands and strong consistency of the maximum
likelihood estimator. Ann. Statist. 24 A996). 1298-1315.
109] C. Hess. On the almost sure convergence of random sets: martingales and extensions. Cahiers Math, de la
Decision, No. 9743, CEREMADE A997).
ПО] С Hess, Conditional expectation and martingales of random sets. J. Pattern Recognition 32 A999), 1543-
1567.
Ill] C. Hess. The distribution of unbounded random sets and the multivalued strong law of large numbers in
nonreflexive Banach spaces, J. Convex Anal. 6 A) A999), 163-182.
112] С Hess, The Doss integral for random sets. Comparison with the Aumann integral. Proceedings of the 8th
International Conference IPMU. Madrid. July 3-7 B000), 515-520.
113] F. Hia'i, Radon-Nikodym theorems for set-valued measures. J. Multivariate Anal. 8A978), 96-118.
114] F. Hiai, Multivalued conditional expectations, multivalued Radon-Nikodym theorems, integral
representation of additive operators and multivalued strong law of large numbers. Conference of Catania 1983,
working paper.
115] F. Hia'i, Strong Law of Large Numbers for Multivalued Random Variables. Lecture Notes in Math.
Vol. 1091, Springer, Berlin A984).
116] F Hiai, Convergence of conditional expectations and strong laws of large numbers for multivalued random
variables. Trans. Amer. Math. Soc. 291 A985), 613-627.
117] F. Hia'i and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions.
J. Multivariate Anal. 7 A977), 149-182.
118] W. Hildenbrand, Core and Equilibria of a Large Economy. Princeton University Press A974).
119] W. Hildenbrand and J.P. Mertens. On Fatou's Lemma in several dimensions. Z. Wahrscheinlichkeitsth.
verw. Geb. 17A971), 151-155.
120] C.J. Himmelberg, Measurable relations. Fund. Math. 87 A975), 53-72.
121] C.J. Himmelberg, T. Parthasarathy and F.S. Van Vleek, On measurable relations. Fund. Math. Ill A981),
161-167.
' 122] M. Hukuhara, Sur I 'application semi-continue dont la valeur est un compact convexe. Funkcial. Ekvac. 10
A967), 43-66.
[123] M. Hukuhara, Integration des applications mesurables dont la valeur est compacte et convexe. Funkcial.
Ekvac. 10A967), 205-223.
[124] A.D. Ioffe, Sun-eys of measurable selections theorems. SIAM J. Control Optim. 16 A978), 728-732,
Russian literature supplement.
[125] M.G. Kendall and P.A.P. Moran, Geometrical Probability. Charles Griffin, London A963).
[126] M.A. Khan and Y. Sun, Integrals of set-valued functions with a countable range. Math. Oper. Res. 21 D)
A996), 946-954.
[127] H.A. Klei. Le theoreme de Radon-Nikodym pour des multimesures ? valeurs faiblement compactes, С R.
Acad. Sci. Paris 297 A983), 643-646.
[128] H.A. Klei, Compacite faible de parties decomposables de Ll(E). C. R. Acad. Sci. Paris 296 A983),
965-967.
[ 129] H.A. Klei, A compactness criterion in L[(E) and Radon-Nikodym theorems for multimeasures. Bull. Sci.
Math. B) 112A988), 305-324.
[130] E. Klein and A. Thompson, Theory of Correspondences, Wiley, New York A984).

Set-valued integration and set-valued probability theory: An overview
671
[131] G. Krupa, Limit theorems for random sets. Thesis. University of Utrecht. November A998).
[132] G. Krupa, Convergence of unbounded multivalued supermartingales in the Mosco and slice topologies.
J. Convex Anal. 5A) A998). 187-198.
[ 133] G. Krupa. Komlos theorem for unbounded random sets. Set-Valued Anal. 8 B000). 237-251.
[134] R. Kruse. On the variance of random sets. Math. Anal. Appl. 122 A987). 469-473.
[135] H. Kudo. Dependent experiments and sufficient statistics. Natur. Sci. Rep. Ochanomizu Univ. 4 A954).
151-163.
[136] K. Kuratowski and С Ryll-Nardzewski. A general theorem on selectors. Bull. Polish Acad. Sci. Math. 13
A965). 397-603.
[137] M. Lavie, These, Universite de Montpellier A990).
[138] M. Lavie, Fonction caracteristique d'un ensemble ale'atoire. Applications a la convergence de martingales
et de sommes d'ensembles aleatoires. С R. Acad. Sci. Paris Ser. I Math. 314 A992). 399^402.
[139] M. Lavie, An extension of Korzeniowski's theorem for bounded multivalued martingales, Actas de las III
Jomadas Zaragoza-Pau de Matematica Aplicada. Universidad de Zaragoza B000).
[140] M. Lavie, Characteristic function for random sets and convergence of sums of independent random sets.
Acta Math. Vietnamica 25 A) B000). 87-99.
[141] S.J. Leese. Miiltifiinctions ofSuslin type. Bull. Austral. Math. Soc. 11 A974). 395^411.
[142] S.J. Leese. Measurable selections in normed spaces. Proc. Edinburgh Math. Soc. 19 A974). 147-150.
[143] S. Li and Y. Ogura, Convergence of set-valued and fuzzy-set-valued martingales. Fuzzy Sets and Systems
2498 101 C) A997), 139-149.
[144] D.Q. Luu, Multivalued quasi-martingales and uniform amarts. Acta Math. Vietnamica 7 B) A982), 3-25.
[145] D.Q. Luu, Applications of set-valued Radon-Nikodsm theorems to convergence of multivalued L -amarts.
Math. Scand. 54A984), 101-113.
[146] N.N. Lyashenko, On limit theorems for sums of independent compact random subsets in the Euclidean
space. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 85 A979). 113-128.
[147] N.N. Lyashenko. Statistics of random compact sets of Euclidean space. Zap. Nauchn. Sem. Leningrad
Otdel. Mat. Inst. Steklov 98 A980). 115-139.
[148] G. Matheron. Random Sets and Integral Geometry. Wiley A975).
[149] E.J. Mc Shane. A Riemann-type integral that includes Lebesgue-Stieltjes. Bochner and stochastic
integrals, Mem. Amer. Math. Soc. 88 A969).
[150] K. Meaya, Characterization of Gaussian random sets. Afrika Mat. 7 C) A997). 39-59.
[151] I.S. Molchanov, Limit Theorems for Unions of Random Closed Sets. Lecture Notes in Math., Vol. 1561.
Springer, Berlin A993).
[152] I.S. Molchanov, Statistical problems for random sets. Random Sets. Theory and Applications. J. Goutsias,
R.P.S. Mahler and H.T. Nguyen, eds. IMA Volumes in Mathematics and its Applications, Vol. 97. Springer,
Berlin A997).
[153] U. Mosco. Convergence of convex sets and solutions of variational inequalities. Adv. Math. 3 A971).
510-585.
[154] U. Mosco, On the continuity of the Young-Fenchel transform. J. Math. Anal. Appl. 35 A971). 518-535.
[155] K. Musiat, Topics in the theory of Pettis integration, in: School of Measure Theory and Real Analysis.
Grado. Italy, May A992).
[156] K. Musial, Pettis integral. Handbook of Mesure Theory. E. Pap. ed.. Elsevier. Amsterdam B002). 531-586.
[ 157] J. Neveu, Martingales ? temps discret. Masson et Cie. Paris A972). English translation: J. Neveu. Discrete-
Parameter Martingales. North-Holland. New York A975).
[158] J. Neveu. Convergence presque sure de martingales multivoques. Ann. Inst. H. Poincare В 8 D) A972),
1-7.
[159] H.T. Nguyen and N.T. Nguyen, A negative version on Choquet's Theorem on Polish spaces. East-West J.
Math. 1A998). 61-71.
[160] T. Norberg. Convergence and existence of random set distributions, Ann. Probab. 12 C) A984), 726-732.
[161] ? Norberg. Random capacities and their distributions, Probab. Theory Related Fields 73 A986). 281-297.
[162] R. Pallu de la Barriere, Quelques proprietes des multimesures. Sim. Anal. Convexe de l'Universite de
Montpellier. No. 11 A973).
[163] N.S. Papageorgiou. On the theory of Banach space valued miiltifiinctions I. Integration and conditional
expectation. J. Multivariate Anal. 17 A985). 185-206.

672 С. Hess
[164] N.S. Papageorgiou, On the theon; of Banach space valued multifunctions 2. Set valued martingales and
set valued measures,]. Multivariate Anal. 17 A985), 207-227.
[165] N.S. Papageorgiou. On measurable multifunctions with applications to random multivalued equations.
Math. Japon. 32 A987), 437^464.
[166] M.L. Puri and DA. Ralescu. Strong law of large numbers with respect to a set-valued probability measure,
Ann. Probab. 11 D) A983). 1051-1054.
[167] M.L. Puri and D.A. Ralescu, Limit theorems for random compact sets in Banach spaces. Math. Proc.
Cambridge Philos. Soc. 97 A985). 151-158.
[168] M.L. Puri and D.A. Ralescu. Gaussian random sets in Banach spaces. Theory Probab. Appl. 31 C) A986),
526-529.
[169] P. Raynaud de Fine. These. Universite de Montpellier A990).
[170] P. Raynaud de Fine, Theoreme ergodique ponctuel et lois fortes des grands nombres pour des points
aleatoires d'un espace metrique a courbitre negative. Ann. Probab. 25 B) A997). 738-766.
[171] H. Richter, Verallgemeinerung eines in der Stati.stik benotingen Satzes der Masstheorie, Math. Ann. 150
A963), 85-90 (corrections pp. 440-441 in the same volume).
[172] H.E. Robbins. On the measure of a random set I. Ann. Math. Stat. 15 A944). 70-74.
[173] H.E. Robbins. On the measure of a random set II. Ann. Math. Stat. 16 A945), 342-347.
[174] R.T. Rockafellar. Integral Junctionals, normal integrands and measurable selections. Non-Linear
Operators and Calculus of Variations. Lecture Notes in Math., Vol. 543. Springer. Berlin A976).
[175] R.T. Rockafellar and R. J.-B. Wets. Variational systems, in multifunctions and integrands, G. Salinetti, ed..
Lecture Notes in Math., Vol. 1091. Springer. Berlin A984). 1-54.
[176] D. Ross, Random sets without separability, Ann. Probab. 14 C) A986), 1064-1069.
[177] J. Saint-Raymond, Topologie sur I'ensemble des compacts поп vides dun espace topologique se'pare.
Seminaire Choquet, annee 1969/70, No. 21.
[ 178] J. Saint-Pierre and S. Sajid. Parametrized integral of multifunctions in Banach spaces. J. Math. Anal. Appl.
239 A999), 49-71.
[179] G. Salinetti. W. Vervaat and R. Wets. On the convergence in probability of random sets (measurable
multifunctions). Math. Oper. Res. 11 C) A986). 420-422.
[180] G. Salineni and R. Wets. On the comergence of sequences of convex sets infinite dimensions, SIAM Rev.
21A) A979), 18-33.
[181] G. Salinetti and R. Wets. On the convergence of closed-valued measurable multifunctions. Trans. Amer.
Math. Soc. 266 A) A981). 275-289.
[182] G. Salinetti and R. Wets. On the convergence in distribution of measurable multifunctions (random sets).
normal integrands, stochastic processes and stochastic infimo. Math. Oper. Res. 11 C) A986).
[183] L. Santalo. Integral Geometry and Geometric Probability. Addison-Wesley. Reading, MA A976).
[184] H.H. Schaefer, Topological Vector Spaces. Springer. Berlin A971).
[ 185] D. Schmeidler. Fatou 's Lemma in several dimensions, Proc. Amer. Math. Soc. 24 A970), 300-306.
[186] D. Schmeidler. Convexity and compactnesss in countably additive correspondences. Differential Games
and Related Topics, North-Holland, Amsterdam A971).
[187] M. Schmin and J. Mattioli, Morphologic mathematique. Rapport ASRF-92-2. Laboratoire Central de
Recherche Thomson - CSF. Domaine de Corbeville 91404 Orsay cedex A992).
[188] J. Serra. Image Analysis and Mathematical Morphology. Vol. I. Academic Press. London A982).
[189] K. Schiirger. Ergodic theorems for subadditive superstationary families of convex compact random sets,
? Wahrscheinlichkeitsth. verw. Geb. 62 A983). 125-135.
[190] Y. Sonntag and С Zalinescu, Set convergences: An attempt of classification. Proceedings of the Intl. Conf.
on Diff. Equations and Control Theory. Iasi, Romania. August 1990.
[191] D. Stoyan, WS. Kendall and J. Mecke, Stochastic Geometry and its Applications, 2nd edn. Wiley A985).
[192] R.L. Taylor and H. Inoue, Laws of large numbers for random sets, in: Random Sets, Theory and
Applications. J. Goutsias. R.P. Mahler and H.T Nguyen, eds. IMA Volumes in Mathematics and its
Applications. Vol. 97, Springer, Berlin A997), 347-360.
[193] D.S. Thiam, Multimesures positives. С R. Acad. Sci. Paris Ser. A 280 A975), 993-996.
[194] D.S. Thiam. Integrate de Daniell a valeurs dans un semi-groupe ordonne. C. R. Acad. Sci. Paris Ser. A
281A975), 215-218.

Set-valued integration and set-valued probability theory: An overview 673
[195] D.S. Thiam. Applications, a /'integration multivoque. de Vintegrale de Daniell dans un semi-groupe
ordonne, C. R. Acad. Sci. Paris Ser. A 280 A975). 955-958.
[196] D.S. Thiam, Integrates multivoques monotones. С R. Acad. Sci. Paris Ser. A 280 A975). seance du 15
decembre 1975.
[197] M. Tsukada, Convergence of best approximations in a smooth Banach space. J. Approx. Theory 40 A984),
301-309.
[198] T. Uemura, A law of large numbers for random sets. Fuzzy Sets and Systems 59 A993), 181-188.
[199] M. Valadier, Integration de convexesfermes. notamment d'epigraphes. R.I.R.O. R2 A970), 57-73.
[200] M. Valadier, Multi-applications mesurables a valeurs convexes compactes. J. Math. Pures Appl. 50 A971),
265-297.
[201] M. Valadier, Surl'esperance conditionnelle nmltivoque поп convexe. Ann. Inst. H. Poincare 16 B) A980),
109-116.
[202] M. Valadier, Conditional expectation and ergodic theorem for a positive integrand. J. Nonlinear Convex
Anal. 1 E) B000), 233-244.
[203] B. Van Cutsem, Martingales de multiapplications ? valeurs convexes compactes. C. R. Acad. Sci. Paris
269A969), 429^432.
[204] B. Van Cutsem, Elements aleatoires a valeurs сот-exes compactes. Thesis, Grenoble A971).
[205] B. Van Cutsem, Martingales de convexesfermes aleatoires en dimensionfinie. Ann. Inst. H. Poincare В 8
D) A972), 365-385.
[206] К. Vind, Edgeworth-allocations in an exchange economy with many traders. Intemat. Econom. Rev. 5
A964), 165-177.
[207] R.A. Vitale, On Gaussian random sets. Stochastic Geometry, Geometric Statistics, Stereology, R.V
Ambartzumian and W. Weil, eds, Teubner, Leipzig A984), 223-224.
[208] D.H. Wagner, Surveys of measurable selections theorems. SIAM J. Control Optim. 15 E) A977), 859-903.
[209] D.H. Wagner, Surveys of measurable selections theorems: on update. Measure Theory at Oberwolfach
1979, D. Kolzow, ed.. Lecture Notes in Math., Vol. 794, Springer, Berlin A980).
[210] D.H. Wagner, Integral of a convex-hull-valued function. J. Math. Anal. Appl. 50 A975), 548-559.
[211] Z.P. Wang and X.H. Xue, On convergence and closedness of multivalued martingales. Trans. Amer. Math.
Soc. 341 B) A994), 807-827.
[212] W Weil, An application of the central limit theorem for Banach-space-valued random variables to the
theory of random sets. Z. Wahrscheinlichkeitsth. verw. Geb. 60 A982), 203-208.
[213] R.A. Wisjman, Ccm-ergence of sequences of convex sets, cones and functions. Bull. Amer. Math. Soc. 70
A964), 186-188.
[214] R.A. Wijsman, Convergence of sequences of convex sets, cones and functions II. Trans. Amer. Math. Soc.
123A966), 32^45.
[215] N. Yannelis, Fatou's Lemma in infinite-dimensional spaces. Proc. Amer. Math. Soc. 102 A991), 303-310.
[216] H. Ziat, Convergence des ensembles aleatoires et applications ? ?? loi forte des grands nombres
multivoque. These de doctorat d'Etat, Universite Sidi Mohamed Ben Abdellah, Fes, Maroc B000).

CHAPTER 15
Density Topologies
Wladyslaw Wilczynski
Faculty of Mathematics, University ofLodi. Bimaiha 22. PL-90-238 Lodz. Poland
Contents
Introduction 677
1. Points of density of linear sets 678
2. Density topology 682
3. Approximately continuous functions 686
4. Density topology in Euclidean space 691
5. ? -density topologies on the real line 693
6. Another local property of measurable sets 700
References 700
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
675

This page intentionally left blank

Density topologies 677
Introduction
The purpose of this chapter is to present basic properties of the density topology on
Euclidean spaces with the special emphasis on the one-dimensional case. The limited
length of the chapter does not allow to consider numerous differentiation bases which
appear if the dimension of the space is greater than one. so we restrict ourselves to ordinary
basis consisting of squares with sides parallel to the axes and to strong basis consisting of
intervals (rectangles with sides parallel to the axes) in the plane E2.
A good reference on differentiation bases, Vitali covering and density theorem is
Bruckner A971) and de Guzman A975). Abstract density topologies are treated in Martin
A964), Lukes et al. A986), Zajicek A986-87).
A density topology on Euclidean «-space for general measure is studied in Troyer and
Ziemer A963). The existence of the lower density operator in a finite measure space has
been proved in Maharam A958). The family of 2' non-homeomorphic density topologies
is considered in Aniszczyk and Frankiewicz A986). A good reference on density topology
is Lahiri and Chakrabarti A991). Topologies generated by liftings are treated in Ionescu
Tulcea and Ionescu Tulcea A969). Other generalization of the density topology one can
find in Rusu A995).
The notions of density point and approximately continuous functions have been defined
at the beginning of XX century. The density topology appeared much later and very quickly
became an important concept.
In this chapter we start the presentation with the properties of single density points of a
set and the lower density operator, proceed to study these properties of density topology
which can be formulated and proved using the sets of density points only, then go to
the properties of density topology which need the notion of approximately continuous
functions (all on the real line). The fifth section is devoted to the brief presentation of the
ordinary and strong density topologies on the plane. The next is again one-dimensional:
we present some results concerning a wide class of topologies similar to density topology
and constructed with the use of theorems of Taylor A959. 1960). At last we present a local
property of measurable sets discovered by Lusin. Although it does not generate a density-
type topology, it is interesting by itself and probably will not appear in another place in the
Handbook.
Throughout this chapter С will denote the ? -algebra of Lebesgue measurable subsets of
the real line, I - the ? -ideal of null sets, ?* and ?* - the outer and inner Lebesgue measure,
respectively. ? + ? = {t + ?: t e E),xE = {xt: t e E} for ? с R and ? e R. ??, Ъ and
?? will stand for the ?-algebra of Lebesgue measurable sets, ?-ideal of null sets and the
Lebesgue measure on the plane R-.
If ? is a topology, then the interior of a set A, closure of a set A and so on shall be
denoted by r-int(A), r-cl(A) etc. We shall omit ? for a natural topology.
The rest of denotations is the same as in the whole Handbook.

678
W. Wilczynski
1. Points of density of linear sets
Let ? с ? be a Lebesgue measurable set. We say that a point jc() e ? is a density point of
? if and only if
k(En[xQ-h,xQ + h]) ,
lim = 1. A.0)
ft—0+ 2h
We say that *? e ? is a dispersion point of ? if and only if xo is a density point of the
complementary set ?' = ? \ E. Equivalently, xq is a dispersion point of ? if and only if
lim ЦЕП[*о-*.ДГо + *])=0. A.0?
л—о+ 2й
Let Ф(?) = {? е ?: .? is a density point of E} if ? e ?. Then we have ?(?) ? ?(? \
?) = 0, hence
Ф(?)\?с(Е\?)\Ф(Е\?). A.1)
The most important property of Ф(?) is described in the following theorem, called the
Lebesgue Density Theorem:
THEOREM 1.1. For any measurable set ? С ?. ?(???(?)) = 0.
PROOF (According to Oxtoby A980)). By virtue of A.1) it is sufficient to show that
?(? \ ?(?)) = 0. Assume also that ? is bounded. We have ? \ ?(?) = Uf >o Ae< where
?(??[? -lux + h])
? e ?: lim inf < 1 — ?
л^о+ 2/7
Observe that Af] С Ae, if ?? > ?:. Suppose that X*(AF) > 0 for some ? > 0. Then there
exists a bounded open set G D A f such that ? (G) < л/_!*/'. Let f be the family of all closed
intervals / such that / с G and ?(? П /) ^ A - ?) ¦ ?(/). Then ? includes arbitrarily short
intervals around each point of Ae (which means that ? is a Vitali cover of Af) and for any
disjoint sequence {/и}яен of elements off we have
r(AAU7")>0- (L2)
Indeed, ?*(?? П Ц, /») < ?„ ?(? П /,,) <: A - ?) ?„ Ml·,) ? A - ?)?@ < ?*(?).
Inductively we construct a special disjoint sequence {/„}„ег; of elements of ?. Let
do be the least upper bound of the measures (= lengths) of members of ? and choose
/? e ? such that ?(/|) > do/2. After choosing /| /„. let Z„ be the set of all
elements of ? that are disjoint to /| /„. From A.2) it follows that ?„ is non-empty.
Let d„ = sup{k(I): I e ?„}. Choose /,,+ i e Zn such that ?(/„+?) > d„/2. Put В =

Density topologies
679
Ae \ U(J /„. We have ?*(?) > 0 by A.2). Hence there exists a positive integer m such
that
f>/„)<^. A.3)
и=ш+|
For each ? > m let J„ denote the interval concentric with /„ such that ?G„) = 3?(/„). The
family of intervals {Jn}n>o do not cover В by virtue of A.3). Take ? e В \ U,TL,„-h J»-
Since ? & Ae \ Un'=i Ai, there exists an interval / e ?,„ centered at ? (recall that ? is a
Vitali cover of ??). There exists ? > m such that / П /„ ? 0. Indeed, in the contrary case
?(/) ^ d„ < 2?(/„+?) for all n, which is impossible since ?^1| ?(/?) ^ ?@ < +oo. Let
A: be the least integer for which / П h ? 0. Then к > m and ?(/) ? dk- \ < 2k(h). Hence
? as the center of / belongs to ? - a contradiction with .v ? U«lm+1 ^« ·
So ?(??) = 0 for each ? > 0 and finally ?(? \ ?(?)) = 0.
The "density theorem" is often proved similarly as above by means which are more
or less equivalent to Vitali's covering theorem (see Blumberg A918-19). Hobson A927),
Lusin and Sierpinski A917), Khintchine A922), Sierpinski A923) or Zajicek A979)). It is
worth mentioning that Lusin attributes the theorem to A. Denjoy (see Lusin A951), p. 220).
It is also possible to use the Lebesgue Differentiation Theorem (Lebesgue, 1904, p. 124)):
If F(x) = /av f(t)dt, where /: [a. b] -» R is summable. then F'(x) = f(x) a.e. on [a, b].
Indeed, observe that ? is a density point of Л с [а. b] if and only if
lim = 1 A.4 )
ft—o+ h
and
г 4AO[x~h,x])
lim = 1. A.4 )
Л—0+ h
If A.4') (respectively, A.4")) is fulfilled we say that ? is a right-hand (respectively, left-
hand) density point of A. Now apply the Lebesgue Differentiation Theorem to f = ??.
From A.4') and A.4") it follows immediately that x is a density point of А с К if and only
if
к(АГ)[х-И:.х + 1ц])
lim = 1. A.5)
hx — 0+ h\ + hi
Л? —0+
h\ +/n >0
The condition A.5) is sometimes written in the form
.. ?(??/„)
lim =0.
«-3C ?(/„)

680
W. Wilczyiiski
for each sequence {/,;}„ек of intervals convergent to ? (which means that ? e P|„ /„ and
?(/„) ,,^c 0), For a detailed analysis of the exceptional set in the Lebesgue Density
Theorem (that is, of the set where the density of a given set differs from 0 and 1), see
Besicovitch A957). It is natural to understand limits in A,0') and A.0") in the sense of
Cauchy (?-? language) or, if we assume AC in the sense of Heine:
For each sequence {/г„}„е^ of positive numbers convergent to 0
k(An[x-h„,x+h„])
hm = 1, (lo)
и-» ос 2/г„
for each sequence {/i„}„epj of positive numbers convergent to 0
k(An[x-h„,x + hn]) ,
hm =0, A.6 )
н-юс 2h„
THEOREM 1,2. //{?„}„,=n is a decreasing sequence convergent to 0 such that sup,, j^- =
с < oo, then ? is a dispersion point of A if and only if
k(An[x-t„,x+tn])
hm =0. A./)
PROOF, If ? is a dispersion point of A. then A,7) is obviously fulfilled.
Suppose now that A,7) holds and take an arbitrary sequence {/г„}леК of positive numbers
convergent to 0. For each ? e N there exists w„ e N such that t,„n + \ ^ h„ < t„,n. We have
also ????,,^?? mn = oo. Since
0 < ?<?? [*-*»¦* +h„]) < ЦАП[х-!,„„,?+ tm„])
1hn ^ 2i,„,] + |
_ k(An[x-tm,:,x + tmJ) r,„„ < ?(?? [*-?„,„,.?+ ?,„„]) c
2tm„ tin,, ¦*- ? 2?„,n
we have lim,,^ ^n|.r-ft,,-v+ft„|) = о, so A,6") holds.
Of course, similarly one can prove that a single sequence suffices for a density point.
However, the condition sup,,^^, -^ < oo is essential, as the following example shows:
Let {i,i}nepj be a decreasing sequence convergent to 0 such that sup,,^^ j^- = oo .Choose
a subsequence {m-heN f°r which limbec r^- = oo and put
A=( UK+l'A+i ·4])?( Ut"A+i -Ч.-4+?] ¦
Then we have (for ? = 0)
?(??[-?„,?„])
hm =0,
?— ? 2t„

Density topologies
681
while
Hm ??[-?+? ¦4.A+i-'»J) = ,
i—oc 2y/t,lli + \ -t„k
So Theorem 1.2 together with the above remark can be formulated in the following form:
A.0") is equivalent to A.7) if and only if sup,, ™- < oo.
From Theorem 1.1 if follows immediately that if ? is a Lebesgue measurable set, then
Ф(Е) is also measurable. In fact, Ф(?) is a Borel set, moreover, Ф(?) е F„s- Indeed,
? e Ф(Е) if and only if for each ? e N there exists к eN such that for each h e ]0, k~' [
wehaveX(?n[;t-/2,jt + /2])/B/2) ^ 1 -и-1. Hence
ад=Пи ? ?*
zieN A-eN /re J0.A--' [
б^(ЕП[х-и + *])^_д.,
2h
Since the set
ueR: ^ 1 - ?
2h
is closed for each ? e N and /г > 0, the conclusion follows.
If ? с К is an arbitrary set (not necessarily measurable), then we say that .vo e ? is an
outer density point of ? if and only if
,. k*iEn[x-h.x + h]) ,
hm = 1.
/1—0+ 2/?
where ?* denotes, as usual, the outer Lebesgue measure. Let ?*(?) = {? e ?: ? is an
outer density point of ?}. Then we have ?*(?) = ?(?), where D D ? is a measurable
cover of ?. The above equality follows from the fact that for each interval / e ? we have
?*(? П /) = ?(?) П /). So taking into account that ? \ Ф*(?) сО\ ?{?) we obtain
from Theorem 1.1 that ?(? \ Ф*(?)) = О, which means that almost each point of an
arbitrary set ? с ? is the point of outer density of ?. However, one cannot expect that
a\so ?(?*(?)\?) = 0, this is impossible for any non-measurable set ? С ?, since ?*(?)
(= ?(?)) is measurable.
Consider the following sequence of equivalent statements for a set A e C:
(A) 0 is a density point of A;
(B) Нт|,_>эсЛ(АП[-л-|,л-|])/B|1-|)= 1;
(C) ?????_>00?((?·?)?[-1,1]) = 2;
(D) {x(„.A)n[-i.i)},,eN convergestox[_U] in measure;
(E) for every increasing sequence {итЬ»ег; of positive integers there exists a
subsequence {птр}р€щ such that lim/,_>TCXi(H,„/),4)p,[-i.n) = X[-i.ij a-e-
The equivalences of (A) to (D) are straightforward, while the equivalence of (D) and (E)
follows from a classical theorem of Riesz concerning convergence in measure on a finite

682
W. Wilczynski
measure space. Observe that in the condition (E) the measure is absent and the only
important thing is the ?-ideal of null sets. This gives a possibility to define, among others, the
category analogue of the density topology. For details see Wilczynski A984-85), Ciesiel-
ski et al. A994). Compare also Tall A976) to see why it is not a "precise analogue" of the
density topology.
Using the condition (E) and taking into account that a sequence of non-measurable
functions may converge almost everywhere to a measurable function one can define a
density-type topology not included in С
This was done in Ciesielski and Larson A993) and Wilczynski A984) leading to a
topology with quite different properties.
2. Density topology
Let ? (A) again denotes the set of all density points of a Lebesgue measurable set А С К.
We shall also write A ~ ? if and only if ?(???) = 0 for measurable sets A.Scl.
THEOREM 2.1. The mapping ? : С -> 2" has the following properties:
@) for each A eC, Ф(А)еС,
A) for each A eC, А~Ф(А),
B) for each A, Be С, if A~ B, then Ф(А) = Ф(В),
C) Ф@) = 0, Ф(К)=К,
D) for each A, BeC, Ф(АП В) = Ф(А)ПФ(В).
PROOF. @) In fact Ф(А) e F„s, so ? (A) is even a Borel set.
A) This is the Lebesgue Density Theorem.
B) If A ~ ?, then for each interval/ с Ш wehaveX(A„n/) =?(??\?) and the equality
follows.
C) Obvious.
D) Observe first that it А, с A2, A,, A2 e C, then Ф(А\) с Ф(А:), so ?(? ? ?) С
? (А) П ?( ?). To prove the opposite inclusion, use the equality
?(/??) ?(/??) ?(/????)
— - + — - 1 ^ — -.
НП НП HD
which holds for each interval / с К.
The mapping ? having the above properties is usually called a lower density. Observe
also that ? is idempotent, i.e., ?(?) = ?(?(?)) for A e С. This follows immediately
from A) and B). Now we are ready to define the density topology. Put ? = {A e C: Ac
?(?)}.
THEOREM 2.2. ? is a topology in the real line R.

Densit\ topologies
683
PROOF. 0, R e ? by virtue of C). From D) it follows that is closed under finite
intersections. The only trouble is to prove that ? is closed under arbitrary unions (recall
that ? с С and С is closed only under countable unions).
Take a family {At}t€T с Т. We have obviously А, с ?(?,) for each t e T. Choose a
sequence {г„}„ем such that for each r e ? we have ?(?, \ \J^L] aO = 0· It is possible
because the Lebesgue measure on R fulfills the countable chain condition. Then we have
Ф(А,) с ?I |Ja,„ J. for each re T.
Hence
ОС / ОС \
[JAtnc\jA, C\J0(A,)C<P( \JA,,X
n=\ t?T teT \n=l /
The first and the last set in the above sequence of inclusions differ by a nullset and both
are measurable, so {jt€T A, e С Also LUr A' c ^'LUr A>) by virtue of the central
inclusion and the monotonicity of Ф, so finally \Jt€T A, &T. ?
The density topology was defined in Haupt and Pauc A952, 1954). However, these
papers had almost no impact. Some years later two papers: Goffman et al. A961) and
Goffman and Waterman A961) attracted the interest to the density topology. It is worth
mentioning that approximately continuous functions (the topic of our next paragraph) were
introduced by A. Denjoy A915. 1916) without relation to any topology on the real line.
The density topology can be also described in the following form: ? = {Ф(А)\Р: А е
С and ? e I}. Indeed, if A e T, then А с ? (A), so A = ? (A) \ (?(?) \ A) and
?(?(?) \ A) = 0. We have also ?(?) \ ? с Ф(Ф(А) \ ?), since ?(?(?) \ ?) =
?(?(?)) = ?(?) (compare Oxtoby A980)).
THEOREM 2.3. ? is essentially stronger than a natural topology on the real line.
PROOF. For each open interval ]a. b\ we have ]a. b[ с Ф( ]a,b[). To find a T-open set
which is not open in the natural topology consider a set of the form R\ E, where ? is a
nullset.
THEOREM 2.4. A point ? e R is a T-limit point of a set А С R (not necessarily
measurable) if and only if
k*(An[x-h.x+h])
lim sup > 0.
/1^0+ 2/2
PROOF. Follows from the definition of a T-limit point.

684
W. Wilczynski
THEOREM 2.5 (Ostaszewski A982)). For an arbitrary set А С R
IntTA = ???(?).
where В is a measurable kerne! of A.
PROOF. Let ? e IntTA. Then there exists a set U e ? such that и e U and U С A.
So ? e Ф([/). Since С/ \ В с Л \ В and С/ \ S e ?, we have ?(?/ \ ?) = 0. Hence
<P(U) = 0(U П ?) с ?(?) and ? e ? ? ?(?).
Suppose now that jc e А П ?(?). Then ? U {x} С A and ? U {x} e ?, so ?(? U {.r}) =
?(?). Hence ? e (? U {*}) ? ?(? U {.?}). Since ?((? U {*}) ? ?(? U {.?})) = ?(? U
{*}) П Ф(Ф(В U {*})) = ?(? U {*}), the set (? U {*}) ? ?(? U [?]) is T-open. So there
exists a T-open set including ? and included in A. hence ? e Intx A. D
THEOREM 2.6 (Scheinberg A971)). A set A e ? is 7 -regular open if and only if A =
?(?).
PROOF. Let A be T-regular open, i.e., A = IntT(Clr(A)). Then A = ClT(A) ?
Ф(С1Т(Л)) = Clx(A) П Ф(А) = ?(?). The first equality follows from Theorem 2.5,
the second from the fact that C\T(A) ~ A (because C\T{A) = R \ IntT(R \ A) and
Intx(R \ A) ~ R \ A) and from Theorem 2.1 B). the third from the inclusion Ф(А) с
Clx(A) (compareTheorem 2.4).
Suppose now that A = Ф(А). We have IntT(ClT(A)) = C1T(A) ? Ф(С1Т(А)) =
Clx(A) П Ф(А) = ?(?) = ?. The verification of the first, the second and the third
equalities is exactly as above, the fourth is simply our assumption. D
The set ? e R is a T-neighbourhood of a point ? e R if and only if there exists a set
Av e ? such that ? e AA and Ax с ?. From the inequality ?*(? П [? - h,x + h]) ^
?( ?? П [? - h, ? + /?]), which holds for all h > 0, it follows that ? is a T-neighbourhood
of ? if and only if
,. ?*(??[?-?,? +h])
THEOREM 2.7. ? is a T-neighbourhood of each of its point if and only if ? e Т.
PROOF. Suppose that ? is a T-neighbourhood of each ? e ? and let AA e ? be the set
such that ? e Av and А, с ?. Then ? = {JxeL Ax and ? e ? by virtue of Theorem 2.2.
The converse is obviously true.
THEOREM 2.8 (Oxtoby A980), Tall A976)). I = {А С R: A is 7'-nowhere dense set] =
{А С R: A is ?-first category set] = {А С R: A is ?-closed ?-discrete set].
PROOF. The first equality. If A e I. then R \ A e T, so A is T-closed. We have Int(A) =
А П Ф(А) = А П 0 = 0, hence A is T-nowhere dense. Suppose now that A is T-nowhere

Density topologies
685
dense. Then Cl-j-(A) is also T-nowhere dense, T-closed (and Lebesgue measurable). So
IntT(ClT(A)) = C1T(A) П Ф(С1Т(А)) = 0, hence C1T(A) e I (compare Theorem 2.1
A)) and finally A el.
The second equality. The family of T-nowhere dense sets is a ?-ideal, so it coincides
with the family of T-first category sets.
The third equality. If А с I, then for ? e A the set (R \ A) U {x} is a T-neighbourhood
of x, so А П ((R \ A) U {?}) = {?} is а Тд-neighbourhood of ? (Тд is a subspace topology
on A) and A is discrete. A is also closed, as we saw earlier.
Suppose now that A is T-closed T-discrete and A ? I. Then there exists a non-
measurable set ? с ?. ? is not T-closed (neither Тд-closed), because T-closed (and
Тд-closed) sets are Lebesgue measurable - a contradiction. ?
THEOREM 2.9 (Scheinberg A971)). ? ?-algebra ofT-Borel sets coincides with С
PROOF. If A e ?, then A = (А П ?(?)) U (A \ ?(?)). The first set belongs to T. while
the second is the nullset, hence a T-closed set. Hence each Lebesgue measurable set is a
T-Borel set. Since T-open sets are measurable, the same is true for T-Borel sets. Finally
the family of T-Borel sets coincides with ? -algebra of Lebesgue measurable sets. D
THEOREM 2.10. If ? cRisa 7-compact set, then ? is finite.
PROOF. Suppose that ? с R is infinite. Let ?() с ? be a countable infinite subset of E.
Then {(R \ ?()) U {jrH.vet,, is a T-open cover of ? without finite subcover. D
THEOREM 2.11. The space (R. T) is neither first countable, nor second countable, nor
Lindelof, nor separable.
PROOF. Take ? e R and let {En}„€\-, be a sequence of T-open neighbourhoods of .v. For
each ? e N choose xn e E„ \ {x} and put ? = E\ \ {x„: « e N). Then ? is also a T-open
neighbourhood of ? which does not include any E„, so (R. T) is not first countable. Hence
it is not second countable, too. Let С be the Cantor set. Then {(R \ C) U {.v}}vec is a
T-open cover of R without any subcover of the power less than continuum, so (R. T) is
not Lindelof. Finally, it is not separable because all countable sets are T-closed. D
THEOREM 2.12. (R, T) is a Baire space.
PROOF. The countable union of T-nowhere dense set is the set of Lebesgue measure zero
(by virtue of Theorem 2.8).
A surprising and very useful (e.g., in proving monotonicity theorems) property of density
has been discovered by O'Malley.
THEOREM 2.13@'Malley( 1972)). //Ac]0, 1 [ is a set oftype F„ which has left density
equal to 1 at all its points. Then there exists xq e [0. 1 [ \A such that A has right density 1
atx0.

686
W. Wilczymki
Much more set-theoretic properties of the density topology one can find in Tall A976)
and Lukes et al. A986).
3. Approximately continuous functions
If / is a real function of a real variable, then we say that / is approximately continuous at
a point xq if and only if there exists a set АЛA е С such that
х0еФ(АХи) and /(*,,) = Шг^ /(л) C.1)
ле.4,„
(Denjoy, 1915, 1916). The approximate upper limit of / at x(> (limsupapv_A.()/(*)) is
the greatest lower bound of all the numbers v(+oo included) for which xq is a point of
dispersion of the set {x e R: /(л) > у}. Similarly one can define liminfapA^V|i f(x).
When these extreme limits are equal, their common value is called the approximate limit
of / at Jto and denoted by limapv_v /(-*)· It is natural to say that / is approximately
continuous at xq if and only if
/(*o)= Um ap/U). C.2)
x-*xu
This definition was introduced also by Denjoy A916).
THEOREM 3.1. The above two definitions of the approximate continuity at a point are
equivalent.
PROOF. Observe that f(xo) = limapv_V|i f(x) if and only if for each ? > 0 xo is a density
point of the set {x e R: \f(x) - f(xo)\ < ?]. Suppose now that there exists a set AV|) e С
fulfilling C.1), then for each ? > 0 there exists ? > 0 such that {x e R: \f(x) - f(xo)\ <
?] D AXoC\ ]xo - ?, xo + <5[ and C.2) follows. To prove the converse implication we shall
need the following fact (which is also interesting by itself): if {Е„},,ец is a decreasing
sequence of measurable sets and xq is a density point of each ?„, then there exists a strictly
decreasing sequence {?„}„е^ of positive numbers convergent to 0 such that xq is also a
density point of the following set:
ОС
A*u = U ( E" \ ^° ~k"-x0 + kn[)-
n = \
This condition is called the condition (J2) of Jedrzejewski A973/74).
Let {
?»}neN be a sequence of positive numbers less than 1 convergent to 0. For each
? e N there exists <5„ > 0 such that for h e ]0, <5„ [ we have
k(E„n]xo-h,x0+h[)
2h >1"?"·
Choose a decreasing sequence {h,,},,^; convergent to 0 such that h„ e]0, <5,,[ for each
? e N and put A:,, = ?„ -h„+\. If h e [h„^\.h„[ for some ? e N then we have

Density topologies
687
k(AXon]xo-h,x0 + h[) ?((?„ П ]дг() - h.xQ + h[)- ]xq - k,„xo + k„[)
2/2 " 2/2
_ ?(??„? ]x0- h.xo + h[)- 2s„h„+i
2/2
^ ! ?„ > 1 - 2?„
2/г
and the conclusion follows. (See also Bruckner A978). Theorem 5.6, p. 23; Thomson
A985), Ex. 14.1, p. 27.) Suppose now that for each ? > 0 xq is a density point of the
set {x e R: \f(x) - f(x0)\ < ?}. Put E„ = {x e R: \f(x) - f(x0)\ < j;} and consider the
set A.V|) described above. Then
/(*o)= Hm /(*),
.veA,()
so / is approximately continuous in the sense of Denjoy. ?
The second definition of approximate continuity has a strict connection with the density
topology T. Namely, a real function / of a real variable is approximately continuous at a
point xq if and only if it is continuous at a point xq as a function from the topological space
(R, T) into a real line equipped with the natural topology. As usual, we shall say that a
function /: R —> R is approximately continuous if it is approximately continuous at each
point. In this case we have obviously /_l (G) e ? for each open (in the natural topology)
set G с R.
There is a strict connection between measurability of real function and approximate
continuity on a big set. To this aim we shall need a classical theorem (Lusin A912); see
also Oxtoby( 1980)).
THEOREM 3.2. A real function f defined on R is measurable if and only if for each ? > 0
there exists a set ? e С such that ?(?) < ? and the restriction of f toR\E is continuous.
We can require in the above theorem that the set R \ ? is closed (or perfect).
THEOREM 3.3 (Denjoy A916), Stepanoff A924)). A real function f defined on R (or on
a set ? e ?) is measurable if and only if f is approximately continuous almost everywhere
on R (on E, respectively).
PROOF. If / is a measurable function, then there exists a sequence {F,,},,^- of closed sets
such that f\F„ is continuous for each ? e N andX(R\F„) < «"'.Then / is approximately
continuous on the set U„eN(F„ ni»(F„)) and k(R \ \Jll€U F„ n<P(F„)) = 0.
Suppose now that / is approximately continuous almost everywhere. Take a e R and put
В = [?: f(x) < a}. We want to show that В is Lebesgue measurable. If С is the set where
/ is approximately continuous, then we have В = (В П С) U (? \ С) and ?(? \ С) = 0.
So it is sufficient to show that В П С is measurable. If ? e В П С, then there exists a set
Av e С such that ? e ?(??) and f(x) = lim;-. f(t). Since f(x) < a, we can choose

688 W. Wilczyiiski
Ax С В (taking, if necessary, АхГ\]х — ?, ? + ?[ for sufficiently small ? > 0), moreover,
we can assume that A.v с С (recall that ?(? \ С) = 0). Then В П С = \JxeBncAi and the
measurability follows as in Theorem 2.2. ?
Theorem of Lusin (Theorem 3.2) cannot be improved. The statement: "if /:R —>¦ R
is measurable, then there exists a set ? such that ?(?) = 0 and /|R \ ? is continuous"
is false. Indeed, let A e С be the set such that A and R \ A have positive measure in
every interval. Then for / = ?? the restriction of / to the complement of any nullset is
discontinuous at each x e R.
To study further properties of the density topology we shall need following facts (Denjoy
A916), see, for example, Bruckner A978)). We shall formulate theorems for functions
defined on R but they are obviously true also for functions defined on intervals (closed,
open or half-open).
THEOREM 3.4. // /:R —>¦ R is appmximately continuous and locally bounded, then
f(x) = F'(x) for each x e R, where F(x) = f* f (t) dt for fixed a e R.
PROOF. Since / is approximately continuous, it is measurable by Theorem 3.3. From local
boundedness it follows that F is defined on the whole real line R. Fix *? e R· We want to
show that
f(x0)=\im- f(t)dt. C.3)
A-o h J_Xu
Let AV() e С be the set such that xo e Ф(АХ{)) and /|AV|) is continuous at *o· Let
M = sup{|/(f): ге[дг0-1,*о+1]}.
Then we have
I 1 [,x«+h | 1
f° f(t)dt-f(xQ)U | f \f(t)dt-f(x0)\dt
•Avo I " ^[л-|,.Л|,+А]ПД1()
/
•/|.v()..V(.
+ ^/ |/(i)dr-/(jto)|dr.
The first term on the right-hand side of the above inequality tends to 0 with
h —>· 0, because |/(?) - f(xo)\ tends to zero, while the second tends to zero because
?([*?,^o + h]\ Axo)/h tends to 0 and the function under the integral sign is bounded
by 2M (for h < 0 [xq, xq + h] means obviously [xq + h. xq]). ?
THEOREM 3.5. /// is approximately continuous at xq and g is continuous at f(xo), then
g о / is approximately continuous at xq.
PROOF. Apply the definition of approximate continuity.
THEOREM 3.6. ///:R —>¦ R is approximately continuous, then it is Darboux-Baire one
function (/ eDB\).

Density topologies
689
PROOF. If / is locally bounded, then it is a derivative by Theorem 3.4. It is well-known
that a derivative is Darboux-Baire one and the thesis follows. If / is not locally bounded,
then let h be a homeomorphism of R onto @, 1). Since ho f is bounded and approximately
continuous, by Theorem 3.5, so ho f e DB\.But / = /?_l o(hof) and finally / e DB\.D
THEOREM 3.7. The family of ?-connected set coincides with the family of sets connected
in the natural topology.
PROOF. We shall prove that each bounded open interval (in the natural topology) is T-
connected. For remaining types of intervals (bounded or not) the proof is similar. Suppose
that ]a, b[ is not T-connected. Hence ]a, b[= A2 U A2, where A \, A2 e T. Put f = ??].
A function / is obviously approximately continuous and has not the Darboux property - a
contradiction with Theorem 3.6.
To finish the proof observe that since a density topology is strictly greater than a natural
topology, the classes of connected sets fulfill the converse inclusion and the equality
follows. ?
It is obvious that (R, T) is a Hausdorff space.
To establish that the space (R, T) is completely regular, we shall need the theorem which
is traditionally connected with the names of Lusin and Menchoff.
THEOREM 3.8 (see Bruckner A978)). Let ? С R be a Lebesgue measurable set (or a
Borel set) and let F с ? be a closed set such that F с ? (?). There exists a perfect set ?
such that F С ? С ? and F с Ф(Р).
THEOREM 3.9 (Zahorski A950)). Let E\.E2, ? be pairwise disjoint subset of Ж such
that E\ U F2 U # = R, E\ U #, F2 U # e ? and both sets are of type F„. Then there
exists an approximately continuous function /:R —>¦ R such that f(x) = 0 for x e E\,
0 < f(x) < 1 for x e ? and f(x) = 1 for x e F2.
The method of the proof reminds that of the Urysohn lemma.
THEOREM 3.10 (Goffman et al. A961)). The space (R, T) is completely regular.
PROOF. Let F с R be a T-closed set and xo i F. Let A be a G& set such that А э F and
?(? \ F) = 0. Put F| = A, E2 = {x0} and Я = R \ (F| U F2). Then F, U ? = R\ {x0} and
F2 U ? = R \ A are both T-open and Fs. Hence by virtue of Theorem 3.9 there exists an
approximately continuous function such that f(x) = 0 for x e E\ (E\ э F) 0 < f(x) < 1
for x e ? and f(x0) = 1. ?
THEOREM 3.11. The space (R, T) is not normal.
PROOF. Suppose that this space is normal. Let F|. F2 be disjoint sets which are dense in
R and ? (F1) = ?( F2) = 0. Then E\.E2 are T-closed. By virtue of Urysohn lemma there

690
W. Wilczxmki
exists an approximately continuous function / such that f(E\) = {0} and /(?2) — {1}·
Then / is discontinuous everywhere - a contradiction with the fact that / is Baire one. D
It is worth mentioning that if A, В с R are disjoint countable sets (possibly dense), then
it is possible to find disjoint T-open sets C. D such that А с С and В с D (Foran, 1991).
There are other topologies related to the density topology and to some subclasses of the
family of approximately continuous functions. A set Л с К is called ambivalent if it is both
an Fa set and a Gs set. Finite unions, finite intersections and complements of ambivalent
sets are ambivalent sets. A function /: R —>¦ R is ambivalent if and only if for each a e R
the sets {x: f(x) > a] and {x: f(x) < a] are ambivalent. Obviously each ambivalent
function is Baire one, however, the converse is not true. A function / is Baire* one if
and only if for every non-empty closed set С there exists an open interval (a. b) such that
]а,МПС^0 and the restriction of / to C, f\c, is continuous on ]a. b[ DC. A function
/ is approximately differentiable at xq if and only if there exists limA _>Л|1 ар -'(Л()~-/(Л|>) and
is finite. Of course, if / is approximately differentiable at xQ, then it is also approximately
continuous at this point. At last, let Л be the family of all sets from ? which are ambivalent.
We shall denote by r (after O'Malley) the coarsest topology making the approximately
differentiable functions continuous.
THEOREM 3.12 (O'Malley A977)). The family ? is a basis for r. Moreover, r is
simultaneously the coarsest topology for which either the Baire* one approximately
continuous functions or ambivalent approximately continuous functions are continuous.
O'Malley A977) has defined another topology in the following way: a set A e ? is
called almost open if and only if ?(?) = X(Int(A)).
THEOREM 3.13 (O'Malley A977)). The family of almost open sets forms a topology
(calleda.e. topology).
THEOREM 3.14 (O'Malley A979)). The a.e. topology is the coarsest topology for
which approximately continuous functions which are almost everywhere continuous are
continuous.
Theorem 3.15 (O'Malley A977)). a.e. ^r^T.
THEOREM 3.16 (O'Malley A977, 1979)). Both topologies r and a.e. are completely-
regular but not normal.
For further properties of topologies a.e. and r see the above quoted papers of O'Malley.
One more "density type" topology has been studied by M. Filipczak and ? Fil-
ipczak A988). They considered the family %? = {4>(G) \ P: G is open in the natural
topology and Pel].
THEOREM 3.17 (Filipczak and Filipczak A988)). %,p is a topology essentially stronger
than a natural topology, essentially weaker than a densit}· topology, essentially stronger
than a.e. topology and incomparable with the r topology.

Density topologies
691
An interesting example of topology has been given by Wojdowski A989). He defined
an operator Фс on the class of B„ sets having the Baire property in the following way:
ФС(А) = <P(G(A)), where G(A) is a regular open part of A. Recall that the set A e B„
if and only if A = GAP, where G is open (in the natural topology) and ? is of the
first category. Among all possible representations of this type there is one for which the
set G is the greatest and then it is regular open set, which we have denoted above by
G(A). The topology of Wojdowski equals the Hashimoto topology generated by the a.e.
topology and the ?-ideal of sets of the first category (that is the topology of the form
{A \ P: A e a.e. and ? is of the first category).
Sarkhel and De A981) while studying integrals of the Perron type have introduced a
notion of sparse set. A set ? с R is said to be sparse at a point ? e R on the right if and
only if for each ? > 0 there exists к > 0 such that each interval ]a. b[ с ]x. ? + k[ with
a - ? < к ¦ (b - x) contains at least one point у such that ?*(? П ]x. y[) < ?(? - ?).
Sparseness on the left is defined similarly. A set ? is said to be sparse at ? if and only if it
is sparse at ? on the right and on the left.
We shall apply this notion to measurable sets only. Let Sr(E) = {x e R: R\E is sparse
atjc).
THEOREM 3.18 (Filipczak( 1988-89)). For each ? e ?, Ф(Е) С S,,(E).
THEOREM 3.19 (Filipczak A988-89)). The operator S,, :? -> 2?- is a lower density
operator and the family 7V, = {E e ?: ? С Sp (?)} is a topology stronger than a density
topology.
For further properties of the topology 7}, see Filipczak A988-89). Most of them are
similar to that of the density topology.
4. Density topology in Euclidean space
Tall A976) says: "Many of the topics we touch upon can be treated in more general
measure-theoretic structures than the real line, but this does not appear to be particularly
fruitful topologically". However, in Euclidean space of dimension greater than one several
differentiation bases (see de Guzman A975)) are considered, from which at least two seem
to be interesting for the construction of density topologies. For simplicity of denotation we
shall present basic facts in R2, the generalization to R" (n ^3) being (in general) obvious.
We say that a point jto, }'o e R1 is an ordinary density point of the set A e ?2 if and only if
,. k2(Ar)([xo-h,xo + h]x[y0-h.yi)+h]))
lim ? = 1. D.U )
A—o+ 4h-
The condition D.1) is equivalent to the following one
к2(АГ\([хо-И„,Х(,+И„]
lim
n^oc 4h„k„
?2(??)([?0 -h,,,xQ +h„] ? [yo-k„. yo+?„]))
lim = 1. D.0 )

692
W. Wilczyiiski
for each pair of sequences of positive numbers {й„}„еь:, {?»ЬеК tending to 0 and for
which there exists a number a e]0, 1 [ (called the parameter of regularity) such that
a < h„ ¦ k~' < a~' for each ? e N.
Let ??(?) denotes the set of all ordinary density points of a set A e ?2 and let
7г = {A e ?2·' А с Фо(А)}. It turns out that Фо is a lower density operator and T> is
a topology on the plane, which means that two-dimensional versions of Theorems 2.1
and 2.2 are true. The (ordinary) approximate continuity of a real function of two real
variables at a point (xq, )?) defined in a similar way as in one-dimensional case either
as a restrictional (or "path") continuity or as continuity with respect to Ъ- Again both
definitions are equivalent. Also all remaining theorems from Section 2 and all but two
(Theorem 3.6 and Theorem 3.7) from Section 3 have their exact analogues is Euclidean
space of dimension greater than one. In Theorem 3.6 the first Baire class is the only one
which can be similarly proved for Ш". ? ^ 2. In Theorem 3.4 one should consider the
derivative of two-dimensional rectangle function (indefinite integral).
THEOREM 4.1. If f :R- —> Ш is (ordinarily) approximately continuous, then it is Baire
one function.
In the literature there are numerous (not equivalent) definitions of Darboux property for
functions of several variables (see, for example, Gibson and Natkaniec A996-97)). One of
the most natural says that a function / has the Darboux property if and only if it transforms
connected sets onto connected sets. However, the exact analogue of Theorem 3.7 does
not hold. For example, the linear segment / = [0.1] ? {0} is connected in the natural
topology on the plane, but it is not in topology Ъ. Indeed, / = (АП/)и(ВП/), where
A = {(x, у): у ?. 0 or (у > 0 and (у < -?2 or у > ?2))} and В = {(?, у): у > 0 and
—?2 < у < ?2} are disjoint 72-open sets. However, the following theorem holds:
THEOREM 4.2 (Goffman and Waterman A961)). Every open (in the natural topology on
R") connected set is T2-connected.
We say that a point (jt0, y0) e R2 is a strong density point of the set A e ?2 if and only if
Ь2(АГ\([хо-Ь,х0 + Ь]х[у0-к,уо+к]))
Inn — 1. D.1)
A—0+ Ahk
Obviously if (jto, yo) is a strong density point of A. then it is also an (ordinary) density
point of A. A point @,0) is a density point of the set A in the above decomposition of /,
but it is not a strong density point of A.
Let Ф.АА) denotes the set of all strong density points of a set A e ?2 and let % =
{A e ?2: А с ФХ(А)}. Again Ф5 is a lower density operator and % is a topology on the
plane (called the strong density topology) stronger than a natural topology and weaker
than 72, so theorems similar to Theorems 2.1, 2.2 and 2.3 are true in this case. However,
the proof of the Lebesgue Density Theorem is essentially different, (see Riesz A934),
Saks A937)). The reason is that there exists a set A e ?2 and a family of intervals A such
that A covers A in the Vitali sense (i.e., for each ? e A there exists a sequence {/,i}„eN

Density topologies
693
of intervals from ? such that ? e I„ for ? e N and diam(/„) -- 0) having the property
that for each sequence {?-keN of pairwise disjoint intervals from ? ?(? \ \Jk Л) > О
(see Caratheodory A927) or Jeffery A962)). Than means that the analogue of Vitali
covering theorem does not hold for intervals (without the assumption on parameter of
regularity). All counterparts of Theorems 2.4-2.12 are also true for Ts, essentially with
similar proofs.
The situation is different for analogues of theorems from Section 3. Again one can
introduce two definitions of strong approximate continuity at a point - one restrictional
(or "path") and the second topological (with respect to %). Observing that (л"п, vo) e K2 is
a strong density point if and only if
?-,(??)?)
lim — - = 1,
diam(/)->O ?2(/)
(л-0..П))е/
where / denotes the interval centered at (ло.уо). it is possible to repeat the proof of
Theorem 3.1 with necessary changes. We have theorems similar to Theorems 3.3, 3.4
(the proof uses the theorem of Jessen et al. A935) on strong derivation of the indefinite
integral), Theorem 3.5 (for a real function g of a real variable), Theorem 3.6 (only the part
concerning Baire one class). Since % is included between the natural topology in the plane
and T>, the same can be said on classes of connected sets with respect to these topologies
(in the inverse order).
However, % behaves much worse with respect to separation axioms than 7?.
THEOREM 4.3 (Goffman et al. A961)). The topology % is not regular.
Hence % is not the coarsest topology relative to which each strongly approximately
continuous function is continuous.
Let % = {A e dr. А с ФЛ(А) and for each л. у e ? the sets A.v = {у: (x.y) е
A], Ay = {x: (x,y) e A) are Lebesgue measurable (as linear sets), moreover Ax с
?(??), ?? СФ(АУ)}.
Theorem 4.4 (Goffman et al. A961), O'Malley A972)). % is a topology on E2
strictly coarser than % and such that each strongly approximately continuous function
is continuous with respect to %.
It is not known whether % is completely regular. O'Malley A972) observed, that the
product of two density topologies on the real line (which is obviously coarser than %) is
too small to make all strongly approximately continuous functions continuous.
5. ?-density topologies on the real line
S. Ulam in the Scottish book in 1937 has posed the problem: if ? e ?, does there exist a
real function ? :E+ —>¦ E+ (dependingon ?), increasing, such that lim,_o+ ^@ = 0 a°d

694 W. Wilczyiiski
lim Mrn/) =0 E.1)
?,/,^?+?(/)·^(?(/))
.ve/
a.e. on ? (here / denotes a closed interval). The affirmative answer to this question has
been given by Taylor A959, 1960).
In the sequel С will denote a class of continuous increasing functions ? : R+ —>¦ R+
such that lim,^?+^@ =0.
For technical reasons it is more convenient to consider the following condition:
k(ET)[x-h,x+h]) n ,„ .,.
lim =0. E.1)
/1^0+ 2hVBh)
It does not change the scope of studies because of the following fact
THEOREM 5.1 (Wagner-Bojakowska and Wilczynski B000)). For each A e С and each
? eRthe condition E.1) is fulfilled for a function ? eC ifandonly //E.?) isfiilfilledfor
a function ?\ e С defined by the formula ?\(?) = ?B~? ¦ t).
Theorem 5.2 (compare Taylor A959), Terepeta and Wagner-Bojakowska A999)). For
each ? e С there exists a function ? e С such that E.Г) isfiilfilledfor all ? e ? except
for a subset of Lebesgue measure zero.
PROOF. Observe that E.?) is fulfilled if and only if for each к eN there exists ? e N
such that for each/2 e]0.p-'[ we have ;-(?'n|-'~/'-v+/l" sC k'2. If we put
? k(E'f)[x-h.x+h]) . , _,r
Ek.,, = \x e E: ^ к - for each h e JO. ? [
then
? П Ф(?)=р|У ?*.,,. E.2)
k=\ ,,= \
Since the set
[ _ ?(?'?[? -h,x + h]) ,_-. , , п. t г 1
Ak.p =\xeR: — < к - for each /2 e JO, ? ' [
is closed for each k, ? e N and ?*.,, = ? ? ?*./;, all sets Ek.,, are measurable. Moreover,
for each k, ? e N, ?*.,, с ?*.,,+ ! С ?. By virtue of E.2) we have ?({]^=, ?*.,,) = ?(?),
so \im,,->x WEk.,,) = ?(?) for each к е N. Choose an increasing sequence {р*Ьек of
Then we have
natural numbers such that k(Ek ,,,) > ?(?) - 2 k for each ieN.
?( f) ?,-.„, ) > ?(?) - JT2-'= ?(?) - 2-(Ь
w'=)t / ?=*

Density topologies
695
SO
/ DC DC
- un-
\A=I/=A
k(\immfEk.pk)=k[ I Jf }Ej.lh )=k(E).
Let^:R+^R+ be a function defined in the following way: ? B- pk . ') = (k - l)_l,
linear and continuous on the intervals [2 · /»^i|.2 · pj~]] for к = 2.3 and any
continuous increasing function on the half-line [2 · p-T1 · +oo[.
Take jc e liminfA Fa.w· There exists кц e N such that ? e F,.№ for all / ^ ?(). if
/г е [pr+'p ?"'], then ?B?) ^i'x and
Л(?'П[дс-й,дс+й]) /-: ._,
< r=/ . ???/^??.
2?·?B?) /-'
Hence iimA^0+M?nj^±M=Oa.e. on F. ?
At the end of this section we shall see also that the continuity of ? is not essential for
the construction of a density-type topologies.
However, the problem whether does exist a function ? independent on ? has the
negative solution (Taylor, 1959)
THEOREM 5.3. For an arbitrary function ? e С and a fixed number a e]0, \[ there exists
a perfect set ? С [0, 1] such that k(E) =? and
k(E'C\[x-h,x + h\)
lim sup = +oo,
,;^o+ 2h4>{2h)
for each ? e E.
PROOF. Put ak = ?(? ¦ 2~k). A sequence {ak}k€\\ decreases to zero. Let {?·}?·???|0| be a
decreasing sequence convergent to 0 such that ro = 1 - a and lirru-^ rk · ??~_!, = oo. Let
Fq — [0, 1]. Suppose that we have constructed F(). F| Fa·, where i: e N U @) and the
set Fk consists of 2* pairwise disjoint closed intervals of equal length. To obtain F;._n we
subtract from each component of Fk a concentric open interval of length (r; — ?+?) ·2~ .
So by induction we have defined a descending sequence of closed sets {?*?·}?-6??|??-
Put ? = ????^·· ^ 's clearly a Cantor sets of measure a. Simultaneously we have
k(Fk)=a + rkmdk(Fk\E) = rk.
Take ? e F. Let P> be a component of Fk such that .v eft for к e N U {0}. Then
{Fi-}*eN is a decreasing sequence of closed intervals. From the construction it follows that
k(Pk) = (a+ rk) ¦ 2'k and ? (Fa \ F) = rk ¦ 2'k. Since Нгпа^-х П = 0, there exists A:, e N
such that for к ^ к\ we have rk < a. Hence for к ^ к\ the following inequality holds:
?((? + rk)- 2'k) <: ? (a ¦ 2~ik-]') = a*.,. So
?(?'??*) гА-2"* ^ rk
к(Рк)-Ф(к(Рк)) (а+гк)-2-к-Ф((а + гк)-2-к) 2а ¦ ak.

696
W. Wikzxn.sk)
for к ^k ?. Let Pk = [x—hk,x + hk] be the smallest closed interval including Pk for к eN.
Then {PjthgN is a decreasing sequence of closed intervals such that {?} = ??? ^ (since
X(Pk) <^2X(Pk)).SoioTk^kl we obtain
ЦЕ'ПРк) > к(Е'ПРк)
2hk-VBhk) 2к(Рк)-ФBк(Рк))
П ¦ 2-k rk
2 · (? + rk) ¦ 2~к ¦ ? ((а + ?) ¦ 2~к) ?? ¦ ак-\
Hence
X(E'n[x-hk,x+hk])
hmsup =+oo. ,-.
k^J 2hk-WBhk) ?
Let ? eC.Wc say that ? is a ? -density point of ? e С if and only if
,. k{E'n[x-h,x + h]) n
hm = 0.
A^o+ 2h4>Bh)
The set of all ? -density points of ? we shall denote ??(?).
THEOREM 5.4 (Terepeta and Wagner-Bojakowska A999)). For each ? eC the mapping
?? : С —> 2Ш has the following properties:
@) for each AeC, ?? (A) e ?,
A) for each A e ?, ФФ(А)сФ(А),
B) for each А, В е С, ifA~B, МепФФ(А) = ФФ(В),
C) ??@) = 0, ??(?) = ?,
D) for each ?,? eC, ??(?? ?) = ??(?)???(?).
PROOF. @) similarly as for the operation ? one can prove that ??(?) e F„s, for each
AeC (the proof does not use the assumption of continuity of ?).
A), B) and C) follow from the definition.
D) follows from the inequality ?((? ? ?)' ??[? - h,x + h] ? ?(?' ? [? - h, ? + h]) +
k(A'n[x-h,x + h]).
From Theorem 5.3 it follows that ?? is not a lower density operator for any function
? e C, because the Lebesgue Density Theorem is not fulfilled. However, with each
function i^eCwe can associate a topology which we shall call ? -density topology in
the following way:
Theorem 5.5. For each ? e С the family ?? = {A e С: А С ?? (A)} is a topology on
the real line, stronger than the natural topology and weaker than the density topology T.
PROOF. From Theorem 5.4 A) we conclude that ?? с Т, which implies that the union
of an arbitrary subfamily of ?? is a Lebesgue measurable set. The rest of axioms of
topological space is easy to verify.

Density topologies
697
Each set of the form Ш \ E, where ? is dense in R and ?(?) = 0, belongs to ?? but
not to the natural topology. The set ? ? ? (?), where E is constructed in the proof of
Theorem 5.3, belongs to T, but not to ??. ?
The set ? с К is a ?? -neighbourhood of a point ? e ? if and only if there exists a set
Ax e ?? such that jc e Ал and Av С ?. Hence ? is a ?? -neighbourhood of ? if and only
if
,. X*(ET)[x-h,x + h])
lim = 0.
/i^o+ 2h4>Bh)
Similarly as Theorem 2.7 one can prove:
THEOREM 5.6. ? is ? ?? -neighbourhood of each of its points if andonly if? e ??.
The formula for ?? -interior of an arbitrary set А с К is much more complicated
than in the case of the density topology. To state the theorem we need the following
denotation for a measurable set А: Ф$ (A) = A; if a ^ 1 is a countable ordinal having the
predecessor, ? =/3+1, then Фф(А) = ФФ(Ф^(А)) and if ? is a limit countable ordinal,
then ??(?) = ?[?<?<???(?). It turns out that a transfinite sequence {??(?)}\<??<?]
of sets is descending. Moreover
Theorem 5.7 (Wagner-Bojakowska and Wilczynski B000)). Fix ? e C. For each set
А С К there exists a countable ore
В С A is a measurable kernel of A.
А С К there exists a countable ordinal ? ^ 1 such that Int^ (A) = А П ??(?) where
An ordinal ? is the smallest ordinal such that ??(?) = ??(?) for each countable
ordinal ? > ?. In connection with this we shall say that a countable ordinal <*o A ^ ao <
?\) is the ? -order of the set A e С if and only if ? ? (A) = ? ?'(A) for each a > a() and
???(?) \ ???°(?) ? 0 for each ?, 1 <; ? < a0.
Theorem 5.8 (Wagner-Bojakowska and Wilczynski B000)). Fix ? e С For each
countable ordinal a ^ 1 there exists a set A e С with the ?-order equal to a.
Let ? с IR be a perfect nowhere dense set of positive Lebesgue measure such that
??(?) = 0 (as in the proof of Theorem 5.3). Then ? is ??-closed and Intx^(?) = 0, so
? is ??-nowhere dense, so Theorem 2.8 does not hold in this case. However, the following
property is true (compare Theorem 2.4):
THEOREM 5.9. A point ? e R is a ??-limit point of a set А сШ (not necessarily
measurable) if and only if
?*(An[x-h,x + h])
lim sup > 0.
,,^o+ 2h4>Bh)

698
W. Wilcztfski
For each ? e С the topology ?? has a lot of properties similar to that of T. We shall
mention only some of them:
Theorem 5.10 (Terepeta and Wagner-Bojakowska A999)). If ? cR is ? ??-compact
set, then ? is finite.
Theorem 5.11 (Terepeta and Wagner-Bojakowska A999)). The space (R. ??) is neither
first countable, nor second countable, nor Lindelof nor separable.
Theorem 5.12 (Terepeta and Wagner-Bojakowska A999)). The family ofTi* -connected
sets coincides with the family of sets connected in the natural topology (and in T).
PROOF. Follows from the inclusion: (top. nat.) с ?? с Т.
There are also essential differences between the properties of ? and ??.
THEOREM 5.13 (Wagner-Bojakowska B001)). The space (R. ??) is of the first category.
PROOF. Let ? be a perfect nowhere dense set of positive measure such that ??(?) = 0.
Put D = \J,.€Q(E + r). Then it is not difficult to show that ?(? \ D) = 0. So R =
(R \ D) U U,- e q (? + r) is of the first category as the union of countably many ?? -nowhere
dense sets.
To establish that the space (R, ??) is completely regular for each ? &C we shall need
the notion of ? -approximate continuity. As in Section 3 we can introduce the notion of
? -approximate continuity of a function ? : R —>¦ R either using the "restricted" definition
or the "topological" definition (compare definitions of Denjoy).
THEOREM 5.14. Both definitions oj ?-approximate continuity at a point are equivalent.
The proof uses the fact that for ? -density points the condition (J?) of Jedrzejewski is
also fulfilled (Wagner-Bojakowska B001)).
Theorem 5.15 (Terepeta and Wagner-Bojakowska A999)). A function /:R -+ R is
measurable if and only if there exists a function ? e С such that f is ? -approximately
continuous almost everywhere.
PROOF. Observe that if {f,,},^^· is a sequence of measurable sets, then there exists a
function ? eC such that ?(?„ \ ФФ(Е,,)) = 0 for each ? e N. Indeed, from Theorem 5.2
for each ? e N there exists a function ?„ e С such that ?(?„ \ ??? (?,,)) = 0. If ? e С
is a function for which there exists <5„ > 0 such that ?(?) > ?„(?) for all t e]0, <$„[,
then ???(?) с ??(?) for each measurable set A and the conclusion follows. Now let
{]йнА<[)неМ be a basis for the natural topology in R and put E„ = /~' (]a„,b„[). If
D(f) is the set of all points where / is not ? -approximately continuous, then D(f) с
U/JeN(?„ \?? (?н))· Hence ?(?)(/))=0 for a function^ е С described above. D

Density topologies
699
So the class of all measurable functions coincides with the union of all classes of
functions which are ? -approximately continuous almost everywhere. However, the union
of all ? -density topologies does not behave so nicely.
Theorem 5.16 (Wagner-Bojakowska and Wilczynski B00..)). U* eC ?* § ?-
Recall that the Hashimoto topology on the real line with respect to I is the family
O* = {Up: U is open in the natural topology and Pel}. (Hashimoto, 1976.)
Theorem 5.17 (Terepeta and Wagner-Bojakowska A999)). П*еСТ* = °*-
The family {??}?^ of all ? -density topologies is partially ordered by inclusion. The
structure of this partially ordered set is rather complicated. Observe that if ?\ (f) ^ ?2(?)
fori e]0,<5[, where <5 is some positive number, then ??, с ??,. For each sequence {<^„}„е;;
of functions from С there exist two functions ?. ? e С such that for each /; e N there exists
?„ > 0 such that ?(?) < ?„(?) < ?(?) fori e]0.<5„[.
Hence for each sequence {??„}„€\., of ? -density topologies there exist two topologies
?? and T= such that ?? с ??„ с Т= for all /; е N. Simultaneously different functions
from the class С can generate the same topology. This is the case, for example, for arbitrary
function ? e С and a function ?\ e С defined by the formula ?\(?) = a ¦ ?(?), where
aeR+.
Let ?\, ?2 e С and к e N. Put
A+ = {x eR+: ?{B?) < к'1 ?2B?)),
Bk+ = {x eE+: ?2B?) <k~l ?{B?)}.
Ak = ?^ U (~??), Ви = Bf U (-Bf). The following theorem gives necessary and
sufficient condition for ??, = ??,.
Theorem 5.18 (Wagner-Bojakowska and Wilczynski B000)). Let ?\,?2 e С The
topologies ??, and ??: are equal if and only if\m\k-*xBk = linu·-»^ Щ = 0, where
k(Akf)[-h,h]) j ,. k(Bkn[-h.h])
Sk = lim sup and m = lim sup — :———— ·
ft^o+ 2/2-^,B/2) A_o+ 2h4t2Bh)
From the above theorem it follows (among others) that there is not need to use increasing
(not necessarily continuous) functions ? : Ш^ -> R+ such that lim, _o+ ^ @ = 0, because
in this way we shall not obtain any new topologies. Namely, we have the following
THEOREM 5.19 (Aversa and Wilczynski B000)). // <^:E+ -> K+ is an increasing
(not necessarily continuous) function such that lim,_*o-r ^@ = 0> then there exists two
functions ?\,?2<?0 such that ?\ (t) ^ ? (t) sC ?2(?) for t e @, 2] and ??, = ??:.

700
W. Wilczynski
6. Another local property of measurable sets
Lusin in this dissertation has proved the following theorem
Theorem 6.1 (Lusin A951), Wheeden and Zygmund A977)). For each set A e C,
А С [0, 2?], the integral
???(? + {)-??(?-{)?(= Hm rXA(X+t)-XA(X-t)dt F1)
? ' F-0+Л t
exists and is finite for almost every ? e [0. 2тг].
The above theorem shows a kind of symmetry of a measurable set, because at the
same time in Lusin A951, Commentary 117), one can find an example of a set A e ?,
А С [0,2?], for which
f '*'(*+0-*'<*-OI</, =+00. 1ога.е.л-е[0.2л].
Jo '
Obviously, if F.1) holds at some point x, then ? can be neither a density, nor a dispersion
point of the set A. Conversely, there exists a set A e С and a point ? e ?(?), for which
the integral F.1) is infinite (Lusin, 1951. Commentary 118).
One can find interesting density theorems independent of Lebesgue Density Theorem in
Cs6rnyeiA998).
References
Aniszczyk, B. and Frankiewicz. R. A986). Non-homeomorphic density topologies. Bull. Acad. Pol. Sci. 34 C-4).
211-213.
Aversa, V. and Wilczynski. W. B000). ?-density topology for discontinuous regular functions. Atti Sem. Mat.
Fis. Univ. Modena 48, 473^*79.
Besicovitch, A. A957). On density of linear sets. J. London Math. Soc. 25. 170-178.
Blumberg, A. A918-19). A theorem on linear point sets. Bull. Amer. Math. Soc. 25. 350-353.
Bruckner. A.M. A971), Differentiation of integrals. Amer. Math. Monthly. No. 12 of the Herbert Ellsworth
Slaught Memorial Papers.
Bruckner, A.M. A978), Differentiation of Real Functions. Lecture Notes in Math.. Vol. 659, Springer. Berlin.
Caratheodory, С A927), Vorlesungen iiber reelle Funktionen. Leipzig.
Ciesielski, K. and Larson, L. A993). Refinements of the density and I-density topologies. Proc. Amer. Math. Soc.
118B), 547-553.
Ciesielski, K., Larsen, L. and Ostaszewski. K. A994), 1-density continuous functions. Mem. Amer. Math. Soc.
515.
Csomyei, M. A998), Density theorems revisited. Acta Sci. Math. (Szeged) 64. 59-65.
Denjoy, A. A915), Memoire sur les derives des functions continues. J. Math. Pures Appl. 1. 105-240.
Denjoy, A. A916). Sur les functions derivees sommables. Bull. Soc. Math. France 43, 161-248.
Filipczak, M. and Filipczak. T. A988). On some topology related to metric density. Rad. Mat. 4. 299-307.
Filipczak, T. A988-89), On some abstract density topologies. Real Anal. Exchange 14 A). 140-166.
Foran, J. A991), Fundamentals of Real Analysis. Dekker. New York.
Gibson, F.G. and Natkaniec. T. A996-97), Darboux like functions. Real Anal. Exchange 22 B). 492-533.

Density topologies
701
Goffman, C. Neugebauer, C.J. and Nishiura. T. A961). The density topology and approximate continuity. Duke
Math. J. 28, 497-506.
Goffman, С and Waterman. D. A961). Approximately continuous transformations. Proc. Amer. Math. Soc. 12,
116-121.
de Guzman, M. A975), Differentiation of Integrals in R-. Lecture Notes in Math.. Vol. 481. Springer. Berlin.
Hashimoto, H. A976), On the * topology and its application. Fund. Math. 91. 5-10.
Haupt, O. and Раис. С A952), La topologic approximative de Denjoy em-isagee comme vraie topologic. C. R.
Acad. Sci. Paris 234, 390-392.
Haupt, O. and Раис, С A954). Uber die durch allgemeine Ableitungsbasen bestimmten Topologien. Ann. Mat.
Рига Appl. Ser. IV. 36, 247-271.
Hobson, E.W. A927). The Theory of Functions of a Real Variable and the Theory of Fourier Series. Vol. I. 3rd.
edn, Cambridge.
Ionescu Tulcea, A. and Ionescu Tulcea. С A969). Topics in the Theory of Lifting. Springer, Berlin.
Jedrzejewski, J.M. A973/74), On limit numbers ofreal functions. Fund. Math. 83 C), 269-281.
Jeffery, R. A962). The Theory of Functions of a Real Variable. University of Toronto Press.
Jessen, В., Marcinkiewicz. J. and Zygmund, A. A935). Note on the differentiability of multiple integrals. Fund.
Math. 25. 217-234.
Khintchine. A. A922). А пек proof of the basic theorem on metric set theory, Proc. of Ivanovo-Vozn. Technical
Institute. No. 6, 39^1 (in Russian).
Lahiri, B.K. and Chakrabarti. S. A991), Density topology. Math. Student 59, 89-108.
Lebesgue. H. A904). Legons sur Г integration et la recherche des functions primitives. Paris.
Lebesgue, H. A910). Sur I 'integration des fonctions discontinues, Ann. Ecole Norm C) 27. 361^50.
Lukes. J., Maly\ J. and Zajicek, L. A986), Fine Topology Methods in Real Analysis and Potential Theory. Lecture
Notes in Math.. Vol. 1189. Springer. Berlin.
Lusin. N.N. A912), Sur les proprietes des fonctions measurables. C. R. Acad. Sci. Paris 154, 1688-1690.
Lusin, N.N. A951). Integral and Trigonometric Series. Moskwa (in Russian).
Lusin, N.N. and Sierpiiiski, W. A917). Demonstration elementaire du theoreme fundamental sur la densite des
ensembles. Rend. Circ. Mat. Palermo 42. 167-172.
Maharam. D. A958). On a theorem of von Neumann. Proc. Amer. Math. Soc. 9. 987-994.
Martin. N.F.G. A964), A topology for certain measure spaces. Trans. Amer. Math. Soc. 112. 1-18.
O'Malley. R.J. A972), Density properties of functions and sets. Thesis. Purdue University.
O'Malley. R.J. A977). Approximately differentiable functions: the r topology. Pacific J. Math. 72 A). 207-222.
O'Malley. R.J. A979), Approximately continuous functions which are continuous almost everywhere. Acta Math.
Acad. Sci. Hungaricae 33 C-4). 395^02.
Ostaszewski. K. A982). Continuity in the density· topology. Real Anal. Exchange 7 B). 259-270.
Oxtoby, J.С A980). Measure and Category. Graduate Texts in Mathematics. Vol. 2. Springer. Berlin.
Riesz. F. A934), Sur les points de densite au sens fort. Fund. Math. 22. 221-225.
Rusu, D. A995). Density-type topologies. An. StiinJ. Univ. "Al. I. Cuza" Mat. 41. 269-285.
Saks. S. A937). Theory of the Integral. Monografie Matematyczne. Warsaw.
Sarkhel, D.N. and De. A.K. A981). The proximalIу continuous integrals. J. Austral. Math. Soc. Ser. A 31. 26-45.
Scheinberg, S. A971). Topologies which generate a complete measure algebra. Adv. Math. 7. 231-239.
Sierpinski, W A923). Demonstration elementaire du theoreme sur la densite des ensembles. Fund. Math. 4.
167-171.
Stepanoff. WW A924). Sur tine propietecaracte'ristique des fonctions measurable. Mat. Sb. 31 D). 87^89.
Tall. F.D. A976), The density topology. Pacific J. Math. 62 A). 275-284.
Taylor, S.J. A959), On strengthening the Lebesgue Density Theorem. Fund. Math. 46, 305-315.
Taylor, S.J. A960). An alternative form of Egoroff's theorem. Fund. Math. 48. 169-174.
Terepeta, M. and Wagner-Bojakowska. E. A999), ?-density topologies. Rend. Circ. Mat. Palermo 48. 451^76.
Thomson, B.S. A985). Real Functions. Lecture Notes in Math.. Vol. 1170. Springer. Berlin.
Troyer, R.J. andZiemer. WP. A963), Topologies generated by outer measures. J. Math. Mech. 12C). 485^94.
Wagner-Bojakowska, E. B001). Remarks on ?-density topology. Atti Sem. Mat. Fis. Univ. Modena 49. 79-87.
Wagner-Bojakowska. E. and Wilczynski. W. B000). The interior operation in ? ?-density topology. Rend. Circ.
Mat. Palermo 49. 5-26.

702
W. Wilczynski
Wagner-Bojakowska. E. and Wilczynski. W B000). Comparison of ?-density topologies. Real Anal. Exchange
25B). 661-672.
Wagner-Bojakowska. E. and Wilczynski. W. B00..). The union of all ?-density topologies, in preparation.
Wheeden. R. and Zygmund. A. A977). Measure and Integral. Marcel Dekker.
Wilczynski. W. A984). Remarks on density topology and its category analogue. Rend. Circ. Mat. Palermo B)
Suppl. No. 5. 145-153.
Wilczynski. W. A984-85). A category analogue of the density topology, approximate continuity and the
approximate derivative. Real Anal. Exchange 10. 241-265.
Wojdowski. W. A989). Density topology involving measure and category. Demonstratio Math. 22 C). 793-812.
Zahorski. Z. A950). Sur la premiere derivee. Trans. Amer. Math. Soc. 69. 1-54.
Zajicek. L. A979). An elementary proof of the one-dimensional density theorem. Amer. Math. Monthly 86. 297-
298.
Zajicek. L. A986-87). Porosity. X-densitv topology and abstract density topologies. Real Anal. Exchange 12.
313-326.

CHAPTER 16
FN-Topologies and Group-Valued Measures
Hans Weber
Universita di Udine, Dipartimento di Matematica e Infornuitica. via delle Science 206. 33100 Udine. Italy
E-mail: weber@dimi.imiitd.it
Contents
Introduction 705
1. Definition and generation of FN-topologies 706
2. Exhaustivity 709
3. ?-submeasures and completeness 713
4. ?-order continuity and order continuity 716
5. Extension of FN-topologies and of measures 719
5.1. Extension of ?-order continuous FN-topologies and ?-additive measures 719
5.2. Extension of exhaustive FN-topologies and exhaustive measures 724
6. The completion of topological Boolean rings and the system FN,. (/?) of exhaustive FN-topologies . . 726
7. More on the ? -topology 732
8. Decomposition of exhaustive measures 734
9. Connectedness 738
10. Vitali-Hahn-Saks and Nikodym theorems 739
References 741
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
703

This page intentionally left blank

FN-topologies and group-valued measures
705
Introduction
In this chapter we present the basic results about the FN-topologies on Boolean rings and
the applications in the theory of group- or vector-valued measures.
A Frechet-Nikodym topology (FN-topology) is a topology generated by a measure in
such a way that the measure becomes continuous. As an illustration, let ? be a positive
real-valued measure on a ring of sets. Then the topology generated by ? is induced by the
semimetric defined by d(A, ?) = ? (A AS).
Frechet A919/20) introduced a metric in the space of measurable functions such that
convergence with respect to the metric is equivalent to convergence in measure. Identifying
characteristic functions with sets the subspace of characteristic functions becomes a
?- algebra and the metric on this subspace is equivalent to the metric induced by a measure
described above. Later on, this metric space was studied by Nikodym. This metric space is
complete and therefore a Baire space. Saks was the first to use a Baire category argument
to prove the Vitali-Hahn-Saks theorem. It was one of the first important application of
FN-topologies. Since that time FN-topologies have now and then been used in measure
theory. But only since the beginning of the seventies, the Frechet-Nikodym approach has
been used in a systematical way starting with Drewnowski A972a). Today FN-topologies
are considered an elegant and powerful tool in measure theory.
The Frechet-Nikodym approach has a particular advantage when the range space of the
measures plays a minor role. This is the case, e.g., for Vitali-Hahn-Saks and Nikodym
theorems, for extension theorems and for decomposition theorems.
In the most known proofs of the Vitali-Hahn-Saks and Nikodym theorems, these are
obtained as an easy consequence of the fact that the domain of the measures is a Baire
space with respect to a suitable FN-topology (see Section 10). As already mentioned this
idea goes back to Saks.
The extension theorems presented in Section 5 are obtained extending the measures by
continuity. So the measure extension theorems are equivalent to extension theorems of
FN-topologies. This approach was used among others by Drewnowski A972a), Herer
A976), Lipecki A974,1987), Maharam A987) and Weber A976,1980) and was originated
by M.H. Stone as pointed out by Maharam A987).
A first unified approach to the Lebesgue decomposition theorem and the Hewitt-Yosida
decomposition theorem was presented by Drewnowski A973b). Stimulated by a question
posed by Drewnowski A973b), Traynor A976) proved that an exhaustive measure can
be decomposed with respect to an arbitrary FN-topology; different choices of the FN-
topologies yield different decomposition theorems. The method to get decomposition
theorems presented here was given by Weber A982a, 1984a). The idea is to extend the measure
to a suitable completion, to study the extensions and then to transfer the results to the
original measure (for details see the beginning of Section 6). The same method is also useful
to prove other results, e.g., the control measure theorem of Bartle-Dunford-Schwartz.
Throughout let R be a Boolean ring. As usual, we denote the symmetric difference
(addition), infimum (multiplication), supremum, difference, natural ordering by ?, ?, ?, \, ^,
respectively. If R is a Boolean algebra, i.e.. Boolean ring with unit, we denote the unit by e
and the complement of an ? e R by x' := e \ x.
G stands for a commutative additively written group.

706
? Weber
1. Definition and generation of FN-topologies
FN-topologies are particular group topologies on boolean rings generated in a natural way
by a measure or by a family of submeasures.
Let us fix our terminology. A pseudonorm on G is a function | |: G —>¦ [0, +oo]
satisfying |0| =0, |-*| = |.v|. \x + у I ^ |.v| + |y|. A seminorm is a real-valued
pseudonorm. A norm is a seminorm satisfying |0| = 0 only if ? = 0. If | \:G-> [0, +oo]
is a pseudonorm (seminorm, norm), then d(x.y) := |.v - y| defines a pseudometric
(semimetric, metric), which induces on G a group topology, called the | |-topology. By
the inequality ||jc| - |y|| ^ \x - y|, a pseudonorm is uniformly continuous; here [0, +oo]
is endowed with the pseudometric (a. b) н> \a - b\ and |(+oo) - (+oo)| := 0. If | | is a
pseudonorm on G, then ||jc|| := min{|.r|. 1} defines a seminorm on G generating the same
topology as | |.
DEHNITION 1.1. A function ? : R -> [0. +эо] is called a submeasure if ?@) = 0 and /j is
monotone and , i.e., ?(? ? у) ^ /j(.v) + ?()-) for all .v. у е /?.
Example 1.2. Let | | be a pseudonorm on G and ?-.R^-Ga. measure, i.e., a finitely
additive function. Then the semivariation of ? defined by
||?||(?) := 5ир{|д(;с)|: Яэ.х^а}
is a submeasure on R.
The total variation of ?, defined by
II
lMl(a):=sup]P|M(A-j)|.
i=l
where the supremum is taken over all finite disjoint sequences xt in R with .?, ^ ?, is a
[0, +oo]-valued measure and therefore a submeasure.
It is easy to see that a function is a submeasure on R iff it is a monotone pseudonorm on
(/?, ?). In particular, a submeasure on /? induces on (R. ?) a group topology, which has a
0-neighbourhood base ([/„)„ef; consisting of ,vo//iisets, i.e., .r ^ у e [/„ implies .v e Un.
Definition 1.3. A group topology on (R. A) which admits a 0-neighbourhood base
consisting of solid sets is called a Frechet-Nikodym topology or FN-topology. We denote
by FNG?) the set of all FN-topologies on R.
This now used terminology goes back to Drewnowski A972a). Vladimirov A972) used
the term monotone topologies.
FN-topologies can be characterized by their 0-neighbourhood systems.

FN-topologies and group-valued measures
707
PROPOSITION 1.4. (a) If и e FN(tf), then for any 0-neighbourhood U in (R. u) there is
a 0-neighbourhood V in (R, u) such that
x, у e V and ? ^x ? ? imply zeU. (*)
(b) If U is a filter on R with the property that for any U el] there is a V e U with (*),
then U is the 0-neighbourhood system for a unique FN-topology on R.
PROOF, (a) Let U be a 0-neighbourhood in (R.u) and V a solid 0-neighbourhood in
(R, u) with VAV cU.lf x.yeV and ; ^ ? ? у, then ? л ? ^ ? e V, z\x ^ у е V,
hence ? л ?, ? \? е V and ? = (? ?.?)?(; \?) e VAV C U.
(b) Let U, V e U with (*). Then VA V с U. Therefore there is a unique group topology
и on (/?, A) such that U is the 0-neighbourhood system of (R. u). Moreover, V с U* : =
{x e R: у e U for any у ^ х} с U, hence U* is a solid 0-neighbourhood contained in
u. a
PROPOSITION 1.5. (a) FNG?) is a distributive complete lattice with the discrete topology
as the greatest element and the trivial topology as the smallest element.
(b) If u,v e FNG?) and U and V are the 0-neighbourhood systems, respectively, in
{R,u)andin (R, v), then ji/W: ?/eU, VeV} is a 0-neighbourhood base in (R,uav)
and {U A V: ?/eU, V e V) is a 0-neighbourhoodbase in (R,uv ?).
PROOF. The system of all group topologies on (R. A) form a lattice with the discrete
topology as the greatest element and the trivial topology as the smallest element. If U
and V are the 0-neighbourhood systems, respectively, for group topologies и and v on
(R, A), then {UAV: U e U. V e V) and {U ? V: ?/eU, VeV) are 0-neighbourhood
bases in (R, и ли) and in (R, и ? и), respectively. If U and V are solid subsets of R,
then [/AV = [/vVand[/nv = [/AVare solid. Therefore FN(tf) is a sublattice of
the lattice of all group topologies on (R. A). Moreover, using the proved description of
the 0-neighbourhood bases of и v и and и л ? given in (b), one easily sees that FNG?) is
distributive. (Observe that it is enough to prove one of the distributive laws.)
FNG?) is complete: If, for a e A, ua e FNG?) and U„ is a 0-neighbourhood base
in (R, ua) consisting of solid sets, then the finite intersections C\uef.- Uu (F finite с А,
Ua e Uff) is a 0-neighbourhood base for supae4 ua- ?
PROPOSITION 1.6. Any FN-topology on R makes the operations ?, ?, ?, \ uniformly
continuous.
PROOF. Let о be one of the operations ?, ?, ?, \. Let и e FNG?) and U. V 0-neighbour-
hoods in (R, u) with (*) of Proposition 1.4(a). If х,. у, e R with .v|Ajt2,yiAy2 e V, then
(x\ oy])A(jC2 °yi) € U since (,vi oyi)A(.V2 oy2) ^ (.vi ?.?2) ? (у|Ау2>· П
In particular, any FN-topology is a ring topology. More precisely, a topology on R is
an FN-topology iff it is a ring topology on (R, А, л) which makes the multiplication л
uniformly continuous iff it is a ring topology on (R, А, л) which is bounded in the sense
of topological rings (for the definition of boundedness see Warner A993)).

708
?. Weber
Corollary 1.7. Let и e FN(tf) and S a subring of (R,u). Then the closure of S
in (R,u) is a subring of R.
Nachbin A965, p. 25) called a partial order on a topological space (?, ?) closed if its
graph {(x, y): ? ^ y] is closed in (?. ?J.
COROLLARY 1.8. If и is a Hausdorff FN-topology on R, then the partial order of R is
closed.
PROOF. Let (xY)yer and (yY)yer be nets converging in (/?, и), respectively, to ? and у
and suppose that xy ? yy for any у e ?. Then ? = limK xy = limy xy л yy = ? л у ^ у. D
PROPOSITION 1.9. Let < be a closed order on a topological space (?, ?).
(a) Then the intervals {x e ?: ? ^ a], {x e ?: ? ^ b) and [a, b] := {x e ?: ? ^ ? ^ b]
are closed in (?, ?) for any a, b e X with a ^ b.
(b) If(xy)yep is a decreasing or increasing net converging to ? in (?, ?), then xy ? ?
or xy | ?, respectively.
PROOF, (a) Let (xY)yer be a net converging to л· in (?, ?) and xY ^ a for у e ?. With
yY := a (ye ?) we obtain ? = limy,, ^ lim.vy = x. Hence [?: ? ^ a] is closed. In
the same way one sees that {?: ? ^ b} is closed. It follows that [a, b] is closed as an
intersection of two closed sets.
(b) We only consider the case that (ху)уег is increasing. For any ? e ? we have by (a)
л = lim„^y xa ^ xy. Therefore ,v is an upper bound of {xy: ? e ?]. If b is any upper
bound of {xy: ye ?), then ? ?. b by (a). Hence ? is the last upper bound of (xy )yer- ?
We now define the FN-topology associated to a measure. Let ?: R —> G be a measure
with values in a topological group. It is well-known that the topology of G is generated by
a system (| \y)yer of seminorms, i.e., {{.v e G: \?\??}: ? e ?. ? > 0} is a subbase for
the 0-neighbourhood system of (G, ?). Put
\\?\\?(?):=$\??{\?(?)\?: R э ? ^ a}
for у е Г (see Example 1.2). Then the ?-topology is defined as the FN-topology generated
by the submeasures (||м||у)уег· Here is a description of the ?-topology without using
pseudonorms.
The sets
{a e R: ?(?) e U for any.? ^ ?},
where U is a 0-neighbourhood in G, form a 0-neighbourhood base for the ?-topology.
Any FN-topology is the ?-topology for some group-valued measure. In fact, if и е
FN(tf), put ? := (R, ?, u) and choose for ? the identity map on R; then и is the
?-topology. Since a ?-topology is generated by a system of submeasures. it follows that any
FN-topology is generated by a system of submeasures.

FN-topologies and group-valued measures
709
PROPOSITION 1.10. Let ?: R -> G be a measure with values in a topological group.
(a) If и e FNG?) and ? is continuous at 0 with respect to u, then ? is uniformly
continuous on (R,u).
(b) The ?-topology is the coarsest FN-topology which makes ? continuous.
PROOF. Let | | be a continuous seminorm on G and ||?|| the semivariation of ? with
respect to | |. Then
|?(-*)-???| = |m(jc\3')-m(.v\jc)| ^ |?(.?\ v)[ + |?(}'\-*)|
^2\\?\\(??>-).
Therefore ? is uniformly continuous with respect to the ?-topology.
Let и e FNG?) and ? be continuous at 0 with respect to и. If ? > 0 and U is a solid
0-neighbourhood in (R, u) such that \?(?)\ ^ ? for ? e U, then U С {? € R: \\?\\(?) ^?}.
Therefore the ?-topology is coarser than u. Moreover, ? is uniformly continuous with
respect to и since it is so with respect to the ?-topology. ?
Let N be an ideal in R. Then ?^(?) = 0 (? e ?) and ?,?(.?) = \ (? e R \ ?) defines a
submeasure on R. ? ? is a measure iff jV is a maximal ideal in R iff R \ N is an ultrafilter
in R iff the quotient R/N has two elements. If N is a maximal ideal, we call the
?,?-topology an ultrafilter topology. A measure which induces an ultrafilter topology, is called
an ultrafilter measure. If G is a Hausdorff topological group without elements of order 2,
then a measure ?: R —> G is an ultrafilter measure iff it is a 2-valued measure.
2. Exhaustivity
Exhaustive measures were introduced by Rickart A943) as a common generalization
of bounded real-valued measures on algebras and ?-additive Banach space-valued
measures on ?-algebras. Exhaustive measures are called in Rickart A943) s-bounded
(strongly bounded) since exhaustivity is for Banach space-valued measures a stronger
condition than boundedness (see Corollary 2.7 and Example 2.11). Exhaustivity is an
essential condition for most of the basic results for finitely additive measures; it is a crucial
condition, e.g., for decomposition theorems, see Remark 8.7. Diestel and Uhl A977)
illustrated the importance of exhaustivity (there called strong additivity) in their book
"Vector Measures", 1.6 Notes and remarks.
????????? 2.1. An FN-topology и on R is called exhaustive if x„ -> 0 (u) for any
disjoint sequence (xn) in R. If ? is a submeasure or a measure on R, then a is called
exhaustive if the ?-topology is exhaustive, or equivalently, if a(xn) —> 0 for any disjoint
sequence (x„) in R.
If G is a topological group and ?-R^-Ga measure, then ? is exhaustive iff the
?-topology is exhaustive.
We give two examples of exhaustive measures.

710
?. Weber
Example 2.2. (a) If R is ?-complete, G a topological group and ?: R —>¦ G a ?-additive
measure, then ? is exhaustive.
(b) If | | is a seminorm on G and ? : R —» (G, | |) a measure of bounded variation,
then ? is exhaustive.
PROOF. Let (Xn)neN be a disjoint sequence in R. In (a), M(sup„eN^;i) = ??=\ ?(¦*«).
hence the series ???? ?(*/?) converges and therefore ?(?„) —>¦ 0 (« —>¦ oo). In (b),
?,^? \?(?>>)\ <supne/j|M|(a) <+oo, hence ? (.*„)-> 0 (л -> oo). D
Any ultrafilter topology and therefore any ultrafilter measure is exhaustive. This follows
from the fact that, for a maximal ideal ? in R. at most one member of a disjoint sequence
in R does not belong to M.
Since for any a e R \ {0} there is a maximal ideal in R not containing a, the supremum
of all ultrafilter topologies on R is Hausdorff. Since the supremum of exhaustive
FN-topologies is exhaustive, we have proved:
PROPOSITION 2.3. The supremum u\, of all exhaustive FN-topologies on R is exhaustive
and Hausdorff.
Proposition 2.4. Let и eFN(R).
(a) и is exhaustive iff every increasing sequence is Cauchy in (R.u) iff every increasing
net is Cauchy in (R, u).
(b) If и is exhaustive, then every decreasing net is Cauchy in (R. u).
PROOF, (a) (i) Let и be exhaustive and (л'^Ьел an increasing net in R. Suppose that
(ха)аел is not Cauchy. Then there is a 0-neighbourhood U in (R, u) such that for every
? e A there is an a ^ ? with xu \ xy ? U. Therefore one can inductively define an
increasing sequence (or,,) in A with dn := xa,l+l \ xu„ $ U. Then (dn)neti is a disjoint
sequence not converging to 0.
(ii) Suppose that any increasing sequence is Cauchy in (R, u). If (xn)nei; is a disjoint
sequence in R, then s„ := V/'=i xi defines an increasing sequence and therefore a Cauchy
sequence. It follows that x„ = sn \sn-\ converges to 0.
(b) Observe that for any decreasing net (ха)аел in R and ? e A the net (?? \??)?^? is
increasing and apply (a). ?
PROPOSITION 2.5. Let и e FN(fi) and S a subring of(R, u). If the relative topology u\s
on S is exhaustive and S is dense in (R, и), then it is exhaustive.
PROOF. Let (а„)„ек be a disjoint sequence in R and U a solid 0-neighbourhood in (R.u).
Let U„ be a sequence of 0-neighbourhood in (R. it) with Uq = U and Un vUn С U„-\ (n e
N). Take b„ e 5" with b„ Aa„ e U„. Then c„ :=b„\ sup, <n fo, defines a disjoint sequence in
S. Therefore eventually c„ eU. Moreover, anAc„ = (a„ \ supl<:„ ai)A(bn \ sup,,,,, b,) ^
supi^n(ajAbi) e U\ ? Ui ? · · · ? Un С U.
It follows that eventually an e UAU. ?
In the following we examine the relationship between exhaustivity and boundedness.

FN-topologies and group-valued measures
711
PROPOSITION 2.6. Let ?:/? —>· (G. | \) be a measure with values in a seminormed
group. Then supve/? \?{?)\ = +oo iff there is a disjoint sequence (xn) in R with
lim„^+0C|MU„)| = +oo.
PROOF, of the nontrivial implication. Let supveK |?(*)| = +oo.
Case 1: Suppose that the semivariation ||?|| of ? is finite. We define inductively a
disjoint sequence xq,x\,xi, ¦ ¦ ¦ in R with \?(?„)\ ^ «. Put xo = 0. If .v, are chosen for
/' < n, put ? = sup''^ x; and take an ? e R with |?(.?)| ^ IImII(c) + "· For .v„ := ? \z we
have \?(?„)\ + \?(? ??)? > ????? > ??(·* ?;)| + /;, hence |?(.??)| ^«.
Care 2: Suppose that /? contains an element a with ||?||(?) = +эо. We define
inductively a disjoint sequence .Vo..V|,.V2 in R with |?(.?„)? ^ " and ||?||(? \
sup/'=0Xi) = +oo. Put jco = 0. Assume that .v, are chosen for /' < ? and ||?||(;) = +oo
where z=a\ sup;<„ ?,. Let ? ^ ? with |?(.?)| ^ |?(;)| +«· Then |?(; \.?)| ^ |?(^)| -
|?(?)| ^ ?· Moreover, ||?||(?-)+ ||?||(: \.?) = +эо. Put.v,, = ? if ||?||(; \jc) = +oo, and
Xn =z \x if ||?||(?-) = +oo. ?
COROLLARY 2.7. ?^? ?: R —>¦ G be an exhaustive measure with values in a topological
group. Then supveK \?(?)\ < +oo for any continuous seminorm \ \ on G.
A subset ? of a commutative topological group G is called bounded in G if one of the
following equivalent conditions is satisfied:
A) supveB |*| < +oo for every continuous seminorm | | on G;
B) for every 0-neighbourhood U in G there is a natural number n and a finite subset F
of G such that В с F+ [/<"> where U{U = U and Ulk+U = U[k) + U (keN).
The equivalence of A) and B) is proved in Hejcmann A959, Theorem 2.5) (cf. also
Weber A986), Section 6). We here use only the first condition. The following theorem is a
reformulation of Proposition 2.6 and Corollary 2.7.
THEOREM 2.8. Let ? : /? —>· G be a measure with values in a topological group.
(a) Then ?G?) is bounded iff [?(?„): ? e ?) is bounded for every disjoint sequence
(a„) in R.
(b) If ? is exhaustive, then ?(R) is bounded.
COROLLARY 2.9. If и isan exhaustive FN-topology on R, then R is boundedin (R, ?, u).
PROOF. Apply Theorem 2.8(b) with G = (R, ?, u) and ?(?) = ? (? e R). ?
Actually, Corollary 2.9 is equivalent to Theorem 2.8(b): Under the assumption of
Theorem 2.8(b), R is bounded with respect to the ?-topology and therefore the uniform
continuous image ?(/?) is bounded in G.
THEOREM 2.10. Let ? e N. For a measure ?-.R^-M." the following conditions are
equivalent:
A) ? is exhaustive.

712
?. Weber
B) ? is bounded.
C) ? is of bounded variation.
PROOF. C) =>· A) =>· B) is contained in Example 2.2(b) and Corollary 2.7.
B) =>· C). It is enough to show that the components ?, of ? = (??,..., ?„) are of
bounded variation. So we may assume that ? = 1, i.e., that ? is real-valued. Let ? : =
supveK ????? and x\,... ,x„ disjoint elements of R. Then ?]"=? |?(-*/)? = ? ?(??) ~~
]?~ ?(??) = ?E\??+ Xj) - ?($\??~ Xj) ^ 2.?, where ]?+ and sup+ (]?~ and sup") is taken
over those /' for which ?(?,) ^ 0 (?(.?,) < 0). Therefore |?|(?) ^ 2s for all a e R. D
There are Banach space valued bounded measures which are not exhaustive:
Example 2.11. Define ?: P(N) -> lx by ?( ?) = ?? where P(N) denotes the power set
of N and ?? the characteristic function of A. Then ? is bounded, but not exhaustive.
On the other hand, there are also infinite dimensional Banach spaces ? with the property
that any ?-valued bounded measure is exhaustive; e.g., any reflexive Banach space, in
particular any Hilbert space, has this property. This follows from the fact that a Banach
space valued measure is exhaustive iff it has a relatively weakly compact range (cf. Diestel
and Uhl A977), Theorem 1.5.2).
By the Dvoretsky-Rogers theorem, for any infinite Banach space ? there is a ?-additive
?-valued measure ?:?(?) —>¦ ? whose total variation is not bounded. Therefore the
equivalences A) ·?> C) and B) ·?> C) fail in any infinite dimensional Banach space.
The following fact is used in Theorem 7.7 to compare the ?-topology and the |?|-
topology.
PROPOSITION 2.12. Let ?: R -> (G, | |) be a measure with values in a seminormed
group. Then \?\ is exhaustive iff \?\ is bounded.
PROOF. We will apply Theorem 2.10 A) о B) for the measure |?|. Therefore we only
have to prove that |?| is real-valued if |?| is exhaustive. Let |?| be exhaustive. Then ? is
exhaustive and therefore ? := supveK |?(.?)| < +oo by Corollary 2.7.
We first show: If a e R and |?|(?) = +oo, then there are disjoint elements b,c e R with
a = bvc, \?\?) ^ ? and \?\(?) = +oc.
If \?\(?) = +oo, there are disjoint elements a\ a„ e R with a = sup,^„a, and
?!'=? 1м(а*I > 3?· Let к be the smallest natural number with ?]f=1 |?(?,)| > ?· We put
b := supi</t a, and с :=a\b = sup^^a,. Then \?\?) > s and
? ? k—\
|д|(с) ^ ]? |?(?,)| = ?\?(?,)\ - |?(?*)| - ]?|?(?,)| > 3? - ? - ? = ?.
i=k + \ i = l ; = l
Moreover, \?\?) = +?? or \?\(?) = +?? since \?\?) + \?\(?) = \?\(?) = +??.
Suppose now that |?|(??) = +°? for some ?? e R. Then, by the proved statement,
we can inductively define sequences (an)„e\: and (b„)nei; in R with ?„-? = b„ va„.

FN-topologies and group-valued measures
713
bn л a„ = ?, |?|(?„) = +oo and |?|(?>„) ^ i. This contradicts the exhaustivity of |?| since
(fo„) is a disjoint sequence. ?
3. ?-submeasures and completeness
? : R —>¦ [0, +00] is called ?-subadditive if ? = sup,!6f;a„ implies /j(a) ^ ??=\ ^(Ял) for
any a,an e R. A monotone function ? : R —>¦ [0. +00] is ?-subadditive iff the implication
DC
a = supa„ =>· /?(a)^ VVa„)
"eN „=i
holds for any disjoint sequence (a„),!et; in /? and a e /?. A ?-subadditive monotone
function /j: /? -> [0, +00] with /j@) = 0 is called a ?-submeasure on /?. Obviously, any
?-submeasure is a submeasure.
Example 3.1. Let | | be a pseudonorm on G and ?: R -> (G.| |) a ?-additive
measure. Then the semivariation ||?|| (defined in Example 1.2) is a ?-submeasure.
PROPOSITION 3.2. For a submeasure ty.R -> [0,+00] the following conditions are
equivalent:
A) ? is a ?-submeasure.
B) If(an)nefi is a Cauchy sequence in (R. ?) and an \ 0, then ?(??) —>¦ 0.
C) If (an),,eN is a Cauchy sequence in (R, ?) and an la. then ?(?„??) —>¦ 0.
D) If(a„)neN is a Cauchy sequence in (R. ?) andan f ?, ?/??>/? ?(????) —>¦ 0.
PROOF. Using the fact that a„ 4, a implies a 1 Act,, 1 a\Aa. an \ a implies a Aa„ 4, 0, one
immediately sees that conditions B)-D) are equivalent,
A) =>. B), Let (a,,)ne^ be a Cauchy sequence in (R. ?) with an 4. 0. Choose a
subsequence (fo„)weN of (а„)„ек with r)(bnAbn+\) ^ 2^" for /7 e N. Since fo„ = sup.^iA \
fy+?) one obtains ??„) ^ ??^?»7(*?' \ fo< + i) ^ 2""*1, Therefore ??„) -+ 0. Since
(a„)nsN is a Cauchy sequence which has a subsequence converging to 0, also (a„)nem
converges to 0 in (/?,/?).
D) =>· A). Let ?,?„ e fi with а = sup„er:a„. We will show that ?(?) ^ Y^=\ ?(?„).
For that we may assume that ??=\ ?(?„) < -boo. Then b„ := sup"=| a, defines a Cauchy
sequence; in fact,
III
??,,,?^,) ??( sup ?, W V] f?(a,) -> 0 (m > n ->¦ +00).
Moreover, b„ f a. Therefore t)(aAfc„) -> 0 by D). Since a = (sup"=1 a,) v (aAb„), we
have
77(a) < ]? ??(?,) + n(aAb„) ? ]? 7?(a,) + il(aAb„).
1 = 1
It follows that ??(?) ^ ]Р?1| /?(а/).
D

714
?. Weber
Condition B) of Proposition 3.2 corresponds to the pseudo ?-Lebesgue property for
locally solid Riesz spaces (see Aliprantis and Burkinshaw A978, p. 115)), A further
characterization of ?-subadditivity is based on the following lemma.
LEMMA 3.3. Let R be ?-complete and ? a ?-submeasure on R. If(xn)net< 's a sequence
in R with ?(????„+\) ^ 2~" for n e ?, ? = liminfjt,, and у = limsupjt,,, thenx,, -> ? (?)
andxn -> у (?).
Proof, (i) For n e N we have
sn := sup^j =xn vsupUi+? \xi).
hence
s„ Ax„ = sn \ xn ^ sup(.vj+i \ Xj),
With the ?-subadditivity of ? we get
?E„??„) ^??(?1 + ] \Xi) iJ2-" + 1.
Since (xn)neN is a Cauchy sequence and - as we have proved - SnAx,, —> 0 (?), also
(sn)neN is a Cauchy sequence. Moreover, sn I y. Therefore sn —> у (?) by Proposition 3.2
A) =>· C). It follows that x„ -> у (?) since Sn ?.?„ -> 0 (?).
(ii) Let ? := sup,!eN;t„ and ;* := ? \ ; for ; ^ J. Then limsup.x* = .v*. Since
?(?*??*+]) ^2~" we get from (i) that, ?* -> ?* (?). It follows that.xn -* ? (?). ?
For a submeasure /;:/?—>¦ [0, -boo] and .?, ? e R we write л· ~,; у iff ?(???) = 0.
It follows from Proposition 1.6 that ~,; is a congruence relation on (R. v. ?. ?, \).
Sometimes we use the following fact: If л„ —>¦ ? (?), then xn —> у (/?) iff ? ~?; }'·
PROPOSITION 3.4. Let ? be a submeasure on R. If R is ?-complete, then the following
conditions are equivalent:
A) ? is a ?-submeasure.
B) // (a„)„eN 's a Cauchy sequence in (R, ?) and liminfa,, = limsupa,, =: a, then
an -+ ? (?).
C) //'(a,!),ieN is a Cauchy sequence in (R, ?), a e R and liminfa,, ~,; a ~,; limsupa,,,
then an^r ? (?).
PROOF. C) =>· B) is obvious, B) =M 1) follows from Proposition 3.2.
A) =>· C). Let (a„)„eK be a Cauchy sequence in (/?./?) and liminfa,, ~,; a ~,;
limsupa,,. Choose a subsequence (x„)net'. of (an)ner; with i)(Xn&Xn+\) ^ 2~". Since
a ~^ liminfa,, ^ ? := liminf.v,, ^ limsupa,, ~,; a, we have ? ~,; .v. By Lemma 3.3,
(xn)neN converges to ? and therefore also to a ? in (R. ?). So (а„)„ек is a Cauchy sequence
which has a subsequence converging to a. It follows that also (an)nen converges to a. D

FN-topologies and group-valued measures
715
PROPOSITION 3.5. Let R be ?-complete and ? a ?-submeasure on R. If (а„)„ек is
a sequence converging to a in (/?,/j). then (a,,)„eu has a subsequence (x,,)net-i with
liminf.*,, ~4 a ~^ limsupjt,,.
PROOF. Let (x„)neN be a subsequence of (а„)„еь: with ?(?„???+?) ^ 2'" for n e N,
? := liminfjt,,, у := limsupjt,,. Then x„ —>¦ ? (?) and л„ —>¦ у (?) by Lemma 3.3.
Moreover, x„ —> ? (?) by assumption. It follows that ? ~,; у ~,; а. П
THEOREM 3.6. Let R be ?-complete and ? a submeasure on R.
(a) If ? is ?-subadditive, then (/?./?) is complete.
(b) If(R, ?) is complete and Hausdorff, then ? is ?-subadditive.
PROOF, (a) immediately follows from Lemma 3.3: If (а„)„ек is a Cauchy sequence in
(/?,/?) and (xn)„eN a subsequence of (a„)„et; with ?(?„???+\) ^ 2~", then (jt„)„eN
converges by Lemma 3.3. Hence (a„)net: converges.
(b) We use the characterization of ?-subadditivity given in Proposition 3.2 A) о B).
Let (a„)„epj be a Cauchy sequence in (R. ?) and an \, 0. We will show that a,, —>¦ 0 (?).
Since (/?, ?) is complete, (а„)пе]ч has a limit a e R. By Corollary 1.8 and Proposition 1.9
we have a„ 4- ?. Therefore ? = 0 and a„ —>¦ 0 (?). ?
COROLLARY 3.7. Le? /? be ?-complete and ? : /? —>¦ G ? ?-additive measure with values
in a pseudonormedgroup. Then R is complete with respect to the ?-topology.
We now give examples of finitely additive, non ? -additive measures inducing a complete
FN-topology. In contrast to the first example, the measures in the second and in the third
example are atomless.
Example 3.8. Let ? be a maximal ideal in P(N) containing all finite subsets of N
and ? the corresponding ultrafilter measure, i.e., ?(?) = 0 for A e ?4 and ?(?) = 1 for
A e P(N) \ M. Then ? is a non ?-additive measure and (P(N). ?) is complete.
Example 3.9. Let ? be the Lebesgue measure on the ?-algebra ? of Lebesgue
measurable subsets of [0. 1] and ?:?([0. 1]) —>¦ [0. 1] be a measure extension of ?
such that ? is dense in (P[0. 1].?). (Such a measure exists by Corollary 5.2.7.) Then
(P([0, 1]),?) is complete since its dense subspace (?. ?) is complete. But under the
continuum hypotheses ? is not ?-additive by Ulam's theorem.
We now modify the last example to avoid the continuum hypotheses.
Example 3.10. Let ? be the Lebesgue measure on the ?-algebra ? of Lebesgue
measurable subsets of [0, 1] and (Nn)ne\-: be an increasing sequence of subsets of [0. 1]
with inner Lebesgue measure 0 and Um=i N» = $· 4· Let ?? be the ideal in P([0. 1])
generated by {jV„: пеЩ and ?0:????^> [0. 1] defined by ??(???) = ?(?) (A e A,
N e ??). Let ? :P([0, 1]) -> [0. 1] be a measure extending ?() such that ???? is dense
in (P([0, 1]), ?). Then (P([0. 1]). ?) is complete. But ? is not ?-additive since ?(?„) =
0(neN)andM(U,?l| W„)=l·

716
?. Weber
The following problem is related to problem 2 on p. 101 of Aliprantis and Burkinshaw
A978): If R is a ?-complete Boolean algebra and и an FN-topology on R generated by a
family of ?-submeasures, does it follow then that (R.u) is sequentially complete?
4. ? -order continuity and order continuity
An FN-topology и on R is called (a)-order continuous if every decreasing net (sequence)
order converging to 0 converges topologically to 0 in (R, u). An FN-topology и on R is
(a)-order continuous iff for every net (sequence) order convergence implies topological
convergence in (R, u). A submeasure ? is called (?)-order continuous if the /j-topology is
(a)-order continuous.
PROPOSITION 4.1. Let ? be a submeasure on R.
(a) If ? is ? -order continuous, then ? is ? -subadditive.
(b) If ? is ? -subadditive and exhaustive, then ? is ? -order continuous.
(c) If R is ?-complete, then ? is ?-order continuous iff ? is ?-subadditive and
exhaustive.
PROOF, (a) immediately follows from Proposition 3.2 B) =>· A).
(b) Let R э an \, 0. Then (a„);ieH is a Cauchy sequence by Proposition 2.4 and therefore
converges to 0 by Proposition 3.2 A) =>· B).
(c) We have only to show that a ?-order continuous submeasure on a ?-complete
Boolean ring is exhaustive. Let (an)„e>: be a disjoint sequence in R. Then sup^a, ^ 0,
therefore ?{??) ^ /j(sup,^„ a,) —>¦ 0 (n —>¦ эс). So ?(?„) —>¦ 0 (n —>¦ oo). ?
COROLLARY 4.2. If R is ?-complete, G a topological group and ? :/?—>¦ G a measure,
then ? is ? -additive iff the ?-topology is ? -order continuous.
PROOF. If the ?-topology is ?-order continuous, then R э an \, 0 implies that (a„)neN
converges to 0 in the ?-topology. hence ????(?,?) = 0 by the continuity of ?.
Vice versa, let ? be ?-additive and (| |,,)r6/' a system of seminorms generating the
topology of G. Then the semivariation ||?||,, with respect to | \Y is a ?-submeasure
by Example 3.1 and exhaustive by Example 2.2(a). hence ?-order continuous by
Proposition 4.1. ?
For any и e FN(tf), the closure N(u) := @)" of {0} in (R. u) is an ideal in R. и is
Hausdorff iff N(u) = {0}. If и is generated by a submeasure ?. then N(u) = ?(?) := {? e
R:n(x) = 0}.
If ? and ? are submeasures on R, we write ? <? ? if ?{?„) -> 0 (n —>¦ oo) implies
?(?„) —>¦ 0 (/; —>¦ oo), i.e.. if the ?-topology is weaker than the /j-topology.
THEOREM 4.3. Let R be ?-complete, ? ? ?-submeasure on R and ? ? ?-order continuous
submeasure on R. Then ? <? ? iff ?(?) С ?'(?).

FN-topologies and group-valued measures
717
PROOF. =>·. The condition that ?(?„) —>¦ 0 implies ?(?„) -> 0 means for constant
sequences exactly ?(?) с N (к).
«=. Suppose that R contains a sequence (a;,)„er.; such that ?(?„) —> 0. but (?(??))„??
does not converge to 0. Passing to a subsequence we may assume that k(a„) ^ ? (? e
?) for some ? > 0. By Proposition 3.5, (а„)„ег; has a subsequence (.v„),ie[.: such
that ? := limsupjt,, ~,; 0, i.e.. ?(?) = 0. Hence ?(?) = 0. On the other hand, ?(?) =
limX(V,^„ ??) ^ ?, a contradiction. ?
COROLLARY 4.4. If R is ? ?-complete and к ?/iii f? агг exhaustive ?-submeasures on R,
then the ?-topology agrees with the ?-topology iff N(k) = ?(?).
Proof. This follows from Theorem 4.3 and the fact that every exhaustive ?-submeasure
is ?-order continuous. D
Let ? and ? be measures on R and и e FN(R). Then ? « м or ? « ? means that
the ?-topology is weaker than и or weaker than the ?-topology, respectively. Moreover,
we put ?(?) = /V^-topology); if ? has values in a Hausdorff topological group, then
?(?) = {ae R: Wx e [0, ?], ?(.?) = 0).
COROLLARY 4.5. Let R be ?-complete, ? a ?-submeasure on R and и а ?-order
continuous FN-topology on R. Then и is weaker than the ?-topology iff ?(?) С N(u).
PROOF. Let (ka)aeA be a family of submeasures generating u. Then и is weaker than the
/j-topology iff ?? <^? for any ? e A iff ?(?) с N(ku) for any ? e A iff ?(?) С N(u). D
COROLLARY 4.6. // R is ?-complete and ? and ? are ?-additive measures on R
with values in pseudonormed groups, then the ?-topology agrees with the v-topology iff
?{?) = ?(?).
This follows from Theorem 4.3 and the fact that the ?-topology and the ?-topology
are ?-order continuous (see Proposition 4.2) or equivalently that the semivariations ||?||
and || ? || are ?-order continuous.
To show the significance of the assumptions in Corollary 4.6, we give some examples of
measures ? and ?, which induce different FN-topologies. although ?'(?) = N(v).
Example 4.7. (a) Let ?:?(?)-> [?, +??] be the counting measure, i.e., v(A) = \A\ for
А с ?, and ? :P(N) -> [0, 1] the probability measure defined by ?.(A) = ?„еД 2'" for
Л С N. Then ? and ? are ?-additive, ?(?) = ?(?) = {0}. ? induces on P(N) = {0, 1}N
the product topology, which is compact, whereas the ?-topology is discrete. Observe
that ? is not exhaustive (cf. Corollary 4.4) and the range space of ? is not a group (cf.
Corollary 4.6).
(b) Let ? be a maximal ideal in P(N) containing all finite subsets of N and ? the
ultrafilter measure defined by к\м = 0 and ?(?) = 1. Let ? be defined as in (a) and
? = ? + k. Then ?'(?) = ? (?) = {0}. ?(?) is compact with respect to the ?-topology, but
not complete with respect to the ?-topology. Observe that ? is not ?-additive.

718
?. Weber
(c) Let ?, ? and ? be defined as in (b). Let ? be an uncountable set with ? ? ? = 0, ??
the ideal of all countable subsets of ? and A = {AUB: AeM, В eAf}U{AUB: А с ?,
N \ A e .?, В с ?, ? \ В е ??]. Then A is an algebra with the property: if a sequence
(A„)/ieN in A has a supremum A in Д, then A = U/Jli ^«· ?(??) := ?(? П ?) and
v(M) := v(M П ?) define ?-additive measures on A with ?(?) = N(v). But Д and ?
induce different FN-topologies on ?. Observe that A is not a ? -algebra.
(d) Let A be the ?-algebra of Lebesgue measurable subsets of [0,1 ] and ? the Lebesgue
measure on A- We denote by ?? the characteristic function of a subset ? of [0, 1]· Then
?(?) = (?(?), ?A) and ?(?) = @, ??) define ?-additive measures on A with values in
G :=R xl|0J| endowed with the product topology. We have ?(?) = N(v) = {0}, but the
?-topology and the ?-topology are different. Observe that G is not metrizable.
The measures Д and ? in Example 4.7(c) are not strictly positive. In fact, it can be
shown that two ?-additive strictly positive real-valued bounded measures on R induce the
same topology even if R is not ?-complete; for more precise information see Weber A994,
Section 4).
THEOREM 4.8. Let R be a complete Boolean algebra and и and ? order continuous FN-
topologies on R. Then и is coarser than ? iff N(v) С N(u).
Proof. =>· is obvious.
·?=. Let (??)??? be a net of submeasures generating v. We may assume that (??)?€?
is increasing. Otherwise, replace (??)??? by (?^) where ?^ := J2aeF'?« ап<* Р is апУ
finite subset of A. xa := supN(^u) e ?(??) since ?? is order continuous and therefore
x := miaeAxa e N(v) с N(u). Since xa 4. x, we obtain xa -+ x (u). hence xa —> 0 (u)
since x e N(u). Let U be a solid 0-neighbourhood in (R.u). We complete the proof
showing that U ? U is also a 0-neighbourhood in (R,v). Let /3 e A with ?? elf and
5= [0,^] where jcl denotes the complement of xp. Since ?(??\?) = {0} С ^V(m|s), we
obtain by Corollary 4.5 that и Is is weaker than /j/j|s-topology. Therefore there is a solid
0-neighbourhood V in (R, ??) such that V ? 5" с U ? 5". We show that V с C/ ? [/: Let
jc e V.Thenjc = (jc \jc/j) v(jc л.х/j) e (V П 5) vi/ с С/ ? С/. It follows that С/? С/ is a
0-neighbourhood in (R, ??) and therefore in (R. ?). ?
COROLLARY 4.9. A complete Boolean algebra admits at most one Hausdorff order
continuous FN-topology.
COROLLARY 4.10. Let R be a complete Boolean algebra. D a disjoint subset of R (i.e..
different elements of D are disjoint) and sup D = e. Let w be a Hausdorff order continuous
FN-topology on R and w(i the relative topology induced by w on [0, d] for d e D. Then
f(x) := (x л d)deD defines a topological isomorphism f : (R, w) -> П(/ед([0, d], w,i).
Proof. Observe first that / is a Boolean isomorphism. Moreover, the product topology
on ??/eD^· ^1 's Hausdorff and order continuous. Therefore its inverse image under / is
Hausdorff and order continuous and agrees therefore, by Corollary 4.9, with w. This means
that / is a topological isomorphism. ?

FN-topologies and group-valued measures
719
COROLLARY 4.11. Let R be a complete Boolean algebra and w a Hausdorff order
continuous FN-topology on R. Then (/?, w) is algebraically and topological!}· isomorphic
to a product Y\aeA(Ra, wa) where Ra is a complete Boolean algebra and wa a metrizable
order continuous FN-topology on Ra.
PROOF, (i) We first prove that for any a e R \ {0} there is an element b e R such that
0 < b ? a and w/, := w|[0./j] is metrizable. Let (иа)аел be afamily of FN-topologies on R
with countable 0-neighbourhood base such that w = supaeA ua. Since w is Hausdorff, a ?
N(ufi) for some ? e A. Let ?? = sup jV(h/j)· Then N(u^) = [0, jc/j] and b := a \ ?? > 0.
Since the restriction of u? on [0, b] is Hausdorff, it coincides with w/, by Corollary 4.9.
Therefore w/, is metrizable.
(ii) It follows from (i) that R contains a disjoint set D such that sup D = e and wd
is metrizable for any d e D. The assertion of Corollary 4.11 now follows from
Corollary 4.10. D
PROPOSITION 4.12. Let и be a Hausdorff FN-topology on R. Then R is a complete
Boolean algebra and и is order continuous iff every increasing net converges in (/?, u).
PROOF. =>·. If {xa) is an increasing net, then (xa) has by assumption a supremum ? in R,
i.e., xa l ?. Hence xa —> ? (и) since и is order continuous.
<=. Let (xa)aeA be an increasing net in R. By assumption, (xa)aeA has a limit ? in
(R, u). It follows from Corollary 1.8 and Proposition 1.9(b) that ? = supxa. Therefore R
is order complete. This argument also shows that xa \ ? implies xa —> ? (и), i.e., и is order
continuous. ?
5. Extension of FN-topologies and of measures
In this section, let S be a subring of R, G be a complete Hausdorff topological commutative
group and ? : S —> G a measure.
Let (| \Y)Yer be a system of seminorms generating the topology of G.
We define the semivariation
||/х||у(йг) := sup{|/z(Ar)|: 5эа$й)
for any a e R and ? e ?.
5.1. Extension ofa -order continuous FN-topologies and ?-additive measures
The topological approach to measure extension theorems is based on the following simple
fact, which follows from Corollary 1.7 and Proposition 1.10.
PROPOSITION 5.1.1. Let и e FN(R) and ?-.S^- G a measure with ? «С и (i.e.,
? <<C и 15). Then the closure ? = S" of S is a subring of R and ? has a unique continuous
extension ? : (Г, и\т) —> G to ?. Moreover, ? is a measure.

720
?. Weber
Starting from a measure ?, we have to construct a suitable FN-topology и to apply
Proposition 5.1.1. It is near at hand to look for an extension и of the ?-topology; this
guarantees the continuity of ? with respect to u. Which other properties of и are needed to
obtain a ?-additive extension of ? on a ?-ring, can be seen by the following proposition.
PROPOSITION 5.1.2. Let R be ?-complete and и be an FN-topology on R generated by
a system of ?-submeasures. Suppose that u\s is exhaustive.
(a) Then the closure ? = S" of S is ? ?-subring of R (i.e., a subring of R closed with
respect to countable suprema).
(b) If ?. «u, then the unique u-continuous extension ? :T —>¦ G of ? is a ?-additive
measure.
PROOF. Let a e R and (a„)HeN be a sequence in Г with supa,, = a. We will show that
a eT. Since Г is a subring of R, we may assume that (а„)„ек is increasing. Since и\т is
exhaustive by Proposition 2.5, (a„)„ef; is a Cauchy sequence by Proposition 2.4. Now it
follows from Proposition 3.2 that a„ —>¦ a (u) and therefore a eT.
If ? : S —> G is a и-continuous measure, then its и-continuous extension ? : ? —>¦ G is
a measure as stated in Proposition 5.1.1. Let (а„)„ек be an increasing sequence in Г and
a = supa,,. As we have seen before, this implies a„ —>¦ a (u) and by the continuity of ?
that ?(a„)->A (?). D
LEMMA 5.1.3. Let R be ?-complete, 0 e А С R and ??:?^ [0, +??] a function with
jjo@) = 0. Then ? : R -> [0, +оо] Лг/шгЛ fcy
/j(-*) = inf
^ijo(a/i): an e A, .v ^supa,,
I /1 = 1
/? ? ?-submeasure and ?? > /?|д. (Яггг inf0 := +oo.) Moreover, ? extends ?? iff ? ?
supa,, implies ??(?) ^ Y^L\ 1o(an)far any sequence (а„)„ек in A and a e A.
THEOREM 5.1.4. Let R be ?-complete. A measure ? : S —> G /?a.v a ?-additive measure
extension to the ?-subring ?(?) of R generated by S iff ? is ?-additive and exhaustive.
PROOF. Suppose that this (obviously necessary) condition holds. For any ? e ?, the
semivariation ||? \\? is an exhaustive ?-submeasure on S and
??(?) :=inf
]? ||?||7(?„): an e S, ? ? supa,,
I /i = l
defines by Lemma 5.1.3 a ?-submeasure extending ||? || ?. Let и be the FN-topology on R
generated by (??)уеГ and Г and ? defined as in Proposition 5.1.2. Then ? is a ?-additive
measure extension of ? and its domain ? contains ?(?). ?
In the proof of Theorem 5.1.4 we used Lemma 5.1.3 with A = S. If one takes for A
a suitable bigger set, the FN-topology и becomes weaker and so the domain S" of the

FN-topologies and group-valued measures
721
measure extension ? becomes bigger. For example, let R be a Boolean ?-algebra, e e S
and (d„)neN be a disjoint sequence in R and
A = Su{.x\ sup dj\ ? eS, ? eN};
then one obtains as in the proof of Theorem 5.1.4 that every ?-additive exhaustive measure
? : S —>¦ G has a ?-additive measure extension to the ?-subring a(S U {ii„: ? e ?}) of /?
generated by S and the disjoint sequence (dn)nei;.
For a more general result (see Theorem 5.1.7 and Corollary 5.1.8) we use the following
lemma.
LEMMA 5.1.5. Let R be ?-complete, S a ?-subring of R, SO = {x e R: ? ? ? e S for
every ? e S], W a well ordered subset of R containing 0 and A = {z\w: ? e So, we W).
Let ?-.S ^ [0, +oo] foi> ? ?-submeasure, ??(?) := supa^xeSk(x) for a e A and ? defined
as in Lemma 5.1.3.
(a) Then ? is a ?-submeasure extending ??.
(b) If ? is exhaustive, then {x e S: ? ^ ?) converges to a in (R, ?) for any a e A, in
particular А С S'>.
PROOF, (a) By Lemma 5.1.3 we have to prove that for a countable index set /,
? e S, aa e A (a e ?), ? ? supaa imply ?(?) ^ ]P 1o(oa)·
«' ael
aa can be written as aa = za\ »« with za e So and wa eW. With the aid of the well order
of W one can define a well order on / such that (wa)aei becomes an increasing net. Since
S is a ?-subring of R, the elements xa := (x л za) \ supp<a ?? belong to S. Moreover,
xa ^ za and
xa ^ ? \ sup г/j ^ sup(z/j \ ш/O \ sup ?? ^ sup(c/j \ w/0 ^ sup ?? \ wa,
?<? pel ?<? ?^? ?^?
hencexa ^za\wa =aa.
We now show that ? ?. supue/Xa. Suppose that ? := ? \ supa6/xa ? 0, Since у ^
supae/ za and / is well ordered, there is a smallest index ye/ with у л ?? ? 0. Therefore
у /\?? =0for/3 < у mdyAzy ^ (x Azy)\supp<yzp = ??. Since улл-у =0, it follows
that у л ?? = 0, a contradiction.
Finally, ?(*) ^ ?ffe/ ?(*„) ^ ??6/ /??(??)·
(b) follows from the following lemma observing that for a e A and jeSwe have
a\x e A and therefore /j(a \jc) = /jo(a \x)· ^
LEMMA 5.1,6. Lei ? : S -> [0, +oo] be an exhaustive function, i.e., ?(?„) -> 0 (n -> oo)
for every disjoint sequence (a,,) in S. Put ?* (?) := supa^.veS k(x)for a e R. Then for any
? e R and ? > 0 г/гггг w аи jc e S ??/'?/г л^ ? and ?*(? \ л) ^ ?.

722
?. Weber
PROOF. Let ? e R and ? > 0. Suppose that ?*(; \?)> ? for any ? e S with ? ^ z. Then
for any ? e S with ? ^ ? there is an у e 5 with у ^ с \ ? and ?()') > ?. Therefore we can
inductively find a sequence (y„) in S with y„ ^ с \ sup, <;i y, andX(y„)> ?, a contradiction
to the exhaustivity of ?. ?
THEOREM 5.1.7. Let S be a subring of ? ?-complete Boolean ring R, W a well ordered
subset of R and ? : 5" —> G ? ?-additive exhaustive measure. Then ? has ? ?-additive
measure extension p.:a(SUW)—>-Gto the ?-subring of R generated by SUW such that
S is dense in a(SU W) with respect to the ?-topology.
Proof. Since R is a ?-subring of a ?-complete Boolean algebra, we may assume that R
has a maximal element e. Replacing W by W U {0} we may assume that W contains 0.
Moreover, in view of Theorem 5.1.4 (and its proof), we may assume that 5" is a ?-subring
of R. We now proceed as in the proof of Theorem 5.1.4. Let
??(?):=??{
^||?||7(?„): ane ?. ? ^supa„
/?=?
where A is defined as in Lemma 5.1.5. By Lemma 5.1.5, ?? isa?-submeasureon R which
agrees on A with ||?||7.
Let и be the FN-topology generated by (??)?^? and ? and ? defined as in
Proposition 5.1.2. Then Д is ?-additive and ? a ?-subring of R. S is dense in ?
with respect to the Д-topology since ? <K u. It remains to show that W с Т. From
Lemma 5.1.5(b) follows that А с Г. Moreover, ceA and e \ w e A for w e W. Therefore
W cT since Г is a ring. ?
COROLLARY 5.1.8. Let R be ?-complete. D a disjoint subset of R and ? : S —>¦ Gaa-ad-
ditive exhaustive measure. Then ? has a ?-additive measure extension ? : ?(? U D) —>¦ G
to the ?-subring of R generated by S U D such that S is dense in a(SUD) with respect to
the ?-topology.
PROOF. Let /эа|->4еОЬеа bijection from / onto D where / is a well ordered
set. If / has a maximal element ?, we put J = I U {? + 1}; otherwise we put J = I.
Then W := {sup^<aify.· a e J] is a well ordered subset of R. Now apply Theorem 5.1.7
observing that da = 5ир^<а+, ?? \ sup^^ ?? and therefore o{S U D) с o{S U W). D
Sometimes the following additional information is of interest.
Remark 5.1.9, With the assumptions and notations of Theorem 5.1.1 we have:
(a) |IAI|y(z)= llMlly(z) for any с e S and ? e ?.
(b) The Д-topology induces on S the ?-topology,
(c) The Д-topology agrees with к17- iff the ?-topology agrees with u\$.

FN-topologies and group-valued measures
723
PROOF, (a) Obviously ||Д||у(с) ^ \\?\\?(?). Let now с e 5" and ? e ? with ? ^ z- We
show that \?(?)\? ^ \\?\\?(?). Let (л„)„е,л be a net in S converging to ? in (Г. и). Then
(?,,??) converges to x, too. Therefore
\?(?)\ =Нт|д(л„лс)| iC ||?||7(;).
(b) follows from (a), (c) follows from (b) and the following proposition. D
PROPOSITION 5,1.10. Let и e FN(tf) and S a dense subring of(R,u). Then ? ы> v\s
defines an order isomorphism from the lattice FN(R. u) of all FN-topologies on R coarser
than и onto the lattice FN(S, u\$).
PROOF. Obviously, r:ww v\s defines a monotone map from FN(R. u) into FNiS, и Is)·
r is surjective: let и e FN(m|s) and 0?„)„ед be a family of submeasures on S generating ?.
Then the continuous extensions rja of ?? on (R.u), a e A. induce on R an FN-topology ?
coarser than и with v\$ = v.
The injectivity of r and the monotonicity of r~' is a consequence of the following
statement: If ? e FN(R, и), then the closures V" in (R. u) of the 0-neighbourhoods in
(S, v\s) form a 0-neighbourhood base in (R, v). In fact, if ? is a continuous submeasure
on (R, u), ? > 0 and V = {x e S: ?(?) < ?], then
{xeR: ?(?)<?} с V" с {? e R: ri(x)^s}, ?
Let ? and ? be measures on R and u.ve FN( R). We write и J_ ? if the infimum of и and
и is the trivial topology, ? J_ ? (and ? J_ v) means that и J_ ? where и is the ?-topology
(and ? is the v-topology). If ? J_ v, we say that ? and ? are singular.
From Theorem 5.1.1, Remark 5.1.9(b) and Proposition 5.1.10 immediately follows:
Corollary 5.1.11. Let и e FN(fi) and S be a dense subring of (R, u). Denote by
a(R, G, u) and a(S, G, u\$) the spaces of all G-valued continuous measures on (R, u)
and (S, и Is), respectively.
(a) Then ? н>- ?|5 defines a group isomorphism from a(R, G, u) onto a(S, G, u\$).
(b) For ?, ? ea(R, G, u) and ? e FN(/?, u) we have:
(i) ? « ? ??"?\5 « v\s, mIuiJmIsI v\s-
(ii) ?«?(^?|5«?|5, ? ±? iff ?\$ ±v\s.
Theorem 5.1.4 was first proved by Sion A969) by a suitable modification of the
Caratheodory process. A topological proof of this theorem is contained in Drewnowski
A972a), Herer A976), Weber A976). D. Maharam A987) observed in a footnote that the
topological approach to obtain a ?-additive extension of a bounded real-valued measure
from a ring to the generated ?-ring was originated by M.H. Stone.
D. Maharam A987) presented a topological approach to extend also [0,+oo]-valued
measures, which in general are not exhaustive. Here, also the following fact observed in
Lembcke and Weber A992, Theorem 3.3.1) is of interest:

724
?. Weber
THEOREM 5.1.12. Let R be ?-complete. Then every ?-additive [0, +oo]-valuedmeasure
on S admits a ?-additive [0, +oo]-valued measure extension to R iff every ?-additive
[0, l]-valuedmeasure does so.
This follows from the fact that every [0, +oo]-valued ?-additive measure is an
(infinite) sum of [0, l]-valued ?-additive measures (see Lembcke and Weber A992,
Corollary 3.1.17)).
Theorem 5.1.7 is essentially contained in Weber A980). It is worthwhile to observe that
the assumption in Theorem 5.1.7 that W is well ordered is essential. The result fails already
for a countable totally ordered set W as observed in the postscript of Lipecki A980a,
1980b). Its Corollary 5.1.8 generalizes results of Bierlein A966), Ascherl and Lehn A977),
Lipecki A980a, 1980b). In Bierlein A966), Lipecki A980a, 1980b), the disjoint system is
countable. In Bierlein A966), Ascherl and Lehn A977), the set functions are probability
measures.
5.2. Extension of exhaustive FN-topologies and exhaustive measures
In contrast to the ?-additive case, any finitely additive vector-valued measure defined on
an algebra ? of subsets of X has a measure extension to the whole power set F(X),
This is an easy consequence of the Hahn-Banach theorem. The result remains true (but is
harder to prove) in the group-valued case, see Carlson and Prikry A982). This is a purely
algebraic result. In measure theory it is more of interest to extend exhaustive measures.
Lipecki A974) proved that any exhaustive measure ?: ? -> G has an exhaustive measure
extension Д :P(X) ->¦ G such that ? is dense in F(X) with respect to the Д-topology.
The significance of the density statement is here in particular to transfer properties from
? to Д. Lipecki's extension theorem for measures is an immediate consequence of an
analogous extension theorem for FN-topologies; actually these two extension theorems
(Corollaries 5.2.6 and 5.2.7) are equivalent. We present in the following also Lipecki's
more general extension theorem for tight measures, cf. Lipecki A987).
Definition 5.2.1. Let К с R. An FN-topology и on 5" is called K-tight if for every
? & S and every neighbourhood U of ? in (S, u) there are elements к е К and ? e U with
у ^ к ^ x, A measure ? : S -> G is called ?-tight iff the ?-topology is ?-tight, i.e., if for
every ? e S and every 0-neighbourhood U in G there are elements к е К and у е S such
that у < к ? ? and ?(?) e U for every ? e [0. ? \ y].
LEMMA 5.2.2. Let ? ; S —> [0, +oo] be a submeasure on S.
(a) Then ?*(?) = inff,^.ve5 ?(?) defines a submeasure on R.
(b) ?*(?) = $???^?€5?(?) defines a monotone function on R with the following
property: ifa,be R and a Ax e S for any ? e S with ? ^ ? ? b, then ?*(? ? b) ^
?*(?) + ?*?).
(c) ?* < ?* and ru\s = ?=
DISPROOF, (а), (с) and the monotonicity of ?* obviously hold. We prove the last statement
of(b).

FN-topologies and group-valued measures
725
Leta,fo e R suchthataAJt e S for any .r e 5??[0, ? ? fo]. Then, for any .r e Sn[0,a vfo],
we have x\a = x\(x ла) e 5??[0, fo], hence ?(?) ^ ^(.v ла) + /j(.v\a) < ?*(?) + /j*C>)·
It follows 77* (? vfc)i?ij*(a) + ij*(fc). ?
LEMMA 5.2.3. Let R be a Boolean algebra, S a subalgebra of R, ?: 5 -> [0, +oo]
an exhaustive submeasure, к e R and ? = {(.? Л к) ? (у Л к'): х, у е S] the Boolean
subalgebra generated by S andk. Put p(a) = ?*(? А к) + ?*(? /\k')for а е Г:
(a) ?/??7? /0 is a submeasure on T and ? ^ p\s ^ 2?\
(b) {jc e S: jc ^ k'} converges to k' in (T. p)\
(c) S is dense in (T, p);
(d) the continuous extension ?: (?, ?) —> [0, +эо] of ? is an exhaustive submeasure
inducing the p-topology;
(e) // К is a sublattice of R,k e К and ? is К-tight, then ? is К-tight, too.
PROOF, (a) p is a submeasure since if and /j*|.w are submeasures, see Lemma 5.2.2.
Let ? e S. Then p(x) = ?*(? л к) + ^*(л л к') ^ ?*(.?) + ?*(?) = 2?(?). We show now
that ?(?) ^ p(x): Let ? > Oand у e 5 with .? А к < у < .? and /j(y) < f?*(.v л ?) + ?. Then
*\у < .xAfc'andijC*)^ /j(y) + ? (? \ у) ^ ?* (? лк)+е + л*(х лк') = ?(?)+?.
(b) Let ? > 0. By Lemma 5.1.6 there is ап.кеХ with .r ^ k' and /j*(^' \ л) ^ ?. Now
observe that /j*(^' \ x) — p(k' \ x).
(c) Sp is an algebra containing S and A:' as proved in (b). Therefore Sp contains ?, the
algebra generated by S and k'.
(d) By (a), ? is continuous on (S. p\s) and its continuous extension ? on (?, p) is a
submeasure satisfying ? ^ ? ?;2?. Therefore ? and /О induce the same topology. The
exhaustivity of ? follows from (c) and Proposition 2.5.
(e) Let a e ? and ? > 0. We write a = (x\ л it) ? (x2 л it') with .v, e 5". By (b) and (d),
there is an xq e S with Jto < k' and rj(k' \xq) < ?. By assumption, there are elements к, е ?
and y, e 5 with у ? < k\ <,x\,y2^k2^ x2 л .то, /j(.ti \yi) ^ ? and ?((?2 ???) \y:) ^ ?.
Then we have у := (yi л к) ? y2 e Г Д := (?? л к) ? к2 е К. у ^к ? a and ?j(a \ у) <
Л(х\ \ У\) + 4@*2 л дсо) \ Уг) + Щк' \ .го) < 3?. ?
THEOREM 5.2.4. Let К be a sublattice of R and и a K-tight exhaustive FN-topology
on S. Then there is a K-tight exhaustive FN-topology ? on the ring ? generated by S U К
such that v\s = и and S is dense in (T. v).
PROOF. Since every Boolean ring can be embedded in a Boolean algebra, we may assume
that R has a unit e. Let (??)?€? be a family of submeasures on S generating и.
lie ^S, let SO := SU {?': ? e S} the Boolean subalgebra of R generated by 5". Then, for
a e A, the restriction ?„ of (?„)* to SO is by Lemma 5.2.2(b) a submeasure which extends
??. Let mo be the FN-topology generated by (??)?€?. One easily sees that mo is exhaustive
and- using Lemma 5.1.6- that S is dense in (So. мц) and мц is ?-tight.
So we may assume that e e S. Consider the class of all families (va)a€A of submeasures
with the following properties:
(i) The va have a common domain D which is a subalgebra of ? containing S.
(ii) va extends ?? for each a e A.

726
?. Weber
(iii) If v denotes the FN-topology generated by (vff )оед on D, then ? is ?-tight and S
is dense in (D, v).
We define a partial order in this family putting (va)a€A < (??)?(?? iff ?« extends va for
every a e A. By Zorn's lemma, this class has a maximal element (/?«)<*,= д. Suppose that
the domain D of ??, a e A, is different from ?. Then there is к e ? \ D. By the preceding
lemma, for any a e ?, ?? has an extension rju on the algebra Д> generated by Z) and A;
to a ?-tight exhaustive submeasure such that {.v e D: ? ^ k'} converges to k' in (Д>, ??„).
Then {* e ?): ? < A;'} converges to k' with respect to the FN-topology ? generated by
the family (rja)a€A. It follows that k' e Dv and finally that S is dense in (Д>. ?). Thus
(т)а)аеА is not maximal, a contradiction. ?
From Theorems 5.2.4 and 5.1.1 follows immediately:
COROLLARY 5.2.5. Let К be a sublattice of R and ?: S -> G a K-tight exhaustive
measure. Then ? has а К-tight exhaustive measure extension ?:? —* G on the ring ?
generated by S U К such that S is dense in ? with respect to the ?-topology.
Every FN-topology and every measure on S is К -tight for К = R. Therefore one obtains
from Theorem 5.2.4 and Corollary 5.2.5 the following two corollaries.
COROLLARY 5.2.6. For every exhaustive FN-topology on S there is an exhaustive FN-
topology ? in R such that v\s = и and S is dense in (R.v),
COROLLARY 5.2.7. Every exhaustive measure ?: S -> G has an exhaustive measure
extension ?: R —* G such that S is dense in the ?-topology.
For the relationship of Theorem 5.2.4 with results of Henry, Lembcke, Bachman and
Sultan, and Dalgas we refer to the introduction of Lipecki A987).
6. The completion of topological Boolean rings and the system FNf(/?) of exhaustive
FN-topologies
Often ?-additive real-valued measures on a ?-algebra are easier to handle than finitely
additive measures. For example, for ?-additive real-valued measures on a ?-algebra we
have:
A) Measures with the same null sets induce the same topology.
B) Several decomposition theorems can be obtained by decompositions of the
?-algebra.
C) The algebraic condition that a measure is atomless implies connectedness properties
of the range.
This is not any more true in the finitely additive case. On sight it may be surprising
that in the study of ?-additive measures on a ?-algebra with values in a non-metrizable
range space there occur similar difficulties as in the finitely additive (real-valued) case; in
particular, A )-C) are not any more true. It turns out that A )-C) are satisfied for exhaustive

FN-topologies and group-valued measures
727
measures which induce a complete FN-topology. So not ?-additivity of a measure ?
but completeness of the ?-topology is often the crucial condition (cf. Theorems 6.5 and
6.16). The idea is now, in order to study exhaustive measures on R, to consider first the
continuous extensions ?, v,... on the uniform completion (R, u) of (R, u) with respect
to a suitable FN-topology u. Results for ?. v.... then yield results for the restrictions
? = ???, ? = v|к This procedure is related to the Stone space argument. In contrast
to the Stone space argument, here the measure ? associated to an exhaustive measure ?
is not only ? -additive but completely additive and its domain is order complete, not only
? -complete.
Let w be a Hausdorff FN-topology on R and R the uniform completion of R^ Then the
ring operations ? and л of R have uniform continuous extensions on R. So R becomes
a Boolean ring and the uniformity of R is induced by an FN-topology w. (R. w) is called
the completion of (R, w).
THEOREM 6.1. Let w be an exhaustive Hausdorff FN-topology on R and (R, w) the
completion of(R, w). Then R is a complete Boolean algebra and w is order continuous.
PROOF. By Proposition 4.12 we have only to show that every increasing net converges in
(R, w). But this follows from Proposition 2.4 since w is exhaustive by Proposition 2.5. D
THEOREM 6.2. Let wbea Hausdorff FN-topology on R. Then w is exhaustive and (R, w)
is complete (as a uniform space) iff R is a complete Boolean algebra and w is order
continuous.
PROOF. =>· is contained in Theorem 6.1.
<^=.Let(fl, w) be the completion of (R, w). We show R= R.Leta e R. Then [0,а]ПЛ
is a closed ideal in (R, w), hence a principal ideal, i.e., [0, ?] ? R contains a maximal
element b. Let c = a\b and ? be the FN-topology on R which induces on [0, c] the trivial
topology and on [0, c'] the same topology as w. ? := v\r is order continuous since ? is
weaker than w. ? is Hausdorff since [0, с] П R — {0}. Therefore ? = w by Corollary 4.9
and v= w by Proposition 5.1.10. It follows с — 0, hence a — b e R. ?
COROLLARY 6.3. Let w e FN(R). Then w is exhaustive and (R. w) is complete iff every
increasing net converges in (R, w).
PROOF. This follows from Theorem 6.2 and Proposition 4.12, if w is Hausdorff.
To conclude the proof we reduce the general case to the Hausdorff case passing to the
quotient (R, w) = (R, w)/N(w) where R = R/N(w) and w is the quotient topology on
R with respect to w. Obviously, (R, w) is complete and exhaustive iff (R, w) is so. It
remains to show that every increasing net converges in (/?, w) if (R, w) has this property.
Let (??)??? be an increasing net in R and xa e ??. «? A. Let ? be the upwards directed
system of all finite subsets of A. Then (supa€f.xa)r€?- is an increasing net in (/?, w),
hence convergent. It follows that (??)?€? is convergent. ?
COROLLARY 6.4. Let w be an exhaustive FN-topology on R such that (R, w) is complete.
Then (R, v) is complete for every ? e FN(R, w).

728
?. Weber
PROOF. Apply Corollary 6.3 observing that every increasing net converges in (R, v) if it
converges in (/?, w). ?
THEOREM 6.5. Let w be an exhaustive Hausdorff FN -topology on R and (R,w) the
completion of(R, w).
(a) Then ? н> v\ r defines an order isomorphism r : FN(/?, w) —> FN(/?, w).
(b) vh>f\ sup N(v) defines an order isomorphism s : FN(/?, w) -> R.
(c) FN(/?, w) й a complete Boolean algebra and sor a Boolean isomorphism from
FN(fl, w) onto R.
(d) Let ?, ? be continuous measures on (R, w) with values in a topological group,
? = Д|к, ? = v\r, ve FN(R, w) and ? = v\r. Then
(i) ? <?u iff ?(?) э N(u), ? ±u iff there is an a e ?(?) with a' e ?(??).
(ii) ? <K ? iff ?(?) D N(v), ? J_ ? iff there is an a e ?(?) with a' e N(v).
PROOF, (a) is contained in Proposition 5.1.10.
(b) follows from Theorem 4.8 observing that - as a consequence of Theorem 6.1 - any
? e FN(/?, w) is order continuous and therefore N(v) = [0, a] where a = sup N(v).
(c) immediately follows from (a), (b) and Theorem 6.1.
(d) follows from (a). Theorem 5.1.1 and Remark 5.1.9(b). ?
We denote the system of all exhaustive FN-topologies on R by FNe(R). Then FNe(R) =
FN(R, we) with the notation of Proposition 2.3.
COROLLARY 6.6. FNe(R) is a complete Boolean algebra.
We now examine the relationship between properties of и (е FN(/?)) and properties of
FN(K, и).
The following fact makes it understandable that the condition of exhaustivity plays a
particular role.
Remark 6.7. Let и e FN(R). Then и is exhaustive iff FN(/?, u) is a Boolean algebra.
PROOF. =>· holds by Theorem 6.5(c).
<=. Let (a„)„eN be a disjoint sequence in /?. For ? e N, put R„ = [0, ? ? ? · · · ? ?„] and
S1,, = {* e /?: jc л у = 0 for every ? e /?„}. The sets U ? S^, where i/ is a 0-neighbourhood
in (/?, и) and teN, form a 0-neighbourhood base for an FN-topology v e FN(/?, u). Let
w be a complement of ? in FN(/?, u). It is easy to see, using that R„ is an ideal in R, that
??|?„ is a complement of и | ?;? in FN(/?„, m|r„). Since ?|?„ coincides with и \цп, we get that
??|?„ is the trivial topology, hence an e N(w) for any ? e N. Therefore (а„)„еи converges
to 0 with respect to w. Obviously, (а„)„ек converges to 0 with respect to ? and therefore
with respect to и since и = ? ? ц\ ?
Dehnition 6.8. Let и e FN(/?). We say that (R,u) is chained if for every 0-neigh-
bourhood U in (/?, и) and every ? e R there is a finite sequence л ? x„ in i/ with
jc = sup^=, x,.

FN-topologies and group-valued measures
729
An equivalent condition is that for every ? .ye R and every O-neighbourhood U
in (/?, u) there is a finite sequence .то,..., л„ in U with лц = ?, ?„ = у and X[Axi-\ e U
(i = 1,..., и). Therefore, our terminology is in accordance with Hejcmann's definition of
chained uniform spaces.
Proposition 6.9. Let и eFN(/?).
(a) The following conditions are equivalent:
A) (R,u) is chained.
B) Every continuous submeasure on (R. u) is real-valued.
C) There is no ultrafilter topology on R coarser than u.
D) (R, u) does not contain any open ideal different from R.
(b) и is an ultrafilter topology iff the trivial topology is the only FN-topology on R
strictly coarser than u.
PROOF, (a) A) =>· B). Let ?: (R.u) -> [0. +oo] be a continuous submeasure and .v e R.
Since (/ := (z e R: ?(?) ^ 1} is a O-neighbourhood in (/?, и), there are x\ x„ e U
with ? ^ x\ ? · · · ? xn. Therefore ?(?) < ??=? ^(-"?') ^ " < +oo.
B) =>· C), Suppose that ? is a maximal ideal in /? such that the corresponding ultrafilter
topology is coarser than u. Then ?(?) = 0 (.? e ?) and /j(.v) = +00 (x e R\ M) defines a
continuous(sub-)measureon (R.u).
C) =>· D). Suppose that / is an open ideal in (R. u) and / ? R. Let ? be a maximal
ideal in R containing /. Then ? is open and therefore the corresponding ultrafilter
topology is coarser than и.
D) =>· A). Let U be a O-neighbourhood in (/?, и) and / the set of elements ? e R such
that there are x\,..., x„ e i/ with .v < .?? ? · · · ? л„. Then / is an open ideal in (/?, и),
hence I = R, i.e., (/?, и) is chained.
(b) =>·. Let и be a non-trivial FN-topology on R coarser than и and V a solid
O-neighbourhood in (/?, v) with V ? V ? R. Then, for any a e V, we have [0,«] ? N(m) с
V ? V ? R. Since N(u) is a maximal ideal, it follows [0. ?] ? N(m) = N(u), hence
a e N(u). Therefore V = N(u) and ? = и.
<=. Suppose that и is not an ultrafilter topology. Then N(u) is not a maximal ideal
and therefore R \ N(u) contains two disjoint elements a and b. Then the sets [0. ?] ? ?/,
where U is a O-neighbourhood in (R.u), form a neighbourhood base for a non-trivial FN-
topology strictly coarser than и. ?
Recall that an atom in a lattice with smallest element 0 is an element a e L such that
there is no ? e L with 0 < ? < a. L is called atomless if L does not contain any atom. L is
called atomic if for every г e i, \ {0) there is an atom a e L with a < z, and atomistic if
every nonzero element of L is a join of atoms.
Corollary 6.10. Let и eFN(R).
(а) и is an ultrafilter topology iff и is an atom in FN(/?) iffFN(R. u) contains exactly
two elements.

730
?. Weber
(b) и is the supremum ofultrafilter topologies iffFN(R. u) is atomistic iffFN(R, u) is
atomic and и is exhaustive.
(c) (/?, u) is chained iff FN(R, u) is atomless.
PROOF, (a) and (c) are reformulations of Proposition 6.9 (b) and (a) A) о C).
(b) If FN(R, u) is atomistic, then и is the supremum of atoms of FN(/?. и), i.e., the
supremum ofultrafilter topologies.
If и is the supremum of ultrafilter topologies, then и is exhaustive, FN( R, и) is a Boolean
algebra by Theorem 6.5(c) and its maximal element is the supremum of atoms in FN(/?, u)
(see (a)). Therefore FN(/?. u) is atomic.
If и is exhaustive and FN(/?. u) atomic, then FN(/?. и) is an atomic Boolean algebra and
therefore atomistic. ?
We used that any Boolean ring is atomistic iff it is atomic. Any atomistic lattice with zero
is atomic; the next proposition shows that FN(/?. u) can be atomic without being atomistic
where и is the discrete topology.
A Boolean algebra is called superatomic iff every subalgebra is atomic. It is easy to see
that A is superatomic iff A does not contain a tree (i.e.. a subset order isomorphic to the
system of intervals {[//2", (/' + l)/2"[: i.n eNU(O), /' < 2"}). The algebra generated by
the finite subsets of a set X is an example of a superatomic algebra. For further information
see Bhaskara Rao and Bhaskara Rao A983), Section 5.3.
PROPOSITION 6.11. Let R be a Boolean algebra.
(a) FN(/?) is atomic iff R is superatomic.
(b) FN(/?) is atomistic iff R is finite.
PROOF, (a) If FN(/?) is not atomic, then there is by Corollary 6Л0(с) a non-trivial FN-
topology и on R such that (R, u) is chained. Then the quotient (R,u) = (R. u)/N(u) is
chained, too. It is easy to see that therefore R contains a tree. Hence R contains a tree and
therefore R is not superatomic.
If R is not superatomic, then R contains a subalgebra A isomorphic to the algebra A
generated by the intervals [//2", (/ + l)/2"[ (/, ? e N U {0}, / < 2"). Let ?: A -+ [0, 1]
be the Lebesgue measure, яМ->Да Boolean isomorphism, ?: R -> [0, 1] a measure
extension of ? ? ? and и the ?-topology. Then (/?, и) is chained. Therefore FN(/?) is not
atomic, see Corollary 6.10(c).
(b) If R is finite, then by Theorem 6.5(c) - which is for finite R obvious - FN(/?) is
isomorphic to R and therefore atomistic.
If FN( R) is atomistic, then the discrete topology is the supremum of ultrafilter topologies
(see Corollary 6.10(b)) and therefore exhaustive. Therefore R does not contain an infinite
disjoint subset. Hence R is finite. ?
We will now characterize FN-topologies satisfying [Bl], i.e., which admit a countable
0-neighbourhood base.

FN-topologies and group-valued measures
731
PROPOSITION 6.12. Let и be an exhaustive Hausdorff FN-topology satisfying [B1 ]. Then
R satisfies the countable chain condition (CCC for short), i.e., every disjoint subset of R is
countable.
PROOF. Let (i/„)neN be a O-neighbourhood base in (R, u) and D a disjoint subset of R.
For any ? & ?, D \ U„ is finite since и is exhaustive. Therefore the union {j)l€i:(D \ Un)
is countable. This union equals to D since и is Hausdorff. ?
THEOREM 6.13. Let и e FJSL(K). Then и satisfies [Bl] iffFN(R, u) satisfies CCC.
PROOF. «=. By Theorem 6.5(c), FN(/?, //) is a Boolean algebra. We now use the following
fact: If Л is a Boolean algebra satisfying CCC and D a dense subset of A (i.e., for every
a e A \ {0} there is a d e D with 0 < d < a). then every element of A is the supremum of a
countable subset of D. Applying that for Л = FN(/?, u) and D = {w e A: u> satisfies [Bl]}
we see that и is the supremum of a countable family of FN-topologies satisfying [Bl] and
therefore itself satisfies [B1 ].
=>·. Passing to the quotient (R,u)/N(u),-we may assume that и is Hausdorff. Let (/?,«)
be the completion of (R,u). Then и satisfies [Bl]. Therefore by Proposition 6.12, R
satisfies CCC. Since R and FN(/?, u) are isomorphic by Theorem 6.5(c), FN(/?, u) satisfies
CCC, too. D
Corollary 6.14. Let ? e FNe(R). if ? satisfies [Bl], then even и е FN(/?, v) satisfies
[Bl].
COROLLARY 6,15, If R isaa -complete Boolean ring satisfying CCC and и is ? ? -order
continuous FN-topology on R, then и satisfies [Bl].
PROOF. CCC of R implies that every subset A of R contains a countable subset В
with sup В = sup A, Therefore R is complete and и is order continuous, hence (R.u) is
complete by Corollary 6,3, Therefore R and FN(R. u) are isomorphic by Theorem 6.5(c),
So FN(/?, u) satisfies CCC and therefore и satisfies [Bl] by Theorem 6,13, ?
THEOREM 6,16, Let и e FN<,(/?) and (R.u) be the completion of the quotient (R,u) =
(R,u)/N(u), Then:
(a) (R, u) is chained iff R is atomless,
(b) и is the supremum of ultrafilter topologies iff R is atomic,
(c) и satisfies [Bl] iff R satisfies CCC,
PROOF. By Theorem 6.5(c), R and FN(^. u) are isomorphic. Moreover, there is a natural
isomorphism between FN(R,u) and FN(R. u). Therefore R andFN(/?,w) are isomorphic.
With this fact, (a), (b) and (c) follow from Corollary 6,10 (c)-(b) and Theorem 6,13, D
Definition 6.17. Let ?; R -> G be a measure. If (R. ?-topology) is chained, we say
that R is ?-chained.

732
?. Weber
This concept corresponds to strong continuity in the sense of Bhaskara Rao and
Bhaskara Rao A983).
From Theorem 6,16(a) and Corollary 3.7 follows:
COROLLARY 6,18, Let R be ?-complete and ?: R -> G ? ?-additive measure with
values in a pseudonormed group. Then R is ?-chained iff ? is atomless (i.e., R|N{?)
is atomless).
7. More on the ?-topology
The ?-topology of a measure ? ;/? -» G depends on the topology taken on G: If
и e FN(/?), G = (/?, ?, u) and ?: R -> G is defined by ?(?) = ?, then the ?-topology
coincides with и. In the next theorem there is given a condition under which the ?-topology
is the same for two different group topologies on G, This theorem has several interesting
consequences,
THEOREM 7.1, Let(G. ?) be a complete topological group, ? a Hausdorff group topology
on G coarser than ? and ?: R -> G a measure exhaustive with respect to ?. Then the
?-topology with respect to ? agrees with the ?-topology with respect to ?.
PROOF. We apply Theorem 6.5 with w = we. With the notation of Theorem 6,5, let
?: (R, w) -> (G, ?) be the continuous extension of ?. Let ? and и be the ?-topology,
respectively, with respect to ? and with respect to ?, and и = u\r, v = v\r. Then
N(v) = ?(?) = N(u), hence ? = и by Theorem 6.5, i.e., the ?-topology with respect
to ? agrees with the ?-topology with respect to ?, see Remark 5.1.9(c). ?
Let ? be a set of measures on R with values in a topological group. ? is called
uniformly exhaustive if for every disjoint sequence (xn)n€\; in R the sequences (?(?„))„€??
converge to 0 uniformly in ? e M. For и e FN(/?), ? is и-equicontinuous if for every net
(xa)aeA converging to 0 in (/?, u) the nets (?(.??))??? converge to 0 uniformly in ? e M.
COROLLARY 7.2. Let и e FN(/?), G be a Hausdorff topological group and ? be a
uniformly exhaustive subset of a (R,G,u). Then ? is u-equicontinuous.
PROOF. We may assume that G is complete; otherwise replace G by its completion.
Let Гэо and ?? be, respectively, the topology of uniform convergence and of pointwise
convergence on GM. Then the measure m: R -> (G'w, Гэс) defined by m(a) = (?(?))?€?
is exhaustive and the m -topology with respect to ?? is coarser than и. By Theorem 7.1, the
m -topology with respect to ?^ agrees with the m -topology with respect to ??. It follows
that the w-topology with respect to ?^ is coarser than и, i.e., ? is и-equicontinuous. ?
COROLLARY 7.3. Let ? be a locally convex linear space, ? its topology and ?: R —>
(E, p) an exhaustive measure. Then the ?-topology with respect to p, the ?-topology with
respect to the weak topology ?(?, ?") and the supremum of the x' о ?-topologies, ?' e ?",
coincide.

FN-topologie.s and group-valued measures
733
PROOF. Let N = {0}p and ? : ? -> ?/? be the canonical map onto the Hausdorff
quotient space E/N. Since ? and ? ? ? induce the same FN-topology, we may assume that
? is Hausdorff. Moreover, since the weak topology of the completion of (E, p) induces
on ? the topology ?(?,?'), we may assume that (E,p) is complete. Therefore, by
Theorem 7.1, the ?-topology with respect to ? agrees with the ?-topology with respect to
?(?, ?"). The last one agrees with the supremum of the ? о ?-topologies, ?' e ?". since
?(?, ?') is the supremum of the ??--topologies, x' e ?". where px- := \x'(x)\ (x e E)
and, moreover, the ?-topology with respect to the px--topology agrees with the ?' о
?-topology. ?
COROLLARY 7.4. Let ? and F be locally convex linear spaces and ?: R -> ? and
?: R —>¦ F be the exhaustive measures.
(a) Then ? <<C ? iff ?' ??« ? for every ?' ? ?'.
(b) ? J_ ? iff ?' ? ? J_ у' о ? for every ?' & ?' and у' е F'.
(c) If R is a subring of a Boolean ring ? and К a sublattice of P, then ? is К-tight iff
?' ? ? is K-tightfor every x' e ?".
(d) (i) R is ?-chained iff R is ?' ? ?-chainedfor every x' e ?".
(ii) IfR is ?-complete and ? is ?-additive, then R is ?-chained iffx'o ? isatomless
for every x' e ?".
PROOF. Besides Corollary 7.3 we use in the proof of (b) that и J_ i; if u. ua, ? e ?Ne(R),
и = supa€Aua and ua ± ? for every a e A. In (c) we use that the supremum of
?-tight FN-topologies is ^-tight. In (d) we use Corollary 6.18 and that (/?, u) is chained
if ua eFN(/?), (R,ua) is chained for every ? e A and и = supa€Aua. ?
COROLLARY 7.5 (Bartle-Dunford-Schwartz). Let (E, p) be a metrizable locally convex
linear space and ?: R -> ? an exhaustive measure. Then there exists a real-valued
measure ? : R —» [0, 1] equivalent to ?, i.e., ?«? and ?«?.
PROOF. Let и be the ?-topology with respect to p. Since ? is metrizable, и has a countable
0-neighbourhood base. By Corollary 7.3, и is the supremum of the ?' ? ?-topologies,
?' e ?". It follows that for some sequence (x'„),i€!· in ?", и is the supremum of the x'n ? ?-
topologies, ? e N. Now put ? = J2r^=\ ?n\x'„ ° Ml where ?„ are suitable small positive real
numbers such that ? becomes [0, l]-valued. Then и agrees with the v-topology. D
If ? is a measure with values in a seminormed group, then ? <<? |?|. We now examine
when |?| <<C ? and therefore ? and |?| are equivalent.
PROPOSITION 7.6. LetG be a seminormed group, ?: R -> G a measure and S a subring
of R which is dense in R with respect to the ?-topology. Then the total variation (?| of ?
extends the total variation of ?\$.
PROOF. Let* e 5" and ? := ? |s. Obviously, \v\(x) ^ |?|(.?). To prove \?\{?) ^ | v|(.v), let
? > 0 and x\,..., x„ be disjoint elements of R with .?, < ? (i — 1,..., n). Let уч е S with

734
?. Weber
||?||U, Ay,·) ^ ?/?2 and Zi := (?, \ sup/<( v7) /\.x. Then
-\?(?? - ?(??)\ ^ ||?||(.?,?;,)= ||?||((.?, \sup*/) Ах\л((у7 \sup>'7) л*)
i
«? ?]||?||(.?7·?.?;-)<?— «? -.
7=1
hence ?''=? lMU-)l «??"=? |v(c#)| + 2? < |?|(.?) + 2?. ?
THEOREM 7.7. Let G be a seminonned group and ?: R -> G an exhaustive measure.
Then the following conditions are equivalent:
A) ??|«?·
B) \?\ is exhaustive.
C) \?\ is bounded.
PROOF. B) о C) is proved in Proposition 2.12. A) => B) is obvious.
B) ^ A). Replacing G by the completion of G/N where N — {x e G: |*| = 0), we
may assume that G is Hausdorff and complete (cf. the proof of Corollary 7.3). We now
apply Theorem 6.5 with w = we. Since, with the notation of Theorem 6.5, ?(\?\) = ?(?),
we obtain that |?| \R <^?. Since by Proposition 7.6 |?| = |?| \r, we have |?| «: ?. ?
8. Decomposition of exhaustive measures
In this section let G be a complete Hausdorff topological commutative group.
We will decompose a measure ? in the form ? = ? + ? where ? J_ v. The next
proposition shows that here ? is ?-additive (or exhaustive or ...) iff both ? and ? are
?-additive (or exhaustive or ...).
PROPOSITION 8.1. Let ?. ? be G-valued measures on R and ? = ? + v.
(a) If ? J_ v, then the ?-topology is the supremum of the ?-topology and the v-topology.
(b) If и e FN(/?), ? «: и and ? J_ и. then the ?-topology agrees with the infimum of и
and the ?-topology.
PROOF. Denote by u(?), и (?). u(v), respectively, the ?-topology, the ?-topology and the
?-topology.
(a) Obviously, ? is continuous with respect to u(k) v«(ii) and therefore и(ц) is coarser
than «(?) ? u(v). Therefore it remains to show that ?«? and ? <*C ? if ? J_ v. We
show that ? <<C ?: Let U and V be 0-neighbourhoods in G with V + V — V с U and
? e R such that ?([0, .?]) С V. We show that ?(.?) e U. Since ? J_ ?, there are disjoint
elements ? ?, ?? e R with .? = x\ ?.??, ?(.?|) e V and v(x2) e V (cf. Proposition 1.5). Now
?(?) = ?(?\) + ?(?2) = ?(?\) + ?.(?2) - v(.X2) e V + V -V cU.

FN-topologies and group-valued measures
735
(b)By (a), ?(?) = u(k)vu(v). Therefore, by the distributivity of FN(/?), we have
u(?) Ли= (?/(?) л и) ? (?(?>) Л и) = «(?)
since ?(?) л и = и(к) and u(v) л и = 0. ?
We now present a general version of the Lebesgue decomposition theorem, which
contains several decomposition theorems as special cases.
In a more familiar formulation of Lebesgue's decomposition theorem, the FN-topol-
ogy и in the next theorem is the v-topology for some measure v.
THEOREM 8.2 (Lebesgue decomposition theorem). Let ?: R -> G be an exhaustive
measure and и e FN(R). Then there exist unique G-valued measures ??, ?? on R such
that ? = ?? +??, ?? «и and ?? J_ и. Moreover, ?,(/?) С ?(R)¦
PROOF. We apply Theorem 6.5 with w = we. Let ? be the infimum of и and the ?-
topology. With the notation of Theorem 6.5, let ? e FN(/?. w) with v\r = ? and ? the
continuous extension of ? on (/?, u>). see Theorem 5.1.1. Put ? = s(v), ?\(?) = ?(? ? ?)
and Дг(л-) = ?(? ? ?') for .? e /?. Then ?? = ??|? and ?? = ??|« are measures with
Ml +M2 = ?· Since ?(??) = [?,?'] = TVCu), we obtain that the ? ? -topology agrees with v,
hence ?? <g u. Since ? e ?(??) and ?' e ?(??), we have ?? J_ v. This implies ?? ±u since
the ?2-topology is weaker than ?-topology.
By the definition of ?, we have ?j(R) С ?/(?) С Д(Л) С ?№-
If ? = ?? + ?? is a further Lebesgue decomposition. 14 <K и and vi J_ и, then ? :=
Д2 — vi = V| — ?? С и and ? J_ и, hence the ?-topology is trivial. It follows that ? = 0
and ?, = ?,- (/' = 1,2). ?
COROLLARY 8.3. Let ?-.R -* G be an exhaustive measure and и an FN-topology on R
coarser than the ?-topology. Then there is a measure ?: R -> G such that и agrees with
the v-topology.
PROOF. Take ? = ? ? of the decomposition of Theorem 8.2. By Proposition 8.1 (b) и agrees
with the v-topology. ?
Remark 8.4. Let ?: R -> G be an exhaustive measure and и е FN(R). Then the
following conditions are equivalent:
A) ? J_ u.
B) If ?: R -> G is a measure with ?«? and ? <<C и, then ? = 0.
C) If Я is any Hausdorff topological commutative group and k:R->H& measure
with ?«? and ?««, then ? = 0.
PROOF. A) =>· C) =>· B) is obvious.
B) =>· A). Let ? be the infimum of и and the ?-topology. By Corollary 8.3, ? is the
?-topology for some measure ?: R -> G. Condition B) implies that ? = 0, i.e., ? is the
trivial topology. ?

736
?. Weber
A measure ? : R -> G is called purely finitely additive if there is no non-zero ? -additive
measure ? on R with values in a Hausdorff topological group such that ? <<? ?.
THEOREM 8.5 (Hewitt- Yosida decomposition). Let ?: R -> G be an exhaustive measure.
Then there are unique G-valued measures ?\, ?? on R such that ? = ?\ + ?2, ?? «
?-additive and ?2 is purely finitely additive. Moreover, ?,(/?) С ?(R) and ?, « ? (? =
1,2).
PROOF. Apply Theorem 8.2 and Remark 8.4 with и = un being the supremum of all
? -order continuous FN-topologies on R. Observe that a measure ? on R is ? -additive
iff ? <<C u„ and ? is purely finitely additive iff ? J_ u„. ?
Another choice of и in Theorem 8.2 yields another decomposition theorem:
THEOREM 8.6. Let R be a subring of a Boolean ring ?, ? a sublattice of ? and
? : R —> G an exhaustive measure. Then there are unique G-valued measures ? ?, ?? such
that ? = ? ? + ?2, ?? is K-tight and there is no non-zero K-tight measure ? : R —> G with
? «; ?2. Moreover, ?, (/?) с ?(?) <я«й/ ?/ «С ? (/' = 1. 2).
PROOF. Apply Theorem 8.2 and Remark 8.4 with и being the supremum of all ?-tight
FN-topologies on R. D
The importance of the exhaustivity assumption in decomposition theorems becomes
clear by the following remark.
Remark 8.7. A measure ?: R -> G is exhaustive iff for every и е FN(/?) there are
measures ?\,?2- R -> G such that ? = ?? + ?;>, ?? <sC и and ?? -L и.
PROOF. =>· is proved in Theorem 8.2.
«=. Let w be the ?-topology and и е FN(R. w). Let ??, ??: R -> G be measures with
? = ?? + ?2, ?? <sC и and ?? -L к. Denote by и, the ?,-topology (/' = 1, 2). It follows
from Proposition 8.1(a) that u\Vut = w and u\ aui=0 and from Proposition 8.1(b) that
u\ = u. Therefore u^ is a complement of и. It follows that FN(/?, w) is a complemented
distributive lattice, hence a Boolean algebra. Therefore w is exhaustive by Remark 6.7. D
For real-valued measures there is another method to obtain a general decomposition
theorem, which is based on the Riesz decomposition theorem. The space ba(R) of all real-
valued bounded measures on R is a Dedekind complete Riesz space. If ? is a band in ba{R)
and ?-1 its disjoint complement, then any ? e ba(R) has a unique decomposition ? =
?\ + ?? with ?? e ? and ?? e ??. These decompositions are exactly the decompositions
which can be obtained by Theorem 8.2. This follows from the next proposition.
PROPOSITION 8.8. (a) For any и e FN(fl). ?(") ;= 1м е ?/?): ? « ") " ? band and
BiuI- = {? e ??): ?-L и}.
(b) For every band В in ba(R) there is an и e FN(/?) ??/с/г ?/??? ? = B(u).

FN-topologies and group-valued measures
737
PROOF, (a) (i) We first prove that B(u) is a band. Obviously B(u) is an ideal in ba(R).
Let (??)??? be an increasing net in B(u) and ? e ba(R) with 0 < ?? | ?. We show that
? «Си. Let ? > 0, ? е /? with ?(?) > sup_e/^(;;) — ? and /3 e A with ?/jfa) > ?(?) —?.
Then for у е /? we have ?(>) ^ ? if у ? ? = 0, and ?(?) — ?/?(?) ^ (? — ?/?)(?) ^ ? if
у < ?. Therefore ?(*) = ?(* \?) + (?(.? ??) - ?/?(.? ??)) +?/?(.? ??)< 2? + ?^(^).
It follows that {? e R: ??(?) ^ ?) С {.? е R: ?(?) ^ 3?). Therefore {? e ?: ?(.?) ?? 3?)
is a O-neighbourhood in (R, и) for any ? > 0. Hence ? <sC и by Proposition 1.10.
(?)???? e b?(/?)and? J_ и. We show that ? e б(и)-1-. Let ? e в(и) and ? — |?| ?|?|.
Then vlu and v«u, hence, ? = 0.
Let ? e б(м)". By Theorem 8.2 there are measures ??, ?: e foa(/?) with |?| =
?? + ?2, ?? « и, ?? J- и and ?,(?) С |?|(?) С [0.+??[, hence ?,- > 0 (/ = 1,2).
?? «Си means ?? е fi(w), and 0 ^ ?? < |?| е б(м)х implies ?? e ?(«)?· Hence ?? =0
and |?| = ?2 -L ?, hence ? J_ и.
(b) Let и be the supremum of the ?-topologies, ? e B. Then ? с В(и). We now show
that B(u) С S. Let ? e в(и). Then there are ?? e В and ?? e ?? with ? = ?? + ?2·
Since ?2 -L ? for any ? e ?, we obtain ?? J_ и. On the other hand ?? <<C ? <SC к. Hence
?2 = 0 and ? = ? ? e В, ?
Generalizing Theorem 8.2 we now give a decomposition of a measure into an infinite
sum of measures.
THEOREM 8.9. Let ?-.R -* G be an exhaustive measure and (ua)a€A be a family in
FN(/?) such that ua J_ up for different indexes ?, ? e A. Then there is a unique family
(??)??? of G-valued measures on R and a measure ?: R —» G such that 53„ед ??(?)
is summable uniformly in ? e R, ? = ? + ???? Iх" and ? J_ ua, ?? <sC ua for any
a e A. Moreover, the ?-topology is the supremum of the ??-topologies and the ?-topology,
?^) c]JXR) and ?„(?) С ?(/?) for a e A.
PROOF (Sketch), We proceed as in the proof of Theorem 8,2, Let ?? be the infimum of ua
and the ?-topology and aa be the corresponding elements in the completion R of (/?, u-v).
Let ? be the continuous extension of ? on R, ??(?) = AaU ? aa) for ? e R and a e A
and ? = ? — ??€? ??- As in Theorem 8,2 one proves that ? and (??)?€? have the desired
properties,
THEOREM 8.10 (Hammer-Sobczyk decomposition). Let ?: R -> G be an exhaustive
measure. Then there is a family (??)?€? pairwise singular G-valued ultrafilter measures
on R and a measure ? : R —» G with the following properties,
A) Х]аед ??(?) is summable uniformly in ? e R,
B) R is ?-chained,
C) ? = ? + ]?^???.
D) The ?-topology is the supremum of the ??-topologies and of the ?-topology,
E) X(R) С ?(/?), ??(?) С ^(R)for a e А.
PROOF. Apply Theorem 8,9 where (иа)а?Л is the family of all ultrafilter topologies on
R and ua ? и? for different indexes ?, ? e A, Then R is ?-chained by Proposition 6,9,

738
?. Weber
since ? J_ ua for every a e A. Since ?? <? /<„. it follows from Corollary 6.10(a) that ??
is an ultrafilter measure or ?? = 0, Now replace (??)?€? by the family (??)?€?' where
A' = {aeA: ????]. D
The Hewitt-Yosida decomposition for group-valued measures and the Lebesgue
decomposition Theorem 8,2 if и is generated by a single submeasure was proved by Drewnowski
A973b). Drewnowski A973b, p, 47) posed the question whether Theorem 8,2 holds true
for an arbitrary FN-topology u. The first proof of Theorem 8,2 was given by Traynor
A976), The approach to decomposition theorems presented here is due to Weber A984a,
1984b). The Hammer-Sobczyk decomposition for group-valued measures was given in
Weber A982a, 1982b),
9. Connectedness
In this section we study connectedness of topological Boolean rings and the range of
measures. It turns out that connectedness of the range of a measure ? : R —* G depends -
in contrast to convexity - only on the ?-topology and not on the (geometric or topological)
properties of the range space.
THEOREM 9,1, Let и be a Hausdorff exhaustive FN-topology on R such that (R,u) is
complete.
(a) Then the following conditions are equivalent:
A) R is atomless.
B) (R, u) is connected.
C) (R,u) is arcwise connected.
(b) The following conditions are equivalent:
A) R is atomic,
B) (R, u) is totally disconnected.
C) For some set A, (R.u) is algebraically and topological!)· isomorphic to the
product {0, 1 }A endowed with the product topology with respect to the discrete
topology on {0, 1},
PROOF, (a) B) =>· A), Let a e R \ Щ. Then [O.a] is connected as the image of (/?, u)
under the continuous map R эн> л л ?. Therefore [O.a] cannot be finite. In particular, a
cannot be an atom,
A) =>· B), Suppose that (R.u) is not connected. Then R contains a clopen
nonempty subset A, such that e ? A. Let С be a maximal chain in A. (c)cec considered
as an increasing net is Cauchy by Proposition 2,4 and has therefore a limit s e A. Then
s = sup С Since s' > 0 and R is atomless, [0. s'] contains a disjoint sequence (d„)„€^ of
nonzero elements. Since и is exhaustive. dn -> 0 and therefore.? vd„ -> s in (/?, u). Since
A is open, s ? dm e A for some m e N. Therefore CU(sv d,„) is a chain in A different
from C, a contradiction to the maximality of С
C) =>· B) is obvious.

FN-topologies and group-valued measures
739
B) =>· C), By Corollary 4,11 and Theorem 6.2 we may assume that и is metrizable.
In this case the implication follows from the fact that any connected, locally connected,
complete metric space is arcwise connected.
(b) A) =>· C), By Theorem 6.2, и is order continuous and (L. ^) is complete. Since
(/?, ^) is complete and atomic, R is algebraically isomorphic to the power set of a set A,
So we may assume that R = {0, 1}A, The product topology on {0. \}A is order continuous
and therefore coincides with и by the uniqueness statement of Corollary 4,9.
C) =>· B) is obvious.
B) =>¦ A). Suppose that R is not atomic. Then there exists an a e R \ {0} such that
[?,?] is atomless. Then [?,?] is connected by (a) (A) =>· B)). Therefore (R.u) is not
totally disconnected. ?
The condition B) of Theorem 9.1(b) implies that (R. u) is compact. On the other hand,
any compact Hausdorff topological ring without total zero divisors is by Anzai A943)
totally disconnected. This shows that a further equivalent condition in Theorem 9.1(b) is
that (/?, и) is compact. This is also contained in the following theorem which is a special
case of Kaplansky's structure theorem for compact semi-simple rings,
THEOREM 9.2 (Warner A993). Theorem 32.8). A Hausdorff topological Boolean ring
is compact iff it is for some set A (algebraically and topologically) isomorphic to the
product {0, 1}A.
All the known proofs of Theorem 9.2 (=>·) are based on the deep result - already used
in Anzai A943) - that any compact Hausdorff topological commutative group admits a
non-trivial character. Only if the topology of the Boolean ring is generated by real-valued
measures, there are also elementary simple proofs of Theorem 9,2 (=>·).
THEOREM 9,3. Let ?;/? —» G be an exhaustive measure with values in a complete
Hausdorff topological group,
(a) If ? is atomless and (R. ?-topology) is complete, then ?(R) is connected.
(b) If R is ?-chained, then ?(/?) is connected,
PROOF. We may assume that the ?-topology is Hausdorff. Otherwise consider ?; R -> G
on the quotient R = R|N(?) defined by ?(.?) = ?(?) for .? e л е R.
(a) Under the assumptions of (a), R is by Theorem 9.1 connected with respect to the
?-topology. Therefore the continuous image ?(/?) is connected,
(b) Let (/?, u) be the completion of (R. ?-topology) and ?: (R.u) -> G the continuous
extension of ? on R. If R is ?-chained, then R is atomless by Theorem 6.16(a), i.e., ? is
atomless. Therefore ?(/?) is connected by (a). Hence ?(R) = ji(R) is connected. D
10. Vitali-Hahn-Saks and Nikodym theorems
We here present a proof of the Vitali-Hahn-Saks and Nikodym theorems based on Baire's
category theorem. The idea of this proof goes back to Saks and is also used in Dunford

740
?. Weber
and Schwartz A957, III.7). With the aid of Drewnowski's Lemma 10.2 the finitely additive
case can be reduced to the ?-additive case, cf. Drewnowski A973b).
THEOREM 10.1. Let R be ?-complete, ?: R -> [0, +oo] ? ?-submeasure, (G, | |) a
seminormed group and ?„ : R —> (G. | |). n e N. a pointwise convergent sequence of
?-continuous measures. Then {?„: n e N) is ?-equicontinuous.
PROOF, Let ? > 0. The sets C,j := {x e R: |?,-(.?) - ?;Or)I ^ ?] and C„ = f|, 7>, cv
are closed in (R, ?). R = (J^=i C„ since (?,,?.?)),,^ converges for any ? e R. Since
(/?, /j) is complete by Theorem 3,6, by Baire's category theorem Cq has for some q e N
an inner point a. Let S > 0 such that for any .v e /? with ?(?) ^? S we have |?,(.?)| ^ ?
for /' = 1,..., q and ? ? ?, a \ ? e Сц, From the equality ?„ (jc) = ?? (jc) + (?„ (???)-
?9(? ? ?)) — (?„(? \ jc) — ?(,(? \ ?)) follows that |?„(.?)| ^ 3? for any n e N and ? e R
with ?(?) ^ ?. This shows that (?,,),,^; is /j-equicontinuous. ?
LEMMA 10,2 (Drewnowski's lemma). Let R be ?-complete, ?-.R^ [0,+oo] an
exhaustive submeasure and (a„)/ieN a disjoint sequence in R, Then there is an infinite
set I С N such that the restriction of ? on the ?-subring ?({??: n e /}) generated by
(an)n€i is ?-order continuous.
PROOF. We first prove that for any ? > 0 and any infinite subset Л с N there is an infinite
subset ? of Л such that /j(sup,eiJ ?,) ^ ?: Let (An)nei; be a disjoint sequence of infinite
subsets of A. Then the elements sup,^ a;, n e N, are disjoint. Since ? is exhaustive,
f?(sup,eA/; ?,) < ? for some n e N.
Using this fact, one can inductively find a decreasing sequence (/л)л6^ of infinite subsets
of N such that /j(sup,e/; a,) ^ l/н and /'„ :=min/„ iIn+\. // e N. Let / = {/'«: ? e N). It
is easy to see that ? is ?-additive on ?({?„: n e /}) = {sup ](zJ af. Ус/}. ?
THEOREM 10.3. Let R be ?-complete, G a topological group and ?„ : R -> G. n e N.
be a sequence of exhaustive measures converging pointwise to ? : R —> G.
(a) Then {?„: n e N) is uniformly exhaustive: in particular, ? is exhaustive.
(b) // for every n e ?, ?„ is continuous with respect to an FN-topology и on R, then
{?„: n e N) U {?} is u-equicontinuous.
(c) If?? is ?-additive for every n e N, then ? is ?-additive.
PROOF, (a) Since it is enough to prove that (?„) is uniformly exhaustive with respect
to any continuous seminorm on G, we may assume that G = (G. | |) is a seminormed
group. Then the semivariation ||?„|| of ?„ is bounded by Corollary 2.7. Therefore there
are numbers ?„ > 0 such that ? := ?^=? ?„||?„|| is a bounded function. It follows that ?
is an exhaustive submeasure and ?„ is ^-continuous for every //. Suppose that (?„)„,=? is
not uniformly exhaustive. Then there is an ? > 0, a disjoint sequence (a,,)n€i; in R and a
subsequence (v„)„eN of (?„)„?? such that \v„(a„)\ ^ ? for every n e N. By Lemma 10.2,
(a„)„eN has a subsequence (bn)„€\; such that the restriction of ? on S := o{{b„: n e N})
is ?-order continuous. Applying Theorem 10.1 to v„\s one obtains that (v„|5)„e^ is

FN-topologies and group- valued measures
741
/jls-equicontinuous. Since ??„) -> 0, it follows that sup,,, \v,„(b„)\ -> 0 (и -> +oo), a
contradiction to | ?„(?„)| ^ ?.
(b) follows from (a) and Corollary 7,2.
(c) Let и be the supremum of the ?„-topologies, и e N. Then и is ?-order continuous if
?„, ? e ?, are ?-additive (see Proposition 4.2). By (b), ? is и-continuous, hence ?-addi-
tive. ?
PROPOSITION 10.4. Let ? be a set of measures on R with values in a seminormed group
(G, | |), //sup{^(a„)|: ? e M, /ieN}< +<%> for every· disjoint sequence (а„)леК in R,
then 5??{|?(?)|: ? e ?, ? e /?} < +эо.
PROOF. Put ||U;J)/ieM|| :=supMeM |jr,,| for (л·,,) e GM and ? := {(л>)е GM: ||U;/)|| <
+oo). Then ?(?) := (?(?))?€? defines an Я-valued measure on R. Since sup„ ||?(?„)|| <
+00 for every disjoint sequence in R, we have supneK ||v(a)|| < +00 by Proposition 2.6,
i.e., sup{|M(a)|: ? e M, a e /?} < +00. ?
We now deduce - as in Drewnowski A974) - Nikodym's boundedness theorem from
Theorem 10.3(a).
THEOREM 10.5, Let R be ?-complete, ? a set of exhaustive measures on R with
values in a seminormed group (G. | |) and sup eM |?(·?? < +oo for any ? e R. Then
sup{^U)|: ? e ?, ? e R] < +00.
PROOF. Suppose that sup{^(.v)|: ? e ?, ? e R] — +00. Then, by Proposition 10.4,
there is a disjoint sequence (a„)„€i< in R and ?„ e ? such that |?„(?„)| ^ 11, ? e N.
For ? e N, we define GK-valued measures on R by ?,,(?) = @,..,, 0, ??(?), 0,...),
where ?„(?) occupies the «th place. We consider G1 with the seminorm defined by
||(jc„)|| := sup„(l//i)|;c„|. Then the measures ?„, не N. are exhaustive and ?,,(^) -> 0
for any ? e R. Therefore {?„: ? e ?} is uniformly exhaustive by Theorem 10.3(a), Hence
II An (a«) II -> 0, a contradiction to ||A«(an)ll = (\ ? ?)\?„(??)\ > 1, ?
Instead of the ?-additivity of /j it is in Theorem 10.1 enough to require that (R, ?) is a
Baire space. It is natural to ask whether for an exhaustive submeasure ?: R -> [0, +00] on
a ?-complete Boolean algebra (R, ?) is a Baire space (see Diestel and Uhl A977, p, 35)
andLabudaA972,p. 455). The examples contained in Armstrong and Prikry A982), Arias
de Reyna A983), Basile and Weber A986) show that this is not the case.
References
Ajupov, S.A,. Cilin, V,I,, Hadjiev, D. and Sarymsakov, T.A. A983). Ordered Groups, Usbek Academy of Sciences.
Tashkent (in Russian).
Aliprantis. CD. and Burkinshaw. O. A978). Locally Solid Riesz Spaces, Academic Press, New York.
Anzai. H. A943), On compact topological rings. Proc. Imp. Acad. Tokyo 19. 613-615.
Arias de Reyna, J. A983), Non-Baire measure spaces. Collect. Math. 34. 109-114.

742
?. Weber
Armstrong. Т.Е. and Prikry. К. A982). On the semimetric on a Boolean algebra induced by a finitely additive
probability measure, Pacific J. Math. 99. 249-264.
Ascherl. A. and Lehn. L. A977). Two principles for extending probability measures, Manuscripta Math. 21, 43-
50.
Basile, A, A984), A Lebesgue-type decomposition theorem for topologies on rings of sets. Rend. Accad. Sci. Fis.
Mat. Napoli 51 D), 61-65.
Basile, A. and Weber, H. A986). Topological Boolean rings of first and second category. Separating points for a
countable family of measures, Rad. Mat. 2. I 13-1 25.
Bhaskara Rao, K.P.S. and Bhaskara Rao, M. A977), Topological properties of charge algebras. Rev. Roumaine
Math. Pures Appl. 22. 363-375.
Bhaskara Rao, K.P.S. and Bhaskara Rao, M. A983), Theory of Charges. A Study of Finitely Additive Measures,
Pure Appl. Math., Vol. 109, Academic Press, New York.
Bierlein, D, A966), Uber die Fortsetzung von Wahrscheinlichkeitsfeldem, Z. Wahrscheinlichkeitstheorie verw.
Gebiete 1, 28^46.
Bogdan, V.M. and Oberle, R. A. A978). Topological rings of sets and the theory of vector measures. Dissertationes
Mathematicae 154,5-70.
Brook, С A984). Decompositions of submeasures, Canad. J. Math. 36, 577-590.
Carlson. T. and Prikry. K, A982). Ranges of signed measures. Period. Math. Hungar. 13. 15 1-155.
Constantinescu, С. A976а), The range ofutomless group valued measures. Comment, Math. Helv. 51 B). 207-
213,
Constantinescu, С A976b), Atoms of group valued measures. Comment. Math. Helv, 51 B), 19 1-205,
de Lucia. P, A985). Funzioni finitamente additive a valori m un gnippo topologico. Quademi dell'Unione
Matematica Italiana, Vol. 29, Pitagora Editrice, Bologna.
Diestel, J. and Uhl. J.J. Jr. A977), Vector Measures. Math. Surveys. Vol. 15, Amer. Math, Soc. Providence, RI.
Dobrakov, I, A974), On submeasures I. Dissertationes Mathematicae 112. 5-39.
Dobrakov, I. and Farkova, J. A980). On submeasures II. Math. Slovaca 1. 65-81.
Drewnowski, L. A972a), Topological rings of sets, continuous set functions, integration I. II. Ill, Bull. Acad.
Polon. Sci. Ser, Sci. Math. Astronom. Phys. 20. 269-276. 277-286, 439-445.
Drewnowski, L. A972b), Equivalence of Brooks-Mvett. Vitali-Halm-Saks and Nikodym theorems, Bull. Acad.
Polon. Sci. Ser, Math. Astronom. Phys. 20, 725-731.
Drewnowski, L, A973a). Uniform boundedness principle for finitely additive vector measures. Bull. Acad. Polon.
Sci. Ser. Math. Astronom. Phys. 21, 115-118.
Drewnowski. L. A973b), Decomposition of set functions, Studia Math. 48, 23^48.
Drewnowski. L. A974), On control submeasures and measures. Studia Math. 50, 203-224.
Drewnowski, L. A974-1975). On complete submeasures. Comment. Math. Prace Mat. 18. 177-186.
Dunford, N. and Schwartz, J.T. A957), Linear Operators. Part I: General Theory, Interscience. New York.
Frechet, M. A919-1920), Sur divers modes de convergence d'une suite de fauctions d'une variable. Bull, Calcutta
Math. Soc, 11, 187-206.
Hejcmann, J. A959), Boundedness in uniform spaces and topological groups, Czechoslovak J. Math. 9, 544-562.
Herer, W, A976), On the extension of measure with values in a topological group. Comment. Math. Prace Mat.
19, 73-80.
Hoffmann-J0rgensen, J. A97 I), Vector measures. Math, Scand, 28. 5-32.
Khurana. S.S. A976), Extensions of exhaustive submeasures. Bull. Acad. Polon. Sci, Ser. Sci. Math. Astronom.
Phys, 24, 213-216.
Khurana, S.S, A979), Submeasures and decomposition of measures, J, Math. Anal. Appl, 70. I I I-1 13.
Labuda. I. A972), Sur quelques generalisations des theoremes de Nikodym el de Vitali-Hahn-Saks, Bull, Acad.
Polon. Sci. Ser. Sci. Math. Astronom. Phys. 20, 447-456.
Labuda, I, A975), Sur les mesures exhaustives el certaines classes d'espaces vectoriels topologiques consideres
par W. Orlicz et L, Schwartz. C. R. Acad. Sci. Paris Ser. A-B 280, Ai. A997-A999.
Landers, D. A973), Connectedness properties of the range of vector and semimeasures, Manuscripta Math. 9,
105-112.
Lembcke, J, and Weber, H, A992). Decomposition of group-valued and [0,oc)-valued measures on Boolean
rings. Measure Theory (Oberwolfach, 1990). Rend, Circ. Mat. Palermo B) Suppl, No. 28, 87-1 16.

FN-topologies and group-valued measures
743
Lipecki, Z. A974), Extensions of additive set functions with values in a topological group. Bull. Acad. Polon.
Sci. Ser. Sci. Math. Astronom. Phys, 22, 19-27.
Lipecki, Z. A980a), A generalization of an extension theorem of Bierlein to group-valued measures. Bull. Acad.
Polon. Sci. Ser. Sci. Math. 28, 441-445.
Lipecki, Z. A980b), The Countable Chain Condition and Restrictions of Group-Valued Additive Set Functions,
Bull. Acad. Polon. Sci. Ser. Sci. Math. 24, 363-365.
Lipecki, Z. A987), Tight extensions of group-valued quasimeasures, Colloq. Math. 51, 213-219.
Maharam, D. A987), From finite to countable additivity, Portugal Math. 44, 265-282.
Nachbin, L. A965), Topology and Order, D. Van Nostrand Company, New York.
Nikodym, O.M. A929), Sur les functions d'eseinples, Comptes Rendus du I Congres des Math, des Pays Slaves,
Warsaw, 304-313.
Nikodym, O.M. A930), Sur une generalisation des integrates de M.J. Radon, Fund. Math. 15, 131-179.
Rickart, C.E. A943), Decomposition of additive set functions, Duke Math. J. 10, 653-665.
Sion, M. A969), Outer measures with values m a topological group, Proc. London Math. Soc. 19, 89-106.
Traynor, T. A972), A general Hewitt-Yosida decomposition. Canad. J. Math. 24, 1164-1169.
Traynor, T. A973), S-bounded additive set functions, Vector and Operator Valued Measures and Applications
(Proc. Sympos., Alta, Utah, 1972), Academic Press, New York, 355-365.
Traynor, T. A976), The Lebesgue decomposition for group valued set functions. Trans. Amer. Math. Soc. 220,
307-319.
Vladimirov, D.A. A972), Boolesche Algebren, Mathematische Lehrbucher und Monographien, B. 29, Akademie-
Verlag, Berlin. (Translated from the Russian and edited by Gunther Eisenreich.)
Warner, S. A993), Topological Rings, North-Holland Math. Studies. Vol. 178, North-Holland, Amsterdam.
Weakley, C.B. A987), On the lattice of Frechet-Nikodym topologies, J. Math. Anal. Appl. 122, 385-392.
Volkmer, H, and Weber, H. A983), Der Wertebereich atomloser Inhalte. Arch. Math. (Basel) 40 E). 464-474.
Weber, H. A976), Fortsetzung von Massen mil Werten in unifonnen Halbgruppen. Arch. Math, 27, 412^423.
Weber, H. A980), Ein Fortsetzungssatz fur gruppenwertige Masse, Arch. Math. 34, 157-159.
Weber, H. A982a), Die atomare Struktur topologiscber Boolescher Ringe und s-beschriinkter Inhalte, Studia
Math. 74, 57-81.
Weber, H. A982b), Vergleich monotoner Ringtopologien und absolute Stetigkeit von lnhalten, Commentarii
Mathematici Universitatis Sancti Paul! 31, 49-60.
Weber, H. A984a), Topological Boolean rings. Decomposition of finitely additive set functions, Pacific J. Math.
110,471^495.
Weber, H. A984b), Group- and vector-valued s-bounded contents. Measure Theory (Oberwolfach, 1983), Lecture
Notes in Math., Vol. 1089, Springer, 181-198.
Weber, H. A986), Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and
Nikodym's boundedness theorem, Rocky Mountain J. Math. 16, 253-275.
Weber, H. A994), Minimal ring topologies, Adv. Math. 109, 140-15 I.

CHAPTER 17
On Products of Topological Measure Spaces
Stratos Grekas
Department of Mathematics, Section of Mathematical Analysis, University of Athens, Pimepistemiopolis,
15784 Athens, Greece
E-mail: sgrekas@math.uoa.gr
Contents
Introduction 747
1. A central problem in topological measure theory 747
1.1. Review of terminology 748
1.2. On product measures 748
1.3. Fremlin's example and related results 751
2. On the 'product-like' structure of the Haar measure on a compact group 752
3. Topological liftings for product measures 755
3.1. Baire, Borel and Strong liftings 756
3.2. On the existence of Baire strong liftings 757
Acknowledgements 762
References 762
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
745

This page intentionally left blank

On products of topological measure spaces
?4?
Introduction
One of the most important achievements in Lebesgue's theory is related to the double
integral, Euler, in 1780, had already used a reduction formula of a double integral to two
single successive ones. Although Riemann's integral was limited, mainly, in continuous
functions of two variables, there were no restrictions in Lebesgue integral; that is exactly
Fubini's theorem A907), Lebesgue theory was completed by Radon's monograph, which
is presented in Caratheodory A918),
The author does not intent to refer to the history of the 'successive integration' notion or
the 'product measure' notion; nevertheless, he notes that, from that time up to nowadays,
every integration theory which was established, was also accompanied by some appropriate
notions of product measure.
In the duration of this procedure, a question arisen; which properties of each one of
the-finite or infinite-factors can also be valid in a product measure (this means they are
productive properties)?
The present chapter is, mainly, concerned with this question in the framework of
topological measure theory. The article is organized in three parts. In Section 2 the question
whether the product of topological measure spaces is again a topological measure space
is discussed. Section 3 studies, particularly, the Haar measure on a compact group and
emphasizes its connections with a usual product measure on a product of compact metric
spaces; its object is the far reaching conjecture of J,R, Choksi, that complete analogies of
certain measure-theoretic properties of such a product space are valid for compact groups.
Section 4 deals with topological (Baire, Borel, strong) liftings for a product measure; the
discussion yields a unified measure-theoretic treatment of Baire strong liftings for a class of
locally compact ?-compact groups and for a class of products of separable metric spaces.
The choice of these three precise themes, instead of a - usually seen - extensive survey
(on product measures) is not only due to my own interest, but also to the completeness of
Fremlin B00?) (a very interesting work) which (as far as I know) will sufficiently overcover
such a survey.
The reader will notice that the main point of this chapter is the conjecture of J,R. Choksi;
this conjecture led my occupation with this area. So, I am pleased to dedicate the chapter
to Jal Choksi.
1. A central problem in topological measure theory
The main problem in this section is whether the product of a family of topological
(probability) spaces is a topological measure space. Although the (vague) description of
the problem is quite simple, it includes some particular questions, like the following ones:
• What kind of measures do we consider?
• Is such a product well defined?
• Is the (finite or infinite) product of such measure spaces of the same nature like the
one of each of the factors?
For convenience (and simplicity), only ?-additive probability measures on completely
regular Hausdorff spaces are considered (Varadarajan, 1965; Gardner, 1975; Fremlin,

748
S. Grekas
200?). The reader will easily check which of the results hold for quasi-Radon measure
space (Fremlin, 1982). For those who are interested in a higher level of generality, see
Fremlin A984), Gardner and PfefferA984).
1.1. Review of terminology
Let (?, ?,?) (simply denoted by (?,?)) be a probability measure space, where X is
a completely regular space and ? a ?-algebra of subsets of X, The family of Baire,
respectively Borel, sets in X is the ?-algebra ??o(X), respectively ??(X), generated by
the zero, respectively closed, sets. A Baire measure ? on X is called completion regular if
every open set in X is ?-measurable. Note that the terminology is due to Halmos A950),
but the concept of completion regularity was discussed earlier by Kakutani; see Choksi
A984), also Theorem 1.3, this section.
A topological probability space is a probability measure space (?, ?) - or just ? - such
that every open set in X is measurable. A topological probability space (?, ?) is called
?-additive, respectively completion regular, if for every family (i/,),e; of open sets in X,
? ( [J t/,¦ J = supj ? ? [J U,j: J С I countable J.
respectively its restriction to the Baire sets is completion regular. For further information
on ?-additive and completion regular measures, see Knowles A967), Kirk A969). Gardner
A981) gives is a detailed exposition with an extensive bibliography.
1.2. On product measures
Suppose that (?,?',?), (?. ?, ?) are (?-additive) probability spaces. One can form
product measures in various ways. A natural product measure notion is the one of simple
(completed) product measure ? = ? ? ?, that is, the completion of the usual product
measure ? = ? ? ?, defined on the product ?-algebra ? <g> ?. This measure is identical
with the ordinary (completed) product measure, derived by Caratheodory's method from
the outer measure ?* (see, e.g., Fremlin A984. A6K)). The simple (completed) product
?,?/?, on Y\j€i Yi, for any family {(У,, ?,). /' e /} of probability spaces, can also be
similarly defined; cf. Halmos A950).
Since ?, ? are assumed to be ?-additive. ? does have a unique ?-additive extension
? <g> ? to the Borel sets in ? ? У (we will take it that ? <g> ? is to be the completion of its
restriction to B(X ? ?)); cf. Shwartz A973), Fremlin B00?).
Let now (X),e/ be a family of topological spaces and X = П/е/ %>¦ ^ 's clear
that ??(X), respectively Bq(X) contains the product ?-algebra <g)j€l B(X,). respectively
<8>,e/ Д)№), but in general the inclusion may be strict (cf, Gardner A981), Fremlin et al.
A996) and references there).
The following result was proved in a special case by Ressel A977), while it was proved
by Fremlin A982) in full generality in the quasi-Radon measures wider field (see also
Gryllakis and Koumoullis A990)).

On products of topological measure spaces
749
THEOREM 1.1. If ?, is ? ?-additive probability measure on X,-, /' e /, then there exists
a unique ? -additive measure ? on X, extending the simple product ?,<=/?,.
? is called the ?-additive product of the family (м,),е/ and we write ? = ®,?/^'
(if X, all are compact spaces, then ? is the Radon product of the ?,, see Bourbaki A965),
FremlinA989)).
The problem will be discussed in this sub-section is the below mentioned.
PROBLEM 1.2. Underwhich (topological or measure-theoretic) conditions, the -finite or
infinite -product of topological probability spaces is also a topological probability space
(and consequently, when the product of completion regular spaces is completion regular)!
The story of this problem started in 1943. when Kakutani A943a. 1943b) presented the
following
THEOREM 1.3. Let (?,,?,), / e /, be an uncountable family of strictly positive
probability spaces, where the X, are all compact metric spaces. Then every Borel set in
X = ???/ ^' 's ®ie/ ??-measnrable.
The key point of the proof of Theorem 1,3 is the following assertion, which will be
mainly discussed in Section 4,
ASSERTION 1.4. Every open set in the product space has an open Baire cover of the same
measure.
In A958a), Maharam, using a very special well ordered martingale/derivation argument,
proved the following significant result.
THEOREM 1.5. There exists a strong, (completion) Baire lifting for the usual product
measure on the Cantor space {0, \}r, ? arbitrary.
Thus, the product measure on {0, l}r satisfies Assertion 2.4 and so. Theorem 1.5
provides a different proof of Theorem 1.3. Evidently, Theorem 1.5 indicates some
connection between strong lifting property and completion regularity; however, as Choksi
A984) says, this connection is somewhat elusive (detailed information is also provided in
Fremlin B00?)).
In A976), Fremlin "making ingenious use of an earlier result of Erdos and Oxtoby
A955), showed that the product of the hyperstonian space of Lebesque measure on [0, 1]
with itself is not completion regular" (see comments in Choksi A984)). On the other
hand, Choksi and Fremlin A979), in the course of a study concerning completion regular
measures, generalized Kakutani's theorem as follows:
THEOREM 1.6. Let ?, ?,, / e /, be completion regular Radon probability measures on X,
Yj, i e /, respectively, where X is any compact space and the У, are compact metric
spaces. If the ?, are, all, strictly positive, then the Radon product measure ? <g> (<S>,e/ ?')
is completion regular

750
S. Grekas
In the same article, the authors posed the question whether the product of two completion
regular measures, each on a product of compact metric spaces, is completion regular. The
answer was an affirmative one (Gryllakis, 1988),
Grekas and Gryllakis A991), using Fubini's theorem, obtain an extension of
Theorem 1,6:
THEOREM 1,7, Let X bea compact space, (Yj)jei a family of compact metric spaces and
?, у completion regular measures on X, Y\.€j Yj, respectively. Then, subject to Martin's
axiom and the negation of the continuum hypothesis, ?® ? is completion regular.
Gryllakis and Koumoullis A990) connected the notion of completion regularity and that
of ?-additivity and showed the more general theorem.
THEOREM 1,8. Let (X,),e/ be a family of separable metric spaces, X = fl,e; X; and ?
a probability Baire, respectively Borel measure on X.
(a) If ? is completion regular, then ? is ?-additive ,
(b) The following conditions are equivalent',
(i) ? is completion regular:
(ii) for every ?-additive probability Borel measure space (?, ?), (? ? ?, ? ? ?) is
a topological measure space;
(iii) for every ?-additive completion regular probability Baire, respectively Borel
measure on a completely regular space Y, the ?-additive product ? <g> ? is
completion regular.
Remark 1,9. In case that ? is a (?-additive) Borel measure, then the three conditions
are equivalent to Assertion 1,4,
In the same paper, the authors verify the following interesting fact: a ?-additive product
measure ®,e/ M/ on апУ product П/е/ %> ls completion regular, provided that each finite
sub-product is completion regular. Thus, Theorem 1,3 of Kakutani could be, in some sense,
extended to the general setting of arbitrary - not, necessarily, metrizable - topological
spaces; for the precise statement, see Gryllakis and Koumoullis A990, Theorem 2,9),
The research was extended to quasi-dyadic spaces. Following Fremlin and Grekas
A995), a topological space is quasi-dyadic if it is the continuous image of a product of
separable metric space. The next two referred results are the main ones of that paper,
THEOREM 1.10, Let (?, ?) be a completion regular topological probability space. If X
is quasi-dyadic, then ? is ?-additive,
THEOREM 1.11. Let (?,?) and (Y,v) be topological probability spaces; suppose
that ? is quasi-dyadic, (Y, v) is completion regular and that (?, ?) is ?-additive. Then,
(? ? ?, ? ? у) is a topological measure space.
The proof of both results, which apply many combinatorial arguments, exploits the
simple fact that the inverse image (under the corresponding continuous surjection) of any

On products of topological measure spaces
751
Baire set in the quasi-dyadic space is effectively determined by a "small" (in fact countable)
set of coordinates; cf. Fremlin B000).
Remark 1.12,
A) In Fremlin and Wajch A993) there are interesting examples which show that the
class of compact quasi-dyadic spaces is not identical to the one of dyadic spaces.
B) As far as we can see, Assertion 1.4 characterizes the completion regularity, provided
that the underlying topological space is (homeomorphic to) a product of separable
metric spaces. It is not clear, at least for the author, when same thing happens in the
(general) case of quasi-dyadic spaces,
C) Section 2,2 is, essentially, an investigation into the structure of product measures;
that is what is discussed here and not the applications in other mathematics areas.
However, Problem 1,2 is not of purely measure-theoretic interest; it is also closely
related to important problems in Functional Analysis and Probability theory; for
instance, it worths to refer the notions of "stability" and "R-stability" as they are
described in Fremlin A983), Talargand 1984). In Talagrand A987a, 1987b) are also
presented various interesting related results.
1.3. Fremlin's example and related results
Let (S, y) be the hyper-stonian space of power Lebesgue measure ?" on [0, 1]". Fremlin
has therefore shown that not all the open subsets of S ? S are (? ? v)-measurable. By
Urysohn's embedding theorem, S can be identified with a compact subset of [0, l]y,
? = w(S) and ? is then a measure on [0, \]Y with suppv = S, It is, therefore, obvious
that the measure ? = \v + ^?? has
PROPERTY 1.13, The open sets in [0, l]y ? [0, 1]'' arenotall (? ? \x)-measurable.
In spite of the fact that 5иррд = [0, l]y, there is insufficient information about the
relation that ? and ?? have; we do not even know how the projections of ? on each
topological factor [0, 1], behave. This lack of sufficient information led Gryllakis and
Grekas A999), to use, expect of all the others, many of Fremlin's arguments and thus to
construct a Radon probability measure ? on [0, 1]'', ? ^ с (:= the continuum), satisfying
the following conditions:
(ii) supp$ = [0.1F;
(ii) ? satisfies Property 1.13;
A3) for every F с ? finite, the projection of ? on [0, 1]^ is a measure equivalent to the
F-dimensional Lebesgue measure ?^.
Remark 1.14, As it has already been noticed, on [0, l]y, ? ^ c, we can construct -
in various methods - measures with Property 1.13, The author does not know what the
situation is for ? < с (under Martin's axiom).
In the second part of Gryllakis and Grekas A999), the above construction is generalized,
in a certain way, by replacing the space [0. \]Y with an arbitrary compact group G, The

752
S. Grekas
main result of their paper is the following "compact group analogue" of the construction
in[0, IF.
THEOREM 1.15, Ifw(G) ^c, then there exists a Radon probability measure ? onG with
the properties:
(ii) supp? = G;
(ii) ? satisfies Property 1.13;
Оз) far every compact Lie factor group G/H, the projection of ? on G/H is - as
a measure - equivalent with the normalized Haar measure on G/H.
2. On the 'product-like' structure of the Haar measure on a compact group
The above mentioned observations show that the study of the product measure on a product
of compact metric spaces has played a key role in the evolution of the topological measure
theory. But, the development of the theory was connected with the investigation about the
Haar measure, in the same time. Since 1940s there has been a systematic effort - as it is
obvious by the bibliography - for searching after common properties for the Haar measure
and such a product measure.
Kakutani and Kodaira A944) had already proved, by an ingenious adaptation of the
original proof of Theorem 1.3, that the Haar measure on a locally compact ?-compact
group is completion regular (for information on the completion regularity of the Haar
measure, the interested reader could consult Halmos A950). Mongomery and Zippin
A955), Hewitt and Ross A963) and Choksi A984), a reference of particular interest). One
can say, roughly speaking, that this result is the group analogue of Theorem 1.3 valid for
product measures. This analogy does not derive only by the measure-theoretic properties of
the Haar measure; it seems to be closely connected with the fact that a locally compact ?-
compact group is the projective limit of (a particular family of) metrizable groups indexed
by closed normal subgroups (the reader worths to pay attention at Choksi A984), p. 84).
Indeed, in A973), Choksi, applying in the area of compact groups arguments valid for
generalized cubes (cf. Maharam A958a), Choksi A972). Graf A982)) proved the following
remarkable result.
THEOREM 2.1. If ? is a Baire measure on a compact group G, then every automorphism
of the measure algebra of ? can be induced by an invertible, completion Baire measurable
transformation of G.
The proof uses an interesting method of combining the classical Weil-Pontrjagin
theorem with ideas of Choksi A972).
Choksi and Simha A978) remarked that the proof of Theorem 2.1 "seems to depend
on topological properties of homogeneity of the space". Confirming this vague assertion,
these authors proved, among others, the following theorem.
THEOREM 2.2. Let G be a locally compact ?-compact group, L a closed subgroup
of G and ? a ?-finite Baire measure on (the homogeneous space) G/L. Then, every

On products of topological measure spaces
753
automorphism of the measure algebra of ? is induced by an invertible, completion Baire
measurable point transformation ofG/L.
It is worthwhile to note, here, that the purpose of this section is not to survey the whole
work on measures on compact groups, but, mainly, to focus the reader's interest on the
similarity that exists between Haar measure and product measure.
In A979), Choksi and Fremlin wrote an article in which a large class of results in
topological measure theory, usually proved for Polish spaces, was shown to hold in any
uncountable product ???/ %i °f compact metric spaces. Two Radon probability measures
on П/е/ ^i'> whose measure algebras are homogeneous of the same Maharam type, where
shown to be (completion) Baire isomorphic. Also, the authors, combining their results with
Maharam's famous theorem, arrived at some surprising theorems, some of whose proofs
do not assume any set-theoretic hypotheses. The main of the latter are included in the
statement (for a precise statement, see Choksi and Fremlin A979, Theorem 5)):
PROPERTY 2.3. For a large, in fact cofinal, class of cardinals I, any two completion
regular measures on П,е/ ^>· are completion Baire isomorphic.
Choksi A984) asks "whether analogues of the results of Choksi and Fremlin - without
additional set-theoretic assumptions - are valid for compact groups (and possibly even
for homogeneous spaces)". This important conjecture was based on the one hand, on
Kakutani and Kodaire theorem, and on the other on Theorems 2.1 and 2.2. Furthermore,
this conjecture implies the necessity to find an intimate connection between Haar measure
and product measure.
Choban A994) had already proved that every compact group is Baire isomorphic to
a product of compact metric spaces. In a subsequent paper, the author rediscovered the
same result and proved that such a Baire isomorphism can be constructed so that it takes
Haar measure to a product measure (Grekas, 1994). Thus, the Haar measure on a compact
group plays a similar role to that of a product measure in the structure theory of measure
algebras on a product of compact metric spaces (similar results can be proved for compact
homogeneous spaces (Grekas, 1995)).
Shortly before the previously mentioned, the author attempted, bearing in mind Choksi's
question, to obtain "compact group analogues" of the results of Choksi and Fremlin A979).
Grekas A992a) deals with the isomorphisms of measure algebras on compact groups and
their inducing point maps. Modifying the proof of Theorem 2.1, the author first proves that
two (Radon probability) measures on the group, whose measure algebras are homogeneous
of the same type, are completion Baire isomorphic. The rest of the paper, as Choksi and
Fremlin did, studies how many non-isomorphic completion regular measures can exist
on the group. Using approximation by Lie groups, it is shown that the compact group
analogue of Property 2.3 is valid (see Theorem 5.A and Remark 5.13C) there); note
that to carry out the extension to compact groups, it is used, in a crucial way, the fact -
incidentally established there - that the Haar measure on any uncountable compact group G
is homogeneous of Maharam type w(G). The author states nextly the following theorem.
THEOREM 2.4. IfG is a compact group with w(G) ^ ?, then the Maharam type of G is
equal to the Maharam type of {0, 1}",G).

754
S. Grekas
Although the statement was correct, there were some gaps in the proof; see also
corrigendum (Grekas, 1994) (a detailed account of these problems is given in Godin
B000)).
Remark 2.5. Theorem 2.4 leads to an affirmative answer to Choksi's question, not only
regarding the results free on additional set-theoretic assumptions but also regarding the
remaining results of Choksi and Fremlin.
A correct proof of Theorem 2.4 was done in Grekas and Mercourakis A998), where
the measure-theoretic structure of topological groups is further more investigated. More
precisely, Theorem 2.4 is an immediate consequence of the following structure theorem
(Grekas and Mercourakis, 1998, Theorems 1.1 and 1.4):
THEOREM 2.6. Let G be a compact group with w(G) = a ^ ?. There exist two families
Gf, ??, ? < a, of compact metric groups (each having at least two points if a ^ ?+) and
two continuous open surjections
such that
where ??, ?, ?$ denotes the (normalized) Haarmeasure on Gf, G, Щ, respectively.
Roughly speaking, the measure ? is compressed between 0^ ?? and 0^ ?? so that
the maps /, g are very close to being continuous epimorphisms (see Remarks 1.2
and Corollaries 1.3, 1.6 in the same paper). Theorem 2.6, whose proof is based on
classical results of Pontrjagin-van Kampen and Mostert (see Price A997); also references
given in Grekas and Mercourakis A998), extends and refines the well-known structure
theorem of Kuzminov A959): every compact group is a dyadic space (Comfort et
al., 1992) contains many interesting insights into the structure of topological groups).
The remaining part of Grekas and Mercourakis A998) deals with the structure of the
Jordan algebra of the Haar measure. As it is already mentioned, the Haar measure
on a compact group G is Baire isomorphic to the Haar measure on some product
X = П$<и.(С)^? °f comPact Lie groups. The authors, therefore, obtain a stronger
result, in fact that the isomorphism can be chosen to be a "Jordan isomorphism"
(see Definition 2.6 and Theorem 2.13); this, particularly, implies that the space of
Riemann integrable functions on a compact group is isometric to that of Riemann
integrable functions on some product of compact Lie groups (Corollary 2.15 there).
Additionally, several applications of the above results are given. Among these, the
authors obtain a direct proof of the existence of a strong - not necessarily invariant
- lifting for the Haar measure on a compact group and a rather illuminating proof of

On products of topological measure spaces
755
the existence of uniformly distributed sequences on a compact separable group (Veech,
1971).
Remarks and questions 2.7.
A) Grekas A995) showed, using Furstenberg's A963) theorem that the phase space of
any minimal distal flow is Baire isomorphic to a product of compact metric spaces;
the isomorphism takes some invariant, completion regular probability measure to a
product measure. According to the previously referred, the author believes that this
isomorphism can be chosen to be also a Jordan isomorphism.
B) It is easy to verify that if two Radon probability spaces have isomorphic Jordan
algebras, then they have isomorphic measure algebras are hence the same Maharam
type (Grekas and Mercourakis, 1998). The converse is not true, as it follows from
Examples A) and B) in the same paper. In view of these examples, it is natural
to ask whether there exists a strictly positive measure ? on X = {0, \}a ,? ^ ?>+,
homogeneous of Maharam type a, such that its Jordan algebra is not isomorphic
to the Jordan algebra of the Haar measure on X (of special interest is the
case ?+ ^ a ^ c; notice that a similar question has been posed in Mercourakis
A996)).
C) Associate with any topological space X is the Boolean algebra ?(?) of Baire sets
modulo sets of first category. Given an automorphism ? of ?(?) it may not be
possible to find a point transformation of X that induces ? (Choksi et al., 1987).
However, when X is a product of complete separable metric spaces, it is shown in
Maharam A979) that any automorphism of ?(?) can be realized by an invertible,
Baire measurable transformation of X. By combining Maharam's result with ideas
of Grekas and Mercourakis A998), it is possible to prove the same result in the case
when X is any compact group (it is a natural question whether category results are
valid for compact groups).
D) Methods which were developed in the proof of Theorem 2.6 can lead to some results
concerning the "homeomorphic measure problem" for compact connected
topological groups (Grekas and Mercourakis, 2002). For more information on this problem,
see references in Prasad A981), Morris and Peck A983, 1984), Gryllakis A989).
3. Topological liftings for product measures
Theorem 1.5 is, perhaps, the first published result which connects the meanings among
them, lifting, topology and product measure. Clarifying elements about the history of this
theorem can be found in Choksi et al. A987), cf. Remark 4.5 in Fremlin A989).
This section emphasizes on topological (strong, Borel, Baire) liftings for product
measures. As far as simple - not necessarily topological - liftings, the reader has to refer
to A. lonescu Tulcea and С lonescu Tulcea A969), Fremlin B00?), Strauss et al. B002).
The terminology and notation are the same with the ones of Section 2, but, in some cases,
the considered measure spaces (which from the beginning are supposed to be localizable)
are not necessarily bounded. All unexplained notions can be found in A. lonescu Tulcea
and С lonescu Tulcea A969).

756
S. Grekas
Let (?, ?) be a topological measure space and ? a Boolean subalgebra of the ?-algebra
? ? =??(?) of ?-measurable sets in X. A Boolean homomorphism ?: ? -> ?? is called
a lifting for ? if ? ~ ? A and ? ~ В implies ? A = pB for ?, ? e ? (where ~ denotes
equivalence modulo negligible sets). The lifting ? is called strong, respectively Borel,
Baire if U с pU for every open set U e ?, respectively pA e B(X),Bo(X) for all
A e ?. If ? is a lifting for ?/;, then we say that ? is a lifting for ? (if there is a strong
lifting for ?, we also say that (?, ?) - or, just ? - has the strong lifting property SLP).
The notion of (multiplicative) lifting /, linear lifting, etc., for (the algebra of bounded
measurable functions) ?/°°(?,?), in the sense of lonescu Tulcea, can also be similarly
defined (p and / are related, in a biunique way to each others, by фрл = 1(Фа), where ?
denotes characteristic function).
3.1. Baire, Bore! and Strong liftings
Problem 1.2 can get, in the liftings language, various forms, as, e.g.:
(Qi) Is the SLP a productive property? (Note that, in this point, product means ?-ad-
ditive product measure.)
(Q2) Which topological measure spaces have the "product-strong lifting property"
(PSLP)? (This means their product with any space having the SLP has again the
SLP.)
Qi is an open question, even if we consider two Radon measures on compact spaces
(Kupka, 1983, Question 3.3). However, since the completion regularity of measures on
products of compact metric spaces is a productive property, it is natural to ask:
(Q3) If ? and ? are completion regular strictly positive measures on products of
compact metric spaces (of weight > ?+) and if ?, у both have the SLP, then
does ? ® ? has the SLP?
As far as Q2 is concerned in the setting of compact spaces, there are in the literature very
special examples of measure spaces (?. ?) having this property:
• X is any product X = JT X, of compact metric spaces and ? = 0 ?, with supp ?, =
X, for all/;
• X is a compact group and ? the Haar measure.
The first one (Grekas, 1987) is an (easy enough) extension of a noticable result,
Theorem 5 in A. lonescu Tulcea and С lonescu Tulcea A969, Chapter VIII). A crucial
step in its proof was the following "extension theorem" from the one factor to the product
A. lonescu Tulcea and С lonescu Tulcea A969, Chapter VIII, Theorem 4); further
generalizations of the lonescu Tulcea theorem are given in Fremlin B00?), Strauss et al.
B002).
THEOREM 3.1. Let Z = T ? X, where ?, ? are compact spaces and let ?, ? be (positive)
Radon measures on T, X respectively, with supp ? = T, supp ? = X. Suppose that:
(i|) X is metrizable;
A2) r is a strong lifting for v.
Then, there is a strong lifting ? for ? <g> ? such that p(f <g> g) = rf <g> g, for all
f e M°°(T, v), g e C(X) (in particular, ? is an inverse lifting of r under the canonical
projection, in the sense of Strauss et al. B002)).

On products of topological measure spaces 757
We will afterwards see that the extension procedure of a lifting from the one factor to
the product has some additional interest.
If one seeks to obtain similar results about Borel or Baire liftings, must bear in mind
that the non-existence of Borel (linear) liftings (for many non-complete probability spaces
(У, В, у)) is consistent with ZFC. This was established by an articles series, in the center
of which is Shelah A983); there was proved that the usual Borel measure on [0, 1]
consistently does not have a lifting. Various improvements of that result followed, see
Burke and Just A991), Burke A993b), Burke and Shelah A992). In the latter are presented
three examples of particular interest:
A) ? = [0, 1]A, B = B(Y), ? is the usual product measure;
B) ? = {0, 1}A, B= B(Y), ? is Haar measure;
C) (?,?, ?) is any probability space, ? = [0. 1] ? ?.?= ?([0?])® ?.
The following problem, which is related to this discussion, arises (see, e.g., Fremlin
A993)):
(F) Can there be a lifting r of ([0, 1]-, B([0, 1])-, mj) ( where mi is the restriction of
Lebesgue planar measure) which is consistent, i.e., such that if ?, F e B([0, 1]),
then r(E ? F) is of the form ?" ? F'l
Notice that, by a theorem [CH] of Mokobodzki, any probability space (?,?,?),
|ГК К2, admits a lifting (Fremlin, 1989).
Remark 3.2.
A) Consistent (not necessarily strong) liftings for complete probability spaces always
exist. Talagrand A982, 1988) introduced them, proved their existence and has
shown that these objects give a simple and efficient way to deal with measurability
problems for a class of stochastic process. On the other hand, any consistent lifting
for an hyperstonian space is not strong, see Strauss et al. B002), where an interesting
variation of the notion of consistent lifting, the so called admissibly generated, is
extensively discussed.
B) Burke A995) and Fremlin B00?) introduced the notion of lifting r respecting
coordinates, for a product (X = f],e/ ^,·, ? = Xie/M;) as following: for every
УС/ there is a lifting rj for х,еуд, such that r is an inverse lifting of rj under
the canonical projection. Concerning topological probability spaces, is interesting,
of course, the r to be strong too; this is the case for any translation invariant lifting
for the usual product measure on {0, 1}" (this fact, due to Fremlin B00?), is the
main step for showing that if the (?,, ?,) are Maharam-homogeneous spaces, then
there is a lifting respecting coordinates for ?. The - not so important and without
any difficulty - observation, at this point, is that a translation invariant lifting r for
the Haar measure on any locally compact group G "respects coordinates", in the
following sense: for every closed normal subgroup ? of G, there is some lifting r#
for the Haar measure on G/H such that r is an inverse lifting of гн under the
canonical projection.
3.2. On the existence of Baire strong liftings
Throughout this subsection, the framework is that of ?-additive product measures. The
first part deals with - more or less - known positive results on the existence of Baire

758
S. Grekas
strong liftings. In the second part, the discussion is about one more existence theorem. The
approach developed, essentially based on Assertion 1.4, provides a unified lifting-theoretic
treatment for a class of ? -compact groups and for a class of products of separable metric
spaces.
One of the reasons which make the existence of a Baire strong lifting interesting is that
such a lifting assures 'good' measurable sections for the continuous functions onto the
underlying topological space; cf. Talagrand A978), also final remarks in Losert A980).
The latter concerns the existence and, mainly, the non-existence of measurable selections;
Losert, following a transfinite induction argument, concludes in:
THEOREM 3.3 (CH). Let X, be compact metric spaces, ?, probability measures on X,-
with supp ?, = X, (i e I) and assume that \l\ ^ Ni. Then there exists a Baire strong lifting
/ог(8>,-е/Л,-.
Grekas and Gryllakis A992), using Losert's arguments and additionally some
characteristics properties of completion regular measures, saw that the conclusion of Theorem 3.3
remains valid, if X, is a product of separable metric spaces, w(X,) ^ с and ?;
completion regular (/' el). Note that, one of these properties is Assertion 1.4; ideas in their proof
suggested to the author to introduce the following concept (Grekas, 1992b).
Let (У, y) be a ?-additive probability space. Consider the condition:
(O2) for every ?-additive probability space (?,?) and every open set U in ? ? ?, there
are a separable metric space ? and a continuous open surjection ? : ? —> ? such
that U ~ p_i pU, where p:XxY->XxZis defined by p(x,y) = (?,?(?)).
(?, у) - or just у - is called an Ch-space, if it satisfies (??). Clearly, every Cb-space
(satisfies Assertion 1.4 and) is completion regular; the converse is also true, concerning
measures on products of separable metric spaces. Also, the Haar measure on any compact
group satisfies {От) (Grekas, 1992b, Propositions 1.3 and 1.4). At this point, some
comments are necessary. The structure theorem of Kakutani and Kodaira (Choksi, 1984,
p. 84) made easy the proof of the latter. If, now, 'Haar measure on a compact group'
is - in the language of Section 3 - the (compact group) analogue of 'product measure
on a product of compact metric spaces', then Proposition 1.4 in Grekas A992b) is the
analogue of Theorem 1.6 of Choksi and Fremlin. Another analogue (of Losert's theorem)
about liftings, is the fact, remarked in Grekas and Gryllakis A994), that (subject to CH)
the Haar measure on any compact group of weight ^ N2 admits a strong Baire lifting (this
lifting, of course, cannot be translation invariant, see, e.g., Burke A993a)).
It is obvious that the existence of a lifting 'compatible" with the structure of a product
?-algebra, can be examined in many ways. Concerning complete, e.g., measure spaces, the
construction of a consistent lifting is always possible (Remarks 3.2). Although, concerning
non-complete ?-algebras, the problem seems to be very fine and remains open (e.g.,
problem (F)).
If now the requirements become weaker: the compatibility to refer to the one out of the
two factors, that is 'the product lifting' (we wish to construct) extends a given lifting for the
one factor, then (we saw) many positive results, although concerning non-complete spaces
(especially measures on non metrizable topological spaces), the study seems to be more
difficult.

On products of topological measure spaces
759
The investigation which follows, motivated by Losert's result, studies the construction
of inverse Baire strong liftings for products. The relative results show that the class of
(?2-spaces is a convenient setting for such an investigation.
LEMMA 3.4 (CH). Let (У, v) be a completion regular probability space, with supp ? = ?
and w(Y) ^ с If (?, ?) satisfies Assertion 1.4, then there is a Baire strong lifting for v.
PROOF. Cf. Remark 3 in Grekas and Gryllakis A994). D
Let now ? and ? be strictly positive measures on X and ? respectively, with w(X) ? ? ?,
w(Y) ^ ? ?. The next lemma is based on techniques, essentially, due to Losert.
LEMMA 3.5. If both ?, ? satisfy (СЬ) and if ? is a Baire strong lifting for ?, then there
is an inverse Baire strong lifting under the canonical projection, for ? <g> v.
PROOF. This is made in two steps.
A) Let W = ? ? ?, ? = ? ? ? and ? be the subalgebra of By (W), consisting of those
subsets of W which are of the form D ? ?, for some D с X. For A e B>.(W) define
d(A) :-\J[rBr\U: ???, ?/openinW, k(BDU\A) = 0}.
Since any open set U in W is ?-equivalent to p~' pU - where ? is as in (От) - it is easy to
check that there are countably many choices for U and so d(A) is a Baire set. Furthermore,
d has the properties (P) in Losert A980, p. 157).
B) The second step is Lemma 3 in Losert A980). Therefore, exists a Baire strong lifting
? for ? such that d(A) с ? (А) с (d(Ac)Y for all A eB,,(W). ? is the required one. D
Consider now a family (?,-,?,-), ? e /, of supporting СЬ-spaces, ? = Y\j€lMi,
? = 0/e/ ?/. The proof of the following result is similar to Grekas and Gryllakis A992,
Lemma 4.6).
LEMMA 3.6. Let {/„}„ек be an increasing sequence of subsets of ?, ? the subalgebra of
BK{M) consisting of those subsets of ? which depend on I„ for some n. Assume that r is
a strong Baire lifting for ?. For A e B,.(M) define d(A) := \J{rB f)U: В e ?, U open
in ?, ?(? П U\A) = 0}. Then d has the properties (P) in Losert A980).
THEOREM 3.7 (CH). Ifw(Mj) ^ c, i e /, and if a = |/| iC c+, then there exists a Baire
strong lifting for (?,?).
PROOF. There is nothing else to do, except almost to repeat arguments in Losert's paper.
Here follows an outline.
There is a directed set ? = {Fj·, j e J} of finite sub-products, of cardinal a such that
projy lim Fj = M. Enumerating ? as [F^, ? <a} and taking Hq = Fq, Hy = Hs ? Fy if
? = ? + 1 (for some ? < a) and Hy = proj5<), lim Hs otherwise, we define, by transfinite
induction, a family (??)?<? of sub-products of ?, e/??, such that ? =projylim Hy.

760
S. Grekas
For ? < a let ?? be the sub-algebra of B,.(M) of those sets which are the form pr~' (D)
with ? < ?, where pr^ : ? -> ? ? is the canonical projection. The lifting pY for i2y if
defined by induction on ? such that ??/?? = ?? for /3 < у. If ? is a limit ordinal with
uncountable cofinality, then the definition of pY follows by the fact that ?? = 1]?<? ? ?.
In the other cases, the lifting py is defined with the help of Lemmata 3.5 and 3.6. D
The question that, naturally, arises is how much is the Ch-spaces class wider than the
one of completion regular probability measures on products of separable metric spaces?
The СЬ-condition leads our thought to an already know topological spaces class: the
class of inverse limits of separable metric spaces; although, such kind of spaces have been
studied in depth, during the last 20 years' period of time by he Russian school of topology,
a great part of this literature has not been translated in English.
More terminology can be fixed: let ? = {Xa. a e A} be an inverse, or projective, system
('spectrum', in the Russian language) of topological spaces, where A is some directed set,
??? ¦' ?? -*¦ ??, с* < /3, are the bonding maps and pa : X -> Xa (a ? A) the (canonical)
projections (we assume throughout that the projections are surjections). From the fact that
a product space is the limit of the family of its countable sub-products, it follows that in
this case
(S) for every increasing sequence {a,,}„et; in A, there exists a e A such that Xa =
lim Xan.
This means that the spectrum is, roughly speaking, closed under countable sub-limits.
We generalize, in some sense, this property, by saying that (any spectrum) ? is ?-closed
if it satisfies (S) and the bonding maps are open. If ? is ?-closed, then a subset В of
X = lim Xa {? 0) is said to be determined by countable many coordinates if there is a e A
such that В = p~\ paB; then, for simplicity, we say that В depends on a (the terminology
has been created just in needs of the text).
The next auxiliary result is an easy application of Scepin A981, Proposition 1.8) (cf.
Tkacenko A981, Lemma 12) or Grekas and Karolis A999, Theorem 2.1)).
PROPERTY 3.8. If the limit space X of a ?-closed spectrum is ccc, then every Baire set
in X is determined by countable many coordinates.
The following examples of inverse limits of ?-closed spectra are of measure-theoretic
interest:
• Every product of separable metric spaces (the product space is the limit of its
countable sub-products).
• A locally compact, ?-compact group (the spectrum is that of metrizable subgroups
described in Choksi A984, p. 84).
• A (Baire subset of a) Dugundji space of weight ^ ?\ (the spectrum is, here, the
corresponding "Haydon system"; see Uspenskii A988), cf. Shapiro A985)).
Of course, a Dugundji space is dyadic and a ?-compact group is quasi-dyadic (Fremlin
and Wajch, 1993). The next discussion, which is adapted from Engelking A997), shows
that the limit of a ?-closed system of compact metric spaces need not be dyadic.
Remark 3.9. For any compact (Hausdorff) space Z, 2Z denotes the space of its
compact subsets with the Vietoris topology (Engelking, 1997, 2.7.20, 3.12.26). Consider

On products of topological measure spaces
761
the spaces D = {0, 1}, ? = D"'-, X = 2Z. Since ? = lim^ Z„, where {Z„} is the system
of countable sub-produces of D"-\ we must have: X = lim^ Xa, where X„ = 2X". Then,
it is immediate that {Xa} satisfies (S); moreover, its projections are open surjections. Note
that the compact space X is not dyadic (Shapiro, 1976). On the other hand, since there are
quasi-dyadic compact spaces which are not dyadic, it is natural to ask whether this space
is quasi-dyadic.
The following proposition shows that the completion regularity in ?-closed spectra
setting is a productive property. The condition that the weight of the one factor is
considered as "relatively low", simply means that the author could not find the way to
prove it without this term.
PROPOSITION 3.10. Let ? = {Ya,a e E} be a ?-closed spectrum of separable metric
spaces and ? a completion regular probability measure on ? = lim^ Ya (ф И). //
w(Y) ^ co+, then (Y, y) is an Oi-space.
PROOF. It combines ideas in Fremlin and Grekas A995) with techniques used in Grekas
A992c). Summary steps are sketched as follows:
A) Construct (by induction on ? < ?\) a sub-system ?' = {??. ? < ?\} of ? with
lim ZY = ?. If (?, ?) is a topological probability space and ? = ? <g> у, then it is not so
hard the СЬ-condition to be seen as a consequence of the assertion. For any ?-measurable
set A, there is ? < ?\ such that ?(? П (U ? ?^] psV)) = 0 for all couples of sets (t/, V)
where U is open in X, V is open in У, ?(? ? (U ? ?)) > 0 and ?(? ? (U ? V)) = 0.
B) Supposing, if possible, that the assertion is not valid for some closed and ?-
supporting set A, we find (by transfinite induction) two families UY, VY, ? < ?\ where
each VY is an open set in У, depending on ? + 1 and each UY is an open set in X, such that
A f) (Uy ? VY) = fi and ?(? ? (UY ? ?~] ?? VY)) > 0, ? > ?,. A)
Nevertheless, A) combined with techniques which have been developed in Grekas
A992c) leads to contradiction. (Note that Grekas A992c) deals with compact groups; the
arguments, however - which are there presented in full details - can also be used in any
limit space like ?.) ?
Remarks 3.11.
A) The above mentioned lead to the following conclusions, about the product (?, ?) of
family (?,, ?,), i e /, of supporting, completion regular probability spaces, where
each Xi is the limit of some ?-closed spectrum of separable metric spaces and
w(Xi) ???+:
(C|) (?, ?) is completion regular;
(Ci) [CH] If |/| ^ c + , then there is Baire strong lifting for ?.
In the case when the X,- are locally compact, ?-compact groups, (C2) is the
?-compact group analogue of a result about product of separable metric spaces;
see Grekas and Gryllakis A992, Remark 5.3).
B) Concerning compact spaces, the things lead to known situations. If, for instance,
in Proposition 3.10, ? is compact, then ? is Dugundji, hence dyadic and so

762
S. Grekas
Theorem 1.11 yields a different and direct proof of the proposition. The same would
happen if ? was, just, quasi-dyadic (it is not clear, for the author, if the limit of
a ?-closed system of separable metric spaces is quasi-dyadic).
Acknowledgements
I would like to thank the editor of the Handbook E. Pap of his kind support and
understanding; J.R. Choksi for precious information about homeomorphic measures;
N.D. Macheras, K. Musiat, W. Strauss for sending a copy of their work and S. Mercourakis
for his valuable suggestions.
References
Bourbaki, N. A965), Integration. Chapitres I-IV, Actualites Sci. Indust., Vol. 1175, Hermann, Paris.
Burke, M.R. A993a), Liftings and the property of Baire in locally compact groups, Proc. Amer. Math. Soc. 117,
1075-1082.
Burke, M.R. A993b), Liftings for Lebesgue measure, Israel J. Math.. Conference Proc., Vol. 6.
Burke, M.R. A995), Consistent liftings. Preprint.
Burke, M.R. and Just, W. A991), Liftings for Haar measure on @. 1}A', Israel J. Math. 73, 33^4.
Caratheodory, С A918), Vorlesungen iiber reelle Funktionen. Teubner. Leipzig. Berlin.
Choban, M.M. A984), Baire isomorphisms and Baire topologies. Solution of a problem of Comfort, Soviet Math.
Dokl. 30, 780-784.
Choksi, J.R. A972), Automorphisms of Baire measures on generalized cubes I, Z. Wahrsch. 22, 195-204. //, ibid.
23,97-102.
Choksi, J.R. A973), Measurable transformations on compact groups, Trans. Amer. Math. Soc. 184, 101-124.
Choksi, J.R. A984), Recent developments arising out of Kakutani's work on completion regularity of measures,
Contemp. Math., Vol. 26, Amer. Math. Soc., Providence, RI, 81-94.
Choksi, J.R., Eigen, S.J., Oxtoby, J.C. and Prasad, VS. A987), The work of Dorothy Maharam on measure theory,
ergodic theory and category algebras, Contemp. Math., Vol. 94, Amer. Math. Soc., Providence, RI, 57-71.
Choksi, J.R. and Fremlin, D.H. A979), Completion regular measures on product spaces, Math. Ann. 241, 113-
128.
Choksi, J.R. and Simha, R.R. A978), Measurable transformations on homogeneous spaces. Studies in Probability
and Ergodic Theory, Adv. in Math. Suppl. Stud., Vol. 2, 269-286.
Comfort, W.W., Hofmann, K.H. and Remus, D. A992), Topological groups and semigroups. Recent Progress in
General Topology, M. Husek, J. Van Mill, eds, Elsevier.
Engelking, R. A997), General Topology, PWN, Warszava.
Erdos, P. and Oxtoby, J.C. A955), Partitions of the plane into sets having positive measure in even- non-null
measurable product set. Trans. Amer. Math. Soc. 79, 91-102.
Fremlin, D.H. A976), Products of Radon measures: a counter example, Canad. Math. Bull. 19, 285-289.
Fremlin, D.H. A982), Quasi-Radon measure spaces, Note of 2.6.82.
Fremlin, D.H. A983), Stable sets of measurable functions. Note of 17 May.
Fremlin, D.H. A984), Consequences of Martin's Axiom, Cambridge Univ. Press.
Fremlin, D.H. A989), Measure algebras. Handbook of Boolean Algebra, J. Monk, ed., North-Holland,
Amsterdam.
Fremlin, D.H. A993), Problems, 9 March.
Fremlin, D.H. B000), Sets determined by few coordinates. Preprint.
Fremlin, D.H. B00?), Measure Theory, to appear.
Fremlin, D.H. and Grekas, S. A995), Products of completion regular measures. Fund. Math. 147, 27-37.

On products of topological measure spaces
763
Fremlin, D.H., Johnson, R.A. and Wajch, E. A996), Countable network weight and multiplication of Borel sets,
Proc. Amer. Math. Soc. 124, 2897-2903.
Fremlin, D.H. and Wajch, E. A993), Quasi-dyadic spaces. Version of 25.10.93. Preprint.
Furstenberg, H. A963), The structure of distal flows, Amer. J. Math. 85, 477-515.
Gardner, R.J. A975), The regularity of Borel measures and Borel measure compactness, Proc. London Math. Soc.
30,95-113.
Gardner, R.J. A981), 77k· regularity of Borel measures. Oberwolfach 1981. Lecture Notes in Math.. Vol. 945,
Springer, Berlin.
Gardner, R.J. and Pfeffer, W.F. A984), Borel measures. Handbook of Set-Theoretic Topology, K. Kunen and J.E.
Vaughan, eds, North-Holland, Amsterdam.
Godin, V. B000), Isomorphic measures and point realization of set tr/msformations on compact topological
groups and compact transformation groups. These de projet. McGill University.
Grekas, S. A987), Remarks on the strong lifting property for products. Israel J. Math. 58. 198-204.
Grekas, S. A992a), Isomorphic measures on compact groups. Math. Proc. Cambridge Philos. Soc. 112. 349-360.
Grekas, S. A992b), Onproducts of completion regular measures, J. Math. Anal. Appl. 171. 101-110.
Grekas, S. A992c), Note of April.
Grekas, S. A994), Structural properties of compact groups with measure-theoretic applications, Israel J. Math.
87, 89-95.
Grekas, S. A995), Measure-theoretic problems in topological dynamics. J. Anal. Math. 65, 207-220.
Grekas, S. and Gryllakis, C. A991), Completion regular measures on product spaces with application to the
existence ofBaire strong liftings. Illinois J. Math. 35, 260-268.
Grekas, S. and Gryllakis, C. A992), Measures on product spaces and the existence of strong Baire liftings,
Monatsh. Math. 114, 63-76.
Grekas, S. and Gryllakis, С A994), A remark on a theorem ofLosert, Monatsh. Math. 117, 95-102.
Grekas, S. and Karolis, P. A999), Notes on projective limits of separable metric spaces. Preprint.
Grekas, S. and Mercourakis, S. A998), On the measure theoretic structure of compact groups, Trans. Amer. Math.
Soc. 350, 2779-2796.
Grekas, S. and Mercourakis, S. B002), Homeomorphic measures on compact connected groups, in preparation.
Gryllakis, C. A988), Products of completion regular measures. Proc. Amer. Math. Soc. 103, 563-568.
Gryllakis, C. A989), Homeomorphic measures on products of intervals. Math. Ann. 284, 377-389.
Gryllakis, С and Grekas, S. A999), Onproducts of Radon measures, Fund. Math. 159, 71-84.
Gryllakis, C. and Koumoullis, G. A990), Completion regularity and ?-additivity of measures on product spaces.
Compositio Math. 73, 329-344.
Halmos, P. A950), Measure Theory, Van Nostrand. Products of completion regular measures. Proc. Amer. Math.
Soc. 103, 563-568.
Hewitt, E. and Ross, K. A963), Abstract Harmonic Analysis. Vol. I, Springer. Berlin.
Ionescu Tulcea, A. and Ionescu Tulcea. С A969). Topics in the Theory of Liftings, Ergebn. Math. Grenzgeb.
B. 48, Springer, Berlin.
Kakutani, S. A943a), Notes on infinite product measure spaces I, Proc. Imper. Acad. Tokyo 19, 148-151.
Kakutani, S. A943b), Notes on infinite product measure spaces II, Proc. Imper. Acad. Tokyo 19, 184-188.
Kakutani, S. and Kodaira, ?. (?944), Uber das Haarsche Mass in der local bikompacten Gruppe, Proc. Imper.
Acad. Tokyo 20,444-450.
Kirk, R.B. A969), Locally compact, B-compact spaces. Indag. Math. 31, 333-344.
Knowles, J.D. A967), Measures on topological spaces. Proc. London Math. Soc. 17, 139-156.
Kupka, J. A983), Strong liftings with application to measurable cross sections in locally compact groups, Israel
J. Math. 44, 243-261.
Kuz'minov, V. A959), On a hypothesis of PS. Alexandroff in the theory of topological groups, Dokl. Akad. Nauk
SSSR 125, 727-729 (in Russian).
Losert, V. A980), A counterexample on measurable selections and strong lifting. Measure Theory, Oberwolfach
1980, Lecture Notes in Math., Vol. 794, Springer, Berlin.
Macheras, N.D. and Strauss, W. A994), On strong liftings for projective limits. Fund. Math. 144, 209-229.
Macheras, N.D. and Strauss, W. A996), On products of almost strong liftings, J. Austral. Math. Soc. Ser. A 60,
311-333.

764
S. Grekas
Maharam, D. A942), On homogeneous measure algebras, Proc. Nat. Acad. Sci. USA 28, 108-111.
Maharam, D. A958a), Automorphisms of products of measure spaces. Proc. Amer. Math. Soc. 9. 702-707.
Maharam, D. A958b), On two theorems of Jessen, Proc. Amer. Math. Soc. 9, 995-999.
Maharam, D. A958c), On a theorem of von Neumann, Proc. Amer. Math. Soc. 9, 987-994.
Maharam, D. A979), Realizing automorphisms of category algebras of product spaces. Gen. Topol. Appl. 10,
161-174.
Mercourakis, S. A996), Some remarks on countabh determined measures and uniform distribution of sequences,
Monatsh. Math. 121,79-111.
Montgomery, D. and Zippin, L. A955), Topological Transforation Groups, Interscience, New York.
Morris, A.S. and Peck, C.V. A983), A note on homeomorphic measures on topological groups, Proc. Edinburgh
Math. Soc. 26, 169-171.
Morris, A.S. and Peck, C.V. A984), On the homeomorphic measure property, Colloq. Math. 49 A), 5-10.
Prasad, V.S. A981), A sur\-ey on homeomorphic measures. Measure Theory, Oberwolfach 1981, Lecture Notes in
Math., Vol. 945, Springer, Berlin, 150-154.
Price, J.? A977), Lie Groups and Compact Groups, Cambridge Univ. Press.
Ressel, P. A977), Some continuity and measurability results on spaces of measures. Math. Scand. 40, 69-78.
Scepin, E.V. A981), Functors and uncountable powers of compacta, Uspekhy Mat. Nauk 36, 3-62 (in Russian).
Shapiro, L.B. A976), The space of closed subsets of D^2 is not a dyadic bicompact, Soviet Math. Dokl. 17,
937-941.
Shapiro, L.B. A985), On Baire isomorphisms of spaces of uncountable weight, Soviet Math. Dokl. 32, 113-117.
Shelah, S. A983), Lifting problem of the measure algebra, Israel J. Math. 45, 90-96.
Shwartz, L. A973), Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures, Oxford Univ.
Press.
Strauss, W., Macheras, N.D.and Musial, K. B002), Liftings, Handbook of Measure Theory, E. Pap, ed., Elsevier,
Amsterdam, 1131-1184.
Talagrand, M. A978), Non existance de certaines sections et applications a la theorie du relevement, C. R. Acad.
Sci. Paris Ser. I Math. 286, 1183-1185.
Talagrand, M. A982), Closed convex hull of set of measurable functions, Riemann-measurable functions and
measurability of translations, Ann. Fourier Grenoble 32, 36-69.
Talagrand, M. A984), Pettis Integral and Measure Theory, Mem. Amer. Math. Soc., No. 307.
Talagrand, M. A987a), The Glivenko-Cantelli problem, Ann. Prob. 15, 837-870.
Talagrand, M. A987b), A 'regular oscillation' property of stable sets of measurable functions. Contemp. Math.,
Vol. 94, Amer. Math. Soc, Providence. RI. 309-313.
Talagrand, M. A988). On liftings and the regularization of stochastic process. Probab. Theory Related Fields 78,
127-134.
Tkacenko, M.G. A981). Some results on inverse spectra I. II. Comment. Math. Univ. Carolin. 22. 621-633.
819-841.
Uspenskii, V. V. A988), Why compact groups are dyadic'!. General Topology and its Relations to Modem Analysis
and Algebra, Z. Frolik, ed., Proc. Sixth Prague Topological Symp., 1986. Heldermann.
Varadarajan, V.S. A965), Measures on topological spaces. Amer. Math. Soc. Transl. Ser, Vol. 48. 161-228.
Veech, W.A. A971), Some questions of uniform distribution. Ann. of Math. 94 B). 125-138.

CHAPTER 18
Perfect Measures and Related Topics
Doraiswamy Ramachandran
Department of Mathematics and Statistics, California State University, 6000 J Street, Sacramento.
CA 95819-6051, USA
E-mail: chandra@csiis.edu
Contents
Introduction 767
0. Preliminaries 767
1. Compact measures 769
2. Perfect measures 771
3. Measures on product spaces 773
4. The marginal problem 775
5. Monge-Kantorovich duality spaces 776
6. Regular conditional probabilities 778
7. Independence and Blackwell spaces 779
8. Standard measure spaces 780
9. Mixtures of perfect measures 781
10. Disintegrations 781
References 783
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
765

This page intentionally left blank

Perfect measures and related topics
767
Introduction
The fascinating class of perfect probability spaces was introduced by Gnedenko and
Kolmogorov A954) "to achieve complete harmony between the abstract theory of
measures and the theory of measures in metric spaces." The two volume monograph of
Ramachandran A979) brought together various developments involving perfect measures,
in a unified manner, showing that perfect spaces are technically the most pleasing class
of probability spaces. In this article we summarize the classical results and outline the
progress involving perfect measures since the late seventies. While the references contain
an almost exhaustive list of relevant papers, we can only provide an overview of some of
the major developments and key results. For proofs, additional details, and open problems
not mentioned herein the reader should consult the references.
0. Preliminaries
We follow the customary measure theoretic terminology and notation (see Billingsley
A986), Halmos A950), Neveu A965)).
All measures considered in this chapter are probabilities. The word function, when used
without qualification, will always mean a real-valued function.
Let С be a collection of subsets of a set X. By Cs. С'1, С" and C& we denote respectively
the smallest class containing С and closed under finite unions, finite intersections,
countable unions and countable intersections. By alg(C) and a(C) we denote respectively
the algebra and the ?-algebra generated by C. a(C) will be called the ?-algebra generated
by С and С will be called a generator of a(C). In case a ?-algebra A has a countable
generator, A will be called a countably generated (henceforth abbreviated as e.g.)
?-algebra. A ?-algebra В С A will be called a sub ?-algebra of Л. А ?-algebra with a generator
of cardinality к ^ Ki will be called an ? ?-generated ?-algebra.
A measurable space is a pair (X. A) where X is a nonempty set and Л is a ?-algebra
of subsets of X. If ? с X then ? ? ? = [А П ?: A e A} is a ?-algebra of subsets of
? and is called the trace of A on Y. If A is generated by С then ? ? ? is generated by
С П ?. If ? is a measure on (X, A) (recall that P(X) — 1 for all measures considered) such
that P*(Y) > 0, where P* denotes the outer measure induced by P, then Py defined on
(?,???) by
??(??)?) = ?*(??\?)/?*(?)
is a probability on (У, А П Y) and is called the trace of ? on (?. ? ? ?).
If (X, A) is a measurable space then a nonempty subset A e A is called an atom of A
(or Д-atom) if
В с А, ВеА => ?=???? = ?.
A is said to be atomic if every nonempty set in Л is a union of atoms. Clearly any two
distinct atoms are disjoint and the atoms of an atomic ?-algebra form a partition of X. Any

768
D. Ramachandran
e.g. ?-algebra is atomic. A ?-algebra A is called separable if it is e.g. and contains all
singletons. If Л is separable and if У с X, then ? ? У is separable. A (X, A) is separable
if Л is a separable ?-algebra.
Let ? be a probability on (?, A). A set A e A is called a P-atom if P(A) > 0 and
for every В е А, В с A either P(S) = 0 or P(B) = P(A). The measure ? is said to
be nonatomic if there are no P-atoms. The measure ? is called discrete if there exists a
sequence {A,,} of pairwise disjoint P-atoms such that ?^_| P(A„) = 1. A measure ? on
(X, Л) where A contains all singletons is called continuous if P({.v}) = 0 for every ? e X.
If A is separable then ? on (X, A) is nonatomic iff it is continuous.
If Ao is an algebra of subsets of X then a set function ?? on (X, Ao) is called a charge
if ?? is nonnegative, finitely additive and ??(?) = 1. The charge ?? is a measure if it is
countably additive.
Two measurable spaces (X,A) and (У, B) are said to be isomorphic if there is a
1-1 map / from X onto ? such that both / and /"' are measurable. Two probability
spaces (X, A, P) and (Y,B,Q) are said to be isomorphic if there is an isomorphism /
from (X,A) to (Y, B) such that Q = P/; they are said to be almost-isomorphic if
there exists X\ eA,Y\ eB with P(X,) = Q(Y\) = 1 such that (X|, ? ? ?\, PXl) and
(У|, В П У|, <2?) are isomorphic.
We say that (Z,C, P) is a thick subspace of(Y,B, Q) and write (Z.C, P) с (У, ?, Q)
if ? с ?,? П ? =C,Q*(Z) = I, and ? = Q*z
If ? is a metric space then we denote by ?? the Borel ?-algebra of X, namely the
?-algebra generated by the open subsets of X. Any subset of the real line ?)? will be given
the usual metric when considered as a metric space. The Borel ?-algebra of any separable
metric space is separable.
If X is a metric space then a measure ? on (?, ?), where ? D ??, is called outer regular
if for every В eB
P(B)=inf{P(U). i/open in X, Set/}
and ? is called inner regular if for every В e В
P(B)=sup{P(K). A: compact, К С В).
Any measure on the Borel subsets of a metric space is outer regular.
A Polish space is a complete, separable metric space. A measurable space (X, A) is
called a standard Borel space if (X, A) is isomorphic to (У,В) where У is a Borel subset
of a Polish space and ? is the Borel ?-algebra of У. If (X, Л) and (Y,B) are standard Borel
spaces and if / is a 1-1 measurable map from X into У then the forward images of Borel
sets under / are measurable; in particular, /(X) e В and / is an isomorphism of (X, A)
onto the standard Borel space (/(?),? П /(X)) (see Kuratowski A966, Theorem 1,
p. 489)).
Let X be a Polish space and let В be the ?-algebra of the Borel subsets of X. A subset
У of X is called analytic if it is a continuous image of the irrationals. The complement
of an analytic subset of X is called a coanalytic set. If У is a an analytic (respectively
coanalytic) subset of X then (У, В П У) as well as any measurable space isomorphic to

Perfect measures and related topics
769
it will be called an analytic space (respectively coanalytic space). For details concerning
analytic and coanalytic sets we refer the reader to Kuratowski A966), Bhaskara Rao and
B.Y Rao A981) and Cohn A980).
For the definition and properties of the Lebesgue measure, to be denoted by ? on the
real line and on the unit interval we refer the reader to Section 3 of Billingsley A986) or
Section 5 of Halmos A950).
The Axiom of Choice will be assumed throughout.
We will specify whenever we assume the Continuum Hypothesis and denote it by (CH).
1. Compact measures
Probability measures on the Borel subsets of a Polish space are well-behaved. They are,
for instance, inner regular, i.e., approximable from inside by compact sets (see Neveu
A965, p. 64)). But, in a general probability space, to start with we do not have a notion
of compactness. In this section we discuss the notion of compact classes of sets and
of compact measures, both due to Marczewski. We derive basic properties of compact
measures.
A collection С of subsets of a set X is called a compact class if for every sequence
(Ci)cC the following condition holds:
P| Ck? 0 for every ? > 1 => f|Ct/H.
k=\ k=\
Recall that a sequence {A„, ? > 1} of subsets of X is said to possess the finite
intersection property if, for every и > 1, ?"=? Ak ? 0- Thus a collection С of subsets of X
is a compact class if every sequence of sets from С having the finite intersection property
has nonempty intersection. The collection of compact subsets of any topological space is a
compact class. Every subcollection of a compact class is compact. If С is a compact class
then so are Cs and С.
Let ?? be a charge on an algebra Ao of subsets of a set X. We say that a class S of
subsets of X ^-approximates Aq if for every ? e Ao and for every ? > 0 there exists a
set S? eS and a set Ee e Ao such that
Ee С Se С ? and ??(? — ?[)<?.
Obviously, a class ? С ^-approximates Л) iff for every ? e Ao and for every ? > 0
there exists a set FE e ? such that
Fe С ? and ??(? - Fe) < ?.
PROPOSITION 1.1. If ? is a measure on (X, A) where A is ? ?-algebra and if ? ?-??-
proximates A then T& П A also ?-approximates A-

770
D. Ramachandran
Marczewski A953) defined a charge ?? on (?,??) where Ao is an algebra to be
compact if there exists a compact class С which ??-approximates Ao and showed that
every compact charge is indeed a measure, that is, it is countably additive; hence such a
compact charge ?? has a unique extension as a measure to ?(??)-
The class of compact measures is sufficiently wide. With the usual compact sets forming
an approximating class, we have
CI. Every measure on a standard Borel space is a compact measure.
PROPOSITION 1.2. Suppose ?? is a compact measure on (X. Ao) where Aq is an algebra.
Then ?, the extension of ?? to ?(.4?) is a compact measure; further, there is a compact
class contained in ?(??) which ^-approximates ?(.4?).
In the above setting, Cs П ?(??) is a compact class ?-approximating ?(??)-
C2. If ? is a compact measure on a ?-algebra A then there is a compact class К, С А
which ?-approximates A.
It is not known whether a similar property holds for compact measures on algebras: if
?? is a compact measure on an algebra Ao, can one find a compact class 1С С Ao which
??-approximates Ao"!
PROPOSITION 1.3. If ? is a nonatomic measure on an algebra Ao, then every ? e Ao
with ??(?) > 0 contains a set N of cardinality с such that ?*(?) = 0.
Sierpinski A956, p. 81) has proved using (CH) that there is an uncountable subset
D С [0,1] such that every subset of D which has Lebesgue measure zero is at most
countable; ?? on (D, B\o, \ j П D) is therefore noncompact.
For further results on Sierpinski sets and related results see Miller A984), Shortt A986),
Jasiriski and Weiss A991).
The following properties of compact measures are useful. For their proofs refer to
Ramachandran A979):
C3. Every discrete measure is compact.
C4. If ? on (X, A) is compact and if ? e A with P(A) > 0 then ?? on (?. ? ? ?) is
compact.
C5. If ? on (X, A) is compact and Q on (X, A) is such that Q <?. ? then Q is compact.
C6. ? on(X, A) is compact iff Pon(X. A) is compact where (X, A. P) is the completion
??(?,?,?).
C7. Let (Z,C,P) с (Y,B.Q), i.e., (Z,C. P) is a thick subspace of (Y.B,Q). If ? is
compact, then Q is compact.
C8. If (X, A) and (Y, B) are two measurable spaces and /: X -> У is an A. ?-measurable
map. If ? is a compact measure on (Y.A), then P/~' is compact on (Y.B).

Perfect measures and related topics
771
For a characterization of standard Borel spaces using the notion of compact classes see
Bjornsson A980). We will specify additional properties of compact measures in the sequel.
2. Perfect measures
Perfect measures are more general than compact measures and the two notions coincide on
e.g. ?-algebras.
Definition 2.1. A measure ? on (?. A) is perfect (equivalently, the probability space
(?, A, P) is perfect) if, for every Д-measurable function / on X, and for every subset С
of the real line for which /_1C e A there is a Borel set В с С such that P(f~l В) =
P(/-'C).
The following fact is useful in verifying perfectness of a measure.
PI. A measure ? on (X, A) is perfect iff for every Д-measurable function / on X there
is a Borel set Bf с f(X) such that P(/-'B/) = 1.
Ryll-Nardzewski A953) defined ? on (X. A) to be quasi-compact if, for every sequence
{An, ? ^ 1} с A and for every ? > o, there exists Ae e A with P(Af) > 1 - ? such that
{An П Ae, ? ^ 1} is a compact class. Every compact measure is quasi-compact; further,
quasi-compactness and perfectness are equivalent. Compactness, perfectness and quasi-
compactness are all equivalent on countably generated ?-algebras. Ryll-Nardzewski's
notion brings perfectness as an intrinsic property of the probability space (?,?,?) while
the notion of perfectness as in Definition 2.1 and PI involves real-valued measurable
functions on (X,A).
The basic properties of perfect measures are:
P2. Every compact measure is perfect.
P2 shows that the class of perfect measures is rich. Consider the Lebesgue measure ? on
([0, l],?[o.ij).LetM c[0, 1] be such that ?*(?) = 1 and ?*(?) = 0. Since ?*(?) =0,
taking / to be the identity function and using PI, we see that ?? is nonperfect (and hence
noncompact) measure on (M, #[o, ij П M).
Musiat A976) has constructed a perfect measure which is not compact with the
underlying ?-algebra being ? ? -generated, thereby showing that the converse of P2 is false.
P3. The restriction to any sub ?-algebra of a perfect measure is perfect.
The corresponding result for compact measures is
C9. The restriction to any sub ?-algebra of a compact measure is compact.
This was established by Pachl A978) using the notion of disintegrations which we will
discuss later.
P4. A measure on (X, A) is perfect iff its restriction to every e.g. sub ?-algebra is perfect.
P5. If ? on (X, A) is perfect then, for every ? e A with P(Y) > 0, ?? on (?, ? ? ?) is
perfect.

772
D. Ramachandran
P6. (?, A, P) is perfect iff (X, A, P) is perfect.
P7. Every discrete measure is perfect.
P8. Let (Z,C, P) с (?,В, Q), i.e., (?,С, ?) is a thick subspace of (Y,B, Q). If ? is
perfect, then Q is perfect.
P9. Let X с ?)?. Every measure on (X, #;)( П X) is perfect iff X is universally measurable.
THEOREM 2.1. Lef (Х,Л) foe a measurable space. Then the following statements are
equivalent:
(a) For every sub ?-algebra С С A, every measure on (X, C) is perfect.
(b) For every e.g. sub ?-algebra V С A every measure on (X, T>) is perfect.
(c) The range of every real-valued ?-measurable function f is universally measurable.
Kallianpur A959) introduced the notion of D-space as a separable space in which the
range of every measurable function is a universally measurable set and showed (b) => (c)
part of Theorem 2.1. Sazonov A962) independently showed (b) => (c) => (a). Blackwell
A956) started the study of connections between perfectness and analytic subsets of the
real line. Darst A969, 1971) constructed a D-space which is not analytic and a universally
measurable space which is not a D-space. Additional results in this direction can be found
in the above references and in Shortt A984).
If X is a separable metric space, ?? its Borel ?-algebra, and ?? is the class of compact
subsets of X. Then a measure ? on (X,A) where Ad ?? is perfect iff ? is inner
regular, that is, C\ P-approximates A. If, further, X is a Polish space, then a measure
on (X, A) where A D ?? is perfect iff ?? P-approximates A; according to Gnedenko
and Kolmogorov this property confirms the reasonableness of the definition of a perfect
measure. Pachl A979) has extended these properties to metric 2D-spaces (spaces no closed
discrete subspace of which has a 2-measurable cardinal).
THEOREM 2.2. Let ? be a measure on the Borel ?-algebra of a metric ID-space. Then
the following are equivalent:
(i) ? is inner regular, i.e., the collection of topological compact sets ?-approximates
the Borel ?-algebra.
(ii) ? is compact.
(iii) ? is perfect.
The above result can also be found in Koumoullis A981). For results on general
(not necessarily finite) perfect measures on topological spaces, the reader is referred to
Koumoullis A981) and the survey article of Gardner A982).
The notion of weakly compact measures (see Erochin A961)) is shown to be equivalent
to compact measures by Pachl A975). Mallory A975) defined a collection K, of subsets of
X to be a monocompact class if for every decreasing sequence {K„, ? ^ 1} С /С such that
?,??? Kn = 0, there is a natural number N such that K,\ = 0. If ? is a measure on (X, A)
and the monocompact class /C P-approximates A, then we say that ? is a monocompact

Perfect measures and related topics
11Ъ
measure. The notion of a monocompact class is weaker than that of a compact class. Tops0e
A979) showed that every monocompact measure is perfect. Recently, Dekiert (nee Remy)
A993) has shown that for K|-generated ?-algebras compactness and monocompactness
are equivalent. Since Musiat's example of a noncompact perfect measure is defined on ? ? -
generated ?-algebra, it follows that there is a perfect measure which is not monocompact.
Related open questions are raised in Dekiert A993).
A measure ? on (X, A) is called purely noncompact (respectively purely nonperfect)
if, for every С e A with P(C) > 0. Pc is noncompact (respectively nonperfect). Remy
A988) has established the following results:
(i) Every probability ? on (X, A) has a unique representation as ? = a \ P\ + <*? Pi +
?^??, where
(a) a, ^ 0, a \ + ai + aj = 1;
(b) P\ is compact;
(c) Pi is perfect and purely noncompact, and
(d) P?, is purely nonperfect.
(ii) If {P„, n^ 1} is a sequence of measures on (X.A) and ? = ??= ? a» ?" wnere
a„ ^ 0 for all ? > 1 with ??= ? ?» = ' <tnen ? is Perfect iff Pn is perfect for every
и ^ 1. 'Perfect' can be replaced by 'compact', 'purely nonperfect,' and 'purely
noncompact' in the assertion,
(iii) Let.M be the set of all probabilities on (X.A) and let M,. Mp,Mpnc and Mpnp
represent the set of all compact, perfect, purely noncompact and purely nonperfect
measures. Then Mr С Мр С ? and Mpnp С MP„r С М. Each is a convex
set and the set of extreme points of each is the set of 0-1 valued measures. Since
every 0-1 valued measure is compact (by C3), Mpnp and M,,,u- do not contain
any extreme point.
As a consequence of (iii) above, if {P„. ? ^ 1} is a sequence of compact measures on
(X, A), then there is a common compact class /С с A such that /C P„-approximates A for
each ? ^ 1 (by taking /C to be a compact class P-approximating A for ? = ?^=\ 2" Pn)-
Aniszczyk A986) has constructed a family {Pa. a e /} of compact measures on a
measurable space (X, A) for which there is no such common compact class
K,/^-approximating A for each a e /, answering a question raised by Pachl A979).
3. Measures on product spaces
In this section we show that perfect measures provide a natural setting for the existence of
product measures. Let {(?/, ?), ? e /} be a family of measurable spaces. Let Ao be the
algebra of subsets of П/е/ %> generated by finite dimensional cylinders. Recall that
?(?))=®?
i€l
is the product ?-algebra. The classic example of Sparre-Anderson and Jessen A948)
shows the existence of a charge ? on (["[,¦ e/ -??. Ao) such that each marginal P, = Pott,
on (Xj,Ai), where ?,- is the canonical projection, is countably additive while ? is not
countably additive. This phenomenon cannot occur if each marginal is perfect.

774
D. Ramachandran
Theorem 3.1 (Marczewski A953);respectively,Ryll-Nardzewski A953)). Let {(?,, ?,
i e /} be a family of measurable spaces and let ? be a charge on (П,е/ ?'?··??) such that
every marginal Pi = ???^] on (Xj.Ai) is compact (respectively perfect). Then
(i) ? is countably additive; and
(ii) the unique extension of ? to ®i€! Д is compact (respectively perfect).
Note that the Daniell-Kolmogorov consistency theorem for the existence of a product
measure is a consequence of Theorem 3.1. In view of РЗ (С9) we have
P10 (CIO). A measure on a product space is perfect (compact) iff each marginal is perfect
(compact).
By using a countable bilateral product of nonperfect spaces Nadkarni and Ramachandran
A978) have answered a question of D. Maharam A978) concerning tail fields.
Let (X, A) and (У, B) be two measurable spaces and let 11 = [A x В: A e А. В е В}
be the semialgebra of measurable rectangles. Let ? be a charge on the semialgebra with
?-additive marginals ? ? ??* on (X, A) and ? ???] on (?, ?), respectively.
The following theorem due to Marczewski and Ryll-Nardzewski A953) is important
and has many useful applications. Due to its significance it should be included in texts on
measure theory.
THEOREM 3.2. A charge on the semialgebra of measurable rectangles of the pmduct of
two measurable spaces is ?-additive if at least one of the marginals is perfect.
An example of Ryll-Nardzewski in the same paper shows that the assumption of
perfectness of at least one marginal is essential in Theorem 3.2 for a charge on the
measurable rectangles to be countably additive.
Ramachandran A996) has the following generalization of Theorem 3.2:
THEOREM 3.3. Let {(X/, Д,), ;' e /} be a family of measurable spaces. If ? is a charge on
the semialgebra TZ of measurable rectangles in ? e / %i w'tn countably additive marginals
such that all but perhaps one of its marginals is perfect, then ? is countably additive.
Let Qj be a measure on (X, C,), / = 1. 2. where each C, is an algebra. A measure Q on
a/g({Ci,Ci}) is called a common extension of Q \, Qi if Q\c, = Q\ for/ = 1.2. A common
extension Q is called a splicing if Q(C\ ПС:) = Q\(C\)Qi(Ci). For results on common
extensions and splicings refer to Stroock A976). Kallianpur and Ramachandran A983),
Schmidt and Waldschaks A991), Hackenbroch A992), Ramachandran A996).
For projective limits of perfect measure spaces see Musiat A980).
The following theorem of Pachl A979) gives two more characterizations of perfect
measures.
THEOREM 3.4. Let (X, A, P) be a probability space. Then the following are equivalent:
(i) ? is perfect.
(ii) If(Y,B, Q) is a probability space and ? is a charge on alg({A ? В: А е А. В е B\)
with marginals ? and Q then ? is countably additive.

Perfect measures and related topics
775
(iii) If (Y, B, Q) is a probability space and ? is a measure on А® В with marginals ?
and Q, then ?*(? ? F) = Q^F) for every F CY.
An example due to Pachl A981) shows that condition of (ii) of Theorem 3.4 cannot be
replaced by the following weaker condition: "Any charge ? on alg({A\ ? ??. A,- e A,
i = 1,2}) with both marginals ? is countably additive."
4. The marginal problem
The marginal problem of Strassen A965) can be formulated as:
Let (X, A, P) and (У, B, Q) be two probability spaces. Given 5" с ? ? ? what are the
conditions that would ensure the existence of a probability ? on (? ? ?, A <g> B) with
marginals ? and Q such that ?*(S) = 1?
If such a ? exists then we say that 5" is marginalizable.
A general formulation of the problem and its importance in applications can be found in
Hoffman-J0rgensen A987). Hansel and Troallic A978, 1986), Lembke A982), Kellerer
A964a, 1964b) and Shortt A983) treat this problem under topological restrictions.
Definition 4.1. Let S be a semialgebra of subsets of ?. A set S с ? is called S-en-
closable if
5"* = (C\{E e a(S): ? D S}\ e S*s.
In the context of a product space and the semialgebra TZ of measurable rectangles we
shall simply use the term enclosable instead of TZ-enclosable.
THEOREM 4.1. Let (X,- ,Ai,Pi), i = 1, 2, be two probability spaces where at least one of
the measures P\ and P2 is perfect. A subset S С Х\ х Хг is marginalizable if
(a) 5" и enclosable; and
(b) {?? xX2)n5"C(X| xA2)nS^ Pi(Ai) <C P2(A2).
Theorem 4.1 unifies and extends beyond the context of analytic and separable spaces
the results contained in Theorem 1 of Shortt A983) and Proposition 3.8 of Kellerer A984).
Recently Plebanek A989) has used the above approach together with a result concerning
charges by Hansel and Troallic A986) to establish the following general solution of the
marginal problem for two-dimensional products.
Let (Xi,Ai, Pi), i = 1,2, be two probability spaces. For ? с ?? ? Хг define
?(?) = sup{P|(A|) + P2(A2): (?? ? А2)П? = 0}.
THEOREM 4.2. Let (X,-, Ai, Pi), i = 1,2, be two probability spaces and let at least one
of P\ and P2 be perfect. Let d ^ 0. If D e A\ <g> Аг is enclosable then the existence of
a probability ? with marginals P\ and P2 such that P(D) > 1 — d is equivalent to the
condition ?{?) ^ 1 +d.

776
D. Ramachandran
In view of Definition 4.1, using Theorem 4.2, we can recast Theorem 4.1 as the following
COROLLARY 4.1. Let I = {1,2} and let at least one of P\ and Рг be perfect. A subset
S С ?? ? Хг is marginalizable if
(a) 5" и enclosable; and
(b) 4(S*) ^ 1.
Let (Xi,Ai, Pi), i e /, be a family of probability spaces. Consider the product space
(П,е/ ^?>?>?€/ -4')- Analogous to Theorem 4.1 we have
THEOREM 4.3. A subset S С Пге/ %i IJ niarginalizable if the following conditions hold:
(a) All but perhaps one of the Pi are perfect.
(b) For each finite subset J С I and for every choice {gj, j e J] where gj on X j is
Aj -measurable and bounded for each j e J
jeJ jeJJxJ
(c) 5" is enclosable.
For additional interesting results in this direction see Plebanek A989), Ramachandran
A996).
5. Monge-Kantorovich duality spaces
The measure theoretic version of the classical transportation problem dating back to
Monge A781) and developing into infinite-dimensional linear programming in the work of
Kantorovich A940, 1942) concerns the validity of a duality theorem which is formulated
below. General treatment of Monge-Kantorovich duality theorems can be found in
Ruschendorf A981), Kellerer A984), Levin A984), Rachev and Ruschendorf A998),
Ramachandran and Ruschendorf A995). They arise in the study of
(i) probabilities with given marginals and given support, stochastic ordering (Strassen,
1965; Sudakov, 1975; Hoffmann-J0rgensen, 1987;Dall'Aglioetal., 1991;Ruschen-
dorf et al., 1996; Benes and Stepan, 1997);
(ii) probability metrics, central limit theorems and asymptotic analysis of algorithms
(Rachev, 1991; Rachev and Ruschendorf, 1998);
(iii) equlibria in assignment models in economics (Gretsky et al., 1992; Ramachandran
and Ruschendorf, 1997);
(iv) operator algebras (Arveson, 1974; Haydon and Shulman, 1996) and many others.
Let M(P\, Рг) = {? on ?? <8> Аг: ? has marginals Pi and Рг) = ?1^,®^2(??, Рг).
?,¦: ?? ? Хг —> Xi denote the canonical projections for i = 1,2. The abbreviation ®g, is
used for X]i=i gi ? ?,-. For a measurable space (X, A) the notation / e A indicates that /
is a real-valued, bounded Л-measurable function on X.

Perfect measures and related topics
777
For h e ? ? <g> Аг, the marginal problem is concerned with
S(ft) = sup| / ???: ? e ? (?,, ?2) = 5??^(?)
UXixX2 J
while the dual problem deals with
/(ft) = inf V f ?,-dP/: A/er'^OandAicffiA, =/?^,(?).
[i = \Jx< i I
The transportation problem seeks the validity of the duality
S(h) = I(h). (D)
The main duality theorem of Kellerer A984) deals essentially with second countable or
metrizable spaces X,¦, i = 1,2, with tight (or Radon) probabilities defined on the Borel sets
in which case (D) is shown to hold for a suitably large class containing all the bounded,
measurable functions. The following result of Ramachandran and Riischendorf A995) is
the most general duality theorem of this type.
THEOREM 5.1. If at least one of the underlying probability spaces is perfect then (D)
holds for all h e A\ <g> Аг-
The direct converse of Theorem 5.1 does not hold; that is, we can construct two
nonperfect probability spaces in the product of which (D) holds for all h e A\ <g> Аг (see
Example 1 in Ramachandran and Riischendorf A996).
In order to study probability spaces which ensure the validity of the general duality
theorem we introduce:
Definition 5.1. A probability space (X\, A\, Pi) is called a duality space if, for every
(X2, Аг, Рг), the duality (D) holds for all heA\® Аг-
A natural strengthening of the notion of a duality space arises if we postulate additionally
that for any sub ?-algebra Сг С Аг such that h e ? ? <g> Сг the optimal value .^д, ®_д2 (h) =
¦SUiS^CO (°r equivalently, /а,®.42(А) = АД|®С2(^))' i-e-. the optimal value remains the
same with refinements on the second space as long as the measurability conditions on h
are fulfilled. In other words the value of the transportation problem remains the same for
only 'technically' different formulations of the problem. Ramachandran and Riischendorf
A995) have shown that perfect spaces have this property which motivates the
Definition 5.2. A probability space (X\, A\, P\) is called a strong duality space if
(i) it is a duality space; and
(ii) for every sub ?-algebra Сг С Аг and for every h e A\ <g> Сг the condition
U^A2(h) = Ui®C2(h) (SD)
holds.

778
D. Ramachandran
Perfect spaces have the following extension property for measures (see Ramachandran
A996, Theorem 9)) which arises naturally in the marginal problem.
Definition 5.3. We say that (X\,A\,P\) has the extension property (respectively
charge extension property) if for every (X2, Ai, Pi) and for every sub ?-algebra Ci С ??,
if ? e M(P[, Pi\c2) then ? /~? e ??(?\, Pi) (respectively ? is a charge on A\ <8>Ci with
marginals Pi and P\c2)< i-e> ? extends to a measure (respectively charge) ? on A\ <g> Ai
with marginals Pi and P2.
It is well known that the projection of a Borel set in the product of standard Borel
spaces is an analytic set and therefore universally measurable (see Hoffmann-j0rgensen
A970), Cohn A980)). This property is useful in descriptive set theory. There is no measure
theoretic analogue for the projection of a measurable set in the product of two probability
spaces. The following gives a suitable measure theoretic definition of the projection
property.
Definition 5.4. A probability space (X\,A\, P[) is said to have the projection property
if for every (?? ? ??,?? ®Ai) and for every С e A\ ®Ai there exists ? ? = ? ? (С) е А\
with />i(Ai) = 1 such that щ{С ? (?? ? ?2)) еА22 ¦
Ramachandran and Riischendorf B000) have characterized strong duality spaces in
THEOREM 5.2 (Equivalence Theorem). Let (X\,A\, P\) be a probability space. Thenthe
following are equivalent:
(a) (X1, A1, P\) is a strong duality space;
(b) (X\,A\,P\)isperfect;
(c) (X1, ? ?, Pi) has the extension property;
(d) (X1, A\, P[) has the projection property.
(e) (X1, ? ?, Pi) has the charge extension property.
The equivalence of the notions of strong duality and perfectness enables us to obtain
several new properties based on PI to P10 that strong duality spaces inherit from being
perfect spaces. These properties are not easily established by direct arguments.
6. Regular conditional probabilities
Let (X, A, P) be a probability space and let ? be a sub ?-algebra of A. A function ?(?. A)
defined on (X, A) is called a regular conditional probability (r.c.p. for short) given В if
?(?, A) satisfies the following conditions:
(CP1) ?(?, ¦) is a probability on A for each fixed ? e X;
(CP2) ?(-, A) is ?-measurable for each A e A; and
(CP3) ? (? ? ?) = fB ?(?, A)dP for every A e А, В е В.
A r.c.p. ?(?, A) given В is proper at jto e X if
(CP4) ?(?0. ?) = 1 whenever x0 e В еВ.

Perfect measures and related topics
779
Classic example of Dieudonne using a non-Lebesgue measurable subset ? с [0,1]
shows that reps need not always exist. However, if ? on (?, ?, ?) is perfect and A
is e.g. for any sub ?-algebra В С A, r.c.p. ?(?, A) given В such that each ?(?, ·) is proper
at each ? ? ?. Blackwell and Ryll-Nardzewski A963) have shown that the exceptional
set ?, in general, cannot be removed even in standard Borel spaces while giving conditions
under which the set N can be removed. For related results see Ramachandran A979, Part I,
Chapter 4). It is not known whether a probability space (X, A, P) in which there is a r.c.p.
given В for every sub ?-algebra ? of Л is necessarily perfect.
For sufficient conditions of a purely measure-theoretic nature for existence of r.c.p.
see Pfanzagl A969), Musiat A972). For results concerning r.c.p. in Doob's sense see
Ramachandran A979, 1981). For further generalizations of the notion of a r.c.p. and their
connection to perfect measures, see Faden A985), Sokal A981).
7. Independence and Blackwell spaces
There are two definitions of independence of two real random variables - one due to
Steinhaus and the other due to Kolmogorov. An example due to Doob A948), Jessen
A948) using a nonperfect probability space shows that Kolmogorov's definition is more
restrictive; however, in perfect probability spaces the two notions are equivalent (see
appendix by Doob in Gnedenko and Kolmogorov A954)). Recent work concerning the
necessity of perfectness for the equivalence of the two notions of independence leads to the
study of measures on Blackwell spaces (see Ramachandran A979), Shorn A987a, 1987b)).
Let (X, A) be a measurable space where A is separable. (X, A) is said to be a Blackwell
space if, for any separable sub ?-algebra A\ of A, we have A\ = A. (X, A) is said to
be strongly Blackwell space if any two e.g. sub ?-algebras of A with the same atoms
are identical. The two definitions of independence are equivalent in strongly Blackwell
spaces. Ryll-Nardzewski has shown the existence of a non-Lebesgue measurable subset
X* С [0, 1] with ?*(?*) = 1 such that (X*,B[0,\] ? ?*) is strongly Blackwell; this
nonperfect space shows that perfectness is not necessary for the equivalence of the two
definitions of independence.
A probability space (X, A, P) is called an independence space if the two definitions
of independence are equivalent in (?,?, ?). A measurable space (X, A) is called a
universal independence space if for every probability ? on (X, A), the space (X, A, P)
is an independence space. Strongly Blackwell spaces are universal independence spaces.
Shortt and Bhaskara Rao A987) have shown, under (CH), the existence of a Blackwell
space which is not a universal independence space answering a question of Ramachandran
A975, P930) in the negative.
A probability space is called almost surely (as.) Blackwell (respectively as. strongly
Blackwell) if for some ? e A with P(Y) = 1, the space (Y, А П Y) is Blackwell
(respectively strongly Blackwell).
Ramachandran A979) has shown that separable perfect spaces are as. strongly
Blackwell and that as. strongly Blackwell spaces are independence spaces. Shortt
A987a, 1987b) has shown, under (CH), the existence of a nonperfect independence
space which is not even as. Blackwell (answering P931 in Ramachandran A975) in the

780
D. Ramachandran
negative). Note that concepts such as universally measurable space, strongly Blackwell
space, universal independence space and Blackwell space are set-theoretic concepts while
concepts such as perfect probability space, as. strongly Blackwell space, independence
space and a.s. Blackwell space are measure-theoretic concepts. For implications involving
the above mentioned, see papers by Shorn A987a, 1987b) and Shortt and Bhaskara Rao
A987). Imposing set-theoretic conditions like the strong Blackwell property does not avoid
measure theoretic pathologies such as being a non-independence space. Shortt A984/85)
has shown the existence of a strongly Blackwell space in which there is no r.c.p. given a
e.g. sub-?-algebra. These findings reinforce the fact that perfect probability spaces are the
best candidates for providing a technically most pleasing model for probabilistic measure
theory.
8. Standard measure spaces
A probability space (?, ?, P) is called a standard measure space (separable measure
space) if there exists X\ e A with P(Xi) = 1 such that (X\, А П X\) is a standard
Borel space (separable space). Lebesgue spaces studied by Rohlin A949) are completion
of standard measure spaces and standard measure spaces are separable measure spaces
where ? is a perfect measure. Every standard measure space with a nonatomic measure
is almost isomorphic to ([0,1],#[?,?],?). For detailed classical results about standard
measure spaces we refer to Rohlin A949) and Ramachandran A979).
We mention the following results:
Let (X, A, P) be a standard measure space and let ? с Л be a e.g. sub ?-algebra. A set
A e A is called a measurable partial selector for В if ? ? ? contains at most one point for
every ?-atom B.
THEOREM 8.1 (Rohlin). There is a decomposition ofX of the form X = Mo U M\ U ¦ ¦ ¦ U
M„ U ¦ ¦ ¦ , where
(i) M„ is a measurable partial selector for В for every ? ^ 1;
(ii) M„ is a set of maximal measure among all measurable partial selectors for В which
are subsets of (U?=i ?i)c for every ? ^ I; and
(iii) Mq contains no partial selector for В of positive measure.
A decomposition of X as in Theorem 8.1 is called a maximal decomposition. Since ?
is perfect there is a r.c.p. ?(?, A) on XxA given В which is almost everywhere proper; it
can be shown that ?(?, -)\м0 is nonatomic a.s. [??] andM(;c, Mn) ^ ?(?, ??+\) as. [??]
for every ? > 1. The following theorem summarizes Rohlin's results on the existence of
independent complements in standard measure spaces.
THEOREM 8.2. Let (X, A, P) be a standard measure space. Let В С A be a e.g. sub-
?-algebra and let X = U/^=o Mn be a maximal decomposition. Then the following are
equivalent:
(i) В has an independent complement which is e.g. i.e., there exists a e.g.
sub-?-algebra B* of A such that В and В* are independent with ?({?, ?*}) = A as. [P].

Perfect measures and related topics
781
(ii) ?(?, M„) = const, as. [??] for each ? ^ 1.
(iii) Every maximal decomposition of X consists of sets ?-independent of В.
(iv) There is a maximal decomposition ofX consisting of sets ?-independent of В.
Corollary 8.1. Let (X, А, Р) be a standard measure space. Let В С Л be a e.g. sub
?-algebra such that [Pg]-almost all measures ?(?, ¦) are continuous where ?(?, A) is any
version of the re.p. given B. Then В has an independent complement which is e.g. What is
more, there is an almost isomorphism of(X, A, P) with the unit square equipped with the
planar Lebesgue measure under which В can be identified with the ?-algebra of vertical
cylinders.
For another proof of Rohlin's theorem on independent complements see Ershov A977).
For classification of e.g. sub ?-algebras of a standard measure space based on the structure
of the corresponding reps see Ramachandran A979).
9. Mixtures of perfect measures
Let (X, A) and (У, В) be two measurable spaces and let ?(?, ?) be a transition probability
on ? ? ?. Let ? be a measure on (X, A). The measure ?? on В defined by
??(?)= ? ?(?,?)??, in В е В
is called the ?-mixture of ?(?, ¦); ? is called the mixing measure, ?(?, ¦) are called
mixand measures and ?? is called mixture measure. Rodine A966) initiated the study
of the role of perfectness in the mixture problem and conjectured that a perfect mixture
of perfect measures is perfect. Ramachandran A974, 1979a, 1979b) (see these papers for
details) showed that the conjecture is false and that all the possible cases can arise in the
mixture problem with respect to perfectness of measures and gave sufficient conditions
for perfect mixtures of perfect measures to be perfect. Since perfectness and compactness
are equivalent, all the possible cases involving compact measures can arise in the mixture
problem. Ramachandran A979a) also showed that a perfect mixture (and hence a compact
mixture) of discrete measures is perfect. The following question is still open: Is a compact
mixture of discrete measures compact!
10. Disintegrations
When working with ?-algebras which are not countably generated reps need not exist
even when the space is perfect. Hence we consider the notion of disintegration (see Valadier
A973), Edgar A975), Maharam A975), Pachl A978)):
Definition 10.1. Let (X,A) and (У, B) be two measurable spaces and let R be a
probability on(X xY,A®B). Let Q be the marginal of R on (У, В). Suppose that for each

782
D. Ramachandran
у e У there is a ?-algebra Ay С AonX and a probability P{y, ¦) on (X, Ay) satisfying
the conditions:
(Dl) For each A e A there is NA e В with Q(NA) = 0 such that A e Ay for every
у ?Na and the function P(-, A) on (У — Мд) is ? П (У — Мд)-measurable; and
(D2) For every AeAandB eB
f P(y,A)dQ(y) = R(AxB)
Jb
(in view of (Dl) this integral is well defined).
The family {(Ay, P(y, )}уеЛ is then called a ^-disintegration of R.
THEOREM 10.1 (Pachl A978)). A probability space (?,?, ?) is compact iff for every
complete probability space (У, B, Q) and for every joint probability R on (? ? ?,?<8> ?)
with marginals ? and Q there exists a Q-disintegration of R.
Pachl's characterization of compactness was used to establish properties C7-C10.
Ramachandran A979) introduced a more general notion of disintegration of a probability
space.
Definition 10.2. Let (Z,C, P) be a probability space and let В be a sub ?-algebra of A.
Suppose that for each ? e ? there is a sub ?-algebra Cz of С and a probability P(z, ¦) on
(Z, Cz) satisfying the following properties;
(Dl') For each С eC there exists Nc e В with P(Nc) = 0 such that С eCz for every
z?Nc and the function P(-, C) on (Z - Nc) is ? П (Z - Nc)-measurable; and
(D2') For every В е В and С е С,
I P(z,C)dP = P(Br\C)
Jb
(in view of (Dl') this integral is well-defined).
The family {(Cz, (??, )}zez is then called a ?-disintegrationof (Z,C, P).
The above definition generalizes the notion of a r.c.p. to ?-algebras that are not
necessarily e.g.
If (Z, C, P) С (У, В, Q) (as a thick subspace) then let C* = ?({?, ?}) and let P* on
(У, C*) be defined by P*(C*) = P(C* ? ?), С* е С*.
Definition 10.3. We say that a probability space (Z,C,P) is disintegrable if for every
(У, B, Q) such that (Z, С, Р) С (У, В, Q) and for every sub ?-algebra B* of C* there is a
^-disintegration of (У, С*, />*).
We have the following unified version of Pachl's results:

Perfect measures and related topics
783
THEOREM 10.2 (RamachandranA979)). For a probability space (Z,C, P) the following
are equivalent:
(i) (Z, С, Р) is compact.
(ii) (Z, C, P) is disintegrable.
(iii) For every compact probability space (Y,B, Q) such that (Z, С, ?) С (?, ?, Q) and
for every sub-?-algebra B* ofC* there is a B*-disintegration of(Y, С*, Р*).
(iv) For some compact probability space (Y,B, Q) such that (Z,C, P) С (?, ?, Q)
there is а В-disintegration of(Y,C*, P*).
Blackwell and Maitra A984) have introduced the notion of factorization; in
Definition 10.1, if Ay = Л and P(y, ¦) is a charge on (X, A) for each у е Y, we say that
{Р(У> )¦ У е У} is a <2-factorization of P.
Analogous to Theorem 10.1, Maitra and Ramakrishnan A988) and have established the
following characterization of perfectness.
THEOREM 10.3 (Maitra and Ramakrishnan). Let (X, A, P) be a probability space where
A is ? ?- gene rated. The following are equivalent:
(i) ? is perfect.
(ii) For any probability space (У, B, Q) and for any measure R on (? ? ?, ? <g> В)
with marginals ? and Q, there is a Q-factorization of R.
(iii) For any probability space (Y, B, Q) with В e.g. and for any measure R on
(? ? Y,A®B) with the marginals ? and Q, there is a Q-factorization of R.
(iv) For any probability Q on (91, By) and any measure R on (? ? Ш, А х Вщ) with
marginals ? and Q, there is a Q-factorization of R.
While (iv) => (i) is true for any ? -algebra A, it is not known whether the reverse
implication holds if ? is not ? ?-generated.
Adamski A986) has shown that for any e.g. ?-algebra A (i) ? (?): For any probability
space (У, B, Q) and for any measure R on {? ? У, Л® В) with marginals P and Q, there
is ? ? -additive Q-factorization ofR.
For additional results in these directions refer to Adamski A986), Blackwell and Maitra
A984), Pachl A978, 1979, 1981), Ramachandran A979), Remy A988).
References
Adamski, W. A986), Factorization of measures and perfection, Proc. Amer. Math. Soc. 97, 30-32.
Aniszczyk, B. A986), A note on "Two classes of measures" by J.K. Pachl, Colloq. Math. 50, 231-232.
Arveson, W. A974), Operator algebras and invariant subspaces, Ann. Math. 100, 433-533.
Banach, S. A948), Onmeasures in independent fields, Studia Math. 10, 159-177.
Benes, V. and Stepan, J. (eds) A997), Distributions with Fixed Marginals and Moment Problems, Kluwer,
Dordrecht.
Bhaskara Rao, K.P.S. and Rao, B.V. A981), Borel spaces, Dissertationes Math. (Rozprawy Mat.) 190, 1-63.
Billingsley, P. A986), Probability and Measure, 2nd edn, Wiley, New York.
Bjomsson, O.J. A980), A note on the characterization of standard Borel spaces, Math. Scand. 47, 135-136.
Blackwell, D. A956), On a class of probability spaces, Proc. Third Berkeley Symp. Math. Stat. Probab., Vol. 2,
1-6.

784
D. Ramachandran
Blackwell, D. and Maitra, A. A984), Factorization of probability measures and absolutely measurable sets, Proc.
Amer. Math. Soc. 92, 251-254.
Blackwell, D. and Ryll-Nardzewski, C. A963), Nonexistence of everywhere proper conditional distributions,
Ann. Math. Stat. 34, 223-225.
Cohn, D.L. A980), Measure Theory, Birkhauser, Boston.
Dall'Aglio, G., Kotz, S. and Salinetti, G. (eds) A991), Advances in Probability Distributions with Given
Marginals, Kluwer, Dordrecht.
Darst, R.B. A969), Properties ofLusin sets with applications to probability, Proc. Amer. Math. Soc. 20, 348-350.
Darst, R.B. A971), On universal measurability and perfect probability, Ann. Math. Stat. 42, 352-354.
Dekiert, M. A993), Two results for monocompact measures, Manuscripta Math. 80, 339-346.
Doob, J.L. A948), On a problem of Marczewski, Colloq. Math. 1, 216-217.
Edgar, G.A. A975), Disintegration of measures and the vector valued Radon—Nikodym theorem, Duke Math J.
42, 447^450.
Erochin, V.D. A961), Zametka к teorii mery, Uspekhi Mat. Nauk 16, 175-180.
Ershov, M.P. A977), On Rohlin's theorem of the independent complement, Uspekhi Mat. Nauk 32, 187-188 (in
Russian).
Faden, A. A985), The existence of regular conditional probabilities: necessary and sufficient conditions, Ann.
Probab. 13, 288-298.
Gardner, R.J. A982), The regularity of Borel measures, Oberwolfach Measure Theory Conference Proceedings
1981, Lecture Notes in Math., Vol. 945, Springer, Berlin, 42-100.
Gnedenko, B.V. and Kolmogorov, A.N. A954), Limit Distributions for Sums of Independent Random Variables,
Addison-Wesley, Cambridge.
Gretsky, N.E., Ostroy, J.M. and Zame, W.R. A992), The nonatomic assignment model, Econ. Theory 2, 103-127.
Hackenbroch, W. A992), Conditionally multiplicative simultaneous extension of measures. Measure Theory
Proceedings, Oberwolfach 1990, D. Kolzow, D. Maharam-Stone, S. Graf, eds, Supplemento ai Rendiconti
del Circolo Matematico di Palermo, Series II 28, 49-58.
Halmos, PR. A950), Measure Theory, Von Nostrand, New York.
Hansel, G. and Troallic, J.P A978), Mesures marginales et theoreme de Ford-Fulkerson, Z. Wahrscheinlich-
keitsth. verw. Geb. 43, 245-251.
Hansel, G. and Troallic, J.P. A986), Surle probleme des marges, Probab. Theory Related Fields 71, 357-366.
Haydon, R. and Shulman, V A996), On a measure-theoretic problem of Arveson, Proc. Amer. Math. Soc. 124,
497-503.
Hoffmann-J0rgensen, J. A970), The Theory of Analytic Spaces, Various Publications Series, No. 10, Aarhus
Universitet Matematisk Institut, Aarhus.
Hoffmann-J0rgensen, J. A987), The general marginal problem. Functional Analysis II, S. Kurepa, H. Kraljenic
and D. Butkovic, eds, Lecture Notes in Math., Vol. 1242, Springer, Berlin, 77-367.
Jasinski, J. and Weiss, T. A991), Sierpinski sets and strong first category, Proc. Amer. Math. Soc. Ill, 235-238.
Jessen, B. A948), On two notions of independent functions, Colloq. Math. 1, 214-215.
Kallianpur, G. A959), A note on perfect probability, Ann. Math. Stat. 30, 169-172.
Kallianpur, G. and Ramachandran, D. A983), On the splicing of measures, Ann. Probab. 11, 819-822.
Kantorovich, L.V A940), A new method of solving some classes of extremal problems, Comptes Rendus
(Doklady) Acad. Sci. USSR 28, 211-214.
Kantorovich, L.V. A942), On the translocation of masses, Doklady Akad. Nauk SSSR 37, 199-201. Translated
into English and reprinted in Manage. Sci. 5 A958), 1^4.
Kellerer, H.G. A964a), Masstheoretische Marginalprobleme, Math. Ann. 153, 168-198.
Kellerer, H.G. A964b), Verteilungsfunktionen mit gegebenen Marginalverteilungen, Z. Wahrscheinlichkeitsth.
verw. Geb. 3, 247-270.
Kellerer, H.G. A984), Duality theorems for marginal problems, Z. Wahrscheinlichkeitsth. verw. Geb. 67, 399-
432.
Koumoullis, G. A981), Onperfect measures. Trans. Amer. Math. Soc. 264, 521-537.
Kuratowski, K. A966), Topology I, Academic Press, New York.
Lembcke, J. A982), On simultaneous preimage measures on Hausdorff spaces. Measure Theory Proceedings,
Oberwolfach 1981, D. Kolzow, D. Maharam-Stone, eds, Lecture Notes in Math., Vol. 945, Springer, Berlin,
110-115.

Perfect measures and related topics
785
Levin, V.I. A984), The mass transfer problem in topological space and probability measures on the product of
two spaces with given marginal measures, Soviet Math. Doklady 29, 638-643.
Maharam, D. A971), Consistent extensions of linear junctionals and of probability measures, Proc. Sixth
Berkeley Symp. on Math. Stat, and Probab., Vol. 2, 127-147.
Maharam, D. A975), Strict disintegration of measures, Z. Wahrscheinlichkeitsth. verw. Geb. 32, 73-79.
Maharam, D. A978), An example of tail fields, Oberwolfach Measure Theory Conference, 1977, Lecture Notes
in Math., Vol. 695, Springer, Berlin, 151-152.
Maitra, A. and Ramakrishnan, S. A988), Factorization of measures and normal conditional distributions, Proc.
Amer. Math. Soc. 103, 1259-1267.
Mallory, D. A975), Extension of set functions to measures, Canad. Math. Bull. 18, 547-553.
Marczewski, E. (Szpilrajn) A938), The characteristic function of a sequence of sets and some of its applications.
Fund. Math. 31, 207-223.
Marczewski, E. A948), Ensembles independants et leurs applications ? la theorie de la mesure. Fund. Math. 35,
13-28.
Marczewski, E. and Ryll-Nardzewski, C. A953), Remarks on compactness and non-direct products of measures.
Fund. Math. 40, 165-170.
Miller, A.W. A984), Special subsets of the real line. Handbook of Set-Theoretic Topology, North-Holland,
Amsterdam, 201-233.
Monge, G. A781), Memoire sur la theorie des deblais et ramblais, Mem. Math. Phys. Acad. Roy. Sci. Paris,
666-704.
Musial, K. A972), Existence of proper conditional probabilities, Z. Wahrscheinlichkeitsth. verw. Geb. 22, 8-12.
Musial, K. A976), Inheritness of compactness and perfectness of measures by thick subsets, Oberwolfach
Measure Theory Conference 1975, Lecture Notes in Math., Vol. 541, Springer, Berlin, 31—42.
Musial, K. A980), Projective limits of perfect measure spaces, Fund. Math. 110, 163-189.
Nadkarni, M.G. and Ramachandran, D. A978), On a question of D. Maharam concerning tail fields, Sankhya,
Series A 40, 313-318.
Neveu, J. A965), Mathematical Foundations of the Calculus of Probability, Holden Day, London.
Pachl, J. A975), Every weakly compact probability is compact. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom.
Phys. 23, 401^405.
Pachl, J. A978), Disintegration and compact measures. Math. Scand. 43, 157-168.
Pachl, J. A979), Two classes of measures, Colloq. Math. 42, 331-340.
Pachl, J. A981), Correction to the paper "Two classes of measures", Colloq. Math. 45, 331-333.
Pfanzagl, J. A969), On the existence of regular conditional probabilities, Z. Wahrscheinlichkeitsth. verw. Geb.
11, 244-256.
Plebanek, G. A989), Measures on two dimensional products, Mathematika 36, 253-258.
Rachev, ST. A991), Probability Metrics and the Stability of Stochastic Models, Wiley, New York.
Rachev, S.T. and Ruschendorf, L. A998), Mass Transportation Problems. Part I: Theory; Part IT. Applications,
Springer- Verlag, New York.
Ramachandran, D. A974), Mixtures of perfect probability measures, Ann. Probab. 2, 495-500.
Ramachandran, D. A975), On the two definitions of independence, Colloq. Math. 32, 227-231.
Ramachandran, D. A979), Perfect Measures, Vols 1, 2, ISI - Macmillan Lecture Notes Series, Vols 5, 7,
Macmillan, New Delhi.
Ramachandran, D. A979a), Perfect mixtures of perfect measures, Ann. Probab. 7, 444—452.
Ramachandran, D. A979b), Existence of independent complements in regular conditional probability spaces,
Ann. Probab. 7, 433^*43.
Ramachandran, D. A981), A note on regular conditional probabilities in Doob 's sense, Ann. Probab. 9, 907-908.
Ramachandran, D. A996), Marginal problem in arbitrary product spaces. Distributions with Fixed Marginals
and Related Topics, L. Ruschendorf et al., eds, IMS Lecture Notes - Monograph Series, Vol. 28, 260-272.
Ramachandran, D. and Ruschendorf, L. A995), A General Duality Theorem for Marginal Problems, Probab.
Theory Related Fields 101, 311-319.
Ramachandran, D. and Ruschendorf, L. A996), Duality and perfect probability spaces, Proc. Amer. Math. Soc.
124, 2223-2228.

786
D. Ramachandran
Ramachandran, D. and Ruschendorf, L. A997), Duality theorems for assignments with upper bounds.
Distributions with Fixed Marginals and Moment Problems, V. Benes and J. Stepan, eds, Kluwer, Dordrecht,
283-290.
Ramachandran, D. and Ruschendorf, L. B000), On the Monge—Kantorovich duality theorem. Theory Probab.
Appl. 45, 403^409.
Remy, M. A988), Disintegration and perfectness of measure spaces, Manuscripta Math. 62, 277-296.
Rodine, R.H. A966), Perfect probability measures and regular conditional probabilities, Ann. Math. Statist. 37,
1273-1278.
Rohlin, V.A. A949), On the fundamental ideas of measure theory. Mat. Sb. 25 F7), 107-150 (in Russian). Amer.
Math. Soc. Transl. Series 1, Vol. 10, 1-54.
Ruschendorf, L. A981), Sharpness of Frechet-bounds, Z. Wahrscheinlichkeitsfh. verw. Geb. 57, 293-302.
Ruschendorf, L., Schweizer, B. and Taylor, M. (eds) A996), Distributions with Fixed Marginals and Related
Topics, IMS Lecture Notes - Monograph Series, Vol. 28.
Ryll-Nardzewski, С A953), On quasi-compact measures, Fund. Math. 40, 125-130.
Sazonov, V A962), On perfect measures, Izv. Akad. Nauk SSSR Ser. Mat. 26, 391—414 (in Russian). Amer.
Math. Soc. Transl. Series 2, Vol. 48, 229-254.
Schmidt, K.D. and Waldschaks, G. A991), Common extensions of positive vector measures, Portugalia Math. 20,
155-164.
Shortt, R.M. A983), Strassen's marginal problem in two or more dimensions, Z. Wahrscheinlichkeitsth. verw.
Geb. 64, 313-325.
Shortt, R.M. A984), Universally measurable spaces: an invariance theorem and diverse characterizations. Fund.
Math. 121, 169-176.
Shortt, R.M. A984/85), Products of Blackwell spaces and regular conditional probabilities. Real Anal. Exchange
10,31^41.
Shortt, R.M. A986), Sets with no uncountable Blackwell subsets, Czechoslovak Math. J. 37 A12), 320-322.
Shortt, R.M. A987a), Borel dense Blackwell spaces are strongly Blackwell, Colloq. Math. 53, 35—41.
Shortt, R.M. A987b), Notions of independence for random variables, Probab. Math. Stat. 8, 81-88.
Shortt, R.M. and Bhaskara Rao, K.P.S. A987), Generalized Lusin sets with the Blackwell property. Fund. Math.
127, 9-39.
Sierpinski, W. A956), Hypothese du continu, 2nd edn, Chelsea, New York.
Sokal, A.D. A981), Existence of compatible families of proper regular conditional probabilities, Z.
Wahrscheinlichkeitsth. verw. Geb. 56, 537-548.
Sparre-Anderson, E. and Jessen, B. A948), On the introduction of measures in infinite product sets, Danske Vid.
Selbskab. Mat. Fys. Medd. 25 D), 1-7.
Strassen, V A965), The existence of probability measures with given marginals, Ann. Math. Stat. 36, 423^439.
Stroock, D. A976), Some comments on independent ?-algebras, Colloq. Math. 35, 7-13.
Sudakov, V.N. A975), Measures on subsets of direct products, J. Soviet Math. 3, 825-839.
Tops0e, F. A979), Approximating pavings and construction of measures, Colloq. Math 42, 377-385.
Valadier, M. A973), Disintegration d'une mesure sur un product, C. R. Acad. Sci. Paris Ser. A 276, 33-35.

CHAPTER 19
Riesz Spaces and Ideals of Measurable Functions
Martin Vath
Department of Mathematics, University of Wiirzburg, Am Hubland, D-97074 Wiirzburg, Germany
E-mail: vaeth@mathematik.uni-wuerzburg.de
Contents
1. Preliminaries 789
2. Riesz spaces of measurable functions 790
2.1. Riesz spaces 790
2.2. Dedekind completeness and support 792
2.3. Order convergence and topology 796
2.4. Bibliographical remarks 797
3. Locally solid ideals of measurable functions 798
3.1. Ideals and bands in Riesz spaces 798
3.2. Ideals of measurable functions 800
3.3. Comparison of various sorts of convergence 803
3.4. Bibliographical remarks 805
4. Ideal spaces 805
4.1. Preideal spaces 805
4.2. Fatou property and perfectness 808
4.3. Completeness 810
4.4. Convergence theorems 811
4.5. Bibliographical remarks 813
5. Order duals 815
5.1. The associate space 815
5.2. Applications to ideal spaces on product measures 817
5.3. Order duals of a Riesz space 818
5.4. Duals of ideals of measurable functions 820
5.5. Bibliographical remarks 822
References 822
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
787

This page intentionally left blank

Riesz spaces and ideals of measurable functions
789
1. Preliminaries
Throughout this chapter, (S, ?, ?) denotes a measure space with a positive measure ?. To
simplify notation, we will assume that this measure space is complete, i.e., that subsets of
null sets are again null sets.
It will be convenient to consider functions with values in У where У is a Banach space,
or ? = Ш, or where У is a Banach space with an additional element oo. By \y\, we
denote the norm respectively absolute value of у e ?. Of course, we put | ± oo| := oo.
We equip ? with the canonical uniform structure, i.e., the restriction to bounded sets is
the uniform structure induced by the norm, and y„ ->· oo if and only if \y„\ —>¦ oo. Since
this uniform structure has a countable base, the classical metrization theorems imply that
it is induced by a metric d. If У is a Banach space, we may choose d(x, y) = \x — y\, and
for У = R, we may choose d(x. y) = | arctan.v - arctan;y| (with the natural conventions
for the case x, у = ±oo). Also if У is a Banach space with an additional element oo, an
"explicit" formula for d can be given by means of the stereographic projection (see, e.g.,
Proposition 2.2.1 of M. Vath A997)):
ж л уУA + Ы2)-уA + М2I2 + (И2-1у12J , , .
d(x,y):= г (?, у ? ос),
A + И2)A + Ы2)
d(x, oo) :=d(oo, x) : = ^ (jc^oo), d(oo, oo) :=0.
1 + |jc|2 ^
Since we are not only interested in integrals, it will be convenient to enlarge the class of
measurable functions slightly compared to the definition of Chapter 17: We call a function
?: S —>¦ ? measurable if the restriction to each finite set is ?-measurable in the sense
of Chapter 17, i.e., if for each set ? e ? with ?(?) < oo the restriction x\e can be
approximated almost everywhere (with respect to d) by a sequence of iT-step functions
(for the terminology see Chapter 17); it may always be arranged that these approximating
functions attain only finite values. For functions with ?-finite support the two definitions
are equivalent, of course. Note that (Bochner) integrable functions have automatically
?-finite support (because {s: \x(s)\ ^ и-1} must have finite measure), and so this
difference plays no role in the context of Chapter 17.
If two measurable functions x,y:S-*Y coincide almost everywhere, we tacitly
identify them unless something else is said. By ?ul((S, ?, ?), У), we denote the set of
all (equivalence classes of) such functions. For brevity, we usually write just S in place
of (S, ?, ?) and do not mention У in case У = Ш. In particular, Wl(S) is the same as
m((S, ?, ?),?).
Recall that if S is ?-finite, then there is a normalized measure v, i.e., (S, ?, v) is a
measure space with the same null sets as (S, ?, ?) but v(S) = 1 (in particular, v(S) < oo).
In this case, we consider on X с 9ЛE', У) the metric
p(x,y):= inf (m + v(\s: d(x(s),y(s)) > m\))
0<m<x

790
?. Vath
or
f d(x(s),y(s))
p(x,y): = I dv(s).
Jsd(x(s),y(s))+\
We say that a sequence xn e Wl(S, Y) converges in measure to ?: S ->· ? on a set ? e ?
if for any ? > 0 the relation m*({s e ?: d(jt„0), ;c(i)) > ?}) -> 0 holds where ?* denotes
the outer measure.
PROPOSITION 1. Let S be ?-finite. For ?,,,? e 9K(S, ?) we /lave ? (?,,,*) -> 0
(p(x„, x) —> 0) if and only if xn -^ ? on each set of finite ?-measure. In particular, ?
and ? induce the same topology on Wl(S, Y), and this topology does not depend on the
particular choice of v.
The following classical results carry over.
THEOREM 1 (Egorov). Let S be ?-finite. If xn e VK(S, Y) converges almost everywhere
to ? : S —>¦ ? (with respect to d), then ? e 9JiE", Y), and there is an increasing sequence
E\ с Е2 с ··· with ?(??() < oo and ?E \ (jEk) =0 such that d(x„(s),x(s)) -+ 0
uniformly on each E^. In particular, p(x„,x) —> 0 (and p(x„, x) —> 0).
THEOREM 2 (Riesz). If xn e Wl(S, Y) converges in measure to some x, then there is a
subsequence which converges to ? almost everywhere (with respect to d). If S is ?-finite,
the same conclusion holds if x„ —> ? in measure on each set of finite measure (i.e., if
p(x„ ,x)^0or p(x„ ,*)_>()).
We will mark all places explicitly where the axiom of choice is used in an uncountable
form which goes beyond the so-called axiom of dependent choices.
2. Riesz spaces of measurable functions
2.1. Riesz spaces
Definition 1. A partially ordered vector space is a real vector space X endowed with a
partial order ^ with the following properties:
A) ? ^ у implies ? + ? ^ у + ? for all ? e X.
B) ? > 0 implies ?? > 0 for all ? > 0.
The set of all leX with ? ^ 0 is the positive cone and is denoted by X+.
Of course, ? ^ у if and only if у — ? e X+. The notion "positive cone" is explained by
the fact that a subset С с X is a positive cone for some order on X which turns X into a
partially ordered vector space if and only if С is a cone, i.e.,
A) С is convex with С + С с С, and
B) СП(-С) = {0).

Riesz spaces and ideals of measurable functions
791
Definition 2. An element ? of a partially ordered set X is an upper bound of a nonempty
subset ? с X if у ^ ? for all у e ?. A /ower footmii is defined analogously. If ?? has an
upper/lower bound, we call it order bounded from above/below. We call ? order bounded
if it is bounded from above and below.
If ? is an upper bound for ? and simultaneously a lower bound for the set of all upper
bounds of ?, then ? is called the supremum of ?, and is denoted by sup ?. The infimum
inf ? is defined analogously.
If supAi or infAi exists, then it is uniquely determined.
Definition 3. A partially ordered space X is called a lattice, if ? ? у : = sup{*, у] and
? л у := inf{j:, у) exists for any Jt, у е X. A Riesz space is a partially ordered vector space
which is simultaneously a lattice.
A partially ordered vector space X is a Riesz space if and only if sup{;c, y) exists for any
x,y e X. Indeed, then inf{;c, y) automatically exists and is the same as — sup{— x, —y).
Definition 4. Let X be a Riesz space. Then the Riesz operations are defined by
x+ := ? vO, x~ := (-x) лО, |jc | := (—?) ? ? for each ? e X.
The lattice operations can be obtained by the Riesz operations:
Proposition 2. We always have
0^л±^|л|, x =x+ — x~ and \x\=x+ + x~.
Moreover,
xv у = -(x + y + \x-y\) and хлу = -(х + у-\х-у\).
The order of a Riesz space X is completely determined by each of the three Riesz
operations. A linear subspace U с X with the inherited order is a Riesz space if it is
invariant under one (and thus all) of the three Riesz operations.
Definition 5. For a sequence x„ in a partially ordered space X, we use the notation
x„ ? ? if ? ? ^ X2 ^ · · · and ? = inf{jt|, xi,...}. x„ | ? is defined analogously.
For later usage, we already mention a related more general notion:
Definition 6. A set ? с X is called directed upwards if for any x, у еМ there is some
? e ? with ? ^ ? and ? ^ y. If ?? is directed upwards and jc = sup ? exists, then we write
? 1 ?. ? ? ? is defined analogously.
To compare the two definitions, the following notion is useful.

792
?. Vath
DEnMTlON 7. A partially ordered space X is called order separable if any ? с X which
posses a supremum or infimum contains an (at most) countable subset which has the same
supremum respectively infimum.
PROPOSITION 3. Ifx„ | x, then {x„: n] | x. Conversely, if ? | ? and either X is order
separable or ? is countable, then ? contains a sequence with xn | x. A corresponding
result holds for xn l x.
We are mainly interested in the lattice Wl(S, R). Clearly, 9JiE", R) is a lattice with
respect to the order
? ? у «=> x(s) ^ y(s) for almost all s e S.
Unless something else is said, we always consider this order. Note that we have xn | ? if
and only if x„(s) | x{s) for almost all s e 5". The subspace W(S) = Wl(S, R) is a Riesz
space whose order is determined by the Riesz operation \x\(s) = \x(s)\-
In view of Proposition 2, we call a subset X с Wl(S) a Riesz space of measurable
functions (with respect to our canonical order) if it is a subspace such that ? e X implies
|*| eX where \x\(s):= \x(s)\.
Example 1. If5" = [0, 1] (with the Lebesgue measure), then C([0, 1]) is a Riesz space of
measurable functions in the above sense while C1 ([0, 1]) is not. Nevertheless, C1 ([0, 1])
is a Riesz space with respect to a different order (defined, e.g., by ? ^ у «=> *@) ^ y@)
and x'(t) ? y'(t) for all t e [0, 1]).
It is noteworthy that any Riesz space X of measurable functions (with the canonical
order) is Archimedean, i.e., for any ? e X+ we have /i~'jc \. 0. This excludes many
pathologies arising in the general theory of Riesz spaces.
2.2. Dedekind completeness and support
We intend now to study certain order properties of Wl(S) in connection with the algebra of
measurable sets: Let ^(S) = *B(S, ?, ?) denote the algebra of all measurable subsets of
S where we identify two sets when they differ only by a null set. Then *B(S) becomes a
lattice with respect to the order
Z)<C? «=> D с ? UN for some null set N. A)
Note that D„ | D in 55(S1) means that up to null sets the relations D\ с ZJ <? · ¦ · and
D = \JD» hold.
Dehnition 8. Given ? e 9JiE"), the support of ? (denoted by supp*) is defined by
supp? := {s e 5": x(s) ? 0). Given some ? с Wl(S, ?, ?), we say that ? has a support
if there is some ? & ? with the following properties:

Riesz spaces and ideals of measurable functions
793
A) For any ? e ? the set supp ? \ ? does not contain a set of positive measure.
B) For each OC? with ?(?)) > 0 there is some ? e ? such that D П supp ? contains
a set of positive measure.
In this case, the set E, considered as an element of 25E), is called the support of ?
(denoted by suppAi).
PROPOSITION 4. ? has a support if and only if the set
(De!8(S): D ^ supp ? for some ? eM) B)
has a supremum in *B(S). In this case, suppAi is this supremum. In particular, suppAi is
uniquely determined.
Recall that the elements of 97tE) and 55E) are actually equivalence classes (because
we decided to ignore null sets). Hence, the union U of the set B) is in general not well-
defined. But even if we fix representative sets D, it may happen that U is non-measurable.
Moreover, even if U is measurable, it need not be the support: it is only clear from the
definition that if U is measurable it is an upper bound, but it need not necessarily be
the least upper bound. Of course, if ? is countable then all these problems do not arise.
However, for uncountable ? the set supp ? need not exist, in general.
Example 2. Let N, So, S\ be pairwise disjoint sets with uncountable So, S\. Put S : =
N U So U S\, and let ? be the system of all ? с 5" with the property that either ? or its
complement Ec is countable. In the first case, we let ?{?) be the number of elements in
E\N, and in the second case ?(?) = oo. Let ? be the set of all measurable functions
which vanish on So- Then a set ? с S satisfies the two properties of Definition 8 if and
only if ? = S\ U No for some No <? N. Since none of these sets belongs to ?, ? does not
have a support.
Example 2 shows additionally that if we would drop the requirement ? e ? in
Definition 8, we had difficulties to speak of the uniqueness of a support (the set N
considered in Example 2 is not a null set but only a so-called local null set, see Theorem 4
below).
Dehmtion 9. We call a lattice X
A) Dedekind complete if for each subset ? ? X which is order bounded from above
(below) sup ? (inf ?) exists;
B) ?-Dedekind complete if for each countable С с X which is order bounded from
above (below) sup С (inf С) exists;
C) super Dedekind complete if for each subset ? с X which is bounded from above
(below) there is some countable С <^M such that sup С = sup ? (inf С = inf ?).
Clearly, each super Dedekind complete space is order separable. _
Note that for ordered vector spaces (and similarly also for <8E) and 971E, K)), it suffices
to verify the properties for the suprema, because inf ? = — sup(—M) (for ? с *8E), we
have similarly inf ? = sup{?': ? e ?}'').

794
?. Vath
The spaces 5?E) and 37iE) are always ?-Dedekind complete (with sup С = {J С
respectively supC(i) = {supjt(r): ? e C}). Note that in general this is not true for all
spaces of measurable functions. For example, C([0, 1]) is not ?-Dedekind complete.
Corollary 1. <8E) is Dedekind complete if and only if any set ? с 971E) has a
support. The space 371E) is (super) Dedekind complete if and only ;/971E, R) is (super)
Dedekind complete.
PROOF. The first statement follows from Proposition 4 (for the converse conclusion
consider for ? с «8E) the set {??: ? e ?} с 371E)). For the connection of_97iE) and
371E, R), consider the set {arctan ? ?: ? e ?] C37iE) in place of ? с 371E, R). D
At first glance, one might suspect that the Dedekind completeness of 371E) implies the
Dedekind completeness of 55E), because for any ? с 55E) the set {??: ? e ?} has
a supremum which is evidently of the form ?^ for some ? с 5. However, ? need not
necessarily be measurable: The relation ?? e 371E) just means that ? П F is measurable
for each set F of finite measure. This difficulty does of course not arise in ?-finite measure
spaces. In view of the difficulties discussed after Proposition 4, it may appear somewhat
surprising that in ? -finite measure spaces there arise even no problems at all concerning
Dedekind completeness:
THEOREM 3. //5 is ?-finite, then 37iE,1) (and thus 971E)) and 55E) are super
Dedekind complete. In particular, any ? -Dedekind complete subspace is super Dedekind
complete (and order separable).
PROOF. In view of the above remarks, it suffices to prove that 371E) is super Dedekind
complete. Passing to a normalized measure, it is no loss of generality to assume ? E) < oo.
Thus, let ? с 971E, IR) be given. By considering {arctanojc: ? e M) in place of M, if
necessary, we may assume that ? has integrable upper and lower bounds. For countable
А с ?, define a number m (A) := {5$???(?)??(?). The supremum s of all these numbers
is finite, and there is a sequence A„ с ? with m(A„) —>¦ s. Putting С := {jA,,, we have
m(C) = s. For any у e ? we have m(C U {>¦}) ^ m(C) < oo which implies у ? sup С.
Hence, sup С is an upper bound of ?. In view of С с ??, it is the least upper bound. D
Corollary 2. Let E, ?1, ?) be an arbitrary measure space, and X — Lp(S) @ < ? <
oo). If ? с X is order bounded in X, then supAi and mi ? exist, and there is some
countable С с X with sup ? = sup С and infM = inf C.
PROOF. Let x\,X2 e L,,E) be upper and lower bounds for M, respectively. Since \xj\p
is integrable, it has ?-finite support. Hence, the support of any ? e ? is contained in the
?-finiteset 5o = suppjf| UsuppjC2-Now apply Theorem 3 with 5o in place of 5. D
It turns out that (super) Dedekind completeness of 55E) or 97tE) are intrinsic properties
of the underlying measure space. Sometimes, measure spaces 5 for which 5SE) is
Dedekind complete are called localizable.

Riesz spaces and ideals of measurable functions
795
The Dedekind completeness of ^(S) and Wl(S) is closely related with the finite measure
property introduced in Chapter 17. The following result gives more insight into this
property.
Recall that a set N с S is called a local null set if N П F is a null set for any set F
of finite measure. Clearly, each null set is a local null set. The converse holds precisely in
spaces with the finite measure property:
THEOREM 4. For a measure space (S, ?, ?) the following statements are equivalent.
A) 5" has the finite measure property.
B) Each set in Ъ{Б) is the supremum of its subsets of finite measure.
C) Each local null set is a null set.
D) Any ? e ? with ?(?) = ?? contains some D e ? with 0 < ?(?) < oo.
In view of the last statement, the finite measure property is sometimes also called finite
subset property.
THEOREM 5. Let S have the finite measure property. If'BiS) is Dedekind complete, then
also 9JiE") is Dedekind complete.
The proof of Theorem 5 makes essential use of the axiom of choice.
Dehnition 10. A measure space (S, ?, ?) has the direct sum property if there exists
a family of pairwise disjoint sets Sa of finite measure with S = (J Sa such that each set
? с 5 of finite measure is (up to a null set) contained in a countable union (J Sa„ ¦
One may equivalently replace the assumption that the sets Sa have finite measure by the
assumption that they are ?-finite.
The direct sum property is a rather straightforward generalization of ?-finite sets. It is
characterized by the fact that functions on S (with values in Banach spaces) are measurable
if and only if they are measurable on each Sa (this is Proposition A. 2.1 of Vath A997)).
THEOREM 6. Let S have the finite subset property. If S has also the direct sum property,
then ^(S) (and thus also ffl(S)) is Dedekind complete.
Also the proof of Theorem 6 relies on the axiom of choice.
There is an important exception of an uncountable subset of 9Л E) where it makes sense
to define a pointwise supremum. It is a rather deep result that this pointwise supremum is
actually the order supremum.
THEOREM 7 (Luxemburg-Gribanov). Let T, S be ?-finite measure spaces, and ? : ? ?
S —> ? be measurable with respect to the product measure. Then for any ? с 9ЛE, ?)
the function
zm@'-= sup I \x(t,s)\\y(s)\ds
yeMJS

796
?. Vath
belongs to 9JiE", Ш) and is the order supremum of the set
Z- z(t) = ? \x(t, s)\\y(s)\ds for some у e ? \.
In view of Theorem 3, the statement of Theorem 7 can be reformulated in the form
that there is some countable С с ? with zm = zc almost everywhere: This is how the
Luxemburg-Gribanov theorem is usually formulated.
2.3. Order convergence and topology
We recall that there is a natural notion of convergence and a topology in Riesz spaces.
DEnNlTlON 11. A sequence xn e X is called order convergent to ? if there is some
sequence y„ \. 0 with \x„ — x\ ?. yn.
A set А с X is called order closed if for any sequence x„ e A which order converges to
some ? e X we have ? e A. The system of all complements of order closed sets determines
the order topology of X.
Clearly, each sequence is order convergent to at most one point. Moreover, the order
topology is in fact a topology on X.
We point out that we define order closed sets only with respect to sequences which is
in accordance with the definitions by Luxemburg and Zaanen A971). Occasionally (e.g.,
by Aliprantis and Burkinshaw A977) or by Fremlin A974)) order closedness is defined
instead with respect to more general directed sets which we will not discuss here. The
difference between these two possible definitions is analogous to the difference between
closed and sequentially closed sets in topological spaces. Although the latter appears more
natural from the viewpoint of a general theory of Riesz spaces, it appears to the author that
the above definition is more appropriate for subsets of Wl(S) (because sequences are more
important here than nets).
Surprisingly, it is not true that order convergence means the same as convergence with
respect to the order topology (also not for the alternative definition of the order topology
mentioned before). In fact, Wl(S) is the most important example for which these notions
do not coincide, as we will see now.
Proposition 5. A sequence xn e 9ft(S) is order convergent to ? e Wl(S) if and only if
x„ —> ? almost everywhere.
The order topology can be described in view of Theorems 1 and 2 as follows.
PROPOSITION 6. Let S be ?-finite. Then the order topology on Wl(S) is induced by ?
(and p). In particular, a sequence converges with respect to the order topology if and only
if it converges in measure on each set of finite measure.

Riesz spaces and ideals of measurable functions
797
One could of course ask whether there exists also a topology on Wl(S) (possibly different
from the order topology) such that convergence of a sequence with respect to this topology
is the same as convergence of that sequence almost everywhere. The following observation
implies that such a topology does usually not exist. Moreover, if any such topology exists,
then the order topology has the same property (if S is ?-finite).
PROPOSITION 7. Let S be ?-finite and such that convergence almost everywhere is not
the same as convergence in measure on sets of finite measure (i.e., there is a sequence
x„ e Wl(S) which does not converge almost everywhere but which on each set of finite
measure converges in measure to some ? e ffl(S)).
Then there is no topology on Wl(S) such that convergence of sequences with respect to
this topology is the same as convergence almost everywhere.
PROOF. Assume by contradiction that such a topology exists. But then x„ ->· ? with
respect to this topology and thus almost everywhere. Indeed, otherwise we could find a
neighborhood of ? which does not contain infinitely many x„k . Since a subsequence of xHi
converges to ? almost everywhere and thus with respect to the assumed topology, this is a
contradiction. ?
Any ?-finite measure space which contains a set ? of positive measure with no atoms
(i.e., any subset of ? of positive measure can be divided into two sets of positive measure)
satisfies the assumptions of Proposition 7.
2.4. Bibliographical remarks
Riesz spaces are a widely used concept in functional analysis. In particular, Banach
lattices (which are simultaneously Riesz and Banach spaces with a Riesz norm; see
the next sections) are well developed. A first systematic treatment of such spaces was
initiated by Riesz A930), Kantorovich A936), and Freudenthal A936). An introductory
textbook into this topic was written by Zaanen A997). More advanced monographs
are by Luxemburg and Zaanen A971) and Zaanen A983) as well as by Aliprantis and
Burkinshaw A976, 1985), Fremlin A974), Krasnoselskii et al. A989), Meyer-Nieberg
A991), Nakano A950), and Schaefer A974), where partially the theory is considered from
a somewhat different point of view. Closely related monographs are by Birkhoff A967),
Kutateladze (as editor) A996), Semadeni A971), and Sikorski A964). Particular emphasis
on the connection with measurable functions is put in the monographs by Fremlin A974),
Luxemburg and Zaanen A971), and Zaanen A983).
The Dedekind completeness of ???(?) and thus the existence of a support on
?-finite measure spaces (Theorem 3) is mathematical folklore but goes probably back to
Kantorovich. Slightly different proofs can also be found in Paragraph 34, Theorem 1, of
Zaanen A967) and in Example 23.3(iv) of Luxemburg and Zaanen A971). Theorems 4, 5,
and 6 are Paragraph 34, Theorem 2, Exercise 35.1, and Paragraph 35, Theorem 1 of Zaanen
A967). We note that the Dedekind completeness of *BE") for ?-finite measure spaces 5"
is also an immediate consequence of the exhaustion theorem which is extremely useful

798
?. Viith
in measure and integration theory and which may be considered as a measure theoretic
analogue of Zorn's lemma:
THEOREM 8 (Exhaustion theorem). Let ? с Ъ{Б) be a system of measurable subsets of
? ?-finite set ? e *B(S) satisfying the following two conditions:
A) ? is directed upwards, i. e. for any D \, Di eT there is some ? e ? with D \ U D2 <?
?.
B) For any measurable ? С Г with ?(?) > 0 there is some D e ? with D С Е and
?(?))>0.
Then there is a sequence of sets Tn e ? with Tn | T.
PROOF. Considering ? as a measure space, we equip ? with a normalized measure v.
There is some sequence E„ e ? with v(E„) —>¦ s := sup{v(E): ? e T}. By induction,
we find T„ e ? with Tn+\ э Jn и En. Then T„ | Г. Otherwise, ? \\JT„ would contain
some set D e ? with ?(?)) > 0 which would yield the contradiction v(D U E„) =
v{D) + v(E„) > s for sufficiently large и. ?
The Luxemburg-Gribanov theorem was originally used to obtain a measurability result
for a mixed norm space by Gribanov A970) and Luxemburg A958). We will turn to this
topic later. The current version can be found as Theorem 99.2 and Corollary 99.3 in Zaanen
A983); it has been generalized recently by Vath B00?) for
Z/w@'-=sup I \x(t,s)y(t,s)\ds,
\€M JS
where ? с Ш(Т ? 5", ?) is a subset of the set of all functions of the form y(t, s) =
v(t)u(s) where и e C(S, Y) and ? e V with V being a so-called equimeasurable set of
functions set of functions ? : S —>¦ [0, oo]. We do not go into details but just point out that
if S is a locally compact metric space with a ?-finite Radon measure, then the choice
V = C(S) is possible.
3. Locally solid ideals of measurable functions
3.1. Ideals and bands in Riesz spaces
Dehnition 12. Let X be a Riesz space. A subset ? с X is called solid if the relations
? e ?, у е X, and \y\ ^ |jc| imply that у е М.
A) A solid subspace of X is called an {order) ideal.
B) An ideal / с X is called a ?-ideal if for any countable С с / for which * = sup С е
X exists, we have ? e /.
C) An ideal / с X is called a band, if for any ?? с / for which ? = sup ? e X exists,
we have jc e /.
Note that any ideal is a Riesz space by Proposition 2.

Riesz spaces and ideals of measurable functions
799
The notion "ideal" might arise some confusion, because it is also used as an algebraic
notion for certain subsets of rings. However, in this algebraic notion the crucial property of
ideals is that they are characterized as the null sets of ring homomorphisms (namely of the
quotient maps). An analogous property is satisfied for order ideals.
DEnMTlONl3. An linear map of Riesz spaces /: X \ -> Xi is called a
A) Riesz homomorphism if |/(jc)| = f(\x\)\
B) ? -Riesz homomorphism ifsup /'(C) = /(supC) for any countable С ? X for which
sup С exists;
C) normal Riesz homomorphism if sup/(Ai) = f(supM) for any ? с X for which
sup ? exists.
Recalling Proposition 2, we see that Riesz homomorphisms preserve the linear structure
and order as well as all lattice and Riesz operations (like finite suprema and infima).
If X is a Riesz space and ? с X is an ideal, then the quotient space X/M can be
endowed with a canonical order defined by [jc? ] ^ [??] whenever the equivalence classes
[x·,] contain representatives y, with y\ ? yi-
Propositions. Let X be Riesz space.
A) If I с X is an ideal, then X/I is a Riesz space, and the quotient map X —> X/I is
a Riesz homomorphism with null space I. Conversely, the null space of any Riesz
homomorphism is an ideal.
B) If a Riesz homomorphism is normal (respectively ? ?-Riesz homomorphism) then
its null space is even a band (respectively ?-ideal). The converse holds if the Riesz
homomorphism is surjective (in particular, for the quotient map X —> X/I).
Ideals have very natural properties concerning Dedekind completeness:
PROPOSITION 9. Let X be a (super) (?-) Dedekind complete Riesz space. Then also each
ideal of X is (super) (?-) Dedekind complete.
Bands have a particular importance in connection with the following definition.
Dehnition 14. Let X be a Riesz space, and x, у e X. Then we call ? and у disjoint (and
write X-Ly) if I*| Л \y\ — 0. For ? с X, the disjoint complement is defined by
? := [? e X: for any у e ? we have X-Ly}.
The set M1- is always a band.
PROPOSITION 10. Let X be Dedekind complete, and В с X be a band. Then В = (B^I-
and X = ? ? ?-1. In particular, there is a projection Рв ¦ X -> В onto В with null
space B^. This projection satisfies
0 <; PBx ? ? (? e X+).

800
?. Viith
3.2. Ideals of measurable functions
We are interested in subspaces X с 97tE) which are simultaneously ideals in 97tE), i.e.
which have the property that
? eX, у e 371E), \y(s)\ <c \x(s)\ for almost alb => yeX. C)
In view of Proposition 9, these spaces are always ?-Dedekind complete and, under
reasonable mild assumptions on 5, they are even (super) Dedekind complete (recall
Theorems 3 and 6).
With ? as described at the beginning, the space 37tE, Y) is not a Riesz space, in
general, because it is not ordered. Nevertheless, we also have a "Riesz operation", defined
by \x\{s) :— \x{s)\ almost everywhere. Moreover, we can define a corresponding "order
topology" which we call o-topology. A set ? с 37tE, ?) is closed in this topology if and
only if
x„ eM, у e37iE, У), d(x„(s), y(s)) ^0 for almost alb =>· у е М.
The following analogue of Proposition 6 follows from Theorems 1 and 2:
PROPOSITION 11. IfS is ?-finite, then the o-topology is induced by ? (and p).
We also call a linear subspace X с 371E, Y) an ideal if C) holds (with 371E) replaced
by 37tE, Y)). Note that, since we require that X be a linear subspace, the functions ? e X
are by definition almost everywhere finite.
For most practical purposes, ideals in the above sense come equipped with a topology
which turns them into topological vector spaces and which is compatible with the order
structure in the sense that it is locally solid, i.e. 0 has a neighborhood base consisting of
solid sets. Of course, we call a subset ? of an ideal X с 37tE, Y) solid if
? e M, у e 371E, Y), \y(s)\ ? \x(s)\ for almost all s => yeM.
The (locally solid) ideals of 97?E, ?) and of 97iE) are in a one-to-one correspondence
with each other:
Proposition 12. Let e eY with \e\ = 1.
A) IfX с 97tE, Y) is an ideal (with a locally solid topology), then
XK:={jte97iE): x()e e X}
is an ideal in 97iE) (with a locally solid topology determined by the neighborhoods
0/0
SK:={;te97iE,yK): x(-)eeB\,
where В runs through all neighborhoods ofO in X).

Riesz spaces and ideals of measurable functions
801
B) Conversely, if Xr с tyl(S) is an ideal (with a locally solid topology), then
X :={xe 3Jt(S, Y): \x(-)\ eX}
is an ideal (with a locally solid topology determined by the neighborhoods ofO
B:={xeWl(S,Y): \х(-)\еВл],
where Z?r runs through all neighborhoods ofO in Хк).
In view of the correspondence of X and Xr, we call Xr the real form of X, but unless
the danger of confusion arises, we will notationally not distinguish between them.
Recall that a pseudo-norm on a linear space X is a map || · ||: X —> [0, oo) which
satisfies the triangle inequality ||jc + y\\ ^ ||jc|| + \\y\\ and which is continuous for scalar
multiplication in the sense that ||?.?|| —> 0 as ? —> 0. If X is a Riesz space, we call || · || a
lattice pseudo-norm if additionally
1*1 «? ? => 11*11 <Ы1-
Occasionally, lattice pseudo-norms are also called Riesz pseudo-norms in literature. Recall
that by the classical metrization theorem of uniform spaces the topology of any topological
vector space is determined by the family of continuous pseudonorms. It is not surprising
that for locally solid topologies even lattice pseudonorms will do.
THEOREM 9. Let X be a Riesz space or an ideal in VK(S, Y). Then each locally solid
topology on X is determined by the family of continuous lattice pseudo-norms, i.e. for
each point ? e X a neighborhood base is given by the system of all "solid balls " (i.e. balls
with respect to some continuous lattice pseudo-norm) with center ?.
PROOF. For Riesz spaces, a detailed proof can be found in Section 22C of Fremlin A974).
The case of a Banach space ? reduces to the case ? = Ш in view of Proposition 12 (see
also Feledziak and Nowak A997)). ?
Corollary 3. Let X be a Riesz space or an ideal in Wl(S, Y) with a locally solid
topology. Then 0 has a countable neighborhood base if and only if the topology is
generated by a single continuous lattice pseudo-norm.
PROOF. If 0 has a countable neighborhood base, we find a countable family of continuous
lattice pseudo-norms || · |k whose balls form a base for the neighborhoods of 0. Then
00 II II
^ 1 + II* Ik
k=\ " "*
is the required pseudo-norm.
D

802
?. Vath
In case ? = Ш, we can evidently also interpret the term "ideal" in some algebraic
sense. Indeed, a linear subspace X is an ideal of Wl(S) if and only if it is invariant
under multiplication by functions from LX(S). A similar characterization can be given
for the locally solid topology. One might suspect that in case of a Banach space ? an
analogous characterization would hold when LX(S) is replaced by Loc(S, C(Y)) where
C{Y) denotes the set of bounded linear operators of ? with the operator norm. This
is indeed true, although the proof is not straightforward, because the measurability of
functions in LX(S, C(Y)) is a more restrictive requirement than one might expect at first
glance:
Example 3. For 5" = [0,1] (with the Lebesgue measure) and ? = L|([0, 1]) there
are functions x,y e LodS, Y) with |jc| = |y| such that there is no measurable function
?: S -+ C(Y) with \z(s)\ ^ 1 and y(s) = z(s)x(s).
Indeed, fori e [0, 1] define
f 1 if/ iCs,
and let x{s) := es and y{s) := e\. Assume by contradiction that a measurable function
z-S -> C(Y) with \z(s)\ ? 1 and y(s) = z(s)x(s) would exist. Let / be the bounded
linear map from C(Y) into the dual space Y*, defined by /(А)(и) := /0' (Au)(t)dt. Then
I := / ? ?: S -> ?* is measurable. Since t(s) e Y* can, as any bounded linear functional
on L ? ([0, 1 ]), be written in the form
l(s)u= / ks(t)u(t)dt,
Jo
where ks e Loo([0, 1]) with l^loc = \i(s)\ ^ |/| \z(s)\ ^ 1, the equality l(s)es = 1 implies
that we must have ks = —es almost everywhere (see, e.g., Theorem 1.39(c) of Rudin
A987)). But this implies |?(j|) - e(s2)\ = \ku - kS2\x = 2 for s\ ^s2- Hence, €E0) is
not separable for any uncountable SO <? S, and so I is not measurable, a contradiction.
As we can already guess from Example 3, the dual space (more precisely, the so-called
duality map) plays the essential role in this problem. For this reason, the following result
is of much use.
LEMMA 1. For any Banach space ? and any ? > 0 there is a continuous map F :Y —>¦ Y*
such that F(y)- \y\2 and \F(y)\ ? \y\(\ + ?) for each у е Y.
The proof of Lemma 1 makes essential use of the axiom of choice.
COROLLARY4. Let ? be a Banach space. Then for any measurable functions ?, у: S ->· ?
with \y(s)\ ^ \x(s)\ and any ? > 0 there IS SOYYIC ? ? Ьх E",C(Y)) with \z(s)\ ?\+????
y(s)=z(s)x(s).

Riesz spaces and ideals of measurable functions 803
PROOF. Choose F as in Lemma 1, and put ? CO := y(s)\x(s)\~2F (define ? CO :=0 for
those s with x(s)=0). О
Example 3 shows that Corollary 4 (and thus Lemma 1) is no longer true for ? = 0.
Corollary 4 now implies the desired alternative characterization of ideals:
PROPOSITION 13. Let X be a linear subspace ofWl(S, Y), and Kx denote the unit ball
of L^S, C(Y)).
A) X is an ideal in Wl(S, Y) ifandonly if KXX с X or, equivalently, LX(S, C(Y))X с
X.
B) Let X be also a topological vector space. Then X is an ideal in Wt(S, Y) with a
locally solid topology if and only if for each neighborhood ?/cX ofO the inclusion
KooU с U holds.
PROOF. Only the sufficiency requires a proof. By Corollary 4, we have for any ?/cX and
any ? > 0 that K^lf contains the solid set
U?:={y eWl(S,Y): \y\ «? UKl+?) for some ? eU).
For U = X, we obtain that X is an ideal, and if U is a neighborhood of 0 with U 2 ?^??,
then U э ?? 5 A + ?)-1 U, and so U? is a solid neighborhood of 0 contained in U. D
3.3. Comparison of various sorts of convergence
Probably the most important property of ideals in ?ul(S, Y) with a locally solid topology is
that this topology is compatible with "convergence a.e." in the following sense:
PROPOSITION 14. Let X be an ideal in 9JiE", Y) with a locally solid Hausdorff topology.
Ifx„ —>· ? in X, then for almost all s e S there is a subsequence with x,4 (s) —> .x CO-
PROOF. We prove that D„m := ??1»{·?: \x(s) ~ xk(s)\ > w~') is a nul1 set· ?*1611
Un.m Bum is a null set which implies the statement. We thus have to show that у : =
m~ Vd„.„, vanishes almost everywhere. But since \y\ < \x — Xk\ and хи —> x,y is contained
in any locally solid neighborhood of 0 and thus in any neighborhood of 0 which means
у = 0, because the topology is Hausdorff by hypothesis. ?
Corollary 5. Let X be as in Proposition 14. Suppose that x„ e X is such that for some
(not necessarily finite) у : S —>· ? we have d(x„(s), y(s)) —* Ofor almost all s. Ifx„ —> ?
in X, then y(s) = x(s)for almost all s (in particular, у is finite a.e.).
Under a slightly more restrictive assumption than in Proposition 14, a much stronger
conclusion can be obtained.

804
?. Vath
???????? 15. Let X be an ideal in Wl(S, Y). Then we call a locally solid topology
on X entire if for each xq e X \ {0} there is some ? e X \ {0} with \x(s)\ ^ |jco(^)| and
some continuous pseudo-norm || · || on X with ||>>|| > 0 for each у e 97t(S, ?) \ {0} with
|y(i)l^|jc(i)|.
Clearly, each entire topology is Hausdorff (because ||jco|| ^ \\y\\ > 0). Conversely, the
notion entire appears only as a slight generalization of the Hausdorff property which is
more appropriate for comparisons with the order. We point out that metrizable topologies
are always entire. In fact, if a locally convex topology on X is Hausdorff and 0 has a
countable neighborhood base, then Corollary 3 implies that there is a continuous pseudo-
norm || · || on X with ||jc|| =0 if and only if ? — 0.
THEOREM 10. Let X be an ideal in Wl(S, Y) with a locally solid topology. Let S be
?-finite, and 9JiE", Y) be endowed with the topology of Proposition 1. Then the embedding of
X into W(S, Y) is continuous if and only if the topology on X is entire. This is in particular
the case if X is Hausdorff and 0 has a countable neighborhood base.
Recall that the topology on UR(S, Y) considered in Theorem 10 is nothing else but the
o-topology (since S is ? -finite).
We remark that if, in addition to the hypothesis of Theorem 10, the space X is metrizable
and complete (i.e., a so-called F-space, see, e.g., Rudin A990)), then Theorem 10 follows
from Proposition 14. In fact, in this case one can apply the closed graph theorem: The
embedding of X into ?ul(S, Y) has trivially a closed graph in view of Corollary 5 (and
Riesz' Theorem 2).
By considering the restriction of X to functions which vanish outside a given set E, we
obtain a corresponding result for measure spaces which are not necessarily ?-finite.
Corollary 6. Let X be an ideal in Wl(S, Y) with an entire locally solid topology (e.g.,
X is Hausdorff and 0 has a countable neighborhood base). Ifx„ —> ? in X, then on each
set of finite measure the sequence x„ converges to ? in measure.
Corollary 6 is a very surprising result, because our definition of ideals and locally solid
topologies is apparently not at all related to the values of the measure: the measure ? only
appears implicitly in the form of null sets.
By Riesz' Theorem 2, Corollary 6 implies a remarkable strengthening of Proposition 14
for ?-finite measure spaces:
Corollary 7. Let X be as in Corollary 6. If S is ?-finite and x„ -> ? in X, then a
subsequence of x„ converges almost everywhere to x.
The converse question (for which sequences xn does "convergence almost everywhere"
imply convergence in X) will be discussed in Section 4.4.
Although the topology in X is "compatible" with the o-topology (even if oo e Y), this
"compatibility" has its limits. For example, the following property is not always satisfied
(examples will be given later).

Riesz spaces and ideals of measurable functions
805
Dehnition 16. We say that a locally solid ideal X in 9JiE", Y) has the Fatou property
if each point has a neighborhood base of sets which are closed in Wl(S, ? U {oo}) with
respect to the o-topology.
The name will become clear later. We point out that our definition slightly differs from
the definition used sometimes in literature (e.g., by Aliprantis and Burkinshaw A977) or
Fremlin A974)), mainly because we consider sequential closedness. On ?-finite measure
spaces, this makes no difference, because Wl(S) is order separable (recall Theorem 3).
3.4. Bibliographical remarks
Spaces of measurable functions are together with spaces of continuous functions the most
natural examples of Riesz spaces. Using the axiom of choice, standard results state that any
Riesz space X satisfying certain natural assumptions is Riesz isomorphic to (a subspace of)
C(K) where К is the set of all maximal (order) ideals, equipped with a certain topology
(analogously to the Gel'fand representation theorem for Banach algebras). Similar results
establish a Riesz isomorphism to L^iS). For such representation theorems, we refer to
Luxemburg and Zaanen A971), Semadeni A971), Schaefer A974), and Zaanen A983),
In particular, one may also establish Riesz isomorphisms of ideals of Wl(S) to C(K).
However, these homomorphisms do not immediately imply, e.g., Theorem 3. Moreover,
ideals in Wl(S) are among the best understood classes of Riesz spaces, and so it makes not
much sense to represent them in a less known form.
The precise connection of Example 3 with the duality map is revealed in Theorem 2.1.1
of Vath A997). Lemma 1 is from Dirr and Vath B000).
Proposition 14 is apparently new, although it was known already in the special case of
preideal spaces (Lemma 2.2.1 in Vath A997))· The notion of entire topologies is related
with the so-called support of the topology, see, e.g., Feledziak and Nowak A997). For
the latter, see also Aliprantis and Burkinshaw A977, 1980a, 1980b). In these references,
the reader will also find other conditions than metrizability which ensure that a Hausdorff
topology is entire (on ? -finite measure spaces, the Fatou property is such a condition, for
example, in view of the remarks following Definition 16 and by the results of Aliprantis
and Burkinshaw A977)). Theorem 10 was proved for scalar functions by Wnuk A986); the
case of vector functions follows from this straightforwardly in view of Proposition 12 (see
Feledziak and Nowak A997) for details). In the context of preideal spaces, the important
Corollary 6 was obtained first independently by Zabreiko A974) and by Luxemburg and
Zaanen A963d). (The arguments of Luxemburg and Zaanen are completely different.)
4. Ideal spaces
4.1. Preideal spaces
Henceforth, we will restrict our attention to locally solid ideals of Wl(S, Y) which are
generated by a single (quasi-)norm. Recall that for a linear space X, a map || · ||: X —>¦

806
?. Vath
[0, ??) is called a quasi-norm if ||jc|| =0 implies ? = 0, ||?*|| = |?|||.?||, and if \\x + y\\·^
C{\\x\\ + \\y\\) for some constant С < oo.
If X is an ideal of 97tE, Y), we call || · || a lattice (quasi-)norm if it has additionally the
property
? eX, ye 971E), \y(s)\ ^ \x(s)\ for almost alb => ||y|| ^ \\x\\. D)
It will be convenient to define \\x || := oo for any ? e 97tE, ?) which does not belong to X.
With a slight misuse of notation, we nevertheless call this extended map || · || a (quasi-)
norm.
With this convention (which we assume throughout), the relation ||*|| < oo is a
reformulation of the statement ? e X. If we interpret D) in this extended sense, any
subspace X с 97?E, ?) with a lattice (quasi-)norm is automatically an ideal in 97?E, ?).
Dehnition 17. A subspace X с 97tE, ?) with a lattice (quasi-)norm is called a (quasi-
normed) preideal space. If X is also complete, it is called ideal space.
PROPOSITION 15. Any (quasi-normed) preideal space X is a locally solid Hausdorff ideal
in 971E, Y). Its real form is also a (quasi-normed) preideal space with the (quasi-)norm
determined by
III*IIIxr = IWI·
Henceforth, we will notationally not distinguish between X and its real form.
Analogously to Proposition 13, one obtains the following result.
PROPOSITION 16. A (quasi-)normed linear subspace ? o/97tE, ?) is a (quasi-normed)
preideal space if and only if its unit ball is invariant under multiplication of functions by
the unit ball of Loo(S, C(Y)).
Recall that Corollary 4 (which was used for the previous result) essentially depends on
the axiom of choice.
Let us give a number of examples which show that most spaces of measurable functions
occurring in applications are actually ideal spaces (we will see later how the completeness
is proved).
Examples 4.
A) The Lebesgue-Bochner spaces Lp(S, Y) @ < ? ? oo) are quasi-normed ideal
spaces. They are normed for ? ^ 1.
B) The sequence spaces lp(Y) @ < ? ^ oo) are obtained as the special case 5 = N
with the counting measure.
C) The space со of null sequences (considered as a subspace of lx) is an ideal space.
D) The space cfin of all sequences which are eventually 0 (considered as a subspace of
cq) is a preideal space but no ideal space.

Riesz spaces and ideals of measurable functions
807
E) Lebesgue-Bochner spaces weighted by some measurable function w : S —> @, oo]
are ideal spaces. Here, the space consists of all measurable functions ? : S ->· ? for
which y(s) := w(s)x(s) belongs to Lp(S, Y).
F) Orlicz spaces are important ideal spaces. We briefly recall the definition. Let
Ф:Ш —> [0, oo] be a Young function, i.e., convex, even with Ф@) = О, and
not almost everywhere oo or 0. Then the Orlicz space Lcp{S, Y) consists of all
measurable functions ? : S —> ? for which the Luxemburg norm
is finite. For the choice Ф(г) = \t|'', one obtains the spaces Lp, and also L^ is a
special case.
G) One may also generalize the Orlicz spaces (and Lp spaces) by choosing a
Young function ? which also depends on s. Under natural assumptions, one still
obtains ideal spaces, see, e.g., Appell and Zabreiko A990). One may consider this
dependence on s as a generalization of a weight function.
(8) Lorentz and Marcinkiewicz spaces are ideal spaces. They belong to a particular class
of preideal spaces whose norm depends only on the decreasing rearrangement of
the function. In other words: The norm does not change if the function is replaced
by an equimeasurable function. For this reason, these spaces are usually called
rearrangement invariant, or sometimes also symmetric.
One of the difficulties in the abstract theory of ideal spaces is that one does not know
whether the spaces contain the constant functions, even if the underlying measure space
is finite (think of weighted Lebesgue spaces or on the more involved implicit weighting
of Orlicz spaces described above). For this reason, for example, one cannot prove that
uniform convergence of a sequence implies convergences in the (quasi-)norm. The example
of weighted Lebesgue spaces (or also, e.g., of ЬР(Щ) suggests that instead of a constant
function one should consider some strictly positive function which belongs to the space (in
the abstract notion of Riesz spaces such a function plays the role of a so-called weak unit).
The following result ensures the existence of such a function for complete spaces, and a
reasonable substitute for incomplete spaces. It is needed for the proof of almost all results
concerning ideal spaces.
THEOREM 11. Let X be a (quasi-normed) preideal space with ? ? -finite support. Then
there are sets E„ | supp X with ??? e supp X. If additionally X is complete, there is a
function ? e X with supp ? = supp X.
PROOF. The first statement follows from the exhaustion Theorem 8. For the second
statement, choose a sequence ?„ > 0 such that ? ^nC" \\??„ II converges (C is the constant
of the quasi-norm). The partial sums of ? ???,, are then a Cauchy sequence of X, and so
the sequence converges in norm to a desired element ? e X (apply Proposition 14). D
The example Cf\„ shows that, in general, apreideal space X does not contain a function ?
with suppjc = supp X.

808
?. Vdth
For ideal spaces, an important strengthening of Proposition 14 holds which is useful in
connection with the dominated convergence theorem discussed later:
THEOREM 12. Let x„ —>· ? in a {quasi-normed) ideal space X. Then there is some у е X
and some subsequence with \x„k | ^ | у | such that x„K —> ? almost everywhere.
Note that, in contrast to Corollary 7, it is not required for Theorem 12 that the underlying
measure space be ?-finite.
The sequence xn = n~]e„ in ct\„ (with e„ being the canonical «th unit vector) shows
that, in general, Theorem 12 does not hold for preideal spaces.
4.2. Fatou property and perfectness
Dehnition 18. Let 1 ^ a < oo, and X be a (quasi-)normed preideal space. Then X is
called
A) semi-perfect if the relations 0 < x„ | ? in ?ul(S, R) and sup ||*„|| < oo imply ? e X.
B) a-almost perfect if the relations 0 ^ x„ | ? e X imply \\x\\ ? a lim || xn \\.
C) ?-perfect if the relations 0 ^ x„ | ? in 9Jt(S\ R) imply \\x || ^ a lim \\x„ \\.
In case a = 1, we do not mention a.
In view of our convention that ||jc || < oo means ? e X, we have that X is perfect if and
only if it is semi-perfect and almost perfect.
Note that the above definition is actually only a requirement on the real form of X. The
properties may be considered as abstract analogies of the monotone convergence theorem.
Examples 5.
A) The spaces Lp(S, Y) @ < ? ^ oo) as well as their weighted forms are perfect (by
the monotone convergence theorem).
B) Orlicz spaces are perfect.
C) Neither cq nor ct\„ are semi-perfect. However, they are almost perfect.
Further examples will follow, soon.
We may reformulate the properties of Definition 18 in a way which does not make use
of the real form of X. These equivalent formulations remind of the lemma of Fatou:
PROPOSITION 17. Let X be a (quasi-)normedpreideal space and 1 ^ a < oo.
A) X is semi-perfect if and only if for any sequence x„ e X with sup \\x„ || < oo and
d(xn(s),x(s)) -+0 a.e. for some ? eWl(S, ? U {oo}) we have ? e X.
B) X is a-almost perfect if and only if for any sequence x„ e X with \x„(s) — x(s)\ —> 0
a.e. for some ? & X we have
||*|| ^ orliminf ||*„||. E)
n—>oo

Riesz spaces and ideals of measurable functions
809
C) X is a-perfect if and only if for any sequence x„ e X with d(x„(s),x(s)) —> 0 a.e.
for some ? e Wl(S, ? U {oo}) we have E).
PROOF. For necessity, consider xn := inft>„ |**|, and for sufficiency, consider Xn '.— Xfj€
witheey, \e\ = 1. " ?
PROPOSITION 18. Let X be a (quasi-normed)preidealspace.
A) X is perfect if and only if its unit ball is closed in W(S, ? U {oo}) with respect to
the o-topology.
B) X is almost perfect if and only if its unit ball is closed in X with respect to the
o-topology.
C) IfX has the Fatou property, then X is ?-perfect for some a. The converse holds if S
is ?-finite.
PROOF. All statements are consequences of Proposition 17. We prove the last one: X
has the Fatou property if and only if its unit ball К is contained in some o-closed set
? с Wl(S, ? U {oo}) which in turn is contained in a ball of some radius a. This implies
the necessity of ?-perfectness, but also the sufficiency if S is ?-finite. Indeed, if S is ?
-finite, the o-topology is induced by a metric (namely p), and so the set ? of all o-limits of
sequences of the unit ball is o-closed. ?
Proposition 18 explains the name "Fatou property".
The properties considered here are not independent of each other:
THEOREM 13. Any semi-perfect (quasi-normed) preideal space is ?-perfect for some a.
The following examples show that one cannot always choose a = 1, and that no other
implications for the properties of Definition 18 hold, even in the case of ideal spaces.
Examples 6.
A) The space of sequences ? = (?„),, determined by
||*|| :=sup|?„| + Xlimsup |?„|
? ?—>oo
(for fixed ? e [0, oo)) is an ?-perfect ideal space for a = 1 + ? but not for any
smaller a.
B) The space of all double sequences ? = (?,,.^,,? determined by
||*|| :=sup |?„.(t|+sup(fc limsup |?«.*|)
п.к к V и-юс '
is an ideal space which fails to be ?-almost perfect for any a (and thus also fails to
be semi-perfect).

810
?. Vath
C) The space of all integer-indexed sequences * = (?;-)* ez determined by
II^H := |sup)teZ|^| +Xlimsup^3C |&| if linu·^-^· =0,
I oo otherwise
(for fixed ? e [0, oo)) is an ideal space which is ?-almost perfect for a — 1 + ? but
for no smaller a and which fails to be semi-perfect.
Definition 19. Given some (quasi-normed)preideal space X, the corresponding Lorentz
space Xl consists of all ? e Wl(S, ? U {oo}) for which
||* Hi. :=inf{lim||*„||: 0 ^ *„ | |*|} F)
is defined and finite.
PROPOSITION 19. We always have X с XL and \\x\\L ^ ||.x||. Moreover, X = Xl (assets)
if and only if X is semi-perfect, and
II*Hz. ^<*_l ||*|| (* e X)
if and only if X is almost ?-perfect. In particular, X = Xl with the same norms if and only
if X is perfect.
It is not immediately clear whether || · ||z. is a (quasi-)norm, namely whether ||*||i. = 0
implies that * = 0 (almost everywhere). However, for ?-finite measure spaces, this is a
consequence of the following result which is obtained by combining Egorov's Theorem 1
with Theorem 11 and a fancy diagonal argument.
THEOREM 14. If ? has ?-finite support, then the infimum in F) is actually a minimum.
Moreover, for any sequence 0 ^ *„ | |*| the relation \\x„\\l t II* Hz. holds.
Corollary 8. Let S be ?-finite. Then Xl is a perfect (quasi-normed) ideal space in
which X is continuously embedded (with embedding constant at most 1). The quasi-norms
are equivalent on X if and only if X is almost ?-perfect for some a.
In other words, Xl is the "perfect hull" of X.
The only statement of Corollary 8 which does not follow from the previous results is the
completeness of Xl which however follows from Corollary 10 below.
4.3. Completeness
We have seen that any preideal space has a perfect hull which is an ideal space. Since any
preideal space has a completion, one might hope that this completion might be written in a
canonical way as an ideal space. This is an especially natural conjecture if one recalls that
the completion of the preideal space ct\„ is the (not perfect!) ideal space со- The following
discouraging example shows that this is not true, in general.

Riesz spaces and ideals of measurable functions
811
Example 7. Let X be the preideal space of sequences ? = (?„)„ defined by the norm
oo
IW|:=532~"l&l+limsuPlS«l-
n=l
Then the sequence
*„:=A,..., 1,0,0,...)
// times
is a Cauchy sequence, but in any preideal space containing X the only candidate for a limit
is ? := A, 1,...) (Corollary 5). But we have x„, ? e X and ||jr„ — *|| -^O.
Dehnition 20. A (quasi-normed) preideal space X has the Riesz-Fischer property if the
relation 0 ^ x„ | ? in 9Jt(S\ R) for some Cauchy sequence x„ e X implies that ? e X.
THEOREM 15. A (quasi-normed) preideal space X is complete if and only if it has the
Riesz-Fischer property.
Corollary 9. X is a (quasi-normed) ideal space if and only if its real form Xr is a
(quasi-normed) ideal space.
COROLLARY 10. Any (quasi-normed) perfect preideal space is a (quasi-normed) ideal
space.
4.4. Convergence theorems
Given D e ? and ? e 371E", Y), we define PDx e OTE, Y) by
PDx(s):=<pD(s)x(s) = \ y '
10 if s ? D.
Dehnition 21. Let X be a (quasi-normed) preideal space. We say that a subset ? с Х
has equicontinuous norm if the relations
lim sup sup ||/??||=0, G)
inf sup || ? ?' ? || = 0 (8)
hold. The regular part X,- of X is defined as the subset of all ? e X for which \x) has
equicontinuous norm. If X — X,, the space X is called regular.
Roughly speaking, condition (8) means that the functions ? e ? become "uniformly
small" (in norm) outside of large sets of finite measure. This condition is automatically

812
?. Valh
satisfied if ?(?) < oo. Similarly, condition G) means that the functions ? e ? become
"uniformly small" on sets of small measure. This condition is automatically satisfied if the
measure space consists only of atoms of a positive minimal measure (e.g., for the counting
measure). Regularity may be considered as an abstract analogue of Lebesgue's dominated
convergence theorem:
PROPOSITION 20. If ? с ? has ?-finite support, then ? has equicontinuous norm if and
only if
Dn ; 0 in <8E) => sup ||PDiix\\ I 0.
PROPOSITION 21. X, is always a closed ideal in X. In particular, ifX is a (quasi-normed)
ideal space, then also X, is a (quasi-normed) ideal space.
Examples 8.
A) The spaces Lp(S) and /,, @ < ? < oo) are regular in view of the dominated
convergence theorem. In contrast, ^oc([0, 1]) is not regular (because G) fails), /30
is not regular (because (8) fails), and LX(R) is not regular (because G) and (8) both
fail). The regular part of l^ is cq.
B) Orlicz spaces (assuming ?(?) < oo) are regular if and only if the generating Young
function satisfies a so-called A^-condition, see, e.g., Rao and Ren A991). If the
generating Young function is everywhere finite, their regular part is the closure of
the family of iT-step functions. Orlicz spaces without /i^-condition are the most
important class of spaces for which the regular part is strictly smaller than the space
but not trivial.
THEOREM 16 (Dominated convergence theorem). Let X be a (quasi-normed) preideal
space, andx,,,x e X be such that xn —> ? almost everywhere or at least in measure on each
set of finite measure. If there is some у e Xr with \xn\ ?. у and \x\^y, then \\xn — ? || —> 0.
Proposition 20 implies that the functions in X, are even characterized by the fact that
they have the property of Theorem 16:
Corollary 11. Let a quasi-normed preideal space X have ? -finite support. Then ? e X
is regular if and only if the relation \x \ ^ x„ I 0 in ffl(S) implies \\x„ \\ I 0.
COROLLARY 12. Any regular quasi-normed preideal space X is a-almost perfect where
a = С is the constant of the quasi-norm.
PROOF. If 0 ^ xn | x, then Цл- - jc„|| 4, 0, and so ||jc|| ^ C(||jc - x„\\ + \\x„\\) ->
Clim ||jr„||. ?
THEOREM 17 (Vitali's convergence theorem I). Let X be a (quasi-normed) preideal
space, and x„ e Xr. Then the following statements are equivalent:

Riesz spaces and ideals of measurable functions
813
A) x„ —> 0 in measure on each set of finite measure, and {x„: n] has equicontinuous
norm.
B) ||*n||-»0.
In complete spaces, we have an analogous result also for convergence to nonzero
elements (this is what is usually called Vitali's convergence theorem):
THEOREM 18 (Vitali's convergence theorem II). Let X be a (quasi-normed) ideal space,
and x„ e Xr and ? e Wl(S, Y) be almost everywhere finite. If S is ?-finite, the following
statements are equivalent:
A) x„ —> ? in measure on each set of finite measure, and {x„: n] has equicontinuous
norm.
B) ? e X and \\x„ - ? || -> 0.
In general (i.e., if S is not necessarily ?-finite), the first of the following statements implies
the second:
A) xn —у ? almost everywhere, and {xn: n] has equicontinuous norm.
B) ? eX and \\xn -x\\ -> 0.
4.5. Bibliographical remarks
Unfortunately, there is no unique terminology in the literature concerning ideal spaces.
Sometimes, they are called Kothe spaces (although, historically, G. Kothe was mainly
interested in a special class of such spaces which were sequence spaces), sometimes they
are called Banach function spaces, and sometimes more or less restrictive requirements are
made for the definition. A systematic treatment was given in a celebrated series of papers
by Luxemburg and Zaanen A963b, 1963c, 1963d, 1963e, 1963f, 1963g, 1963h, 1964a,
1964b, 1964c, 1964d, 1964e, 1964f), and by Luxemburg A965a, 1965b, 1965c, 1965d,
1965e, 1965f) (see also Luxemburg and Zaanen A963a)) where the case of scalar functions
is covered almost completely, and simultaneously the related theory of Riesz spaces is
developed in much depth. Some surveys (containing also new results) were written by
Zaanen A967), Zabreiko A974), and the author A997), where only the latter covers
also the vector-valued case. The vector-valued case was also considered by Bukhvalov
A972, 1978), Macdonald A973a, 1973b, 1974, 1976), and by Feledziak and Nowak
A997). The quasi-normed case was particularly studied by Povolotskii and Kalitvin A991)
and Vath B000).
Many special ideal spaces arise naturally in interpolation theory, see, e.g., Bergh and
Lofstrom A976), Krasnoselskii et al. A958), Krein et al. A978), and Lindenstrauss
and Tzafriri A979). In particular, Lorentz and Marcinkiewicz spaces play a special role
there, see Lorentz A950, 1951, 1961), Marcinkiewicz A939), Milman A978), and O'Neil
A963). Rearrangement invariant function spaces were studied by Luxemburg A967) and
Chong and Rice A971). Introductory monographs to the theory of the mentioned Orlicz
spaces are by Krasnoselskii and Rutickii A958) and Rao and Ren A991).
Preideal spaces are closely related with so-called modular spaces which are generated
by certain functionals (called modulars). In particular, in the case of an Orlicz space

814
?. Vath
Ъф, the corresponding modular is given by m(x) = fs<P(\x(s)\)ds. Such a modular
induces a topology which is not necessarily the same as the topology we discuss
here. For comparisons of these topologies (in case of Orlicz spaces), see, e.g., Nowak
A984a, 1984b, 1993).
Proposition 16 suggests the following definition: call a Banach subspace X с W(S, Y)
ideal* if it has the property that its unit ball is closed with respect to multiplication by
functions from the unit ball of Loo(S, R).
In case ? — R, this is just a characterization of ideal spaces in view of Proposition 16
(because C{Y) = R). However, in general, this class of spaces is much larger.
Unfortunately, this class lacks in general the important properties of Section 3.3 if У has infinite
dimension, see Example 3.1.1 of Vath A997). For finite-dimensional Y, it still has
analogous properties, but the proofs are rather cumbersome. By means of an example, consider
a function ? :R" —>¦ [0, oo) which is convex and even (but not of the form ? (и) — Фп(|и|)
for some Young function Фо). Then the set of all measurable ? : S -*¦ R" for which the
norm
|?|:={?>0:/5?(^),^?}
is finite is an important example of an ideal* space which is not an ideal space, in general.
Ideal* spaces and the above generalizations of Orlicz spaces were considered by Nguyen
A987, 1991) and by Nguyen and Zabreiko A990).
Theorem 13 builds on a result of Amemiya A953), see, e.g., Paragraph 65, Theorem 2,
ofZaanen A967) or Theorem 3.2.1 of Vath A997). The result has been formulated there
only for the normed case, but the proof of the quasi-normed case is similar: one mainly
has to insert an appropriate power of the quasi-norm constant С in order to ensure the
convergence of the series in the proof of Theorem 3.2.1 of Vath A997) (similarly, as we
did in the proof of Theorem 11).
Theorems 12 and 15 are proved simultaneously in Theorem 3.26 of Vath B000). The
proofs for the claims in Example 6 is given in Examples 3.2.3, 3.2.4 and 3.2.5 of Vath
A997). Theorem 14 follows from Lemma 3.2.9 and Corollary 3.2.6 of Vath A997) (see
also Paragraph 66, Theorems 2 and 4 of Zaanen A967)): the result is claimed there only
for the seminormed case, but the proof for a quasi-norm carries over unchanged.
The question whether a preideal space can be completed to an ideal space is rather
involved and has been discussed by Luxemburg and Zaanen A963d, 1964e). We mention
only the following result.
THEOREM 19. Let X be a preideal space with ?-finite support. Then X can be completed
to an ideal space if and only if for any Cauchy sequence xn I 0 the relation \xn II X 0 holds.
Theorems 16, 17, and 18 follow from Theorems 3.22, 3.18, and 3.20 of M. Vath B000)
and from Corollary 6. For further refinements of Vitali's convergence theorem (e.g., if the
limit function is not assumed to be finite a priori), we refer to Vath A997).
Straightforward variants of the Lebesgue and the Vitali convergence theorem can be
proved also in the more general context when X is only an ideal in W(S, Y) with a locally

Riesz spaces and ideals of measurable functions
815
solid topology: the results then are formulated in terms of a generating family of pseudo-
norms. For details, see Feledziak and Nowak A997).
5. Order duals
5.1. The associate space
In this section, we assume that У is a Banach space, and that Y* is the corresponding dual
space.
Dehnition 22. Let X с 97?E, Y) be a preideal (normed!) space with supp X = 5". Then
we define the associate space X' с 97tE, Y*) by the lattice pseudo-norm
IIУIIr := sup / |;y(s)||.x(j)|dj.
\\x\K\JS
PROPOSITION 22. X' is a perfect ideal space. For any у e 971E, ?*), ? e 971E, Y), we
have the generalized Holder inequality
/ \y(s)x(s)\ds ^ / |yE)||^E)|di ^ 1М1НИ1·
The name Holder inequality is indeed justified:
Example 9. Let 5 have the finite measure property. Then X = LP(S, Y) A ^ ? ^ oo)
satisfies suppX = 5, and we have X' = Lp>(S, Y*) with \/p+ \/p' = 1.
We could have equivalently used the pairing of Y* and ? for the definition of the
associate space:
THEOREM 20. For any у e97iE, ?*) we have
IIУIIx' = sup / y(s)x(s)ds.
ll-rKI JS
Theorem 20 appears rather natural. Nevertheless, its proof involves some surprising
technical difficulties if 5 fails to be ?-finite.
If X is complete, we have a characterization of the elements of X' which is simpler to
verify for applications:
THEOREM 21. Let X be an ideal space. Then a function у e 97tE, Y*) satisfies \\y\\x> <
oo if and only if у ? e L\(S)for any ? e X.

816
?. Vath
We have not excluded so far that X' is the trivial space. In fact, we cannot do this, in
general.
Example 10. Let S be such that each set which is not a null set has infinite measure.
Then, for X = LooCS"), we have X' = {0}.
It is a rather deep result that for ? -finite measure spaces the problem of Example 10
cannot happen. In this case, the associate space is never trivial and actually has full support.
It follows then easily that this result holds also under a weaker assumption on the measure
space:
THEOREM 22. Let S have the finite measure property. Then supp X' = S.
The proof of Theorem 22 is based on the separation theorem in the Hubert space Li(E)
for an appropriate set ?CX.
We consider now the second associate space X" := (X1)'. For simplicity, we put now
? =R.
PROPOSITION 23. X is continuously embedded into X" with embedding constant 1, i.e.,
\\x\\x" ^ \\x\\for any ? e X.
It is a natural question to ask whether X" is strictly larger than X. By a refinement of the
proof of Theorem 22, one can obtain the following result which (for ? -measure spaces)
answers this question completely:
THEOREM 23 (Lorentz, Luxemburg). Let ? e X be such that supp ? is ?-finite. Then
\\x\\x" = \\x\\l,
where \\x \\ ? is given by F).
Theorem 22 is actually a simple consequence of Theorem 23 and Corollary 8.
COROLLARY 1 3. Let the preideal space X satisfy supp X = S and assume that any ? eX
has ?-finite support. Then X" = X l with the same norms.
Together with the properties of the Lorentz space, we obtain:
COROLLARY 14. Let S bea -finite, and Xbea preideal space with supp X = S.
A) X is semi-perfect if and only if X = X" as sets.
B) X is a-almost perfect if and only if \\ ¦ \\?" is an equivalent norm on X with
a~l\\x\\<\\x\\x»^\\x\\ (xeX).
In particular, X is perfect if and only if X" = X with the same norms. Historically, this
property is the reason for our terminology "perfect", because G. Kothe used this term for
this property.

Riesz spaces and ideals of measurable functions
817
5.2. Applications to ideal spaces on product measures
In this section, let Sj (i = 1,2) be ? -finite measure spaces, У, be Banach spaces, and
?(Y\, Yl) be the corresponding set of bounded linear operators. Moreover, let X,- с
ЗЛE,-, Yi) be preideal spaces. The following result is a motivation to define a certain class
of preideal spaces on the product space S = S\ ? Si.
PROPOSITION 24. For any ? e Wl(S, C(Y\, Yi)) the function y(si) : = \\x(si, ·)||?< is
measurable. If in addition \\y\\x-, < 00, then the integral operator
Az(s2):= x(s2,s\)z(s\)ds (si e Si)
Js,
acts from X\ into X2 with \\A\\ ^ ||}'||??.
PROOF. The measurability of у is a consequence of the Luxemburg-Gribanov Theorem 7.
The remaining statements follow from the generalized Holder inequality. ?
????????? 23. Let [X| -> X2] denote the set of all ? e 9JiE", Y) for which the quantity
1И1[х,—х2] := ||-S2 ь> Iл-E7, ·)|? ??, (9)
is defined and finite. Then [X| —> Xi] is called the waerf «orw .vpace of X| and Xi.
Proposition 24 states that in case ? = C(Y\, Yi) the number ||jc ||(?'_?,] is a bound for
the norm of the integral operator Ae?(X|,XT)with kernel ?.
It is not clear whether [X| —> Xi] is a preideal space, in general; moreover, it is not
even clear whether [X| —>· Xi] is a linear spaces. The difficulty is that the function
у (si) := ||* E2, )llxi may fail to be measurable for measurable x. It is not easy to find
such an example, but by nonstandard methods it is possible to obtain some, see Luxemburg
A963).
On the other hand, the Luxemburg-Gribanov Theorem 7 implies that if X| is an
associate space, then у is always measurable. If X1 is almost perfect, then the embedding
of X| into X" is norm preserving by Corollary 14. Hence, we obtain:
THEOREM 24 (Luxemburg-Gribanov). If X\ is almost perfect, then [X\ -> Xi] is a
preideal space.
For completeness, let us collect some further properties of mixed norm spaces.
THEOREM 25. Suppose that X := [X, -> Xi] is a preideal space (e.g., X1 is regular or
almost perfect). Then we have:
A) suppX=suppX| xsuppXi.
B) If X\ and X2 are semi-perfect, then X is a semi-perfect ideal space.

818
?. Vath
C) If X? is a\-almost perfect and Xi is ai-almost perfect, then X is a\a2-almost
perfect.
D) If X\ and X2 are regular, then X is regular.
E) [X^ —>¦ X!,] is defined and the same as X' (with the same norm).
5.3. Order duals of a Riesz space
We return now to the general theory of Riesz spaces (and consider only ? = R). We
consider in the following linear operators A : X —>¦ ? for a Riesz space X and a Dedekind
complete Riesz space Z. Actually, we are only interested in the case ? = R, but the results
hold for more general Riesz spaces as well (provided ? is Dedekind complete).
Dehnition 24. The linear operator A : X -> ? is called
A) positive if Ax > 0 for ? > 0;
B) regular if A = A+ — A~ for two positive operators A± :X—>Y;
C) order bounded if A maps order bounded sets into order bounded sets.
The space of all order bounded operators A: X —>¦ ? is denoted by ?/,(X, Z). In case
? = R, the space X~ := Cb(X, R) is called the order dual of X.
Any positive operator and thus any regular operator is order bounded.
The space Сь(Х, ?) is partially ordered by the relation
? ?; ? «=> В — A is positive.
THEOREM 26 (Jordan decomposition). The space ?/,(X, Z) is a Dedekind complete Riesz
space which consists only of regular operators. The corresponding Riesz operations are
given by
A+x = sup Ay — sup Ay,
0<v<jr+ 0<v<.r"
A~ ? = inf Ay — inf Ay,
0<v<.r- 0<v<.r +
|A|*= sup Ay— sup Ay = sup \Ay\ — sup \Ay\.
|?|<?·+ |.??.? 1уКл+ I.vK-?-
If X is equipped with a lattice norm, let X* denote the usual dual space, i.e., X* : =
?(X,E).
THEOREM 27. Let ?, ? be equipped with lattice norms. Then we have:
A) X* с X~, a/jii X* is even an ideal in X~. The norm ofX* is a corresponding lattice
norm.
B) IfX is complete, then Cb(X, Z) с ?(X, Z); in particular X* = X~.

Riesz spaces and ideals of measurable functions
819
We note that for ? ? R the set ?(X, Z) is in general not contained in Cj,(X, Z). In
case X = ? = ?.2([0, 1]), for example, there exist integral operators A e ?(X, Z) which
fail to be regular (and thus fail to be order bounded). The first example of such an integral
operator was given by Mytiagin, see Example 4.3 in Krasnoselskii et al. A958).
Theorem 27 implies that in order to study the dual space X*, it is sufficient (and even
better) to study the order dual X~.
DEHNITION 25. An operator A e (X, Z) is ?-order continuous if x„ 4 0 implies
inf{| Ajc„|: n} = 0. Similarly, A is called order continuous if ? 4 0 implies inf{| Ajt |: ? e
M}=0.
The set of all ?-order continuous operators A e ?/>(X, Z) is denoted by CC(X, Z), and
the set of all order continuous operators is denoted by ?„ (X, Z).
In case ? = R, the elements of X~ := ?((X,R) are also called integrals, and the
elements of X~ := ?,,(X, R) are called normal integrals.
The term "integral" will be explained later. Proposition 3 immediately implies:
PROPOSITION 25. We have ?„(X, Z) с ?(.(Х, Z) with equality ifX is order separable.
The reader should be warned that order separability is not necessary for ?„(X, Z) —
?C(X, Z) (even if ? — R). The question whether X~ and X~ differ is rather delicate, see,
e.g., the discussion in Paragraph 87 of Zaanen A983). The following result gives a different
characterization of the set of (a-)order continuous operators.
THEOREM 28. ?C(X, Z) and ?H(X, Z) are bands in ?/>(X, Z). In particular, we havefor
any operator A e ?^(?, ?):
A) A is ? -order continuous if and only if\A\ has this property. This is the case if and
only if xn 4 0 implies \A\x„ 4 0.
B) A is order continuous if and only if\A\ has this property. This is the case if and only
if ? 4,0 implies |A|M 4,0.
Since ?C(X, Z) and ?,,(X, Z) are bands in Cb(X, Z), Proposition 10 implies that the
corresponding disjoint complements ?tJ(X, Z) := ?( (X, Z)L respectively ?,«(X, Z) :=
?„(X, ZI- are bands, and
Cb(X, Z) = ?C(X, ?) ? ?„(X, Z) = C„(X, ?) ? C„S(X, Z). A0)
The operators in ?C.?(X, Z) are called singular, and the operators in C„.,(X, Z) are called
normal singular.
To describe the corresponding projection, it suffices to consider positive operators,
because any A e Сь(Х, ?) is the difference of two positive operators. Moreover, since
? = x+ — x~ for any ? e X, it suffices, by linearity, to describe the projection for positive
values. In this sense, the following result gives an "explicit" formula for the corresponding
projection.

820
?. Vath
THEOREM 29. Let A e Сь(Х, ?) be positive, and Ac and A„ be the corresponding
component in ?C(X, Z) and C„(X, Z), respectively. Then
Acx =inf{supA;c„: 0^ x„ | x\ {? ^ 0),
/I
A,,jc = infjsup A(M): X+ 2 ?? | jc } (jc > 0).
If A e?fc(X,Z), then thcnull ideal is defined as the set {jc: |A||jt| =0}.lf A e ?,(X, Z)
or A e ?n(X, Z), then the null ideal is a ?-ideal respectively a band. The converse does
not hold in general, even in case of functionals (Z = R). But slightly less does hold.
THEOREM 30. For any A e?(.(X, Z) (A e?„(X,Z)) the null ideal [x: \A\\x\ =0} is a
?-ideal (respectively a band).
Conversely, if A e ?/>(X, Z) /s smc/i f/ш? for any positive operator В e ?/>(X, Z)
vv/fft 0 ^ В ^ |A| f/ie null ideal {x: B\x\ = 0} /s ? ?-ideal (respectively a band), then
A e CC(X, Z) (respectively A e ?„(X, Z)).
Corollary 15. We have CC(X, Z) = Сь(Х, ?) if and only if for any positive operator
В е Сь(Х, Z) the null ideal is ? ?-ideal. Similarly, ?„(X, Z) = ?/,(X, Z) //a/id ои/у //
for any positive operator В е Сь(Х, Z) the null ideal is a band.
In case ? = R, we denote the band of singular and normal singular functionals by
X~ := ?„(X, Z) and X~s := ?„.S(X, Z), respectively. If X is equipped with the lattice
norm, we have in view of Theorem 27 and A0)
X* = (X* П X~) ? (X* ПХ~) = (X* П X~ ) ? (X* П Х~ ). A1)
Proposition 10 immediately implies:
PROPOSITION 26. The spaces occurring /и A1) are all closed subspaces of X*. and the
corresponding projections are positive and have norm 1 (if they are not trivial).
5.4. Duals of ideals of measurable functions
For the rest of this chapter, let S be a ?-finite measure space, ? = R, and X be an ideal in
Wl(S). To simplify notation, we will assume throughout that supp X = S which is actually
no loss of generality (consider the measure space S := supp X instead of S, if necessary).
Since X is an ideal in ffl(S), and Vft(S) is super Dedekind complete by Theorem 3,
it follows that also X is super Dedekind complete and in particular order separable.
Proposition 25 thus implies:
PROPOSITION 27. For any Dedekind-complete Riesz space ? we have the equality
?,(X,Z) = ?„(X, Z).

Riesz spaces and ideals of measurable functions
821
In particular, all integrals on X are normal. The following result characterizes the
integrals and is the reason for the term "integral": Similarly to the classical calculation of
the dual space of Lp, the proof of this result is essentially based on the Radon-Nikodym
theorem.
THEOREM 31. Afunctional f e X~~ belongs to X~ = X~ if and only if there is some
у e 9JiE") such that у ¦ ? is integrablefor each ? e X and
f{x)= ? y(s)x(s)ds (xeX). A2)
Moreover, у is (almost everywhere) uniquely determined by this property, and
f±(x)= ? y±(s)x(s)ds (xeX), A3)
\f\(x) = f\y(s)\x(s)ds (xeX). A4)
The formulas A3) and A4) are straightforward consequences of the corresponding
formulas in Theorem 26.
COROLLARY 16. Let X be also a preideal space. Then the bounded integrals f e
X~ ? ?* = ?~ ? ?* are in a canonical one-to-one correspondence with the elements
у ofthe associate space X' which is given by A2). Moreover, \\f\\ = \\\f\\\ = \\y\\x'·
Corollary 11 implies that if X is a regular preideal space, then any positive functional
/ e X* is ?-continuous, and thus any / e X* is an integral. Using the Hahn-Banach
theorem, it can be proved that this trivially sufficient condition is actually necessary. An
analogous result holds for the corresponding trivially sufficient condition for reflexivity.
THEOREM 32. Let X be also a preideal space.
A) An element ? e X belongs to the regular part X,· if and only if f(x) = 0 for any
singular f e X*. In particular, X is regular if and only if any f e X* is an integral.
B) X is reflexive if and only if X and X' both are regular and X is perfect.
Theorem 32 is the natural generalization of the classical result that (in the canonical
way) Lp([0, 1])* = Lp>([0, 1]) with l/p+l/p' =\ if and only if 1 ^ ? < oo and that
Lp([0, 1]) is reflexive if and only if 1 < ? < oo.
It is a curiosity that, without the axiom of choice, the necessity of the conditions in
Theorem 32 completely fails: If we do not assume the axiom of choice (only the weaker
so-called axiom of dependent choices), we cannot exclude that X* = X' holds for any
preideal space (regular or not), and consequently any perfect ideal space is reflexive, see
VathA998).
The connection of the regular part X,· of X with singular functionals is emphasized by
the following result.

822
?. Vath
THEOREM 33. Let X be also a preideal space and suppose that supp Xr = S (= supp X).
Then afunctional f e X* is singular if and only if f vanishes on Xr. Moreover, X* = X'
in the sense of A2).
Note that, conversely, if X is a preideal space for which suppX, is strictly smaller
than S, then any nonzero function у e X' with supp у П suppXr = 0 (such a function
exists by Theorem 22) determines a nontrivial integral / e X* via A2) which vanishes on
Xr (hence, also ?* ? ?').
5.5. Bibliographical remarks
The omitted proofs of Section 5.1 can, e.g., be found in Section 3.4 of Vath A997).
Slightly different proofs for the most important Theorems 22 and 23 are in Paragraph 71,
Theorems 4(a) and 2, of Zaanen A967). For an immediate proof of Theorem 22 (without
applying Theorem 23), see Zabreiko A974).
Theorem 24 was the original motivation for the Luxemburg-Gribanov theorem, and this
is what was actually proved by Luxemburg A958) (for perfect spaces). For the proof of
the various statements in Theorem 25, see Sections 4.1 and 4.3 of Vath A997). For some
applications, it is useful to consider also the case where the space ? ? in (9) may depend
on 52- If this dependence is not "too bad" (i.e. continuous or at least measurable in a certain
sense), many results carry over by the earlier mentioned generalization of the Luxemburg-
Gribanov theorem by Vath B00?). However, the reader should be warned that even if each
of the spaces ? ? (S2) and also Xi have full support, it may happen that the corresponding
"mixed norm space" is trivial. Such an example, based on the continuum hypothesis, is
given in Example 4.3.3. of Vath A997).
The theory of positive operators is one of the earliest results in the theory of Riesz spaces.
Correspondingly, Theorem 26 already occurs in the papers of Riesz A930), Kantorovich
A936), and Freudenthal A936), at least for functionals. For the proof of Theorems 27
and 28, see Zaanen A983), Theorems 85.6 and 83.12, Lemma 84.1 and Theorem 84.2.
Theorems 29 and 30 are reformulations of Theorems 22.9 and 22.3 of Zaanen A997), and
the main part of Theorem 31 can be found as Theorem 86.3 of Zaanen A983). The proofs
of Theorems 32 and 33 can be found in Paragraphs 72 and 73 of Zaanen A967).
For ideal spaces X с Wl(S, Y) when У is a Riesz spaces, some of the results in
Section 5.4 have been generalized by Mullins A974, 1976).
References
Aliprantis, CD. and Burkinshaw, O. A976), Locally Solid Riesz Spaces, Academic Press, New York.
Aliprantis, CD. and Burkinshaw, O. A977), On universally complete Riesz spaces. Pacific J. Math. 71, 1-12.
Aliprantis, CD. and Burkinshaw, O. A980a), Minimal topologies and Lp-spaces, Illinois J. Math. 24, 164-172.
Aliprantis, CD. and Burkinshaw, O. A980b), On the structure of locally solid topologies, Canad. Math. Bull. 23,
185-191.
Aliprantis, CD. and Burkinshaw, O. A985), Positive Operators, Academic Press, New York.
Amemiya, I. A953), A generalization of Riesz-Fischer's theorem, J. Math. Soc. Japan 5, 353-354.

Riesz spaces and ideals of measurable functions
823
Appell, J, and Zabreiko, PR A990), Nonlinear Superposition Operators. Cambridge Univ. Press, Cambridge,
Bergh, J, and Lofstrom, J. A976), Interpolation Spaces -An Introduction. Springer, Berlin,
Birkhoff, G, A967), Lattice Theory, 3rd edn, Amer. Math, Soc. Coll, Publ,, Providence, R.I,
Bukhvalov, A,V. A972), Vector-valued function spaces and tensor products (in Russian). Siberian Math, J, 13
F), 1229-1238.
Bukhvalov, A.V. A978), Geometric properties of Banach spaces of measurable vector valued functions. Dokl,
Akad, Nauk SSSR 239 F), 1279-1282.
Chong, K,M, and Rice, N,M. A971), Equimeasurable rearrangements and functions. Queen's Papers in Pure and
Applied Math., No, 28, Queen's University, Kingston Ontario, Canada.
Dirr, G, and Vath, M, B000), Continuity ofnear-dualin maps and characterizations of ideal spaces of measurable
functions. Recent Trends in Nonlinear Analysis, Appell, J., ed.. Festschrift Dedicated to Alfonso Vignoli on
the Occassion of His Sixtieth Birthday, Birkhauser, 139-148.
Feledziak, K, and Nowak, M. A997), Locally solid topologies on vector valued function spaces. Collect. Math.
48 D-6), 487-511.
Fremlin, D,H, A974), Topological Riesz Spaces and Measure Theory. Cambridge Univ. Press, Cambridge.
Freudenthal, H, A936), Teilweise geordnete Moduln. Proc. Netherl. Acad, Sci, 39, 641-651.
Gribanov, Yu,I. A970), On the measurability of a function, lzv. Vyssh. Uchebn, Zaved. Mat. 3, 22-26 (in
Russian),
Kantorovich, L,V, A936), Concerning the general theory of operations in partially ordered spaces, Dokl, Akad,
Nauk SSSR 1,271-274,
KrasnoselskiT, M,A,, Lifshits, Je,A, and Sobolev, A.V. A989), Positive Linear Systems. Heldermann, Berlin,
KrasnoselskiT, M.A, and Rutickil, Ya.B. A958), Convex Functions and Orlicz Spaces. Fizmatgiz, Moscow (in
Russian), English translation! Noordhoff, Groningen, 1961.
KrasnoselskiT, M,A,, Zabreiko, PR, Pustylnik, E.I, and Sobolevskil, RE. A958), Integral Operators in Spaces of
Summable Functions, Nauka, Moscow (in Russian). English translation: Noordhoff, Leyden, 1976.
KreTn, S.G,, Petunin, Ju,I, and Semenov, E.M. A978), Interpolation of Linear Operators. Nauka, Moscow (in
Russian), English translation: Providence, RI, Amer. Math. Soc., 1982.
Kutateladze, S,S, (ed,) A996), Vector Lattices and Integral Operators, Kluwer, Dordrecht.
Lindenstrauss, J, and Tzafriri, L, A979), Classical Banach Spaces II: Function Spaces. Springer, Berlin.
Lorentz, G.G, A950), Some new functional spaces. Ann. Math. B) 51 A), 37-55.
Lorentz, G.G. A951), On the theory of spaces ?. Pacific J. Math. 1, 411^429.
Lorentz, G.G. A961), Relations between function spaces. Proc. Amer. Math. Soc. 12 A), 127-132.
Luxemburg, W,A,J. A958), On the measurabilirv of a function which occurs in a paper by A.C Zaanen, Proc.
Netherl. Acad. Sci. (A) 61, 259-265 (Indag. Math. 20, 259-265).
Luxemburg, W,A,J, A963), Addendum to On the measurability of a function which occurs in a paper by A.C.
Zaanen. Proc, Netherl. Acad, Sci, (A) 66, 587-590 (Indag. Math. 25 A963), 587-590).
Luxemburg, W.A.J, A965a), Notes on Banach function spaces XIV A. Proc. Netherl, Acad. Sci. (A) 68, 229-239
(Indag. Math, 27 A965), 229-239).
Luxemburg, W,A,J. A965b), Notes on Banach function spaces XIV в. Proc. Netherl. Acad, Sci. (A) 68, 240-248
(Indag, Math, 27 A965), 240-248).
Luxemburg, W.A.J, A965c), Notes on Banach function spaces XV \. Proc. Netherl. Acad. Sci. (A) 68, 415^429
(Indag, Math, 27 A965), 415^429).
Luxemburg, W,A,J, A965d), Notes on Banach function spaces XVB, Proc, Netherl. Acad. Sci. (A) 68, 430-446
(Indag. Math. 27 A965), 430-446).
Luxemburg, W.AJ. A965e), Notes on Banach function spaces XVIA. Proc. Netherl. Acad. Sci. (A) 68 A), 646-
657 (Indag. Math. 27 D) A965), 646-657).
Luxemburg, W.A.J. A965f), Notes on Banach function spaces XVIB. Proc. Netherl. Acad. Sci. (A) 68 A), 658-
667 (Indag. Math. 27 D) A965), 658-667).
Luxemburg, W.A.J. A967), Rearrangement invariant Banach function spaces. Queen's Papers in Pure and Appl.
Math., No. 10, 83-144.
Luxemburg, W.A.J. and Zaanen, A.C. A963a), Compactness of integral operators in Banach function spaces.
Math. Ann, 149, 150-180.
Luxemburg, WA.J, and Zaanen, A.C. A963b), Notes on Banach function spaces I. Proc. Netherl. Acad. Sci. (A)
66, 135-147 (Indag. Math. 25 A963), 135-147).

824
?. Vath
Luxemburg, W.A.J, and Zaanen, A.C. A963c), Notes on Banach function spaces II, Proc. Netherl. Acad. Sci. (A)
66, 148-153 (Indag. Math, 25 A963), 148-153).
Luxemburg, W.A.J, and Zaanen, A.C. A963d), Notes on Banach function spaces III, Proc, Netherl. Acad. Sci.
(A) 66, 239-250 (Indag. Math. 25 A963), 239-250).
Luxemburg, W.A.J, and Zaanen, A.C. A963e), Notes on Banach function spaces IV, Proc. Netherl. Acad. Sci.
(A) 66, 251-263 (Indag. Math. 25 A963), 251-263).
Luxemburg, W.A.J, and Zaanen, A.C. A963f), Notes on Banach function spaces V, Proc. Netherl. Acad. Sci. (A)
66, 496-504 (Indag. Math. 25 A963), 496-504).
Luxemburg, W.A.J, and Zaanen, A.C. A963g), Notes on Banach function spaces VI, Proc. Netherl. Acad. Sci.
(A) 66, 655-668 (Indag. Math. 25 A963), 655-668).
Luxemburg, WA,J. and Zaanen, A.C. A963h), Notes on Banach function spaces VII, Proc. Netherl. Acad. Sci.
(A) 66, 669-681 (Indag. Math. 25 A963), 669-681).
Luxemburg, W.A.J, and Zaanen, A.C. A964a), Notes on Banach function spaces VIII, Proc. Netherl. Acad. Sci.
(A) 67, 104-119 (Indag. Math. 26A964), 104-119).
Luxemburg, W.A.J, and Zaanen, A.C. A964b), Notes on Banach function spaces IX, Proc. Netherl. Acad. Sci.
(A) 67, 360-376 (Indag. Math. 26 A964), 360-376).
Luxemburg, W.A.J, and Zaanen, A.C. A964c), Notes on Banach function spaces X, Proc. Netherl. Acad. Sci. (A)
67, 493-506 (Indag. Math. 26 A964), 493-506).
Luxemburg, W.A.J, and Zaanen, A.C. A964d), Notes on Banach function spaces XI, Proc. Netherl. Acad, Sci.
(A) 67, 507-518 (Indag. Math. 26 A964), 507-518).
Luxemburg, WA.J. and Zaanen, A.C. A964e), Notes on Banach function spaces XII, Proc. Netherl. Acad. Sci.
(A) 67, 519-529 (Indag. Math. 26 A964), 519-529).
Luxemburg, W.A.J, and Zaanen, A.C. A964f), Notes on Banach function spaces XIII, Proc. Netherl. Acad. Sci.
(A) 67, 530-543 (Indag. Math. 26 A964), 530-543).
Luxemburg, W.A.J, and Zaanen, A.C. A971), Riesz Spaces, Vol. I, North-Holland, Amsterdam.
Macdonald, A.L. A973a), Vector valued Kothe function spaces I, Illinois J. Math. 17, 533-545.
Macdonald, A.L. A973b), Vector valued Kothe function spaces II, Illinois J. Math. 17, 546-557.
Macdonald, A,L. A974), Vector valued Kothe function spaces III, Illinois J. Math. 18, 136-146.
Macdonald, A.L. A976), ? weak theory of vector valued Kothe function spaces, Illinois J. Math. 20, 410-424.
Marcinkiewicz, J. A939), Sur I 'interpolation d'ope'rateurs, C. R. Acad. Sci. Paris Ser. I Math. 208, 1272-1273.
Meyer-Nieberg, P. A991), Banach Lattices, Springer, Berlin.
Milman, M.M. A978), Some new function spaces and their tensor products. Bull. Austral. Math. Soc. 19, 147-
149.
Mullins, C.W A974), Linear junctionals on vector valued Kothe spaces, Proc. Madras Conference Functional
Analysis 1973 (Berlin), Garnir, H.G., Unni, K.R. and Williamson, J.H,, eds. Lecture Notes in Math., Vol. 399,
Springer, Berlin, 380-381.
Mullins, C.W. A976), Order vector valued Kothe spaces, J. London Math. Soc. B) 13 A), 34-40.
Nakano, H. A950), Modulared Semi-Ordered Linear Spaces, Maruzen, Nihonbashi, Tokyo.
Nguyen, H.T. A987), The superposition operator in Orlicz spaces of vector valued functions, Dokl. Akad. Nauk
BSSR 31, 197-200 (in Russian).
Nguyen, H.T. A991), The theory of semimodules of infra-semiunits in ideal spaces of vector-valued functions,
and its applications to integral operators, Dokl. Akad. Nauk SSSR 317, 1303-1307 (in Russian), English
translation: Soviet Math, Dokl. 43A991), 615-619.
Nguyen, H.T, and Zabrelko, P.P. A990), Cones of vector functions in Orlicz spaces of vector functions (in
Russian), VescT Akad. Navuk BSSR Ser. FIz.-Mat. Navuk, No. 3, 30-34.
Nowak, M. A984a), On the finest of all linear topologies on Orlicz spaces for which ?-modular convergence
implies convergence in these topologies. Bull. Acad. Polon. Sci. Ser. Sci. Math. 32, 439^t45.
Nowak, M, A984b), On the modular topology on Orlicz spaces, Bull. Acad. Polon. Sci, Ser. Sci. Math. 36,41-50.
Nowak, M, A993), Order continuous seminorms and weak compactness in Orlicz spaces. Collect. Math. 44,
217-236,
O'Neil, R. A963), Convolution operators arid L(p,q) spaces, Duke Math. J. 30, 129-142.
Povolotskil, A.I. and Kalitvin, A.S. A991), Nonlinear Partial Integral Operators, Leningrad, 1991 (in Russian).
Rao, M,M. and Ren, Z,D. A991), Theory of Orlicz Spaces, M. Dekker, New York.

Riesz spaces and ideals of measurable functions
825
Riesz, F. A930), Sur la decomposition des operations fonctionnelles line'aires, Atti Congr. Intemaz. Mat. Bologna,
1928, Vol. 3, 143-148,
Rudin, W. A987), Real and Complex Analysis, 3rd edn, McGraw-Hill, Singapore.
Rudin, W. A990), Functional Analysis, 14th edn, McGraw-Hill, New York.
Schaefer, H.H. A974), Banach Lattices and Positive Operators, Springer, Berlin.
Semadeni, Z. A971), Banach Spaces of Continuous Functions, Vol. I, Polish Scientific Publ., Warszawa.
Sikorski, R. A964), Boolean Algebras, Springer.
Vath, M. A997), Ideal Spaces, Lecture Notes in Math., Vol. 1664, Springer.
Vath, M. A998), The dual space of Lx is L\, Indag. Math. 9 D), 619-625.
Vath, M. B000), Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York.
Vath, M. B00?), Some measurability results and applications to spaces with mixed family-norm (submitted).
Wnuk, W. A986), On a continuous embedding into a space of measurable functions. Bull. Acad. Polon. Sci. Ser.
Sci. Math. 34, 413^416.
Zaanen, A.C. A967), Integration, North-Holland, Amsterdam.
Zaanen, A.C. A983), Riesz Spaces, Vol. II, North-Holland, Amsterdam.
Zaanen, A.C. A997), Introduction to Operator Theory in Riesz Spaces, Springer, Berlin.
Zabreiko, P.P. A974), Ideal spaces of functions I, Vestnik Jaroslav. Univ. 8, 12-52 (in Russian).

CHAPTER 20
Measures on Quantum Structures
Anatolij Dvurecenskij*
Mathematical Institute, Slovak Academy of Sciences, Stefa'nikova 49, SK-814 73 Bratislava, Slovakia
E-mail: dvurecen@mat.savba.sk
Contents
1. Introduction 829
2. States on quantum logics 831
2.1. Orthomodular lattices and orthomodular posets 831
2.2. Orthoalgebras and effect algebras 833
2.3. Decompositions of states 836
3. Gleason's measures 840
3.1. Introduction 840
3.2. Generalizations of Gleason's theorem 842
3.3. Gleason's theorem and completeness criteria of inner product spaces 848
4. States on MV-algebras 853
5. Measures on BCK-algebras 856
5.1. Commutative BCK-algebras with the relative cancellation property 856
5.2. Measures on commutative BCK-algebras with the relative cancellation property 858
6. States on pseudo MV-algebras 860
6.1. Pseudo MV-algebras 860
6.2. States and pseudo MV-algebras 863
6.3. Existence of states 865
7. Conclusion 866
References 866
*The work has been supported by the grant 2/7193/20 SAV, Bratislava, Slovakia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
827

This page intentionally left blank

Measures on quantum structures
829
1. Introduction
D. Hubert, in his famous program presented at the Second International Mathematical
Congress in Paris, 1900, formulated many open mathematical problems which were
stimulating for the development of mathematics and were a fruitful source of very deep
and fundamental ideas. During the whole 20th century, mathematicians and specialists in
other fields have been solving problems which can be traced back to Hilbert's program,
and today there are many basic results stimulated by this program. It is sure that even at
the beginning of the third millennium, mathematicians will still have much to do.
One of his most interesting ideas, lying between mathematics and physics, is his sixth
problem: to find a few physical axioms which, similar to the axioms of geometry, can
describe a theory for a class of physical events that is as large as possible.
We try to present some ideas inspired by Hilbert's sixth problem and give some partial
results which may contribute to its solution, mainly we will concentrate at measure-
theoretical point of view of solving such a problem.
In the Thirties the situation in both physics and mathematics was very interesting.
A.N. Kolmogorov published his fundamental work [48] Grundbegriffe der Wahrschein-
lichkeitsrechnung in which he, for the first time, axiomatized modern probability theory.
From the mathematical point of view, in Kolmogorov's model, the set L of
experimentally verifiable events forms a Boolean ? -algebra and, by the Loomis-Sikorski theorem,
roughly speaking can be represented by a ? -algebra S of subsets of some non-void set ?.
The Kolmogorov model is very important, but it does not describe the situation that
arises in quantum mechanics. A physical system is investigated through a measurement
process. If a measurement reaches the microlevel, non-negligible indeterminacies appear
that must be respected in the model. Therefore, in many cases, the use of classical
probability theory is not sufficient, as follows, for example, from the Heisenberg uncertainty
principle [43] which asserts that the position ? and the momentum ? of an elementary
particle cannot be measured simultaneously with arbitrarily prescribed accuracy. If ?,„ ?
and ?,?? denote the inaccuracies of the measurement of the momentum ? and position ?
in a state m, then
(AlnPJ-(AmxJ^-h2>0, A.1)
where ft = h/?? and h is Planck's constant.
Similar relation holds for the energy-life time
(AlnEJ.(AmtJ^-h2>0,
It is interesting to notify that similar uncertainty relations is possible to find also in other
scientific areas, for example social sciences, psychology, the activity of the human brain,
automata. In psychology it is known that when we examine a patient through questions,
the patient's psychological stress and the accuracy of his answers are roughly speaking
connected via A.1),

830
A. Dvurecenskij
J. von Neumann [71 ] made the first fundamental step towards a rigorous mathematical
apparatus, as well as an interpretational logic of the new mechanics on the basis of a
complex separable Hubert space.
In 1936, G. Birkhoff and J. von Neumann published their paper [6] The Logic of
Quantum Mechanics. In this historic paper the authors showed that the set of assertions
of quantum mechanics has a different algebraic property from a Boolean algebra, and they
suggested the structure of a projective geometry.
The physical system is described through a measuring process which is described by
a state which is a probability measure (in many case countably additive) on the physical
system.
From this, we have that the system L(H) of all closed subspace of a real, complex, or
quaternion Hubert space is of particular interest. G. Mackey [53] in his investigation of the
mathematical foundations of quantum mechanics posed the following problem: describe
the set of all states on the quantum logic L{H) for a separable real or complex Hilbert
space. A.M. Gleason [37] published the answer to Mackey's problem in 1957, and this
result deeply influenced the progress in the Hilbert space quantum mechanics. In addition,
Gleason's theorem influenced also mathematical research and the theorem was extended
to many cases and structures.
A turbulent development of quantum theories started in the Sixties, and there appeared
many important results which were reflected in numerous monographs. There appeared
structures like orthomodular posets and orthoalgebras. Today all these structures which
were inspired by attempts of describing mathematical foundations of quantum mechanics
are called quantum structures, and in the beginning of the Nineties the International
Quantum Structure Association was founded which organizes Biannual Congresses on
which mathematicians, physicists, logicians, and philosophers meet.
In the last decade, Slovak and Italian schools contributed with an important new notion,
D-posets (difference posets), or equivalently, weak orthoalgebras. Later the American
school presented effect algebras. All these equivalent algebraic partial structures combine
algebraic as well as fuzzy set ideas. It was shown that the space of effect operators of a
Hilbert space and also very important MV-algebras can be formalized within this frame.
In 1958, C.C. Chang [8] presented a new axiomatic structure, MV-algebras, which
generalizes the two-valued reasoning, taking only values 0 and 1, to infinitely-valued one
taking values from the whole real interval [0, 1]. Later in 1986, D. Mundici [59] showed
that there is a natural equivalence among MV-algebras and Abelian unital lattice-ordered
groups with strong unit as the interval [0, и], where и is a strong unit. Today MV-algebras
present an important substructure of quantum structures while, for example, a compatible
part (i.e., "classical one") of lattice-ordered effect algebras forms an MV-subalgebra.
Another generalization of Boolean rings are BCK-algebras which were introduced by
Imai and Iseki [44] inspired by prepositional calculus and set theory, and they have been
intensively studied by many authors. Also for the case of commutative BCK-algebras with
the relative cancellation property there exists an ?-group representation, [25].
Recently, G. Georgescu and A. Iorgulescu [35] introduced pseudo MV-algebras which
are a non-commutative generalization of MV-algebras which are a generalization of a
two-valued reasoning. A non-commutative reasoning can be found, e.g., in psychological
processes: in clinical medicine on behalf of transplantation of human organs, an experiment

Measures on quantum structures
831
was performed in which the same two questions have been posed to two groups of
interviewed people:
A) Do you agree to dedicate your organs for medical transplantation after your dead?
B) Do you agree to accept organs of a donor in the case of your need?
If the order of questions was changed in the second group, the number of positive answers
was more higher than in the first group.
For these pseudo MV-algebras, the author [21] presented a representation via intervals
[0, и] where и is a strong unit in an ?-group not necessarily Abelian.
From the measure-theoretical point of view all above structures present also important
algebraic structures where it is possible to built up measure theory as well as probability
theory and which will be done below. We will see how nice mathematical constructions it
was necessary to present to show for example that there are even finite quantum logics
having no state, which in the Kolmogorov model is a trivial fact that there is plenty
of probability measures. Or we hope to be fascinated with the mathematical beauty of
Gleason's theorem to which a monograph was dedicated [16].
2. States on quantum logics
2.1. Orthomodular lattices and orthomodular posets
States on quantum structures are the basic way through which we are able to derive an
information from the physical system under measurement. States are roughly speaking
probability measures and they describe the physical system. For classical Kolmogorov
model, we have plenty of states which are ? -additive probability measures, e.g., the system
of Dirac measures {<5?: ? e ?] determines completely the order on the corresponding ?-
algebra S of subsets of ?. If we take Boolean algebras, they have plenty of finitely additive
states, but there exists an example, [68], of a Boolean ?-algebra having only plenty of
finitely additive states but no ? -additive state.
We recall that an orthomodular poset (OMP, for short) is a poset L with two elements
0 ^ 1 and a unary operation -1, called an orthocomplementation such that, for all a,b e L,
we have
(i) ?±± = ?;
(ii) b1- ^ a1- whenever a ^ b\
(iii) a v a1- = 1;
(iv) a v b & L whenever a ^ b^\
(v) b = a v (b л a^) whenever a ^ b (orthomodular law).
We write a ±b whenever ? ? b-1, and we say that a and b are orthogonal or mutually
exclusive. If an OMP is a lattice, we call it an orthomodular lattice (OML for short); if it is
a ?-complete lattice, we called it a quantum logic; if \/,? ? ?\ exists in L for any sequence
of mutually orthogonal elements {a,} of L, L is said to be an ortho-?-complete logic.
For example, if S is a ?-algebra of subsets of ? ? 0, then putting 0 := 0, 1 := ?,
A1- := ? \ A, and ^ := c, we see that S is a quantum logic.
The most important example of quantum logics is the quantum logic L(H) of a real,
complex or quaternion Hubert space H, where L(H) consists of all closed subspaces ?

832
A. Dvurecenskij
of H, and ?1- denotes its orthogonal complement; the partial order is the set-theoretical
inclusion. This example is of basic importance for the so-called Hubert space quantum
mechanics. An equivalent structure is the system P(H) of all orthogonal projectors on H.
We say that two elements a and b of an OMP L are compatible if there exist three
mutually orthogonal elements a\,b\,c'\TiL such that a = a \ ? с and b = b\ ? с. We recall
that an OMP L is a Boolean algebra iff all elements of L are compatible. This property
shows that a system of mutually compatible elements generates a Boolean subalgebra
of L. Moreover, this property determines areas where we can use classical mechanics,
or a classical Kolmogorov model. Anyway, every our structure has a locally classical part,
and any maximal Boolean subalgebra is said to be a block. It is not hard to show that every
OMP can be covered by the set of all its blocks, but some of them can not interact with
others.
We recall that a state on a OMP L is a mapping m : L —>¦ [0, 1] such that w(l) = 1,
and m(a ? b) =m{a) + m{b) whenever a _L b. If m is ?-additiveon L, i.e., m(\J',· a,·) =
J2i w (a,) for any sequence of mutually orthogonal elements {a,} for which V, ai is defined
in L, we called it a ?-additive state. In a similar way we define a completely additive state.
The problem of existence of states on quantum logics, such as orthomodular lattices or
orthomodular posets, showed that the situation with existence of states on quantum logic is
a complicated problem, and it generated new ways of constructions of finite orthomodular
lattices and posets, respectively. The main contribution is due to Greechie [39,40].
Posets are pictured via Hasse diagrams, which in the case of many elements became very
blind. A more elegant and simple method was suggested by R. Greechie. His method allows
to construct new quantum structures and it is a base for many interesting examples. This
method of construction of quantum logics uses the pasting of Boolean algebras, based on
the familiar assertion that a finite Boolean algebra is completely determined by its atoms.
He supposed that a quantum logic has a special form: for any two blocks A and В of L
only one of the conditions is satisfied:
(i) A = B;
(ii) АПб = {0, 1);
(iii) А П В = {0, 1, ?, ?-1}, where a is an atom in A and B.
Such logics are said to be Greechie logics. The Greechie diagram consists of the set of
points and lines. Points represent atoms in blocks, and atoms of a given block are connected
by a smooth line. An atom which belongs to more blocks lies on more smooth lines which
are "crossing" at the point corresponding to this atom.
We notice that a more general method of logic construction using pasting of so-called
astroids has been suggested by M. Dichtl [11], and a method of pasting of OMPs is used
in [61] by M. Navara.
For example, the Boolean algebra 23 which has three atoms a,b,c has the Greechie
diagram consisting of a line with three atoms while the Hasse diagram has eight points and
twelve lines.
The propositional logic of the model consisting from pasting three Boolean algebras 23,
where each two blocks have only one atom in common, has the Greechie diagram given by
Figure 1 and the corresponding Hasse diagram by Figure 2 (this is an orthoalgebra as we
will see below).
The following Greechie diagram with Figure 3 gives an OMP which has no state.

Measures on quantum structures
833
/
Fig. I.
0
Fig. 2.
II II 1. II
Fig. 3.
2.2. Orthoalgebras and effect algebras
D.J. Foulis and C.H. Randall [33] introduced orthoalgebras which are more general than
OMPs. We recall that an orthoalgebra is a set L with two particular elements 0, 1 @ ? 1),
and with a partial binary operation ?: L ? L —>· L such that for all a, b, с е L we have:
(OAi) if ? © b e L, then b®a e L and ? ? fo — b ? ? (commutativity);
(OAii) if fo ? с e L and ? ? (b ? с) e L, then ? ? fo e L and (? ? fo) ? с е L, and
? ? (fo ? с) = (а ф fo) ? с (associativity);

834
A. Dvurecenskij
(OAiii) for any a e L there is a unique be L such that ? © b is defined, and a ®b—\
(orthocomplementation);
(OAiv) if ? ?? is defined, then a = 0 (consistency).
If the assumptions of (OAii) are satisfied, we write ? ? b ? с for the element (? ? b) ? с =
? ? (fr ? с) in L. We introduce a partial order ^ on L via a ^ fo iff а ф с — fo for some
с е L, and we write a _L fo iff ? ? fo exists in L. A iiaie is a mapping m : L —> [0, \] such
that w(l) = 1 and m(a ®b) = m(a) +m(b).
For example, if we put ? ? b := ? ? b whenever a _L b in an OMP, then we see that any
OMP is an orthoalgebra. Figure 2 gives an example of orthoalgebra which is not OMP
We recall that quantum mechanics is not a unique source of inspiration for quantum
logics theory. We now present one more-less realistic example of an experiment whose
propositional structure satisfies only axioms of orthoalgebras.
EXAMPLE 2.1 {Firefly in a three-chamber box). Consider a firefly in a three-chamber box
pictured in Figure 4.
The firefly is free to roam among the three chambers and to light up to will. The sides
of the box are windows with vertical lines down their centers. We make three experiments,
corresponding to the three windows А, В and C. For each experiment E, we record
1е,ге,пе if we see, respectively, a light to the left, right, of the center line or no light.
It is clear that we can identify гд = /с =¦' e, re = lв ='¦ с, rg — /д —: ?, but now we do not
identify / :=ид, b :=пв, d := nC-
The propositional logic of this model has the Greechie diagram given by Figure 2.
For orthoalgebras we can reconstruct the original ideas of G. Boole [3] who said that for
our reasoning we need to know only disjunction of mutually exclusive elements and not of
all pairs of elements. This is also the requirement for the definition of states.
In the beginning of the Nineties, two my former students, F. Корка and F. Chovanec [49]
introduced a new structure, difference poset, D-poset for short. This structure requires
difference of comparable elements as a primary notion. An equivalent partial structure is
an orthoalgebra introduced by D. Foulis and M.K. Bennett [32], see also [36] with addition
as a primary notion.
An effect algebra is a non-empty set L with two particular elements 0, 1 @^1), and
with a partial binary operation ©: L ? L —>¦ L such that, for all a, b, с е L, we have
/«/V
/ \
/ \
I , \
Fig. 4.

Measures on quantum structures
835
(EAi) if ? © b e L, then b®a e L and ? ? b = b © a (commutativity);
(EAii) if b ? с e L and ? ? (fo ? с) e L, then ? ? fo e L and (? © b) ? с е L, and
? ? (b © с) = (? ? b) ? с (associativity);
(EAiii) for any a e L there is a unique b e L such that ? ? b is defined, and ? ? b = 1
(orthocomplementation);
(EAiv) if 1 ? ? is defined, then a = 0 (zero-one law).
Let a and fo be two elements of an effect algebra L. We say that
(i) a is orthogonal to b and write a _L fo iff ? © fo is defined in L;
(ii) ? is less than or equal to b and write a ^ b iff there exists an element с е L such
that ale and а ? с = b (in this case we also write fo ^ a);
(iii) b is the orthocomplement of a iff b is a (unique) element of L such that b La and
???)=? and it is written as ?~4
We recall that every orthoalgebra is an effect algebra, and an effect algebra L is an
orthoalgebra iff a _L a implies a = 0.
A basic example of effect algebras is the system E(H) of all Hermitian operators on a
Hubert space ? which are between the zero operator and the identity. E(H) contains as a
proper subject the system of all orthogonal projectors, and E(H) is not a lattice. We recall
that E(H) is a basic tool for the so-called Hubert space quantum mechanics.
All above defined algebraic structures are nowadays called quantum structures.
The notion of a state is the same as that for orthoalgebras. A system S on L (where L
is any quantum structures) of states is order-determining iff m(a) ^ m(b) for all m e <S
entails a ? b. The existence of states, e.g., of a system of order-determining ones, enables
to represent any quantum structure as a system of fuzzy sets. For example [23]:
THEOREM 2.2. Let a system of fuzzy sets L с [0, l]x, ? ? 0, satisfy the following
conditions:
(i) ixeL;
(ii) \? — f e L whenever f e L;
(iii) f +g e L whenever f + g ? \?.
Then L = (L; ?,??, 1?,), where f®g is defined ifand only if f + g ^ \x,(f,g e L) and
we put f (&g — f +gt is an effect algebra. In addition, the system S = {mx: ? e X], where
mx: L —>· [0, 1] is defined via mx(f) := f(x), f e L, is an order-determining system of
states.
Conversely, let L = (L; ?, 0, 1) be an arbitrary effect algebra with an order-determining
system of states S. Then L is isomorphic with the system of fuzzy sets L с [0, 1] , where
L = {a e [0, l]5: a(m) :—m(a), m eS, a e L], and L satisfies the conditions (i)-(iii).
Among effect algebras we can meet a very interesting subfamily of effect algebras,
called interval effect algebras, i.e., an effect algebra of the form T(G, u) (where (G, u)
is an Abelian unital po-group with a strong unit u), where ? ? b is defined in T(G, u) iff
a + b e T(G, u) and in such the case ? ? b := a + b (it is clear that T(G, u) is always
an effect algebra); we recall that a positive element и е G is a strong unit (or order-unit)
if, given g e G, there exists an integer ? ^ 1 such that g ^ им. For example, the system
of all effect operators, E(H) on a Hubert space Я is an interval effect algebra, while

836
A. Dvurecenskij
?(?) = ?(?(?), I), where B{H) is the po-group of all Hermitian operators on a Hubert
space H, and / is the unit operator.
2.3. Decompositions of states
In what follows we will study some questions concerning the decomposition theorem for
semigroup-valued measures.
Let (S; 0, +, ^s) be an ordered semigroup, that is, S is a commutative semigroup written
additively, with the neutral element 0, and partial ordering ^5 such that if ? ^5 v, then
?+? ^sv+? for any ? e S. Let S+ denote the set of all positive elements of S, that is,
S+ - {? e 5": 0 ^s Mb We say that 5" is
(i) positive if S — S+\
(ii) naturally ordered if, for ?, ? e S+, we have ? ^s ? iff ? + ? = ? for some ? e S+.
It is clear that an ordered semigroup S is positive and naturally ordered iff
M^sv <s> 3? e S" with ? + ? = v. B.1)
Indeed, if S is positive and naturally ordered, then 0 ^s ? for any ? e S. If now ?+? = v,
????? = ?+0 ^S ?+? = ?, so that ? ^s v. If B.1) holds, then 0 + ? -?, which implies
0 ^s ? for any be S.
A cone of S is a subset С of S+ such that (i) if ??, ?? e С, then ?? + ?? & С,
and (ii) 0 e С. The cone С is ?-closed if, for any bounded chain С in C, the join
\JS С := V5{M·' деС) exists in 5" and is an element of С For more details on ordered
semigroups see [34,5].
We say that an element ? e S+ is singular with respect to the cone С if ? ^s ? f°r some
? e С implies ? = 0. We denote by C: the set of all elements of S+ which are singular with
respect to C.
Now we present the following important result [24]:
THEOREM 2.3 (Decomposition theorem). Let (S; 0, +, ^s) be a commutative, naturally
ordered semigroup and let С a v-closed cone in S. Then, for any ? e .?>+, there exist two
elements ? e С and ? e C~ such that
? = ?+?. B-2)
Suppose that L is an effect algebra. We say that a finite sequence F = {a \,..., an} in
L is (^-orthogonal if ? ? ? · ¦ ¦ ? ?„ exists in L. In this case we say that F has a ®-sum,
?/'= ? ai. defined via
n
??,-=?? ?---??„. B·3)
?=?
It is clear that two elements a and b of L are orthogonal, i.e., a _L b, iff {a, b) is
©-orthogonal.

Measures on quantum structures
837
An arbitrary system G = {a, },e/ of not necessarily different elements of L is
?-orthogonal iff, for every finite subset F of /, the system {a,-},-ef is ?-orthogonal. If G = {a/he/
is ?-orthogonal, so is any {a,-},<=,/ for any J с /. An ?-orthogonal system G = {a,-},¦<=/ of
L has a ф-яиш in L, written as ф/е/ a,-, iff in L there exists the join
/e/ F ieF
where F runs over all finite subsets in /. In this case, we also write фС:= ф,е/ сч¦
It is evident that if G = {a\,..., a„) is ?-orthogonal, then the ©-sums defined by B.3)
and B.4) coincide.
We recall that if G = {?,},€/, ?, = a for any i e / and card / = oo, then ? G exists in L
iff ? =0. Indeed, letao = ®G, thenao = ai(, ® Ф,е/\|/(,]а' =? ®?° which gives ? =0.
In addition, if G is ?-orthogonal, then it does not entail, in general, that a = 0. Really, for
example, let L be the set of all non-zero integers with the usual addition as ©. Then any
system G = {a, he/, at = a for any / e / is ?-orthogonal, but a is not necessarily equal to
0.
Throughout the rest of this subsection, by (L; 0, ©) and (V;0, +, ^y) we mean a effect
algebra with with a zero 0 and a binary operation ? and an ordered semigroup, respectively.
Define the following natural ordering ^„ on VL: ?\ ^„ ?? iff Ml (?) ^ M2(a) for any
a e L.
Example 2.4. Let [0, К] с [0, oo], К > 0, endowed with the usual order ^. Define
a binary operation +v: [0, ?] ? [0, К] —>¦ [0, К] via s +v t = s ? / := max(i, /), s, t e
[0, AT]. Then ([0, AT]; 0, +v, ^) is a positive, naturally ordered semigroup which is not
cancellative.
We say that an element ? e VL is a finitely additive measure if
(i) ?@) = 0;
(ii) ? (? ? b) = ? (?) + ??) whenever ? ? b is defined in L;
if ? : L —> V+ satisfies (i) and (ii), we call ? ? positive finitely additive measure, and then
? ^ fo implies ?(?) <y ?(^).
We denote by a(L, V) the system of all finitely additive measures from VL. If we
define the addition of two measures ?, ? e a(L, V) pointwise, i.e., (? + ?)(?) := ?(?) +
? (?), ? e L, and if o(a) := 0 for all a e L, then (a(L, V); o, +, ^n) is an ordered
semigroup with the positive cone a (L, V)+, the set of all positive finitely additive measures
on L with values in V.
Let / be an arbitrary non-void set, respectively. We denote by T(V) the set of all finite
non-empty subsets of /.
A mapping ? e V^ such that
M@)=0, ?????= V ]??(?,)

838
A. Dvurecenskij
holds whenever ф,е/ ?, is an element of L is said to be a positive completely additive
measure or a positive ?-additive measure if the former equality holds for an arbitrary
or countable index set /. We denote by ca(L, V)+ and aa(L, V)+ the sets of all
positive completely additive or ? -additive measures, respectively, from V+. It is clear that
ca(L, V)+ cCTa(L, V)+ ca(L, V)+ Ca(L, V).
It is worth recalling that a mapping ? e vl· with ?@) = 0 is a ? -additive measure iff
\n = l / n=l i'=l
whenever ?,^, ?„ exists in L.
We say that a mapping ? e a(L, V)+ is
(i) continuous from below if ? ? ^ ?? $5 ¦ ¦ ¦, Vol ? ??/ = ?> ?, ??, ??,... e ?-, imply
{?(??I? ?(?);
(ii) continuous from above at the least element 0 if {? (?,,)} 4- ? @) whenever ? ? ^??^
PROPOSITION 2.5. A finitely additive measure ?&?A, V)+ " ?-additive if and only if
? ;s continuous from below.
? ?-additive measure is not necessarily continuous from above at 0 as we can see in the
following example.
Example 2.6. Let ? be an uncountable set and let L be the system of all subsets A such
that either A or ? \ A is countable. Then (L, 0, ?), where A®S = AUBiffAnS = 0,
is a ? -effect algebra.
Take any non-trivial ordered semigroup, for example ([0, K]; 0, +v, О, К > 1, from
Example 2.4, and define, for any A e L,
JO, cardCA^No,
[ 1, otherwise.
Then ? is a ? -additive measure being not continuous from above at 0.
We can apply Decomposition Theorem to the semigroup S = a(L, V) to obtain Yosida-
Hewitt-type or Lebesgue-type decomposition theorems for positive finitely additive
measures defined on effect algebras with values in a relatively resituated semigroups.
THEOREM 2.7. Let the semigroup (a(L, V)+; o, +, ^„) of all positive finitely additive
measures from an effect algebra L into a Dedekind upwards complete, relatively resituated,
naturally ordered semigroup V be naturally ordered with respect to ^„. Then every
positive finitely additive measure ? ea(L, V)+ can be expressed as a sum
? = ?+?, B-5)

Measures on quantum structures
839
where ? is a positive completely additive measure from VL, and ? is a finitely additive
measure such that if ? ^,, ?, ? e ca(L, V)+, then ? = о.
PROOF. Define 5" :~a(L, V) and the set С := ca(L, V)+. We show that С is a v-closed
cone in a(L, V)+; and consequently, С is a cone.
Further, let {у,-} be a chain in a(L, V)+ with a bound ? ea(L, V)+ with respect to the
natural ordering ^„, and define
У0(с) = \/ Ydc), ceL.
i
Since 0 ^v Yi (c) ? ? у (с), the Dedekind upwards completeness of V implies that yo(c) is
defined correctly on L. Moreover, yo is finitely additive. Indeed, let ? ? b be defined in L.
Then [yi(a)} and {y/(fr)} are equidirected, and ?,(?) f yo(a), ?, ?) ? ю(&)- We conclude
that
уа(а ®b) = Yi (? ? fo)t = (y/ (a) + У, (&)) ? = Yi (a)T + У/ (fo)T
= У0(а)+У0(^)·
Therefore,
Y0 = \JSYi,
i
where V is taken in 5, and conversely, if y^ = \/f у,-, then yo = yo-
Now we claim to show that yo e ca(L, V)+ if {y,} is a bounded chain in C. Let
a = (§)j<zjaj exist in L. Then from the monotonicity of yo we have yo(af) ^v yo(o),
where «f := 0 -ef ay for any F e J'(i).
Assume that yo(af) ^ ? -* for some ? e V and for any F e JTG). Then y, (aF) ? у x for
any / and any F e J"G). The complete additivity of y, entails that y,(a) ^y ? for any /,
so that ??(?) ^v x¦ This gives
Y0(a)= V ?^?
consequently ?? e ca(L, V)+, and ca(L, V)+ is a v-closed cone of a(L, V)+.
Now we can apply Theorem 2.3 to obtain the assertion in question. D
Now we present two Lebesgue-type decompositions of measures on effect algebras.
Let W be another lattice naturally ordered semigroup, and let ? e a(L, V)+, ? e
a(L, W)+ be given. We say that
(i) ? is ?-continuous, and we write ? <ЗС? ?, if for every ? >y 0, ее V+, there is
? >w 0, <5 e W+, such that every a e L with ?(?) <w <5 implies ?(?) <y ?;
(ii) ? is dominated by ?, and we write ? <?C ?, if ?(?) — 0 implies ? (?) = 0;
(iii) ? is ?-singular if, whenever yeafi,, V)+, у <5?? ? and у ^„ ?, then у = о.

840
A. Dvureienskij
THEOREM 2.8. Let the semigroup (a(L, V)+; o, +, ^„) of all positive finitely additive
measures from an effect algebra L into a Dedekind upwards complete, relatively resituated,
naturally ordered semigroup V with the property that, for any и e V, 0 <y и, there
exist u\, И2 e V, 0 <y mi, mi with u\ + uj = u, be cancellative and naturally ordered
with respect to ^„. In addition, if {?,} \. ? for {?,} С a(L, V)+, then (y,(l)} 4- 0· Let
W be a lattice ordered semigroup with the property that if ? \, ?? e W, 0 <w ? ?, ?2, then
0 <w ?\ ?^2· Let ? ea(L, V)+, andke a(L, W)+. Then ? can be expressed in the form
? = ? + ?, where ?, ? e a(L, V)+, ? <iCf ?, and ? is ?-singular.
THEOREM 2.9. Let the conditions of Theorem 2.7 be satisfied. Then, for any pair of finitely
additive measures ?,???^, V)+, there exist two elements ? and ? in a (L, V)+ such that
? = ? + ?, ? «?,
and ? ? ? = ?, where the meet л is taken ina(L, V)+.
Finally we note that in contrast to the classical Yosida-Hewitt decomposition theorem,
for effect algebras we do not have guaranteed the uniqueness of the decomposition. The
uniqueness is guaranteed, for example, for L = L(H) or L = E(S), where E(S) is the
system of all splitting subspaces of an inner product space S; see below Aarnes-type
theorems Theorem 3.7 (for L(H)) and Theorem 3.31.
3. Gleason's measures
In the present section, we describe the state space of the most important quantum logic
L(H) of a real, complex or quaternion Hubert space H\ for more details on Gleason's
theorem see, e.g., [16].
3.1. Introduction
Recall that a state on L(H) is a mapping m : L(H) -> [0, 1] such that
@0 \ ЭС
0M,¦ J =]Гт(М,-)
(here by 0,e7- M, we shall mean the join of a family of mutually orthogonal subspaces
{Mt: t еГ) of L(H)).
It is clear that if ? is a unit vector in H, then mx : L(H) -> [0, 1] defined via
mx(M)=\\PMx\\2, MeL(H),

Measures on quantum structures
841
where Рм is the orthoprojector from ? onto M, is a state on L(H) for any H\ it is said to
be a vector state. More generally, if ? is a positive, Hermitian trace operator1 on H, with
trT = 1, then шг, where
mT(M) = tr(TPM), MeL(H). C.1)
defines a state on L(H).
Clearly there is a very large class of states on the quantum logic L (H) defined by C.1).
In the case that ? is two-dimensional, it is possible to show that C.1) does not describe all
states.
Example 3.1. Let #2 be a two-dimensional, real or complex Hilbert space. If
II if ? = H2,
0 ifM = {0},
0, or 1 if dim ? = 1,
where in the latter case we suppose m (M) + m(M±) = 1, then m is a two-valued state on
L(H2).
Fix a one-dimensional subspace Mo. If we define a state m via Example 3.1 such that
one-dimensional subspaces ?,?1- ? Mq attain the value either 0 or 1 in m, and for
w(Mo) — 1/2, then those states are not expressible via C.1).
A.M. Gleason [37] published the answer to Mackey's problem in 1957:
THEOREM 3.2 (Gleason theorem). If ? is a separable, real or complex Hilbert space,
dim ? ? 2, then for any state m on L{H) there exists a unique positive, Hermitian trace
operator ? on ? with trT = 1, such that
m(M) = tr(TPM), MeL(H). C.2)
Gleason's theorem can be reformulated into an equivalent form: for any state m on L(H)
of a separable Hilbert space H, dim ? ? 2, there exists an orthonormal system of vectors
{?-,} and a system of positive numbers (?,} such that J2i ^< = 1 > and
w(M)=^%m.V;(M), MeL(H). C.3)
Gleason's theorem belongs to the fundamental results of quantum logic theory. The heart
of Gleason's result lies in the care of a real, three-dimensional Hilbert space. The original
proof for this particular case is very complicated, and, consequently, many attempts to
simplify it have been made, which continue to the present day.
1 We recall that an operator ? on a Hilbert space ? is a trace operator if ?f | (Tx,, xj ) | < 00 for any ONB [xj )
of H; the expression trT := J^ (Txj, Xj) is said to be a trace of ?.

842
A. Dvurecenskij
For example, the following anecdote, [16], is told concerning J.S. Bell, the author of
the well known Bell inequalities: When J. Bell became familiar with the Gleason result,
he said that either he would find a more elementary proof or he would leave from this
area. Fortunately, he found a relatively simple proof of the partial result that there are no
two-valued states on L(H), dim ? ? 2.
An elementary proof of Gleason's theorem with the additional assumption that m assigns
the value 1 on some one-dimensional subspace has been presented by С Piron [63].
S. Kochen and E. Specker [47] have found a proof for a special case restricted to a finite
number of closed subspaces. The most elementary proof has been found by the triumvirate
R. Cooke, M. Keane, and W. Moran, [10], which was first presented at a conference in
Cologne, Germany, June 13-16, 1984. Their proof is accessible to undergraduates who
have completed a first course in real analysis.
When we regard the quantum logic of a Hubert space as the quantum logic of all
orthoprojectors of a von Neumann algebra A on H, we see that Gleason's theorem
describes states on the quantum logic of the factor type I (more precisely, for I„, ? =
1, 3,4,..., oo). The Gleason method is not, however, applicable to factors type II and III.
The problem of linearity is the following: if ? is a finitely additive measure on the quantum
logicof a von Neumann algebra A, L^{P, H), i.e., the system of all orthogonal projectors
in A, is it possible to extend it to a bounded linear functional on ?? Conversely, any
continuous linear functional on A restricted to L^(P, H) gives a finitely additive measure
on Lj\(P, H). For states on L(H) this problem is evident due to Gleason's theorem and
the linearity of trace.
The linearity problem is important for non-commutative integration theory and non-
commutative probability theory. The first result generalizing Gleason's theorem is due to
J. Gunson [42], and others are due to J.F Aarnes [1,2]. Then there appeared the paper of
A.A. Lodkin [50], which asserted the correct result but with an given incomplete proof.
Important contributions were made by M.S. Matvejchuk [56,57], A. Paszkiewicz [62]. The
complete proof was presented by E. Christensen [9] and F.J. Yeadon [72,73]; for a lucid
exposition see S. Maeda [55]. For ?-finite measures this problem is studied in [58]. For
complex-valued measures the solution is given by L.J. Bunce and J.D.M. Wright [7].
3.2. Generalizations of Gleason's theorem
In the present part, we show how Gleason's theorem was generalized to signed states and
also to nonseparable Hubert spaces as well as to unbounded measures.
We recall that the three-dimensional version of Gleason's theorem is most important for
applications of Gleason's theorem.
Let Я be a real or complex Hubert space. Let S{H) be the unit sphere in H. We say
that a mapping /: S(H) -> R is a frame function on ? if there is a constant W, called the
weight of /, such that for any ONB {jc, } of ? we have
J2f(xi) = w.
i
It is clear that if / is a frame function on H, then

Measures on quantum structures
843
(i) /(a;c)=/(jc)foranyaeC, \a\ = \,x eS(H);
(ii) f\S(M), where ? is a closed subspace of H, is a frame function on M.
PROPOSITION 3.3. Let m be a completely additive signed measure on L{H). Then the
mapping f,„ defined by
f„(x)=m(sp(x)), xeS(H),
is a frame function on ? with the weight m (H).
A frame function / on ? is
(i) bounded if sup{|/(jc)|: ? eS(H)} < oo;
(ii) semibounded'iiinf{/(jc): ? e S(H)} > —oo;
(iii) regular if there is a Hermitian operator ? on ? such that
f(x) = (Tx,x), xeS(H).
We recall that if / is regular, then Г satisfying the former equation is a unique trace
operator on H.
PROPOSITION 3.4. For any integer ? ^ 2, there exists an unbounded frame function
(in fact infinitely many) on an ?-dimensional Hilbert space H. Equivalently, there exist
unbounded signed measures on L(H).
To see that, it is sufficient to take a discontinuous additive functional ?: R —> К of
Hamel, and let ? ? к I be a Hermitian operator on H, where к is a real, non-zero constant
and / is the identity operator on H. Define a frame function f(x) = ф((Тх,х)), х е
S(H). Then it is unbounded.
We recall that a signed measure on L(H) is any mapping m : L(H) —> M. which is
?-additive. A completely additive state or a signed measure is additive on all joins of any system
of mutually orthogonal elements of L(H). Any finitely additive measure on L(H) is said
to be a charge. A charge is Jordan if it is difference of two positive charges.
If m is a bounded signed measure, then we have the first generalization of Gleason's
theorem by A.N. Sherstnev [67] and the author [14]:
THEOREM 3.5. Let m a bounded signed measure on L(H) of a separable Hilbert space
H, dim ? ^ 3. Then there exists a unique Hermitian trace operator ? on ? such that
m(M) = \i(TPM), MeL(H).
As we have seen, on any L(H) of a finite-dimensional Hilbert space ? there exists an
unbounded frame function. Therefore, the result of Dorofeev and Sherstnev [12] saying, in
particular, that any signed measure on L(H) is bounded whenever dim ? = oo, was very
surprising. It needs the following notion.

844
A. Dvurecenskij
A mapping /: S(S) —> ? is said to be a frame type function on ? if
(i) for any orthonormal system (ONS, for short) {jc,} in S {/(.*,)} is summable; and
(ii) for any finite-dimensional subspace К of S, f\S(K) is a frame function on K.
We recall that condition (i) is equivalent to the requirement that {/(.*,)} is summable for
any sequence of orthonormal vectors {jc,}. It is clear that if / is a frame type function on
? with given S, then if К is a closed subspace of ? such that S П К is dense in ?, then
?? '¦= f\S(S П К) is a frame type function on K.
THEOREM 3.6 (Dorofeev-Sherstnev). Any frame type function on a Hilbert space H,
dim ? = oo, /s bounded.
Now we are able to present a variation of the Yosida-Hewitt decomposition theorem for
signed measures on L(H) of arbitrary dimension.
THEOREM 3.7 (Aarnes theorem). Any Jordan charge m on L(H), dim ? ? 2, is uniquely
decomposed as a sum m —m\ + mi, where m\ as a Jordan completely additive signed
measure and mj is a Jordan charge vanishing on any finite-dimensional subspace of H.
We recall that according to S. Ulam, a cardinal number / is said to be non-measurable
if there exists no probability measure ? on the system of all subsets of a set A whose
cardinality is / such that v{{a}) = 0 for any a e A. In the opposite case / is said to
be measurable. For example, any finite cardinal, No, К ?, ??, ?? are non-measurable
cardinals, and also с (the cardinal of the continuum) (under the continuum hypothesis) is
also a non-measurable cardinal. If J ? I and / is a non-measurable cardinal, so is J.
The existence of non-measurable cardinals is a well known problem in set theory and
measure theory. The class of all non-measurable cardinals is huge, and it possesses, roughly
speaking, all cardinals which usually appear in mathematical problems.
M. Eilers and E. Horst [31] andT. Drisch [13] proved that Gleason's theorem also holds
for a non-separable Hilbert space whose dimension is a non-measurable cardinal ? 2. The
author [15] extended this result to the following one.
THEOREM 3.8. Any finite (?-additive) measure m on L(H) is a Gleason measure if and
only if the dimension of a Hilbert space ? is a non-measurable cardinal ? 2.
For completely additive states and signed measures we have the following generalization
of the result of Maeda [54], [16].
THEOREM 3.9. Let m be a completely additive signed measure on L(H), dim Я = oo.
Then there exists a unique Hermitian trace operator ? on ? such that
т(М) = 1т(ТРм), MeL(H).
Now we present a generalization of Gleason's theorem for measures which can also
attain improper values. For example, m (M) := dim ?, ? e L(H), is such one. We show
that even for this case we can find a special formula which for finite measures converts into

Measures on quantum structures
845
the Gleason formula. The first result in this direction was obtained by G.D. Lugovaja and
A.N. Sherstnev [52].
First we extend the notion of frame function. A mapping /: S(H) —>· RU {-00} U {+00}
is said to be г. frame function if
(i) f{X) = f(Xx), ? e C, ? e S(H);
(ii) there is a constant W (can be ±00), called the weight of / such that ?,- f(xi) — W
for any ONB {*,¦} in tf.
From the definition there follows that / can attain at most one of the improper values
±00. A frame function / has afiniteness property if | ???? /(-*<)! < °° f°re some ^NS
{*,¦: i e 1} entails that f\S(G) is a frame function on G = 0ie/sp(.Xj) with a finite weight.
It is clear that any frame function with a finite weight has the finiteness property. A frame
function / is regular if there exists a symmetric bilinear form t with D(t) = {x e ?: ? ?
{0}, I/(л:/H*||)I < 00} U {0} such that f(x) = t(x,x), ? e S(H) ? D{t).
Let ? be a cardinal number. We shall say that a frame function / is ?-finite if there
exists an ONB {xa: a e A) in ? such that A = (J/e/ ^ь where А, П Ay = 0 for / ? j, and
? ???? f(xi)\ < °° f°r апУ j e ^. anci the cardinal of / is n. For example, if ? = H,„ we
shall say that / is ?-finite.
Without loss of generality we shall assume that m : L{H) —> ]—00, 00].
First we present the basic result for L(E3) which has the same importance as that for
finite states, [52]:
LEMMA 3.10 (Lugovaja-Sherstnev). Let m : L(E3)^· ]-oo,oo] be a finitely additive
charge with m (??) — oo, and let there be a two-dimensional subspace Q of a finite
measure. Then if ? is a one-dimensional subspace of finite measure, then ? <zQ.
Using this basic result, it is possible to prove the following result, [16]:
THEOREM 3.11. Let ? be the quantum logic of a real or complex Hilbert space H,
dim Я ? 2. Let ? be a cardinal number and let m be an ?-finite measure. The following
assertions are equivalent:
A) There exists a (unique) positive, symmetric bilinear form t with the domain D(t)
dense in ? such that
m(M) =
tT(toPM) iftoPMeTT(H),
00 otherwise.
B) m has a support, i.e., such an element M,„ that m (N) = 0 if and only if N _L Mm.
C) m is a completely additive measure.
PROPOSITION 3.12. Any positive self-adjoint operator ? defines by C.4) a Gleason
measure, where t is defined by t(x, y) = (?1/2*, T^/2y) with D(t) = D(T]/ ).
Now we show that the situation is more complicated when we use bilinear forms which
are not determined by positive a self-adjoint operator.

846
A. Dvurecenskij
Example 3.13. Let {e„\ be an ONB for an infinite-dimensional Hilbert space ? which is
a part of a Hamel basis {e,},€/ for ? consisting of unit vectors from #.Fix an element e,0,
/'o e /, which does not belong to the ONB {e,,}, and define a linear operator ? : ? —>¦ ?
and a linear transformation /: ? —>¦ С via
T ( ? "'' e' ) = ?'? e'o and / ( ?!?'e/ ) = ?'°'
? ^ i
where <*,-„ is the scalar corresponding to e,-0 in the decomposition ? = ?? ai еь x e #' w'tn
respect to the given Hamel basis. Then \\T\\ = oo = ||/||, and let t(f, g) - (Г/, Tg), then
t defines via C.4) only finitely additive measure attaining infinite values.
G.D. Lugovaja [51] showed that on a separable Hilbert space any singular positive
symmetric bilinear form does not define any ?-finite Gleason measure by C.4). For that
reason the following important result has been proved by B. Simon [69]:
THEOREM 3.14. Let t be a densely defined positive symmetric bilinear form on a Hilbert
space H. Then there exist two positive symmetric bilinear forms tr and t5 with D(tr) =
D(t) = D(ts) such that
t = tr+ts,
where tr is the largest closable bilinear form less than t.
The following result of Lugovaja gives only finitely additive measures:
THEOREM 3.15. Let t be a singular, positive symmetric bilinear form with the dense
domain D(t) in H. Then the mapping m : L(H) -> [0, oo] defined by C.4) is not a measure
on L(H) of an infinite-dimensional Hilbert space H.
The affirmative answer is also due to Lugovaja:
THEOREM 3.16. A positive symmetric bilinear form t with a dense domain D(t) defines
through C.4) a Gleason measure on L(H)for every infinite-dimensional Hilbert space ?
if and only if for any ? e L(H),
(to PM)r eTr(tf) implies to Рм еТт(Н),
where (t о Рм )r is the regular part of the closure t о Рм-
Now we shall study the question of which kind of functions is defined by C.4). It will be
shown that it is precisely the set of all P(#)-regular finitely additive measures on L(H)
which can also attain even infinite values. We recall that a finite charge m on L(H) is

Measures on quantum structures
847
P(H)-regular if given ? > 0 and any ? e L(H) there is a finite-dimensional subspace N
of ? such that
\m(M nW)! <?.
For finite measures we have the following characterization:
THEOREM 3.17. Let m be a finite Jordan charge on L(H), dim ? ? 2. Then m is P(H)-
regular if and only if there is a Hermitian trace operator ? on ? such that C.1) holds.
For infinite measures we have this characterization:
THEOREM 3.18. Let m be ? ?-finite, ?'(H)-regular, finitely additive measure on L(H),
dim ? ? 2 of a separable Hilbert space H. Then there exists a unique bilinear form t with
a dense domain such that C.4) holds. Conversely, for any positive bilinear form t on ?
such that D(t) is dense in H, the mapping mt: L(H) —>¦ [0, oo] given by
mt(M) ¦¦
H(toPM) iftoPMeTT(H),
oo otherwise,
is ? ?-finite ?(?)- regular finitely additive measure.
Finally, we show some convergence properties of finite signed states on L(H). We recall
that a Gleason measure we understand any signed measure m on L (H) such that C.1) holds
for some Hermitian trace operator ? on H.
THEOREM 3.19 (Nikodym theorem). Let {m„} be a sequence of finite signed measures
on L(H) of a Hilbert space H. If there is a finite limit m(M) = lim„ m„ (M) for any
? e L(H), then m is a finite signed measure on L(H), and {m,,} is uniformly ? -additive
with respect to n.
THEOREM 3.20. The space of all finite completely additive signed measures on L(H) is
sequentially weakly complete for any Hilbert space H.
In addition, the space of all finite Gleason signed measures is sequentially complete.
We say that a system of finite charges {v,: / e /) on L(H) is uniformly absolutely
continuous with respect to a finite, finitely additive measure m if given ? > 0 there is a
S > 0 such that m (?) < ? implies | v,- (M) | < ? for any / e /.
THEOREM 3.21 (Vitali-Hahn-Saks). Let [v„] be a sequence of semi bounded, completely
additive signed measures on the quantum logic L(H), dim Я ф 2, such that every v„ is
absolutely continuous with respect to a completely additive measure m. If there is a finite
limit v(M) = lim„ v„(M), ? e L(H), then {v,,} is uniformly absolutely continuous with
respect to m.

848
A. Dvurecenskij
For uniform convergence of bounded signed measures we have the following statement
combining this convergence with the convergence of trace operators, [45]:
THEOREM 3.22. A sequence of Gleason signed measures m„() = tr(r„P(.)) converges
uniformly to 0 if and only if
????(?„) = 0,
II
where ? is the trace norm of trace operators on H.
3.3. Gleason's theorem and completeness criteria of inner product spaces
In this section, we show how Gleason's theorem can be used even for characterization of
completeness inner product spaces (= Hubert spaces).
Let S be a real or complex inner product with an inner product (·, ¦)· We recall that for
any MCSwe put M1- = [x e S: (x,y) = 0 Vy e ?}, and by ? we mean the completion
ofM.
Denote by
F(S) = [M с 5": М = М±±},
i.e., the set of all orthogonally closed subspaces of S, and
E(S)= [M C5: M + M^ = S],
i.e., the set of all splitting subspaces of S. It is easy to see that E(S) с F(S). Moreover,
5" is complete iff E(S) — F(S), which is a consequence of the Amemiya-Araki theorem.
We present now two simple completeness criteria, [17]:
THEOREM 3.23. A real or complex inner product space S is complete iff there is a unit
vector ? e S and a positive number W > 0 such that, for any MONS {.v,} in S,
0< W ^J2\(x,Xi)\2
i
holds.
THEOREM 3.24. A real or complex inner product space S is complete iff there is a unit
vector ? e S and a positive number W > 0 such that, for any MONS [xi} in S,
0< W^^|(jc,jc,)|2
holds.

Measures on quantum structures
849
Slight generalizations of the above facts are the following ones. A mapping /: S(S) :—
{x e S: ||jr|| = 1} -> E+ is said to be a frame function iff there is a constant W (called the
weight of /) such that
i
holds for any MONS {*,¦} in 5".
A mapping / :S(S) —>¦ E+ is said to be a weak frame function iff
(i) there is a positive constant W such that
0< W ^Y^f(xt)< oo
i
holds for any MONS {*,¦ } in S; and
(ii) f\S(M) is a frame function for any finite-dimensional subspace ? of S.
From (ii) we have that f(Xx) = f{x) for any |?| = 1.
It is evident that any frame function is a weak frame function, and the converse holds,
too, as we shall see below, which will prove the completeness. We recall that if ? is a
unit vector in S or in S, then f:z ь* ? (?,*) ?2, ? e S(S), is sometimes (iff S is complete)
a special type of a frame function or a weak frame function.
THEOREM 3.25. An inner product space S is complete iff there is at least one weak frame
function f on S(S).
COROLLARY 3.26. Any weak frame function is a frame function.
A mapping / : S(S) := [x e S: \\x || = 1} —>¦ ? is said to be a signed frame function iff
there is a constant W (called the weight of /) such that
i
holds for any MONS {*,¦} in 5".
A mapping / :S(S) —>¦ ? is said to be a weak signed frame function iff
(i) for any MONS {*,} is S, {/(*/)}/ is summable;
(ii) there is a positive constant W such that
o< ?^??-?-*')! < oo
i
holds for any MONS {*,-} in S; and
(iii) f\S(M) is a signed frame function for any finite-dimensional subspace ? of S.
From (iii) we have that f(Xx) = f{x) for any |?| = 1.
It is evident that if / is a signed frame function with a non-zero weight, then / is a weak
signed frame function.

850
A. Dvurecenskij
THEOREM 3.27. An inner product space S is complete if and only if there exists at least
one weak signed frame function f on S(S).
What is important for us is that applying the Gleason theorem for bounded signed
measures on finite-dimensional Hilbert spaces we have that there is a bounded bilinear
from; on S ? S such thatt(x,x) — f(x), ? e S(S).
We have seen above that it has been shown that S is complete iff there is at least one
non-zero signed measure frame function on S(S). Using the property f(x) = m(sp(x)),
? e S(S), we have the following measure-theoretical characterization of the completeness
of inner product spaces.
A completely additive signed measure (state) on E(S) or F(S) is any completely
additive function m on E(S) or F(S) which preserves all existing joins of mutually
orthogonal elements of ?E') or F(S), respectively.
THEOREM 3.28. A real or complex inner product space S is complete if and only if on
E(S) or F(S) there exists at least one nonzero completely additive measure.
For the reader's convenience, we collect some completeness criteria of inner product
spaces. For other explanation of new notions, see, e.g., [16]; we recall only that:
A) D(S) is the system of all Foulis-Randall subspaces of 5", i.e., of all subspaces ?
of 5" for which there is an ONS {и,} in ? such that М={и^^, which is a complete
orthoposet. Any ? of D(S) possesses at least one local complement M'', i.e., such
an element ?' e D(S) for which M' _L ? and ? ? ?' = S.
B) R(S) is the set of all subspaces ? of S such that ? - [u^1- for any MONS {и,}
in ?, which is a poset.
C) V(S) is the set of all subspaces ? of 5" such that ? = {ui}^1- and M1- = {v^1-
for all MONSs {и,} and {uy} in ? and M^, respectively, which is an orthocomple-
mented poset.
D) C(S) is the set of all subspaces of S of finite or cofinite dimension, which is an
orthocomplemented, modular lattice.
E) P(S) is the set of all finite-dimensional subspaces of S, which is a lattice.
F) W(S) is the set of all closed subspaces of S.
THEOREM 3.29. Let S be a real or complex inner product space. The following criteria
are equivalent.
A) S is complete.
B) E(S)=W(S).
C) F(S) = W(S).
D) For any proper closed subspaces ? of S, ?1- ? {0}.
E) If f is a continuous linear functional on S, there exists у e S such that f(x) —
(x,y)forallx e S.
F) For any non-zero continuous linear functional f on S, (Кет f) ? {0}.
G) For any continuous linear functional f on S, Ker/ e E(S).
(8) F(S) is orthomodular
(9) E(S) = F(S).

Measures on quantum structures
851
A0) E(S) is a complete lattice.
A1) E(S) is ? ? -lattice.
A2) E(S) is a ?-orthoposet (= quantum logic).
A3) ?E") possesses the join of any sequence of mutually orthogonal one-dimensional
sub spaces ofS.
A4) ? (S) has the subsequential interpolation property.
A5) ?(S) has the strong subsequential completeness property.
A6) Any modular pair in W(S) is dual-modular
A7) D(S) isanOML.
A8) R(S)=F(S).
A9) D(S) = E(S).
B0) К (S), if К e {C, E,V, R, D, F, W], possesses at least one non-zero, completely
additive signed measure.
B1) 5" possesses at least one non-zero frame function.
B2) Every MONS in S is an ON В in S.
B3) There exists a unit vector у e S such that у = ?V:(y, jc, )jc, for any MONS [xj } in S.
B4) F(S) (W(S)) possesses at least one Jordan P(S)-regular, non-zero charge.
B5) К (S), where К e [E,V, R} and dimension of S is a countable cardinal, possesses
at least one non-trivial, strongly ?(S)-regular, finitely additive measure.
B6) К (S), where К e [D, F, W}, possesses at least one state having a support.
B7) K(S), if К e {D, F, W] and the dimension of S is a non-measurable cardinal,
possesses at least one non-trivial signed measure.
B8) К (S), where К e {F, W}, possesses at least one signed measure non-vanishing on
P(S).
B9) К (S), where К e [F, W} and S is Dacey. possesses at least one signed measure
non-vanishing on P(S).
C0) К (S), where К e {F, W], possesses at least one finitely additive state with a finite-
dimensional support.
C1) K(S), where К e {F, W], possesses at least one finitely additive state with a P(S)-
regular support.
C2) Any P(S)-regular state on E(S) has a support in E(S).
C3) For any sequence {jc,} of orthonormal vectors in S and all positive numbers {?/},
with J2i ?; = 1, the state J2i ?/ mXi has a support in E(S).
C4) For any sequence {jc,} of orthonormal vectors in S, the state 5Z,- /ял, /2' has a
support in E(S).
C5) There is a strong system ? of states on E(S), dim S ? 2, such that any counter on
Con(.M) is ? -expectational.
C6) Any P(S)-regularfinitely additive state on E(S) has the ?-Jauch-Piron property.
C7) Any ?(S)-regular finitely additive state on E(S) has the ?-weak Jauch-Piron
property.
C8) Any mx, ? eS(S), has the ? -weak Jauch-Piron property.
C9) Sis Dacey.
D0) D(S) is an OMP
D1) (S(S), T{S)), where T(S) is the setofallMONSs in S, is an algebraic test space,
[27].

852
A. Dvure(enskij
D2) There is a unit vector ? e S (x e S) and a constant W such that 0 < W ?
?,? I С*,*/) I2 holds for any MONS {x,-} in S.
D3) There is a weak frame function on S(S).
D4) There is a weak signed frame function on S(S).
Finally we present another generalization of Gleason's theorem to describe regular
finitely additive measures on E(S). We note that on E(S) there exist plenty of finitely
additive states for a non-complete inner product space S, but no completely additive. For
example, let ? be a unit vector in S, then
mx(M)=\\xM\\2, MeE(S),
where ? = хм +хм±,хм e ?, хм^ е М^, is a finitely additive state on E(S). Moreover,
the system of all such states describes the order on E(S).
We say that a charge m on E{S) (F(S)) is ?(S)-regular if given ? e E(S) (F(S)) and
given ? > 0 there is a finite-dimensional subspace N of ? such that
\m(M nW)! <?.
It is clear that ? (S) -regularity is equivalent to the statement that given Me E(S) we
find a non-decreasing sequence [M„} of finite-dimensional subspaces of ? such that
m(M) = \im m(M„).
PROPOSITION 3.30. Let ? e ??EJ be a non-zero Hermitian operator, then тт defined
via
mT(M) = \i(TPg), MeE(S), C.5)
is a P (S)-regular Jordan charge on E(S).
The following result generalizes Aarnes' theorem and we give a Yosida-Hewitt-type
decomposition for positive measures; here the role of Gleason measures will be played by
? (S) -regular charges.
THEOREM 3.31. Any Jordan charge m on E(S), dim S ? 2, can be uniquely decomposed
as a sum m —m\ + mj, where m\ is a P(S)-regular charge, and mi is a Jordan charge
vanishing on all finite-dimensional subspaces ofS.
THEOREM 3.32. A Jordan charge m on E(S), dim 5" ? 2, is P(S)-regular if and only if
there is a Hermitian operator ? e TrE") such that
т(М) = 1т(ТРй), MeE(S).
By Tr(H) we denote the system of all Hermitian trace operators on H.

Measures on quantum structures
853
As wee see, formula C.5) describes all measures on E{S) which can be described by
Gleason's formula, i.e., using a Hermitian trace operator on ? — S. We recall, that E(S)
can be embedded into L(S) via ? ь* ?, ? e E(S).
Problem 3.33. Finally we note that the analogue of Nikodym's theorem for the space
of all ?(S)-regular Jordan charges on E(S), dim S ? 2, seems to be open. Similarly, the
existence of at least one finitely additive state on F(S) for an incomplete S seems to be
open, too.
4. States on MV-algebras
MV-algebras are many-valued analogues of a two-valued logic, and they have were
introduced by Chang [8]. The importance of MV-algebras for quantum structures
follows from an important property that they model a classical situation with unsharp
measurements. We recall that they form a counterpart to Boolean algebras in orthomodular
poset models.
We recall that according to Mundici [59], they can be characterized as follows. An
MW-algebra is a non-empty set ? with two special elements 0 and 1 @^1), with a
binary operation ?: ? ? ? —>¦ ? and with a unary operation *: ?? —>¦ ?? such that, for all
a,b,ce ??, we have
(MVi) ? © b = b © a (commutativity);
(MVii) (? ? b) ? с — ? ? (b ? с) (associativity);
(?Viii) ? ? 0 = a;
(MViv) ? ? 1 = 1;
(MVv) (a*)* = a;
(MVvi) a ©a*= 1;
(MVvii) 0*= 1;
(MVviii) (a*®b)*®b= (a®b*)*®a.
We recall that the above axioms are not independent, e.g., (MVvii) follows from (MVvi)
and (MViii) for a = 0.
We define binary operations ©, v, л as follows, for all a, b e M:
aQb:=(a*®b*)*, a v b:= (a* ®b)* ? b, а Л b := (a* v b*)*.
For a, b e M, we define
a ^ b o· a = a l\b,
then ^ is a partial order on M, and (?; ?, ?,?, 1) is a distributive lattice with the least
and greatest elements 0 and 1, respectively. We recall that ? ?? biff b®a* = 1.
Let (G; +, 0, ^) be an Abelian ?-group with a strong unit и, i.e., given ? e G, there is
an integer/г > 1 such that —nu^v^nu.
Define
r(G,u)={geG: O^g^u]
D.1)

854
A. Dvurecenskij
and set for all g\ ,g2,g? T(G, u)
g\®g2-{g\+gl)/\u, g\Og2 = (g\+gl-u)vO, g* = u-g.
Then (f(G, и); ?, ?,* , 0, и) is an MV-algebra. The famous Mundici result [59] says that
given an MV-algebra ? there is an Abelian ?-group G with a strong unit и such that ? is
isomorphic with some T(G, u). In addition, ? defines a categorical equivalence between
the category of unital ?-groups and the category of MV-algebras.
The MV-algebra of the real interval [0, 1], i.e., (T(R, 1)), is of particular interest.
We recall that the categorical equivalence of MV-algebras with unital ?-groups implies
that if ? is an order- and +-preserving mapping from ? = T(G, u) into a po-group H,
then there exists a unique group homomorphism f:G —>¦ ? such that ? {?) = ? (a),
ae M.
Given an MV-algebra M, we can introduce a partial binary operation + via a + b is
defined iff a ^ b*, and in this case we put
a +b:=a ®b.
It is easy to see that a+0 = 0 + a = afor any a e M, and + is commutative, i.e., if a + b
is defined in M, so is b+a,anda +b = b+a\ + is associative, i.e., if (a +b), (a+b)+c
are defined in ? so are defined b + с and a + (b + c), and (a + b) + с = a + (b + c).
Identifying the MV-algebra ? with r{G,u) via D.1), we can see that our partial operation
+ coincides with the group addition +.
A non-void subset / of ? is said to be an ideal of ? if
(i) x,ye I imply ? ? у е /;
(ii) ? e /, у ^ ? imply у е I.
A proper ideal A of ? is said to be maximal if there is no proper ideal of ? containing A
as a proper subset. Let ? (?) denote the set of all maximal ideals of ?. Then ? (?) ? 0.
Denote by
Rad(M) := [\\A: A e M(M)},
and we call Rad(Ai) the radical of M. An MV-algebra ? is said to be semisimple if
Rad(M) = {0}.
Since if ? is an MV-algebra, then ? with the partial addition and 0 and 1 is an effect
algebra, we define a state on an MV-algebra in the same way as that for effect algebras.
A state on an MV-algebra ? is a mapping m :M —>¦ [0, 1] such that w(l) = 1, and
m(a + b) = m(a) +m(b) whenever a + b is defined in M. Denote by S(M) the set of
all states on M, then S(M) ? 0.
Let (G, u) be an Abelian unital ?-group. A state on (G, u) is any mapping s : G -> К
such that
(i) i(«)=l;
(ii) s(gi +g2) = i(gi)+i(g2), gi,g2eG; and
(iii) s (g) > 0 for each g e G+.
Due to the above mentioned categorical equivalence, we have that there exists a one-to-
one equivalence between the states on T(G, u) and (G, и); this correspondence is given

Measures on quantum structures
855
by the property that if s is a state on (G, и), then m := s\T(G, u) is a state on T(G, и),
and conversely, each state m on T(G, u) can be uniquely extended to a state s on (G, и).
Moreover, extremal states on T(G, и) correspond to extremal states on (G, и), and vice
versa.
For example, the MV-algebra T(R, 1) possesses a unique state m, namely m{t) = t,
re [0,1].
We write that a net {ma} of states on ? converges weakly to a state m iff m„(a) —>· m(a)
for any a e M. Then 5(??) is a convex compact Hausdorff topological space.
Let Ext(Ai) denote the system of all extremal states on M. A state m is said to be state-
morphism \im{a®b) — min{m(a) +m(b), 1).
THEOREM 4.1. Let ? be an MV-algebra. The following statements are equivalent:
(i) m is an extremal state on M.
(ii) m is a state-morphism on M.
(iii) m(x Ay) = min{m(jt), m(y)}, x, у е М.
The set of all extremal states is in a one-to-one correspondence with maximal ideals
of ?. This correspondence is given by
m «*· K(m):= [aeM: m(a) = 0}, m e Ext(<S(M)). D.2)
THEOREM 4.2. Let ? be an MV-algebra. Then S(M) is a non-empty compact convex
Hausdorff space with respect to the weak topology, and the space of all state-morphisms
on X is a non-void compact Hausdorff space. Any state is a weak limit of convex linear
combinations of the set of extremal points ofM.
For every a e ?, we put
M(a):= {I eM(M): a ?1].
Then(i)M(O)=0, ? (а) с ? ?) whenever ? ? b, ? (а лЬ) = ? (?) ? ? ?), a, be ?,
? (? ? b) = ? (a) U ? ?), a,b e ?, and {? (a): a e M] is the base of the so-called
hull-kern topology Т_м on M(M).
PROPOSITION 4.3. Let ? be an MV-algebra. Then the hull-kern topology defines a
compact Hausdorff topology such that the closed subspaces ofM(M) are exactly of the
form
C = C(J):= {I eM(M): I^j],
where J is an ideal of M. Similarly, every open set О is of the form
0= 0(J):=[I eM(M): /2 7}.

856
A. Dvurecenskij
THEOREM 4.4. Let ? be an MV-algebra. Then the correspondence D.2) defines a
homeomorphism of the space Ext(M) of all extremal states on ? and the space ?4 (M) of
all maximal ideals of M.
Now we show how order-determining systems of states on ? V-algebras determine they
character.
THEOREM 4.5. Let ? be an MV-algebra. The following statements are equivalent:
(i) ? is semisimple.
(ii) ? possesses an order-determining system of states.
(iii) ? possesses an order-determining system of extremal states.
(iv) ? is isomorphic to a system of fuzzy sets on a non-void set.
(?) ? is isomorphic to a system of continuous fuzzy sets on a non-void compact
Hausdorffset.
For other aspects of states on MV-algebras we recommend also the paper of B. Riecan
and D. Mundici [66] in this Handbook.
5. Measures on BCK-algebras
BCK-algebras, introduced by Imai and Iseki [44], constitute a generalization of Boolean
algebras, arising from the many valued logic of Lukasiewicz in the same manner as
Boolean algebras arise from two-valued logic. In particular, bounded commutative BCK-
algebras are equivalent to MV-algebras.
5.1. Commutative BCK-algebras with the relative cancellation property
We recall that a BCK-algebra is an algebra (X; *,0) of type B,0) such that, for all
x, y, ? e X, we have
(i) ((x * y) * (x * z)) * (z * y) — 0;
(ii) {x * (x * y)) * у = 0;
(iii) ? * ? = 0;
(iv) ? * у = 0 and у * ? = 0 imply ? = )·;
(?) 0*jt = 0.
In every BCK-algebra X = (X; *, 0) we can define a partial order ^ via ? ^ у iff ? * у = 0.
X is said to be bounded if there exists the greatest element 1 in X.
A BCK-algebra (X; *, 0) is said to be commutative if
? * (x *y) = у * (у * ?), x,yeX,
and in this case, хлу=х*(х*у).

Measures on quantum structures
857
Example 5.1. Let X = {0, 1,2,3,4} and * operation be given by the table. Then
(X; *, 0) is a bounded non-commutative BCK-algebra.
*
0
1
2
3
4
0 1
0 0
1 0
2 1
3 3
4 4
2 3 4
0 0 0
0 0 0
0 1 0
3 0 0
4 4 0
In [25], an important class of commutative BCK-algebras was studied: we recall
that a commutative BCK-algebra (X; *,0) has the relative cancellation property if, for
x,y,a e X, such that ? > а, у ^ a and x*a = y*av/c have ? = у. Such BCK-algebras
form a variety which is a proper subvariety of the variety of commutative BCK-algebras.
In this case we can introduce a partial binary operation + on X as follows a + b is defined
in X and equals с iff с > a and b = c*a. For the basic properties of + we recall only that
+ is commutative, associative and with a neutral element 0.
For example, ([0, oof; *r, 0), where
s *r t = max{0,i — t],
s,te [0, oo), is an example of a commutative BCK-algebra with the relative cancellation
property.
Suppose that (G; +, ^,0) is a lattice-ordered group with positive cone G+ = [g e G:
g > 0). Then (G+; *g,0) is a commutative BCK-algebra with the relative cancellation
property, where *g is defined via
и *g ? '¦— (u — t>) ? 0, u, ? e G+.
More generally, if Go is a non-void subset of G+ such that u, ? e Go imply и *g v e Go,
then (Go;*g,0) is a commutative BCK-subalgebra of (G+;*g,0) having the relative
cancellation property. In addition, the derived partial sum in X = Go coincides with the
sum in G, provided that this is again in Go-
The latter example is in some sense the archetype of a commutative BCK-algebra with
the relative cancellation property, owing to the following basic representation theorem for
commutative BCK-algebras, proved in [25]:
THEOREM 5.2. Let (X; *,0) be a commutative BCK-algebra with the relative
cancellation property. Then there exists a lattice-ordered group (G; +, ^, 0) with positive cone G+
and an injective BCK-homomorphism h:X—> G+ such that the pair (G, h) is the
universal group for X: that is, for any partial ordered Abelian group (K; +, ^,0) andanyorder-
and -^-preserving mapping g.X —> К there is a homomorphism of Abelian ordered groups
g' :G —> К such that g—g'oh.

858
A. Dvurecenskij
5.2. Measures on commutative BCK-algebras with the relative cancellation property
A mapping m : X —>· [0, oo) such that, for all x, у е X.
(i) m(x * y) = m(x) — m(y) whenever у ?. ? is said to be a measure;
(ii) if 1 e X and m is a measure with m A) = 1, then m is said to be a state;
(iii) m(.x * y) = max{0, m(x) — m(y)} is said to be a measure-morphism;
(iv) if 1 e X and m is a measure-morphism with m{\) = 1, w is said to be a state-
morphism.
An /dea/ of a BCK-algebra (X;*,0) is a non-empty set / of X such that, for all
x,yeX,
(i) 0e/;
(ii) ? * у e I and у e I imply jc e /.
A proper ideal of X which is not included as proper subset in any proper ideal of X is
said to be maximal; we denote by M(X) the set of all maximal ideals on X. In contrast to
bounded commutative BCK-algebras (= MV-algebras), it can happen that X possesses no
maximal ideal. Similarly, there exists a commutative BCK-algebra which has only measure
identically equals to 0, [28]:
Example 5.3. Let Z+ = {0, 1,2,...} and let ? = (Z+)K. Set К = {/ e H: f has a
countable support}. Then К is an ideal of the upwards directed commutative BCK-algebra
H. Put X := H/K, then X is a Dedekind ?-complete3 with M(X) — 0, and any measure
on X is identical 0.
IfM(X)^0, the set
Rad(X):=f^|{M: MeM(X)}
is called the radical of X. If Rad(X) = {0}, X is said to be semisimple.
If m is a measure on X, then
Ker(w) := {x e X: m(x)=0}
is an ideal of X.
In view of the categorical equivalence of commutative BCK-algebras with the relative
cancellation property, any measure m on X can be uniquely extended to a measure on the
corresponding group G; we recall that a measure on an Abelian ?-group G is any mapping
s : G —> ? which preserves + and the positivity on X.
PROPOSITION 5.4. Let m ? апатг be two non-zero measure-morphisms on a commutative
BCK-algebra (X;*,0) with the relative cancellation property. Suppose Ker(wi) —
Ker(w2). If there is an element xq e X such that m\(xo) = mi(xo) > 0, then m\ = mi-
Otherwise, there exists a constant с such that m\ = стг-
It means, e.g., that the corresponding ^-groups is so.

Measures on quantum structures
859
THEOREM 5.5. Let (X; *, 0) be a non-trivial commutative BCK-algebra with the relative
cancellation property. Then there is a one-to-one correspondence between the set of all
measures-morphisms on X and the set of maximal ideals ?4(?) given by m -o· Ker(m).
The correspondence is unique up to a constant с
In particular, if(X; *, 0) is a bounded commutative BCK-algebra, then ? (?) ? 0, and
there is a one-to-one correspondence between ?4(?) and the set of all state-morphisms
onX.
We can say more for BCK-algebras with a quasi strong unit. We recall that an element и
of X is a quasi strong unit of X if the ideal of X generated by и is equal to X. It is possible
to show that и is a quasi strong unit for X iff и is a strong unit in the corresponding ?-group
G representing u. Therefore, X possesses a measure m such that m(u) = 1. Denote by
SU(X) the space of all measures m on X such that m(u) — 1.
The following result extends Theorem 4.2 to commutative BCK-algebras which need
not be bounded.
THEOREM 5.6. Let и be a quasi strong unit of a non-trivial commutative BCK-algebra
(X; *, 0) with the relative cancellation property. Then S„(X) is a non-empty compact
convex Hausdorff space, and the space of all measure-morphismsfrom <S„ (X) is a non-void
compact Hausdorff space. Any measure from <S„ (X) is a weak limit point of the convex hull
of the set of extremal points ofSu (X). In addition, the following statements are equivalent:
(i) m is an extremal measure from <S„ (X).
(ii) m is a measure-morphism from S„(X).
(iii) m(x л у) = m'm{m(x),m(y)}, ?, у е X, m(u) = 1.
The notion of an order-determining system is similar as that in previous chapters. We
recall that X is said to be A rchimedean iff, for a, b e ?,?? ^ b for any ? > 1 implies ? = 0.
This notion is equivalent with the Archimedeanicity of the corresponding representation
?-group.
For example, let X be a system of nonnegative functions on ? ?? which is closed with
respect to the requirement /, g e X then / * g e X, where (/ * g)(co) = /(?) *? #(?),
? e ?, is an important example of BCK-algebras.
THEOREM 5.7. Let и be a quasi strong unit of a non-trivial commutative BCK-algebra
(X; *, 0) with the relative cancellation property. The following statements are equivalent:
(i) X is semisimple.
(ii) X is Archimedean.
(iii) X has an order-determining system of measure-morphismsfrom S„(X).
(iv) X has an order-determining system of measures from <S„ (X).
(?) ? is isomorphic to some commutative BCK-algebra of functions on some ? ? 0.
A nonzero element a e X is said to be an atom if b ^ a implies b e [0,a]. We denote
by A(X) the set of all atoms of X. X is said to be atomic if given a nonzero ? e X, there
exists an atom a e X such that a ^ x. Given an element xeX.we define
ja(x) =sup{n >0: /?? is defined in X, na^ x}. E.1)

860
A. Dvurecenskij
THEOREM 5.8. Let a be an atom of an Archimedean commutative BCK-algebra (X; *, 0)
with the relative cancellation property. Then ja defined via E.1) is a non-trivial measure-
morphism on X, and there exists a unique maximal ideal Ma ofX such that a ?
Main addition, X is atomic if and only if{ja ¦ a e A(X)} is order-determining. In this case,
X is semisimple, and X is a subdirectproduct of\\a X/Ma.
We can say more, if X is roughly speaking the direct product of BCK-algebras of the
form {0, a,2a, 3a,..., Ka], where К is a finite integer or A" — oo. Such BCK-algebras are
the following: we say that an atomic commutative BCK-algebra is atomically bounded if
any system {iaa: a e A(X)}, where ;'„ is a non-negative integer such that iaa is defined in
X, has an upper bound in X. For example, any bounded atomic commutative BCK-algebra
is atomically bounded, [26].
THEOREM 5.9. Let (X; *, 0) be a Dedekind complete and atomically bounded
commutative BCK-algebra with the relative cancellation property. Then:
(i) A complete ideal ? of X is a maximal ideal if and only if ? — Ma for some
a e A(X), where Ma is defined in Theorem 5.8.
(ii) Every ja is a completely additive measure-morphism, and {ja: a e A(X)} is order-
determining.
(iii) A non-trivial completely additive measure m on X is a completely additive
measure-morphism if and only ifm — сja for some atom a and a positive constant с
6. States on pseudo MV-algebras
Recently, G. Georgescu and A. lorgulescu [35] introduced pseudo MV-algebras which are
a non-commutative generalization of MV-algebras introduced in 1958 by C.C. Chang [8]
and which are an extension of a two-valued reasoning. A non-commutative generalization
of reasoning can be found, e.g., in psychological processes. J. Rachunek [65] introduced
non-commutative MV-algebras, which in fact are equivalent with pseudo MV-algebras.
Such a noncommutative reasoning we can meet for example in psychological or social
processes.
We present new results of the author concerning states on pseudo MV-algebras.
6.1. Pseudo MV-algebras
Definition 6.1. A pseudo MV-algebra is an algebra (??;?,~,~,0, 1) of type B, 1, 1,
0,0) such that the following axioms hold for all x, y, ? e ? with an additional binary
operation © defined via
(??) ? ® (y ® z) = (x <?> y) <?> z;
(A2) ??0 = 0??=?;

Measures on quantum structures
861
(A3) x® 1 = \®x= 1;
(A4) 1~ = 0; Г =0;
(A5) (гшу-г = (гел_;
(A6) x®x~Oy-y®y~~Ox=xOy-®y = yOx-®x;
(A7) д:0(Г9у) = ие y~) ? у;
(A8) (дг)~=дг.
We shall assume that 0 ? 1. If we define ? ^ у iff jc~ ? у = 1, then ^ is a partial order
such that ? is a distributive lattice with ;tvy=jt©jt~Oy. For basic properties of pseudo
MV-algebras see [35].
For example, if и is a strong unit of a (not necessary Abelian) ?-group G,
Г(Си):=[0,и]
and
? ®y -=(x +y) ли, х~:=и-х,
x~ := — ? + u, ? © у :— (? — и + у) ? ?,
then (T(G, и); ф,_ ,~, 0, и) is a pseudo MV-algebra [35].
Example 6.2. Let G = (? ? ? ? ?; +, @,0,0), ^) be the Scrimger 2-group, i.e.,
(? ? + m2,m\ + k2,ri\ +П2), ifniisodd,
(k\,m\,n\)-\-{k2,m2,nj) : — ,
' (it? + k2,m\ + m2,n{ +п2), if n2 is even.
Then 0 — @,0,0) is the neutral element, and
(—m,—k,—n), if ? is odd,
(k,m,n)
(—k,—m,—n), it ? is even.
and G is a non-Abelian ?-group with the positive cone
G+ = ? ? ? ? Z+0 U Z+ ? ?+ ? {0},
or equivalently, (k[,m\,n\) ? (k2,m2,n2) iff (i) «? < n2, or (ii) «? = n2, k\ ^ b,
m ? ij Ш2.
Then
(k\,m\,n\) ? (k2,m2,n2)
(it|,b,M|), if «1 > «2,
(&i ? it2,wi ? Ш2,И| vna), if«i=«2,
(k2,m2,n2), if «l < «2,

862
A. Dvurecenskij
and и — A, 1, 1) is a strong unit for G. Consequently, the corresponding pseudo MV-alge-
bra has the form
r(G,u) = Z+ xZ+xjOjUZ^i xZ^i ? {1}.
with
(k,m,0)~ = (\ -k,\-m,\), (k, m,0)~ = A - m, 1 - k, 1),
(k,m, 1)" = A -m, 1 -?,?), (?,/я, 1)~ = A - ?, 1 -w,0),
and
{к\,т\,0)® (k2,m2,0) = (к\ +к2,т\ + m2,0),
(?i,mi,0)©(?2,w2, 1) = ((mi +Ь) л l,C«2+*i)A 1, 1),
(kum\, l)®(k2,m2,0) = ((?| +Ь)л1,(ш|+ш2)л1,1),
(*i,i»i|,l)e(*2,W2,l) = (l,l,l)·
We recall that every finite pseudo MV-algebra is an MV-algebra.
Now we show that the pseudo MV-algebras of type T(G, u), where (G, u) is a unital
?-group, are prototypical examples of pseudo MV-algebras in view the following principal
representation of pseudo MV-algebras proved in [21, Theorem 3.9] which generalizes the
famous result of Mundici for MV-algebras.
THEOREM 6.3. For any pseudo MV-algebras ?, there exists a unique {up to isomorphism)
unital l-group G with a strong unit и such that ? = Г(С«). The functor Г defines
a categorical equivalence of the category of pseudo MV-algebras with the category of
unital l-groups. In addition, if h is a given isomorphism of ? with T(G, u), (G, h) is
the universal group for M: that is, for any partially ordered group (K; +, ^, 0) and any
order- and -^-preserving mapping g:M—>-K there is a homomorphism of partially ordered
groups g' :G —> К such that g = g' о h.
We say that an ideal of a pseudo MV-algebra ? is a subset I of ? such that
(i) 0e/;
(ii) if x, у e /, then ? ? у е / and у ? ? е /; and
(iii) if ? e /, у e M, and у ^ x, then у е /.
An ideal / of ? is said to be
(i) normal if ? ? / = / ? ? for any x e ?;4
(ii) prime if ? А у e / entails ? e / or у e /;
(iii) maximal if / is a proper ideal of ? and it is not included into any proper ideal of
? different of/.
We define ? ® / := {? ® ?: ? e /); similarly we define / ?-?.

Measures on quantum structures
863
6.2. States and pseudo MV-algebras
In the present section, we introduce states and state-morphisms on pseudo MV-algebras,
and we show their close connections with normal and normal maximal ideals on pseudo
MV-algebras,
A state on a pseudo MV-algebra ? is a mapping m : ? —>· [0, 1 ] such that
(i) w(l) = 1; and
(ii) m (a + b) = m (a) + m(b) whenever ? + b is defined in M.
If ? is a commutative pseudo MV-algebra (i.e., an MV-algebra), then ? possesses at
least one state. Take now ? = T(G, u) from Example 6.2, and define a mapping m on ?
by m((k,m,0)) = 0 and m((k,m, 1)) = 1. Then m is a unique state on this pseudo MV-
algebra ? which is not an MV-algebra.
The basic properties of states are as follows.
PROPOSITION 6.4. Letm be a state on a pseudo MV-algebra M. Then, for all a,b e M,
we have
(i) m@) = 0.
(ii) If a ^ b, then m(a) ^ m(b) and
m(bQa~) = m(b) — m{a) = m(a~~ ? b).
(iii) m(a~) = 1 — m(a) = m(a~).
(iv) m{a~)=m{a) = m{a^).
(v) m(a ? b) +m(a A b) = m{a) +m{b).
(vi) m(a ? b) + m(b Qa)=m(a) + m(b).
(vii) The set
Ker(w) := [a e M: m(a) = 0}
is a normal ideal of M.
(viii) a/Ker(w) = fo/Ker(m) if and only if m (a) = m(a A b) = m(b).
(ix) There is a unique state m on Ai/Ker(w) such that m([a]) = m(a), [a] ?
?I Ker(w), where [a] is the coset in M/ Ker(w) determined by a e M.
(x) m(a ®b) =m{b®a).
A mapping m from a pseudo MV-algebra ? into the standard MV-algebra [0, 1] such
that, for all a, foe M,
(i) m(a®b)=m(a)®mm(b),
(ii) m(a~) = w(a~) = 1 — m(a), and
(iii) w(l) — 1 is said to be a state-morphism.
THEOREM 6.5. Letm be a state on a pseudo MV-algebra M. Then m is a state-morphism
if and only if Ker(w) is a maximal ideal of M.
Let m\ and mi be two measure-morphisms on a pseudo MV-algebra ? such that
Ker(wi) = Ker(w2). Then m\ = m^.

864
A. Dvurecenskij
Conversely, let I be a normal and maximal ideal of a pseudo MV-algebra M. Then there
is a unique state-morphism m on ? such that Ker(w) = /.
PROPOSITION 6.6. Let m be a state on a pseudo MV-algebra M. Then the following
statements are equivalent:
(i) m is an extremal state on M.
(ii) m is a state-morphism on M.
(iii) m(x Лу) = m'm{m(x),m(y)}, х, у е М.
Denote by S(M) the set of all states of a pseudo MV-algebra ?. Then S(M) is a convex
set, and let ExtE(M)) be the set of all its extremal states. A net {ma} of states on ?
converges weakly to a state m iff ma{a) —>¦ m(a) for every a e M.
We recall that due to the categorical equivalence, there exists a one-to-one
correspondence between states on T(G, u) and states on (G, u).
THEOREM 6.7. Let ? be a pseudo MV-algebra. IfS(M) ? 0, then S(M) is a compact
convex Hausdorff space with respect to the weak topology, and the space of all state-
morphisms on ? is a non-void compact Hausdorff space. Any state is a weak limit of
convex linear combinations of the set of extremal points ofS(M).
For every a e ?, we put
M(a):= {I eAfM(M): a ?/}.
Then (i) M@) = 0, ? (а) с ? ?) whenever я ^ b, ? (a Ab) = ? (?) ? M{b),a,b e M,
M(a ? b) = ? (a) U ? ?), a,be M, and {M(a): a e M) is the base of the so-called
hull-kern topology TtfM on ???4(?).
Then the hull-kern topology defines a Hausdorff topology such that the closed subspaces
of ???(?) are exactly of the form
C = C(J):={I еЛГМ(М): /ЗУ},
where J is an ideal of ?. Similarly, every open set О is of the form
О = O(J) := {I еЛГМ(М): I 2 ·/}·
The following statement shows that extremal states and normal maximal ideals are
homeomorphical.
THEOREM 6.8. The mapping ? : ExtE(M)) -+ ???(?) defined by
0(/и):=Кег(/и), т е Ext(<S(M)),
is a homeomorphism.

Measures on quantum structures
865
6.3. Existence of states
As we have seen in Section 4, every MV-algebra possesses at least one state. For pseudo
MV-algebras, the situation can be more complicated, because as it was shown in [22], there
exists a pseudo MV-algebra having no state on it. Therefore we will now concentrated to
the question when a pseudo MV-algebra has at least one state.
Example 6.9. Let G be the group of all matrices of the form
where ? and a are rational (real) numbers such that ? > 0; the group-operation is the usual
multiplication of matrices. We denote A — (?, a). Then A = (\/?,-?/?), and A,0) is
the neutral element. We define G+ :— {(?, a); where (i) ? > 1, or (ii) ? — 1 and a > 0).
Then G with the positive cone G+ is a linearly ordered ?-group with a strong unit U =
B,0).DefineM = r(G,tf).ThenM = M|UM2UM3,whereM| ={(?,«): 1 <? <2),
Мг = {B, ?): ? ^0), and ?/3 = {(?,?): ? ^ 0). My is a unique normal and maximal ideal
of M, and there is a unique state-morphism m, namely ??((?, a)) = log2(?), (?, a) e M.
THEOREM 6.10. Every linearly ordered pseudo MV-algebra ? possesses a unique state.
We say that a pseudo MV-algebra ? is representable if it can be represented as a
subdirect product of linearly ordered pseudo MV-algebras.
THEOREM 6.11. Every representable pseudo MV-algebra ? possesses at least one state.
Let g be a nonzero element of a pseudo MV-algebra M. According to [35], we say that
a value of g is an ideal V of ? such that g ? V, and V is maximal with respect to this
property. As it was shown in [35, Theorem 2.15] such an ideal always exists in ? and is a
prime ideal. We denote by r(g) the set of all values of g. For any value V of an element g
we have that there is a unique least ideal V* properly containing V; it is called a cover
of V.
? is said to be
(i) finite valued if every non-zero element g e ? has finitely many values;
(ii) normal-valued if, for any g > 0, any values V e r(g) is normal in its cover V*.
THEOREM 6.12. Each normal-valued pseudo MV-algebra ? possesses at least one state.
In particular, //|?A)| = ?, then ? possesses exactly ? state-morphisms.
Finally we recall only that every MV-algebra is representable, every representable
pseudo MV-algebra is normal-valued, and the converse implications do not hold, in
general. Moreover, due to categorical equivalence, we have also built up theory of states
on unital ?-groups not necessarily Abelian.

866
A. Dvurecenskij
7. Conclusion
As we have seen, the sixth Hilbert's problem inspired experts to develop a very nice
theory of quantum structures which today can live independently on the original quantum
mechanics motivation. Nowadays we can find situations very far from quantum mechanics,
where measured data do no fulfill axioms of the Kolmogorov axiomatical model of
probability theory.
Theory of quantum structures is a typical example of interdisciplinary scientific area
where mathematicians, physicists, logicians can exchange their ideas. Today there exists
an extensive list of monographs dedicated to this theory, we mentioned only [4,14,28,41,
46,64,70] and others. In this chapter we showed how deep is theory of states and measure
on these structures. Moreover, we have shown their very close connections with classical
mathematical objects as ?-groups and po-groups, respectively.
Pseudo MV-algebras are probably the first algebraic structures with noncommutative
addition for which measure theory is developed. Other new mathematical structures are
pseudo effect algebras introduced recently by the author and ? Vetterlein [29,30] which
also need measure theory.
Finally we can say that on the theoretical analysis of any measured object, the
relationship between the apparatus of quantum mechanics and quantum structures is
similar to that between statistical physics and probability theory. Recent results show
that the effort of mathematicians and physicists in building up probability theory and
mathematical statistics was successful. Hence, there is a real hope that the investigation
of quantum structures can bring new and useful results for both mathematics and quantum
mechanics.
References
[1] J.F. Aarnes, Physical states on a C*-algebra. Acta Math. 122 A969), 161-172.
[2] J.F. Aarnes, Quasi-states on C*-algebras. Trans. Amer. Math. Soc. 149 A970), 601-625.
[3] G. Boole, An Investigation of the Laws of Thought, Macmillan A854). Reprinted by Dover, New York
A967).
[4] E.G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics, Addison-Wesley, Reading A981).
[5] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., Vol. 25, Providence, RI A967).
[6] G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 A936), 823-834.
[7] L.J. Bunce and J.D. Wright, The Mackey-Gleason problem. Bull. Amer. Math. Soc. 26 A992), 288-293.
[8] C.C. Chang, Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 A958), 467^490.
[9] E. Christensen, Measures on projections and physical states, Comm. Math. Phys. 86 A982), 529-538.
[10] R. Cooke, M. Keane and W. Moran, An elementary proof of Gleason's theorem. Math. Proc. Cambridge
Phil. Soc. 98A985), 117-128.
[11] M. Dichtl, Astroids and pasting. Algebra Univer. 18 A984), 380-385.
[12] S.V. Dorofeev and A.N. Sherstnev, Frame-type functions and their applications, Izv. Vuzov Matematika,
No. 4 A990), 23-29 (in Russian).
[13] T. Drisch, Generalization of Gleason's theorem, Intemat. J. Theoret. Phys. 18 A979), 239-243.
[14] A. Dvurecenskij, Signed states on a logic. Math. Slovaca 28 A978), 33^40.
[15] A. Dvurecenskij, Generalizations of Maeda's theorem, Intemat. J. Theoret. Phys. 25 A986), 1117-1124.
[16] A. Dvurecenskij, Gleason's Theorem and Its Applications, Kluwer, Dordrecht, and Ister Science Press,
Bratislava A993).

Measures on quantum structures
867
[17] A. Dvurecenskij, Frame functions and completeness of inner product spaces. Ann. Inst. H. Poincare Phys.
Theor. 62A995), 429^438.
[18] A. Dvurecenskij, Gleason's theorem in applications. Quantum Measures and Spaces, G. Kalmbach, ed.,
Kluwer, Dordrecht A998), 39-50.
[19] A. Dvurecenskij, Measures on commutative BCK-algebras, Atti Sem. Mat. Fis. Univ. Modena B000), to
appear.
[20] A. Dvurecenskij, Loomis-Sikorski theorem for ?-complete MV-algebras and l-groups, J. Austral. Math.
Soc. Ser. A 68 B000), 261-277.
[21] A. Dvurecenskij, Pseudo MV-algebras are intervals in ?-groups, J. Austral. Math. Soc. 72 B002), to appear.
[22] A. Dvurecenskij, States on pseudo MV-algebras, Studia Logica 68 B000), 301-327.
[23] A. Dvurecenskij, Fuzzy set representation of some quantum structures. Fuzzy Sets and Systems 101 A999),
67-78.
[24] A. Dvurecenskij, P. de Lucia and E. Pap, On a decomposition theorem and its applications. Math. Japonica
44A996), 145-164.
[25] A. Dvurecenskij and M.G. Graziano, Commutative BCK-algebras and lattice ordered groups. Math.
Japonica 49 A999), 159-174.
[26] A. Dvurecenskij and M.G. Graziano, Dedekind complete commutative BCK-algebras. Order 17 B000),
23-41.
[27] A. Dvurecenskij and S. Pulmannova, Test spaces, Dacey spaces, and completeness of inner product spaces,
Lett. Math. Phys. 22 A994), 299-306.
[28] A. Dvurecenskij and S. Pulmannova, New Trends in Quantum Structures, Kluwer, Dordrecht, and Ister
Science, Bratislava B002).
[29] A. Dvurecenskij and T. Vetterlein, Pseudo-effect algebras. I. Basic properties. Intemat. J. Theoret. Phys. 40
B001), 685-701.
[30] A. Dvurecenskij and T. Vetterlein, Pseudo-effect algebras. II. Group representation, Intemat. J. Theoret.
Phys. 40B001), 703-726.
[31] M. Eilers and E. Horst, The theorem of Gleason for nonseparable Hilbert space, Intemat. J. Theoret. Phys.
13A975), 419^424.
[32] D.J. Foulis and M.K, Bennett, Effect algebras and unsharp quantum logics. Found. Phys. 24 A994), 1325-
1346.
[33] D.J. Foulis and C.H. Randall, Operational statistics. I. Basic concepts, J. Math. Phys. 13 A972), 1667-
1675.
[34] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford A963).
[35] G. Georgescu and A. Iorgulescu, Pseudo-MV algebras, Multi-Valued Logic B000), in print.
[36] R. Giuntini and H. Greuling, Toward a formal language for unsharp properties. Found. Phys. 19 A989),
931-945.
[37] A.M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 A957), 885-893.
[38] K.R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Math. Surveys Monogr., Vol. 20,
Amer. Math. Soc., Providence, RI A986).
[39] R. Greechie, On the structure of orthomodular lattice satisfying the chain condition, J. Comb. Theor. 4
A968), 210-218.
[40] R. Greechie, Orthomodular lattices admitting no states, J. Comb. Theor. 10 A971), 119-132.
[41] S. Gudder, Stochastic Methods in Quantum Mechanics, Elsevier, Amsterdam A979).
[42] J. Gunson, Physical states on quantum logics, Ann. Inst. H. Poincare, Phys. Theor. 17 A972), 295-311.
[43] W. Heisenberg, The Physical Principles of Quantum Theory, Dover, New York A930).
[44] Y. Imai and K. Iseki, On axiom systems of prepositional calculi, Proc. Japan Acad. 42 A966), 19-22.
[45] R. Jajte, On convergence of Gleason measures. Bull. Acad. Polon. Sci., Ser. Math. Astr. Phys. 20 A972),
211-214.
[46] G. Kalmbach, Orthomodular Lattices, Academic Press, London A983).
[47] S. Kochen and E.R. Specker, The problem of hidden variables in quantum mechanics, J. Math. Mech. 17
A967), 59-67.
[48] A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin A933).
[49] F. Корка and F. Chovanec, D-posets, Math. Slovaca 44 A994), 21-34.

868
A. Dvurecenskij
[50] ?.?. Lodkin, Every measure on the projectors of a W*-algebra can be extended to a state, Funct. Anal.
Appl. 8 A974). Translated from Funkc. Anal. I Ego Priloz. 8 A974), 54-58.
[51] G.D. Lugovaja, Bilinear forms defining measures on projectors. Izv. Vuzov Matematika, No. 2 A983),
88-88 (in Russian).
[52] G.D. Lugovaja and A.N. Sherstnev, On the Gleason theorem for unbounded measures, Izv. Vuzov
Matematika, No. 2 A980), 30-32 (in Russian).
[53] G.W. Mackey, Mathematical Foundations of Quantum Mechanics, Benjamin, New York A963).
[54] S. Maeda, Lattice Theory and Quantum Logic, Maki-Shoten, Tokyo A980) (in Japanese).
[55] S. Maeda, Probability measures on projectors in von Neumann algebras. Rev. Math. Phys. 1 A990), 235-
290.
[56] S.M. Matvejchuk, Finite measures in an approximative-finite factor, Izv. Vuzov Matematika, No. 5 A976),
79-85 (in Russian).
[57] S.M. Matvejchuk, Extension of measures in approximative factors, Izv. Vuzov Matematika, No. 2 A977),
84-90 (in Russian).
[58] S.M. Matvejchuk, An extension of unbounded measure to weight, Izv. Vuzov Matematika, No. 4 A987),
47-50 (in Russian).
[59] D. Mundici, Interpretation ofAF C*-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 A986),
15-63.
[60] D. Mundici, Averaging the truth-value in Lukasiewicz logic, Studia Logica 55 A995), 113-127.
[61] M. Navara, Two descriptions of state spaces of orthomodular structures, Intemat. J. Theoret. Phys. 38
A999), 3163-3178.
[62] A. Paszkiewicz, Measures on projections of von Neumann algebras, J. Funct. Anal. 62 A985), 87-117.
[63] C. Piron, Foundations of Quantum Physics, Addi son-Wesley, Reading, MA A976).
[64] P. Ptak and S. Pulmannova, Orthomodular Structures as Quantum Logics, VEDA and Kluwer, Bratislava
and Dordrecht A991).
[65] J. Rachunek, A non-commutative generalization of MV-algebras, Czechoslovak Math. J. 52 B002), 255-
273.
[66] B. Riecan and D. Mundici, Probability on MV-algebras, Handbook of Measure Theory, E. Pap, ed., Elsevier,
Amsterdam B002), 869-909.
[67] A.N. Sherstnev, On a notion of a charge in noncommutative scheme of measure theory, Veroj. Metody
Kibem., Kazan, No. 10-11 A974), 68-72 (in Russian).
[68] R. Sikorski, Boolean Algebras, Springer, Berlin A964).
[69] B. Simon, A canonical decomposition for quadratic forms with application to monotone convergence
theorems, J. Funct. Anal. 28 A978), 377-385.
[70] VS. Varadarajan, Geometry of Quantum Theory, Vol. 1, Van Nostrand, Princeton, NJ A968).
[71] J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin A932).
[72] F.J. Yeadon, Measures on projections in W*-algebras of type ll\. Bull. London Math. Soc. 15 A983),
139-145.
[73] F.J. Yeadon, Finitely additive measures on projections in finite W* -algebras. Bull. London Math. Soc. 16
A984), 145-150.

CHAPTER 21
Probability on MV-Algebras
Beloslav Riecan
Mathematical Institute, Slovak Academy of Sciences. Stefanikova 49. SK-814 73 Bratislava, Slovakia
E-mail: riecan@mat.savba.sk
Daniele Mundici
Department of Computer Science, University of Milan, Via Comelico 30-41, 20135 Milan, Italy
E-mail: mundici@mailserwr.uiiimi.it
Contents
1. Background on ?V-algebras 871
Prologue: Operator algebras and MV-algebras 871
1.1. MV-algebras: basic notions 872
1.2. Representation of semisimple MV-algebras 873
1.3. MV-algebras and ^-groups 874
1.4. Tribes and ?-complete MV-algebras 876
1.5. The Loomis-Sikorski theorem for MV-algebras 877
1.6. Bibliographical remarks 878
2. States and observables 879
2.1. States and observables in MV-algebras 879
2.2. Independence 883
2.3. Kolmogorov's construction 884
2.4. Functions of several observables 885
2.5. The MV-algebraic central limit theorem 886
2.6. MV-algebraic laws of large numbers 888
2.7. Bibliographical remarks 889
3. MV-algebras with product 890
3.1. Product and joint observables 891
3.2. Conditional expectation 893
3.3. Upper and lower limits 895
3.4. Individual ergodic theorem 897
3.5. Bibliographical remarks 898
4. Finitely additive measures 899
4.1. Basics 900
4.2. Entropy of dynamical systems 901
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V All rights reserved
869

870 В. Riecan and D. Mundici
4.3. Entropy of full tribes 903
4.4. Bibliographical remarks 904
5. Open problems 905
References 906

Probability on MV-algebras
871
1. Background on MV-algebras
Our aim is to survey the MV-algebraic generalization of notions and results of boolean
algebraic probability theory stemming from the work of Caratheodory and von Neumann.
The former author was motivated by the program of founding probability theory without
mentioning points in the event space ?; the second author was driven by the hope that an
algebra of observables might be built on the lattice structure of the quantum mechanical
counterpart of ?.
MV-algebras (to be defined in 1.2 below) are a non-commutative generalization of
boolean algebras, in a sense that will be made more precise in Theorem 1.1, using the
language of C*-algebras. Readers only interested in MV-algebraic probability theory can
safely skip the following prologue, and proceed to the next section.
Space constraints will not allow us to include many proofs; however, at the end of each
section the reader will find adequate bibliographical references for the proofs of all stated
results.
Prologue: Operator algebras and MV-algebras
In the approach of Birkhoff and von Neumann [3], properties of a quantum system S
are identified with self-adjoint idempotent elements (i.e., projections) in the algebra B(H)
of continuous linear operators on the Hilbert space ? of S. Equivalently, properties are
elements of the family С-ц of all closed linear subspaces of ?. Observables of S are then
built from the projections. The second main important notion is that of a state of S. The
following definition captures the intuitive ideal that a "physical state" should be a sort of
continuous averaging process for these propositions.
A state is a map m : С-ц -> [0, 1] such that m(H) = 1 and whenever ? ?, ??, A3,.., с
Cn are pairwise orthogonal closed linear subspaces of ?, then m(\J^L, A,) = Y^LX w(A,).
Here, V/^i ?? denotes the closed linear subspace of ? generated by the set-theoretic union
U A,, Measure theory for lattice structures associated to Hilbert space projections will be
the object of other chapters in this Handbook.
By contrast, no Hilbert space ?. can be canonically assigned to a system with
infinitely many degrees of freedom, such as those naturally occurring in quantum
statistical mechanics and in quantum field theory. Any such system S typically possesses
many physically inequivalent irreducible Hilbert space representations, paralleling its
macroscopically different phases, or preparations. This happens in particular when S arises
as the thermodynamic limit of finite spin-like systems. The C*-algebraic counterpart of S
is then an AF C*-algebra, i.e., the norm-closure A of the union of an ascending sequence
?\ с f2 ? ¦ · ¦ of finite-dimensional C*-algebras, all with the same unit [1,4].
While all physical information about S is now coded into A, rather than in a Hilbert
space ?, the role of the projections of A is no less important than in the case of B(H).
To see this, one first defines two projections p,q e A to be equivalent if there exists
an element ? e A such that vv* = ? and v*v — q, where v* is the adjoint of v. We
denote by [p] the equivalence class of p, and by L(A) the set of equivalence classes of
projections of A, The Murray-von Neumann order over L (A) is defined by stipulating that

872
В. Riecan and D. Mundici
[p] ^ [?] if ? is equivalent to a projection r such that rq = r, Elliott's partial addition,
denoted +, is the partial operation on L(A) given by adding two projections whenever they
are orthogonal. Then + is associative, commutative, monotone, and satisfies the following
residuation property;
(*) For every projection ? eA, among all classes [q] such that [p] + [q] = [1д] there
is a smallest one,1 denoted -*[p], namely the class [1_д — ?].
Theorem 1.1.
(i) There is at most one extension of Elliott's partial addition to an associative,
commutative, monotone operation © over the whole L(A) having the above
residuation property (*). Such extension © exists if, and only if L(A) is a lattice.
(ii) Let >C(A) — (L(A), [0], ->, ?). Then the map А ь* ?,(?) is a one-one
correspondence between isomorphism classes of AF C*-algebras whose L(A) is lattice-
ordered, and isomorphism classes of countable MV-algebras,
(iii) In particular, up to isomorphism, the map А ь* ?,(?) is a one-one correspondence
between commutative AF C* -algebras and countable boolean algebras. The inverse
correspondence is given by the map clop(X) ь* C(X), the latter denoting the C*-
algebra of complex-valued continuous functions over an arbitrary separable totally
disconnected compact Hausdorff space X, Here clop(X) denotes the boolean
algebra of closed-and-open subsets ofX,
1.1. MV-algebras: basic notions
The above theorem gives a precise formulation to the intuition that (countable)
MV-algebras are a non-commutative generalization of (countable) boolean algebras. Accordingly,
the MV-algebraic probability theory developed in this survey can be thought of as a non-
commutative generalization of classical boolean algebraic probability theory.
Definition 1.2. An MV-algebra ? = (?, 0, 1, -л ?, ?) is a system where ? is
associative-commutative with neutral element 0, and, in addition, -? = 1, -? = 0, ? ? 1 = 1,
x Oy = -¦(-¦ ? ®-\y) and у ® -*(у ® -*х) =х 0-4* ? ~\у), for all ?, у е ?,
Since both the constant 1 and the operation ? are definable from the remaining
operations 0, ->, ?, there no danger of confusion between the present definition, and the
statement of Theorem l.l(ii).
MV-algebras are the models of the equational theory of magnitudes with a distinguished
archimedean unit, in a sense that will be made precise by Theorem 1.7 below. They stand
to the infinite-valued calculus of Lukasiewicz as boolean algebras stand to the classical
two-valued calculus.
An example of MV-algebra is the real unit interval [0, 1] equipped with the operations
-a— 1-х, ? ®y = min(l,;t +y), ? ? у = max@, x + у - 1). A)
?? fact, ~·[?] is the only class [q] satisfying [p] + [q] = [I.4].

Probability on MV-algebras
873
This MV-algebra has the same role for MV-algebras as the two-element algebra {0, 1} has
for boolean algebras. Indeed, Chang's completeness theorem states that if an equation is
satisfied by [0, 1] then the equation is satisfied by all MV-algebras. It follows that every
MV-algebra is a homomorphic image of a subalgebra of a product of copies of [0, 1 ].
In every MV-algebra ? = {?, 0, 1, ->, ?, ?), the binary relation ^ given by a ^ b iff
? ? -<b = 0, is a partial order. Furthermore, the derived operations ? and л given by
avb = -"-(--? ? b) © b and a /\b = --(--? ? -*b) make ? into a distributive lattice with
least element 0 and greatest element 1; this is called the underlying lattice of ?.
Boolean algebras coincide with those MV-algebras such that ? ? ? = ?. When the latter
equation holds, then the ? operation coincides with v, and © coincides with л.
DEFINITION 1.3, An MV-algebra ? is said to be ?-complete if its underlying lattice is
?-complete, i.e., every non-empty countable subset of ? has a supremum in ?.
Every finite MV-algebra ? is ?-complete - indeed, ? is complete, in the sense that
every non-empty subset of ? has a supremum in M. Further examples of ?-complete
MV-algebras are given by ?-complete boolean algebras, and in particular, ?-algebras of
sets. More examples will be given in Section 1.4. In every ?-complete MV-algebra ? we
have the distributivity law
b A (b\ ? Ьг ? · · ·) = (b л b\) ? (b л bi) ? · · ·, B)
1.2. Representation of semisimple MV-algebras
An MV-algebra ? is said to be semisimple if for each non-zero element ? e ? there is
a homomorphism ?:?->[0,\] with ?(?) ? 0. Ideals are kernels of homomorphisms.
Unless otherwise specified, every ideal / of any MV-algebra ? considered in this paper
shall be assumed to be proper, i.e., / ? ?. An ideal / is said to be maximal if there is no
ideal of ? strictly containing / (other than the improper ideal M). Thus, maximal ideals
are automatically proper. We let M(M) denote the set of all maximal ideals of M. Since
/ is maximal iff it is maximal for the property of not containing the unit element 1, it is not
hard to see that M(M) ? 0.
PROPOSITION 1,4. For any MV-algebra ? and maximal ideal I of M, there is a
unique isomorphism i/ of the quotient MV-algebra M/I onto a subalgebra o/[0, 1]. Let
? l: ? —>¦ M/I be the quotient map. Then the map I ь* M/I ь* ?/(?/I) is a one-to-one
correspondence between the set of maximal ideals of ? and the set of homomorphisms
of ? into [0, 1]. The inverse of this map sends any such homomorphism into its kernel.
For every ideal / of M, let 0/ = {K e M{M) \ К 2 /}. Then the collection of all sets
of the form ? ? is a compact Hausdorff topology on ? (?), called the spectral topology.
By abuse of notation, we shall let M{M) also denote the space of maximal ideals of ?
equipped with the spectral topology.
Also known as the hull-kernel topology.

874
В. Riecan and D. Mundici
For any compact Hausdorff space X let us agree to denote by C(X) the MV-algebra of
all continuous [0, l]-valued functions on X with pointwise operations as in A), Following
tradition, let [0, l]x denote the MV-algebra of all [0, l]-valued functions on X. Recalling
Proposition 1,4, for every MV-algebra ? let the map a e ? ь* а* е [0, l]x be defined by
stipulating that, for each / e M(M),
?*(/) = (?/?0,)(?)=?,(?//)?[?, 1], C)
where о denotes composition of functions.
The following theorem gives a concrete representation of semisimple MV-algebras:
THEOREM 1.5. For every semisimple MV-algebra M, the above map а ь* a* is an
isomorphism of ? onto a separating subalgebra M* of C(M(M)), in the sense that
whenever I, J are distinct maximal ideals of ? there are elements a*, b* e M* with
а*A)фЬ*У).
COROLLARY 1.6. An MV-algebra ? is semisimple iff ? is isomorphic to a separating
MV-algebra o/[0, \]-valued continuous functions defined over some compact Hausdorff
space, iff ? is isomorphic to an MV-algebra of[0, \\-valued functions defined over some
set.
1.3. MV-algebras and i-groups
By an ?-group we shall mean a lattice-ordered Abelian group. For any ?-group G, an
element и e G is said to be a strong unit of G if for all g e G there is an integer ? > 1
such that nu ^ g. By a morphism ?: (G, u) -> (G', и') we mean a group homomorphism
0: G —>¦ G' that also preserves the lattice structure and satisfies the condition ф(и) = и'.
THEOREM 1.7. For any ?-group G with a strong unit и let T(G, u) be the unit interval
[0,H] = (/ieG|0^/i^H), equipped with the operations
->g = u-g, g®h = u A(g+h), and g ? h = 0 ? (g +h - u).
Then;
(i) the structure ? = ([0, и], 0, и, ->, ?, ?) = T(G, u) is an MV-algebra;
(ii) letting, for any morphism ?; (G, u) -> (G\ и'), Г (ф) be the restriction of ? to
[0, u], then ? is a categorical equivalence (i.e., a full, faithful, dense functor) from
?-groups with strong unit to MV-algebras;
(iii) the lattice operations on ? agree with those ofG;
(iv) the map J ь* J ? [0, и] is an isomorphism between the lattice of l-ideals
of G {equipped with set-theoretic inclusion) and the lattice of ideals of ?
(also equipped with inclusion). All infinite suprema and infima are preserved by
this correspondence. There is a natural isomorphism between r(G/J, u/J) and
Г(С,»)/GП[0,«]).

Probability on MV-algebras
875
Thus in particular, up to isomorphism, every MV-algebra ? can be identified with the
unit interval of ? unique ?-group G with strong unit u, in symbols,
M = r(G,u), D)
We say that G is the I-group (with strong unit u) corresponding to ?.
COROLLARY 1,8. Let ? bean MV-algebra,
(i) ? is ?-complete iff its corresponding ?-group G is Dedekind ?-complete, in the
sense that every bounded sequence of elements in G has a supremum in G.
(ii) If ? is ?-complete then ? is semisimple, and the set of maximal ideals of ?
equipped with the spectral topology is a basically disconnected compact Hausdorff
space.
Recall that a compact Hausdorff space X is basically disconnected if whenever ? c. X
is an open FCT subset, then the closure of ? is open. This is equivalent to saying that X
is homeomorphic to the maximal ideal space M{B) of a ?-complete boolean algebra B,
A subset of X is meager (also called, a set of first category) if it is the countable union of
subsets of X whose closure has empty interior.
Notation. Throughout this paper we let N = {1,2,,..} denote the set of non-zero natural
numbers.
Suppose aij is a double sequence of real numbers > 0 such that, for each fixed / the
sequence a,j tends to 0 as j tends to infinity. Then for each ? > 0 and every / there is an
index j* = ??(?) such that aiy < ? for all j > j*. It follows that Vai'.<M') ^ ?> whence
°= ? Va'X"·
феЦм i
DEFINITION 1,9, An ?-group G is said to be weakly ?-distributive if it is Dedekind
?-complete, and whenever 0 ^ aiy e G is a bounded double sequence and for each / e N,
a\j > aij+\ and Д aiy = 0, then
°= ? Va<>«>
tj>eNH i
An MV-algebra ? is said to be weakly ?-distributive if it is ?-complete, and whenever
bij e ? is a double sequence and for each / e N, bij > foiy+i and Ду b/j = 0, then
°= Л Vbw)<
From Theorem l,7(iii) we get

876
В. Riecan and D. Mundici
PROPOSITION 1,10, Let ? be an MV-algebra and G its corresponding l-group with
strong unit u. Then ? is weakly ?-distributive iff so is G,
1.4. Tribes and ?-complete MV-algebras
Examples of ?-complete MV-algebras are given by tribes, the latter being a generalization
of ?-algebras of sets:
Definition 1.11. By a tribe over a non-empty set ? we mean an MV-algebra ? of
[0, l]-valued functions on ? such that for each sequence f\, /2,... e ? the pointwise
supremum / = sup, /, also belongs to ?.
While ? is a ?-complete MV-algebra, not every ?-complete MV-algebra is isomorphic
to a tribe (already a ?-complete boolean algebra need not be isomorphic to a ?-algebra
of sets). The totality of ?-complete MV-algebras shall be characterized in Theorem 1.16
below.
Let ? be a non-empty set and S a ?-algebra of subsets of ?. Let ? be the set of all
5-measurable functions from ? to [0, 1]. Then ? is a tribe, and, trivially, all [0, l]-valued
constant functions over ? are members of ? (for short, ? is a fall tribe).3 Conversely we
have:
THEOREM 1.12. Let ? be a tribe over a non-empty set ?. Let S be the family of subsets
of ? whose characteristic function belongs to T. We then have:
(i) S is a ?-algebra of subsets of ? and each function f in ? is S-measurable;
(ii) // ? is full then ? coincides with the set of all S-measurable functions f : ? —>¦
[0, 1].
Notation. For any ? с ? we let ?? : ? -> {0, 1} denote the characteristic function,4
of Y. Following tradition, a sequence a 1, ai, ...of sets, is said to be monotone if a \ Ce2C
• · ·. The notation a„ / a stands for "? ?, ai,... is a monotone sequence of sets whose union
is a". Similarly, for real numbers x\, xi,... the notation xn / x stands for "x\ ^xi ^ · · ·
and x is their supremum". Finally, given elements bi,bi,... in an MV-algebra M, the
notation bn / b stands for "b\ ? bi ^ · · · and b is their supremum, with respect to the
underlying order of ?".
In the light of Theorem 1.12, the following is a generalization of the notion of probability
measure on a ?-algebra of sets.
Definition 1.13. Given a tribe T, by a state on ? we mean a map m:T^>- [0, 1]
satisfying the following conditions:
(i) m(l) = 1;
Also known as a generated tribe [5].
Or indicator.

Probability on MV-algebras
877
(ii) if a, b,c e ? and a = b + c, then m (a) =m(b) + m(c)\
(iii) for all a,a\,a2,... e J7, if a„ /a, then m(a„) / т(а).
The following is an integral representation theorem for tribes with a state:
THEOREM 1.14. Let (?, J7) be a (not necessarily full) tribe with a state m.LetSbe the
?-algebra of all those subsets ? of ? whose characteristic function ?? belongs to T. Let
the map ? : S -*¦ [0, 1 ] be given by P(Y) = m (??). Then ? is a probability measure on S,
and for every f e ? we have
m(f)= f fdP. E)
As a trivial converse, for any probability space (?, S, P) let ? be the full tribe of all
<S-measurable [0, l]-valued functions over ?. Let the state mp :T —> [0, 1] be given by
mp(f) = fn /dP. Then (?, J7) is a full tribe with a state mp. By applying Theorem 1.14
to the triplet (?, ?,mp) we recover the probability space (?, S, P).
1.5. The Loomis—Sikorski theorem for MV-algebras
Definition 1.15. Given two ?-complete MV-algebras F and M, a homomorphism
?: F —>¦ ? is said to be a ? -homomorphism if ? preserves all countable suprema, in
symbols,
ч(Уа«-)=\/(^а'-»
for all sequences a \, ai,... e F.
In particular, when F is a tribe ? over a set ?, and ? a ?-complete MV-algebra,
a homomorphism ?:?—>-? is said to be a ? -homomorphism if for each sequence
/?. /2, /3, · · · e T, letting / = sup, /, be their pointwise supremum, r\(f) coincides with
the supremum V; l(fi) m M.
The following result generalizes the classical Loomis-Sikorski theorem:
THEOREM 1.16. Let ? be a ?-complete MV-algebra. Let ? = M(M) be the set of
maximal ideals of ? with the spectral topology. Then there exists a tribe ? over ? and a
? -homomorphism ? of ? onto M, satisfying the following conditions:
For each a e ? there is precisely one continuous function a* e ? such that ?(?*) = a.
A function in ? has the same image as a* iff it differs from a* on a meager subset of ?.
The above tribe ? precisely consists of all functions / : ? -> [0, 1 ] which differ only on
a meager set from some (necessarily unique) function arising from the representation of ?
as a separating algebra of continuous functions over ? (Theorem 1.5). In general, ? need

878
В. Riecan and D. Mundici
not be full. To ensure fullness one must assume ? to satisfy an additional condition, as
follows:
PROPOSITION 1.17. Let ? be a ?-complete MV-algebra and G its corresponding
l-group with strong unit u. Then the following conditions are equivalent:
(i) ? has no finite quotients;
(ii) for every maximal ideal J of M, the quotient MV-algebra M/J is infinite;
(iii) ? is divisible (in the sense that G is a divisible group);
(iv) for each integer ? ^ 1, there is w e ? such that nw = 1 = u inG, or equivalently,
w ? w ? · · · ? w = -*w;
? — I times
(v) the MV-algebra Q ? [0, 1] is embeddable into M.
For ?-complete divisible MV-algebras the Loomis-Sikorski theorem has the following
formulation:
THEOREM 1.18. Let ? be a ?-complete divisible MV-algebra.
(i) Let ? = M(M) be the compact Hausdorff space of maximal ideals ofM. Then the
map а ь* a* of Theorem 1.5 is an isomorphism between ? and the MV-algebra
?(?) of all continuous [0, \~\-valued functions over ?.
(ii) Let ? be the set of all functions f: ? —> [0, 1] which are equal to a continuous
function g* e ?(?) except for a meager set. Then g* is uniquely determined by f,
? is a full tribe over ?, and the map f ь* g* ь* g is ? ? -homomorphism of ?
onto M.
The proof depends on the following
LEMMA 1.19. Adopt the above notation. Let ? be a ?-complete divisible MV-algebra.
Then the map а ь* a* is an isomorphism of ? onto ?(?) that preserves all countable
suprema, in the sense that for any sequence a\ e ? the element (\J a,-)* is the supremum
of the functions a* : ? —> [0, 1] in ?(?), (which need not coincide with the pointwise
supremum of these functions).
1.6. Bibliographical remarks
See, e.g., [21] for the Hubert space representation of finite systems, and for the C*-alge-
braic representation of systems with infinitely many degrees of freedom, including
quantum spin systems and their AF C*-algebras. Elliott's classification theory originated
in [20]. See [27] for a comprehensive account, using ??-theory and partially ordered
groups with the Riesz interpolation property, of which Abelian ?-groups are a special
important case. Theorem 1.1 was proved in [63] using the ? functor and the main results
of [57].
The monograph [13] gives a comprehensive and self-contained account of MV-algebras.
The latter were introduced by C.C. Chang in [7,8] to give an algebraic proof of the

Probability on MV-algebras
879
completeness of the infinite-valued calculus of Lukasiewicz [45,46], [83, Chapter IV],
[13,Chapter 4].See[9]forfurther historical remarks. All results of Sections 1.1 and 1.2 are
due to Chang. Self-contained proofs are given in [13, Chapters 1 and 3]. The distributive
law B) is a well known fact (the proof given in [13, Lemma 6.6.4] for complete MV-
algebras can be trivially adapted to the present case).
We refer to [26] and [2] for background on ?-groups. The equivalence between MV-
algebras and Abelian ?-groups with strong unit was first obtained in [57]. An elementary
presentation can be found in [13, Chapter 7].
The ? functor has an ubiquitous role in MV-algebraic probability theory, allowing the
application to MV-algebras of various results originally proved for ?-groups. Further, the
very definition of certain basic notions is naturally given by resorting to the ?-group
with strong unit corresponding to M. This is the case, e.g., of partitions of unity, to be
defined in Section 4.2. The purely MV-algebraic definition (which can be obtained from
[57, Definition 3.2, Proposition 3.3]) would not be as useful and immediately clear as
the group-theoretical definition given here. The same applies to the definitions of state,
observable, product given in the next sections. As explained in [57], composition of ?
with Grothendieck's Ко functor provides the link between countable MV-algebras and AF
C*-algebras whose Murray-von Neumann order is a lattice.
Proposition 1.10 is due to Jakubik [33].
The weak ?-distributivity of the underlying ?-group structure of any Riesz space V is
a necessary and sufficient condition for every V-valued measure to be extendable from a
subalgebra of a set to its enveloping ?-algebra (see [79, Chapter 3]).
For a proof of Theorem 1.12(i) see [39,5, Proposition 3.2]. For Theorem 1.12(h) see
[5, Proposition 3.3]. For Theorem 1.14 see [4, Theorem 2.6(c)] and [5]. Proofs can also be
found in [79, 8.1]. A characterization of tribes is given in [41, Theorem 4.3]. Piasecki's
concept of P-measure [67], [79, pp. 140-141] was a pioneering anticipation of the notion
of state on a tribe.
The MV-algebraic Loomis-Sikorski theorem was independently proved in [17,61],
building on [29] via the ? functor. An analysis of divisible MV-algebras can be found
in [19]. See [79, Chapters 3 and 9] for more information on weak ?-distributivity.
2. States and observables
Caratheodory's approach to probability theory is based on the idea of disregarding sets
of zero measure, and considering instead ?-complete boolean algebras equipped with a
distinguished strictly positive probability measure. Our aim is to generalize this approach
from ?-complete boolean algebras to ?-complete MV-algebras.
2.1. States and observables in MV-algebras
The following is a generalization of Definition 1.13:
Definition 2.1. Let ? be a ?-complete MV-algebra. A state is a map m : ? -+ [0, 1]
satisfying the following conditions for all a, b, c, a„,b„ e M:

880
В. Riecan and D. Mundici
(i) w(l)=l;
(ii) whenever b,c e ? and с 0 b = 0 it follows that m(b ? с) =m(b) + m(c);
(iii) if a„ /a, then m(a„) / m(a).
We say that ш is faithful^ if ш(;с) ^0 whenever O^jc gM.
The assumption с © /? = 0 in condition (ii) amounts to asking b ? с = b + с, where +
is addition in the ?-group with strong unit corresponding to M, as given in Theorem 1.7
and D).
Putting together Theorems 1.16 and 1,14 we have the following integral representation
theorem, where о denotes composition of functions:
PROPOSITION 2.2. Let ? be a ?-complete MV-algebra equipped with a state m. Let
? = M{M) be the space of maximal ideals of ? with the spectral topology. Let the tribe
(?,?) and the ? -homomorphism ?: ?' —»· ? be as in Theorem 1.16. Then (?, J7) is a
tribe and moijij a state on it. Let (??, <S, P) be as given by Theorem 1.14 starting from the
triplet (?, ?, m ? ?). Let the map * : ? —>· ? send each a e ? into the unique continuous
function a* e ? such that ?(?*) =a. Then for all a e M,
m(a)= / a*UP.
'?
Generalizing Caratheodory's approach to boolean probability theory, we shall be mainly
concerned with a situation where the state m is faithful, as follows:
DEFINITION 2.3. A probability MV-algebra is a pair (M, m) where ? is a ?-complete
MV-algebra and m is a faithful state on ?. We say that (??, m) is divisible if so is ?. We
say that (M, m) is weakly ?-distributive if so is ?.
Example, Let (?, J7) be a tribe with a state m. Let J be the set of all / e ? such that
m(f) = 0, Then J is an ideal of the MV-algebra ?, and J is closed under countable
suprema. The quotient MV-algebra ? = T/J is ?-complete,6 and the quotient map
?: ? —> ? is a ?-homomorphism. Two functions f,g e ? have the same image under
? iff m(\f - g\) = 0. Let the map m : ? -> [0, 1] be defined by m(d(f)) =m(f) for each
element 9{f) e M. Then the map is well defined and is a faithful state on M, and (M, m)
is a probability MV-algebra.
A variant of Proposition 2.2 can be used to show that this construction yields the most
general possible example of a probability MV-algebra.
A particular case of this example is given by the real unit interval [0, 1 ] equipped with
the state m(x) =x.
As another particular case, let ? be a non-empty, finite or denumerably infinite, set of
elements e\, ei equipped with a corresponding sequence p\, ??, ¦ ¦ ¦ of real numbers
pi > 0 such that ]T; p,¦ = 1. Let ? be the set of all sequences /:?->· [0, 1] equipped
Also called, strictly positive.
Indeed, since ?IJ has a faithful state, it cannot have uncountably many non-zero pairwise disjoint elements.
Thus, arguing as in the boolean case, we see that ?IJ is a complete MV-algebra.

Probability on MV-algebras
881
with termwise MV-operations. Let m : ? -> [0, 1] be defined by m(f) = J2i fiPi- Tnen
m is a faithful state on the complete MV-algebra ?.
A first property of probability MV-algebras is given by the following
PROPOSITION 2.4. Any probability MV-algebra (M, m) is weakly ?-distributive.
PROOF. Let bjj e ? be a double sequence having the property that, for each / e N,
Ьц ^ bij+\ and /\jbij = 0, with the intent of proving 0 = /\фец\\ Vi^/'.tfx/')· Assume
(absurdum hypothesis) 0 < b e ? is a lower bound for each element V, ^/.^аь where 0
ranges over NN. From the assumed faithfulness of w it follows that m(b) > 0. Since for
each fixed /, b л />,y tends to 0 as j tends to infinity, then so does m(b л />,y). There is
j* = ?(?) such that m{b л />,>(')) ^ m(b)/2'+]. By absurdum hypothesis, /? ^ V, ^?.?(?)·
Then from the distributivity law B), we get the contradiction m (/>) = mO л \Ji />,>(')) =
m(\/i(bAb,^(,)))^J2im(bAb,^0)) <m(b). D
As usual, we let R denote the set of real numbers. In the point-free version of probability,
the algebraic counterpart of a random variable is not given by a real-valued function over
the sample space ? but, vice versa, by a function from the ? -algebra B(R) of Borel sets
in R into the ?-boolean algebra of events. The explicit construction of sums and products
of such random variables in terms of boolean operations was successfully carried out by
Caratheodory. In order to generalize his results to MV-algebras we prepare
DEFINITION 2.5. Let ? be a ?-complete MV-algebra. An ?-dimensional observable
of ? is a map ? :B(R") —>¦ ? satisfying the following conditions:
(i) jc(R")= 1;
(ii) whenever А, В e B(W) and А П В = 0, then ? (?) ??(?) = 0 and x(A U B) =
x{A)®x{B)\
(Hi) for all A, A,, A2,... e ?(R"), if A„ / A, then x(A„) / x( A).
When ? = 1 we simply say that ? is an observable.
Thus, condition (ii) above states that, whenever ? ? ? = 0 then x(AUB) =x(A) + x(B)
in the ?-group with strong unit corresponding to ? via Theorem 1.7.
Remark. In the particular case when ? is a ?-complete boolean algebra, an я-dimen-
sional observable ? :B(W) —> ? is a ?-homomorphism into ? of the ?-complete boolean
algebra ??(R") of all Borel sets in R". In general, this need not be true of ?-complete MV-
algebras.
Examples. l.Let (?,?, ?) be a probability space, ? :? —>¦ R a random variable, and J"
the full tribe of all [0, 1 ]-valued 5-measurable functions on ?. Let the map ? : ??(R) —>¦ ?
be defined by x{A) = x^-^A), where /? is the characteristic function of ? с ?. Then ?
is an observable.
2. Let ? = [0, 1 ] with the usual MV-algebraic operations. Let ?: R ->· [0, oof be a
(Borel measurable) density function, in the sense that f^° (p(t)dt = 1. Define ? : В(Ш) ->·

882
В. Riecan and D. Mundici
? by the formula ? (A) = f ? ??, where ? is Lebesgue measure. Then ? is an observable
of M.
3. Again assume ? = [0,1]· Let v\,...,vm be a finite set of real numbers. Let
(p\,...,pm) be an w-tuple of real numbers > 0 such that p\ + ··¦ + pm = 1· Let
у: В(Ш) -> ? be defined by y(A) = ??-ел A· Then У is an observable of M.
4. Given a ?-complete ?V-algebra M, let G be the ?-group with strong unit и = 1
corresponding to M, as given by Theorem 1.7. Then, for any observable ? of M, the
map ? :В(Ш) —> [0, и] с G also defines a group-valued measure, i.e., a positive, additive,
continuous map from В(Ш) to G. Conversely, every group-valued measure у: ? (?.) —* G
such that y(R) = и is an observable of M. The above construction yields the most general
possible observable ? in ?. For instance, the observable of Example 1 arises as a group-
valued measure ? : ? (?.) ->· G, where G is the ?-group of all measurable functions on ?
with the constant function 1 as the strong unit. To construct the observable of Examples 2
and 3, one can similarly let (G, u) = (R, 1).
Having thus described the most general construction of an observable, let us now give a
down-to-earth property of observables, when combined with states:
PROPOSITION 2.6. Let ? be ? ? -complete MV-algebra with an ?-dimensional observable
? : В(Ш") -> ? and a state m : ? -> [0, 1 ]. Define the composite map mx : B(W) -> [0, 1 ]
by the stipulation that, for all X e B(R"),
mAX) = (mox)(X)=m(x(X)). F)
Then mx is a probability measure on B(M").
The proof is immediate. The above map mx has the same role as the probability
distribution of a random variable in the classical Kolmogorov theory. One can for instance
investigate conditions ensuring the existence of the moments of observables in probability
MV-algebras, in the following sense:
Definition 2.7. Let (M,m) be a probability MV-algebra. Let x:M -> B(R) be an
observable of M. Then ? is said to be integrable in (M,m), and we write ? e L]m, if
the expectation
E(x)= I tdmx(t) G)
exists. We say that ? is square integrable, in symbols, ? e L2m, if the dispersion (variance)
?2(?)= \\t-E(x)fdmx(t)= I t2 dm АО - ? (xJ (8)
Jm J&
exists.
Also called the mean.

Probability on MV-algebras
883
2.2. Independence
One of the most important notions in probability theory is independence. Let ? and ? be
two random variables in a probability space (?, «S, P). We then have
• a random vector ?: ? ->· R2 given by ? (?) = (? (?), /j(o>)) for each we ?,
• a ?-homomorphism h: B(R2) ->· 5 given by А ь* Т~' (A),
• a probability distribution Pt:B(R2) -> [0, 1] determined by the stipulation /MA) =
Р(Г-'(А)) = Р(Й(А)).
Two random variables ? and /j are said to be independent if for all Borel sets А, В с R,
? (? "'(?) ?/j (S)) = /'(?"'(?)) -P(/j_l(S)), where · denotes ordinary multiplication.
Since ? ~' (?) ? rf' (?) = ?~' (? ? ?), the last identity can be written
PT(A ? ?) = Pf (A) · ??(?) = (?? x P„)(A х ?),
where ?? ? ?? is the product measure of the measures
/>? = Ро?-' and ?? = ?°?-\ (9)
in agreement with the notation of F). Since ?? ? ?? coincides with ?? on rectangles,
these two measures automatically coincide over all of ??(R2). It follows that Pj = ?? ? ?,.
Since ?? = ? ? h, we can say that ? and /j are independent if and only if there exists a ?-
homomorphism h:??(R2) -> 5 such that Poh = ?? ? ??. The generalization to n random
variables is straightforward. We are thus led to the following
Definition 2.8. Let (M,m) be a probability MV-algebra. We say that the observables
x\,...,x„ are independent (with respect to m) if there exists an и-dimensional observable
h:B(R")^ ? such that
m(h(C\ x ¦ ¦ ¦ x C„)) = w(jc,(C|)) m{xn(Cn)) = mXl{C\) mx„(C„)
forallC|,...,C„ eB(R).
The map h is called the joint observable of x\,.. .,xn (with respect to m) in (M,m).
While h is not uniquely determined, any two mappings h \ and /22, satisfying the
conditions in the above definition automatically satisfy moh\ =mohi- Indeed, moh\ and
m о /г 2 must coincide on all sets of the form C\ ? Ci ? ¦ ¦¦ ? C„, whenever C\,..., C„ e
??(R). Since m о h \ and m о hi are probability measures on ??(R") they must coincide over
allof?(R").
In the particular case when ? is full tribe, ? comes equipped with pointwise
multiplication of functions, by Theorem 1.12(ii). This additional structure allows one to
drop the assumption that ? is equipped with a (faithful) state, and still yields a natural
definition of joint observable. To this purpose one first proves the following key fact:

884
В. Riecan and D. ? undid
LEMMA 2.9. Let (?, T) be a full tribe and x\,...,x„: B(R) -> ? observables. Then
there exists an observable h : B(Rn) —> ? such that
h{C\ x-xCn)=x](C])-x„(C„)
forallC\,...,C„ eB(R).
PROOF. For notational simplicity assume ? = 2. Fix a point ? e ?. For each / = 1,2
let ??., :В(Ш) -* [О, 1] be defined by ??.,(?) = (jt;(A))(a>). Each ??? is a probability
measure. Let ??:?(??) ->· [0, 1] be the product measure ?? = ??,\ ? ??.2- Let the
map h :B(R2) -* [0, \]? be defined by (fc(C))(w) = v„(C) for each С е ?(?2). For all
?, ? e #(R) and ? e ? we can write
(й(А ? ?))(?) = (??.| ? ??.2)(? ? ?) = ??.? (?)??.2(?)
= {?\{?)){?) ¦ (?2(?))(?) = {?? (?) ·?2(?))(?).
The function h(C): ? -+ [0, 1] is measurable whenever С = ? ? ? and ?, ? e #(R),
because /?(C) is then the product of two measurable functions. Further, the family of
those Borel sets D for which h{D) is measurable is a ?-subalgebra of B(M.~), whence
it must coincide with B(R2). In conclusion, h(C) e JF, for all С e B(R2) and /г is a two-
dimensional observable such that, forall A, ? e ?(?),/г(А ? ?) =?\(?)?2(?). The proof
is complete. ?
By a slight abuse of terminology, one can naturally say that h is the joint observable of
x\,...,x„ in the full tribe (?,?) - independently of any state m. When (?,?) happens
to be equipped with a (not necessarily faithful) state m, we have the following equivalent
reformulation of independence:
PROPOSITION 2.10. Letx\,...,x„: B(R) -> ?be observables in a full tribe (?,?) with
a state m. Then ? \,..., xn are independent {with respect to m) if and only if
m(x\(C\)---xn (C,,)) =m(x\(C\))---m (x„(C„))
forall Ci,...,C„ eB(R).
The proof is an immediate consequence of Definition 2.8, together with Theorem 1.12(H)
and Lemma 2.9.
2.3. Kolmogorov 's construction
A main idea of MV-algebraic probability theory is to represent every sequence of
observables in a probability MV-algebra (M,m) by a sequence of random variables. The
construction is straightforward in the particular case when ? \, x2, ¦ ¦. is an independent
sequence of observables, in the sense that every finite subsequence x\,x2,.. .,xn is

Probability on MV-algebras
885
independent (with trivial adaptations for ? = 1). As a matter of fact, let wv,, mXl,... be
their associated probability distributions mXi = m о jc,. One then constructs the infinite
product measure ? of the mXj as follows:
First Step. Let RN denote the set of all sequences of real numbers. Let С be the
collection of all subsets С of RN that can be written in the form (П denoting here Cartesian
product)
C = f]Ci, A°)
where C,· e B(R) for each / e N and the set {/ | С, ф R} is finite. (Note this expression is
unique, unless С = 0.) Then elements of С are called the cylinders of RN.
Second Step. Let a(C) be the ?-algebra generated by the family of all cylinders of
RN. The probability measure ?: a(C) -+ [0, 1] is uniquely determined by the stipulation
(? denoting here multiplication)
Р(С) = \\тХ1(С{), (?)
whenever the cylinder С еа(С) has the above form A0).
DEFINITION 2.11. The above triplet (RN, a(C), P) is called the Kolmogorovprobability
space of the observables jc,- in (M,m). For each ? e N, the canonical projection function
(„:RN^R given by
1„(U\,U2,...) = U„
is called the nth coordinate random variable of (RN, a(C), P). The function 7„ : RN -> R"
given by
7„(И|,И2, ...) = (« I, «2, ¦¦·,"«)
is called the «th coordinate random vector.
2.4. Functions of several observables
Let /: R -+ R be a Borel measurable function and ? : ? (?.) -+ ? an observable. Then one
can naturally define the observable f(x): B(R) -+ ? by stipulating that for each A e B(R.)
(f(x))(A)=x(f-\A)).
Using in a self-explanatory way the symbol о for composition of functions, we can write
f{x)=X0f-\.

886
В. Riecan and D. ? undid
More generally, given independent observables x\,...,x„: В(Ш) ->· ? in a probability
MV-algebra (M, w), together with a Borel measurable function g: R" ->· R, the observable
gC*i, ...,*„) is defined by the formula
where h : В(Ш") —> ?? is the joint observable of ? \,..., x„.
As an example, addition ? + у: ? (?.) ->· ? of independent observables л and у in a
probability MV-algebra (M, w) is defined by stipulating that, for all X e ??(R)
U+y)(X) = /J(plus-|(X)),
where the function plus: R2 —> R is the natural addition operation, plus(w, v) = u + v. This
definition is a counterpart of the classical identities for arbitrary random variables ? and ?,
(? + ?)~\?) = (plus($, ?))~1(?) = (plus о Г) (X) = Г~' (plus (?)).
Similarly, product is defined by (x ¦ y)(X) = h(times"'(X)), where the function
times -. R2 —> R is multiplication.
Further, let x\,..., x„ be independent observables in a probability MV-algebra (M,m)
and let h : B(W) -^ МЫ their joint observable. Let the map average: R" -+ R be defined
by
1 "
average^ ?,..., i„) = - У^'с
for all real numbers t\,..., tn. One can now define the arithmetical mean - X]' = | X\ of
? ?,..., x„ by writing
1 "
- ) x,¦ = h о average-'.
n '—'
2.5. The MV-algebraic central limit theorem
THEOREM 2.12 (Central limit theorem). Given a probability MV-algebra (M,m), let
? ?, JC2, ... be an independent sequence of square integrable observables having the same
probability distribution mXi = mXl = ¦ - ·. Let further a = E(x\) = E(xi) — ¦ ¦ ¦ be their
common expectation, and assume that, for some 0 < ? e R, their common dispersion is
given by ?2. Then for all t e R,
(x\+ Vxn-na . \ 1 [' _.jL
lim ml — (]-oo, t[) = —7= I e ? du.
n^oo \ ajn ') slln J-oc

Probability on MV-algebras
887
PROOF. For each ? = 1, 2, 3,... let the Borel function gn : R"
w\ + h wn — na
be given by
?,,(»!,..., №,,):
? Un
Let further
(a) h„ :??(R") —>¦ ? be the joint observable of x\,X2, - ¦ -,xn\
(b) the observable y„:B(R) ->· ? be given by the stipulation y„ = /?„ о g~l =
g„(x\,...,x„y,
(c) the random variable tj„:Rn ->· R be given by /j„ = g„(i\,i2, ¦ --,'«) = ?>i °ii,
where Г,, is the nth coordinate random vector in the Kolmogorov probability space
(RN,a(C),P).
By definition of joint observable, we then have the identities
m о h„ = mx ? ¦ ¦ ¦ ? m ,
r-l
? ? ? = m_r, ? ¦¦¦ ? mXn =mohn
Poin =Pln=mXn=moxn,
?
Р°лп
¦m о y„.
It follows that
x\-\ \-x„ -na , .
m| — (]-oo,i[)
= w(g,,Ui,...,j:,1)(]-oo,i[))
:m(hn(g„ l(]-oo,i[))) = (w.v, ? ¦¦¦ xw.v„)(gn '(]-oo,i[))
-'(!¦¦
M| +
? у/? \)
¦+i„(u)-na \\
??? *'])¦
The coordinate random variables i\,ii,... in (RN,a(C), P) have their common
distribution Ph = Pn = - · · coinciding with the common distribution mXl =тхг = ¦ ¦ - of the
observables x\,X2, Thus in particular, all i„ are square integrable, with non-zero
common dispersion ?2 and expectation a. A moment's reflection recalling A1) shows that
i ?, B,... is an independent sequence; as a matter of fact,
Р(G1(А|)П.-.П(,71(Ап))=тл-,(А|)--.ш.с„(А,1) = Р1,(А|).--Р(„(А„)
= Р((Г1(А|)).--Р((„-'(А,1)).
By the classical central limit theorem, we have
? !(«) + ¦
lim ?
?—»oo
({..
.¦ + hl(u)-na < ]\ 1 f e-4du
? y/n J/ y/?? J-oc
as required to complete the proof.
D

888 В. Riecan and D. Mundici
2.6. MV-algebraic laws of large numbers
Suppose ?\, 7j2, · · · to be a sequence of random variables in a probability space (?, S, P):
(i) The sequence is said to converge in distribution to a function F :R ->· [0, 1], if for
all t e R,
lim P(/?,7l(]-oo,i[)) = F@.
n—>oo x v "
(ii) The sequence is said to converge to 0 /и measure P, if for each real ? > 0,
(iii) The sequence is said to converge ?-almost everywhere to 0, if
OO OO ОС
p ПиГК'0-?/??/?) =1,
\?=?*=??=?· /
i.e.,
lim lim lim P( ?/,,;'(]-l/p,l/p[) ) = 1.
\)i=k I
The above classical notions have the following MV-algebraic counterparts:
Definition 2.13. Let y\, yi,... be a sequence of observables in a probability
MV-algebra (M ,m).
(i) The sequence is said to converge in distribution to a function F :R —* [0, 1] if for
each t e E,
lim m(y„(]-oo,t[)) = F(t).
(ii) The sequence is said to converge in measure to 0 if for each 0 < ? e R,
lim m(y„(]-e,?[)) = 1.
(iii) We say that the sequence converges to 0 almost everywhere (а.е.), if
/k+i \
lim lim lim m Д yn(]-\/p, \/p[) = 1.
/;^oci;^oc/^oc \' ? /
\n=k
PROPOSITION 2.14. Let x\,x2,..- be an independent sequence of observables in a
probability MV-algebra (M,m), with h„:B(M") —>¦ ? being the joint observable of

Probability on MV-algebras
889
? ?, X2, . .., x„. Let ? be the infinite product measure of the m V| as given by Kolmogorov 's
construction A1). For each ? = 1, 2, 3, ... let g„ : R" —> R be an arbitrary Borel function.
Let further the observable y„ : B(R) —>¦ ? be given by y„ =h„ogn] = g„(x\,..., x„), and
the random variable ?„ : RN —> R be defined by ?„ = g„ G„), where T„ is the nth coordinate
random vector, it follows that ? ? ?~' = m о у„ and
(i) the sequence y\, yi,... converges in distribution to a function F if and only if so
does the sequence ? ?, ??, ...;
(ii) y\, yi,... converges to ? in measure m if and only if ? ?, /j:, ... converges to 0 in
measure P\
(iii) if ?\,??, ¦ ¦ ¦ converges P-almost everywhere to 0, then yi, уз, ... converges m-
almost everywhere to 0.
The following results are MV-algebraic generalizations of two fundamental theorems in
classical probability theory:
THEOREM 2.15 (Weak law of large numbers). Given a probability MV-algebra (M,m),
let x\,X2,... be an independent sequence of integrable observables having the same
probability distribution mXl = wv, = ¦¦·. Let a = E(x\) = fU:) = ¦¦¦· Then the
sequence of observables
a (n = 1, 2, ...)
n
converges in measure to 0.
THEOREM 2.16 (Strong law of large numbers). Given a probability MV-algebra (M,m),
let x\,X2, ... be an independent sequence of square integrable observables such that
??^? ?2(??)/?2 < oo. Then the sequence of observables
x\ - E(x\) + x2- E(x2)-\ \-x„-E(x„)
(n = 1,2, 3, ...)
n
converges m-almost everywhere to 0.
2.7. Bibliographical remarks
Caratheodory's point-free probability theory is expounded, e.g., in [6]. Early papers
containing results about states and observables in MV-algebras are [10,69]. See [68] for
a more recent paper. A more general class of states without the ?-additivity assumption
was investigated in [59,60]. Also see [15]. They shall be discussed in Section 4 below.
Probability MV-algebras generalize the classical notion of "probability algebra", or
"?-complete boolean algebra with a normalized positive measure" (see [23], [24, 322A]
and [31, p. 64]).
The proof of Proposition 2.4 is new here, and is a natural generalization of its well known
counterpart for ?-complete boolean algebras equipped with a faithful state, [24, 325 I-K].

890
В. Rlelan and D. Mundici
When ? is a ?-complete boolean algebra, our definition of observable boils down
to Sikorski's notion of real homomorphism [82, p. 152ff], and also coincides with
Varadarajan's definition of observable [84, p. 14ff].
Up to restriction to unit intervals, both states and observables are instances of a unique
notion: that of a normalized positive ?-additive group homomorphism between Dedekind
?-complete ?-groups with strong unit [29].
Lemma 2.9 is taken from [79, Theorem 8.3.2].
The idea of using the Kolmogorov construction in MV-algebras for a representation
of observables by random variables was first presented in [71]. Further information
can be found in [79, 8.4]. While the construction is used in this chapter only for
sequences of independent observables, the same method can be applied, without assuming
independence, to any sequence of observables having joint observables. This is the case,
e.g., of full tribes with a state (by Lemma 2.9). By Theorem 3.6 below, this is also the case
of all weakly ?-distributive MV-algebras with product (to be defined in 3.1 below).
Our present proof of the MV-algebraic central limit theorem (Theorem 2.12) is a
particular case of [79, Theorem 9.2.6].
The proofs of the identity ? ? ?~] = m о у„, as well as of conditions (i) and (ii) in
Proposition 2.14 are straightforward. Condition (iii) follows from the inequality
(k+i \ /k+i
?^?/?,?/?[)<™ ???-1/?,?/??
n=k I \n=k
(see [79, Proposition 9.3.3]).
A proof of Theorem 2.15 can be obtained from [37] and, in a more general context,
from [75].
Theorem 2.16 is proved in [79, 9.3.4].
3. MV-algebras with product
As shown by Lemma 2.9, given a full tribe (?,?), for any two observables x, y:B(R.)
—>· ? one can always construct a two-dimensional observable h : В(Ш~) —> ? such that
h(C xD)=x(C)y(D). A2)
While ? need not be equipped with any state m, in the above formula a crucial role is
played by pointwise multiplication of the two [0, l]-valued functions ? (С) and y(D). With
a view of giving a purely algebraic generalization of this construction, in this chapter we
shall consider semisimple MV-algebras ? enriched by an additional "product" operation.
Since ? is canonically isomorphic to an algebra of [0, l]-valued functions, we shall
take care of relating the abstract product operation of ? with the natural pointwise
multiplication of functions. We shall freely denote by 1 e ? the strong unit и of the
?-group corresponding to ?.

Probability on MV-algebras
891
3.1. Product and joint observables
Definition 3.1. An MV-algebra with product is a pair (M, ¦). where ? is an
MV-algebra and ¦ is a commutative and associative binary operation on ? satisfying the following
conditions, for all a, b, с e ?:
(i) 1 ¦ a = a;
(ii) a-(bQ-'c) = (a-b)Q -*(a - c).
Since the operation aQd = aO-*d coincides with truncated subtraction (a — d) ? 0 in
the ?-group G with unit и = 1 corresponding to ?, then condition (ii) amounts to requiring
the distributivity of ¦ over (truncated) subtraction.
Examples. Every boolean algebra becomes an MV-algebra with product, by letting the
product operation coincide with the infimum operation. Another example is given by
the unit interval [0, 1] equipped with multiplication. More generally, every MV-algebra
of [0, l]-valued functions (closed under pointwise multiplication and) equipped with
pointwise multiplication is an example of an MV-algebra with product. As we shall see in
Theorem 3.4 below, this is the most general possible example of a semisimple MV-algebra
with product.
The following are almost immediate consequences of the definition:
PROPOSITION 3.2. in every MV-algebra with product (M, ¦) we have:
(i) 0 ¦ a = 0;
(ii) (monotony) ifa^b then с ¦ a < с ¦ b;
(iii) аОЬ^а-Ь^алЬ;
(iv) ifaQb = 0 then с ¦ (? ? b) = (c - ?) ? (с ¦ b) and by (ii), (c ¦ ?) ? (с ¦ b) = 0;
(?) с ¦ (?? b) = (c - ?) ? (с ¦ b)\
(vi) с (а лЬ) = (с -а) л (с ¦ b)\
(vii) the congruences of(M, ¦) coincide with the congruences of the MV-algebra M.
PROPOSITION 3.3. Let (M, ¦) be an MV-algebra with product. Fix an element a & ? and
consider the map ?? : ? —>· ? given by ? ь* а ¦ ?. Then ?? is an MV-homomorphism of ?
into the interval MV-algebra ([0, ?], -?,, ?„, Q0) of all elements of ? between 0 and a,
with a being the unit o/[0, a], and the latter interval being equipped with the operations
--„c = a ? с = a ?->c, b фн с = а Л (b ? с),
Let now ? be a subalgebra of [0, 1]. Assume the operation *:M ? ? —>¦ ? makes
(?, •) into an MV-algebra with product. Fix a e ? and, to avoid trivialities, assume
? ? 0. Then the above map ?? is an MV-homomorphism of ? into the interval MV-
algebra [0, a]. By Theorem 1.7(ii), letting (G, a) be the subgroup of ? corresponding to
[0, a], and (Я, 1) be the subgroup of R corresponding to M, there is an order preserving

892
В. Riecan and D. Mundici
group homomorphism ??:? ->· G sending 1 into a, such that ?? = ?(??). From the
general theory of ordered subgroups of R it follows that ?„ is multiplication by a.
Thus the * operation coincides with ordinary multiplication, and ? is closed under such
operation. Since the congruences of (M, ¦) coincide with the congruences of M, recalling
Theorem 1.5 we have
THEOREM 3.4. Let ? be a semisimple MV-algebra. Then there is at most one operation
¦ making (M, ¦) into an MV-algebra with product, if ? does admit such operation ¦ then,
upon representing ? via the isomorphism а ь* a* onto a separating MV-algebra M*
of[0, \\-v alued functions over the maximal ideal space ofM, it follows that M* is closed
underpointwise multiplication. Further, map а ь* a* also preserves the product operation.
By Theorem 1.12(H), every full tribe can be made into an MV-algebra with product.
On the other hand, boolean tribes provide examples of tribes that can be equipped with
a product, while not being full. More examples are given by the following variant of
Theorem 1.12(H). A proof can be obtained using Theorems 1.18and3.4:
PROPOSITION 3.5. For any ?-complete divisible MV-algebra ? there is precisely one
operation ¦ such that (M, ¦) is an MV-algebra with product.
As expected, the existence of a product operation simplifies the construction of joint
observables. The proof of the following theorem relies on Theorem 1.7, together with
classical results on extensions of group-valued measures:
THEOREM 3.6. Suppose (M, ¦) is a weakly ?-distributive MV-algebra with product. Then
for any two observables x,y:B(R) —> ? there exists an observable hx> ;B(R') —*¦ ?
such that
hxy(CxD)=x(C)-y(D)
for all C,De B(R).
Equipping ? both with a product and with a faithful state, we are led to consider the
following
DEFINITION 3.7. A probability MV-algebra with product (M,m, ¦) is a probability MV-
algebra (M, m) equipped with a binary operation ¦ on ? making (M, ¦) into an MV-algebra
with product8.
Examples. 1. If (M,m) is a probability algebra, and ? is boolean, then we obtain a
probability MV-algebra with product (M,m, ¦) by letting ? ¦ у = ? Л у = ? ? у for all
?, у е ?.
8Since M is complete (see Footnote 6), by a celebrated theorem of Stone and von Neumann, ? can be
decomposed into a product ? ? ?, where A (resp., B) is the MV-algebra of all {0, l)-valued (resp., [0, l]-valued)
continuous functions over an extremally disconnected space X (resp., Y).

Probability on MV-algebras
893
2. Let (?,?,?), be a probability space, and (?,?,??) be the full tribe of all
«S-measurable functions f :? ->· [0, 1], with m(f) = jQfdP. As in the example
following Definition 2.3, let У с f be the ideal of all / e ? with m(f) = 0. Let
? = T/ J. Then ?? is ?-complete. The quotient map ? :\T —> ? is а ?-homomorphism.
The map m : ? ->· [0, 1] defined by m(9(f)) = m(f) is a faithful state on M. The
probability MV-algebra (M,m) can now be enriched with a binary operation ¦ arising via ?
from pointwise multiplication of two functions in ?. A straightforward verification shows
that (M, m, ¦) is a probability MV-algebra with product.
Since by Proposition 2.4, weak ?-distributivity is automatically ensured in all
probability MV-algebras, Theorem 3.6 yields the following characterization of independence:
PROPOSITION 3.8. Let (M,m, ¦) be a probability MV-algebra with product. Then two
observables x,y: В(Ш) —>· ? are independent if and only if
m(x(C) ¦ y(D))=m(x(C))m(y(D))
forallC,DeB(R).
3.2. Conditional expectation
In order to define the conditional expectation ?(*|.?) of observables ? and у in a
probability MV-algebra with product (M,m, ¦), we shall proceed by analogy with the
classical case, making use of the joint observable hxy. Traditionally, given a probability
space (?, S, P), by the conditional expectation of two random variables ? and ? we
understand a Borel function ?(?\?): R -> R such that, for all В е B(R),
? ?(?\?)???= f ???.
JB JirHB)
The latter integral intuitively gives the expected value of ? once the value of ? is
known to lie in B. This suggests that, given observables ? and у in a probability MV-
algebra with product (M,m, ¦), we should define E(x\y) as a Borel function satisfying
fB ?(;c|>')dwv = fy(B)xdm, and this leaves us with the problem of defining faxdm, for
any fixed element a e ?. Again proceeding by analogy with the classical case, for each
integrable random variable ? : ? —> R and С е S we can write
[???=[
tXcdP,
?
whence for each Borel set В we have
, Cnf'(i) ifO^B,
(^Xc)-|(S)=l
(Cnr'(fi))U(i2\C) ifOeS.

894
В. Riecan and D. Mundici
PROPOSITION 3.9. Let (M,m, ¦) be a probability MV-algebra with product. Let a e ?
and ? be an observable of M.
(i) Let the map xa : B(R) -> ? be defined by
Xa(D) = <
\?-(?(?)) + -? ifOeB.
Then xa is an observable, and is integrable whenever ? is integrable.
(ii) Define the indefinite integral of ? over a, in symbols f ? dm, by the stipulation
J' ? dm = I
a JR
tdmXa(t).
Then faxdm = j^xadm = E(xa). For every observable у of ? let the map
к: B(R) -> ? be defined by
k{B)= / xdm = E(xx(B))-
Jy(B)
it follows that к is a (possibly non-normalized) ?-additive measure, and к <К my.
Definition 3.10. Let x, у be observables, with ? integrable, in a probability
MV-algebra with product (M,m, ¦). Then the conditional expectation (of ? given y) is the Borel
measurable function ? (? \ у): R —> R such that
E(x\y)dmy= xdm = E(xy(B)),
JB JviB)
forallBeS(R).
The existence of E(x\y) :R —>¦ R follows from the Radon-Nikodym theorem.
THEOREM 3.11. Let x, у be observables, with ? integrable, in a probability MV-algebra
with product (M, m, ¦). Lethxy be the joint observable of ?, у as given by Theorem 3.6 and
Proposition 2.4. In the probability space (R2, ??(R2), m о hxy) let the random variables ?
and ? be defined by ?(и, ?) = и and /j(m, v) = v. Then Pn = my and E{x\y) = ?(?\?),
my-almost everywhere.
One can now prove the following variant of the martingale convergence theorem. Here
the sequence y\,y2, ¦¦ ¦ of conditioning observables is generated by a fixed observable у
together with a sequence g\, gi,... of Borel measurable functions.

Probability on MV-algebras
895
THEOREM 3.12. Let (M,m, ¦) be a probability MV-algebra with product. Let g\,gi, ...
be Borel measurable functions g„ : Ш —> Ш such that
DC
g (В(Щ) с g^, (B(R)) fl„d |J g-' (?(?)) = B(R).
/?=?
G/ve/г /iow да observable у : ??(R) —> ?? /ef ms write y„ = у о g~' a/id assume my <K w v„
(и = 1,2,...). Then for every integrable observable ? : ? (?.) —>¦ ? the sequence of Borel
measurable functions E(x\y„) converges to E(x\y), m ^-almost everywhere.
3.3. Upper and lower limits
As shown by Theorems 2.15 and 2.16, frequencies A /?) ]?"_, Xi converge to mean values.
When such limits are not known a priori, one can still profitably use liminfs and limsups,
as follows.
Let ?i, ?2. - ¦ ¦ be a sequence of random variables in a probability space (?, S, P). Then
for every ? e ? we have
ОС ОС ЗС
lim sup?„ (?) < t iff ? e (J (J f] ?~
?=\ k = \ n=k
This motivates the following:
Definition 3.13. Given a sequence x\,xi,... of observables in a probability
MV-algebra (M, m), we write ? = limsup,,^^ xn if ? : В(Ш) -+ ? is an observable having the
following property: for every t e R,
/¦V"i i-V^ '4V~-
^Aj v^* v^*
i(]-oo,/[)=VVAx«
/,= l k = \n=k
Note that if another observable у satisfies the above condition then my = m;x. Similarly,
we write lim inf,,^^ x„ = ? if ? is an observable satisfying the condition
DC DC DC
*(]-oo,f[) = \/Л\Л»
p=\k=\n=k
for alii el.
PROPOSITION 3.14. Let x\,X2, ... be a sequence of observables in a probability MV-
algebra (M,m). Suppose that both observables ? and * exist. Then for every feR,
x(]-oo,t[) ^x(]-oo,t[).
—00, t
1
?
-oo,?
?
-00, t -
PL

896
В. Riecan and D. Mundici
Definition 3.15. A sequence ?\,??,... ofobservables in a probability MV-algebra
(M, m) is said to converge m-almost everywhere to an observable ?, if both observables ?
and * exist, and for each ieR,
m(x(]-oo,t[)) =m(x(]-oo,t[)) = m(.x(]-oo, t[)).
Remark. As shown by the following proposition, in the case of probability MV-alge-
bras with product, one recovers Definition 2.13(iii) by letting ? coincide with the zero
observable Ом. The latter is given by
0M(A)= {
[0 ifO^A.
PROPOSITION 3.16. A sequence ofobservables in a probability MV-algebra with product
(M,m, ¦) converges m-almost everywhere to the zero observable Ом if and only if it
converges m-almost everywhere to 0.
The following result is the key tool for the proof of the /.''-completeness Theorem 3.19
and the Individual Ergodic Theorem 3.21 below. Note that we are not assuming our
sequence of observables to be independent. All we need to define g(x\,..., x„) is the
existence of joint observables, and this is ensured by Proposition 2.4 and Theorem 3.6.
THEOREM 3.17. Let ? \, xj,... be a sequence of observables in a probability
MV-algebra with product (Ai,w,·). Let и,'2,··· be the coordinate random variables in the
Kolmogorov probability space (RN, a(C), P). Let g\,gi,-.- be a sequence of Borel
measurable functions, where g„ : R" —> R. If the sequence g\ (? \), g2(i ?, ?),... converges
?-almost everywhere, then the sequence ofobservables g\(x\),g2(x\,xi), ¦ ¦ ¦ converges
m-almost everywhere to an observable. Further, we have the identity
P(^u eRN | Jim^gnGn(u)) < ij) = m(limsupg„(*?,.. .,x„)(]-oo,t[fj
for every ieR.
The following definition generalizes the notion of integrable and square integrable
observable, Definition 2.7:
DEFINITION 3.18. Fix a real number ? ^ 1. Given a probability MV-algebra (M,m, ¦)
with product, an observable ? : ? (?.) ->· ? is said to be in L','„ if the integral
/
Js.
\t\pdmx@
exists.

Probability on MV-algebras
897
The map ?'. ъ\п ? LJjj —>· К is defined by stipulating that, for any ?, у 6 l^m·*
where the joint observable hxy: M.2 —>¦ ? is as in Theorem 3.6, and the function g: R2 ->· R
is given by g(u, v) = \u — v\.
THEOREM 3.19. lf(M,m,)isa probability MV-algebra with product then (L','„, p) is a
complete pseudometric space.
The proof follows from Proposition 2.4, and Theorems 3.17, 3.6, together with the
classical Lp-completeness theorem.
3.4. Individual ergodic theorem
The individual ergodic theorem is traditionally formulated for dynamical systems (? ,<S,
?, ?), where (?,?, ?) is a probability space and ?: ? —>¦ ? is a measure-preserving
map, i.e., ? satisfies the following condition:
whenever ? eS then Г-|(У) e<S and P(T'](Y)) = P(Y). A3)
The theorem states that whenever /: ? —>¦ ? is an integrable function, then for P-almost
all ? e ? the limit
jfi-l
/»= lim -YfoT'(w)
n—>oc ? '—'
?=0
exists. Further, ?(/*) = E(f). To give an MV-algebraic generalization to (M,m, ¦), we
shall replace the random variable / by an observable ? : B(R) -> ?. We shall also replace
the transformation ? : ? —> ? by a mapping ? : ? —> ?? as follows:
Definition 3.20. Let (M,m,-) be a probability MV-algebra with product. Then a
mapping ?:?? —> ?? is said to be an m-preserving transformation, if the following
conditions are satisfied:
(i) г(и) = и;
(ii) if a + b sC u, then ? (? + b) = ? (a) + z{b) for all a, b e ?? (with reference to the
?-group with unit и = 1 corresponding to ?);
(iii) if ?„ /?, then ?(?„) /* ? (?);
(iv) w(r(ai) ¦ ?{?2) ¦ ¦ -x{ak)) = m{a\ ¦ ai ¦ ¦ a*) for all a\, ..., a„ e M.
The above condition (ii) can be equivalently rephrased as follows: whenever a,b e ?
satisfy ? ? b = 0 then ? (?) ? ?(/>) =0 and ? (? ?/?) = ?(?)? ? ?).
The following result is an MV-algebraic extension of the individual ergodic theorem;
its proof follows from Proposition 2.4, and Theorems 3.17,3.6, together with the classical
individual ergodic theorem.

898
В. Riecan and D. Mundici
THEOREM 3.21. Let{M,m,)bea probability MV-algebra with product. Let ? : ? (?.) ->
? be an integrable observable. Let ? : ? —>¦ ? be an m-preserving transformation. Then
there exists an integrable observable x* such that E(x*) = E(x) and
lim ^=^ -=x*
/7—»00 Ц
m -almost everywhere.
3.5. Bibliographical remarks
In defining MV-algebras with product we have followed Montagna's approach [56,
Definition 2.8]. The assumption that there exists a product in a semisimple MV-algebra ?
is then equivalent to saying that ? is multiplicative in the sense of [61, Section 5]. To
see this one notes that a commutative bimorphism ([61, Section 2]) ? of A x A into A
such that ?(\,?) = ? equips the MV-algebra A with a product operation. Conversely,
in any MV-algebra ? with product ·, for each fixed a e ? the map ??:? \-+ a ¦ ?
is a homomorphism of ? into the interval MV-algebra [0, a], and ?\ is the identity
homomorphism over ?.
MV-algebras with product are an interesting object of study in se, owing to their
relationship with real closed fields. (See, e.g., [56].)
For a proof of Proposition 3.2 see [56, Lemmas 2.9 and 2.11]. Proposition 3.3 is now a
consequence of [61, Proposition 2.3]. Theorem 3.4, as well as the analysis of the possible
products in subalgebras of [0, 1] are due to Ioana Leustean (private communication to
D.M.). The result on subgroups of R mentioned before the statement of Theorem 3.4 can
be found, e.g., in [26, Proposition 2, p. 46].
Proposition 3.5 was first proved in [19]. As already noted, any tribe which is closed
under pointwise multiplication is automatically a ?-complete MV-algebra with product.
The structure of all such tribes was characterized in [52] (when the universe is finite or
denumerably infinite) and [53] (in the general case).
One can also find in the literature a weaker notion of product, as follows:
Definition 3.22 ([14,19,34,72]). A generalized product in an MV-algebra ? is an
associative and commutative binary operation * satisfying the following conditions, for
all a,b,c,an,b„ e M:
(I) и*и = и.
(II) If ? ? b = 0 then (c * a) Q (c * b) = 0 and с * (? ? b) = (с * а) ф (с * b).
(Ill) If a„ \ 0 and b„ \ 0, then an *bn\0.
With reference to the addition operation in the ?-group with unit и corresponding
to M, the premise in condition (ii) can be equivalently stated as follows: a + b < u. The
conclusion can be stated as a distributive law, as follows: for every с е М, с * (a + b) =
c*a + c + b.

Probability on MV-algebras
899
As noted by Montagna in [56, p. 98], the ¦ operation in MV-algebras with product
satisfies all conditions of Definition 3.22. Suppose now condition (I) in the definition is
naturally strengthened as follows:
(?) u*x = ? for all ? e M.
Then the resulting operation * is monotone, and condition (III) is automatically satisfied,
together with the inequality ? * у < ? л у.9 As already noted, every MV-algebra with
product satisfies conditions (Г) and (II). Conversely, Jakubik (private communication to
B.R.) proved that any associative commutative operation satisfying conditions (Г) and (II)
automatically satisfies Montagna's conditions in Definition 3.1. As a first step, one derives
from (Г) and (II) the distributivity of * over the ?V-algebraic ? and л operations. Then,
a variant of [61, Propositions 2.2] yields distributivity of * over the ? operation.
Many results about independent and joint observables were originally proved for the
larger class of MV-algebras equipped with the above generalized product.
In the papers [22] and [86] one can find proofs of the extension theorem used in the
proof of Theorem 3.6. The proof of this latter theorem was first given in [72, Theorem 2]
using [79, Theorems 3.4.2 and 9.5.3]. The special case of full tribes had been considered
in [54].
The proof of Proposition 3.9 can be found in [35, Section 2]. For its tribe-theoretic
counterpart see [74, Proposition 1 ]. Theorems 3.11 and 3.12 are proved in [35, Theorems 1
and 2] respectively.
For a proof of the tribe-theoretic version of Proposition 3.14 see [79, Proposition 8.6.4].
The present algebraic version is similarly proved, as a straightforward consequence of
the definitions. A proof of Proposition 3.16 can be found in [78, Theorem 3]. (The tribe-
theoretic variant was proved in [79, Proposition 8.6.6].) Also Theorem 3.17 follows from
its tribe-theoretic counterpart [79, Theorem 8.6.9]. For a proof of the present MV-algebraic
version see [78, Theorem 2].
Under the assumption of weak ?-distributivity, the Lp-completeness Theorem 3.19 was
proved in [76] (while the special case of full tribes with a distinguished state had been
considered in [73]). The proof of Theorem 3.19 follows from the proof of [76], together
with Proposition 2.4.
Under the same redundant assumption of weak ?-distributivity, together with an
additional boundedness assumption, the Individual Ergodic Theorem 3.21 was first
proved in [36, Theorem 4], for probability MV-algebras with product. Actually, the
boundedness hypothesis can be dropped using Theorem 3.17 together with the proof of
[79, Theorem 8.7.2] dealing with the special case of full tribes.
4. Finitely additive measures
In this section we shall relax the ?-completeness assumption in the definition of state, and
apply the resulting finitely additive measures to various topics in probability theory, such as
partitions and entropy of dynamical systems. We shall also relate finitely additive measures
in MV-algebras to tracial states in their corresponding AF C*-algebras. Unless otherwise
specified, throughout this chapter, ? is a not necessarily ?-complete MV-algebra.
In the literature, such operation is sometimes called a "strong product".

900
В. Riecan and D. Mundici
4.1. Basics
DEFINITION 4.1. A (normalized) finitely additive measure*0 on an MV-algebra ? is a
function ? : ? ->· [0, 1] с R satisfying the following conditions:
A) M(D=1,
B) for all a,be ? with ? ? b = 0 we have ?(?) + ?(/>) = ?(? ? />).
As above, we say that ? is faithful if ?(?)^0 whenever 0 ? a e M. We say that ? is
invariant if for every automorphism ? of ? and every aeMwe have ? (?) = ? (? (?)).
PROPOSITION 4.2. Lef ? foe a finitely additive measure on an MV-algebra M. Then we
have:
(i) ? /s a valuation: ?(?) + ??) = ?(? ? b) + ?(? Ob) for all a, b e M;
(ii) ? /s monotone: ifa^b then ?(?) ? ??)\
(iii) if ? is faithful then ? is strictly monotone: a < b implies ?(?) < ??)\
(iv) ? is also a valuation with respect to the underlying lattice order of M; stated
otherwise, for all a, b e M, we have ? (?) + ??) = ?(? ? b) + ?(? л b)\
(?) ? is subadditive, in the sense that ? (a V b) ^ ? (? ? b) ^ ? (?) +??)·
PROPOSITION 4.3. The function ?: ? ? ?? -> [0, 1 ] g/ve/г by
?(?, fo) = ?(\? - b\) = ?((? ? ^b) ®(bQ -a))
/? a pseudometric on M. This is a metric iff ? is faithful.
PROPOSITION 4.4. Every countable semisimple MV-algebra has a faithful finitely
additive measure. Conversely, if a (possibly uncountable) MV-algebra ? has a faithful
finitely additive measure then ? is semisimple.
The finitely additive measures of ? form a convex set, which inherits the topology of
the product space [0, 1 ]M of all [0, 1 ]-valued functions defined over ?.
THEOREM 4.5. Let ? be an MV-algebra. Then the set of extremal finitely additive
measures of ? equipped with the topology inherited by restriction from the product space
[0, \]M is a compact Hausdorff space which is homeomorphic to the space M(M) of
maximal ideals of ? with the spectral topology. Every finitely additive measure of ? is in
the closure of the convex hull of the set of extremal finitely additive measures of M.
Closing a circle of ideas originating in our prologue, we now discuss the relationship
between finitely additive measures in MV-algebras, and "states" on C*-algebras. To this
purpose we prepare:
Dehnition 4.6. Let ? be a C*-algebra.
'Called "state" in [59.60].

Probability on MV-algebras
901
(i) A tracial state on Л is a normalized positive linear (complex-valued) functional s
satisfying s(aa*) = s(a*a) for all a e A. We say that s is faithful if 0 ? a* a e A
implies 0^s(a*a).
(ii) A [0, l]-valued map m defined on the projections of A is called a measure on the
projections of A if m(\j\) = 1 and m(p) + m(q) =m{p + q) whenever pq = 0.
We say that m is faithful if ? ? 0 implies m{p) ? 0. We say that m is invariant if
m(p) =m(a(p)) for every projection ? and every automorphism a of A
THEOREM 4.7. Le? Л foe да AF C*-algebra whose Murray-von Neumann order is a
lattice. Let ? be the corresponding MV-algebra as given by Theorem 1.1 (ii). Then we
have:
(i) Invariant finitely additive measures of ? are in one-one correspondence with
invariant measures on the projections of A. Faithful invariant finitely additive
measures correspond to faithful invariant measures.
(ii) Finitely additive measures of ? are in one-one correspondence with tracial states
of A. Faithful finitely additive measures of ? correspond to faithful tracial states
of A.
4.2. Entropy of dynamical systems
Given a probability space and a partition whose /th component has probability p,, one
defines the entropy of the partition by
- ? Pi log Pi.
This is naturally extended to partitions of unity in MV-algebras, as defined below. The
simplest way to construct joint refinements of partitions of unity is to assume that ? is a
(not necessarily ?-complete) semisimple MV-algebra with product.
Dehnition 4.8. A partition of unity in a semisimple MV-algebra with product (M, ·) is
an и-tuple A = (??,..., a„) of elements of ? such that
a\ ? \-a„=\,
where + is addition in the lattice-ordered group with strong unit corresponding of M. If
В = (b\,..., Ь^) is another partition of unity in M, then the product joint refinement ? ? ?
is the (nk)-tup\c defined by
? ? ? = (a,¦ ¦ bj | / = 1,..., n; j = 1,... ,k).
It is easy to see that ? ? ? is again a partition of unity. Given partitions of unity
? ?,..., A,„ in M, we shall denote by V!'=i A tneir product joint refinement.

902
В. Riecan and D. Mundici
Dehnition 4.9. Let (M, ·) be a semisimple MV-algebra with product. A generalized
dynamical system is a quadruple (??,-,?,?/), where the maps ? : ? —>¦ [0, 1] and
U :M —>· ?? satisfy the following conditions, for all ?, b e ??:
(i) ? is a finitely additive measure on M;
(ii) if a + b ^ и, in the ?-group with strong unit и = 1 corresponding to M, then
I/(a+fc) = I/(a) + I/(fc);
(iii) t/(w) = и;
(iv) ?(?/(?)) = ?(?);
(?) U(a-b) = U(a)-U(b).
Condition (ii) can be equivalently rephrased as follows: whenever ? ? b = 0 then
U(?) ? t/(fo) = 0 and U(a ®b) = U(a) ? U(b).
As usual, we define the entropy function ?:[0, 1 ] —> [0, 1 ] by
@ if jc =0,
<p(jc) = |
I — .xlog.x if jc > 0.
Dehnition 4.10. Let A = (a\,.. .,a„) and В = (b\,. ..,bk) be two partitions of unity
in a semisimple MV-algebra with product (M, ·). Assume ? to be equipped with a finitely
additive measure ?. Then we define the entropy of A by
?
?{?) = ??{?(?{)) A4)
?' = ?
and the conditional entropy
?(?\?) = ±^??])?(>^-?\ A5)
where all summands with ?(?>^) = 0 are omitted.
Dehnition 4.11. Given a partition of unity A = (??,..., a„), we write i/°(A) = A, and
inductively,
?/1(?) = (?/(?,),...,?/(?„)), ?/2(?) = (?/(?/(?,)),...,?/(?/(?„))), ....
Note that i/(A) is a partition of unity.
Upon letting a„ = Я(\/"~о U'(A)), one can prove the subadditive law an+m ^ a„ + a,„
for all и, т е N. This implies the existence of the limit
h(A,U)= lim -HI V i/'(A) |. A6)
\i=0 /

Probability on MV-algebras
903
THEOREM 4.12. Adopt the above notation and assumptions. Suppose that the semisimple
MV-algebra with product (M, ·) is equipped with a finitely additive measure ?. Then for
any two partitions of unity A and С in ? we have the inequality h(A,U) ^ h(C, U) +
НЩС).
4.3. Entropy of full tribes
Let (?,?, ?) be a probability space and, as in A3) above, assume ?:? —>¦ ? to be
a measure preserving transformation, in the sense that for all ? e S, T~\E) e S and
P{E) = P(T~[ (?)). Let (?, ?) be the full tribe of all 5-measurable functions from ? to
[0, 1]. Equip (?, Л with the state m : ?-> [0, 1] given by m{f) = fn fdP.
A partition of unity A = (f\,..., /„) in (?,?) is said to be boolean if all functions
fi are {0, 1 (-valued. We let V denote the set of all boolean partitions of unity in (?,?).
Further, we let U denote the family of all partitions of unity in (?, J7).
Upon defining the map U: ? -> ? by U(f) = f о Г for each / e T, we obtain a
generalized dynamical system (T,m, U) in the sense of Definition 4.9. Recalling A4)-
A6), let us consider the quantity
sup{h(A,U)\AeV]. A7)
This is known as the Kolmogorov-Sinai entropy of (J7, m, U).
Suppose now that the boolean partition A* e V is a generator, in the sense that the ?-
algebra generated by the set IJ^o W (^*) coincides with S. Then the Kolmogorov-Sinai
theorem establishes the identity
sup{h(A,U)\AeV} = h(A*,U). A8)
One can prove the following tribe-theoretic generalization of the Kolmogorov-Sinai
theorem:
THEOREM 4.13. Adopt the above notation, and again assume the boolean partition A*
to be a generator in the generalized dynamical system {T, m,U). Then for every partition
of unity A = (g\,... ,gk) eW we have the inequality
h(A, U) ? h(A*, U)+ [ Y<p(gi)dP. A9)
To obtain from this result the classical Kolmogorov-Sinai theorem, it is enough to note
that, in the particular case when A = (g\,... ,gk) is a boolean partition of unity, then
<p(gj) = 0, whence from A9) we get A8).
Generalizing A7), we can now try to introduce a "tribe-theoretic" notion of entropy, by
considering the quantity
hu=sup{h(A,U)\ AeU},

904
В. Riecan and D. Mundici
and more generally, for each subset Q of T, let
hu.g = sup{h(A, U)\A = {gi,...,gk)eU, gi e Q].
This definition has the following drawback:
PROPOSITION 4.14. If Q с ? contains all constant functions /: ? -> [0, 1], then
hu.g = oo. Thus in particular, h\j = oo.
Hudetz suggested the following alternative notions:
DEnNlTlON 4.15. Let A = (/?, ·.., ?) be a partition of unity in a full tribe (?,?)
equipped with a (not necessarily faithful) state m and a map U as above. Then we let
1 = 1 \i = l /
and
hb(A,U)=Yim ??,1\/?'(?)\.
Further, for each subset Q of ? we let the quantity hv g be defined by
hbug=sup{hb(A,U)\A = (f^...,fk)eU, f eQ].
Using Theorem 4.13 we then obtain the following tribe-theoretic version of the
Kolmogorov-Sinai theorem, with respect to Hudetz entropy:
THEOREM 4.16. Let (?,?,?) be a probability space equipped with a measure
preserving transformation T. Let ? be the full tribe of all functions f : ? —>¦ [0, 1] which
are measurable with respect to S. Let U :T'-> ? be given by U(f) = f oT. Suppose the
boolean partition A* = (g*,..., g?) is a generator ofS. Also assume Q to be a subset of ?
such that g* e Q for each i = l,...,k. Then hbug = hb(A*, U) = h(A*, U).
4.4. Bibliographical remarb
Finitely additive measures on MV-algebras, called "states" by the author, were first
investigated in [59] as averaging procedures for the truth-value of propositions in the
infinite-valued calculus of Lukasiewicz. These measures correspond, via the ? functor,
to normalized positive real-valued homomorphisms (called "states" the literature) of
?-groups with strong unit (see, e.g., [29]). They also find an interesting application in the
probabilistic approach to the Ulam-Renyi game of Twenty Questions with lies [28].

Probability on MV-algebras
905
The correspondence between finitely additive measures on MV-algebras and AF
C*-algebraic tracial states was established in [60]. Proposition 4.3 immediately follows
from Proposition 4.2(i) recalling the basic properties of Chang's distance function [7,13,
Proposition 1.2.5]. Proofs of all the other results of Section 4.1 can be found in [59,60]. In
this latter paper the reader can find further references to measures on projections.
Theorem 4.12 was first proved by [64, Proposition 8] building on [65], for
MV-algebras equipped with the generalized product of Definition 3.22, and without assuming
semisimplicity (compare with [79, Theorem 10.2.14]).
For a proof of Theorem 4.13, Proposition 4.14 and Theorem 4.16, see [79,
Theorem 10.3.4], [79, Proposition 10.3.7] and [79, 10.3.16], respectively.
Entropy as an invariant for dynamical systems was introduced in [42]. The tribe-theoretic
generalization was given in [80] and was further developed in [16,48,55]. See [79, p. 250]
for more information. In [66] the Kolmogorov-Sinai theorem is extended to to dynamical
systems based on general T-norms. In his recent paper [1] Ban, building on previous work
of [ 16], presents an axiomatic approach virtually encompassing the most general dynamical
systems. It turns out that Ban's theory can be reduced to two known cases: the one first
considered in [49], and actually collapsing to the classical boolean case (as shown in [51],
[79, 10.5]), and the one presented here in Section 4.3 (as shown in [79, 10.4]).
According to the notion of partition of unity presented here, (Definition 4.8) a
component a, may be zero; further, a component may occur several times. Thus a partition
of unity is not a set, in general. Further, partitions of unity do not generalize the classical
notion of boolean partitions, because all components of a boolean partition are non-empty.
In his definition [60, Section 5], [62] of MV-algebraic partition on ?, the author assumes a
multiset of components, thus incorporating the integer multiplicity of each a,; components
are further assumed to be linearly independent (with respect to the Z-module structure of
the ?-group corresponding to M). In this way, zero components are automatically excluded,
as in the classical definition of boolean partition. The joint refinability of MV-algebraic
partitions in any MV-algebra is a deep result of Marra [50].
5. Open problems
A) For every probability space (?, S, P), element A eS and ?-subalgebra ? of S,
the conditional probability of A with respect to ? is a T-measurable function
P(A\T)::? -> R such that, for all С eT,
J P(A\T)dP = P(ADC).
Generalize this to the case of a probability MV-algebra (M,m,) with product,
together with an element a & ? and a ?-subalgebra ? of M.
B) Prove a strong version of the martingale convergence theorem leading to a proof
of the Kolmogorov-Sinai theorem on generators in general MV-algebras (compare
with [79, 10.3.3]).
C) Extend Maharam's classification [47] to divisible probability MV-algebras, and to
probability MV-algebras with product.

906
В. Riecan and D. Mundici
D) Given a ?-complete MV-algebra with product (?, ¦), consider the following related
MV-algebras:
• its boolean skeleton B(M), i.e., the ?-subalgebra of ? given by the elements
a e ? such that ? ? a = a;
• its boolean quotient Q(M) which is obtained by canonically representing ?
as a separating MV-algebra of functions over M{M), and then restricting any
such function / to the set of all points corresponding to those maximal ideals
J such that M/J is finite (whence, M/J = {0, 1});
• the quotient D(M), which similarly arises by restricting / to the set of all
maximal ideals J such that M/J is infinite (whence M/J = [0, 1]).
Under which conditions can ? be uniquely recovered (up to isomorphism) from
the above three algebras? For tribes closed under multiplication, see the analysis
in [52,53].
E) Given two ?-complete MV-algebras ? and ?, let us agree to define their ?-tensor
product by the following procedure:
• let R = ? ® ? be their semisimple tensor product as in [61]. In the particular
case when ? and N are boolean, one obtains their "free product";
• in the light of [61, Theorem 4.3], identify R with a separating MV-algebra
of [0, 1]-valued continuous functions over the maximal ideal space M(R) =
M{M) xM(N);
• let Я+ be the smallest ?-complete MV-algebra of [0, l]-valued functions over
M(R) containing R. Warning: already in the boolean case, the maximal ideal
space of M(Rf) need not coincide with M{R): indeed, the product of two
basically disconnected spaces need not be basically disconnected.
Assuming now ? and N to be probability MV-algebras, generalize the classical
theory of "stochastically independent" ?-subalgebras А с ? and В с N as defined
in Fremlin's treatise [24, 325L].
F) Extend to probability MV-algebras with product the classical boolean construction
of free products of probability algebras [24, 325J].
G) Given partitions of unity A and В in an MV-algebra with a finitely additive measure
? (without assuming ? to have a product) investigate alternative notions of "joint
refinement" ? ? ? of ? and В, replace the product preservation condition (v) in
Definition 4.9 by an appropriate "joint u-refinement" preservation property. Then
develop the Kolmogorov-Sinai theory for the corresponding dynamical systems.
References
[1] A.I. Ban, Entropy oj fuzzy ? dynamical systems, J. Fuzzy Math. 6 A998), 351-362.
[2] A. Bigard, K. Keimel and S. Wolfenstein. Groupes et Anneaux Reticules. Lecture Notes in Math.. Vol. 608,
Springer, Berlin A971).
[3] G. Birkhoff and J. Von Neumann. The logic of quantum mechanics, Ann. Math. 37 A936), 823-843.
[4] D. Butnariu, Values and cores of fuzzy games with infinitely many players, J. Game Theory 16 A987),
43-68.
[5] D. Butnariu, E.P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer.
Dordrecht A993).

Probability on MV-algebras
907
[61 C. Caratheodory. Mass und Integral und ihre Algebraisierung. Birkauser, Boston A956). English
translation: Algebraic Theory of Measure and Integration. Chelsea. New York A963).
[7] C.C. Chang, Algebraic analysis of many-valued logics. Trans. Amer. Math. Soc. 88 A958), 467^490.
[8] C.C. Chang, A new proof of the completeness of the Lukasiewicz axioms. Trans. Amer. Math. Soc. 93 A959),
74-90.
[9] C.C. Chang, The writing of the MV-algebras, Studia Logica 61 A998), 3-6. (Special issue on Many-Valued
Logics, D. Mundici, ed.)
[10] F. Chovanec, States and observables on MValgebras, Tatra Mt. Math. Publ. 3 A993), 55-65.
[11] F. Chovanec andE. Rybarikova, Ideals and filters in D-posets, Intemat. J. Theoret. Phys. 37 A998), 17-19.
[12] R. Cignoli and D. Mundici, An invitation to Chang's MV-algebras, Advances in Algebra and Model-Theory,
M. Droste and R. Gobel, eds, Gordon and Breach, Readings, UK A998), 171-197.
[13] R. Cignoli, I.M.L. D'Ottaviano and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Trends
in Logic, Studia Logica Library, Vol. 7, Kluwer, Dordrecht B000).
[ 14] A. Di Nola and A. Dvurecenskij, Product MV-algebras, Multi-Valued Logic 6 A-2) B000). 193-215.
[15] A. Di Nola, G. Georgescu and A. Lettieri, Extending probabilities to states on MV-algebras, Coll. Logicum,
Ann. Kurt-Godel-Soc. 3 A999), 31-50.
[16] D. Dumitrescu, Fuzzy measures and the entropy of fuzz\ partitions, J. Math. Anal. Appl. 176 A993), 359-
373.
[17] A. Dvurecenskij, Loomis-Sikorski theorem for ?-complete MV-algebras and l-groups. J. Australian Math.
Soc. Series A 67 A999), 1-17.
[18] A. Dvurecenskij and S. Pulmannova, New Trends in Quantum Structures, Kluwer, Dordrecht B000).
[19] A. Dvurecenskij and B. Riecan, Weakly divisible MV-algebras and product. J. Math. Anal. Appl. 234 A999),
208-222.
[20] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras.
J. Algebra 38 A976), 29^44.
[21] G.G. Emch, Mathematical and Conceptual Foundations of 20th Century Physics, North-Holland,
Amsterdam A984).
[22] D.H. Fremlin, A direct proof of the Matthes-Wright integral extension theorem, J. London Math. Soc. 11
A975), 276-284.
[23] D.H. Fremlin, Measure algebras. Handbook of Boolean Algebras, Vol. 3, J.D. Monk, ed.. North-Holland,
Amsterdam A989).
[24] D.H. Fremlin, Measure Theory, Available from the author's site in the University of Sussex A995).
[25] R. Fric, On observables. Int. J. Theor. Phys. 39 B000), 673-682.
[26] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, NY A963).
[27] K. Goodearl, Notes on Real and Complex C*-algebras, Shiva Mathematics Series, Vol. 5, Birkhauser,
Boston A982).
[28] B. Gerla, Conditioning a state by a Lukasiewicz event: a probabilistic approach to Ulam game, Theoret.
Comput. Sci. 230 B000), 149-166.
[29] K.R. Goodearl, Partially Ordered Abelian Groups with Interpolation, Amer. Math. Soc., Providence, RI
A986).
[30] S. Gudder. Fuzzy probability theory. Demonstrate Math. 31 A998), 235-254.
[31] PR. Halmos, Lectures on Boolean Algebras, Springer, New York A974).
[32] T. Hudetz, Quantum topological entropy: first step of a pedestrian approach. Quantum Probability and
Related Topics, VIII. L. Accardi et al., eds. World Scientific, Singapore A993), 237-261.
[33] J. Jakublk, On complete MV-algebras, Czechoslovak Math. J. 45 A995). 473^480.
[34] J. Jakublk, On the product MV-algebras with product, Czechoslovak Math. J., to appear.
[35] M. Jureckova, On the conditional expectation on probability MV algebras with product. Soft Comput. 5
B001), 381-385.
[36] M. Jureckova, A note on the individual ergodic theorem on product MV-algebras, Intemat. J. Theoret. Phys.
39 B000), 753-760.
[37] M. Jureckova and B. Riecan, Weak law of large numbers for weak observables in MV-algebras. Tatra Mt.
Math. Publ. 12A997). 221-228.
[38] A. Kacian, Probability on t-rings, PhD thesis. University of Bratislava. Slovak Academy of Sciences
A999).

908
В. Riecan and D. Mundici
E.P. Klement, Construction of fuzzy ?-algebras using triangular norms, J. Math. Anal. Appl. 85 A982),
543-565.
E.P. Klement, R. Mesiar and E. Pap, Triangular norms. Trends in Logic, Studia Logica Library, Vol. 8,
Kluwer, Dordrecht B000).
E.P. Klement and M. Navara, A characterization of tribes with respect to Lukasiewicz t-norm, Czechoslovak
Math. J. 47A997), 689-700.
A.N. Kolmogorov. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue
spaces, Dokl. Akad. Nauk SSSR 119 A958), 861-864 (in Russian).
F. Корка, Boolean D-posets as the factor spaces, Intemat. J. Theoret. Phys. 37 A998), 93-101.
K. Kuriyama, Entropy of a finite partition ofafuzz\ sets, J. Math. Anal. Appl. 94 A983). 38—43.
J. Lukasiewicz, Ologice trojwarkosciowej [On three-valued logic]. Ruch Filozoficzny 6 A920). 170-171.
J. Lukasiewicz and A. Tarski. Untersuchungen iiber den Aussagenkalkiil, Comptes Rendus des seances de
la Societe des Sciences et des Lettres de Varsovie, Classe III. 23 A930), 30-50.
D. Maharam. An algebraic characterization of measure algebras. Ann. Math. 48 A947). 154—167.
P. Malicky and B. Riecan. On the entropy of dynamical systems, Proc. Conf. Ergodic Theory and Related
Topics, II. H. Michel, ed„ Teubner. Leipzig A986). 135-138.
D. Markechova, The entropy of fuzzy dynamical systems and generators. Fuzzy Sets and Systems 48 A992),
351-363.
V. Marra, Every abelian i-group is ultrasimplicial, J. Algebra 225 B000), 872-884.
R. Mesiar. The Bayes principle and the entropy on fuzzy probability spaces. Int. J. General Systems 20
A991), 67-72.
R. Mesiar, Fundamental triangular norm based tribes and measures, J. Math. Anal. Appl. 177 A993),
633-640.
R. Mesiar and M. Navara. Ts-tribes and Ts-measures. J. Math. Anal. Appl. 201 A996), 91-102.
R. Mesiar and B. Riecan, On the joint observable in some quantum structures. Tatra Mt. Math. Publ. 3
A993). 183-190.
R. Mesiar and J. Rybarik. Entropy of fuzzy partitions—a general model. Fuzzy Sets and Systems 99 A998),
73-79.
F. Montagna, An algebraic approach to prepositional fuzzy logic, J. Logic. Lang. Inf. 9 B000). 91-124.
(Special issue on Logics of Uncertainty. D. Mundici. ed.)
D. Mundici, Interpretation of AF C*-algebras in Lukasiewicz sentential calculus. J. Funct. Anal. 65 A986),
15-63.
D. Mundici, Logic of infinite quantum systems, Internat. J. Theoret. Phys. 32 A993), 1941-1955.
D. Mundici, Averaging the truth value in Lukasiewicz sentential logic, Studia Logica 55 A995), 113-127.
(Special issue in honor of Helena Rasiowa.)
D. Mundici, Uncertainty measures in MV-algebras and states of AF C*-algebras, Notas Soc. Mat. Chile 15
A996). 42-54. (Special issue in memoriam Rolando Chuaqui.)
D. Mundici, Tensor products and the Loomis-Sikorski theorem for MV-algebras, Adv. Appl. Math. 22
A999), 227-248.
D. Mundici, Ulam game, the logic of MaxSat. and many-valued partitions. Fuzzy Sets. Logic and Reasoning
about Knowledge. D. Dubois et al., eds, Kluwer. Dordrecht A999). 121-137.
D. Mundici and G. Panti. Extending addition in Elliott's local semigroup. J. Funct. Anal. 171 A993). 461-
472.
J. Petrovicova, On the entropy of dynamical systems in product MV-algebras, Fuzzy Sets and Systems, 121
B) B001). 347-351.
J. Petrovicova, On the entropy of partitions in product MV-algebras, Soft Comput. 4 B000). 41^44.
J. Petrovicova and B. Riecan, On the entropy of T-dynamical systems, Busefal, D. Dubois, ed., 80 A999),
9-13.
K. Piasecki. Extension of fuzzy P-measure by usual measure. J. Fuzzy Math. 7 A987), 117-124.
S. Pulmannova, A note on observables on MV-algebras, Soft Comput. 4 B000). 45^48.
B. Riecan, Fuzzy connectives and quantum models. Cybernetics and System Research '92, Vol. 1, R. Trappl,
ed.. World Scientific, Singapore A992), 335-338.
[70] B. Riecan, Upper and lower limits of observables in D-posets of fuzzy sets. Math. Slovaca 46 A996). 419-
431.

Probability on MV-algebras
909
[71] B. Riecan, On the almost everywhere convergence of obsenables in some algebraic structures. Atti Sem.
Mat. Fis. Univ. Modena44 A996). 95-104.
[72] B. Riecan. On the product MV-algebras, Tatra Mt. Math. Publ. 16 A999). 143-149.
[73] B. Riecan. On the W space of obsenables. Fuzzy Sets and Systems 105 A999), 299-306.
[74] B. Riecan, On the conditional expectation of obsenables in MV-algebras of fuzzy sets. Fuzzy Sets and
Systems 102 A999), 445^450.
[75] B. Riecan. Probability theory on some ordered structures, Atti Sem. Mat. Fis. Univ. Modena 47 A999),
305-315.
[76] B. Riecan, On the W space of obsenables on product MV-algebras, Intemat. J. Theoret. Phys. 39 B000),
847-854.
[77] B. Riecan. On the probability theory on MV-algebras, Soft Comput. 4 B000). 49-57.
[78] B. Riecan, Almost everywhere convergence in probability MV-algebras with product, Soft Comput 5 B001),
396-399.
[79] B. Riecan and T. Neubrunn. Integral, Measure, and Ordering, Kluwer, Dordrecht A997).
[80] E. Ruspini. A new approach to clustering. Inform, and Control 15 A969), 22-32.
[81] J. Rybarik. The entropy of partitions on MV-algebras, Intemat. J. Theoret. Phys. 39 C) B000), 885-891.
[82] R. Sikorski, Boolean Algebras, Springer. Berlin A960).
[83] A. Tarski. Logic. Semantics. Metamathematics. Clarendon Press, Oxford A956), Chapter IV. Reprinted
1983, Hackett, Indianapolis.
[84] V Varadarajan, Geometry of Quantum Theory, Vol. 1, Van Nostrand Reinhold, Princeton, NJ A968).
[85] M. Vrabelova. On the conditional probability in product MV-algebras, Soft Comput. 4 B000), 58-61.
[86] J.D.M. Wright, The measure extension problem for vector lattices, Ann. Inst. Fourier (Grenoble) 21 A971),
65-85.

CHAPTER 22
Measures on Clans and on MV-Algebras
Giuseppina Barbieri and Hans Weber
Universita di Udine, Dipartimemo di Matematka e Infonnatica via delle Scienze 206 33100 Udine. Italy
E-mail: weber<g>dimi. uniud. it
Contents
Introduction 913
1. MV-algebras and clans 914
1.1. Basic properties of MV-algebras 914
1.2. The centre of an MV-algebra 916
1.3. Representation of MV-algebras 918
1.4. Loomis-Sikorski theorem 919
1.5. Decomposition of complete MV-algebras 919
2. Submeasures on MV-algebras 921
3. Real-valued measures on MV-algebras 923
3.1. The space ba(L) 923
3.2. Representation of real-valued measures 927
4. Uniform MV-algebras 927
4.1. The lattice CUA(L) 927
4.2. Decomposition of order continuous uniform MV-algebras 930
4.3. The lattice CUA(L. w) 933
4.4. The uniformity generated by a measure 934
5. Measures on MV-algebras with values in a locally convex space 936
5.1. Representation of measures 936
5.2. Decomposition theorems 938
5.3. Hammer-Sobczyk's decomposition theorem and Lyapunov's theorem 939
5.4. Vitali-Hahn-Saks-Nikodym theorem and Nikodym boundedness theorem 941
5.5. Extension of measures 943
References 944
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V All rights reserved
911

This page intentionally left blank

Measures on clans and on MV-algebras
913
Introduction
The main aim of this chapter is to present in a unified approach the results about a certain
type of fuzzy measures, namely about Tx-valuations on clans of fuzzy sets.
A clan ? of fuzzy sets is a family of [0, 1 ]-valued functions defined on a set ? such that
1 e ? and (/ - g) ? 0 e ? whenever /, g e T. Identifying sets with their characteristic
functions (i.e., {0, 1 (-valued functions), any algebra of sets is a clan of fuzzy sets.
A ???-valuation on a clan С of fuzzy sets is a real-valued function such that ?(/ + g) =
m(/) + M(g) if /, g e ? and / + g < 1. Any finitely additive measure on an algebra of
sets is a ???-valuation. Tx-valuations were introduced by Klement A982) and intensively
studied by Butnariu and Klement A993) in their book "Triangular norm-based measures
and games with fuzzy coalitions". There they treat the real-valued or R"-valued case. Here
we present T^-valuations with values in a locally convex linear space in order to include
also vector-valued measures on algebras of sets. Any ??-valuation is a modular function,
see Proposition 1.1.9. This fact allows us to apply some topological techniques elaborated
by Weber A996,2002) in the study of modular functions. Indeed, we develop a topological
approach to study Гзс-valuation inspired by the method of Weber A984a, 1984b) using
FN-topologies in the study of measures on Boolean algebras. To realize this approach
we need, as a domain of fuzzy measures, instead of a clan of fuzzy sets a more general
structure, which is equationally defined and therefore closed under quotients and uniform
completions. A natural generalization of clan of fuzzy sets is an MV-algebra.
DEnNlTlON 0.1. An MV-algebra (L. +,'; 0, 1) is a commutative semigroup (L,+)
with 0, 1 and a unary operation ': L —>¦ L which satisfies the following axioms:
(LI) jt + 1 = 1,
(L2) {x')'=x,
(L3) 0'= 1,
(L4) (x' + y)' + y = (x+ y')' + ? for every x.yeL.
Actually, we don't use the previous axioms for an MV-algebra; but, after defining ? ? у
iff ?' + у = 1 and у ?? := (? + у')' whenever x ^ у we use the following axioms
(PI) x^x+y;
(P2) x^y^z+x=*yAx<:z;
(P3) у = у Ax +x whenever ? ? у;
(P4) ?" =?.
Incidentally, the first three axioms define a A-?-semigroup, a structure defined in Weber
A997).
MV-algebras were introduced by Chang in 1958 in the theory of multi-valued logic.
"So because of my own shortcomings in following proofs in Polish notation
during a seminar held by Rosser on the Completeness of Lukasiezwicz axioms
for infinite valued propositional logic, MV-algebras came to be." (Chang,
1998)
"It occurred to me that the approach used in the proof of completeness of
the two valued propositional logic via the Boolean (Lindebaum) algebra of

914
G. Barbieri and H. Weber
equivalence classes of formulas and the Boolean maximal ideal theorem might
be another way to do what Rosser did, but avoiding syntactical manipulations
of formulas in Polish notation. This was the beginning of MV-algebras"
(ibidem).
Chang found a proof of the completeness of the 1%3-valued logic (developed by
Lukasiewicz and Tarski and by Rose and Rosser). Since then MV-algebras have found
increasing interest. In 1995 Mundici introduced finitely additive measures (called states)
on MV-algebras with the intent of capturing the notion of 'average degree of truth' of a
proposition.
Here we carry on some measure theory on MV-algebras. This mainly follows Barbieri
and Weber A998), Barbieri A999), Barbieri et al. B001).
The topics we are interested in are representation of measures, decomposition of
measures, extension of measures and Vitali-Hahn-Saks and Nikodym theorems. For a
treatment of measures on MV-algebras from a probabilistic point of view we refer to Riecan
and Mundici B002).
From now on, L stands for an MV-algebra.
1. MV-algebras and clans
1.1. Basic properties of MV-algebras
We always consider on the MV-algebra L the relation ^ defined by
? ^ у iff ? + у = 1.
It can be shown that ^ is a partial order on L and (L, ^) is a distributive lattice (see
Mangani A973), where the lattice operations are given by ? ? у = (? + у')' + ? and
x/\y = (x'v у')').
A basic example of an MV-algebra is an order interval of a commutative ?-group:
PROPOSITION 1.1.1. If (?, ?, ^) is a commutative i-group and e is a positive element
of H, then the interval {x e H: 0 ^ ? ^ e} becomes an MV-algebra with respect to the
operations ? + у := (? ? у) ? e and x' := e — ?; we denote this MV-algebra by Lq(H, e).
Chang A958b) and Mundici A986) have proved that any MV-algebra is of this type.
THEOREM 1.1.2. Let L be an MV-algebra. Then there exists a commutative l-group ?
with strong unit e (i.e., Wg e ? Bn e N with ne ^ \g\) such that L = Lo{H,e). L is
complete iff ? is Dedekind complete and L is ? -complete iff ? is Dedekind ? -complete.
The completeness statement in Theorem 1.1.2 was observed by Jakubik A995).
COROLLARY 1.1.3. If(L, +,';0, 1) is an MV-algebra and ее L, then Le:=[x eL: x^e]
is an MV-algebra with operation ? defined by ? ? у := (? + у) л е and complementation
* defined by ?* := ?' ? е.

Measures on clans and on MV-algebras
915
In the introduction we have already defined у Ax if x, у e L and ? ^ y. We extend ? on
Lx L by ? Ay = (xvy)A(x Ay). With the notations as in Theorem 1.1.2, if L = Lo(H,e),
then xAy = \x — y\.
Every MV-algebra satisfies an important property known as the Riesz decomposition
property.
Proposition 1.1.4 (Riesz decomposition property). If x,x\,xi e L with ? ^ x\ + хг,
then there are f, e L(i = 1, 2) such that f, ^ jc, and ? =t\ +t2,t\ ^ t'r
Proposition 1.1.5.
(a) x' = \Ax, ? +x' = 1.
(b) xAy = Oiffx = y.
(c) xAy^(xAz) + (zAy).
(d) [xA(xAy)] + y = xvy, (x Ay) + (x vy)=x +y.
(e) (y + x)Az^(yAz) + (xA z).
(f) If о denotes one of the operations +, ?, ?, ?, then (? ? z)A(y ? ?) ^ xAy.
(g) If(xa)aeA I x and (yfl)fleB I }'¦ Then xa + у ? \. ? +y.
(h) Let a„,s„,s e L such thats,, = supi>nai ands,, Is. Then a \ As ^ ??=\ ?/???,,+|.
One can deduce these rules from the axioms (P1)-(P3) given in the introduction (see
Weber A997)). Proposition also 1.1.5 follows from Theorem 1.1.2 and the known rules in
?-groups.
Proposition 1.1.6.
(a) If x, у e L, then (x + y)Ay = ? л у'.
(b) If(x„) is a sequence in L with J2"=i x\ -^ x'n+l (n e N), then s„ := ?'?=\ ?' defines
an increasing sequence and s„Asn-\ = x„(n e N).
(c) If (s„)newu{0] is an increasing sequence, then x„ := s„As,,-i defines a sequence
with snAs0 = ?"=| ?, ^ x'n+i (n e N).
We call a subset N of L an idea! if x, у e ?, ? e L, ? ^ ? + у imply ? e ?. ? ?-ideal
is an ideal closed under countable suprema.
The following proposition shows that there is a natural bijection between congruence
relations and ideals in L.
PROPOSITION 1.1.7. Let ~ be a congruence relation with respect to + and ' (and
therefore with respect to ?, ?, ?) on L.
(a) Then N = [x e L: ? ~ 0} is an ideal and ? ~ у iff xAy e N. Conversely, any
ideal N in L by
x~y iff xAyeN
defines a congruence relation with respect to +,', ?, ?, ?.
(b) The quotient L/N := L/ ~ with the induced operations is an MV-algebra.

916
G. Barbieri and H. Weber
DEFINITION 1.1.8. A vector-valued map ?: L —>¦ ? is called a measure if ?(>0 =
?(???) + ?(?) whenever ?, у е Р and x ? у.
PROPOSITION 1.1.9. Let ?: L^- ? be a vector-valued map.
(a) Then ? is a measure iff ?(? + у) = ?(?) + ?(}') whenever ?, у e L and x ? у'.
(b) If ? is a measure, then ? is a modular function, i.e. ?(? А у) + ?(? ? у) =
?(?) + ?-iy) whenever ?, у е L.
PROOF, (a) (=>) If x,y e L and ? ^ y', then (jc + y)A)· = x. Hence, ?(? + у) =
?((? + у) Ay) + ?(>0 = ?(?) + ?(}').
(<=I? x,y e L and ? ? у, then )>Ajc ^ x'. Hence, ?()·) = ?((? ??) + ?) = ?(???) +
?(?).
(b)Let ?, у & L. Since xA(x Ay) ? у', we obtain using (a) ?(?) + ?(?) = ?(? Лу) +
?(??(? л ?))+?{?) = ?(? А y)+?((xA(x ? }¦)) + >') = Д(* ? ?) + ?(? ? )·)· ?
PROPOSITION 1.1.10. Let ? : L -> ? be a vector-valued measure.
(a) Then ?(?) := [? e L: ?(?) = Ofor every ? e [0, ?]} is an ideal in L.
(b) Let ~ be the congruence relation induced by ?(?) according to Proposition 1.1.7.
Then a ~ b iff ?(?) = ?(}')^ ?, у е [а A b, a v b].
(c) For an ideal N С ?(?), define ? by ?(?) := ?(?), ? e a e L/N. Г/геи ? /s ?
measure on L/N and ?(?) = {0} iff ? = ?(?).
PROOF, (a) Let х\,хг e ?(?) and у ^ jc? + X2- By the Riesz decomposition theorem
there are y, e L with y, ^ jc, (/ = 1,2) and у = y\ + yi, y\ ^ y'-,. Therefore ?^) =
?(?\) + ?(?2) = ?.
(b) (=>) If ? ~ b and jc e [а л b, ? ? b]. Then г := (? ? b)Ax ^ ??/> e ?(?), hence,
?(?) = 0 and ?(? w b) = ?(?) + ?(?) = ?(?).
(«=) Let ? ^ aAb. Then ? + (а л fo) e [а л fo, ? ? b], thus ?(? л fo) = ?(? + (а л ?)) =
?(?) +?(? ? fo) by Proposition 1.1.9, so ?(?) =0. It follows that aAb^O and a ~ fo.
(c) follows from (b). ?
1.2. Г/ге centre of an MV-algebra
The centre of L here has a basic role in decomposition theorems and in the representation
of measures. The centre of a lattice R with 0 and 1 is the set of elements e e R such that
for some element f e R the map R э ? н> (jc л e, ? л /) е [0, е] ? [0, /] defines a lattice
isomorphism. In this case / is the unique (!) complement of e. If R is distributive then
C(R) is the biggest Boolean subalgebraof (R, л, v,0, 1).
Here we give some characterizations of the centre.
PROPOSITION 1.2.1. C(L) = {x e L: ? ax' =0} = {x e L: x' is the unique complement
of x] = [x e L: ? has a complement in (L, ^)) = {x e L: 2x = x).
PROOF, (i) By Proposition 1.1.6(a) jc ajc' = 0 holds true iff 2jc = x.
(ii) We will show that C(L) с {jc e L: jc л jc' = 0).

Measures on clans and on MV-algebras
917
Let ? e C{L) and ? the complement of x. Then by Proposition 1.1.5 ?' = 1?* =
(? ? ?)?(? ? 0) ^ ? АО = ?, hence ? лх' ?? ??=0.
(iii)Letjc e L and ? ??' = 0, then jc' is a complement of jc: in fact, by Proposition 1.1.5,
we have ? ? ?' = ? + ?' = 1.
The rest follows from the distributivity of (L, л, ?). ?
Corollary 1.2.2. If L is ?-complete, then a := sup„eti na e С(L) for every a e L.
PROOF. Thanks to Proposition 1.2.1 it suffices to prove that 2<5 = a.
Indeed 25 = sup„eNna +sup„6N/ia =sup„ w6N[(n +m)a] = supneNna =a. D
PROPOSITION 1.2.3. С\L) is regular embedded in L.
PROOF. It is sufficient to prove that if C(L) э xa \, ? in L, then ? e C(L). By
Propositions 1.1.5 and 1.2.1 we have xa = 2xa I 2x, therefore 2x = x, and so by Proposition 1.2.1
xeC(L). О
PROPOSITION 1.2.4. Letae C{L). Then ?(?) := (jc ? ?, jc л ?') defines an MV-algebra
isomorphism between L and [0, a] x [0, a'].
PROOF. Since a e C(L), ?:? н> (jc л а,х л ?') defines a lattice isomorphism from L
onto [0, ?] ? [0,a']. What is left to show is that, with notation as in Corollary 1,1.3,
?(? +y) = ?(?) ? ф(у) and ф(х') = ф(х)* whenever jc , ? e L, i.e., that for all x,y e L
and с е C(L) we have
(i) (x + у) л с = (? л с) + (}> л с);
(п) jc' лс = (jc ас)*.
(i) We have (* лс) + (у лс) = (jc +у) л (jc +с) л (у +с) а(с+с) and (? +с) л (у +с) л
(с + с) = с if с + с = с.
(ii) (jc л с)* = (jc л с)' л с = (jc' ? с') л с = (jc' л с) ? (с' л с) = х' л с. D
The following proposition, which immediately follows from Proposition 1.2.4, makes it
understandable that the centre C(L) plays an important role for decomposition theorems
of measures.
Corollary 1.2.5. Let a e C(L), ? : L —>· ? a measure with values in a linear space.
Then ?(,(?) := ?(? л jc) and ??'(?) := ?(?' ? jc) define two measures on L and ? =
??+??·-
We conclude the section with a remarkable example.
Example 1.2.6. Let ? be a nonempty set, ? an ?-subgroup of Жп such that ?? e ?
and, with the notation of Proposition \.\.\, L = Lq(H, ??)- Then L is a Гэс-clan of fuzzy
sets.
The centre C(L) is the space of {0, 1 (-valued functions contained in H, called the space
of crisp sets, ? := {А С ?: ?? e ?] is an algebra of subsets and ДэЛихл defines a
Boolean isomorphism between Л and C(L).

918
G. Barbieri and H. Weber
1.3. Representation of MV-algebras
A ???-tribe is a ???-clan such that (?„6? /,() л ?? e ? whenever /„ e T.
Its centre C(T) is a ?-algebra. Important is here Artstein's observation that any function
of a ???-tribe ? is measurable with respect to C(T) (see Butnariu and Klement A993,
3.2)). In Proposition 1.3.1 we give a simpler proof of this fact.
For an algebra A of subsets of ? we denote by S(A) = spanfxA: A e A} the linear
space generated by the characteristic functions хд, А еД and by CX(A) the closure of
S(A) in the space /oo(i2) of all bounded functions with respect to the sup norm || ||oo.
Proposition 1.3.1 will be a model for a representation of an MV-algebra.
PROPOSITION 1.3.1. Let ? be a Toe-tribe of fuzzy sets on ? and A = C(T) the
?-algebra of crisp sets in T. Then ? с CX(A).
PROOF. Let / e ? and ALn :={???: f(x) <i/n] for ? e N and / eNU(O).
Then ???? = sup,n6N(w(/ - nf)+ л ??) e ?, hence, A,.„ e A, g„ := ?,"=??'('/")
(Хл/+1.„ - Xa,„) e 5(Л) and 0 < / - g„ < \/n. ?
Now we will represent an MV-algebra. This representation theorem is obtained as a
corollary of a well-known representation theorem for ?-groups. Its proof can be found in
Vulikh A967, Section V, Theorem V.3.1). We recall that the set of all components of an
element e of the positive cone of an ?-group G coincides with the centre of the MV-algebra
{x e G: 0 ^ ? ^ e] (cf. Proposition 1.2.1).
Bernau A963) proved that if G is a Dedekind ? -complete ?-group with strong unit e,
then there exists a set ?, an algebra of sets Ao and a group and lattice isomorphism
f:G^ Go С Loo(Ao) with f(C) С Ao, where С is the set of components of e, and
moreover ?(?) = ?^. Now we prove that Aq can be chosen to be isomorphic to С
THEOREM 1.3.2. Let G be a Dedekind ?-complete l-group with strong unit e, С the
Boolean algebra of components of e. Then there exists A an algebra of subsets of a set ?
and a group and lattice isomorphism ?: G —> Go С СХ(А) in such a way G is regular
embedded in Coo(A) and ф(С) = (хд: A e A] (i.e., f{C) = A if we identify sets with
characteristic functions).
PROOF. By the theorem of Bernau A963) mentioned above we have to show that if
A = f(C), then Go С L^A).
Let / e Go, ? e N. We shall have established the theorem if we prove that there exists
s e S(A) such that ||/ — s\\x ? \/?.
As / e/oo(i2) there exists A: e N such that ||/||зс ^k/n.
We will show that for / e ? there exists A, e A such that
{f<Ln)CA-C{f^\
Let g := (??? - nf)+ л ?? and h := (??? - nf)~ л ?? = (nf - ???)+ л ??. Then
g, h e [0, ??] = {/ e Go: 0 ^ / ^ ??). With the same notations as in Corollary 1.2.2

Measures on clans and on MV-algebras
919
~g and h e C([0, ???)· By Example 1.2.6 g~, h are {0, 1 (-valued functions, so there exist
?,, ?, e Л with g~ = ??? and /? = ???. Moreover, as ~g ^ g and, analogously, h ^ h,
we have {g > 0} С A, (*) and {h > 0} с S, (**); as g л h = 0, we get J л /г = 0 and
consequently А, П ?, = 0 (***).
We have {/ < i/n} с А,:
If /(л) < i/n, then g(;t) > 0, therefore by (*) x e A,.
We have А, с {/ ? i/n}:
Let /(jc) > ;'/n, then /j(jc) > 0, therefore by (**) x e S, and consequently by (***)
x$Aj.
Puti:=Ef=^('/")(XA,+, - хл;). Then ? e SD) and ||/-i Hoc ?? 1/n. ?
THEOREM 1.3.3. Lei L fee ? -complete. Then there exist an algebra A of subsets of a
set ? and a regular embedding ? : L -> Coc(A) such that *P(C(L)) = (хд: А е A} and
?(? +y) = (?(?) + ? (у)) л ?? and ? (?1) = ??- ? (x) far ?, у е L.
PROOF. From Theorem 1.1.2 L is the interval [0, e] of a Dedekind ?-complete ?-group
(G, ?, ^) with strong unit e, where the operations of L can be defined by the ones of G
in the following way: х+у = (х(Ву)ле and x' = e - x. Now put A and ? as in
Theorem 1.3.2 and define ? := ?\? the restriction of ? to L. ?
1.4. Loomis-Sikorski theorem
The Loomis-Sikorski theorem for Boolean algebras says that every ?-complete Boolean
algebra is isomorphic to a ?-complete Boolean algebra of point sets modulo a ?-ideal in
that algebra. A. Dvurecenskij B000) and D. Mundici A999) have independently given a
generalization of this theorem for MV-algebras. Here we give an alternatively and easy
proof of this result based on the representation Theorem 1.3.3.
THEOREM 1.4.1. Let L be ?-complete. Then there exists a tribe ? of fuzzy sets and a
?-order continuous epimorphism <p:T —>¦ L onto L (with respect to the operations of an
MV-algebra).
Proof. We use the same notation as in Theorem 1.3.2. By the classical Loomis-Sikorski
theorem there is a ?-algebra Aq and a ?-order continuous epimorphism ? : Ao —>¦ A such
that N = ?'\??) is a ?-ideal. By 45B of Fremlin A974) ? extends ? : Cx(Ao) ->
?оо(Л) and jt~' @) = ???(?0. By 14L(b) of Fremlin A974) ? is ?-order continuous.
Now put ? := ? (\lr(L)). Then ? is a tribe and there is a ?-homomorphism from ?
into L (with respect to the operations of an MV-algebra). ?
1.5. Decomposition of complete MV-algebras
Any complete Boolean algebra С is isomorphic to the product of an atomless Boolean
algebra and {0, 1}A where A denotes the set of atoms of C. We use this decomposition

920
G. Barbieri and H. Weber
for the centre of L to obtain a decomposition of L (see Theorem 1.5.4). This result will
be applied to obtain a topological decomposition of uniform MV-algebras which itself is
a basic tool to prove the decomposition Theorem 5.3.1 for measures and properties of the
range of measures (see Theorem 5.3.2).
We recall that an atom of a lattice Lo with zero 0, is an element a e Lo\{0) such that
if ? e Lq and 0 ^ ? ? a, then either ? = 0 or ? = a. Lq is atomic if for every ? e Lq\{0]
there exists an atom a of Lo with ? ? ?; Lq is atomless if Lo doesn't contain atoms.
PROPOSITION 1.5.1. Let L be ?-complete and a be an atom of L. Then a is an atom of
C(L). (Notation as in Corollary 1.2.2.)
PROOF. Let c\, сг be disjoint elements of C(L) with c\Vc2=a. We will show that с ? =0
orc2 = 0. Since ? = (а лс\) ?(? лс2) and a is an atom, we have ? лс| = 0ora лег =0.
Suppose that а л сг = 0. Then a ^ c\, hence, a ^c\ =c\ ^ a. It follows that a = c\ and
therefore сг = 0. ?
With notations as in Proposition 1.1.1 let Сх := Lq(R,1) the unit real interval
considered as an MV-algebra and C„+\ := {//и: / = 0,..., ?] the MV-subalgebra of Cx
which consists of ? + 1 elements for n e N.
PROPOSITION 1.5.2. Let L be ?-complete and irreducible (i.e., C(L) = {0, 1}). Then L
is (as an MV-algebra) isomorphic to C„ for some neNU{oo},«^ 1.
PROOF. We use the same notations as in Theorem 1.3.3. Since C(L) contains exactly two
elements, we have A = {0, ?). Hence, S(A) = {???: a e Щ = Cx(-A)< so ??(?) is
isomorphic to R. Therefore we may assume that L is an MV-subalgebra of ??.
If L \ {0} has a minimal element a, then a = \/n for some ? e N and therefore
L = C,l+\.
If L \ {0} has not a minimal element, then L is dense in jCoc and therefore L — jCoc»
since L is ?-complete. ?
We need a generalization of Proposition 1.2.4.
PROPOSITION 1.5.3. Let L be complete and D a set of pairwise disjoint elements of
C(L) with supD = 1. Then ?:? н> (? ? d)(/eo defines an isomorphism between L and
PROOF. By Proposition 1.2.4 ? н> ? л d defines for any d e D an homomorphism
between MV-algebras. Therefore ? is an MV-algebra homomorphism. Since sup D = 1
and therefore ? = ? л sup D = sup(/eD ? Ad, ? is injective. ? is surjective since for any
(Xd)ddD e ??/6?[°' <*] we have 0(sUP(/6D-*rf) = (xd)deD- °
THEOREM 1.5.4. Let L be complete. Denote by AX(L) the set of all atoms a ofС(L) for
which [0, a] is infinite and by Af(L) the set of atoms a ofC(L)for which [0, a] is finite.
Put aoo :=sup^oo(L), a/ :=sup.4f(L) and с := (ax ??/)'·

Measures on clans and on MV-algebras
921
(a) Then L is isomorphic to [?, ? ? ? [0, ax] ? [0, с].
(b) [0, af] is isomorphic to a product Y\aeA ?,,„ where A = Af(L).
(c) [0, ax] is isomorphic to the power C^ where В = Ax (L).
(d) The centre of [0, c] is atomless.
(e) L is atomless iff a) — 0 iff A/ (L) = 0.
(f) L is atomic iff ?? = с = 0.
PROOF. aoc,a/,c e C(L) by Proposition 1.2.3. (a), (b), (c) now follow from
Propositions 1.5.3 and 1.5.2. (d) follows from the fact that С([0,a]) = C(L) ? [?,?] for any
a e C(L). (e) and (f) follow from (a), (b), (c) and Proposition 1.5.1. ?
2. Submeasures on MV-algebras
Let ?-.S —>¦ [0, +00] be defined on an MV-subalgebra of L. ? is called subadditive
if ?(? + у) !? ?(?) + ? (у), ? is called ? -subadditive if ?,?,, e S (и e ?) and ? =
SUP„ ?"=?-*, =·' ??=\?'? ??1?^ П(х) < ??=\ n(x,i)- Л is а (?-) submeasure if 77 is
monotone, (?-) subadditive and ?@) = 0.
Example 2.1. Let (E, p) be a seminormed linear space, S an MV-subalgebra of L and
?: S —> ? a measure.
(a) Then the semivariation \\?\\?: S^> [0, +00] defined by ||?||,>(·*¦)= &??{?(?(?)): t e
[0, x]} is a submeasure.
(b) If ? is ?-order continuous (i.e., for every monotone sequence (x„) in S with order
limit jceSwe have ????(.?„) = ?(?)), then ||?||,, is a ?-submeasure.
PROOF, (a) Let x, jc 1, x2 e 5 with jc ^ л 1 + *2 and t e [0, v]. By the Riesz decomposition
property there are f, e 5 with f, ^ jc, A = 1,2) and t = t\ + л», ? ^ '?- Therefore
?(?) = ?(?,) +?(?2) and ?(?(?)) ? ?(?(?\)) + ?(?(?2)) ? ||?|1,,(-*?) + ?|???/>(*:>)·
It follows that ||?||/;(?) ? \\?\\?(?])+ \\?\\,,(?2).
(b) Let jc, jc„ e S(n eN)and;c = ?^1, x„.Lett e S, t <;c.Then ||?||,,(? л^, *,¦) ^
?!=? II?II/.>(-*,') by (a). Moreover, t л ?!=? -*'' t '· Therefore
?(?@) = ???^???,?? л]Гх, J J ^ Дгт^ ]? ||?||/,№) = ]? ||?||,,(-?„)·
it follows ?^??^-?^?^? Им им*,,)· D
It follows from Example 2.1 that any positive real-valued (?-order continuous) measure
on L is a submeasure (?-submeasure).
The next proposition shows that a ?-submeasure can be extended from a ?-subalgebra
of L to L.

922
G. Barbieri and H. Weber
PROPOSITION 2.2. Let L be ?-complete and ? : S —>· [0, +oo] be ? ?-submeasure on an
MV-subalgebra of L. Then
/?*(*) = inf
defines ? ? -submeasure on L extending ?.
PROOF, ?* extends ? since ? is a ?-submeasure.
Let x,xn e L with jc ^ ??=\ xn- We show that ?*{?) ^ ?,^? /7*(лг„). We may assume
that ?*(??) < +oo for each n e N. Let ? > 0 and x,Lm e S with л:,, ^ ?^=\ Xn.m and
?~=? l(xn.m) < /j*(jc„) + ?/2" (и e N). Then дс <; ??„,= \ xn.m and therefore /?*(*) ^
?»°°..=?4^.-.)^?»"?4*?+?· ?
Let /j: L —>¦ [0, +oo] be a submeasure. It follows from Proposition 1.1.5 thatdtl(x, y) :=
?(???) defines a pseudometric on L and d,((;t oj.yo:)^ d,((.x, y) (?, ?, ? e L) where
о denotes one of the operations +, ?, л, v. Therefore the operations +, ?, л, ? are
uniformly continuous; in particular, ? н> ?' is uniformly continuous since ?' = 1?*.
DEFINITION 2.3. A submeasure ? is called ?-order continuous (order continuous) if
every decreasing sequence (net) with order limit 0 converges in (L, ??) to 0.
A submeasure ? is called exhaustive if every monotone sequence in (L, ??) is Cauchy.
The following proposition shows that any ?-order continuous submeasure is a
?-submeasure.
PROPOSITION 2.4. Let ?: L -> [0, +oo] be a submeasure. Then the following two
condition are equivalent
A) ? is a ?-submeasure.
B) Ifa„ I 0 and (a„)„6pj /s a Cauchy sequence, then a„ —> 0 in (L, ??).
PROOF. A) => B) Let (/>„) be a subsequence of (a„) with ^b„Ab„+\) ^ 2~" for и e N.
Since by Proposition 1.1.6(c) bnAbm+\ = ??=,? biAbi+\ for m ^ и and therefore bn =
?^„ bi ?^?+?, we obtain
+ i
пФп) ^ilMi'i+i) < ?2~' = 2_"
Therefore /j(/>„) —>¦ 0. Since (/>„)„6? is a subsequence of (a„),,6N and (a„)„6pj is a Cauchy
sequence, it follows that also (?„)„6? converges to 0 in (L, dn).
B) => A) Let (a„)„6N e L,s„ = ?-'=l a/(" e N) and s - sup„6Ni„. We shall show
that r](s) ^ ??=\ nian). We may assume that ??=\ ^(?,?) < +°°· Then (???»)«6? is
a Cauchy sequence since ?((sAs,l)A(sAs,n)) ? ??=? l(fli) ^ 0 for m > ? ^ +oo.

Measures on clans and on MV-algebras
923
MoreoversAsn i 0. Hence, ?^??,,) -> 0 by B). Since
II DC
?=? ?=?
and /?(???„) —>¦ 0, we obtain /j(s) ^ ?,?? ???)· ^
THEOREM 2.5. Lei L be a-complete and ? : L —>¦ [0, +oo] fee ? ?-submeasure.
(a) Г/геи (L,dn) is complete.
(b) If(an)nef$ converges to a in (L, ??), then (?„)„6? /?as ? subsequence (fe,i)„6N such
that ? (a A lim sup fe„) = 0.
PROOF. Let (а„)„6щ be a Cauchy sequence in (L,d,j) and (fe,,)„6N a subsequence
of (a„)„6N with /j(fe„Afe„+|) ^ 2'" for и е N. Let fe = limsupfe,,. Since bAb„ ^
?~„/>,?/7, + |, we obtain ij(*A*„) < ?,?„»?(*??*?+?) < 2""+'· Therefore (fe„)
converges to fe. It follows that also (a„)«6N converges to b. Moreover, if (а„)„ещ converges
to ? in (L,dn), then /j(aAfe) = 0. ?
If ?: L -> [0, +00] is a submeasure, then N(/j) := [a e L: /j(a) = 0} is an ideal in L.
THEOREM 2.6. Let L be ?-complete, ? : L -> [0, +oo] a ?-submeasure and ?: L —>
[0, +oo] ? ?-order continuous submeasure on L. Then ?(?) С ? (?) ??[?(?„) —>¦ 0 implies
?(?„) —>¦ 0/or any sequence (a„)„6pj /и L.
PROOF. («=) is obvious. (=>) Suppose that L contains a sequence (?„)„6^ such that
?(?„) —> 0 but (?(?„)),?6? doesn't converge to 0. Then there is a subsequence (fe„)„6N of
(йп)мем and an ? > Osuch thatX(fe„) ^ ? for every ? e N. By Theorem 2.5(b), (fe,i)„6N has
a subsequence (c„)„6pj such that /j(limsupc„) = 0. Let s = lim sup c„ and s„ = sup,^,( c,.
Then lim„^+00 ?(?„) = ?(?) = 0, a contradiction to k(s„) ^ ?(?„) ^ ? (n e N). D
THEOREM 2.7. Lei L fee ?-complete and ? and ? be ?-submeasures on L. If for every
? > 0 there is an element ?? e L such that ?(??) < ? and ?(?^) < ?, then there is an
element a e C(L) such that ?(?') = ?(?) = 0
PROOF. Let ? > 0. For every n e N we can choose x„ e L such that ?(?„) ^ ?/2" and
?(*;,) <; ?/2". Put ? = ??\x"- Then ?(?) ^ ??=\ ?(?,·) <? and ?(·?') < ?«> -»¦ °·
hence, ?(?') = 0. Therefore for every n e N there exists )¦„ e L such that ?()„) = 0 and
?(?'„) <: l/л. Put у := ?~=? >'«· Then ??0 < ?? М}'„) = 0 and /j(y') ^ /?(}',',) -»· 0,
hence, jjO") = 0· By Corollary 1.2.2, a := sup„6N ny e C(L). Then /j(a') ^ ?(?') = 0 and
?(?) =0 since ?(?) is aa-ideal. ?
3. Real-valued measures on MV-algebras
3.1. The space ba(L)
Let us denote by ba{L) the space of all bounded real-valued measures on L.

924
G. Barbieri and H. Weber
Obviously, ba(L) is a partially ordered linear space where the partial order and the
operations on ba{L) are defined pointwise.
The basic tool in this section is the following result.
THEOREM 3.1.1. ba{L) is a Dedekind complete Riesz space and
(i) if ?,? e ba{L) and ? e L, then
(? ? v)(x) = sup^(;t|) 4- v(xi): x\, xi e L\ x\ ? x'y, x\ + xi =¦*}',
(ii) if (??)??? is an increasing pointwise bounded net in ba(L), then (ду)у6г has a
supremum in ba{L) and (sup еГ ??)(?) = sup{My(.x): ? e ?] for ? e L.
PROOF. To simplify the calculation we assume that L is an order interval [0,e] of an
?-group (G, ?, ^) and ? + у = (? ? у) л е.
First observe that for ? \, x2 e L, the condition ? \ ^ x'2 and x\ + x2= ? means exactly
X\ ®X2 = X-
Let ?,? e ba(L), ? e L and ?(?) be defined by the right-hand side of (i). Then ? is
well-defined and bounded. We shall check that ? is a measure.
Let x, у e L with ? ^ y'. If x\,xi\ y\, y2 e L such that x\ ? xi=x and y\ ? y2 = y,
then by additivity of ? and v, we have ?(?\) + v(x2) + ?(}'?) + ?(}'?) = ? С* ? + Л) +
i>(*2 + уг) ^ ?(* 4- у). Hence, ?(?) + k(y) ? ?(? + у). Now let w\,w2 e L such
that w\ ? W2 = ? ? у. Then by the Riesz decomposition property there exist z,, tj e L
(i =1,2) such that w\ = z\ ? t\, w2 = ?? ? t2 and z\ ? ii = x, t\ ? ь = }'¦ Therefore
?(?? ?) 4- ?(??2) = ? (г ?) + ?(? ?) + ?(?2) 4- ?(?2) = ? (? ?) + ?(^2) + ?(?\) + v(t2). Since
? ? ? ?2 = ? and ?| ? ti = у, we have ?{??\) 4- у(иъ) ^ Цх) + ?)·)- So ? (л 4- }') ^
Цх) + Цу).
Obviously, ? is the supremum of ? and v, since it is an upper bound and any upper
bound of ? and ? is necessarily greater than ?.
By the rules valid in any partially ordered linear space we get ? л ? = -[(—?) ? (—?)]
in the sense that if one side exists then the other side exists and they are equal.
To prove the Dedekind completeness of ba(L) it suffices to show that any increasing
pointwise bounded net (?7)/6? has a supremum in ba(L). ТЫ ?(?) :=supyer ??(?) for
every ? e L. We shall prove ? a measure. We have
?(? +y) = $\}???(? + у) = sup (??(?) + ??(у)) = sup ??(?) + sup ?7(H
уеГ уеГ уеГ уеГ
= ?(*) + ??0
whenever ?, у e L and ? ^ y'. Since the pointwise supremum ? of (?7)/6? is a measure,
it is the supremum of (?7)/6? in ba(L). ?
PROPOSITION 3.1.2. For ? e ba{L) denote ?+ := ? ? 0, ?" := (-?) ? 0 and |?| :=
? ? (—?). Г/геи
(а) //?, ? e ba(L), then
(? ? i>)(;t) = inf^(jt|) 4-v(jt2): ??,?? e ?<; ¦*! ^ x->'> ?\ + -*2 =-*}·

Measures on clans and on MV-algebras
925
In particular \?\ ? |?| =0 (in symbols ? J_ ?) iff for every ? > 0 there exists ? e L
such that \?\(?) ^ ?, |v|(jc') ^ ?.
(b) ?+(?) = sup{M(y): Lb у ? ?}.
(c) M~(-x) = -inf{M(H: L3y^x}.
(d) \?\(?) = sup{M(jf|) - ?(?2): ?\, ?? e L; .V| ^ x'2; x\ +x2 = x}-
(e) ||?|| ^ \?\ ^2\\?\\ where \\?\\(?) := $??[\?(}·)\: Еэ у ? ?).
Proof, (a) By the rules valid in any Riesz space we have
(???)(?) = -[(-?)?(-?)](?)
= — sup{— ?(?\) — v(xi): x\,x2 e L\ x\ ^ x'2\ x\ + xi = x\
= ???{?(?\) + v(x2): x\, x2 & L\ x\ ^x'2\ x\ +x2 = x].
(b) and (d) follow from Theorem 3.1.1 (i). (c) follows from (a) and (e) from (d). D
The Jordan decomposition theorem valid in any Riesz space yields the following
decomposition theorem for measures. It generalizes a result contained in Butnariu and
KlementA993).
THEOREM 3.1.3 (Jordan decomposition). Let ? e ba(L). Then ?+, ?~ are the unique
measures in ba(L) such that ? = ?+ — ?~ and ?+ ??~ =0. Moreover, |?| =?+ + ?~¦
The Hahn decomposition theorem for measures on MV-algebras can be derived from the
Jordan decomposition theorem.
THEOREM 3.1.4 (Hahn decomposition). Let L be ?-complete and ?-.L -> ? ? ?-
order continuous measure. Then there exists a e C(L) such that ?(?) ^ 0 for ? ^ a and
?(?) ^ Ofor ? ^ a'. In particular, ?(?) = ????(?,) and ?(?') = 5???(?,).
PROOF. Apply Theorem 2.7 for ? = ?+ and ? = ?". For ? < a we have ?(?) =
?(? л а) = ?+(? л а) - ?"(? л ?) = -?"(? ? ?) ^ 0. Analogously ?(?) ^ 0 for
?^?'. ?
Other decompositions of measures can be obtained as band decompositions applying the
Riesz decomposition theorem to ba(L).
THEOREM 3.1.5 (Riesz decomposition). Let ? be a Dedekind complete Riesz space, F
a band in ? (i.e., an order closed solid Riesz subspace). Then ? is equal to the direct sum
ofF and F1- = {x e E: \x\ л |/| = Ofor every f e F].
We use the Riesz decomposition theorem to transfer the Hewitt-Yosida decomposition
theorem and the Lebesgue decomposition theorem in the setting of measures on MV-
algebras.
In any Dedekind complete Riesz space ? the band generated by a subset A of ? is A-1-1-.
We now describe the band generated by an element ? e ba(L).

926
G. Barbieri and H. Weber
NOTATION 3.1.6. For ?, ? eba(L), we write ? <^? iff \?\{?„) -> 0 implies ?(?„)-> 0;
equivalently, ? <g ? iff for every ? > 0 there exists <5 > 0 such that \?\(?) ^ <5 implies
\?(?)\??.
LEMMA 3.1.7. Let ?, ? eba(L); ?;«? ?/?^? J_ ? /wp/j ? = 0.
PROOF. Since ? J_ ?, for every ? e N there exists jc„ e L such that |?|(^„) ^ \/n and
???-?,',) <: 1J п. Therefore \?\{?'?) -> 0 and |?| (-*,,) -> 0. As ? « ? we get \v\(x'n) -> 0.
Now by additivity of |v|, we have M(l) = \v\(x„) + \v\(x'n) ^ 0, hence, |v|(l) =0 and
consequently ? = 0. ?
PROPOSITION 3.1.8. For ? e ba(L), В := {? e ba(L): ? «С ?] is the band generated
by ?.
PROOF. Obviously, В is an ideal. We will prove that ? is a band, in other words we will
prove that if В э va ^ 0, va \ v, then ? e B.
Let (jc„) be a sequence in L such that \?\(?,?) —>¦ 0. We show v(x„) —>· 0.
Let ? > 0. Then u(l) ^ vp(\) + ? for some index ?. Since vp{x„) -> 0, there exists
/o e N such that for every ? ? (jc, ) ^ ? ?' ^ /о; then for /' ^ /o we have ? (?,¦) = (? - ? ?) (*,¦) +
Vfi(Xj) ? (? - ??)(\) + vp(xj) ^ ? + ?. Soi>(x„)^0.
Of course ? e ?, so {?}-1"", the band generated by ?, is contained in B.
Now suppose ? e S, we shall prove ие(д)и.
By the Riesz decomposition theorem, ? = v\ + v2, where v\ e {?}-1 and V2 e {?} 
As | V| | ^ \v\, we have v\ « и « ? and v\ J_ ?. By Lemma 3.1.7 v\ =0 and ? = ?>2 e
Corollary 3.1.9. Lei ? eba(L) and В be a band in ba(L). Then ? e B1- iff ? e B,
?«? imply ? = 0.
We now are ready to prove the Lebesgue decomposition theorem.
THEOREM 3.1.10 (Lebesgue decomposition). Let ?, ? e ba(L). Then there are unique
measures ?, eba(L) (i = 1, 2) smc/? ?/??? ? =?\ + ??, ?\ <&? and ?? -L ?.
PROOF. Apply the Riesz decomposition theorem for F = {?}-1. ?
Analogously, applying the Riesz decomposition theorem for F = ca(L), where ca(L)
is the set of all real-valued ?-order continuous measures on L, which is a band in ba(L),
we get the Hewitt-Yosida decomposition theorem.
We need the following definition.
DEFINITION 3.1.11. We say that ? eba(L) is purely ???-?-order continuous if the zero
measure is the only ? e ca{L) such that ? <? ?.

Measures on clans and on MV-algebras
927
THEOREM 3.1.12 (Hewitt-Yosida decomposition). Let ? e ba(L). Then there are unique
measures ?, e ba(L) (/ = 1, 2) such that ? = ?\ + ??, ?? e ca(L) шЛ ?? риге/у пои
?-order continuous.
Now we leave this algebraical approach and we shall develop a topological approach in
order to study measures on MV-algebras.
3.2. Representation of real-valued measures
In this section we show that Butnariu's integral representation theorem (see Butnariu and
Klement A993, 6.2)) is an easy consequence of the Hahn decomposition theorem.
Theorem 3.2.1. Let ? be а Тх-тЬе of fuzzy sets on ? and A = C(T) the ?-algebra of
crisp sets in T. Then the restriction ?: ? н> ?| д defines a linear isome try from the Banach
space (ca(T), || ||oo) onto the Banach space (ca(A), \\ \\x)- For ? e ca(T) and f e ? we
have ?(/) = / /??, where ? = ?\?.
PROOF. Obviously, ? is linear and, by Theorem 3.1.4, norm preserving, in particular
injective. The surjectivity of ? follows from Proposition 1.3.1: Let ? e ca(A). Define
v(f) := f f dv for / e CX(A). ? is ?-order continuous by Lebesgue's convergence
theorem. By Proposition 1.3.1, ? = ?|? is well-defined and ?(?) = ?. D
4. Uniform MV-algebras
4.1. The lattice CIAA(L)
We denote by CUA{L) the system of all uniformities on L which makes + and ' (and
therefore ?, л, v) uniformly continuous. If и e CUA(L), we call (L, u) a uniform MV-
algebra. If ? : L -> [0, +oo] is a submeasure, then (L,dn) is a uniform MV-algebra, see
Section 3.
If и e CUA(L), then the system F = и@) of 0-neighbourhood in (L,u) has the
following property
VFeF3GeFV;ceLVy,zeG, xsiy + z => x e F. (*)
We denote by TNA{L) the set of all filters F on L with property (*). If F and G are
taken according to (*), then s(G) := {x e L: By e G ? ^ }'} is a solid subset of L with
G cs(G)c F. So, equivalently, we can define TNA(L) as the set of filters F on L such
that F has a basis of solid sets and F + F e F for each F e?.
The symbol TJ\fA(L) is to remind one of FN-topologies defined on Boolean algebras.
The next theorem shows in particular that any и е CUA(L) is uniquely determined by
its 0-neighbourhood system.

928
G. Barbieri and H. Weber
THEOREM 4.1.1. (а) ^:ми и@) defines an order isomorphism from CUA(L) onto
TNA(L), where и@) denotes the ^-neighbourhoodsystem in (L, u).
(b) For и e CUA(L) the sets FA := {(x, y) e L2: ? Ay e F), F e и@), form a basis
ofu.
PROOF, (i) Obviously ? is well-defined, i.e., и@) е FNA{L) for и е CUA(L) and ? is
monotone.
(ii) We show that ? (?) с V(i>) implies и с и for и, и e CUA(L). Let ?/ e и. We may
assume that ?/ is symmetric. Choose ?/' e и with U' + U' cU and symmetric V, W e и
with V@) с U'@) and WvlVAWc V. We show that W С ?/ ? U. Let (.?, у) е W
and г = ? ? y. Then (zAjc,0) = ((у,*) v (x,x))A(x,x) e (W ? 1V)AW С V, hence
??* e V@)c i/'@)and (z,Jt) = (?:,?:) + (???:,0) e U' + U' С ?/. Analogously we have
(z, y) e i/. It follows that (x, y) eU oU.
(iii) V is surjective. Let F e TAfA(L). Then the sets FA, F e F, form a basis of filter u.
One easily sees using Proposition 1.1.5 that и is a uniformity and +, ?, л, ? are uniformly
continuous. Hence, ? e CUA(L). Obviously ?(?) = F. ?
COROLLARY 4.1.2, Any homomorphism between MV-algebras continuous at 0 is
uniformly continuous.
Proposition 4.1.3. TNA(L) is a distributive complete lattice. If?\, F2 e TAiA{L),
then the sets F\ ? F2 and Ft + F2 (F/ e F,) form a basis of?\ ? F2 and ?\ ? F2,
respectively.
PROOF. Let F„, a e A, be a family in TNA(L). Then the system of finite intersection of
sets of UaeA ?a is a basis of a filter F e TAfA(L). Therefore F is the supremum of the
family (?a)aeA in TNA(L). Since TAfA(L) contains a smallest element, namely {0, L),
it follows that TMA{L) is a complete lattice.
We now prove the statement about F| л F2. Let F, e F,-. Choose G, e F, such that for
every ? e L and for every y,, zi e G,, the inequality ? ^ y,- + Zi implies j: e F,. Using the
decomposition property Proposition 1.1,4 one sees that if y, ? e G\ +G2 and Lsx^y+z
then ? e F\ + F2. Therefore the filter F generated by the sets F\ + F> (F, e F,) belongs
to TNA(L). Since every filter of TNA{L) coarser than F| and Ft is coarser than F, we
obtain F = F, aF2.
It easily follows from the description of F| ? Ft and F| л Ft that TAfA(L) is
distributive. ?
COROLLARY 4.1.4. ?UA(L) is a distributive complete lattice.
We have already observed that any submeasure on L generates a uniformity of CUA(L).
We now show that, vice versa, any uniformity of CUA(L) is generated by a family of
submeasures.
LEMMA 4.1.5. Let (S, +) be a semigroup with 0 and let С := (?/„)n€N be a family of
subsets of S such that U\ := S, 0 e U„ and U„+\ + U,l+\ + Ull+\ С U„ for every ? e N.

Measures on clans and on MV-algebras
929
Let g : S -> R defined by
g(x) = 0 ifxef^iU.-.neN},
g(x) = 2~" ifxeU„\U„+i.
Let ? : S -> R defined by
?
?(?) = inf
]Pg(Zi): z\ +zi-\ l·Zp
11 = 1
Then ? is subadditive, ? ^ 0 and the system {x ? S: ?(?) < ?}, ? > 0, generates the same
filter as (i/„)„6N-
In the case of S being a group, the proof can be found in Warner A989); without
modifications it works also in this case.
THEOREM 4.1.6. Every uniformity и ? CUA(L) is generated by a system (??)??? of
submeasures defined on L, i.e., the sets {(x, y) ? L2: ??(???) < ?} (a ? ?, ? > 0)form a
subbasis for u.
PROOF. Since for any и ? OAA(L) there is a family (ua)aeA of uniformities of OAA{L)
with countable basis such that и = supaeAua, we may assume that already и has a
countable basis. Then there is a sequence U\, U2, ¦ ¦ ¦ of solid O-neighbourhood in (L, u)
with U,i+\ + U,i+\ + ?/,,+ ? с U„ (n e N) and U\ = L. Define ? as in Lemma 4.1.5. Then
? is subadditive and ?@) = 0. Moreover, it follows from the Riesz decomposition property
that ? is monotone. Therefore ? is a submeasure and [x e L: ?(?) < ?], ? > 0, is a basis
for the O-neighbourhood system of (L, u), i.e., ? generates и see Theorem 4.1.1. D
PROPOSITION 4.1.7. Let и ? CUA(L) be Hausdorff and (xa)aeA be an increasing
(decreasing) net converging to ? in (L, u). Then ? = supxa (x = infxa).
PROOF. We consider only the case that (ха)аел is increasing. Then ((xa ? х)Ах)аел is
an increasing net converging to (? ? ?) Ax = 0. Therefore (xa ? ?) Ax = 0, hence, xa ^ ?
for any a ? A; i.e., ? is an upper bound of (ха)аел- Let now у be an arbitrary upper bound
of (Xa)aeA- Then ? = \\mxa = Yimxa л у =х л у, hence, ? ^ у. ?
We now introduce some important properties for uniformities of CUA(L).
DEFINITION 4.1.8. A uniformity и е CllA(L) is called ?-order continuous (order
continuous) if every decreasing sequence (net) with order limit 0 converges in (L, u) to 0.
A uniformity и is called exhaustive if every monotone sequence in (L, u) is Cauchy.
It is easy to see that a uniformity и of CUA(L) is order continuous (?-order continuous)
iff order convergence implies topological convergence in (L, u) for any net (sequence)
in L; and that и is exhaustive iff every monotone net in (L, u) is Cauchy.

930
G. Barbieri and H. Weber
The following proposition shows that, for L being a Boolean algebra, и е CUA(L) is
exhaustive iff every disjoint sequence converges to 0 in (L, u).
PROPOSITION 4.1.9. Let и е ?UA(L). и is exhaustive iff every sequence (x„)„m in L
with the property ??=\ ?; ^ x'n+i(n € N) converges to 0 in (L, u).
PROOF. (=>) If (хп)пец is a sequence in L with s„ := ?'?=\ Xi ^ x'n+i (n e N), then
*„+? =?„+|??„ by Proposition 1.1.6. Therefore (jc„)„6n converges to 0 since the
increasing sequence Cs„)«sN is Cauchy.
(«=) Suppose that (a„)„6N is an increasing sequence in L which is not Cauchy with
respect to u. Then (a„)„6pj has a subsequence (fe,i)rt6N such that (Ь„+\АЬ„)„ещ does
not converge to 0 in (L, u). Let bo = 0 and jc„ = b„Abn-\. Since ?"_| -*; ^ -*,',_,_, by
Proposition 1.1.6, the sequence converges to 0 by assumption, a contradiction.
If (a„)„6pj is a decreasing sequence, then (a;'()„6pj is increasing, therefore Cauchy in
(L,m) as we have already proved. It follows that (а„)„6щ is Cauchy since a„Aam =
a'nAa'm. О
The property formulated in Proposition 4.1.9 is equivalent to Гзс-disjointness in
Butnariu and Klement A993, p. 37).
4.2. Decomposition of order continuous uniform MV-algebras
Let L\, L2 be uniform MV-algebras. A topological isomorphism ?: L\ —> L2 is an
isomorphism such that ? and ?'] are continuous. By Corollary 4.1.2, ? is then also a
uniform isomorphism, i.e., ? and ?'' are uniformly continuous.
PROPOSITION 4.2.1. Let L be complete, D a disjoint subset of С\L) with sup D = 1 and
и an order continuous uniformity of ?UA(L). Let Ud = и|[о.</] be the relative topology
induced by и on [0, d]. Then ?:?\-^>- (? A d)ljeo defines a topological isomorphism from
(L,u)onto(Y\deD([0,d],ud).
PROOF. By Proposition 1.5.3 ? is an isomorphism, ? is continuous since ? н> х л d is
continuous for any d e D. We show that ф'] is continuous at 0 (and therefore continuous
by Corollary 4.1.2). Let U and Uq be solid 0-neighbourhood in (L, u) with Uq v Uq С ?/.
Let J" be the system of all finite subsets of D directed by inclusion and sf '¦= sup(Z) \ F)
for F e ?. Since sf 4- 0 and и is order continuous, jf0 e ?/o f°r some Fo e .F. Let и = | Fq\
be the cardinality of Fo and V a 0-neighbourhood in (L, и) such that ? ? ? · · · ? ?„ e Uq if
zi,...,z„e V.Then W := {(^) e П</6о[°' rfl· x<i ? V for d e F0) is a 0-neighbourhood
in П([0'^]' "</)· For any (д-^) e W we have 0~'((¦*</)) = suprf6D^/ ??/. ?
PROPOSITION 4.2.2. For и e CUA{L) the intersection N(u) of all 0-neighbourhood in
(L, u) is a closed ideal in L. If L is complete and и order continuous, then N(u) = [0, a]
for some a e C(L).

Measures on clans and on MV-algebras
931
PROOF. Obviously, N(u) is an ideal. N(u) is closed since any point of a uniform space
has a neighbourhood basis of closed sets. To prove the second statement, consider N(u)
as an upwards directed set. Since и is order continuous this directed set converges to
a := sup N(u). Since N(u) is closed, we obtain a e N(u), hence N(u) = [0, a]. Since
2a e N(u) and therefore 2a = a, we have a e C(L) by Proposition 1.2.1. D
Тн EOR EM 4.2.3. Let L be complete.
(a) Then CUA(L) contains at most one order continuous Hausdorff uniformity.
(b) If и e CUA(L) and и is Hausdorff and order continuous, then (L, u) is (topol-
ogically and algebraically) isomorphic to a product ?\??/\(^?, m) of metrizable
MV-algebras.
PROOF. Let ? be the finest order continuous uniformity of CUA(L), и be an arbitrary
Hausdorff order continuous uniformity of CUA(L) and Л be a set of submeasures
generating u. For ? e ? there is by Proposition 4.2.2 an aA e C(L) with ?(?) = [0, ??].
Since и is Hausdorff, we have ????€? aA = 0 and 5ирЯбЛ ?· = 1. Let (<?0л.6л be a disjoint
family in C(L) with d\ < ?'? for any ? e Л. Let u,_ be the uniformity generated by ? on
Lx := [0, dx]. For any order continuous submeasure ? on L we have N(/?k,.) Э {0} =
?(?-\?*?)- Therefore, by Theorem 2.6, the d,,-uniformity is coarser than их. Since this
holds for any order continuous submeasure ? on L, we obtain that v\lx С их. Obviously
v\Lk D u\l-k Dux. Therefore ? |^? = u\l} = их. It follows with Proposition 4.2.1 that и = ?
and (L, u) is isomorphic to ????^?- их). П
THEOREM 4.2.4. Let L be complete. Let u,v e CUA{L) order continuous. Then N(v)
С N(u) iff и С и.
PROOF. (<=) Is obvious. (=>) By Proposition 4.2.2 ? = supN(w) e C(L). The
uniformities ?|[?,?'] and и|[о.й'] are order continuous and Hausdorff and therefore equal by
Theorem 4.2.3(a). ?|[?.?] is the trivial uniformity and therefore coarser than u|[o.«]. It follows
from Proposition 4.2.1 that и с v. ?
Theorem 4.2.5. LetueCUA(L).
(a) If L is complete as a lattice and и is order continuous, then и is exhaustive and
(L, u) is complete as a uniform space.
(b) If и is exhaustive and (L, u) is complete as a uniform space and Hausdorff, then L
is complete as a lattice and и is order continuous.
PROOF, (a) Obviously, order continuity implies exhaustivity.
Let a = supN(u). Then (L,u) is isomorphic to ([0, а], и|[0.л]) х ([0> a'L и|[0.л'))· и
induces on [0, a] the trivial uniformity and on [0, a'] a Hausdorff uniformity. Therefore, by
Theorem 4.2.3(b), ([0, а'], и\[0м']) is isomorphic to a product Плел^' и^ of metrizable
MV-algebras. Since и is order continuous and L is complete, uA is order continuous and Lx
is complete (as a lattice) and therefore (Lx, их) is complete uniform space by Theorem 2.5.
Since a product of complete uniform spaces is complete, it follows that (L, u) is complete.

932
G. Barbieri and H. Weber
(b) Let (xa) be an increasing net. Then (xa) is a Cauchy net in (L,u) since и is
exhaustive and therefore has a limit ? in (L, u) since (L, и) is complete. By
Proposition 4.1.7 ? = sup jc„ . It follows that L is a complete lattice and и is order continuous. D
THEOREM 4.2.6. Let и be an exhaustive Hausdorff uniformit}' of CUA(L) such that
(L,u) is complete.
(a) Then the isomorphisms of Theorem 1.5.4 (a)-(c) are also topological isomorphisms.
Here [0, af], [0, ax], [0, c] are endowed with the uniformity induced by u; ?„ and
Coo with the real Euclidean uniformity and the products have the product uniformity.
(b) [0, ay] is totally disconnected and compact, [0, ax] is arcwise connected and
compact, [0, c] is arcwise connected and not compact ifc^O.
PROOF, (a) follows from Proposition 4.2.1 and the uniqueness of the Hausdorff order
continuous uniformity of CUA(C„), neNU {oo} \ {1}.
(b) [0, aj] is topologically isomorphic to the totally disconnected and compact space
TiaeA^-'ia- [0>азс] is topologically isomorphic to the arcwise connected and compact
space ??,. Suppose с ? 0. The centre С of [0, c] is an atomless Boolean algebra, and
closed in (L, u) since С = {? e [0, с]: ? =2?], hence, complete (as a uniform space).
Therefore there is by Volkmer and Weber A983, Satz 1.4) a continuous map y.I^-C
defined on the real closed unit interval / with y@) = 0 and ? (I) = с. For any a e [0, c],
a(t) := y(t) A a defines then a continuous map <*:/—>· [0, c] with <*@) =0and a(\) = a.
Therefore [0, c] is arcwise connected.
Since С is connected, С cannot be compact (see Kaplansky A947)). Therefore [0, c] is
not compact. ?
Under the assumptions of Theorem 4.2.6, L is compact or totally disconnected or
connected, respectively, iff с = 0 or с = ax — 0 or at = 0. This leads to the following
characterization of compact, totally disconnected or connected MV-algebras.
COROLLARY 4.2.7. Let и be an exhaustive Hausdorff uniformity of CUA(L) such that
(L,u) is complete.
(a) Then L is atomic iff (L,u) is totally disconnected iff (L,u) is isomorphic and
topologically isomorphic to a product Y[aeA C„a for a e A and na e N \ {1}. In
particular if (L, u) is totally disconnected, then (L, u) is compact.
(b) L is atomless iff {L, u) is arcwise connected iff (L, u) is connected.
COROLLARY 4.2.8. Let ? a Hausdorff topology definedon L which makes continuous the
operations + and' ofL. Then (L, ?) is compact iff(L, ?) is isomorphic and topologically
isomorphic to a product Cf^ ? Па6д?„ц for a e A and na e N.
Proof. (=>) Since ? is compact, ? is induced by a uniformity u, and + and ' are uniformly
continuous with respect to u. Therefore и е CUA(L). Moreover, и is exhaustive by Weber
A991/1993, 6.1) and complete. Therefore и satisfies the assumptions of Theorem 4.2.6,
and so L is isomorphic to [0, c] ? ?^ ? П„6д?„а, and с = 0 since и is compact. D

Measures on clans and on MV-algebras
933
4.3. The lattice CUA(L, w)
We will study uniformities as well as measures on L passing to a suitable uniform
completion of L. This is motivated by the fact that the used uniform completion is easier
to handle since it is complete as a lattice and its uniformity is order continuous.
PROPOSITION 4.3.1. Let w e CUA(L) be Hausdorff. Then (L,w) is a dense MV-
subalgebra of a Hausdorff complete uniform MV-algebra (L, w).
Proof. Let (L,w) be the uniform completion of (L,w). Then the addition and the
complementation ' of L have a continuous extension on L. So (L, w) becomes a uniform
MV-algebra. ?
(L, w) of Proposition 4.3.1 is called the completion of (L, w).
PROPOSITION 4.3.2. Let L be a dense subspace of a uniform MV-algebra (L, w). Then
w is exhaustive iff w\t is exhaustive.
PROOF. (=>) is obvious. («=) By Theorem 4.1.6, it is enough to prove that any continuous
submeasure ?^on (L, w) is exhaustive if ?\? is exhaustive. Let (?,?),?6? be an increasing
sequence in L, ? > 0 and b„ e L with /?(?„?/?„) ^ ?1~". Then c„ := sup,-^„ b\ defines
an increasing sequence in L and is therefore a Cauchy sequence with respect to d,t.
Moreover, /j(c„Aa„) = /j(sup,<„ bjA supl<:„ ?,) ^ ??'=? ?????,) ? ?. It follows that
(an) is a Cauchy sequence with respect to dn. ?
From Proposition 4.3.2 and Theorem 4.2.5 follows:
COROLLARY 4.3.3. Let w e ?UA{L) be Hausdorff and exhaustive and (L, w) be the
completion of(L, w). Then (L, $5) is a complete lattice and w is order continuous.
For w e CUA(L) we put CUA(L, w) := {u e CUA(L): и с w].
The following proposition can easily be verified using Theorem 4.1.1.
PROPOSITION 4.3.4. Let w e CUA{L). Denote by a the equivalence class of L : =
L/N(w) containing a and let U := {a: a e U}for U с L. Then, for и е ?UA(L, w), the
sets U(U e u@))form the 0-neighbourhood basis of a uniformityие CUA(L). ?:«и«
defines an order isomorphism from CUA(L, w) onto CUA(L, w).
THEOREM 4.3.5. Let w be an exhaustive uniformity ofCUA(L), (L, w) the completion
of {L,w) := (L,w)/N(w). Then r:u н> u\i defines an order isomorphism from
?UA(L, w) onto CUA(L, w). s :u \-^> supN(u) and s о г~' о q define a dual order
isomorphism from CUA(L, w) onto C(L) and from CUA(L, w) onto C(L), respectively.
PROOF. The statement about r follows from the fact that for и е CUA(L, w) the set of all
continuous submeasures ?: (L, w) —> [0, +oo] such that the restriction ?\? is continuous
with respect to и generates the unique uniformity и e ?UA(L, w) with r(u) = u.

934
G. Barbieri and H. Weber
It follows with Proposition 4.2.2 and Theorem 4.2.4 that s is a dual isomorphism. With
Proposition 4.3.4 one obtains that ior'o^ isa dual order isomorphism. D
Remark 4.3.6. Let w e ?UA(L) be exhaustive and (L, w) be the uniform completion
of (L, w). Then L is complete as a lattice, C(L) is a complete Boolean algebra and
OAA{L, w) is the lattice of all order continuous uniformities of CUA(L).
PROOF. The first two statements follow by Theorem 4.2.5 and Proposition 1.2.3. For the
last statement, let и be order continuous and a = supN(u). If ? e CUA(L, w) such that
s(v) = a, then we have, since N(u) = N(v), by Theorem 4.2.4 и = v. D
Remark 4.3.7. By the notation of Theorem 4.3.5, let ф(и) be the FN-topology induced
by и on C(L). Then и н> ф(и) is an order isomorphism from CUA{L, w) onto the space
of all order continuous FN-topologies on C(L).
Corollary 4.3.8. If w is an exhaustive lattice uniformity ofCUA(L), then CUA(L, w)
is a complete Boolean algebra.
COROLLARY 4.3.9. By the notation of Theorem 4.3.5 «н>й:= с(Г> " an or^er
isomorphism between ?UA(L, w) and ?UA(C(L), w|C(?,)·
4.4. The uniformity generated by a measure
In this section let ? : L -> ? be a measure with values in a locally convex linear space.
This measure raises a uniformity in a natural way.
Definition 4.4.1. With the notation as in Example 2.1. Let pa, a e A, be a family
of seminorms generating E. The uniformity generated by the submeasures (||?||/,11)?6? is
called ?-uniformity.
The ?-uniformity can also be described without using seminorms on E. In fact the sets
U* := {x e L: ?([0,.?]) С U], where U is a O-neighbourhood in ?, form a filter base of
TNA(L). The ?-uniformity is the uniformity which corresponds to this filter according
to Theorem 4.1.1. This description of the ?-uniformity immediately yields that ?(?) = ?
(?-uniformity).
PROPOSITION 4.4.2. For и е CUA(L) the following conditions are equivalent:
A) ? : (L, u) —> ? is continuous at 0.
B) ? : (L, u) —> ? is uniformly continuous.
C) ?-uniformity is weaker than u.
PROOF. Denote by ? the ?-uniformity. Obviously, B) is equivalent to C), i.e., to и с и;
and A) is equivalent to u@) С i/@).But и@) С и@) iff и С и by Theorem 4.1.1. D

Measures on clans and on MV-algebras
935
By Proposition 4.4.2, the ?-uniformity is the weakest uniformity of ?UA(L) which
makes ? uniformly continuous. In Barbieri and Weber A998, Section 3.1) it is observed
that the ?-uniformity is also the weakest lattice uniformity that makes ? uniformly
continuous. Therefore Definition 4.4.1 is in accordance with the definition of the
?-uniformity given by Weber A996).
DEFINITION 4.4.3. (a) ? is order continuous if xa I 0 implies ?(??) -> 0 for each net
(xa) in L.
(b) ? is exhaustive iff {?{?„))„?? is a Cauchy sequence for each monotone sequence
(*„) in L.
PROPOSITION 4.4.4. (a) ? is exhaustive (?-order continuous, order continuous) iff the
?-uniformity is so.
(b) ? is exhaustive iff for every sequence (x,,),ieN in L with J2"= \ ¦*/ ^ x'll+i (n e N) the
sequence (?(?,?))??? converges to 0.
PROOF, (a) («=) Follows from the uniform continuity of ?.
(b) (=>) Let (jc„)„6n be a sequence in L with s„ := ?"=? ¦*/ ^ x'„+\(n e ^)· Then
?(??+\) = ?(?„+|) — ?(?„) —>¦ 0 since (?(?,())„6? is a Cauchy sequence.
Suppose that ?(?„) —>¦ 0 whenever ?"=] Xi ^ x'n+\(n e N). We show that the
?-uniformity is exhaustive using Proposition 4.1.9.
Suppose that there is a sequence (>?)„6? in L such that ?"_| >? ^ 3-',',+|(" e ^) anc'
(yn)neN doesn't converge to 0 with respect to the ?-uniformity. Then there is a sequence
(¦*n);ieN such that x„ ? y,,(n e N) and (мСхя))леМ does not converge to 0, a contradiction
since ?"=\ *? < ?"=? У· < ?,+? < <?+? ·
The fact that the ?-uniformity is (a)-order continuous if ? is (a)-order continuous can
be found in Weber A996, 3.5). ?
PROPOSITION 4.4.5. Let ?-.L —>· ? be continuous with respect to a uniformity w e
CUA(L) and let и be the ?-uniformity.
(a) If ? is defined on L := L/N(w) by ?(?) = ? (?) for a e a e L, then ti defined as in
Proposition 4.3.4 is the ?-uniformity.
(b) If (L, w) is a dense subspace of a uniform MV-algebra (L, w) and ? = ?|?. is the
restriction of a continuous measure ?: (L, w) —> E, then и is the restriction of the
?-uniformity.
PROOF. We may assume that ? is a Banach space.
(a) follows from the fact that ||?||(?) = ||?||(<3) for a e a e L, where ||?|| denotes the
semivariation of ? (cf. Example 2.1).
(b) It is enough to show that ||?||(?) = ||?||(?) for ? e L. Obviously, ||?||(?) ^ II? II (a):
let jc e L andjc ^ a. Let (xa) be a net in L converging to ? in (L, w). ThenJcaAa converges
to jc, too. Therefore
\?(?)\ = \\?\\?(?? ла)\ ?. ||?||(?).
It follows ||?||(?) $5 ||?||(?).
D

936
G. Barbieri and H. Weber
NOTATION 4.4.6. Let u, ? e CUA(L) and ?, ? measures defined on L. We write и ±v
iff и Л и is the trivial uniformity. If и = ?-uniformity, we write also ? J_ ? instead of
и J_ u; if moreover ? = ?-uniformity, we write ? J_ ? instead of и 1 ?. ? « и means that
?-uniformity is weaker than v, or equivalently by Theorem 4.1.1, ? is continuous with
respect to v. If ? = ?-uniformity, we write ?«?; instead of ? <<C ? and we say that ? is
v-continuous.
5. Measures on MV-algebras with values in a locally convex space
In this section, let (?, ?) be a complete Hausdorff locally convex linear space.
5.1. Representation of measures
In Theorem 5.1.3 we give an isomorphism between the space of all ?-valued exhaustive
measures on an MV-algebra and the space of all ?-valued order continuous measures on
a complete Boolean algebra. This isomorphism allows us to transfer results known for
measures on Boolean algebras to the case of measures on MV-algebras.
PROPOSITION 5.1.1. Let Abe an algebra of sets.
(a) Then ? н>- ??, T?(f) = J f ?? defines a linear isomorphism between ba(A, E),
the space of all ?-valued bounded finitely additive measures defined on A and the
space L((Coo(A), || Hoc), E) of all ?-valued linear continuous maps defined on
Coo{A).
(b) A measure ? eba (A, E) is (?-) order continuous iff ?? : Cx (A) —>¦ ? is (?-) order
continuous.
PROOF, We only prove (b) in the order continuous case. Identifying sets with their
characteristic functions, A becomes a regularly embedded sublattice of Coo(A) (cf.
Theorem 1,3,3 and Proposition 1.2.3) and ? is the restriction of ?? to A. This immediately
yields the proof of («=),
(=>) Since ? can be embedded in a product of Banach spaces, we may assume that ?
is a Banach space. Let ? e ba(A, E) be order continuous. Then the semivariation ||?|| of
? is by Proposition 4.4.4 order continuous, too. Let (fa)aeA be a net in CX(A) with
fa I 0. Then, for any a e A, there is a sequence (/г„.„)„6м in S(A) with han \. fa
and ||/г„,„ — /?||?? < l/n. With the order defined on ? := ? ? ? componentwise we
have /г/3 ; 0. Let ? > 0. Then {hp ^ ?) ; 0, hence ||?||({/?/3 ^ ?}) -> 0. Therefore (hp)
converges in measure to 0. By the dominated convergence theorem, ^??? —> 0. It
follows that / /„ ?? -> 0. ?
THEOREM 5.1.2. Let L be ?-complete. Then ?:?? Mlc(L) defines a linear
isomorphism from the space ca(L, E) of all ?-order continuous ?-valued measures on L onto
ca(C(L), ?), ? is order continuous iff ?\c^L) is order continuous. Moreover, ???(?,) =
со?@\L)) for ? e ca(L, E).

Measures on clans and on MV-algebras
937
PROOF. Since C(L) is regular embedded in L by Proposition 1.2.3, the restriction r (?) :=
/J-lc(L) belongs to ca(C(L), E) for any ? e ca{C{L), E). Obviously r is linear.
r is injective. Let ? e ca(L, E) with ?(?) = 0. Then for any continuous linear
functional x' on ? we have ?' ? ?|(;(?.) = 0 and therefore .?' ? ? = 0 by Theorem 3.1.4. It
follows that ? = 0.
r is surjective. By Theorem 1.3.3 we may assume that L is a Гзс-clan of fuzzy
sets regular embedded in CX(A) for some algebra A of sets and that A = C(L).
If ? e ca(C(L), E), then the integral operator Tv defined in Proposition 5.1.1 is ?-
order continuous by Proposition 5.1.1. Therefore ? := Tv\l e ca(L, E) and ?(?) = v.
Proposition 5.1.1 (b) shows that ? is order continuous iff ? is order continuous.
To prove the last statement, observe that for ? e ca(L, E) and ? = ?(?) we have
со ?(?) = [f f dv: f e S(A)} (cf. Diestel and Uhl, 1979, proof of Lemma IX. 1.3, p. 263).
Therefore ?(?) с ?(?,) с {f f dv: f e CX(A)} С {j f dv: f e S(A)} = ???(?).
Hence, ???{?) = ???(?). ?
THEOREM 5.1.3. Let w be an exhaustive uniformity of CUA(L) and (L,w) be the
completion of(L, w)/N(w).
(a) Then for any continuous measure ? : (L, w) —> ? the function ?: L —>¦ ? defined by
?(?) = ?(?), ? e x e L, has a unique continuous extension ?: (L, w) —> E. Denote
by ?: C(L) -+ ? the restriction of ? to the centre C(L) of L. Then ?(?,) = fi(L)
???~???{?) = cop.(C(L)).
(b) ди/i defines an isomorphism from the linear space of all continuous measures
? : (L, w) —> ? onto the linear space of all ?-valued order continuous measures on
C{L).
(c) If F is a complete locally convex Hausdorff linear space and ? : (L, w) —> ? and
?: (L, w) —> F are continuous measures, we have ?«?; iff ? «? and ? J_ ? iff
? J_ у. Moreover, if ? : ? -^> F is a continuous linear map and ? = ? ? ?, then
? = ? ? ?, ? = ? ? ? and ? = ? ? ?.
PROOF, (a) ? has a unique continuous extension ? to (L, w) since ? is uniformly
continuous with respect to w. Then the range of ? is dense in the range of ?, therefore
?(?,) = ?(?,) = fi(L) and со?(?,) = cofi(L). Now observe that, by Corollary 4.3.3,
L is a complete lattice and w is order continuous, therefore ? is order continuous and
cofL(C(L)) = ~???(?) by Theorem 5.1.2.
(b) Obviously, ? н> ? (? н> ?) defines a linear 15отофЬ15т between the space of
all w-continuous measures on L (of all ю-continuous measures on L) and w-continuous
measures on L, respectively. To show that ? t->- ? defines a linear isomorphism from the
space of all w-continuous measures on L and the space of all order continuous measures
defined on C{L) use Theorem 5.1.2 and the fact that a measure on L is order continuous
iff it is ю-continuous. If ?: L —>¦ ? is an order continuous measure, then the ?-uniformity
is order continuous by Proposition 4.4.4 and therefore by Remark 4.3.6 weaker than w,
i.e., ? is ю-continuous, see Proposition 4.4.2. As already observed in the proof of (a), any
ю-continuous measure is order continuous since w is order continuous.
(c) The first statement follows from Corollary 4.3.9 and from the fact that if и is the
?-uniformity, then й is the ?-uniformity. To show the last statement, we have to show that

938
G. Barbieri and H. Weber
the correspondent elements of C(L) with respect to the isomorphisms of Theorem 4.3.5
are equal. Thanks to Proposition 4.4.5 it is sufficient to prove that for a e C(L) we have
?(?) = 0 for every ? e L with ? < a iff ?(?) = 0 for every ? e C{L) with ? < a. To
prove («=), we put v{z) := ?(? ? ?) for ? e L. From ? = 0, we have ? = 0 and hence
? = 0, i.e., ?(*) = 0 for every ? e L with jc < a.
The second statement of (c) is obvious. ?
5.2. Decomposition theorems
With the aid of Theorem 5.1.3 we now transfer a general decomposition theorem for
measures from the Boolean case to the case of MV-algebras.
THEOREM 5.2.1. Let ? be exhaustive and и е CUA(L). Then there are unique measures
? ?, ??: L —>· ? such that ? = ? ? + ??, ? \ <? и and ?? J_ и. Moreover, the ?-uniformity
is the supremum of the ? \ -uniformity and the ??-uniformity.
PROOF. Existence. Let w be the finest exhaustive uniformity of ?UA(L), i.e., the
supremum of all exhaustive uniformities of CUA(L). Replacing и by и л w, we may
assume that и is coarser than w. We use the notation of Theorem 5.1.3. By Weber B002,
9.2) ? has a decomposition of the form ? = ? ? + ?2, where ? ? and ?2 are ?-valued
//-continuous measures on C(L)X| « и, Аг 1 и and ? is the supremum of the ? ?-uniformity and
the X2-uniformity. By Theorem 5.1.3 there are w-continuous measures ?, : L —>· ? with
?,- = ?,· decomposing ? according to Theorem 5.2.1.
Uniqueness. If ? = ? ? + ?? = ?\ + ?>2 are two measures decompositions with ? ? <g и,
v\ <K и and ?2 -L и, ^2 -L и, then ? := ? ? - v\ = ?>2 - M2 J- " and ? <? и, hence, ? = 0. D
As in the Boolean case, different choices of и yield different decomposition theorems.
If и is the yo-uniformity for some measure p, then Theorem 5.2.1 becomes the Lebesgue
decomposition theorem.
If we take for и the supremum of all ?-order continuous uniformities of CUA(L), then
Theorem 5.2.1 yields the Hewitt-Yosida decomposition theorem.
THEOREM 5.2.2 (Hewitt-Yosida decomposition). If ? is exhaustive, then there are unique
measures ? \, ?2: L —>¦ ? such that ? ? is ?-order continuous and ?? is purely поп ? -order
continuous.
Here ? is called (in accordance with Definition 3.1.11) purely non-?-order continuous
if the zero measure is the only ?-order continuous measure v: L —>¦ ? with ? <5? ?. In
this case there is also no non-zero ?-order continuous and ?-continuous measure with an
arbitrary range space. This follows from
Remark 5.2.3. If ?: L -> ? is an exhaustive measure and и е CUA(L) is coarser than
the ?-uniformity, then there is a measure ? : L -> ? such that и is the ?-uniformity.

Measures on clans and on MV-algebras
939
PROOF. Let ? = ?\ + ?? be the decomposition according to Theorem 5.2.1, ? the ?-
uniformity and u,- the ?,-uniformity. Then и л v\ =v\ and и л V2 = 0, hence и = и ? ? =
и л (и ? ? V2) = (и л ?\) ? (и л V2) = v\, i.e., и is the ? ? -uniformity. ?
5.3. Hammer-Sobczyk's decomposition theorem and Lyapunov's theorem
THEOREM 5.3.1. Let (L, <) be complete, ? order continuous and ?(?) = {0}. Then
there is a ?-continuous measure v. L —>· ? and there are ?-continuous positive measures
pa : L —>· R a/id ?/,: L —>· R a/id elements ya, Zb ^ ? (a e A, b e B) with the following
properties:
A) (/0а(л:)>'а)а6д and (ai,{x)zb)b<iB ore summable uniformly in ? e L; ? = ? + ? +?
where p(x) := J2aeA Ра(х)Уа anda(x) := Y^heBob{x)Zb\
B) pa{L) = C„Jor SOW6 H(i eN\(l), ob(L) — Coo (a e A, b e B); p(L) is compact;
a(L) is convex and compact; cov(L) = cov(C(L)), v(L) is arcwise connected;
C) the restriction v\c(L) is an atomless measure;
D) ? =0 iff L is atomless.
PROOF. We use the notation as in Theorem 1.5.4. Put A = A/(L) and В = A<x(L). For
d e A U ?, the interval [0, d] is an irreducible complete ?V-algebra. Therefore there is by
Proposition 1.5.2 an isomorphism kj: [0, d] -> ?„(/ where nj e ? \ {1} for d e A and
nj = oo for d e B. We put v(x) = ?(? л с), ра(х) = Ki(x ? ?), ?/>(.?) = ?/,(^ л b),
ya = ?(?), ?/, = ??) ??? a e A, b e В and jc e L. Let d e AU B. Since ?(* л d) =
??·(? ??)?(?) for any ? e C(L), the equality holds by Theorem 5.1.2 even for any ? e L.
Therefore ?(? л ?) = А,(-х);уа and ?(* л />) = a/,(;c)z/, for ? e L, ? e A and /? e S.
We now verify the properties from A) to D).
A) Let || ? II,, be the semivariation with respect to a fixed continuous seminorm ? on E.
For any finite subset F of A and ? e L we have
?(? ла/)-^ pa(x)ya = ?(? ? af) - ]??(.? ? ?) = ?(.? ? (?/ \ sup F)),
aeF aeF
therefore ?(?(? л af) - ??<=?- Ра(х)Уа) < II?II/Да/ \ sup F). Since ||?||,, is order
continuous, the right-hand side converges to 0. Therefore {ра{х)Уа)ачА is summable
uniformly in ? e L and p(x) := ???? Ра(х)Уа = ?(? л af).
In the same way one sees that (ob(x)zb)beB is summable uniformly in ? e L and ? (*) =
?(^ л ???). Therefore v(x) + р(х) + ? (?) = ?(? Ac) + ?(? Aaf) + ?(? ??,?) = ?{?).
a(L) is the image of the compact convex set C^ under the continuous affine map
Ub)beB ^ iLbeB^Zb and therefore it is compact and convex. Similarly we obtain that
p{L) is compact. Since [0, c] is arcwise connected by Theorem 4.2.6, v(L) = ?([0, с]) is
arcwise connected. By Theorem 5.1.2 we have cov(L) = cov(C(L)). C), D) and the last
statement of B) hold obviously. ?
The last result allows us to transfer results about the range of measures from the Boolean
case to the case of MV-algebras. If in Theorem 5.3.1 v(C(L)) is compact and convex,

940
G. Barbieri and H. Weber
then v(L) = v(C(L)) and therefore ?(?,) is compact as a sum of three compact sets (see
Theorem 5.3.1B)); if, moreover L is atomless, then ?(?,) is convex. This shows that the
classical Lyapunov theorem for measures on Boolean algebras yields the following version
of Lyapunov theorem for measures on MV-algebras.
THEOREM 5.3.2. Let L be ?-complete, ? e N and ?: i, -> R" a ?-order continuous
measure.
(a) Then ?(?,) is compact.
(b) If ? is atomless, then ?(?,) is convex.
For the proof, we can apply Theorem 5.3.1 to the quotient measure ?: L/?^?) —>¦ К"·
With a quite other method Butnariu and Klement A993) have proved Theorem 5.3.2(b) for
?-order continuous measures on tribes. A finitely additive version of Theorem 5.3.2(b) is
contained in Barbieri B001b).
Another consequence of Theorem 5.3.1 is the decomposition Theorem 5.3.5 for
exhaustive measures on MV-algebras, which generalizes the Hammer-Sobczyk decomposition
theorem for real-valued bounded measures on Boolean algebras (see Bhaskara Rao and
Bhaskara Rao A983)).
We recall the concept of chained uniform space. A uniform space (X,u) is called
chained if for x, у e X and U e и there are jco, x„ e X with xq = x, x„ = у and
(*,-_ \,Xi)eUfori=l,...,n (see Hejcmann A959, p. 547) and Bourbaki A966, p. 204)).
PROPOSITION 5.3.3. (a) Every connected space is chained.
(b) // ? is a dense subspace of a uniform space X, then X is chained iff ? is chained.
The following proposition shows in particular that a charge ? on a Boolean algebra A is
strongly continuous in the sense of Bhaskara Rao and Bhaskara Rao A983) iff A is chained
with respect to the ?-uniformity.
PROPOSITION 5.3.4. Letu e CUA(L). Then (L, u) is chained iff every O-neighbourhood
U in (L,u) contains a finite number of elements x\,..., x„ eU such that ??'=? ?? = 1·
PROOF. (=>) Let U be a solid O-neighbourhood in (L,u) and ao,...,a„ with ao = 0,
an= 1 and ?(·??,-_? eU (i = \,...,n). Put st := sup{ao, · · · .a/} for /' > 0 and *,- :=
SjAsj-\ for/ ^ l.Then ]T"=| ¦*; =s„ = \ and jc, = (j,-_i ??,-)?(?/_? ??,-_?) ^ ?(-??,-_?,
hence, *; eU (/ = 1,..., n).
(<=) Let U be a solid O-neighbourhood in (L, u) and x\,...,x„ e L with ]Г"=| л-,· = 1
and x,- e U (i = ?,.,.,?). Let a,b e L. Put jc0 = 0, a, = a ? ?'^?^' and b' =
b л ?'^?^' (' = 0, ...,и). Then ?,-??,-? ^ *,-, hence, ?,??,-? e i/ and similarly
b;Abj-\ e i/. Therefore for (zo, · · · ,Z2«) := (a;i. ...,a\,0,b\, ...b„) we have zo = a,
Z2„ =b and ?,??,-1 e U (i = 1, ...,2n). ?
THEOREM 5.3.5 (Hammer-Sobczyk decomposition), //? ? exhaustive, then there is ? ?-
continuous measure ?: L —> ? and there are ?-continuous positive measures pa : L —>¦ ?
andelements ya e ?(a e Л) лмс/г that

Measures on clans and on MV-algebras
941
A) (ра(х)Уа)аеА " summable uniformly in ? e L; ? = ? + ? where ? := ?? ел РаУа',
B) L is chainedwith respect to the ?-uniformity;
C) pa(L) = C„a for some na e ? \ {1} (a e A); ^(L) и relatively compact; k(L) is
connected.
PROOF. Let w be the ?-uniformity, (L, w) the completion of (L, w) = (L, w)/N(w) and
?: L —>¦ ? the continuous extension of ?, where ? is defined as in Proposition 4.4.5.
Then (L, $5) is complete and ? is order continuous. Let ? = ? + ? + ? and ? =
???? РаУа the decomposition of ? according to Theorem 5.3.1 and ? = ? + ?. Define
?: L —>¦ ? by v(x) = ?>(?) if ? e L and ? e ? e L, and analogously ?, ?? and ?. Then
obviously A) holds.
We will prove B): let и be the ?-uniformity, и the ?-uniformity and и = и|?. By the
notation of Theorem 1.5.4, let ax = supAoc(L),a f = sup A/ (L) and с = (?? ??/)'. In
the proof of Theorem 5.3.1 it is mentioned that v(x) = ?(? л с) and ?(?) = ?(? ла^с)
for ? e L. Therefore ?(?) = ?(? л (с ? ах)), ? е L. Since the interval [0, с ? ???] is,
by Theorem 4.2.6, a connected subspace of (L, w) and ? is ю-continuous, L is connected
with respect to u. By Proposition 5.3.3, (L, u) and its dense subspace (L, w) are chained.
Since (L,u)/N(u) = (L,??), by Proposition 4.4.5, (L, и) is chained, too. ^ ^
We will prove C): pa(L), p(L) and ?(?,) are dense subsets, respectively, of pa(L), p(L)
and^X(L). By Theorem 53.1B), pa(L) = ?„,, for some nu e ? \ {1} and therefore finite,
p(L) is compact and ?(?,) is connected as a sum of the connected set v(L) and cr(L).
Therefore pa(L) = pa(L) = ?„o, p(L) is relatively compact and ?(?,) is connected. D
5.4. Vitali-Hahn-Saks-Nikodym theorem andNikodym boundedness theorem
We here transfer Vitali-Hahn-Saks-Nikodym convergence theorem and Nikodym
boundedness theorem from the Boolean case to the case of measures on MV-algebras. With
another method P. de Lucia and E. Pap have given theorems of this type in a more general
setting. For details, we refer to de Lucia and Pap B002).
Definition 5.4.1. A sequence (м,я)тем of measures is called uniformly exhaustive if
for each monotone sequence (a,,)neN in L the sequence (?,„ (a„))n6pj is Cauchy uniformly
with respect to m e N.
THEOREM 5.4.2. Let ? : L —>· (?, ?) be an exhaustive measure and ? a Hausdorff linear
topology on ? weaker than ?. Then ?-uniformity with respect to ? coincides with ?-
uniformity with respect to ?.
Proof. Let ?:(?, ?)-> (?,?) the identity map. We apply Theorem 5.1.3 with w being
the ?-uniformity with respect to ?. Put ? := ? ? ?. Then by Theorem 5.1.3(c), ? = ? ? ?.
As ? (?) = ? (?), we have ? <<C ? and ? <<C ? and by Theorem 5.1.3(c) ? <? ? and ? <? ?,
i.e., the ?-uniformity with respect to ? coincides with the ?-uniformity with respect
toa. ?

942
G. Barbieri and H. Weber
Proposition 5.4.3. Letwe CUA(L) ????,,: (L, w) -> (?, ?), ? e ?, be a pointwise
bounded, uniformly exhaustive sequence of continuous measures. Then ?,,.??,») —>
(?, ?), ? eN, ? equicontinuous.
PROOF. We denote by 1~?(?) the space of all ?-valued bounded sequences, by ?-»
the topology of uniform convergence on lx(E) and by ?? the topology of pointwise
convergence on loo(E). Then ? := (?„)„6? is a measure on L with values in /«;(?).
Denote by Wcc the ?-uniformity with respect to ?? and by w,, the ?-uniformity with
respect to ??. As ?: L —>· A?(?), ??) is exhaustive, we obtain by Theorem 5.4.2
Юоо = wp. Since ?„ : (L, w) —>¦ (?, ?) is, for every ? e N, continuous, we have wp С w.
It follows Woo С w, i.e., (?„)?6? is equicontinuous with respect to w. ?
PROPOSITION 5.4.4. Let (?„)„6? ^e ? pointwise bounded sequence of exhaustive
measures defined on L with values in E. By the same assumptions and notations as
in Theorem 5.1.3, // {?„: ? e ?) is uniformly exhaustive, then (??)„6? is uniformly
exhaustive.
PROOF. Let loo(E) and ?^ be defined as in the proof of Proposition 5.4.3.
Suppose {?„: ? e ?} uniformly exhaustive. Then ? := (/x„)neN: C(L) -+ (?N, ??) is
exhaustive. Since (?„)«6? is pointwise bounded by assumptions, ? attains values in loc(E).
Moreover, by Proposition 5.4.3 {?„: ? e ?) is equicontinuous with respect to w, i.e.,
? : (C(L), w) —>¦ loc(E) is continuous; here w = w\C{ir Let ? : (L, w) -+ lx(E) be the
measure which corresponds to ? according to Theorem 5.1.3. Applying Theorem 5.1.3(c)
to the projections (x„),!6n ?-» xn from ??(?) into ? we obtain ? = (?„)„6?· Since
?: (L, w) —> loo(E) is continuous and hence exhaustive, (?„) is uniformly exhaustive. D
THEOREM 5.4.5. Let L be ?-complete and ?„ : L -> E,n e N, a sequence ofa-order
continuous measures, which converges pointwise to ? : L —> E.
(a) Г/геи (?„)?6? " uniformly exhaustive and ? is ? ?-order continuous measure.
(b) ///or every и e ?, ?„ /j continuous with respect to ? ? ?UA(L), then {?„: ? e ?)
U {?} /? equicontinuous with respect to v.
PROOF. Let w the uniformity generated by {?„: ? e ?). As ? is the subspace of a product
of Banach spaces, we may assume ? is a Banach space.
Then (L, w) is complete by Theorem 2.5. Passing to the quotient (L, w) := (L, w)/
N(w), we can suppose w Hausdorff. So (L, w) = (L, w) = (L, w).
By classical Vitali-Hahn-Saks-Nikodym theorem the restrictions
An:=Mnlcu), neN,
are uniformly exhaustive. Hence ?„, ? e ?, are uniformly exhaustive by Proposition 5.4.4
and therefore {?„: ? e ?) U {?} is uniformly exhaustive. Hence, by Proposition 5.4.3,
{?„: ? e ?} U {?} is equicontinuous with respect to ? if ?„ <«C ? for any ? e N.
Applying this fact for ? = w we obtain ? ? -order continuous. ?

Measures on clans and on MV-algebras
943
THEOREM 5.4.6. Let L be ?-complete and ? a pointwise bounded set of ?-order
continuous measures with values in E. Then ? is uniformly bounded.
PROOF. As in the proof of Theorem 5.4.5 we may assume ? a Banach space.
Since a subset A of ? is bounded iff all countable subsets of A are bounded, we may
assume ? countable, i.e., ? = {?„: ? еЩ.
Let w the uniformity generated by {?„: ? e ?). Then (L,w) is complete by
Theorem 2.5. Passing to the quotient (L,w) := (L,w)/N(w), we may assume w
Hausdorff. So (L, w) = (L, w) = (L, w).
By classical Nikodym boundedness theorem, {?„: ? e ?) is uniformly bounded, i.e.,
for some positive real number r we have Д„(С(?)) с {? e ?: \\x\\ < r} for every ? e N.
By Theorem 5.1.3 ?,,(?,) с cojln(C(L)) С {? e ?: \\x\\ ^ r]. Hence, {?„: ? e ?) is
uniformly bounded. D
5.5. Extension of measures
One of the basic results of classical measure theory is the extension theorem for measures
from an algebra to the generated ? -algebra. We will generalize this theorem for measures
on MV-algebras. An extension theorem in the real-valued case you can find in Jureckova
A995).
THEOREM 5.5.1. Let L be ?-complete, S an MV-subalgebra of L and ?:? -> ? ?
measure. Then ? has a unique ?-order continuous measure extension to the ?-complete
MV-subalgebra ?(?) of L iff ? is ?-order continuous and exhaustive. {Here, sup and inf
are always taken in L.)
PROOF. (=>) is obvious. (<=) Existence. Let ? be a system of seminorms generating the
topology of E. Then, for ? e P, the semivariation ||?||? of ? with respect to ? is an
exhaustive ?-submeasure on S by Example 2.1 and Proposition 4.4.4. Therefore
/j„(a):=inf
]Р||МЫ*/.): х„ eS", a s^J2x"
/1=1 /1 = 1
defines a ?-submeasure on L extending ||?||,, for ? e ? by Proposition 2.2. Let и be the
uniformity generated on L by {??)?<??, ? the closure of S in (L, u) and ? the continuous
extension of ? on ?. Obviously, ? is a measure. We now show that ? is ?-complete
(hence ?(?) с ?) and ? is ?-order continuous. Let (a„) be an increasing sequence in ?
and a = supa,,. Since и\т is exhaustive by Proposition 4.3.2, (a„) is a Cauchy sequence;
therefore (aAan) is Cauchy, moreover ???„ | 0, hence (aAa„) converges to 0 in (L, u)
by Proposition 2.5, i.e., (a„) converges to a. It follows that a e ? = ? and ?(?„) -> ? (?)
by the continuity of ?.
Uniqueness. For / = 1, 2 let ?>, : ?(?) —> ? be a ?-order continuous measure extension
of ?, ? := ?| — V2, and и the v-uniformity. Since и is ?-order continuous, the closure S oi S

944
G. Barbieri and H. Weber
in (a(S), u) is a ?-complete subalgebraof a(S), hence S = a(S). Since ? is continuous
with respect to и and ? vanishes on S, we get ? = 0, i.e., ?>? = ?>?· ?
References
Barbieri, G. A999), Misure su A-l-semigruppi e su MV-algebre. Ph.D. thesis. Naples.
Barbieri, G. B001a), A note on fuzzy measures. J. Electrical Engrg. 52. 67-70.
Barbieri, G. B001b), Lyapunov's theorem for measures on D-posets, Intemat. J.Theoret. Phys. (submitted).
Barbieri, G., Lepellere, M.A. and Weber, H. B001), The Hahn decomposition theorem for fuzzy measures and
applications, Fuzzy Sets and Systems 118 C), 519-528.
Barbieri, G. and Weber, H. A998), A topological approach to the study of fuzzy measures. Functional Analysis
and Economic Theory, Abramovich, Avgerinos. Yannelis, eds, Springer, Berlin, 17^46.
Basile, A. and Traynor, T. A991), Monotonely Cauchy locally solid topologies. Order 7,407^416.
Bhaskara Rao, K.P.S. and Bhaskara Rao, M. A983), Theory of Charges, Academic Press. San Diego.
Bemau, J. A963), Unique representation of Archimedean lattice groups and normal Archimedean lattice rings,
Proc. London Math. Soc.3A5),599-631.
Bourbaki, N. A966), General Topology, Part I. Hermann, Paris.
Butnariu, D. A983), Decomposition and range for additive fuzzy measures, Fuzzy Sets and Systems 10, 135-155.
Butnariu, D. A985), Non-atomic fuzzy measures and games, Fuzzy Sets and Systems 10, 39-52.
Butnariu, D. A987), Values and cores for fuzzy games with infinitely many players, Intemat. J. Game Theory 16,
43-68.
Butnariu, D. and Klement, E.P. A993), Triangular Norm Based Measures and Games with Fuzzy Coalitions,
Kluwer, Dordrecht.
Chang, C.C. A958a), Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467^490.
Chang, C.C. A958b), A new proof of the completeness of Lukasiewicz axioms. Trans. Amer. Math. Soc. 93,74-80.
Chang, C.C. A998), The writing of the MV-algebras, many-valued logics, Studia Logica 61 A), 3-6.
Cignoli, R., D'Ottaviano, I.M.L. and Mundici, D. B000), Algebraic Foundations of Many-Valued Reasoning,
Trends in Logic, Studia Logica Library, Vol. 7, Kluwer, Dordrecht,
de Lucia, P. and Pap, E. A995), Nikodym convergence theorem for uniform space valued functions defined on
D-posets, Math. Slovaca 45 D), 367-376.
de Lucia, P. and Pap, E. B002), Comergence theorems for set functions, Handbook of Measure Theory, E. Pap,
ed., Elsevier, Amsterdam, 125-178.
Diestel, J. and Uhl, J.J. Jr A979), Vector Measures, Amer. Math. Soc.. Providence, RI.
Di Nola, ?., Georgescu, G. and Lettieri, A. A999), Extending Probabilities to States of MV-algebras, Collegium
Logicum, Annales of the Kurt-Godel-Society, Vol. 3. 31-50.
Drewnowski, L. A972), Topological rings of sets, continuous set functions, integration I, II, III, Bull. Acad.
Polon. Sci. Math. Astr. Phys. 20, 269-286, 439-445.
Dunford, N. and Schwartz, J.T. A957), Linear Operators. Parti, Interscience, New York.
Dvurecenskij, A. B000), On Loomis-Sikorski's theorem for MV-algebras and BCK-al gebras, Contribut. General
Algebra, Vol. 12 (Vienna, 1999). Heyn, Klagenfurt, 165-180.
Fleischer, I. and Traynor, T. A980), Equivalence of group-valued measures on an abstract lattice. Bull. Acad.
Polon. Sci. 28, 549-556.
Fremlin, D.H. A974), Topological Riesz Spaces and Measure Theory, Cambridge Univ. Press, Cambridge.
Jakubik, J. A995), On complete MV-algebras. Czechoslovak J. Math. 45, 473^480.
Jureckova, M. A995), The measure extension theorem on MV?-algebras. Fuzzy Sets '94 (Liptovsky Jan, 1994),
Tatra Mt. Math. Publ. 6, 55-61.
Hejcmann, J. A959), Boundedness in uniform spaces and topological groups, Czechoslovak J. Math. 9, 544-562.
Kaplansky, I. A947), Topological rings, Amer. J. Math. 69, 153-183.
Klement, E.P. A982), Characterization offuzz\ measures constructed by means of triangular norms, J. Math.
Anal. Appl. 86 B), 345-358.
Lacava, F. A979), Sulla struttura delle C-algebre, Accademia Nazionale dei Lincei Estratto dai Rendiconti della
Classe di Scienze fisiche matematiche e naturali, serie VIII LXVII, 275-281.

Measures on clans and on MV-algebras
945
Mangani, P. A973), Su certe algebre connesse con logiche a piii valori. Boll. Un. Mat. Itai. 8, 68-78.
Mundici, D. A986), Interpretation of AF C*-algebras in sentential calculus, J. Funct. Ana). 65, 15-63.
Mundici, D. A995), Averaging the truth-value in Lukasiewicz logic, Studia Logica 55 A), 113-127.
Mundici, D. A999), Tensor products and the Loomis-Sikorski theorem for MV-algebras, Adv. Appl. Math. 22,
227-248.
Riecan, B. and Mundici, D. B002), Probability on MV-algebras, Handbook of Measure Theory, E. Pap, ed.,
Elsevier, Amsterdam, 869-909.
Rose, A. and Rosser, J.B. A958), Fragments of many-valued statement calculi. Trans. Amer. Math. Soc. 87, 1-53.
Schmidt, K.D. A989), Jordan Decomposition of Generalized Vector Measures, Pitman Research Notes in
Mathematics Series, Vol. 214, Longman, Essex.
Traynor, T. A976), The Lebesgue decomposition for group-valued set functions. Trans. Amer. Math. Soc. 220,
307-319.
Volkmer, H. and Weber, H. A983), Der Wertebereich atomloser Inhalte, Arch. Math. (Basel) 40 E), 464-474.
Vulilch, B.Z. A967), Introduction to the Theory of Partially Ordered Spaces, Wolthers-Nordhoff, Groningen.
Warner, S. A989), Topological Fields, North-Holland, Amsterdam.
Weber, H. A984a), Topological Boolean Rings. Decomposition of finitely additive set functions. Pacific J. Math.
110 B), 471-495.
Weber, H. A984b), Group- and vector-valued s-bounded contents. Measure Theory (Oberwolfach, 1983), Lecture
Notes in Math., Vol. 1089, Springer, Berlin, 81-198.
Weber, H. A991/1993), Uniform lattices I: A generalization of topological Riesz spaces and topological Boolean
rings; Uniform lattices. II: Order continuity and exhaustivity, Ann. Mat. Рига Appl. 160, 347-370; 165, 133-
158.
Weber, H. A996), On modular functions, Funct. Approx. 24, 35-52.
Weber, H. A997), An abstraction of clan of fuzzy sets, Ricerche Mat. 46 B), 457^472.
Weber, H. B002), FN-topologies and group-valued measures. Handbook of Measure Theory, E. Pap, ed., Elsevier,
Amsterdam, 703-743.
Zadeh, L.A. A965), Fuzzy sets. Inform. Control 8, 338-353.

CHAPTER 23
Triangular Norm-Based Measures
Dan Butnariu*
Department of Mathematics, University of Haifa, 31905 Haifa, Israel
E-mail: dbutnaru @ math2. haifa.ac. il
Erich Peter Klement
Department of Algebra, Stochastics and Knowledge-Based Mathematical Systems,
Johannes Kepler University, Linz, Austria
E-mail: ep.klement@jku.at
Contents
1. Introduction 949
2. Triangular norms, fuzzy subsets 949
3. ?-tribes 956
4. 7"-measures and their representation by Markov kernels 958
5. Integra) representation of 7"l -measures 964
6. Integra) representation of monotone 7\-measures 970
7. Decomposition of monotone 7\-measures 973
8. Jordan decomposition of bounded 7"L-measures 983
9. Jordan decomposition of finite ^-measures 988
10. Absolute continuity of 7"l-measures 991
11. Vector 7"l-measures with Darboux property 992
12. Nonatomic 7"l-measures 997
13. A Liapounoff type theorem for 7"l-measures 1004
14. A Liapounoff type theorem for 7? -measures 1006
References 1007
The first author gratefully acknowledges the support of the Israel Science Foundation founded by the Israeli
Academy of Sciences and Humanities.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. AU rights reserved
947

This page intentionally left blank

Triangular nonv-based measures
949
1. Introduction
Triangular norm-based measures, or simply ?-measures, are special real valuations defined
on Г-tribes, the latter being classes of fuzzy sets (generalizing the concept of a ?-algebra)
based on a triangular norm ?. They attracted the interest of scientists because of their
applications in fields like mathematical statistics (see [28,29]), pattern recognition [80,75],
possibility and plausibility theory [81,40] and many-valued logics [31,32,69]. Butnariu and
Klement [17, Chapter IV] have used ?-measures to identify classes of games for which
Aumann-Shapley values in the spirit of [2] exist. Among the basic tools employed in this
respect are integral representations of ?-measures.
This survey first presents the necessary preliminaries about triangular norms and fuzzy
sets, including a concepts of disjointness, emphasizing the special role of Frank t-norms ??
[30]. Next, Г-tribes are introduced, and their most prominent properties are listed. Three
different representations for ?-measures (for measures with respect to the minimum Гм
and to the Lukasiewicz t-norm Tl, and for general T^-measures) as integral with respect to
a suitable Markov kernel follow, as well as a decomposition of monotone ?? -measures.
The next group of sections concentrates on measures with respect to the Lukasiewicz
t-norm Tl and some of their most important properties: Jordan decomposition, absolute
continuity, Darboux property, nonatomicity.
The two last sections provide a Liapounoff type theorem concerning the compactness
and convexity of the range of vector measures, first for T^-measures (this turned out to be
one of the basic tools to show the existence of an Aumann-Shapley value on the space of
games with fuzzy coalitions spanned by positive integer powers of monotone Tl-measures
[17, Chapters IV-VI]) and then for general Гд-measures, as recently proved by Barbieri
and Weber [4].
2. Triangular norms, fuzzy subsets
Fuzzy sets, as introduced by Zadeh [79], are a many-valued generalization of Cantorian
sets. For a Cantorian set A the possible truth values of the statement 'x is an element of
A' are 0 and 1 (false and true), whereas for a fuzzy set A the truth values of the statement
'дг is an element of A' can be any number in the interval [0, 1]. We shall describe fuzzy
sets in more detail below - first we want to provide the tools for extending operations such
as intersection and union of sets.
In order to do so properly, we consider suitable functions Г: [0, 1]" —>¦ [0, 1] and
S:[0, l]2 —>¦ [0, 1], whose restrictions to {0, l}2 coincide with conjunction and disjunction,
respectively, and which satisfy such general properties like associativity, commutativity
and monotonicity.
Such functions are known as triangular norms and triangular conorms, respectively; they
proved to be useful in the theory of probabilistic metric spaces, as initiated by Menger [56]
and extensively studied by Schweizer and Sklar [73]. In such spaces the distance between
two points is measured by probability distribution functions rather than by real numbers,
and the triangular norms allow the triangle inequality to be properly generalized.

950
D. Butnariu and E.P. Klement
In continuum physics Menger [59,58,57] interpreted triangular norms as rules for
generating new probabilistically determined objects from existing ones in the psychophysical
continuum space. In accordance with one of the main ideas underlying the foundation of
modern game theory in Neumann and Morgenstern [66], we shall use triangular norms as
models for interaction rules of economic agents in a way which is similar to the modelling
of the interaction of physical particles.
Triangular norms are also closely related to the still unsolved general associativity
functional equation [21]. In the setting of many-valued logics with truth values in the
whole interval [0, 1] (generalizing the Boolean two-valued logic whose set of truth values
equals {0, 1}), triangular norms are natural models for the interpretation of the conjunction
(naturally extending the classical conjunction л of the Boolean logic) [31,32,55,50].
Dehnition 2.1.
(i) A function T:[0, l]2 —>¦ [0, 1] is called triangular norm{t-norm for short) if it
satisfies the following conditions for all ij,ze [0, 1]:
(Tl) T{x,\) = x, {boundary condition)
(T2) T(x,y) ???(?,?) whenever y^z, (monotonicity)
(T3) T(x, y) = T(y, x), (commutativity)
(T4) T(x, T(y,z)) = T(T(x, y),z). (associativity)
(ii) If ? is a triangular norm, then the corresponding triangular conorm (t-conorm) is
the function 5": [0, 1 ]2 -> [0, 1 ] defined by
(TCO) S(x,y)=l-T(l-x,\-y).
It is obvious that, given a t-norm T, the corresponding t-conorm 5 of ? satisfies
conditions (T2), (T3) and (T4) and, dually to (Tl),
(SI) S(x, 0) = x. (boundary condition)
A function 5": [0, l]2 -> [0, 1] fulfilling (SI) and (T2)-(T4) is called a triangular conorm
(t-conorm). It is easily seen that given a t-conorm 5, then ?: [0, l]2 —>¦ [0, 1] defined by
T(x,y)=l-S(l-x,l-y).
is a t-norm whose corresponding t-conorm is exactly the t-conorm 5 we started with.
Example 2.2. Some of the most important pairs of t-norms and corresponding t-conorms
are the minimum Гм and the maximum 5м, the product Tp and the probabilistic sum 5p,

Triangular norm-based measures
951
the Lukasiewicz t-norm 7l and the Lukasiewicz t-conorm 5l, and the drastic product Tq
and the drastic sum SO given by, respectively:
Tu(.x,y) = \
nin(x, y),
Tp(x,y)=x ¦ y,
TL(x, у) = тах@, x +y - 1),
Ых,у) =
? ify=l,
У if x = 1,
0 otherwise,
Sm(*,>') = i
ъах(х,у).
SP(x, y) = ? + у - ? ¦ у,
SL(x,y) = min(l,x + y),
So(x,y) =
? if у = 0,
у ifx=0,
1 otherwis
From the boundary condition (Tl) and the monotonicity (T2) it readily follows that Тц
is the smallest and Гм is the largest t-norm, i.e., for each t-norm Г we have
To < ? < Гм-
A t-norm is a rather special semigroup operation on the unit interval [0, 1 ]. It is therefore
possible to consider additional algebraic properties such as the Archimedean property and
strictness.
A t-norm ? is called Archimedeanii for each (x, y) e ]0, 1[2 there is an ? e N such
»
that x{T < y, where xyT ' is the и-th power of ? with respect to ? defined inductively by
m
= 1 and
>+n
¦7D"'·*)
A t-norm ? is called stricni it is continuous and if it satisfies T(x, y) < T{x,z) whenever
? e]0, 1] and y,z e [0, 1] with у <z-
Obviously, each strict t-norm is Archimedean. The Lukasiewicz t-norm 7l is continuous
and Archimedean but not strict, the drastic product Tq is Archimedean but not continuous
(and hence not strict). The minimum Гм is continuous but not Archimedean (and hence
not strict), and the product T? is strict (and therefore Archimedean).
A very special role is played in this survey by the family of Frank t-norms (Г^)лб(О.эс]
(they were called fundamental t-norms in [17]). The starting point for the investigation of
Frank t-norms was the following problem in connection with the associativity of duals of
copulas [65,74] in the framework of distribution functions and probabilistic metric spaces
[73]: characterize all continuous (or, equivalently, nondecreasing) associative functions
F: [0, l]2 —>· [0, 1], which satisfy for each ie[0,l] the boundary conditions F@, x) =
F(x,0)and F(x, 1)= F(l,x) = x, such that the function G : [0, l]2^ [0, l]givenby
G(x,y) = x
F(x,y)
is also associative. Frank [30] has shown that then F has to be an ordinal sum [20,54,72] of
members of the following family of t-norms. In particular this means that each pair G\, S\)
of mutually corresponding Frank t-norms and t-conorms satisfies the functional equation
T(x,y) + S(x,y) = x +y.
A)

952
D. Butnariu and ?.P. Klement
Example 2.3.
(i) The family of Frank t-norms Gх)х6[о.зс| is given by.
Ых,у) =
Тм(х,у)
Tp(x,y)
TL(x,y)
log.fl
(?-?-1)·(?-?-1)
if ? = 1,
if ? = oo,
ifXe]0,oo[\{l]
?- 1
(ii) The family of Frank t-conorms E;0ле[0.зс| is defined as follows:
Sx(x,y) =
Sm(x,)>)
Sp(x,y)
Sh(x,y)
l-logx(l
(?
I-л
1)·(?
l-v
1)
1
ifX = 0,
if ? = 1,
if ? = oo,
ifXe]0,oo[\{l]
All Frank t-norms are continuous, and a Frank t-norm ?? is strict if and only if
? e ]0, oof. Recall that Tx (= 7l) is Archimedean but not strict, and that Го (= Гм) is
neither Archimedean nor strict. The family (Г>.)л6[о.зс| is a continuous family in the sense
that for each ?? e [0, oo] we have
lim ??
??0.
A nontrivial result is that the family of Frank t-norms (??)?6??.??] is strictly decreasing
with respect to the parameter ?, i.e., for ?|, ?? e [0, oo] we have ?·?? < ??? if and only
if ? ? > k2- A first (and rather complicated) proof of this monotonicity of the family of
Frank t-norms was given by Butnariu and Klement [17, Proposition 1.12], a significantly
shorter proof based on the additive generators of the Frank t-norms was published
in [47, Example 3.2]. Many more details about Frank t-norms are contained in [30] and
in Section 4.4 of Klement et al. [48].
Turning back to an arbitrary t-norm ?, the associativity (T4) allows us to extend this
binary operation in a unique way to an /i-ary operation in the usual way by induction,
defining for each и-tuple (дг|, дгг,..., дг„) e [0, 1]":
?\, ?2, ···,*«)·
The fact that each t-norm ? is bounded from above by Гм makes it possible to extend it to
a (countably) infinitary operation, putting for each sequence (*;);'6N of elements of [0, 1]:
OO II
? Xj = lim ? Xj.

Triangular norm-based measures
953
Note that the limit on the right-hand side always exists since the sequence
(? ?
\i=\ /,,6N
is nonincreasing and bounded from below.
Coming back to sets, a well-known fact about Cantorian sets (for which we shall use the
term crisp set to distinguish them from the fuzzy sets to be defined below) is that, given
a universe of discourse X, each subset A of X can be identified with its characteristic
function A:X —>¦ {0, 1), assigning the value 1 to all elements of X which belong to A, and
the value 0 to all remaining elements of X.
It was exactly this concept of a characteristic function which was generalized by
Zadeh [79] who replaced the two-element set {0, 1} by the unit interval [0, 1], therefore
considering a continuum of truth values rather than the two classical truth values true and
false (which correspond to 1 and 0, respectively) only.
Therefore, given a (crisp) universe of discourse X, a fuzzy subset A of X (or, briefly,
? fuzzy set A) is characterized by its membership function
A:X->[0, 1],
where for ? e X the value A{x) is interpreted as the degree of membership of ? in the fuzzy
set A. The class of all fuzzy subsets of X will be denoted [0, \]x.
In a logical setting, the degree of membership A(x) can also be seen as the truth value
of the statement 'x is element of A'.
Since the membership function of a fuzzy subset of X generalizes the characteristic
function of a crisp subset of X, each crisp subset of X is just a special case of a fuzzy
subset of X, and its characteristic function is a special membership function (assuming the
values 0 and 1 only). Subsequently, the universe X and the empty set 0 are identified with
the constant functions mapping each ? e X to 1 and 0, respectively.
Given a fuzzy subset A of a universe X we can associate with it several crisp subsets
of X, among them the kernel ker(A) and the support supp(A) defined by, respectively,
ker(A) = {x eX: A(x) = l}, B)
supp(A) = {xeX: A(x)>0}. C)
Let ? be a ?-norm and S its corresponding r-conorm. In order to define the intersection
and the union of two fuzzy subsets A and В of a universe X, we extend the binary
operations ? and S on [0, 1] to [0, \]x pointwise, i.e.,
(ATS)U) = T(A(x), B(x)), (ASB)(x) = S(A(x), B(x)).
Finite and countable intersection and union are introduced in a straightforward way. Note
that the duality of ? and S implies the validity of the De Morgan laws, i.e., for a sequence
(A„),i6N of fuzzy subsets of X we have

954
D. Butnariu and E.P. Klement
@0 \ OO / OO \ ОС
? a„ =SCa,„ С Sa„) = TCa„,
n=\ / n=\ \n=\ / n=\
where the complement CA of a fuzzy subset A of X is defined by CA(x) = 1 - A(x).
Restricted to crisp sets, these operations coincide with intersection, union and
complement, respectively, regardless which t-norm and t-conorm are considered. The class [0, 1 ]x
of all fuzzy subsets of X, together with either one of the operations ? and S, forms a
partially ordered commutative semigroup having 0 as smallest element and X as largest
element. However, [0, l]x provided with the operations T, S and the complement С is not a
Boolean algebra. In general, ? and S are not distributive with respect to each other, and
aTCa may be different from 0 and A S С A may be different from X.
In order to classify ?-measures and to distinguish between ?-measures and other kinds
of valuations which can be defined on ?-tribes (see, for instance, [71]), we also need
a concept of disjointness. There are different approaches to disjointness (mostly in the
context of fuzzy partitions) [22,25-27,37-39,42,51,63,70,76]. Our concept is based on
Butnariu [11] (see also [17]).
Dehnition 2.4. Let X be a nonempty set, Г a t-norm and S its corresponding t-conorm.
(i) A finite family (? ?, ??,..., A„) of fuzzy sets in [0, 1 ]x is said to be ?-disjoint if
for each к e {1,...,«}
(TD) (S АЛТA* = 0.
(ii) A countable family (?„)?6? <? [?, \]x is called ?-disjoint if, for each ? e N, the
finite family (? ?,..., A„) is ?-disjoint.
(iii) A sequence (A„)„6n ? [0, l]x is called a T-partition of a fuzzy set A if it is
?-disjoint and if
OO
(TP) A= S A„.
Each subfamily (A„)n6/ of an at most countable ?-disjoint family (An)nej is also
?-disjoint: it suffices to show this for finite index sets J and /?. Indeed, for each
/ e / we get
( S ?„)??,<;( S а„)та,=0.
vne/\li] ' ??|?] '
Also, the definition of the ?-disjointness does not depend on the order in which the fuzzy
sets (A„)n6pj are numbered, i.e., if ? is a permutation of N and the countable family
(An)n6pj is ?-disjoint, then the permuted family (A,t(„))„6n is ?-disjoint too.
Different t-norms may lead to different concepts of disjointness. However, for some
classes of t-norms the corresponding disjointness concepts do not depend on the choice of
the t-norm in that class.

Triangular norm-based measures
955
Example 2.5. Let (А„)„ещ be a countable family of fuzzy subsets of X.
(i) If all A„ are crisp subsets of X, then for each t-norm ? the ?-disjointness is
equivalent to the pairwise disjointness of the crisp sets,
(ii) For each t-norm ?, the ?-disjointness implies pairwise ?-disjointness, but the
converse is not generally true: if we take the family of constant fuzzy sets
(A |, A2, A3) given by А|(дг)= A2(x) = Аз(дг) = 0.5, then A\, A2, A3 are pairwise
TL-disjoint, but the family (A\, A2, A3) is not TL-disjoint.
(iii) For ? e [0, oof we get: (А„)пещ is ??-disjoint if and only if each elements e X is
"contained" in at most one Ak (i.e., if and only if Ак(х) > 0 for at most one к е N).
(iv) The ??-disjointness of (A„)„6pj means that for each ? e X exactly one of the
following conditions holds:
(a) there is exactly one к e N such that А* (дг) = 1, and A„ (x) = 0 for all ? ? к;
(b) there are at most two indices k, I e N such that {А*(дг), А/(х)} С ]0, 1[, and we
have A„(x) = 0 for ? <?{k,l].
The following results (see Lemma 4.4 and Corollary 4.5 in [17]) emphasize the crucial
role of the functional equation A) with respect to disjointness.
LEMMA 2.6. Let ? be a t-norm and S be its corresponding t-conorm such that the
functional equation A) holds, and let (A1, A2- ¦¦, A„) be a finite family of fuzzy subsets
of X. Then for each ? ^ 2 the following are equivalent:
(i) the family (A\, A2,..., A„) is ?-disjoint;
(ii) for all к e {2,..., n] we have (Sf=i' A,) ? Ak = 0;
(iii) for all к e [2,.. .,n) we have Si=\ A, = ?/=? ^ь
(iv) for each subset I с {1,2,...,«} containing at least ? - 1 elements we have
biel Ai = ?/6/ A-i-
Corollary 2.7. Let ? be a t-norm and S its corresponding t-conorm such that the
functional equation (I) holds, and let ( A„)„6n be a countable family of fuzzy subsets ofX.
Then the following are equivalent:
(i) the family (А„)„ещ is T-disjoint;
(ii) for all k^lwe have (S*=|' A/) T Ak = 0;
(iii) for all k^lwe have Si-\ A, = ?/=? ^';
(iv) for each finite subset I с N we have Si6/ A, = ?]i6/ A,-.
As an immediate consequence we have that an at most countable family of fuzzy sets
(A„)„eJ is rL-disjoint if and only if J2nej A„ ^ 1.
From Lemma 2.6 and Corollary 2.7 we know that if Г and its corresponding t-conorm
5" satisfy A) and if (A„)neJ is ?-disjoint, then necessarily J2„eJ A" ^ '· However, the
converse is not generally true (see Example 2.5 (iii)). Also, the requirement that Г and S
satisfy A) cannot be dropped in Lemma 2.6 and in Corollary 2.7. For instance, for the
t-norm To and its corresponding t-conorm So the conditions (i), (ii) and (iii) are no longer
equivalent.

956
D. Butnariu and E.P. Klement
3. ?-tribes
We now study certain subclasses of [0, \]x, the class of all fuzzy subsets of a (crisp)
universe of discourse X, which are closed under intersections and unions based on t-
norms and t-conorms, respectively. In many situations it is also desirable to have also
a kind of ?-completeness. The following concept of ?-tribes was presented in [17], it
basically was introduced by Klement [46] for arbitrary t-norms ?, in [44] for ? = Гм and
by Butnariu [11] for ? = TL (see also [23]).
For all lattice-theoretical notions we usually refer to Birkhoff [8], in the context of
algebras and ?-algebras of crisp sets to Halmos [35] and Bauer [5].
Dehnition 3.1. Let ? be a t-norm.
(i) A subclass С of [0, l]x is called a T-clan on X if the following properties are
satisfied:
(TCI) 0eC;
(TC2) for all A e С we have CA e C;
(TC3) for all A, fie С we have AT ? eC.
(ii) A ?-clan ? с [0, l]x is called a T-tribe if
DC
(TT) for all sequences (A„)„6n m ? we have ? A„ e T.
Trivially, each algebra of crisp subsets of X [35,5] is a ?-clan with respect to an arbitrary
t-norm ?. Moreover, a subclass Д of ? (X), i.e., a class of crisp subsets of X, is a ? -algebra
if and only if it is a ?-tribe with respect to an arbitrary t-norm ?.
Each Гм-clan is a De Morgan algebra [41], and each Гь-clan is a semi-simple MV-
algebra as introduced by Chang [18] (compare also [6,7,24,41]). Also, each Гм-tribe is
a ?-complete De Morgan algebra, and each Tl-tribe is both a ?-complete semi-simple
MV-algebra and a clan in the sense of Wyler [78].
Obviously, not each ?-clan is a ?-tribe (a rather simple counterexample is given by the
TL-clan {A: for all ? e X: A(x) e Q) which is not a TL-tribe).
Example 3.2.
(i) Define for each ? e N and к e {1,2 ? - 1} the fuzzy subset A*/„ of X by
Ak/n(x) = k/n. Then the class T„ = {0, A\/n, Аг/„ ?(„_?,/„,?} is both a
Гм-clan and а Гь-с1ап, but it is never a ?-clan when Г is a strict t-norm.
(ii) Given a Borel measurable t-norm Г and a ? -algebra ? of crisp subsets of X, then
the class Лл of all fuzzy subsets of X with Д-measurable membership functions is
a ?-tribe. Note that Ал was called a generated (T-)tribe in [17,44,46], and that it
was used in [80] as the class of fuzzy events,
(iii) Given a ?-tribe T, the set Tv of all crisp sets in ? is a ?-algebra, and hence a
?-tribe.

Triangular norm-based measures
957
(iv) The class ? consisting of all fuzzy subsets of X, whose membership functions are
either constant or have all their values in the interval [1/3,2/3], is both a Го-tribe
and а Гм-tribe, but no ?-tribe whenever Г is a continuous Archimedean t-norm
[46].
(v) If in (iv) we additionally require that all the membership functions of the elements
of ? be continuous, then ? is a Го-tribe, but no longer a Гм-tribe [46].
(vi) Consider a crisp subset ? of X. Given a ? -algebra A of crisp subsets of X, the
class Ay of all fuzzy subsets of X, whose membership functions are Д-measurable
and assume on ? only the values 0 and 1, is a ?-tribe with respect to an arbitrary
Borel measurable t-norm ?. Note however that, up to the case ? = X, Ay does
not contain any fuzzy subset of X with a nontrivial constant membership function
(compare Theorem 3.3(iii)). The tribe Ay was called a semigenerated (T-)tribe by
Mesiar [60].
The following result for tribes based on Frank t-norms (see Propositions 2.7 and 3.3 and
Theorem 3.2 in [17]) demonstrates the special role of the Frank t-norms in this context.
The crucial argument is, of course, that the Frank t-norms and their dual t-conorms are
solutions of the Frank functional equation A).
THEOREM 3.3. For all ? e ]0, oo] the following assertions hold:
(i) Each ??-tribe is a Tj^-tribe (and, hence, ? ?-complete De Morgan algebra) as well
as a T^-tribe {and, hence, ? ?-complete semi-simple MV-algebra).
(ii) For each ??-tribe ? we have ? с (??)?, i.e., each element in ? has a Tv-
measurable membership function.
(iii) A ??-tribe ? is generated if and only if it contains all fuzzy subsets of X which
have a constant membership function.
The interesting problem to characterize all subclasses of [0, \]x which are ?-tribes with
respect to an arbitrary Borel measurable t-norm ? was solved by Navara [64] (compare also
Mesiar [60,61] and Mesiar and Navara [62]). The result turns out to be a generalization of
the fact that each class of crisp subsets of X is a ? -algebra if and only if it is a ?-tribe with
respect to an arbitrary t-norm ?, and of Theorem 3.3(i) (for the definition of supp(A) and
ker(A) see C) and B)).
THEOREM 3.4. Let X be a nonempty set and ? a subclass of [0, 1 ]x. Then the following
are equivalent:
(i) ? is a T-tribe for all Borel measurable t-norms ?;
(ii) there is ? ? e ]0, oof such that ? is a ??-tribe;
(iii) ? is a weakly generated tribe, i.e., there is ? ? -ideal V ofTv such that
T= (??(??)?: supp(A) \ker(A) e V).
This means that a ??-tribe for ? e]0, oof on X is not only a ? -complete lattice
(in particular, it is closed under Гм) and a ?-complete semi-simple MV-algebra (in
particular, it is closed under Tl), but that it is also closed under each Borel measurable

958
D. Butnariu and E.P. Klement
t-norm. Observe that this is not true for 7l-tribes: for example, the Гь-clan mentioned in
Example 3.2(i) is a TL-tribe, but no ?-tribe if ? is a strict t-norm (see also Example 3.6
below).
We already mentioned that each TL-tribe is a ?-complete semi-simple MV-algebra.
In fact, each semi-simple MV-algebra is isomorphic to some Гь-с1ап (see Belluce [6,7],
a complete characterization of all MV-algebras, involving nonstandard real numbers, can
be found in Di Nola [24]). Another characterization of ^-tribes was given by Klement
and Navara [49]. Both results are summarized below.
THEOREM 3.5. Let ? be a subclass o/[0, \}x. Then the following are equivalent:
(i) ? is a T^-tribe;
(ii) ? is ? ?-complete semi-simple MV-algebra;
(iii) there is ? ? -algebra ? of crisp subsets of X and a sequence (?n)neN of ?-filters
in ?, with ?m с ?n whenever ? is a divisor of m, such that
Т={АеЛл: {? e X: п-А(х)еЩ e?„forallneN}.
Combining all these results, it is possible to give, as a simple example, a rather complete
picture of all possible tribes with respect to Frank t-norms, to arbitrary strict t-norms, and
to the weakest t-norm To, respectively.
Example 3.6. Let {xo} be a singleton. Then the only generated tribe on {x0}, let us call
it %o, is just isomorphic to the unit interval [0, 1]. There are two semigenerated tribes,
namely, %o and 7j (which is isomorphic to {0, 1}). Since {xo) is finite, these are also the
only weakly generated tribes. Due to Theorem 3.4 each ??-tribe with ? e ]0, oof equals
either Too or T\, the same being true for each strict t-norm. The only 7l-tribes on {xq}
are Та and T„ with ? e N (as mentioned in Example 3.2(i)). Each Гм-tribe on {xq} is
isomorphic to some closed subset of [0, 1] which contains 0 and which is symmetric with
respect to the midpoint 0.5, and each Го-tribe on {xq} is isomorphic to some arbitrary
subset of [0, 1] which contains 0 and which is symmetric with respect to 0.5.
4. ?-measures and their representation by Markov kernels
Dehnition 4.1. Let ? be a t-norm, S its corresponding t-conorm, let ? be a ?-clan and
let m: ? —> [—oo, oo] be a function which assumes at most one of the values — oo and oo.
(i) The function m is called a T-valuation on Tif it satisfies the following conditions:
(TM1) m@) = O.
(TM2) m(ATB) + m(ASB) = m(A)+m(B).
(ii) The function m is said to be finitely ?-additive if it satisfies (TM1) and
(TM3) m(ASS) = m(A) + m(S) whenever ??? = 0.

Triangular norm-based measures
959
(iii) The function m is said to be monotone if
(TM4) m(A)<m(B) whenever ? ? ?.
Remark 4.2.
(i) If m:T —>¦ [-00, 00] is a ?-valuation on the ?-clan T, then m is also finitely
Г-additive, the converse not being generally true: if, for instance, ? consists
of all the constant functions in [0, \]x and if ? e [0, oof, then because of the
absence of any nontrivial ??-disjoint elements in the 7\-clan T, each function
m:T —>· [-00,00] which satisfies (TM1) is finitely TA-additive without being
necessarily a ??-valuation. This shows that ?-valuations are particular additive
functions in the sense of Schmidt [71, p. 558] and, consequently, if they are finite,
they can be represented as differences of monotone Г-additive functions (compare
[71, Theorem 2.2]). However, it is not possible to conclude directly that the
?-valuations have Jordan decompositions,
(ii) If ? is a Гь-clan and m is a finite T^-additive function on T, then m is also a
Tl-valuation,
(iii) If ? is an algebra of crisp subsets of X and if ?* is some t-norm, then the finite
r*-additive functions are ?-valuations for each t-norm ?.
?????????4.3.
(i) Let ? be a ?-clan. A function m: ? —> [—00, 00] is called a T-measure if it is a
?-valuation and if it is continuous from the left in the following sense:
(TM5) lim m(A,,) = m(A) whenever
n—»зо
(A„)„6n с ?, (?„)„6? / A and A e Т.
(ii) If a ?-measure m on ? assumes only values in [0,1 ]and if m(X) = 1, we shall call
m г. probability ?-measure.
(iii) Let ? be a Г-tribe. The function m: ? —> [—oo, 00] is said to be countably
?-additive if it satisfies (TM1) and if for each sequence (A„)„6n с ? we have
@0 \ °°
S A„ I = ]Рт(А„) whenever (А„)„6к is Г-disjoint.
It is clear that countably Г-additive functions are always finitely Г-additive. Moreover,
Г-measures on Г-tribes are countably Г-additive since if we take a Г-disjoint sequence
(Ai)neN in the Г-tribe ? and define Bn = S"=i An, then (S„)„6n is a sequence in ?
which converges nondecreasingly to S ^=1 A„ and, therefore,
m( S ?=??( lim B„) = lim m(Bn) = lim Ут(А„) = ) m(A„).
\ „_| / VH^DC ' ;l-»30 H-»30 '—' '—'

960
D. Butnariu and E.P. Klement
Remark 4.4. In general, for arbitrary t-norms ? and ?-tribes T, the countable ?-
additivity does not imply the continuity from the left condition (TM5), and thus countably
?-additive functions are not necessarily ?-measures. For example, if ? e [0, oof and
? = [0, l]x, then for each fixed xoe X the function m: ? -> [-oo, oo] defined by
[ 0 otherwise,
is countably ??-additive, but it is not a Ti-measure since it does not satisfy (TM5).
However, it will be shown later (see Proposition 8.1) that countably 7l-additive
functions are 7l-measures, that is, the countable TL-additivity and the continuity from
the left are equivalent in this case. The same is true when ? is an arbitrary t-norm and ?
consists of crisp subsets of X only. If ? is a ?-algebra of crisp subsets of X, then each
(?-additive) measure on ? is a Г-measure with respect to any t-norm ?. In this case, the
countable ?-additivity and the continuity from the left are equivalent too.
The following result will turn out to be useful when we represent ?? -measures by
Markov kernels.
LEMMA 4.5. If ? is a t-norm and if ? is both a T-clan and a Тм-clan, then each T-
valuation on ? is also a Тм-valuation on T.
PROOF. For А, В e ? we get
m(ATMS) +m(ASMS)
= m((ATMS)T(ASMS))+m((ATMS)S(ASMS))
= m(A ? ?) + m(A S ?) = m(A) + m(fi),
showing that m is indeed a Гм-valuation. ?
Remark 4.6.
(i) Lemma 4.5 shows that if m is a Г-measure on ? (? being both а Г- and а Гм-
clan), then m is also a Гм-measure.
(ii) Taking into account Theorem 3.3(i), each m is a ??-measure on a ??-tribe ? with
? e ]0, oo] is also a Гм-measure.
(iii) The converse of Lemma 4.5 does not generally hold: for instance, let ? be
the family of all Borel measurable fuzzy sets on X = [0, 1]. Then the function
m:T —» [—oo, oo] defined by
m(A)= / A+A(jt))d*
J\A>0}
is a Гм-valuation (even a Гм-measure) but not a T^-valuation.

Triangular norm-based measures
961
The Гм-measures play a fundamental role in the following exposition. In order to give
an integral representation for them we need the notion of Markov kernels (for more details
see, e.g., Bauer [5]).
Dehnition 4.7. Let (X, A) be a measurable space, В be the family of all Borel subsets
of [0, 1], and B\ = В П [0, 1[. A function К : ? ? ?\ -> R is called an ?-Markov kernel
if it satisfies the following conditions:
(i) for each ? eX, the function K(x,):B\ -> R defined by K(x,-)(B) = K(x,B) is
a probability measure on B\;
(ii) for each ? e B\, the function K(-, B) :X -> R defined by ?(·, B){x) = K(x, B) is
Д-measurable.
From Example 3.2(ii) we know that if (X,A) is a measurable space, then the family
Ал of all Д-measurable functions from X to [0, 1] is a Гм-tribe. The next result, from
Klement [43], shows that Гм-measures on Ал can be represented as integrals of Markov
kernels in the following way.
THEOREM 4.8. Let A be ? ?-algebra of crisp subsets of X and Ал the family of all
?-measurable functions in [0, \]x. Ifm is a finite monotone ?^/?-measure on Ал, then
there exists a unique measure m on A, namely, the restriction ofm to A, and an m-almost
everywhere uniquely determined ?-Markov kernel К : ? ? ?\ —> ?. such that for all A e A
m(A) = / K(x, [0, A(x)[)dm(x). D)
PROOF. The proof is carried out in several steps. First, observe that if D) holds for some
measure m on A, then m has to be necessarily the restriction of m to A since for ? e A
equality D) gives
m
i(M)= / K(x,[0, l[)dm(jt) = m(M). E)
Without loss of generality we may restrict our proof to the case when m(X) = 1.
CLAIM 1. For each rational number a e [0, 1 ], the function m„ : A -> [0, 1 ], defined by
ma(M) = m(a-M), F)
is a nonnegative finite measure on A, and it satisfies m„ ^ m.
Obviously, m„@) — 0 and m„ is finite and monotone because of the corresponding
properties of m. To prove ?-additivity of m„, let (A„)„6pj be a sequence of mutually
disjoint sets in A. Then
m« ( [J A„ ) = mi sup I a · ? [J A, \ I: ? e N

962
D. Butnariu and E.P. Klement
= sup mi a · I [J A,
sup ^т(а.Д,·)
ОС
]Гт(а· А„).
neN
? eN
n=l
The last equality follows from the monotonicity of m. Therefore, Claim 1 is proved.
Claim 1 implies that each measure m„ defined by F) is m-absolutely continuous. Hence,
for each a e [0,1] П Q there exists a Radon-Nikodym derivative
fa =
dm«
dm
Since the Radon-Nikodym derivative of an m-absolutely continuous measure is m-a.e.
unique, it follows that for all a e ]0, 1 [
/o = 0, m-a.e.,
/? = 1, m-a.e.,
fa =sup{ft: /3eQn[0,a[}, m-a.e.
G)
(8)
(9)
Therefore, changing (if necessary) the values of the functions fa on an m-null set, we may
assume that G) holds everywhere on X, and we still have
ma(M)= / /„dm.
Jm
For an arbitrary (not necessarily rational) a e [0, 1], define
ga = sup{fp: /3eQn[0,<*[}.
A0)
A1)
Since ga is the supremum of a countable family of Д-measurable functions, it is also A-
measurable, and for each ? e A we again have
Ша(М)
= I ga
Jm
dm
A2)
because of G) and the Monotone Convergence Theorem. Furthermore, for each ? e X the
function hx : [0,1] -> R defined by
hx(a)=ga(x)
A3)

Triangular norm-based measures 963
is a probability distribution function which determines a unique probability measure qv on
([0,1[,#|) which, whenever0 ^ a < ? ^ 1, satisfies
4*([?,0[)=?0)-??).
Claim 2. The function ? : ? ? ?, -> R <te/wi<;d fey
K(*,B) = qu(B) A4)
и an ?-Markov kernel, i.e., K(x,-) is a probability measure on B\ for each ? e X and
K(-, B) is A-measurable for each В &B\.
The first statement of the claim is trivial. The second one will be proved now. Consider
the set V = {С е B\: K(-,C) is Д-measurable}. This is a Dynkin system (see, e.g., [5])
on [0, 1 [. Indeed, [0,1 [ e V since for each ? e X
K(*,[0, l[)=q,([0, l[) = l.
Also, if C, D e V and С с D, then for each jc e X
K(x,D\C) = q.r@) - q.v(Q = K(x, D) - *(*, С),
and this shows that K(x, D\ C) is Д-measurable too. Finally, if (C,,)„6n is a sequence of
pairwise disjoint sets in V, then we have for each ? e X
(OO \ / OO \ OO 00
*. U c")=чЛ ^. U c")=Sit(Cn)=? ^^' c"}'
n = l / V n = l / ;?=? ;?=?
and, subsequently, ?(-,U,^i C„) is also measurable. Thus, ? is a Dynkin system on
[0, 1 [. Now, observe that for all ?, ? e [0, 1 ] with a < ? and for each ? e X v/c have
and this implies that ?(·,[?,?[) is Д-measurable, i.e., [a,fi[e V. Thus the Dynkin
system V contains all intervals [?, ?[ с [0, 1[ which means V = B\, and so Claim 2 is
proved.
Now we have to show that the Л-Markov kernel К defined in A4) indeed satisfies D).
To this end, let A be an Д-measurable step function A = ]T"=| <*; · A. where all a,- e [0,1]
and the sets A \, A2, · · ·, A„ form a partition of X. Then
ш(А) = У]т(а,-А,) = У" / gaf dm = Y] / K(*,[0,a/[)dm(*)
= ?]? ^(•i,[0,a,[)A,(;c)dm(jc)= f К(дг, [0, A(x)[)dm(x)-

964
D. Butnariu and E.P. Klement
If A is an Д-measurable fuzzy set and (?„)„6? 1S a nondecreasing sequence of A-
measurable step functions converging pointwise to A, then D) holds because of
m(A) = sup{m(A„): ? e N}
= sup|/ K(x,[0, A„(x)[)dm(x): n e N
= / K(x,[0,A(x)[)dm(x).
Jx
Finally, we have to show the uniqueness of K. Since for Д-measurable functions
«,v:X->Rwe can conclude и = ? m-a.e. if for all ? e ? we have
/ и dm = / ? dm,
Jm Jm
it follows that К is m-a.e. unique in the sense that any others-Markov kernel satisfying D)
may differ from К on an m-null subset of X only. ?
Taking into account Theorem 3.3(i), Proposition 4.5 and Remark 4.6(ii) we can restate
Theorem 4.8 in the following form for measures based on Frank t-norms.
COROLLARY 4.9. IfTi is a Frank t-norm with ? e [0, oo], ? is a generated ??-tribe and
m is a finite monotone ??-measure on T, then there exists a unique measure m on Tv,
namely, the restriction o/m to Tv, and an m-a.e. uniquely determined Tv-Markov kernel
К : ? ? B\ -> ? such that the representation D) holds for all A e T.
5. Integral representation of 7l-measures
Theorem 4.8 and Corollary 4.9 show that monotone finite measures based on Frank t-norms
?? and defined on generated ??-tribes can be represented as integrals of Markov kernels.
It is clear that this holds for ? = T^ too. However, in this particular case the result is true
even if the underlying TL-tribe ? is not generated. This is based on the following properties
proved by Butnariu [14], which are specific for TL-measures.
PROPOSITION 5.1. If ? is a T^-tribe and ifm is a finite T^-measure on T, then we have:
(i) m и monotone if and only ifm is nonnegative;
(ii) г/m is nonnegative, then m is continuous in the sense that
lim m(A,,) = m(A) whenever (А„)„ец С.Т and lim A„ = A. A5)
n—*oo ii—>oc

Triangular norm-based measures
965
PROOF. In (i) the "only if" part is trivial. Assume that m is nonnegative, А, В e ? and
A sC B. Then В = A SL(B TL С A) and A JL(B TL С A) = 0. Thus
m(B) = m(ASL(STLCA))+m(ATL(BTLCA))=m(A) + m(STLCA),
i.e., m(B)^m(A).
In order to prove (ii), let (A„)„6pj be a nonincreasing sequence in ? converging
pointwise to A. Then
lim m(A„) = m(X)- lim m(CA„) = m(X) - m(CA) = m(A)
because of the continuity from the left of m and since for all С е T,C and CC are
TL-disjoint and satisfy CS|_Cc = X. Now, if (A„)„6n is a (not necessarily monotone)
sequence in ? converging to A, then
ОО/СЮ \ ОС / ОС \
Tm I Sm A„ I = A = Sm I Tm A„ I.
k = \ \ n=k ) /t = l V n=-k /
and ? is a Го-tribe (compare Theorem 3.3(i)). Therefore, we have
mi Tm Ak\ ^ m(A„) ^ ml Sm Ak \
and, taking here the limit ? -> oo, we obtain m(A) = lim,,^^ m(A„). ?
The following representation of TL-measures is essentially Theorem 2.6(c) of Butnariu
[15]. We present it here with an alternative proof (see Butnariu and Klement [17,
Theorem 6.2])
THEOREM 5.2. If ? is a T^-tribe and m is a finite nonnegative T^-measure on it, then
there exists a unique measure m on Tv, namely, the restriction ofm to Tv, such that for
each A in ?
m(A)= / A(jc)dmU). A6)
PROOF. It is clear that E.2) implies that m has to be the restriction of m to Tv.
Claim 1. // A e ?, ?, ? e [0, 1] and a < ?, then the set
Aa.p = {xeX \a< A(x) ^ ?} A7)
belongs to Tv, and the product A ¦ ??_? = AT|_ ??.? e T, and
?-??(??.?)^ m(A· Aa_p). A8)

966
D. Butnariu and E.P. Klement
The first assertion follows from Theorem 3.3(H). The second results from the fact that
for each ? e Tv we have A ¦ ? = A T|_ ? e T. Now, in order to prove A8), it is sufficient
to show that this inequality holds for all A in ? and for all ?, ? e [0, 1], such that a < ?
and a has a finite binary representation
k
2'
i = l
with ieN and a,- e {0, 1} for each i e {1,..., к]. Suppose that A8) holds whenever ? has
a finite binary representation. For all 0 < a < ? ^ 1 we can find a sequence (a„)„6pj which
is nonnegative, nonincreasing and converges to a, and such that each an has a finite binary
representation as above. Using the continuity of m (see Proposition 5.1 (ii)) we get
a ¦ m(Aa a) = lim a„ · m(A„„ «) < lim m(A · A„ «) = m(A · ??_?),
since (A„n,/j) gN \ A„,/j. Now assume that a has the form as above. If a = 0 there is
nothing to prove. Suppose a > 0. In this situation we proceed by induction upon the
number к in the binary representation of a. If к = 1, then a = 0 or a = \. In the first
case A8) clearly holds. In the second case we have ??_? = (A · A„./j)Sl(A · A„,/j), and
this implies
m(Aa./j) = m(A„./0
= m(A · ??_?) + m(A · ??.?) - m[(A · A„./j)TL(A · ??_?)]
^ 2 · m(A · ??.?),
which is exactly A8) with a = \. Suppose that A8) holds for all A e ? and for all ?, ? in
[0, 1] such that a < ? and a has a binary representation with at most к binary digits. Put
k + \
2'
with a, e {0,1} for each ? e {1,..., m + 1} and suppose that a*+| ^ 0. Then a = 8/2
where
k + \
;=i
Case 1.1. 5 < 1 and ? < \. Then Aa.p = (AS|_ А),5.2/з, and using the inductive
assumption for the set (ASlA)^,2/j (this is possible since the sum in A9) has at most
к nonzero terms in our case) we get S · m(Aa,/j) ^ m[(AS|_ A) · ??,?]. Observing that

Triangular norm-based measures 967
(A SL A) · A„,/j = (A · A„,/j) SL(A · A„./j), we obtain
5 · m(Aa.p) sC m[(A SL A) · ??,?]
^m[(A· A„./j)SL(A· A„./j)]
= 2 · m(A · A„./j) - m[(A · A„./j)TL(A · Aff./j)]
< 2 · m(A · ???)
by the monotonicity and additivity of m (see Proposition 5.1(i)). This implies A8) in this
case.
Case 1.2. S < 1 and ? = +. Consider a sequence (y„)neN in ]<*, ?[ which converges
increasingly to ?. The sets A„, Yn satisfy the conditions of Case 1.1. Thus we have for each
a-m(Aa.Yn) ^m(A-Aa.Y„).
B0)
The continuity of m (compare Proposition 5.1(H)) and the convergence (Aa-Yil)„e^ /
Sm,^i AcYn imply
?·??( Sm Aa.y„ J ^m
n=\
A ¦ ( ом Aa,y„
;?=?
B1)
where
( U A«-r„ ) U {* e X: А(дс) = ?} = ???.
Since the sets in the latter union are disjoint, we may write
a ¦ m(Aa.p) = a ¦ ml [J Aa,Y„ J +a ¦ m({A = ?))
? A-^Sm AaA + <*-m({A = /*}).
С m
B2)
Since for ? = J the fuzzy sets B\ and Bz defined by
Bi(x) = B2(x) =
0 ???(?)??,
? ???(?) = ?,
are Tl-disjoint and are contained in T, we have m(S| S|_S2) = 2 · m(/3 · {A = /3})
(compare Remark 4.2(i)) and S| Sl St = ? = /3}. Hence,
m
{?-{? = ?))=?- m({A = /3}) > <* · m({A = ?\).

968
D. Butnariu and E.P. Klement
Combining this with B2) and using the TL-additivity of m we deduce
? -?\(??_?) ^ m
m
A-(Sm AaA + ??(?-[? = ?})
А-({А = Р)и(?лАаЛ\
= m(A· ??.?).
This proves A8) in this case.
Case 1.3. S < 1 and/3 > ±. Then A · ??.? = {A-Aa.|/2)SL(A · A|/2./j), where the fuzzy
sets on the right hand side are 7l -disjoint. Hence, according to Remark 4.2(i) we have
m(A · ??_?) = m(A · AaA/2) + m(A · ?\/2.?).
The first term on the right-hand side satisfies the conditions of Case 1.2 and the second one
meets the inductive assumption. Hence,
m(A· А„./з) >а-т(А|/2./0+ ? · ??(?\/2.?) > <* ¦ [т(А„.|/2) + ??(?\/2.?)]
= a ¦ т(А„./з)
showing that A8) holds in this case too.
Case 1.4. S > 1. In this case a can be written as a = (? + l)/2 where
??'+? гг. ??
-?- e [0, 1].
?=?
We also have A„,/j = [ATL A]^2p with ? = ? - 1/2. Using the inductive assumption for
the set [A T|_ A] - we obtain
1 ? ? ?
a ¦ т(А„./з) = - · m(A„.^) + - · m(Aa.p)
= --m(Aa.fl)+?--m([ATLA]?2p)
^-•m(Aa./j) + --m([ATLA]?2p)
= - · &(??.?) + ? · т(Ас.р ¦ [ATL A]e 2p).
B3)

Triangular norm-based measures 969
Observe that Aa.p · (ATLA) = DJlCC with D = ??.? and С = ??.? ¦ (CaSlCA).
According to Remark 4.2(i), it follows that whenever E, F e ? and ? ^ F then
m(?TLCF)=m(?)-m(F). B4)
Since we clearly have D > C, B3) combined with B4) gives
a ¦ m(Aa.fl) < 3 · m(Aa.p) + 5 · [m(O) - m(C)] = m(A„./)) - \ ¦ m(C). B5)
Now, taking into account <5 > 1, we deduce
C = (CA-A0./i)Sl(CA-A0./») and (CA-Ao./j)Tl(CA-Ao./j)=0.
Thus m(C) = 2 · m(CA · A„./j) by the TL-additivity of m. Since
CA · ??? = ??.? TL C(A · A„./j),
we get m(C) = 2 · [т(А„./з) - m(A · A„./j)] because of B4). Substituting this into B5) we
obtain A8), and Claim 1 is completely proved.
Claim 2. If A e T, 0 < a < ? ^ 1 and
Ao.fl = {xeX\ot <?(?)<?) B6)
then ??,? e Tv, A · Aa^ e ? a/id
m(A-Aa.p)^fi-m(Aa_p). B7)
The first two assertions are obvious. We only have to prove B7). To this end, observe
that
А0./, = СЛ>.0. B8)
where a' = 1 - a and /3' = 1 - ?. Then
m[C(A · ??,?)] = m(X) - m(A · A„./j)
= т(А„./з) + т(СА„./з) - m(A · ??_?)
= m(CA · A„./j) + m(CA„./j)
= m(CA · СА/з-,?') + m(CA„./j)
>/3'· m(X) + т(САм)
= т(Х)-/3-т(А„^),
where the forth equality is a consequence of B8), and the inequality follows from Claim 1.
This implies B7), and Claim 2 is proved.

970
D, Butnariu and E.R Klement
In order to complete the proof, choose A e T. For a given ? e N denote
{xeX: A(jt) = 0} if i = 0,
"¦' ' A(,--i)/2»,,72» iff e {1,2, ...,2я},
_ ??)/2-.(,4-?)/2- if/e{0, l,...,2"-l},
{jceX: A(;t) = 1} iff = 2".
From Claims 1 and 2 it follows that the step functions
2" . 2" .
i=l ?'=?
are all Tv-measurable. It is clear that (s„) / A and (t„) \ A. Taking into account A8)
and B7) we deduce
у ? 1 ~
/ s„ dm = 22 —^- ¦ m(G„.i) < 2jm(A · G,,.,) = ?
2"
; — ? .—.
:m(A)
i = l " i=0
and
111 "}/t
? ?„ dm = J2 ^ ¦ MH„j) > j^miA ¦ HnJ) = m(A).
? = ? ?=0
Taking the limit ? -» oo in the last two inequalities we obtain A6), completing the proof
of the theorem. ?
Comparing Theorem 5.2 with Corollary 4.9 it is easily seen that the 7l-measures on a
generated ?? -tribe with ?? a Frank t-norm are exactly those ?? -measures for which the
corresponding Markov kernel K, in the representation D), for each ? e X can be written
in the form
?(?,[?,?[)=?-??.
This suggests that the concept of Lebesgue integral could be axiomatically defined as a
7l -measure.
6. Integral representation of monotone ?? -measures
Theorem 5.2 not only drops the condition that ? be generated in Corollary 4.9 in the case
? = 7l, but also specifies the form of the Markov kernel occurring in the representation D)
in this particular case. For Frank t-norms ?? with ? e ]0, oof and generated 7\-tribes it is
also possible to specify the form of the Markov kernel in D). To be precise, we get the
following result of Klement [45].

Triangular norm-based measures
971
THEOREM 6.1. If T\ is a Frank t-norm with ? e ]0, oof, ? is a generated ??-tribe and
m is a monotone finite ??-measure on T, then there exists a unique measure m on Tv,
namely, the restriction of m to Tv, and an m-a.e. uniquely determined Tv-measurable
function f: X -> [0, 1] such that for all A e ?
m(A)= / [/ + (!_/). a] dm. B9)
|A>0)
'=/" [/ + (l-/)-A]dm
./|A>0)
PROOF. First, observe that if B9) holds for some measure m on Tv and for some Tv-
measurable function / on X, then, necessarily, m has to be the restriction of m to Tv.
Now let us show that the required function / exists. To this end, consider the Tv-Markov
kernel К corresponding to m in the representation D). For each a in [0, 1] define the
function fa : X -> [0, 1 ] by
fa(x)=K(x,[0,a[). C0)
Each fa is Tv-measurable since К is a Tv-Markov kernel. We claim that there exists a
function /: X -> [0, 1] such that for all a,be]0,\[ with a + b ^ 1,
fa+b = fa + fb- f m-a.e. C1)
In order to prove that, consider the sequences (a„)„6pj and (Ьп)„ещ defined as follows:
a\ =a, b\ = b,
a„+\ = S\(a„,b„), bn = Tx(an,b„).
Since ? is generated, these sequences belong to ? (compare Theorem 3.3(iii)). Exactly
as in the proof of Theorem 3.3(i) we show that the sequence (??)?6? IS nondecreasing,
(bn)neN is nonincreasing and converges to zero, and for each ? e N we have a„ + b„ =
a + b. Define the Tv -measurable function
/ = inf{//,„:«eN}. C2)
For each ? in Tv, the fuzzy sets A„ = an · ? and Bn = b„ ¦ ? are contained in ?
(compare Theorem 3.3(i)), and (An)neN / A| Sl B\ and (S„)„sN \ 0. Also, for each
? e N, we get A„ + S„ = (a + b) ¦ M. Induction upon ? yields that m(A„) + m{B„) =
m(a ¦ M) + m(b ¦ M) for all ? e N. Consequently, we have
/ fa+b dm = m[(a + b)-M]= m(sup{ A„: ? e N}) = sup{m(A„): ? e N}
= m(a ¦ M) + m(b ¦ M) - inf{m(B„): пеЩ
= ? {fa + ?) dm - inf{m(B„): ? e N}

972
D. Butnariu and E.P. Klement
by C0), D) and the continuity from the left of m. From C0), D) and the Monotone
Convergence Theorem we also have
f{m(S„): ? e N} = infl ? fbn dm: ? e N
= ( inf{//,„:«eN}dm= f /dm.
Jm Jm
Hence,
/ fa+bdm = / (fa + fb- /)dm
Jm Jm
for each ? in Tv. This implies C1), and our claim is proved. Equation C1) is a Gaussian
functional equation (see Aczel [1]) with the border conditions f\ = 1 and /o = /· The
unique solution of such an equation has to be /, = / + A - /) · a m-a.e. for all a in ]0, 1 [.
Taking into account C0) this gives
K(x,[0,a[)=f + (\-f)a m-a.e.
for a e [0, 1]. Inserting this into D), we get B9) and the proof is complete. ?
Theorem 6.1 is proven for 7\ -measures on generated 7\-tribes when ? e]0, oof.
Combining this result with Theorem 5.2 we get the following corollary.
COROLLARY 6.2. IfTi is a Frank t-norm with ? e ]0, oo], ? a generated ??-tribe and m
a monotone finite ??-measure on T, then there exists a unique finite measure m on Tv,
namely, the restriction o/m to Tv, and an m-a.e. uniquely determined Ty-measurable
function f : X -> [0, 1], such that for all AeT the equality B9) holds.
It is obvious (compare Theorems 3.3(i) and 5.2) that Гь-measures on 7\-tribes with
? e ]0, oof are also 7\-measures (even if the ??-tribe is not generated). However, not each
??-measure is necessarily a TL-measure, even on generated ??-tribes. The following shows
that a ??-measure on a generated 7\-tribe is a TL-measure (and hence an integral) only if
it is continuous from the right at 0 (and then, clearly, at each A e T).
COROLLARY 6.3. Let ? e ]0, oo[, ?? be the corresponding Frank t-norm and ? be ? ??-
tribe. A monotone finite ??-measure топ ? is a ?^-measure on ? if and only if it satisfies
the following condition:
lim m(A,,)=0 whenever (Ап)пец С.Т and (A„)ne^\&. C3)
> oo
PROOF. From Proposition 5.1 we already know that TL-measures are continuous and,
therefore, satisfy C3). Conversely, assume that m is a ??-measure on the generated 7\-tribe
? satisfying C3). Then m can be represented in the form B9) and for each ? in Tv the

Triangular norm-based measures
973
sequence (M„)neN = (^ · M)„6n is contained in T. Using the Monotone Convergence
Theorem we deduce that for each ? e Tv
0= lim f Г/ + A-/).Л dm = ? /dm.
"^¦^JmL n\ J M
This implies that / = 0 m-a.e., and rewriting B9) in this case we obtain for A e ?
m(A) = / Adm = / A dm
J\A>0\ JX
l[A>0) JX
which immediately implies that m is a 7^ -measure. ?
7. Decomposition of monotone 7\-measures
Theorem 6.1 shows that for ? e]0, oof finite monotone Гл-measures m which are
defined on generated tribes ? differ from 7l-measures (i.e., from integrals according to
Theorem 5.2) only by functions of the form
At—> / /dm,
which are themselves monotone finite 7\ -measures on T. The question we deal with in this
section is how much a ??-measure defined on a nongenerated 7\-tribe differs from a 7l-
measure (i.e., from an integral). The results are drawn mostly from Butnariu and Klement
[16], some important completions of the proofs there are due to Mesiar [60] and Mesiar
and Navara [62] (see also Section 8 in Butnariu and Klement [17]).
PROPOSITION 7.1. Let ? e ]0, oo], ? be a ?-,.-tribe and m: ? -> ? a finite monotone ??-
measure on T. Then there exists a unique pair (m^, m-A) of functions from ? to [0, oof
such that:
(i) mL is a T^-measure on T;
(ii) niA и а ??-measure on T\
(iii) m = mL + m;u
(iv) mL is the greatest ??^-measure in the sense that we have mL > ? whenever
?: ? —> [0, oof и a T^-measure such that m — ? is monotone;
(v) There exists a unique measure m on Tv, namely, the restriction ofm to Tv, and
an m-a.e. unique Tv -measurable function f: X -> [0, 1] such that for all A e ?
mL(A)= / (l-/)Adm C4)
Jx
and for all ? e Tv
???(?)= /" /dm. C5)

974
D. Butnariu and E.P. Klement
PROOF. Let M(m) be the set of all TL-measures p:T -» [0, oof, such that m - ? is
monotone (and, therefore, a Ti -measure due to Remark 4.6(H) and Corollary 6.3). The set
M(m) is nonempty since it contains the zero measure. M{m) is naturally equipped with
the partial order
? ^ q if and only if p(A) sC q( A) for all AeT. C6)
CLAIM 1. There exists a maxima! element in _M(m).
If (Pj)jej is a chain in M(m), then the function ?: ? -н>- [0, oo] defined by
p(A) = sup{py(A): j e j} C7)
is a ^-valuation. Indeed, p@) = 0 clearly holds and for all А, В e ? we have
p(ASLfi) + p(ATLfi) =sup{p,(ASLfi): j e J} +sup{p,(ATLB): jej]
= sup{p7(ASLS)+py(ATLS): jeJ}
= sup{py(A) + py(S): jeJ}
= ?(?)+?(?)
where the second and the fourth equality follow from the monotonicity of the chain
(Pj(C))jej for each С e T. It is clear that ? is monotone. Hence, 0 ^ p(A) ^ m(A) ^
m(X) for all A&T, implying that ? is also finite. If (??)«6? is a nondecreasing sequence
in T, then
lim p(A„) = sup{p(A„): ? e Nl = sup{sup{py(A„): ;' e J}: ? e Nl
= sup{sup{py(A„)·· ? e N}: j eJ}= supl^liir^p^A,,): j e J)
= p( lim A„),
showing that ? is continuous from the left. Hence, ? is a finite nonnegative TL-measure
on T. Moreover, m — ? is also monotone since for all A, В e ? with ? ? ? it follows
from pj e M(m) that
(m - p)(A) = inf{(m - py)(A): j e J} < inf{(m - py)(S): jeJ]
= (m-p)(S),
thus showing that ? e M(m). In other words, each chain in M(m) has an upper bound
in M(m) and, by Zorn's Lemma, M(m) contains a maximal element denoted mL. The
7l-measure mL is also a 7\-measure, and so is the difference
??? =m-mL.
C8)

Triangular norm-based measures
975
Because of niL e M(m) we know that m^ is monotone and finite, and the pair (niL, m^)
satisfies properties (i)-(iii). The following claim [60,62] shows that M(m) is a directed
set, thus immediately implying that m^ (and subsequently ш;) are uniquely determined.
Claim 2. For all qi, Ц2 e M(m) there exists a q e .M(m) smc/i ?/??? qi ^ q and q2 ^ q.
Since qi, q2 are ^-measures, we may write for i e {1, 2}
q,(A) = / Adq,,
Jx
where the finite ?-additive measure q, is the restriction of q, to Tv . Then qo = qi + q2
is also a finite-additive measure on Tv, and q, is absolutely continuous with respect to
q for i e {1,2}. Then the Radon-Nikodym Theorem provides Tv-measurable functions
h ?, /12: X -> R such that
-l·"
q,(L)= / /i,-dqo
for all L e Tv and ? e {1,2}. Define the Tv-measurable function /i:X -> R by /i =
max(/i ?, /12) and the finite ?-additive measure q on Tv by
q(L)= / ftdqn.
Then the function q: ? -+ [0, oof given by
q(A)= /"Adq
is a finite monotone TL-measure such that q, < q for i e {1,2}. The monotonicity of m - q
can be seen as follows. Consider the crisp sets ? — {h = h \} and Q = [h > h \}. Obviously
P, Q e Tv and ? U Q = X and ? ? ? = 0. Note that for any fuzzy set С е [0, \]x and
for any crisp subset L с X we always have CT/ L = С · L. Therefore, using the fact that
for fuzzy sets D, ? e ? with Z) < ? and ? < Q we always have q(D) = qi (D) and
q(?) = q2(?), we obtain for all А, В e ? with A < S
(m - q)(A) = (m - q)(ATA P) + (m - ?)(??? ?)
= (??-4|)(????) + (??-42)(????)
<(??-4|)(????)+(??-?2)(????)
= (m - q)(STA P) + (m - q)(ST/ ?)
= (m-q)(S),
showing that m - q is monotone. This completes the proof of Claim 2 which together with
Claim 1 implies that (mL, ш;) satisfies (iv) too.

976
D. Butnariu and E.P. /Clement
In order to prove (v), we take m for the restriction of m to Tv. We have to find a Tv-
measurable function /: X -> [0, 1] such that C4) and C5) hold. To this end, observe that
? is a TL-tribe by Theorem 3.3(i). Hence, according to Theorem 5.2, the ^-measure mL
can be written as
mL(A)= f AdmL C9)
Jx
for each A e ? where riiL is the restriction of mL to Tv. From mL ^ m it follows that riiL
is absolutely continuous with respect to m. Because of riiL < m there is a Radon-Nikodym
derivative
d(mL)v
dm
which maps X into [0, 1]. Putting / = 1 - driii/dm, then / is also a Tv-measurable
function with values in [0, 1], and C9) implies C4). Now, taking into account C8), we can
write for all crisp sets L e Tv
mA(L) = m(L)-mL(L) = m(L)- /*(l-/)dm= I /dm,
and therefore, C5) also holds. Formulae C4), C5) and C8) imply that no other choice
for m (and, subsequently, for /) is possible. ?
Remark 7.2.
(i) The component m^ of a TA-measure m, which exists due to Proposition 7.1, is
a pure ??-measure in the sense that it has a zero TL-component: (т^)ь = О. In fact,
assuming the contrary contradicts the maximality of mL in Proposition 7.1.
(ii) If the ??-tribe ? is generated, then C5) somehow holds for each A e ? (and not
only for crisp sets in the ? -algebra Tv). To be precise, we always have
J\A>C
?»?(?)= / /dm.
'|A>0)
This follows from B9) in combination with C4) and C5), and the m-a.e. uniqueness
of / in B9).
Let ? e ]0, oof and ? be a T^-tribe. If ? is a ?-additive measure on the ?-algebra Tv and
if g, h : X ->¦ [0, oo] are two Tv-measurable functions, then the function m: ? -> [0, oo]
defined by
m(A)= / (g+A-A)dp D0)
J[A>0\
is a monotone 7\-measure on T. Indeed, monotonicity is obvious as is m@) = 0;
continuity from the left follows from the Monotone Convergence Theorem, taking into

Triangular norm-based measures
977
account that for a nondecreasing sequence (?„)„6? <? ? which converges pointwise to A
we have
/1
\J[A„>0} = {A>0}.
The 7\-additivity is shown as follows:
??(????)+??(?3??)
[g+h(AJxB)]dp+ [ [g+h(ASAB)]dp
,?>0)?|?>0) J[A>0)U[B>0)
= ? [2-g+A.((ATxB) + (ASiB))]dp
J[A>0)n\B>0}
+ [ [g + h(AS/iB)]dp+ ? [g+h(ASAB)]dp
J\A>0}n\B=0) J\A=0)n\B>0)
= / [2-g+h-(A + B)]dp+ I (g + h-A)dp
J\A>0}n\B>0) J[A>0)n[B=0)
+ / (g+h-B)dp
J[A=0}D\B>0}
= / (g + h-A)dp+ ? (g + h-B)dp
J[A>0} J\B>0}
= m(A) + m(B).
Note that the measure ? involved in the representation of m via D0) does not necessarily
coincide with the restriction of m to Tv.
Dehnition 7.3. ? ??-measure m on ? which can be represented in the form D0) for
some nonnegative ? -additive measure ? on Tv and for some ??-measurable functions
g,h:X^f [0, oo] is said to be generated (by p. g and h).
Theorem 6.1 shows that if ? is generated, then all finite monotone 7\-measures on ?
are generated. From Theorem 5.2 we already know that Tb-measures on ? are always
generated, even if ? is not generated. Thus, it is natural to ask whether, in general, ??-
measures on T^-tribes are automatically generated. In order to answer this question we
introduce the following notion.
Dehnition 7.4. ? ?? -measure m on a ??-tribe ? is called monotonically irreducible if
it is monotone and if the zero measure on ? is the only generated 7>. -measure q on ? such
that m — q is monotone.
It is obvious that a T\-measure m is generated if and only if it can be extended to a
?? -measure on the generated 7\-tribe (??)?: if m is generated, then it is always defined
¦/

978
D. Butnariu and E.P. Klement
on the whole generated tribe (??)?; the converse follows from Theorem 6.1. On the other
hand, monotonically irreducible ?? -measures, with the exception of the zero measure, are
not generated and, therefore, cannot be extended to (??)?.
THEOREM 7.5. Let ? e]0, oo], ? be a ??-tribe and m a finite monotone ??-measure
on T. Then m can be uniquely decomposed into the sum of a monotonically irreducible and
a generated ??-measure, i.e., there exist a unique monotonically irreducible ??-measure
monT, a measure m on Tv (namely, the restriction ofm to Tv) and two m-a.e. uniquely
determined Tv -measurable functions g,h:X—> [0, 1], such that for all A e Tv
m(A)=m(A)+ / (g + h-A) dm. D1)
PROOF. If ? = oo, then the result follows from Theorem 5.2 taking for g and/i the constant
functions g(x) = 0 and h(x) = 1, respectively, and for m the zero measure. Assume,
therefore, that ? e ]0, oof. In this case we proceed in several steps.
CLAIM 1. If ? is a finite monotone ??-measure on T, then there exists a finite monotone
??-measure ? on the generated ??-tribe (??)? which is maximal in the sense that the
difference ? — q is also monotone on (??)? whenever q is a ??-measure on (??)? such
that ? — q is monotone on T.
In order to verify this, let JV(p) be the set of all monotone 7\-measures q on (??)?
such that ? — q is monotone on T. This set is partially ordered by the dominance relation
defined by
q^>r if and only if q — г is monotone on (Tv) . D2)
Let (qj)jej be a chain in jV(p), and for each A e (??)? define
q(A) = sup{q,-(A): j ej}.
Exactly as in the proof of Proposition 7.1, it follows that q is a monotone finite TA -measure
on (??)?. For A,B eT with A ^ В we have
p(A) - q(A) = inf{p(A) - qy (A): ; e J} < inf{p(S) - qy(S): j e J}
= p(B)-q(B),
i.e., ? - q is monotone on T. For each к e J and for А, В e (??)? with A < B, the fact
that (qj)jej is a chain induces
(q - q*)(A) = sup{(q; - qk)(A): je^ sup{(qy- - q*)(B): jej}
= (q-q*)(B),

Triangular norm-based measures 979
which means that q — q* is monotone for each к e J, i.e., q ^> q*. Hence, each chain
in jV(p) has an upper bound in JV(p) and, by Zorn's Lemma, this implies that Л/"(р),
equipped with the dominance relation ^>, contains a maximal element, completing the
proof of Claim 1.
The following claim shows that ??(?) is also a directed set, implying that the maximal
element is indeed the unique greatest element.
Claim 2. For all q\, q2 e Л/"(р) there exists a q e Л/"(р) such that q » qi and q » q2-
Let qi, q2 be the restrictions of qi and q2 to Tv, respectively, let (P, Q) be a Hahn
decomposition of the ?-additive signed measure ?| - q?, and define q: (??)? -> [0, oo]
by
4(?) = 4,(????)+42(????).
Obviously, q@) = 0 and q is continuous from the left. To see that q is also a
revaluation, take into account that ? and Q are crisp subsets of X, and then we get for
all A, Be (??)?:
q(ATAS)+q(ASAB)
= 4,(???????)+42(???????)
+ qi((AS,S)T,P)+q2((AS,S)T,e)
= 4?((????)??(????))+4?((????K?(????))
+ q2((AT, ?)??(??? Q)) +q2((AT, Q) S,(STA Q))
= 4?(????)+4,(????)+42(????)+42(????)
= q(A) + q(fi).
The ??-measure q is monotone since for fuzzy sets А, В e ? with A ^ В we have
q(A) = (| (??? ?) + 42(??? Q) <? (| (??? ?) + qi^T;, ?) = q(S).
The monotone 7\-measure q is an element of A/"(q) since for fuzzy sets А, В е (??)?
with A < В we get
(p-q)(A) = p(A)-q(A)
= ?(??? ?) +?(??? ?) - (| (??? ?) - 42(??? Q)
= (?- 4,)(??? ?) + (? - ?2)(??? ?)
^(?-4,)(????) + (?-42)(????)
= ?(?)-4(?)
= (p-q)(B),

980
D, Butnariu and E.P. Klement
where the inequality is a consequence of q ?, q2 e Л/"(р). То complete the proof of Claim 2,
it remains to show that q J?> qi andq^> q2,i.e., q —qi andq — q2 are monotone on (??)?.
For each A e (??)? we get
D-4?)(?)=4?(????)+42(????)-4?(????)-4?(????)
= D2-4,)(????)>0,
showing that q - q ? is nonnegative. This also implies that the restriction of (q - q ?)v to the
?-algebra of all crisp sets of the form L П Q with L e Tv is nonnegative and, therefore,
monotone. Applying Theorem 6.1 to the 7\-measure qi — qi which is monotone on the
??-tribe of all fuzzy subsets of Q contained in (??)?, we get for all А, В e (??)? with
A ^ В and for a suitable ??-measurable function k:X->[0.l]:
(q-qi)(A) = (q2-qi)(ATxj2)= [ {k + A - k) · A) d(q2 - q, )v
J\A7;.Q>0]
= [ (* + (l-*)-A)d(q2-qi)v
J[A>0)nQ
?? [ (? + A -A:)-B)d(q2-qi)v
J[A>0}nQ
+ 1 (* + (l-*)-S)d(q2-qi)v
J\A=o\n[B>o\nQ
= [ (A: + (l -Ar)-B)d(q2-qi)v
J\B>0)nQ
= D2-4\){BTkQ)
= (q-qi)(B).
Monotonicity of q — q2 is shown by complete analogy, interchanging the roles of q ? and
q2 as well as of ? and Q, thus completing the proof of Claim 2.
Now let in be the greatest ?? -measure (with respect to the dominance relation ») in
jV(m) resulting from Claims 1 and 2. Since in is defined on (??)?, Theorem 6.1 yields
the representation
m(A)= [ [/-(l-/).A]d(in)v D3)
where (in)v is the restriction of in to ((??)?)? = Tv and /: X -> [0,1] is a suitable Tv-
measurable function. Remark 7.2(H) tells us that the unique decomposition ((in)^, (т)л)
of in (compare with Proposition 7.1) is given by
(in)L(A)= /"(l-/).Ad(m)v, D4)
Ух

Triangular norm-based measures
981
(mMA)= [ (l-/)d(in)v. D5)
J[A>0\
Let (ть, шл) be the unique decomposition of m resulting from Proposition 7.1, let rii be
the restriction of m to Tv, and let f: X -> [0, 1] be the Tv-measurable function such
that C4) and C5) hold.
Claim 3. (in)L = mL.
From in e ??(?\) it follows that the function г: ? -н>- R defined by г = m - in is
a monotone ??-measure. Using m - (m)L = г + (??)?, the 7\-measure m - (m)L is
monotone too. Since niL satisfies Proposition 7.1(iv), we get
mL ^ (m)L. D6)
Conversely, note that m - ть is a monotone ??-measure on T, which, together with the
fact that in is the greatest element in Л/"(т), implies that in ^> m^, from which we deduce
that in - niL is monotone, showing that niL e .M(in). Since (in)L is the greatest element
of .M(in), we get (in)b ^ niL and, together with D6), (т)ь = ть-
CLAIM 4. The monotone ??-measure m: ? —> [0, oof defined by
m = m - in D7)
is monotonically irreducible.
In the case in = m the claim is proved by definition. Suppose that in ? m, i.e., m is
different from the zero measure. Then Claim 3 yields
m = m-in = mL+m/- (in)L - (??)? = ??? - (??)?
= ??? - (???)~ + (???)~ - (??)?.
If we are able to show that
(???? = (??)?, D8)
then the proof is complete because of the maximality of {тхУ in jV(mx). In order to
prove D8), note that (тя)ь is identically zero on (??)? because of m = mL + (m/)L +
(ша)а and the maximality of mL in Л/"(т). Similarly, ((???)~)?, is the zero measure on
(??)?. Observe that
m - (mx)~ = mL + (??? - (???)~),
the ?? -measure on the right-hand side being monotone on T. Since in is the greatest
element in JV(m), we have in » (m\)~. From the monotonicity of m = m^ - (??)? and

982
D. Butnariu and E.P. Klement
the maximality of (ша)~ in ?{?\?), we deduce that (m/)~ - (in)/ is monotone and,
subsequently, (т;)~ ^ (??)?. Assume that {m\)~ ? (in)/, and define the monotone T^-
measure m: (??)? -> [0, oof by m = (in)L + (mi)". Then we have
m - in = (m)L + (???? - m = (m)L + (??)? - (m)x + (mA)~ - m
= (???? - (??)?, D9)
the latter being monotone, showing that m ^> in. On the other hand, Claim 3 yields
m-m = mA - (шаГ,
which implies that m — m is monotone too, i.e., m e Af(m). Therefore, m = in which,
together with D9), contradicts the assumption (тд)" ф (in)/, thus completing the proof
of Claim 4.
Now D7) shows that m = m + in, where m is monotonically irreducible and in is
generated by definition. It remains to show that D1) holds. From Claim 1 we know that
(m)v < m and, therefore, (m)v is absolutely continuous with respect to m. Thus we may
assume that the Radon-Nikodym derivative
d(in)v
dm
has values in [0,1] only. Taking into account D5) we obtain for each A e (??)?
(m)k(A)=[ ld(m)v=f /.^-dm.
J[a>0] J[a>o\ dm
Writing
d(in)v
g=l- —r^-
dm
and taking into account Claim 3 and C4), we have for all A e (??)?
in(A) = (m)L(A) + (in)A(A) = mL(A) + (??)?(?)
= f (l-/)-Adm+ /" gdm= [ [g + (l-/)-A]dm.
J[A>0} J[A>0} J[A>0}
Putting h = 1 - / gives D1). Since in is unique and /, / and the Radon-Nikodym
derivative
d(in)v
dm
are m-a.e. uniquely determined, so are g and h. ?

Triangular norm-based measures
983
A combination of the results of Theorems 4.8 and 7.5 leads to the following corollary.
COROLLARY 7.6. If ? e ]0, oo], ? is a ??-tribe and m is a finite monotone ??-measure
on T, then there exists a unique finite nonnegative ?-additive measure ? on Tv, a p-
a.e. uniquely determined Tv -Markov kernel К : ? ? ?\ —> Ш and a unique monotonically
irreducible ??-measure топ ? such that for all A e ?
m(A)=ih(A)+ / K(jt,[0,A(jt)[)dp(jt).
•/?
From the results in Mesiar [60] and Mesiar and Navara [62] it follows that for ? e ]0, oo]
each ??-measure m, which is defined on a 7\-tribe on a countable set, is a generated
measure, i.e., m equals the zero measure.
8. Jordan decomposition of bounded T^-measures
It was shown in Theorem 7.5 that monotone Frank triangular norm-based measures can be
decomposed into a sum consisting of a generated measure and a monotonically irreducible
measure. Moreover, if the tribe on which the Frank triangular norm-based measure is
defined is a generated one, then the monotonically irreducible component is necessarily
zero (compare Theorem 6.1).
It is an interesting open question whether the monotonically irreducible component
of a Frank triangular norm-based monotone measure on a nongenerated tribe is always
identically zero. We shall prove that in the particular case of the ^-measures, this happens
even if the TL-measure is not monotone.
The fact that ^-measures are Jordan decomposable can be deduced directly from a
result in Schmidt [71]. However, we present the proof due to Butnariu [14] because some
technical details of this proof will be used later.
It is natural to ask in which sense ^-measures differ from other Frank triangular norm-
based measures. In order to answer this question, we recall that ?-measures are always T-
countably additive functions but, in general, ?-countable additivity does not characterize
?-measures (see Remark 4.4). However, 7l-countably additive functions are necessarily
7l-measures. To be precise, we have the following result.
PROPOSITION 8.1. Ifm is a finite T^-countably additive function on a T^-tribe T, then
m и a T^-measure on T.
PROOF. Observe that
(ATLC(ATLS))TLB = 0, (ATlC(ATlS))SlS = ASlS.
Therefore, using the TL-additivity of m we get m( A S|_ B) = m(A TL C(A T|_ B)) + m(S).
But A = (ATLC(ATL S))Sl(ATl S), and the fuzzy sets ATLC(ATL B) and ATL В are
7l -disjoint. Hence,
т(А) = т(АТ|_С(АТ|_б))+т(АТ|_б),

984
D. Butnariu and E.P. Klement
and this implies
m(ASLS) = m(A) - m(ATL B) +m(B),
showing that m is a ^-valuation on T. Finally, we show that TL-countably additive
functions are left continuous. Let (A„)neN be a nondecreasing sequence in ? which
converges to A. Define Aq = 0 and, for each ? e N, B„ = A„ TL С A„_ ?. Then for all ? e N
(Sl S,)tlZ?„+i = A„Tl(A„+|TlCA„) =0,
showing that the sequence (В„)пец is Tl-disjoint. Since for each ? e N we have
we
A„
obtain
A =
= Sl
k =
oo
= Sl
,Bk,
1
B,u
and, consequently,
lim m
n—>oo
(A„) = lim m( Sl Si I = Hm Fm(Bt) = Vm(fi„)
v*-' / k = \ n=\
= тЫ^В,Л=т{А).
Hence, m is a ^-measure. ?
Proposition 8.1 allows us to approach Tl-measures on T^-tribes in a way similar to the
treatment of (?-additive) measures on ?-algebras of subsets of X.
Dehnition 8.2. Let ? be a TL-tribe, m: ? -> [-oo, oo] a function such that m@) = 0,
and AeT.
(i) The function m+: ? -> [0, oo] defined by
m+(A) = sup{m(B): BeT(A)} E0)
is called upper variation of m, where T(A) denotes the class of all fuzzy subsets
of A contained in T, i.e., T(A) = {B: BeT,B^A].
(ii) The function m~ : ? -> [0, oo] defined by
m~(A) = sup{-m(B): В eT(A)} E1)
is called lower variation of m.

Triangular norm-based measures
985
(iii) The function m* : ? -» [0, oo] defined by m* = m+ + m is called total variation
of m.
Obviously, these functions are all monotone, and they satisfy
m+ = (-mr, пГ = (-т)+, m+@) =m_@) = m*@) =0. E2)
To simplify notation, from now on we shall use the following shortcut: for fuzzy sets
А, В e ? we shall write
AQB = AlLZB.
Obviously, each 7l-tribe ? is closed under this operation.
Proposition 8.3.
(i) Ifm is finitely T^-additive, then m+, m~ and m* are finitely T^-additive too.
(ii) Ifm is a T^-measure, then m+, m~ and m* are monotone ?^-measures.
PROOF. When proving (i), it is sufficient to show that m+ is 7l-additive (compare with
E2)). Let ? and F be TL-disjoint fuzzy sets in T. For each В е T(E SL F) the fuzzy sets
С = ? ?|_ ? and D = ? ? С are elements of T, and they form a TL-partition of B. Since
D e T(E) and С е T(F), we have
??(?) = m(C) + m(D) < m+(F) + m+(F),
which implies
m+(F SL F) sC m+(F) + m+(F).
For each ? > 0 there exist fuzzy sets A e T(E) and В eT(F) such that
m+(F)<m(A)+^, m+(F) < m(S) + |.
From A Tl S ^ F Tl F we deduce
m+(F) + m+(F) ^ m(A) + m(B) + ? = m(A SL ?) + ? < m+(F SL F) + ?.
Taking the limit i^Owe obtain m+(F) + m+(F) < m+(F Sl F), and this completes
the proof of (i).
In order to show the validity of (ii), assume that m is a TL-measure. According to
Proposition 8.1 and equations E2), it is sufficient to prove that m+ is TL-countably
additive. Let (А*)*ем be a TL-disjoint sequence of fuzzy sets in T. Then the sequence
(S„)„sN defined by
B„ = Sl Ak

986
D. Butnariu and E.P. Klement
is nondecreasing and, since m+ is 7l -additive, it follows that for each ? e N
?
т+(б„) = ]Гт+(А*).
Thus we have
lim m+(Bn) = Ym+(Ak).
n—>oo ·—>
k=\
Since for each ? e N
oo
Bn ^ Sl ^*
k = \
and since m+ is monotone, we obtain
@0 4°°
S_LA*b]Tm+(A*). E3)
m _„ .. ,
\ I— I /
* = l
For each
agt(sIa*
we have
(OO \ / ОС \ ОС
Sl аЛ = ATM ( Sm S» ) = Sm (ATM B„) = lim (ATM B„).
k = \ / V n=\ / »=l "^°°
The sequence (АТм В„)пещ is nondecreasing and m is left-continuous since it is a 7l-
measure. Hence we have
m(A) = m( lim (АТмб,,)) = "т т(АТмй„) ^ lim m+(B„)
\n—>oov '/ и—>oc и—>oo
because of ATM B„ e T{B„), which implies
oo
т(А)^]Гт+(А*).
Therefore, we obtain equality even in E3). and the proposition is proved. ?

Triangular norm-based measures
987
Proposition 8.4.
(i) A finitely 7l-additive function m.T —> [—00, 00] is bounded if and only if m* is
bounded.
(ii) If the finitely T^-additive function m: ? —> [—oo, 00] is bounded, then we have
m = m+-m~, E4)
i.e., the pair (m+, m_) is a Jordan decomposition ofm.
PROOF. We first show (i). If m* is bounded, then for each AeT
|m(A)| sC sup{m*(B): В e T] < 00,
i.e., m is bounded. Conversely, if m is bounded, then
max(m+(A),m~(A)) sC sup{|m(B)|: В eT(A)}
sCsup{|m(B)|: В e T} < 00
for all A eT. Hence, m+ and m~ are both bounded, and, consequently, so is m*.
In order to prove (ii), assume that m is bounded. Then m+, m~ and m* are bounded too,
and for A e ? and ? e ? (A) we have m(M) = m(A) - m(A QM). Since ? ? ? e T(A),
it follows that m+(A) > m(A ? ??) and m~(A) > -m(A ? ??). Hence,
m(A) - m+(A) ^ m(A) - m(A ? ??) ^ m(A) + m~(A)
and, consequently,
m(A) - m+(A) < m(M) sC m(A) + m~(A).
From E0) and E1) we deduce
m~(A) sC m+(A) - m(A), m+(A) < m(A) + m~(A),
and this completes the proof. ?
A straightforward combination of Proposition 8.4, Theorem 5.2 and Theorem 3.3(H)
yields the following corollary.
Corollary 8.5. Let m be a bounded T^-measure on a T^-tribe T. Then we have:
(i) m и continuous in the sense of A5);
(ii) for all AeT,
m(A) = f A(;t)d((m+)v - (m-)v), E5)
Jx
where (m+)v and (m~)v are the restrictions ofm+ and m- to Tv, respectively;

988
D. Butnariu and E.P. Klement
(iii) (m+)v and (m )v are the upper and lower variation o/m, the restriction ofm to
Tv, and, therefore,
m = (m+)v-(m-)v. E6)
9. Jordan decomposition of finite 7l-measures
In Section 8 we have shown that bounded ^-measures on T^-tribes have Jordan
decompositions and, therefore, are continuous (in the sense of A5)) and can be represented
by integrals (see E5)). The aim of this section is to show that finite 7l -measures on 7l-
tribes are necessarily bounded and, consequently, have Jordan decompositions by finite
TL-measures, a result which is due to Butnariu [14].
LEMMA 9.1. Let m be a finite T^-measure on a T^-tribe T. The following are equivalent:
(i) m и bounded;
(ii) m+(X) < oo and m~(X) < oo.
PROOF. The implication (i) =>· (ii) is trivial. Assume that (ii) holds. The monotonicity
of m+ and m~ implies that they are both bounded. Thus m* and, consequently, m are
bounded (see Proposition 8.4). ?
THEOREM 9.2. Ifm is a finite T^-measure on a T^-tribe T, then there exist nonnegative
finite T^-measures m+ and m~ on ? such that
m = m+ —m~, E7)
i.e., (m+, m~) is a Jordan decomposition ofm.
PROOF. Let m be a finite TL-measure on T. According to Proposition 8.4 and Lemma 9.1,
it suffices to show that m+(X) < oo and m~(X) < oo. In order to show m+(X) < oo, a
fuzzy set ? & ? such that m+(M) = oo will be called unbounded.
CLAIM. If ? is unbounded, then there exists a nonincreasing sequence (Ап)„ещ of
unbounded fuzzy sets in T(M) such that m(A„) ^ ? for all ? e N.
Let ? be unbounded and assume that there exists an no e N such that m(A) < no for
all unbounded fuzzy sets A e T(M). Since ? is unbounded, there exists an M\ e T(M)
such that m(M\) ^ no, which means that M\ is not unbounded, i.e.,
sup{m(A): ???(??,)} <oo. E8)
Choose an arbitrary В e T(M) and denote B\ = ? ?? ?\, ?? = ? ? Si. It is clear that
?, TL B2 = 0, Si SL B2 = S, Si e T(M|) and B2 e T(M ? M\). Therefore, m(S) =
m(B|)+m(B2)and
m+(M ? M\) > m(B2) = m(B) - m(S,) > m(S) - m+(Ai,).

Triangular norm-based measures
989
Since this is true for all В e T(M), since ? is unbounded and since E8) holds, we have
m+(M ? ? ?) ^ m+(M) - m" (?,) = oo,
i.e., ? ? ? ? is unbounded. Then, by assumption, m(M Q M\) < щ, and there exists
a fuzzy set C, e ?(?? ? ?,) such that m(C|) ^ 1. Put M2 = ?, SL C|. Then ?, TL С ? ^
Ai|TL(MeM|) = 0and
m(M2)=m(M|) + m(C|)^ii0+1. E9)
Using the assumption again yields that Mi is not unbounded. A similar reasoning as above
shows that ? ? ??2 is unbounded too. Hence, m(Ai ? Mi) < no and m(Ci) ^ 1 for
some C2 e ?(?? ? M2). Denote M3 = Мг^Ст. Since Ai2TL C2 < ??2?,_(?? ? ?2) = 0
it follows from E9) that we have т(ЛГ?) = m(M2) + m(C2) ^ no + 2, i.e., My, is not
unbounded. In this way we obtain fuzzy sets M\, Mi, My,,... in T(M), none of which are
unbounded, and C\, C2, C3,... such that C„ e T(M ? M„) and m(C„) > 1 for all и е N.
Moreover, we can show that for all и e N
?
]Гс, <?-??|. F0)
i = \
Indeed, if ? = 1 this is true because of the definition of C\. If F0) holds for all к е
{1, 2,...,«}, then we have for each ? e X:
C„+| (x) < max@, ? (л) - ?„+\(?))
= Щх)-М„+\(х)
= М(х)-(Мп(х) + Сп(х))
= М(х) - (??„_? (л) + Ся_| (.?) + С„(дс))
= М(дс)-М|(дс)-5]С1-(дс).
i-l
Hence, F0) is proved by induction. An immediate consequence of F0) is that the sequence
(Cn)neN is Tl-disjoint which, together with m(C„) ^ 1, yields
(oo \ °°
m( Sl C„ ) = 2^m(C„) = oo.
n=\
Since m is finite, this contradicts the assumption that m(A) < no for all unbounded fuzzy
sets A e T(M). Therefore, there exists an unbounded fuzzy set ? ? e T(M) such that

990
D. Butnariu and E.P. Klement
m(A ?) ^ 1. Since A | is unbounded, there exists an unbounded fuzzy set A 2 e T( ? ?) such
that m(A2) ^ 2, and so on, showing that the claim holds.
To complete the proof of the theorem, assume that X is unbounded and choose the
nonincreasing sequence (A„)neN of unbounded fuzzy sets in ? which exists according to
the claim. For each ? e N denote
n-l
S„=Sl(A*©A*+i).
k = \
Clearly, since (A„)n6N is nonincreasing, the sequence (?* ? A^+\ )km is 7L-disjoint, and
therefore,
m(B„) = ]Tm(A* ? Ак+\) = m(A,) - m(A„).
k=\
The sequence (В„)пещ is nondecreasing which leads to
m( lim B„) = lim m(B„) = m(A\)— lim m(A„) = -oo,
\il—>OC / И—>ЭС И—>3C
where the latter equality follows from m(A„) ^ и and m(A|) < oo. This contradicts the
finiteness of m. Therefore, the assumption that X is unbounded is false and, consequently,
m+(X) <oo.
The inequality m~ (X) < oo is shown in complete analogy, m is bounded by Lemma9.1,
and the Jordan decomposition exists because of Proposition 8.4. ?
Since each bounded TL-measure is automatically finite, in the proof of Theorem 9.2 we
have implicitly proven the following result.
Corollary 9.3. A T^-measure топа T^-tribe ? is bounded if and only if it is finite.
Combining Corollary 8.4 and Theorem 9.2 we realize that finite TL-measures on
TL-tribes are exactly (Lebesgue) integrals.
COROLLARY 9.4. Let m be a finite T^-measure on a T^-tribe T. Then we have
(i) m и continuous in the sense of A5);
(ii) Ifim+У and (m~)v are the restrictions o/m+ and m~ to Tv, respectively, then
for all AeT
m(A) = [ A(;t)d((m+)v - (m-)v); F1)
Jx
(iii) (m+)v and (m~)v are the upper and the lower variation, respectively, ofm, the
restriction ofm to Tv, and
m = (m+)v-(m-)v; F2)

Triangular norm-based measures
991
(iv) If(P, Q) is a Hahn decomposition of X with respect to the ?-additive measure m
on Tv, then for all AeT
m(<2) = -??-(?) ^ m(A) < m+(P) = m(P). F3)
10. Absolute continuity of ^-measures
TL-measures on not necessarily generated 7l -tribes will play an essential role in the
following exposition. Some of the important properties of such Tb-measures are the
absolute continuity and the singularity of 7l -measures. These will be studied in this
section.
DEnNlTlON 10.1. Let m and ? be TL-measures on a TL-tribe T. The measure ? is called
m-absolutely continuous (briefly ? <<c m) if for each A e ?
(AC) n(A) = 0 whenever m*(A) = 0.
Absolute continuity is a reflexive and transitive relation. Moreover, if ? ? <*C m and
П2 <K m, then also ? ? ± 112 « m. There are several characterizations of the absolute
continuity which are rather easy to prove (see Lemma 11.2 of Butnariu and Klement [17]).
LEMMA 10.2. Let m and ? be T^-measures on a T^-tribe T. The following are
equivalent:
(i) ? « m;
(ii) for all A e T: n(A) = 0 whenever m*(A) = 0;
(iii) for all A e T: n+(A) = 0 = n~ (A) whenever m*(A) = 0;
(iv) for all A e T: n*(A) = 0 whenever m*(A) = 0;
(v) n* «cm*;
(vi) for each ? > 0, there exists a <5F > 0 such that for each AeT
n*(A) < ? whenever m*(A) < <$F;
(vii) for each ? > 0, there exists an ?? > 0 such that for each A e ?
|n(A)| < ? whenever m*(A) < ??\
(viii) ? <? m;
(ix) (?)* «(m)*.
A property which is dual to absolute continuity is that of singularity. Some
characterizations of the singularity which are dual to the characterizations of the absolute continuity
and which are listed below (see Lemma 11.4 of Butnariu and Klement [17]).

992
D. Butnariu and E.P. Klement
Dehnition 10.3. Let m and ? be FL-measures on a FL-tribe T. The FL-measure ? is
said to be m-singular (briefly ? _L m) if there exists a fuzzy set A e ? such that
n*(A) = m*(CA)=0.
LEMMA 10.4. Let m and ? be T^-measures on a T^-tribe T. The following are
equivalent:
(i) ? _L m;
(ii) m _L n;
(iii) there exists a fuzzy subset A of X such that AeT and m*(A) = n*(CA) = 0;
(iv) n+ _L m and n" _L m;
(v) n* _L m*;
(vi) ? _L m;
(vii) (n)*_L(m)*.
The following properties of the absolute continuity and the singularity of finite T^-
measures will be used later.
Proposition 10.5.
(i) If п\,П2 and m are finite T^measures on a ??^-tribe ? such that ? ? _L m and
П2 -L m, then n\ ± 112 _L m.
(ii) Ifm and ? are finite T^-measures on a T^-tribe ? such that ? <K m and ? _L m,
then ? equals the zero measure.
PROOF. The proof of (i) is trivial. In order to show (ii), let m and ? be TL-measures on
? such that ? <^ m and ? _L m. Since ? <K m, it follows that for each к eN there is a
&\/k > 0 such that n*(F) < l/k whenever F e ? and m*(F) < 5|/*· Since nlm, there
is a set A e ? such that n*(CA) = m*(A) = 0. Clearly, n*(A) = 0 since m*(A) < 5|/* for
all k e N. If ? e T, then
n*(F) = n*(FTLA)+n*(FTLCA)=0
because of F Tl A eT(A), FTlCA e T(CA) and the monotonicity of n*. D
11. Vector /^-measures with Darboux property
In Section 9 we proved that finite Tl-measures on FL-tribes have bounded range (see
Corollary 9.3(iv)). The question is now whether or under which conditions the range of
a finite Гь-measure on a Гь-tribe (generated or not) is exactly an interval. This problem
is studied here in a more general context, i.e., we ask whether or under which conditions
a finite vector valued rL-measure on a rL-tribe has a convex and compact range. For
measures defined on a ? -algebra of crisp subsets of X (which are particular FL-measures)
this problem was approached in Poprugenko [67], and it was solved in Liapounoff [52].
In Halmos [33,34], Blackwell [9], Chernoff [19], and Lindenstrauss [53], Liapounoff's

Triangular norm-based measures
993
result was reproved and/or extended. In view of Corollary 9.4, in Dvoretzki, Wald and
Wolfowitz[29] it was shown that the range of a vector ^-measure on a generated TL-tribe
is necessarily convex and compact. We intend to generalize this result by showing that
nonatomic ^-measures on TL-tribes (generated or not) have convex and compact range.
To this end, we first study the class of ^-measures with Darboux property.
In this section ? denotes a Гь-tribe on X. In what follows, all ^-measures are assumed
to be defined on T. An ?-vector T^-measure on ? is a function m = (Ш|, m2,..., m„)
from ? to R" such that each m, is a finite TL-measure. If m is an ?-vector TL-measure,
we denote the range Ran(m) of m by
Ran(m) = {m(W): W eT}.
Clearly, 0 e Ran(m) and Ran(m) is bounded in R" (compare Corollary 9.3).
DEHNITION 11.1. An ?-vector ^-measure m = (Ш|, Ш2,..., m„) has the Darboux
property if for each pair (E, F) of fuzzy sets in ? with ? ^ F there exists a family
(Wr)re[0.\] in ? such that for all r e [0, 1]
? <; wr <: F, F4)
m(Wr) = (l-r)-m(E) + r-m(F). F5)
This is just an extension of the notion introduced in Poprugenko [67]. A typical example
of a measure having the Darboux property is the Lebesgue measure on the Borel subsets of
[0, 1 ]. By Corollary 9.4, finite TL-measures on generated tribes have the Darboux property.
PROPOSITION 1 1.2. Let m = (mi, m2,..., m„) be an ?-vector T^-measure on T. Then
m has the Darboux property if and only ifm is semiconvex in the sense that for each F eT
there exists a fuzzy set W eT{F) such that
m(W) = t.m(F).
PROOF. Assume that m has the Darboux property. Then for each F eT and for ? = 0
there exists a family (Wr)r6lo.ij с ? such that F4) and F5) hold. In particular, W =
W\/2 e T(F) and m(W) = j ¦ m(F), and the semiconvexity of m follows. Conversely, let
m be semiconvex, consider E, F eT such that ? ^ F and denote Wq = Vq = ? and W\ =
F. Then there exists a fuzzy set U\ e T(W\ ? Vn) such that m(U\) = \ ¦ m(W\ ? V0).
Denoting V| = Vo S|_ U\ and using the TL-disjointness of Vo and U\, we obtain
m(V|) = m(Vo) + m(i/i) = ? · (m(W|) + m(W0)).
By the semiconvexity again, there is a fuzzy set Ui e T(W\ ? V|) such that m(i/2) =
{ ¦ m(W| ? V|). Denote V2 = V, SLU2, then we get m(V2) = m(V,) + m(i/2) = \ ¦
m(W|) + g · m(Wo). Continuing in this way we can generate sequences (U„)„e^ and

994
D. Butnariu and E.P. Klement
(Vn)neN in ? such that (t/„)neN is TL-disjoint, (V„)iieN is nondecreasing, and for each
? e N we have
m(V„) = ^ ¦ m(W0) + ( 1 - — ) · m(W,),
04)
m(t/„) = --m(W|©V,,_|)=^-(m(W|)-m(V,,_i)).
Choose an arbitrary r e ]0, 1 [ and let
oo
? Pi
2'
i = \
be its (unique) nonterminal binary representation (i.e., the binary representation with
Pi e {0, 1} and pi ? 0 for infinitely many i e N). For each к eN define
r*=Ef' Zri=WoSL(SLPi-u\
Clearly, Zn eT(W,)and
m(Zrt) = m(W0) + ??·- т(^") = r* · m(w\) + (!-»·*)· m(Wo)·
i = l
The sequence (Zrt)igN is nondecreasing. Hence, the limit Wr = Нт*-»-»^. exists,
belongs to T( W\) and satisfies
m(Wr)= lim m(Zn) = r ¦ m(W\) + A - r) -m(W0).
Obviously for each r e [0, 1] we have W0 ^ Wr ^ W\, and thus the proposition is
proved. ?
PROPOSITION 11.3. An ?-vector T^-measure m= (mi,m2, ...,m„) on ? has the
Darboux property if and only if for each pair (E, F) of fuzzy sets in ? with ? ^ F there
exists a family of fuzzy sets (Wr)r6[0.i] Я Т such that F4), F5) and
Wr ? Ws whenever 0 ^ r < s ? 1 F6)
are satisfied.

Triangular norm-based measures
995
PROOF. Sufficiency trivially holds. Assume that m has the Darboux property and start
with W0 = ? and W\ = F. Then there exists Z|/2 e T(W\ ? Wo) such that m(Z|/2) =
2 · m(W| ? Wo) (see Proposition 11.2). For W\p_ = Wo S|_ ?\? we have
m(W,) = i-m(W|) + i-m(W0).
Using again the semiconvexity of m, we obtain a fuzzy set ? ?/4 e T(W\/2 ? Wo) and
a fuzzy set Z3/4 e T(W| ? W|/i) such that
m(Z|/4) = ? -111(^1/2© W), m(W3/4)=2-m(W| ©W|/2).
Denote W|/4 = Wo Sl ?1/4 and W3/4= W1/2SLZ3/4. It is clear that
WoiJ W|/4<W|/2<W3/4< W|
and
m(W|/4) = I · m(W,) + I · m(Wo), m(W3/4) = | · m(W,) + J · m(W0).
In this way we obtain for each ? e N a finite family (Wk/2")ken 2 2"-?? °^ fuzzv sets m
? such that
Wo ?? W\pn ij W2/2" < · · · < WB--|)/2" < W,
and
m(Wk/2n) = ^ · m(W,) + ? - ?? . m(w0)
for all A: e {0, 1,..., 2"}. Each r e ]0, 1 [ can be written as r = lim„^oo rk with rk as above.
The fuzzy set Wr = Hindoo Wn is contained in ? and satisfies F5). A simple computation
shows that the family (Wr)r6[o, 1] also fulfills F4) and F6). ?
COROLLARY 11.4. An ?-vector TL-measure m = (Ш|,..., m„) on ? has the Darboux
pmperty if and only if for each ? e ? there exists a family of fuzzy sets (Wr)r6[0.1] in ?
such that the condition F6) holds and for each r e [0, 1 ]
m(Wr) = r-m(?). F7)
Remark 11.5. Halmos [34] called и-vector measures m (defined on a ?-algebra A of
crisp subsets of X) convex measures if the following property holds:
For each ? e A the set /C(m, E) of all (А П ?)-measurable functions g:E->
[0, 1 [, such that m([x e E: g(x) < r}) = r · m(?) for all r e [0, 1 ], is convex.

996
D. Butnariu and E.P. Klement
Here Ad ? denotes the restriction of the ?-algebra A to F, i.e.,
АПЕ = {АПЕ: АеА}.
Combining Lemma 4 from [34] and Corollary 11.4, it follows that each convex measure
has the Darboux property. Since the convexity of an и-vector measure m implies its
semiconvexity (compare [34, Lemma 2]), an и-vector measure m (defined on a ?-algebra
of crisp subsets of X) is convex if and only if it has the Darboux property.
Now we are able to state the main result of this section, namely, a convexity theorem of
Liapounoff type for vector 7l-measures with Darboux property:
THEOREM 11.6. If m = (m\,..., m„) is an ?-vector T^-measure on ? with Darboux
property, then its range Ran(m) is convex.
PROOF. It suffices to prove that for each pair (?, F) of fuzzy sets in ? and for all r e [0, 1]
there exists a fuzzy set G e ? such that
m(G) = r-m(E) + (l-r)-m(F). F8)
First assume that FTLF = 0. Then there exist Er e T(F) and F|_r e T(F) with
m(Fr) = r ¦ m(F) and m(F|_r) = A - r) · m(F) (compare Corollary 11.4). Denote
G = Er S|_ F| _r. Since Er T\_ F\ _,· = 0, we have proven F8) in this case.
Consider the case that ? and F are not 7l -disjoint. Then, however, A = ? ?
(FTL F),B = F ? (FTl F) and С = FTl F are TL -disjoint. Hence, there exist Ar e
T(A)and B\-r e ?(?) such that ArTLS|-r = 0 and
m(ArSLS|_r)=r-m(A) + (l -r)m(B).
Denote G = (Ar SL В, _r) SL С. Then
m(G) = r ¦ m(A) + A - r) · m(B) + m(C) F9)
since Ar, ?|_? and С are 7l-disjoint. From m(A) = m(F) — m(C) and m(B) = m(F) -
m(C) it follows that F8) holds in this case too (see F9)). ?
In the special case of the finite 7l-measures Theorem 11.6 can be restated as follows:
COROLLARY 11.7. Ifm is a finite T^-measure on ? having the Darboux property and if
(P, Q) is a Hahn decomposition of X with respect to the ?-additive measure m on Tv,
then for each ? e ?
(m(W): W e T(F)} = [m(Q ¦ F), m(F · ?)].
In particular, we have Ran(m) = [m(<2), m(P)].

Triangular norm-based measures
997
PROOF. For ? e ? and W e T(?) we get
m(? · Q) = -m"(?) < -m-(W) < m(W) ^ m+(W) < m+(?) = m(P · ?).
If r e [m(? · Q), m(? · P)] then there is some s e [0, 1] such that
_lsm(EP) ifr^O,
?"|?·??(? Q) ifr<0.
Hence, in both cases there exists a fuzzy set W e T(?) such that m(W) = r (see
Corollary 11.4). ?
Remark 11.8. Taking into account Corollary 11.7, a finite 7l-measure m on ? has the
Darboux property if and only if for each ? e ? the set {m(W): W e T(?)} is convex, i.e.,
if and only if this set is an interval. Generally speaking, if m = (mi,..., m„) is an и-vector
7l -measure on ? with ? ^ 2 and if for each ? e ? the set
Ran?(m) = {m(W): W e T(?)}
is convex, then m has the Darboux property (compare Proposition 11.2). However, we do
not know whether the converse also holds. Clearly, for ? = X the set Ran?(m) is convex
by virtue of Theorem 11.6. But the proof of Theorem 11.6 can not be applied for ? ? ?
since, in general, we do not have AS|_ В e T(?) for each pair (?, ?) in T(?). Such an
implication, however, is substantially involved in the proof.
12. Nonatomic /^-measures
In the preceding section we have shown that the range Ran(m) of an и-vector 7l-measure
m = (mi, ??2,..., m„) with Darboux property is convex. Under which conditions for the
components m, of m can we deduce the Darboux property for m? Note that if m has the
Darboux property, each of its components Ш|, mi,..., m„ necessarily has the Darboux
property. In this section we show that the converse is also true. This is one of the results
of Butnariu [12,13], and it involves ideas contained in the proof of the Liapounoff theorem
given in Halmos [34].
Let ? be a 7l-tribe. All 7l-measures considered in this section are assumed to be
defined on T. If m = (mi, ni2,..., m„) is an и-vector 7l-measure, then define the finite
7l -measure m': ? -> R by
m' = mf + ni2 + h m*.
Dehnition 12.1. Let m = (mi, Ш2,..., m„) be an и-vector 7l-measure on T.
(i) A 7l-measure ? on ? is called m-absolutely continuous if it is m'-absolutely
continuous, i.e., if n<K m'.

998
D. Butnariu and ?.P. Klement
(ii) A 7l -measure ? on ? is called atomic if there exists a fuzzy set A e ? with
n(A) ^ 0 such that for each В e T(A) we have either ?(?) = 0 or ?(?) = n(A).
(iii) A T^-measure ? on ? is called nonatomic if it is not atomic, i.e., if for each AeT
with n(A) ? 0 there exists а В e T(A) such that ?(?) ? {?, ?(?)}.
(iv) The и-vector 7l -measure m is called nonatomic if each of its components
Щ|, ??2,..., m„ is nonatomic.
THEOREM 12.2. Let m = (mi,m2, ...,m„) be an n-vector T^-measure on T. The
following are equivalent:
(i) m has the Darboux property;
(ii) for each i e {1, 2, ..., n] the component m/ has the Darboux property,
(iii) m is nonatomic.
PROOF. The implications (i) =>· (ii) and (ii) =>· (iii) are trivial. It remains to show that
(iii) =>· (ii) and (ii) =>· (i). The proof of these implications will be done in the following
sequence of lemmas. ?
Lemma 12.3. Assume that E, F e ? with El\_ F = 0, and let
W=(W|,W2,...,W„)
be a nonnegative ?-vector T^-measure on ? with Darboux property and ? a finite
T^-measure on ? such that n«w. Then there exists a family of fuzzy sets (Hr)re[0.\]
in ? such that for each r e [0, 1]
w(Hr) = r -w(?) + (l -r)-w(F), G0)
H0 = F, Я,=?. G1)
Moreover, for the function /„ : [0, 1] -> R given by fa{r) = n(#r) we have
/n is continuous. G2)
PROOF. If w(?) = 0 or w(F) = 0, then such a family (#r)r6[o. i] exists by Corollary 11.4.
Assume that w(?) ? 0 and w(F) ? 0. Since w has the Darboux property, by Proposition
11.3 there are two families (Wr)re[o. i] and (Zr)r6[o. и in ? such that for r, s e [0, 1] with
r < s we have
Wo = Zo = 0, W,=?, Z,=F, G3)
Wr < Ws, Zr^Zs, G4)
w(Wr) = r -w(F), w(Zr) = r-w(F). G5)
Since FTLF = 0 it follows that WrlLT\-r = 0 for all r e [0,1]· Denote Hr =
Wr SL Z,_r. Then G0) and G1) are clearly satisfied. Choose s,re[0,l] with s > r. Then
l/n(r) - /n(i)l = ln(Z|_r © Z|_.5) - n(Ws ? Wr)|

Triangular norm-based measures
999
^ \n(Ws ? Wr)\ + |n(Z,_r ? Z\-s)\. G6)
For a given ? > 0 there exists a ?? > 0 such that for each A e ? we have
, . ?
|n(A)| < - whenever w (A) < <5?, G7)
because of n<Z? w1. Since w is nonnegative, we get
W(WS ? Wr) = W(WS) - w'(Wr) = (s- r)-w'(E)
and
W(Zi-r ez\-s) = W(Z\-r)-W{Zi-s) = (j - r) · W(F)
because of G4) and G5). Denote
. (, ?? ?? \
?? = nun 1, — , — .
V 'w'(F) W(F)J
If 0 < s - r < ?? then we deduce (compare with G7)) |n(Ws ? Wr)\ < ?/2 and
|n(Z|_r ? Z\-s)\ < ?/2. Taking into account G6) it follows that |/„(r) - fn(s)\ < ?
whenever 0 < s — r < ?(?). Hence, /„ is continuous. ?
LEMMA 12.4. Ifw = (wi, wi,..., w„) is a nonnegative ?-vector T^-measure on ? which
has the Darboux property, then for each finite T^-measure ? with ? <? w the (n + \)-vector
T^-measure w' = (wi, W2,..., w„, n) has the Darboux property.
PROOF. It is sufficient to show that w' is semiconvex (see Proposition 11.2). Suppose
that A eT satisfies w(A) ? 0. Then there exists an ? e T(A) with w(?) = 5 · w(A).
If n(?) = 2 ¦ n(A), we are done. Assume that n(?) ? \ ¦ n(A) and denote F = ? ? ?.
Clearly, j ¦ n(A) is contained in the interval with the boundaries n(?) and n(F). Since
? and F satisfy Lemma 12.3, there exists a family (#r)r6[0.1] ш Т sucn mat G0), G1)
and G2) hold. Hence, there is an ro e [0, 1] such that
n(//ro) = /n(r0) = i-n(A)
because of /n@) = n(F), /n(l) = n(?) and the continuity of /„. For ? - НГA we deduce
thatn(W) = J n(A) and
yv(H) = r0 ¦ w(?) + A - r0) · w(F) = i · w(A),
i.e., w'(W)= ^ -w(A).
D

1000
D. Butnariu and ?. P. Klement
LEMMA 12.5. Ifm and ? are finite T^-measures on T, if ? <g m and ifm is nonnegative
and has the Darboux property, then ? also has the Darboux property. Moreover, m ± ? are
finite T^-measures with Darboux property.
PROOF. Indeed, if in Lemma 12.4 we take ? = 1 and W| = m, then the 2-vector measure
(m, n) has the Darboux property, and the result follows. ?
LEMMA 12.6. Ifm = (m|,mi,...,m„) is a nonnegative ? -vector T^-measure on ? with
m, <g m,_ | for all i e {2, 3,..., n) and ifm\ has the Darboux property, then m itself has
the Darboux property.
PROOF. For ? = 1 the result is trivial. Assume that the result holds for each к < n.
In order to show its validity for к = ? consider the (n - l)-vector T^-measure w =
(mi, ??2,..., m„_ ?) which has the Darboux property. Since
m„ «; m„_| « mi + m2 ? h m,,-i = v/,
it follows that m„ «; w. Hence, (w, m„) = (mi, mi,..., m„) has the Darboux property by
Lemma 12.4. ?
LEMMA 12.7. If ? is a finite nonatomic nonnegative T^-measure on T, then for each
AeT with n(A) ? 0 there exists a sequence (А/0/teN,, in T such that for all к e N
(i) А = А0>А\>А2>--->Ак,
(ii) 0<?(?*)<;1·?(?*_|),
(iii) Hindoo Ak =0.
PROOF. Put So = A and assume that we have already obtained fuzzy sets Bq ^ B\ ^
B2 > ¦ ¦ ¦ > Bk in ? such that 0 < ?(?,) < A/2') · n(A) and ?(?,) < { ¦ ?(?,-?) for
i e {1,2, ...,k]. Then there exists В e T(Bk) with 0 < ?(?) < n(Bk) because of the
nonatomicity of n. Define ?* + ? by
_ \B ifn(B)<;-n(Bt),
Bk + \ = \
[ Bk ? ? otherwise.
Clearly
0<?(?*+?)<?·?(?*)
and ?* + ? < Bk. Moreover, we have
?(?(+?)^·?(?).
The sequence (Bk)ke^ с ? obtained in this way is nonincreasing and its limit W =
Hindoo Bk belongs to ? (see Theorem 3.3(i)). Since n(Bk) < ^ · n(A) for all к eN,

Triangular norm-based measures 1001
it follows that n(W) = 0 because of the continuity of n. If we define Ak = Bk ? W for
к e N and Ao = A, then the sequence (Ак)кец0 satisfies the properties (i)—(???). ?
LEMMA 12.8. If ? is a nonnegative nonatomic finite 7l measure on T, then ? has the
Darboux property.
PROOF. We prove that ? is semiconvex. For each A eT with n(A) ? 0, there exists a
sequence (A*)*eN0 - ^ sucn tnat (i)-(i») ш Lemma 12.7 are satisfied. Denote Wk =
Ak-\ ? Ak for each к e N. From (i) it follows that (W,t)/ieN is a ^-disjoint sequence
in T(A). For ieN and re@,1) define the fuzzy set
@ ifr=0.
Ifre[0, l]and
ЭС
k = \
its unique nonterminal binary representation, denote
t*W = SLWk.ll.
k=\
Clearly, t *W eT(A) and
ОС
n(t*W) = J2tk -n(Wk).
k = \
We claim that the function g : [0, 1 ] -> R defined by g(t) = n(i * W) is continuous. Indeed,
if 0 ^ s < t !? 1, then there exists a smallest jeN such that f, = 1 and sy- = 0. Hence,
ос эс
g(t)-g(s) = n(Wj)+ ? (tk-sk)-n(Wk)>n(Wj)- ? n(Wk).
k=j+\ k=j + \
Since the latter sum equals n(Aj), it follows that g(t) - g(s) ^ n(Aj-\) - 2 · n(Ay) ^ 0
because of Lemma 12.7(ii). Now observe that
ОС
g(t) - g(s) ? ?("< - J*> · n(A^-i) < 2 · (i - i) · n(A),
k=\
and the continuity of g follows. The function g satisfies g@) = 0 and
g(l)=n(?lwk\=n(A).

1002
D. Bulnariu and E.P. Klement
Hence, there exists a t° e [0, 1] such that g(t°) = \ · g(l), i.e., ?(?) = j · n(A) and
В e T(A) for В = t° * W, and ? is semiconvex. ?
LEMMA 12.9. If П\,П2,...,п„ are nonnegative finite T^-measures on ? having the
Darboux property, then ? = ? ? + П2 + h n„ has the Darboux property.
PROOF. We prove that ? is nonatomic. Assume that AeT and n( ?) ? 0. Then ?, (?) ? 0
for at least one index i. Since n, has the Darboux property, there exists a fuzzy set
В e ? {A) such that ?,-(?) = \ · ?;(?) < ?, (A). Hence, 0 < ?(?) < n(A) since all n,
are monotone (see also Proposition 5.1). ?
Lemma 12.10. //m= (mi,m2, ...,m„) is a nonnegative ?-vector T^-measure on ?
and if each of the T^-measures Ш|, m2,..., m„ has the Darboux property, then m itself
has the Darboux property.
PROOF. For i e {1,2,..., n} denote m^ = Y?k=i mu. Each 111° has the Darboux property
by Lemma 12.9. Observe that m° «; m^, whenever i e {2,3,...,и}. Thus m° =
(m", 1П2,..., m,^) has the Darboux property by Lemma 12.4. Let / = (/1,..., /„) be the
linear operator /: E" -> R" defined by
?
f,(y\,y2,.-.,yn) = J2yk-
It is clear that / is one-to-one and onto (its associated determinant is different from zero).
Therefore, /~' exists and is linear. From m° = / о m it follows that m = /~' о m° and,
subsequently, that m is semiconvex. ?
Now we are ready to show that, in Theorem 12.2, property (ii) implies indeed
property (i).
LEMMA 12.11. If m = (mi, 1112,..., m„) is an ?-vector T^-measure on ? and if each
component Ш|, m2,..., m„ has the Darboux property, then m itself has the Darboux
property.
PROOF. For each index i e {1,2,..., n) there exists a Hahn decomposition (/>,-, Q;) of
X with respect to m, (the restriction of m, to Tv). Denote E® = Pi and EJ = Q;. For
a = (??,?2,...,?„) e {0, 1}" denote ? (?) = ?"=? ET ¦ Clearly, ? (a) e T. Define the
finite Гь-measures m", m",..., mjj by
mii(A) = (-l)a'-m,(ATL?(a)).
Each m? is nonnegative since
? Qi if a,- = 1,
ATL?(a)<
I Pi ifai=0.

Triangular norm-based measures
1003
We can even show that each m" has the Darboux property: if A e ? and m"(A) ? 0, then
there exists а В е T(A) such that
та(В) = (-\У' ¦mi(BJLE(a)) = \-(-\f· ¦ml(ATLE(a)) = \-mc;(A),
since m, is semiconvex and E(a) is a subset of X. Hence, m" = (m", m",..., m°) is a
nonnegative и-vector ^-measure with Darboux property (compare Lemma 12.10). Since
[E(a) | a e {0, 1}"} is a partition of X, we have
m= ]T (-l)fl-m°, G8)
«6@.1)"
where the и-vector Tb-measure (—1)" · m" is given by
(-1У · mfl = ((-1 )<" · m'l, (-1)"- ¦ m$,...,(-1)"" · m?).
By the Darboux property of each m", for each AeT and for each a e {0, 1}", there exists
an A(a) e T(A) such that ma(A(a)) = \ ¦ m"(A). Denote B(a) = A(a)TL ?(a) and
B= ? ?(?).
я6{0.1)"
Then because of G8) and the TL-disjointness of the family (S(a))«6jo. i)" we have
m(B)= ? m(B(a))= ? ("D" т"(А(а))
аб@.|)" дб{0.1)"
= \· ? (-l)"-ma(A) = ^-m(A).
яб@.1)"
This shows that m is semiconvex. ?
The following lemma verifies implication (iii) =>¦ (ii) of Theorem 12.2, and this will
complete the proof of this theorem.
LEMMA 12.12. Ifm = (???,???, ...,m„) is a nonatomic ?-vector TL-measure on ? then
each component Ш|, ni2,..., m„ has the Darboux property.
PROOF. If m is nonatomic, each of its components m, is finite and nonatomic. If each
in; is nonnegative, then the result follows from Lemma 12.8. Assume that m/ has also
some negative values. If we can show that m^ and m~ are also nonatomic, then they
have the Darboux property (compare Lemma 12.8) and, consequently, the 2-vector T^-
measure (m+,m~) (compare Lemma 12.10) and the TL-measure m,- = m+ - m~ also
have the Darboux property due to m, = /(m+, m~) and to the fact that the linear function

1004
D. Bulnariu and E.P. Klement
f :R2 -> R defined by f{y\,yi) = }'\ — yi preserves semiconvexity. If A e ? with
m+(A)^0, then
m,(P,TLA)=m+(A)^0
for each Hahn decomposition (/>,-, <2,) of X with respect to m, (compare Corollary 9.4).
Since m, is nonatomic, there exists a fuzzy set В еТ(Р, T|_ A) such that
0 ? m, (?) ? m/ (?, TL A) = m+(A).
As a consequence, m+(S) = iii,(P,7lS) = m,(Z?) ? {0, m+(A)}, i.e., m,+ is nonatomic.
The nonatomicity of m~ can be seen analogously. ?
Theorem 12.2 together with Corollary 9.3 allows Theorem 11.6 to be restated as follows:
COROLLARY 12.13. // m = (m|,...,m„) is a finite nonatomic ?-vector T^-measure
on T, then its range Ran(m) is convex and bounded. In the case ofn= 1, moreover,
Ran(m) = [m(<2),m(P)],
(P, Q) being a Hahn decomposition of X with respect to the ?-additive measure m.
13. A Liapounoff type theorem for /^-measures
In Section 12 we have shown that the range Ran(m) of a nonatomic TL-measure m on a T^-
tribe is convex and bounded (see Corollary 12.13). In this section we show that Ran(m) is
also compact. The idea of this proof is extracted from the proof of the Liapounoff theorem
by Halmos [34]. Throughout this section ? will denote a T^-tribe, and all Гь-measures are
assumed to be defined on T.
THEOREM 13.1. //m = (mi,..., m„) is a nonatomic ?-vector T^-measure on T, then its
range Ran(m) is convex and compact.
PROOF. Convexity and boundedness of Ran(m) follow from Corollary 12.13. Without loss
of generality, we may assume that the affine dimension of Ran(m) equals n. Otherwise,
all the following arguments can be carried over from R" to the linear hull of Ran(m). It
remains to show that Ran(m) is closed. To this end, it is sufficient to show that
? ? cl(Ran(m)) с Ran(m), G9)
where cl stands for the topological closure, for each supporting hyperplane ? of Ran(m)
(see Rockafellar [68, Theorem 11.6.2 and Corollary 11.7.1]). Let ? be such a supporting
hyperplane. Then there exists an r el and a linear function /:R" -> R such that
? = [? e R": f(x) = r} and /(m(?)) > r for all ? e T. The proof is carried out in
several steps.

Triangular norm-based measures 1005
CLAIM 1. For each crisp set AeTv the function mA : ? -> R defined by
mA(?) = m(CA?)-m(A?) (80)
is a nonatomic n-vector ?^-measure and Ran(mA) = Ran(m) — m(A).
Indeed, a simple computation shows that mA is a /i-vector TL-measure. For ? e ? we
have
mA(E) = m((CA · ?)SL(A · C?)) - m(A),
i.e., Ran(mA) с Ran(m) - m(A). Conversely, since A e Tv, we get
m(?) - m(A) = mA((CA · ?)SL(A · C?)),
i.e., Ran(m) — m(A) с Ran(mA).
To show the nonatomicity of mA, observe that for ? e ? there exist TL-disjoint fuzzy
sets F| e T(CaTl ?) and F2 e T(ATL ?) such that
m(F|)= ^m(CATL?), m(F2) = i, m(ATL?),
since m is nonatomic (see Theorem 12.2 and Proposition 11.2). Therefore,
шд (Fi SL F2) = mA(F,) + mA(F2) = m(F,) - m(F2) = !, ¦ mA(?),
showing that mA is nonatomic.
Claim 2. 77?еге ejcww aset Qe Tv ii/c/г ?/??? ??(?) e ? ? Ran(m).
Define the finite ^-measure u:T->Rbyu = /om. Observe that
r = inf{u(?): ?еГ},
and let (P, Q) be a Hahn decomposition of X with respect to the ?-additive measure u.
By Corollary 9.4 (iv), we have u(Q) = r, and this proves Claim 2.
Now we are in the position to show G9). Observe that / ? ???(?) > 0 for all ? e T,
and define uq = f ? ???. Let L be the linear subspace L = {x еШ": f{x) = 0}, i.e.,
L = H - m(<2). By Claim 1, G9) is equivalent to
Lncl(Ran(me)) cRan(me). (81)
Denote w = m'Q (see Definition 71). Clearly, each coordinate of mg is w-absolute
continuous. We prove (81) by induction upon the number ? of components of the vector
rL-measurem. If и = 1 then (81) follows from Corollary 12.13. Assuming that (81) is true
for all A; < n, we want to prove it for к = ?. Then, by Corollary 60(H) and Halmos work [36,

1006
D. Butnariu and ?. P. /Clement
p. 125], there exists a crisp set В e Tv such that ???(?) = 0 and the и-vector TL-measure
wb defined by wb(E) = w(B ¦ E) is ???-absolutely continuous. Define the и-vector T^-
measureswi and W2 by W|(?) = ???(?-?) and \??(?) =iiiq(CB E),respectively.Since
for each ? e ? we have
/oW|(?) = /ome(B-?) = uQ(B ¦ E) = 0,
we also get Ran(wi) с L. Hence, the affine dimension of Ran(wi) is less than or equal
to ? — 1, implying that Ran(W|) is compact due to the nonatomicity of wi and the
inductive assumption. Let ? be a point in L П cl(Ran(mg)). Then there exists a sequence
(E/)/eN Я: Т with lim/^oo ???(?7) = ?, and therefore,
0 = f(x) = f( lim mQ(Ei)) = lim ???(?,).
By the ???-absolute continuity of wi we get lim/^эс wi(?/) = 0. Note that ??? = W| + wi
which implies
lim mn(?;)= lim W|(?/)=^,
or, in other words, ? e cl(Ran(W|)) =Ran(wi) с Ran(m?). Hence, (81) holds. D
Theorem 13.1 obviously extends the Liapounoff Theorem [52]. Observing that 7l-
measures on generated tribes are necessarily nonatomic (see Corollary 9.4 (ii)), Theorem
13.1 contains the theorems of Dvoretzki et al. [29] stating that the range of a vector integral
is convex and compact as a special case.
14. A Liapounoff type theorem for ??-measures
Most recently, Barbieri and Weber [4] have proved a Liapounoff Theorem for general
?? -measures. In order to state these results, some preparatory steps are necessary [4].
PROPOSITION 14.1. Let ? be a TM-c!an and m: ? -> R be a function satisfying (TM 1)
and (TM2) (with respect to Гм). Then the function dm:T -> [0, oo] defined by
dm(A, B) = sup{\m(C) - m(D)\: C,DeT, ATM В с С с ?) с ASM 6}
« ? pseudo-metric on the T^-clan T.
Definition 14.2. Let ? be a TM-clan and m:T -> R be a function satisfying (TM1)
and (TM2) (with respect to Гм). Then ? is said to be m-chained if, for all А, В e ? and
for each ? > 0, there exist Co, C|,..., C„ in ? with A = Co <? С ? с · · · с С„ = S and
dm(Ci-1, d) < ? for all i e {1,2,..., n).

Triangular norm-based measures
1007
We only mention that, in general, whenever ? is m-chained then m is nonatomic, but not
conversely. However, a TL-measure m on a Гд-tribe ? (and, therefore, its restriction m to
the ?-algebra of crisp sets Tv) is nonatomic if and only if ? is m-chained or, equivalently,
if Tv is m-chained. Also note that, in the case of a measure m on a ?-algebra ? of crisp
subsets of X, the pseudo-metric space (A, dm) is always complete. The following result is
also due to [4].
THEOREM 14.3. LetT bea ??-tribe andm.T'-> Ш" a nonatomic n-vector ??-measure
on ? for some ? e ]0, oof. Then we have:
(i) if ? is m-chained, then Ran(m) is convex;
(ii) if the space (T, dm) is complete, then Ran(m) is compact.
Theorem 14.3 does no longer hold if we drop the additional assumptions in (i) and (ii),
as the following counterexamples show [4].
Example 14.4.
(i) Consider the (rA-)tribe [0, l](v,I of all fuzzy subsets of the singleton {*o}. Then the
function m: [0, l](l,I -> [0,2] defined by
,лч @ ifA(*0) = 0,
m(A) = 1
[ A(xo) + 1 otherwise,
is a nonatomic TA-measure for each ? e ]0, oof whose range, however, is not convex,
(ii) IfX:??—» [0, 1] denotes the usual Lebesgue measure on the ?-algebra В of all Borel
subsets of [0, 1 ], then the function m: Вл -> [0, 1 ] given by (compare C))
m(A)=X(supp(A))- / Adk
./{0.1]
is a rA-measure for each ? e ]0, oof and Вл is m-chained, but Ran(m) is not
compact.
Most results of this section can be generalized: the values of the measures in question
may be not only numbers in the (extended) real line, but in an arbitrary normed
commutative group (see Weber [77] and Barbieri and Weber [3,4]).
References
[ 1 ] J. Aczel, Lectures on Functional Equations and their Applications, Academic Press, New York A966).
[2] R.J. Aumann and L.S. Shapley, Values of Non-Atomic Games. Princeton University Press, Princeton A974).
[31 G. Barbieri and H. Weber, A topological approach to the study of fuzzy measures. Functional Analysis and
Economic Theory, Y. Abramovich. E. Avgerinos and N.C. Yannelis. eds. Springer. Berlin A998). 17^46.
[4] G. Barbieri and H. Weber, A representation theorem and a Lyapunov theorem for 7", -measures: The solution
of two problems of Butnariu and Klement. J. Math. Anal. Appl. 244 B000). 408^424.
[5] H. Bauer, Probability Theory and Elements of Measure Theory. Academic Press, London A981).

1008
D. Butnariu and ?.P. /Clement
[6] L.P. Belluce, Semi-simple algebras of infinite valued logic and bold fuzzy set theory, Canad. J. Math. 38
A986), 1356-1379.
[7] L.P. Belluce, Semi-simple and complete MV-algebras, Algebra Universalis 29 A992). 1-9.
[8] G. Birkhoff, Lattice Theory, Amer. Math. Soc.. Providence. RI A973).
[9] D. Blackwell, The range of certain vector integrals, Proc. Amer. Math. Soc. 2A951). 390-395.
[10] L. Borkowski (ed.), J. Lukasiewicz: Selected Works, Studies in Logic and Foundations of Mathematics,
North-Holland, Amsterdam A970).
[11] D. Butnariu, Additive fuzzy measures and integrals. I, J. Math. Anal. Appl. 93 A983). 436-452.
[12] D. Butnariu, Decomposition and range for additive fuzzy measures. Fuzzy Sets and Systems 10 A983),
135-155.
[13] D. Butnariu, Non-atomic fuzzx measures and games. Fuzzy Sets and Systems 17 A985), 39-52.
[14] D. Butnariu, Fuzzy measurability and integrability, J. Math. Anal. Appl. 117 A986), 385^410.
[15] D. Butnariu, Additive fuzzy measures and integrals. Ill, J. Math. Anal. Appl. 125 A987), 288-303.
[16] D. Butnariu and E.R Klement, Triangular norm-based measures and their Markov kernel representation,
J. Math. Anal. Appl. 162 A991), 111 -143.
[17] D. Butnariu and E.P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions,
Kluwer, Dordrecht A993).
[18] C.C. Chang, Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88 A958), 467^490.
[19] H. Chemoff, An extension of a result of Liapounoff on the range of a vector measure, Proc. Amer. Math.
Soc. 2A951), 720-726.
[20] A.H. Clifford, Naturally totally ordered commutative semigroups, Amer. J. Math. 76 A954), 631-646.
[21] A.C. Climescu, Sur Vequation fonctionelle de I'as.sociativite. Bull. Ecole Polytechn. Iassy 1 A946), 1-16.
[22] B. De Baets and R. Mesiar, Г-partitions. Fuzzy Sets and Systems 97 A998), 211-223.
[23] M. de Glas, Representation of Lukasiewicz' many-valued algebras. The atomic case, Fuzzy Sets and
Systems 14 A984), 175-185.
[24] A. Di Nola, Representation and reticulation by quotients of MV algebras, Ricerche Mat. 40 A991), 291-
297.
[25] D. Dubois, M. Grabisch and H. Prade, Gradual rules and the approximation of functions. Proceedings 2nd
International Conference on Fuzzy Logic and Neural Networks, Iizuka A992), 629-632.
[26] D. Dumitrescu, Fuzzy partitions with the connectives T^, Sy_, Fuzzy Sets and Systems 47A992), 193-195.
[27] D. Dumitrescu, Fuzzy measures and the entropy of fuzzy partitions. J. Math. Anal. Appl. 176 A993), 359-
373.
[28] A. Dvoretzky, A. Wald and J. Wolfowitz, Elimination of randomization in certain problems of statistics and
game theory, Proc. Nat. Acad. Sci. USA 36 A950), 256-259.
[29] A. Dvoretzky, A. Wald and J. Wolfowitz, Relations among certain ranges of vector measures. Pacific J.
Math. 1A951), 59-74.
[30] M.J. Frank, On the simultaneous associativity of F(x.y) and x+y- F(x, v), Aequationes Math. 19A979),
194-226.
[31] S.Gottwald, A Treatise on Many-Valued Logic, Research Studies Press, Baldock B001).
[32] P. Hajek, Metamathematics of Fuzz\ Logic, Kluwer, Dordrecht A998).
[33] PR. Halmos, On the set of values of a finite measure. Bull. Amer. Math. Soc. 53 A947), 138-141.
[34] PR. Halmos, The range of a vector measure. Bull. Amer. Math. Soc. 54 A948), 416-421.
[35] PR. Halmos, Measure Theory, Van Nostrand Reinhold, New York A950).
[36] PR. Halmos, Measure Theory, Springer, Berlin A974).
[37] U. Hohle, Fuzzy equalities and indistinguishability. Proceedings EUFIT '93, Aachen, Vol. 1 A993), 358-
363.
[38] U. Hohle, On the fundamentals of fuzzy set theory, J. Math. Anal. Appl. 201 A996), 786-826.
[39] U. Hohle, Many-valued equalities, singletons and fuzzy partitions. Soft Computing 2 A998), 134-140.
[40] U. Hohle and E.P. Klement, Plausibility measures - a general framework for possibility and fuzzy
probability measures, Aspects of Vagueness, H.J. Skala, S. Termini, and E. Trillas, eds, Reidel, Dordrecht
A983), 31-50.
[41] U. Hohle and S. Weber, Uncertainty measures, realizations and entropies. Random Sets: Theory and
Applications, J. Goutaias, R.P.S. Mahler, and H.T. Nguyen, eds. Springer, Heidelberg A997), 259-295.

Triangular norm-based measures
1009
F. Klawonn and R. Kruse, Equality relations as a basis for fuzzy control, Fuzzy Sets and Systems 54 A993),
147-156.
E.P. Klement, Characterization of finite fiizzv measures using Markoff-kemels, J. Math. Anal. Appl. 75
A980), 330-339.
E.P. Klement, Fuzzy ? -algebras and fuzz\ measurable functions. Fuzzy Sets and Systems 4 A980), 83-93.
E.P. Klement, Characterization of fuzzy measures constructed by means of triangular norms. J. Math. Anal.
Appl. 86A982), 345-358.
E.P. Klement, Construction of fuzzy ?-algebras using triangular norms, J. Math. Anal. Appl. 85 A982),
543-565.
E.P. Klement, R. Mesiar and E. Pap, A characterization of the ordering of continuous t-norms. Fuzzy Sets
and Systems 86 A997), 189-195.
E.P. Klement, R. Mesiar and E. Pap, Triangular Norms, Kluwer, Dordrecht B000).
E.P. Klement and M. Navara, A characterization of tribes with respect to the Lukasiewicz t-norm,
Czechoslovak Math. J. 47 A22) A997), 689-700.
E.P. Klement and M. Navara, A sunev on different triangular norm-based fuzzy logics. Fuzzy Sets and
Systems 101 A999), 241-251.
R. Kruse, J. Gebhardt and F. Klawonn, Foundations of Fuzzy Systems, Wiley. Chichester A994).
A.A. Liapounoff, Sur les functions-vecteurs completement additives, Bull. Acad. Sci. URSS Ser. Math. 4
A940), 465^478.
J. Lindenstrauss, A short proof of Liapounoff's convexity theorem, J. Math. Mech. 15 A966), 971-973.
CM. Ling, Representation of associative functions, Publ. Math. Debrecen 12 A965), 189-212.
J. Lukasiewicz, О logice trowartosciowej, Ruch Filozoficzny 5 A920), 170-171. (English translation
contained in [10].)
K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA 8 A942), 535-537.
K. Menger, Ensembles flous et functions aleatoires. C. R. Acad. Sci. Paris Ser. A 232 A951), 2001-2003.
K. Menger, Probabilistic geometry, Proc. Nat. Acad. Sci. USA37 A951), 226-229.
K. Menger, Probabilistic theories of relations, Proc. Nat. Acad. Sci. USA 37 A951), 178-180.
R. Mesiar, Fundamental triangular norm based tribes and measures, J. Math. Anal. Appl. 177 A993),
633-640.
R. Mesiar, On the structure ofTs-tribes, Tatra Mt. Math. Publ. 3 A993). 167-172.
R. Mesiar and M. Navara, Ts -tribes and Ts -measures, J. Math. Anal. Appl. 201 A996), 91-102.
R. Mesiar and J. Rybarik, Pan-operations structure. Fuzzy Sets and Systems 74 A995). 365-369.
M. Navara, A characterization of triangular norm based tribes, Tatra Mt. Math. Publ. 3 A993). 161-166.
R.B. Nelsen, An Introduction to Copulas. Lecture Notes in Statistics, Vol. 139, Springer, New York A999).
J. von Neumann and O. Morgenstem, Theory of Games and Economic Behavior. Princeton University Press,
Princeton A944).
G. Poprugenko, Sur la proprie'te de Darboux des functions continues d'ensemble, Fundam. Math. 12A928),
254-268.
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton A970).
A. Rose and J.B. Rosser, Fragments of many-valued statement calculi. Trans. Amer. Math. Soc. 87 A958),
1-53.
E. Ruspini, A new approach to clustering. Inform, and Control 15 A969). 22-32.
K.D. Schmidt, A general Jordan decomposition. Arch. Math. 38 A982). 556-564.
B. Schweizer and A. Sklar, Associative functions and abstract semigroups, Publ. Math. Debrecen 10A963),
69-81.
B. Schweizer and A. Sklar. Probabilistic Metric Spaces, North-Holland, New York A983).
A. Sklar, Functions de repartition a n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris 8 A959),
229-231.
M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis, Tokyo Institute of Technology
A974).
H. Thiele and N. Schmechel, On the mutual definability of fuzzy equivalence relations and fuzzy partitions.
Proceedings of the International Joint Conference of the Fourth IEEE International Conference on Fuzzy
Systems and the Second International Fuzzy Engineering Symposium, Yokohama, Vol. Ill, IEEE Press
A995), 1383-1390.

1010
D. Butnariu and ?. P. Klement
[77] H. Weber, On modular functions, Funct. Approx. Comment. Math. 24 A996), 35-52.
[78] O. Wyler, Clans, Compositio Math. 17 A966), 172-189.
[79] L. A. Zadeh, Fuzzy sets. Inform, and Control 8 A965), 338-353.
[80] L.A. Zadeh, Pmbability measures of fuzzy events, J. Math. Anal. Appl. 23 A968), 421^427.
[81] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1 A978), 3-28.

CHAPTER 24
Geometric Measure Theory:
Selected Concepts, Results and Problems
Miroslav Chlebik
Department of Mathematics, Comenius University. Mlynska dolina, 842 48 Bratislava, Slovakia
E-mail: chlebik@fntph.uniba.sk
Contents
Introduction 1013
1. Preliminaries from measure theory 1013
2. Structure theory for integral dimensional sets Ю17
3. Densities of measures and rectifiability 1020
3.1. Preiss's theorem 1020
3.2. Related results and problems 1022
3.3. Besicovitch j-problem 1023
4. Sets of finite perimeter 1025
5. Measure-theoretic calculus of variations 1027
5.1. Normal and integral currents 1028
5.2. Closure and compactness theorem 1033
5.3. Regularity of minimal surfaces 1034
References 1035
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1011

This page intentionally left blank

Geometric measure theon:· Selected concepts, results and problems
1013
Introduction
One of the main themes of geometric measure theory is the detailed study of geometric
properties of general sets and Borel measures in R". The first decades after Caratheodory's
fundamental paper in 1914 were spent studying how subsets of R" behave with respect to
m-dimensional measures, mainly Hausdorff-like. Geometric measure theory originates in
the pioneering work done by Besicovitch on the structure of the subsets of the plane having
finite one-dimensional Hausdorff measure.
Besicovitch's work was extended by Federer, Marstrand, and later by Mattila and Preiss,
to m -dimensional subsets and measures in R", whose rectifiability, density, and projection
properties were analyzed.
During the last fifty years the results and methods developed in the study of geometry of
sets and measures by many researchers have fused with modern techniques from functional
and harmonic analysis, and the calculus of variations. This resulted in producing new and
very efficient tools: the theory of generalized surfaces and varifolds, the theory of real flat
chains, normal and integral currents of Federer and Fleming, the (partial) regularity theory
of solutions of measure-theoretic variational problems (De Giorgi, Fleming, Reifenberg).
From the recent trends one should mention results in fractal geometry (Falconer), works
on quantitative notions of rectifiability and their applications to singular integral operators
(Jones, David, Semmes). The very new is the progress in the results from geometric
measure theory in metric spaces (Ambrosio, Kirchheim, Cheeger).
This chapter gives an exposition of selected concepts, results and problems from
geometric measure theory. For more comprehensive treatment we refer to the list of
references. Nevertheless we hope that the choice of topics presented here can show the
character of this modern piece of mathematics.
General references. The most standard reference for the geometric measure theory is
Federer's extensive book [19]. Other monographies covering much of the topic are due to
Rogers [37], Simon [39], Giusti [22], Hardt and Simon [23], Morgan [32], Ziemer [44],
Evans and Gariepy [13], Mattila [30]. The mathematical theory of fractals is covered by
books of Falconer [14,15].
A good account of open problems in geometric measure theory is the list edited by
Brothers [8]. Manila's book [30] discusses many open problems and provides an excellent
list of references. A nice treatment on the topic is an expository paper by Federer [20].
1. Preliminaries from measure theory
We shall work mainly in Euclidean and closely related spaces, but most of measure theory
treated here is valid in much more general setting. To introduce the basic notations, let X
be a metric space with a metric d. The closed and open balls with center a e X and radius
r > 0 are denoted by
B(a,r) = {xeX: d(a,x) <; r}, U(a,r)=\x e X: d(a,x)<r].

1014
?. Chlebt'k
In measure theory usually a "measure" means a non-negative countably additive set
function defined on a ?-algebra of subsets of X. In our exposition we use the notion
of "measure" as a convenient abbreviation for "outer measure", the notion derived from
Caratheodory's one.
A set function ?: {А: А С X} -> [0, +oo] is a measure on a set X if ?@) = 0 and
? (A) sC ??\ H(Ai) whenever А с U?i A,¦ С X.
A set А С X is ?-measurable if
?(?) = ?(???) + ?(?\?) whenever Г С X.
It is rather convenient to have measures defined for all subsets of X. On the other hand,
a measure ? gives a countably additive set function when restricted to the ?-algebra of
?-measurable sets.
With each measure ? on X and each subset В of X we associate the measure ? L В
on X, the restriction of ? to ?, defined by
(???)(?) = ?(????) for А С X.
A measure ? on a metric space X is called a Borel measure if all Borel sets are ?-
measurable. It is Borel regular if it is a Borel measure and if for every А с X there is a
Borel set В С X such that А С ? and ?(?) = ?(?).
It is well known that a measure ? on a metric space X is a Borel measure if and only if
?(? U ?) = ?(?) + ?(?) whenever ?, ? С X with dist(A, ?) > 0.
Further, ? is a Radon measure if it is a Borel measure and
A) ?{?) < oo for compact sets К с Х,
B) ?(?/) = sup^(?): ? cU,K compact} for open sets U С X,
C) ?(?) = inf^(A): А С U, U open} for all А С X.
In R", a measure ? is a Radon measure if and only if it is Borel regular and finite on
bounded sets. More generally this holds for any ?-compact metric space.
For any two positive integers m < ? there are many distinct m -dimensional Borel regular
measures on Ш" that are invariant under the Euclidean group and coincide with the surface
w-measure, when restricted to "nice" m-dimensional subset. Some of such measures are
based on the integral geometry approach, another ones on coverings or packings.
The following very general construction can yield, in particular, many m-dimensional
measures.
Caratheodory's construction. Let (X, d) be a metric space, ?' a family of subsets of X
and ? a non-negative function on У.
For 0 < S ? oo we construct the size S approximating measure <ps as follows: <ps($) = 0
and whenever Ъф А с ?, ?&(?) is the infimum of the set of numbers (possibly +oo)

Geometric measure theory: Selected concepts, results and problems
1015
corresponding to all countable (possibly finite) families & with
У С {Е: Ее ^ and diam(?) ^ <$} and ACU^.
The fact that ?& ^ ?? whenever 0 < ? ^ ? ^ oo implies the existence of ?(?) =
Unburn (A) = supi>0^(A).
The measure ? is called the result of Carathiodory's construction from ? on ?. One
finds that ??, can be highly non-additive but, on the other hand, ? is always a Borel measure.
If all members of & are Borel sets, ? is a Borel regular measure.
Hausdorff measures. Using Hausdorff measures is the most natural way to measure lower
dimensional objects in higher dimensional space.
For every non-negative real number m we define a(m) = ?A /2)'"/ Г(т/2 + 1). For a
fixed metric space (X, d) we apply Caratheodory's construction with
,9 = {A: Ъф AcX} and ? (A) = a(wJ"'"(diam A)'".
The resulting measure ? is called the m -dimensional Hausdorff measure on X and
denoted by Ж'". One obtain the same measure by letting & be the family of all
nonempty closed subsets of X, or the family of all non-empty open subsets of X. It follows
that Жт is Borel regular. In case X is a normed vector space, the same measure results
when the members of & are further restricted to be convex.
For a fixed subset A of (X, d) only the restriction of a metric d to the set A is relevant to
determine Жт (A), regardless of the ambient space. In fact, Ж™ (А) = Ж"' (A) whenever
AcX and X isometrically embeds in Y.
The integral dimensional Hausdorff measures are of a special importance. One finds that
Ж° is the counting measure over X,
Ж°(А) = the number (possibly +oo) of elements of A.
We recall (see for instance [24, Lemma 6(i)]) that if X is an m-dimensional vector space
and ?@, 1) is its unit ball, then Ж"'(В@, 1)) is a dimensional constant independent of the
norm of X, and equal, in particular, to a(m).
For m = 1, Ж' has a concrete interpretation as a generalized length measure. More
generally, if m is an integer, 1 ^ m < n, and ? is a sufficiently regular (e.g., С' smooth)
m -dimensional surface in R", then the restriction Жт\—М coincides with the surface
measure on M.
For m = n, Ж" coincides in R" with the Lebesgue measure Jif". The proof is rather
difficult and relies on the isodiametric inequality
?"'(A) sC 2""a(«)(diam(A))" whenever 0 ф А с R",
see [19, 2.10.33].

1016
?. Chlebik
Invariant measures. The orthogonal group 0{n). Let 0(n,m) be the space of all
orthogonal injections (i.e., inner product preserving linear maps) of R"' into R". Moreover,
0{n) = 0(n, n) be the orthogonal group of R", with composition as a group operation.
From the general theory of invariant measures recall the well-known existence result of a
Haar measure on a compact topological group to ensure that there exists a unique (left and
right) Haar measure 9„ on 0(n) normalized with ?[0(??)] = 1.
The Grassmannian G(n,m). For any positive integers m < и the Grassmannian G(n, m)
consists of all m-dimensional linear subspaces of R". The action (g, V) ь* gV (g e 0(n)
and V e G(n,m)) is transitive and induces a unique normalized invariant measure y„.„,
on G(n,m). The special cases m = 1 or m = n — 1 are much easier to handle, as the
measures ?„.\ and ?„.„-\ can be easily represented using the surface measure on the unit
sphereS".
Other measures that agree with the surface m -measure when restricted regular torn-
dimensional sets are the integralgeometric measures ^'" A ^ t ^ oo).
Integralgeometric measures. Let us start with Crofton's classical method for finding the
length of a plane curve A: for each line L count the number of points in the intersection
А П L and integrate this varying number over all lines L in R2. Doing this in R" with affine
(n — m)-dimensional subspaces of R" we can define the m-dimensional integralgeometric
measure (with parameter 1) on R" as the Borel regular measure such that
S['\A)= f f Ж0(АГ)Р-\у))<МГ"(у)а-уп,п(У)/Рап,т)
JGU'.m) JV
for every Borel set А с R", where Pv : R" -> V is the standard orthogonal projection on
V e G(n,m), and
is the normalizing constant. In the above formula for J"\A) we integrate the varying
number of points of intersection of A with arbitrary (и — m )-dimensional affine subspaces
Py ' (y) corresponding to all pairs (V, y) with V e G(n, m) and ye V.
It turns out that J?"' can also be defined via Caratheodory's construction. For this
purpose we can choose У to be the family of all Borel subsets of R" and we let ? be
the function defined on Borel sets by
?(?)= ? JPm(Pv(B))dY„.m(V)/fii(n,m).
JG(n.m)
More generally, to get a one-parameter family ,_?"' A ^ t ^ oo) of m-dimensional
integralgeometric measures, we let ?, be the function defined on Borel sets by
b(B)=\\V»jr,(PviB))\\L,lYHM)/fr(,i,m),
where fit(n, m) is a properly chosen normalization constant.

Geometric measure theory: Selected concepts, results and problems
1017
One can observe that А с R" is ./'/"-null if and only if A is contained in a Borel set В
with Ж'"(Ру(В)) = 0 for y„.„, almost all V e G(n, m).
Densities. Given a measure ? on a metric space (X, d) and ? e X, the иррег and /ower
m-dimensional densities of ? at ? are respectively defined by
в^,Ху.= Ш«*<±1», ^V,^:=lim^^.
/-40 a(w)r'" 7?0 a(w)r"'
In the case ?*'"(?, ?) — ?"'(?, ?), this common value is denoted by ?'"(?,?) and termed
the m-dimensional density of ? at ?.
If ? = Жт^А for a set А С X, we use simpler notation ?*'"(?,?), ??(?,?) and
?"'(?,?).
In this chapter we do not treat the case of non-integral m. From now on, m is a positive
integer. The case m = 0 could be trivially covered with some care.
2. Structure theory for integral dimensional sets
In this section we present parts of the structure theory: a characterization of rectifiable sets
in terms of their projection and tangential properties.
In his pioneering papers [5-7] Besicovitch investigated ways of differentiating between
regular and irregular 1-sets in the plane. It is easy to see that for almost all straight lines
(through the origin) the orthogonal projection of a rectifiable (or regular) 1-set is a set of
positive length. Besicovitch [7] studied the projection properties of irregular 1-sets and
showed that the orthogonal projections onto lines of a purely unrectifiable (or irregular)
1 -set in the plane would have zero length for almost every line.
Concerning tangential properties, a rectifiable 1 -set has an approximate tangent at almost
all its points. In contrast, at almost all points of a purely unrectifiable 1 -set no approximate
tangent exist.
Federer [19] generalized results of Besicovitch to m-dimensional subsets of R*.
His work has extended considerably our understanding of the structure of (integral
dimensional) sets in Euclidean space and clearly illustrates the profound dichotomy which
exists between rectifiable and purely unrectifiable sets. A good account of the Besicovitch-
Federer structure theory, variants of projection theorems and results on approximate
tangent planes, may be found in the books by Federer [19] and Mattila [30].
Let m be a positive integer and ? be a measure on R". A set А с R" is called (?, m)
rectifiable if there are Lipschitz maps f : R'" -> R", / = 1,2...., such that
? A\|Jy;(R''') =0.
i = \
A set А С R" is called purely (?, m) unrectifiable if ? (А П /(R'")) = 0 for every Lipschitz
map /: R'" -> R".
In case ? = Жт we will use terms w-rectifiable and purely m-unrectifiable set instead.

1018
?. Chlebik
This notion of rectifiability very naturally generalizes to a measure-theoretic setting
the classical approach to first order smoothness. Indeed, one can prove (see Federer
[19, 3.2.29]) that А с R" is w-rectifiable if and only if there are w-dimensional C1
submanifolds ?,, M2, ¦. ¦, of R" such that Ж'"(A \ [jfi, M,-) = 0.
Let С с R" with Ж'" (С) < оо be given. It is easy to see that the set С can be
decomposed into w-rectifiable and purely w-unrectifiable part, and that such
decomposition is unique up to Жт null-sets. Indeed, one can show that among the sets of the form
U^i fi№m), corresponding to all sequences {./}}?, of Lipschitz maps /, :E'" -> R",
/ = 1,2,..., there is one, say B, which maximize Ж'"(С ? ?).
Clearly ? is a Borel set (?-compact), С ? В is w-rectifiable and С \ В is purely w-
unrectifiable.
To describe a notion of the (?, w) approximate tangent cone Tan'" (?, -*) at * e R", we
will use the open cones
?(?,?,?) = jyel": \\t(y - x) - v\\ < ? for some? > 0}
with the vertex x, whenever ? e R" and ? > 0.
We define the (?, w) approximate tangent vectors at the point ? as the elements of the
closed cone with vertex 0
Тап'и^,.х) = р){иеК"·. ?*'"^?(?,?,?),?) > 0].
The following deep result was proved first for w = 1 and ? = 2 by Besicovitch, Morse
and Randolph, then for arbitrary dimensions by Federer.
STRUCTURE THEOREM. Let m < ? be positive integers, ? be a finite Borel regular
measure on M." such that
0<?*'"(?,?) <?? for ? almost all ? eR",
and A = {x e R": there is ? e G(n, m - n) such that Tan'"(?, x) П ? С {0}}.
Then:
A) A is ? (?, w) rectifiable Borel set.
B) For ? almost all ? e A, Tan"V,jt) e G(«,w) ????*'"(?,?) = ?"(?,?)-
C) Wie/iever ? С ?, ?(?) > 0 implies Жт(Ру(В)) > 0 for ?„.,„ almost all V e
G(n, m). Consequently, whenever В С А, У"(В) = 0 implies ?{?) = 0.
D) R" \ A is purely (?, w) unrectifiable.
E) JT"(/V(R" \ A)) = 0/or y„.„, ?/woii a// V e G(n, w), /геисе ^"'(R" \ A) = 0.
As a corollary we obtain that the following conditions are equivalent:
(i) R" is ^,w) rectifiable.
(ii) Tan'"(?,*) e G(n, m) for ? almost all ? e R".
(iii) Whenever В с R", </,'" (?) = 0 implies ?(?) = 0.
In particular, for a Borel set С С R" with Ж'"{С) < oo, and ? =Ж'"\- С, the structure
theorem implies that С П A is w-rectifiable and С \ A is purely w-unrectifiable.

Geometric measure theory: Selected concepts, results and problems
1019
An important consequence (see also Federer [19, 3.2.26]) is the following: Жт ^ У"'
and whenever С с R" is such that Ж'" (C) < oo. then
Ж'п (С) = J™ (С) if and only if С is w-rectifiable, and,
У" (C) = 0 if and only if С is purely w-rectifiable.
This is exactly the reason, why we find the Hausdorff measure Ж'" and the integralgeo-
metric measure У"' the two most important w-dimensional measures.
The dichotomy between rectifiable and purely unrectifiable sets concerning their
projection properties can be summarized as follows:
THEOREM. Let Ac R" be an Ж'"-measurable set with Ж"\А) < oo.
A) A is m-rectifiable if and only if Jff"'(Pv(B)) > Ofor ?,,.,,, almost all V e G(n, w)
whenever В С A is an Ж'" -measurable set with Ж"\В) > 0.
B) A is purely m-unrectifiable if and only if Jf'"(Pv(A)) = Ofor ?„,,„ almost all
V eG(n,m).
To describe the dichotomy between rectifiable and purely unrectifiable sets concerning
their tangential properties we need some notation and the definition of the approximate
tangent planes.
If V e G{n, w), ? e R" and j e @,1) we denote
Xix, V,s) = {yeR": dist(}--;c, V) < s \\y - x\\\.
This is a kind of an open cone around the affine in -plane (V + x).
Let А С ?", ? е ?" and V е G in, m). We say that V is an approximate tangent m -plane
for A at ? ???* (A, x) > 0 and for all j e @,1)
\imr-'nJf'"(AC\Bix,r)\Xix, V,s))=0.
THEOREM. Let Ac R" be an Ж'"-measurable set with Ж'" (A) < oo.
A) A is m-rectifiable if and only if for Ж'" almost all ? e A an approximate tangent
m-planefor A at ? exists.
B) A is purely m-unrectifiable if and only if for Ж'" almost all ? e A no approximate
tangent m-planefor A at ? exists.
It is very useful to find weaker sufficient conditions for a set to be w-rectifiable. One
can conclude that A is w-rectifiable from the existence for Ж'" almost every дг е A of so
called "rotating approximate tangent w-plane" for A at x.
THEOREM. Let А с R" be an Ж'"-measurable set with Ж"\А) < oo. Then A is m-
rectifiable if and only if for Ж'" almost all ? e A the following holds: whenever ? > 0
there are ro > 0 and ? > 0 such that for any r e @, rn) there is V e Gin, m) satisfying
Ж'п (AC\ Bix ,r)\{y &R": dist(j - x, V) < ?}) <?"\

1020
?. Chlebik
and
Ж'п(АГ)В(у,г)) >krm for every ye (V + ?) ? B(x, r).
In this theorem an "approximate tangent w-plane" V for A at ? in a ball B(x,r) is
allowed depend on radius r. Therefore it is called a "rotating approximate tangent w-
plane".
The theorem is due to Marstrand [27] and Mattila [29] (see also [30, Chapter 16]). It was
the first powerful method to prove, for m > 1, w-rectifiability, and a simpler precursor to
the Preiss's method of tangent measures, discussed partially in the next section.
3. Densities of measures and rectifiability
In this section we will work with general Radon measures on R" and investigate the relation
between their w-densities and w-rectifiability. First we have to introduce the notion of
rectifiability of measures.
A Radon measure ? on R" is said to be m-rectifiable measure if ? <<C Hm and there
exists an w-rectifiable Borel set ? such that ?(?" \ E) = 0.
The conjecture that Radon measures on E" with positive finite w-density are w-
rectifiable was one of central problems in geometric measure theory for a long time. Special
cases were established over a period of fifty years by Besicovitch [6], Morse and Randolph
[33], Moore [31], Marstrand [28] and Mattila [29]. This conjecture was positively resolved
by Preiss [34].
3.1. Preiss's theorem
Preiss's theorem says that if the behaviour of a Radon measure on small balls settles down
in the positive and finite density then the measure is just given by a locally H'" summable
function which can be non-zero only on countably many С' submanifolds; the measure
of any Borel set A just being the integral of this function over A. We shall now state his
theorem.
THEOREM. Let ? be a Radon measure on W with the property
?(?(?,?))
0 < hm < oo
r|0 Г'"
for ? almost all ? e R". Then ? is m-rectifiable.
For Hausdorff measures restricted to sets this Theorem gives the following:
COROLLARY. Let A be a subset ofR" with 7i"\A) < oo and such that the density
?'" (A, x) exists for almost all ? e A. Then A is m-rectifiable.

Geometric measure theory: Selected concepts, results and problems
1021
The proof of the theorem of Preiss is very complicated and we shall only discuss some
ideas behind it.
The very important notion introduced and effectively used in his proof was that of
tangent measure.
Given a Radon measure ? on R" and a point ? el", the set ???(?,?) of tangent
measures to ? at ? consists of the non-zero measures obtained from weak limits of scaled
push-forwards of ? by sequences of expansive homotheties around x. More precisely,
? e Тап(д, jc) if and only if ? is a non-zero Radon measure on R" and there exist sequences
(с,-) and (r,) of positive numbers, such that r, 4- 0, and
?,??.??#? -> ? weakly as / -> oo.
Here TXJ is the map given by TXJ(y) = (y — x)/r and the image ???#? of ? under Txr
is defined by
??.*?(?) = ?(??+?), А С М".
It should not be too hard to believe that the assumption that ? has positive finite w -density
almost everywhere will result in the fact that for ? almost all ? e R" every tangent measure
? e Тап(д, х) is m-uniform in the following sense:
There exists a positive number с = c(x) such that
v(B(y,r)) = cr'" for all у e sptv and all r > 0.
Anyone would think that every w-uniform measure on R" is just the one given by a
constant multiple of W" restricted to an w-plane. Such a measure will be called w-flat.
If this was true this would imply that for ? almost every ? e spt ? inside small enough
balls B(x, r) the measure ? would be concentrated close to an w-plane through ?; ? would
have at ? so called "rotating approximate tangent w-plane". Due to methods developed
earlier by Marstrand [27] and Mattila [29] and mentioned in the previous section, this
would imply rectifiability.
But this was found to be true for w < 3 only! For w ^ 3 let
K = {xeR'"+]: *f=jcf+*?+*;}.
Then TV" L? is an w-uniform measure. Moreover, Kowalski and Preiss [25] proved that
it is effectively (up to rotations, translations and multiplication by a constant) the only
possible example of a non-trivial w-uniform measure in R'"+l.
The structure of w-uniform measures in higher codimensions is open.
What was important in the proof of Preiss's theorem was a partial classification of
w-uniform measures into two distinct types, those "flat at infinity" such as the w-flat
examples, and those "curved at infinity" like Ti.'"\—K.
This classification of possible limiting behaviour of w-uniform measures resulted
eventually in the proof that at ? almost every ? e R" every tangent measure ? e Тап(д, х)
is flat that is enough to conclude the proof.

1022
?. Chlebik
Firstly, pushing methods developed by Marstrand [28] to the tangent measure setting
one concludes that at ? almost every ? e R" there exists tangent measure ? e Тап(д, х)
that is flat.
Secondly, if Тап(д, х) contains tangent measures of both types, flat and curved, then a
proper choice of blow-up sequence would generate a tangent measure which is neither flat
nor curved; which contradicts with the partial classification of m-uniform measures. This
is how the proof works.
3.2. Related results and problems
For the special case of the Hausdorff m -measure on R" restricted to sets of locally finite
measure, and with the m -density equal to 1, some results had been proved earlier.
THEOREM. If А с R" is a set with the property that the density ?'" (A, x) exists and equals
1 for Jif" almost all ? e A, then A is m-rectifiable.
The converse of this implication is well known and not difficult. The statement of the
theorem was proved for m = 1 and ? = 2 by Besicovitch [5], for m = 1 and all ? by
Moore [31], for m = 2 and ? — 3 by Marstrand [27], and for general m, ? by Mattila [29].
The case m = 1 is simpler an a nowadays proof can be found in the book by
Falconer [14]. Marstrand's method (extended by Mattila) by which this theorem was
proved for any m is based on a relatively simple geometric observation. It uses a certain
rotundity property of the Euclidean ball and the fact that the m -density of A is equal to 1
to establish a very strong kind of approximate local symmetry of the set. This additional
information about A allows to prove its m-rectifiability.
The author was able to prove (essentially using the method of Marstrand and Mattila)
that the theorem above holds in any real Hubert space.
On the other hand, it is an open question (except when m = 1) if this theorem generalizes
to all metric spaces.
Question. Given a metric space (X,d) with Jif""(X) < oo, Ж'" being the m-dimen-
sional Hausdorff measure on (X,d).
Suppose X has the property that
hm = 1
r|0 a(m)r'"
for Ж"" almost all points xeX. Does it follow that X is m-rectifiable (in the sense that
Ж'" almost all of X is contained in countably many Lipschitz images of subsets of R'")?
The positive answer would be the ultimate (and very surprising) rectifiability result for
metric spaces. It would also provide the characterization of m-rectifiable spaces, as the
converse implication holds in any metric space by the theorem of Kirchheim [24].
For case m = 1 the positive answer has been given by Preiss and Tiser [35].

Geometric measure theory: Selected concepts, results and problems
1023
As any separable metric space is isometric to a subset of lx, the first interesting open
part of this problem is the case of metric space /^ and m = 2.
One can ask as well if the theorem of Preiss could be generalized to the setting of metric
spaces. But the result that 2-rectifiability follows from the existence of finite and non-zero
2-density fails even in a separable Hilbert space for a restriction of Ж2 to a properly
chosen subset. We may consider an easier metric space example (but it could be embedded
isometrically into /2), namely (X, d) being the real line with metric d(x, y) = \x - y\!/ .
In this space we have for any ball B{x, r)
Jf2(B(x,r))=l-a{2)r2,
hence, 2-density of the measure Ж2 exists everywhere and equals constantly j. On the
other hand, it can be shown easily that the image of any Lipschitz map from a subset of
E2 to (X,d) will have zero Ж2 measure, hence the measure Ж2 on (X,d) is purely
2-unrectifiable.
But in finite dimensions there is a quite natural conjecture generalizing Preiss's theorem.
Conjecture. Suppose X is a finite dimensional normed vector space and ? is a Radon
measure on X with the property that
?(?(?,?))
0 < lim < 00
no r'"
for ? almost all ? e X, then ? is w-rectifiable.
The first result that proves w-rectifiability from an initial condition about m-densities
for m > 1 and outside inner product spaces is the recent theorem of Lorent [26] stating that
locally 2-uniform measures in /^ are 2-rectifiable. We state his theorem that supports the
conjecture above.
THEOREM. Suppose ? is a Radon measure on l^ with the following property: There exist
some S > 0, с > 0 such that for all ? e 5??? and all r e ]0, <5[
?{?(?,?)) = cr~.
Then spt ? is 2-rectifiable.
3.3. Besicovitch ^-problem
Let us recall that a subset A of a metric space (X,d) is called w-set if 0 < Ж'"(А) < oo.
Such a set A is said to be purely w-unrectifiable if Ж"'(? ? ?) = 0 for every m-
rectifiable set B.

1024
?. Chlebik
For an w-rectifiable w-set A in (X, d) we have
?'"(?,?)= 1
for Ж'" almost all jc e A.
This is basically due to Besicovitch for subsets of R", and for subsets of metric spaces
it is due to Kirchheim [24].
At least in Euclidean space a purely w-unrectifiable w-set А с R" have remarkably
different behaviour, namely, for Жт almost all jc e A
??(?,?) ? ???(?")?*'"(?,?) (^ t„,(R")),
where r,„(R") < 1 is some constant. This is due to Preiss [34], but the one of its
consequences, that it is possible to characterize w-rectifiability in R" by means of the lower
w-density ?™, follows also from the earlier results of Besicovitch [5], Martstrand [27] and
Mattila [29].
Let ?,„ (X), for a metric space (X, d), be the smallest non-negative number such that, if
A is a purely w-unrectifiable w-set in X, then
?*(?,?) ? ???(?) for Ж'" almost all ? e A.
As any w-set A satisfies
?*'"(A, x) <: 1 for Ж"' almost all ? e A
(see Federer [19, 2.10.18C)]),clearly ?,,,(?) sC 1.
If a metric space (X, d) is such that ?,,,(?) < 1, one can characterize w-rectifiability in
X by means of the lower w-density ?"'.
It is proved by Besicovitch [5] that
a,(R2)iC 1- 1(?2576,
and in [6] he showed that ?\ (R2) ^ |. He also gave an example of a purely 1-unrectifiable
1-set A inR2 for which ?± (A, x) = \ for Ж' almost all jc e A. In this way he proved that
?? (R2) > J and conjectured that ?\ (R2) = j.
This conjecture is still open. In less formal way it reads as follows:
Conjecture. Suppose А с R2 is such that 0 < Ж' (A) < oo and
??(?,?)>- for Jf1 almost all jc e A.
Then A is 1-rectifiable.

Geometric measure theory: Selected concepts, results and problems
1025
Preiss and Tiser [35] slightly improved the Besicovitch's upper bound on ?\. Their proof
works for subsets of general metric spaces. Namely, for any metric space X,
?, (?) ? B + л/46)/12 % 0.732.
On the other hand, metric spaces with ?\ (X) > \ exist.
The recent results on the problem are due to Farag [16]. He proved that ^ is the correct
upper bound on ?} (A, ·) for purely 1-unrectifiable 1-set A in a Hubert space, under the
additional assumption, that the set A satisfies a measure-theoretic flatness condition.
For higher dimensional densities (m ^ 2) we only know that
sm := sup(a,„(X): ? is a real Hubert space} < 1 for every m.
4. Sets of finite perimeter
Research on the problem of finding the most general and geometrically natural form of the
Gauss-Green divergence theorem has contributed greatly to the development of geometric
measure theory. For a certain time, the theory of sets for which Gauss-Green theorem holds
was developed independently using various approaches. Later it became clear that they are
all equivalent.
The notion of set of finite perimeter was introduced by R. Caccioppoli and developed by
De Giorgi [9,10]. Their approach was based on approximation (with respect to the metric
of convergence in measure) of a given set by a sequence of sets with smooth (or, which
is equivalent, polyhedral) boundary and with equibounded perimeters. Such limiting sets
possess many good properties of sets with smooth boundary. In particular, a version of
Gauss-Green theorem holds for them.
On the other hand, Federer [17] introduced the notion of measure-theoretic exterior
normal to a set and established the Gauss-Green theorem for every open set G С К" with
JF"~\dG) < oo. Some additional observations of Federer [18], based on results of De
Giorgi, and a contribution of Volpert [41], lead to an optimal version of Gauss-Green
theorem valid for sets of (locally) finite perimeter.
A good account on the sets of finite perimeter can be found in Federer [19], Giusti [22],
Ziemer [44], and Evans and Gariepy [13].
Later it became clear that the notion of set of finite perimeter is the simplest particular
case of normal и-dimensional currents in R", which are discussed in the next section.
Nevertheless, here we give an independent exposition for sets of finite perimeter, since in
this case no highly developed apparatus, as in the general theory of currents, is required.
We say that a set А с К" has finite perimeter in an open set ?/cl" if А П i/
is measurable and if there exist a finite Borel measure ? on U and a Borel function
И : U -> S"~' U {0} С К" such that the following generalized Gauss-Green formula holds
f div<pdx= ? ??????
A JU

1026
?. Chlebik
for every smooth (or, equivalently, Lipschitzian) 1 -vectorfield <p:U —» R" with compact
support in U.
Hence, the vector measure (-???) is the derivative of ?? (denoted by ???) in the
sense of distributions on U, while ? = \???\ is its total variation. The perimeter Pu(A)
of A in U is defined by |?? A\(U), and we use the notation P(A) in the case i/ = R".
Otherwise, if A is not of finite perimeter in i/, we write Pu(A) = oo. We further say
that A is of locally finite perimeter in U if Pv(A) < oo for every open set V <ё U.
By Riesz theorem, whenever ? ? U is measurable
P,;(A) = sup| J ??????: 0 e C('(t/, R"), \\?\\ ? 1
Essential boundary. An inequality Ри (A) iC Jf""' (9 А П i/) always holds.
A much deeper (and ultimate) result is due toFederer [19,4.5.6,4.5.11]:
Pu(A) = Жп~' (deA DU) = J"~\beA П U) for all А С Е".
Here 9fA stands for the essential boundary (called also the measure-theoretic boundary)
of A, defined by
deA = {xeR": ?*"(?, x) > 0 and <9*"(R" \ A,x) > 0}.
Measure-theoretic exterior normal. Assuming А с ?" and * e R" we say that и e S"-1
is a measure-theoretic exterior normal of A at л: if
9"({ye A: (y-x)ou >0},jc)=0, and
в"({уе (R"\A): (y-x)ou <0},jc)=0.
There exists at most one exterior normal of A at x. We denote
n(A, x) = и if A has the exterior normal и at x,
n(A, x) = 0 if A has no exterior normal at x.
Reduced boundary. Given А С R", the set
Э*А = {хеШ": n(A,x) e S"_l}
of points at which the measure-theoretic exterior normal of A exists, is called the reduced
boundary of A.
According to results of De Giorgi, Federer and Volpert, if А с R" is a set of locally
finite perimeter in an open set U с R", then the sets
Z)XA(Z?(*,r)) ^gl,_,l
jc e t/: lim -. e э ]
rW\DXA\(B(x,r)) J
3,AU[/, d*ADU and

Geometric measure theory: Selected concepts, results and problems 1027
are up to Jf" null-sets the same, (n - l)-rectifiable, |?>xA|Li/ = Jf" L(9,,An i/),
and
lim— „ =-n(A,x) forJf" almost all ? e de? П ?/.
In particular, we obtain the following:
GAUSS-GREEN THEOREM. If Ac K" is a set of locally finite perimeter in an open set
?/cK", then
/ ?\\???= / 0(jc)on(A,;c)iiJf","l(jc)
7a Ju
for every Lipschitzian 1 -vectorfield <p:U->M." with compact support in U.
Lower semicontinuity, approximation and compactness. Unlike various (n — 1
^dimensional measures of the boundary, the functional А н> P(A) is lower semicontinuous
with respect to local convergence in measure (i.e., L'oc convergence of the characteristic
functions).
For any set A with P(A) < oo there exists a sequence of sets An with smooth
(or polyhedral) boundary locally converging in measure to A and such that ? (A) =
lim,,^oo P(An).
Any sequence of sets with equibounded perimeters admits subsequences locally
converging in measure.
Isoperimetric inequalities. If А С ?" (? > 2), then
Р(А)^па(пУ/"[тт{^"\А),^"'(Шп \А)}]'^\
The constant na(n)^n is the optimal isoperimetric constant. A local counterpart is the
relativite isoperimetric inequality in a ball
Puu.r)(A)>fi(n)[mm{J?"(U(x,r)nA),2"l(U(x,r)\A)}]'^i,
where ?(?) is a positive constant.
5. Measure-theoretic calculus of variations
The methods and results of geometric measure theory have been successfully applied in
the modern calculus of variations. The Plateau problem of finding the minimal surface,
i.e., of finding the surface of least area among those bounded by a given curve, was one
of the most fundamental variational problems. In the thirties a solution was given to the
problem of Plateau in R3 by results of Douglas [12] and Rado [36]. The possibility to

1028
?. Chlebik
generalize their methods even for minimal hypersurfaces in higher dimension was found
very limited. The difficulty to prove, by parametric methods, existence of solution for area
minimization problem in spaces of dimension higher than 2 was the main motivation for
the development of intrinsic theories for this kind of variational problems. After the work
of De Giorgi [9,10] on sets with finite perimeter, Federer and Fleming [21] introduced a
series of new techniques in the calculus of variations, which leads directly to the existence
of (weak, but geometrically well acceptable) solutions for a wide class of variational
problems.
5.1. Normal and integral currents
The approach of Federer and Fleming modeled w -dimensional geometric objects in R" as
currents, i.e., continuous linear functional on differential w-forms. The general concept
of current was first introduced by G. De Rham for use in the theory of harmonic forms.
His work was closely related to the development of distribution theory by L. Schwartz.
Independently, L.C. Young introduced a notion of generalized surface, defined as a
continuous linear functional on the space of continuous parametric integrands.
Perhaps the central result of the theory of rectifiable currents developed by Federer and
Fleming is one known as the Closure Theorem. It says that certain useful classes of integral
currents (currents which, together with their boundaries, correspond to integration over
rectifiable sets against integer valued density functions), form a weakly closed subspace of
the much larger space of normal currents. This closure theorem allows one to solve many
minimization variational problems in a weak sense, while still obtaining solutions which
correspond to rectifiable geometric objects. This is accomplished by passing to the (weak)
limit of (properly chosen subsequence of) minimizing sequence of integral currents.
Almgren [ 1 ] was able to extend many of results of Federer and Fleming to the class of
size bounded rectifiable currents.
We recall here some basic facts from the theory of normal currents and we refer for more
information to Federer and Fleming [21], Federer [19], Simon [39], Hardt and Simon [23]
and Morgan [32].
A logical prerequisite is the knowledge of differential forms and operations with
m-vectors and w-covectors of R". We refer to the book by Federer [19] for a very
comprehensive treatment.
We denote by $m (U) (m being a non-negative integer) the vector space of all infinitely
differentiable w-forms with compact support in an open set U of R". Each of the subspaces
9™{U) = 9'"{U)n{4>: spt0c/n. К с U compact,
is relatively topologized by the seminorms
?'? (?) = sup{ || DJ<p{x) 1: 0 sC j'^ i and ? e К}
corresponding to all non-negative integers i. (Here || · || stands for the comass norm on
covectors.) Their union $'" (U) is endowed with the largest topology making the inclusion

Geometric measure theory: Selected concepts, results and problems
1029
maps from all the spaces 3>'^{U) continuous. Members of the dual space %,(U) =
3>m{U)*, i.e., continuous linear functionals on &>'"(U), are called m-dimensional currents
in U.
If ? e Q)m (U) and V с U is an open set, the mass of ? in V is defined by
Mv(T) = sup{TD>): фе®'"(У), spt<? с V and \\?(?)\\ ? 1 in V].
We write briefly ?(?) instead of Ми(Т).
A current Г e $>m{U) with finite mass, М(Г) < oo, extends naturally as a continuous
linear functional on the space of all compactly supported continuous m-forms endowed
with the sup norm. Consequently, from the Riesz representation theorem we deduce the
existence of a Radon measure \\T\\ on U, and of a || ГЦ-measurable function ? :U ->
?„,(?") satisfying ||f(;c)|| = 1 for \\T\\ almost all xeU, such that
Т(ф) = j(f(x),4>(x))d\\T\\(x) forany0e^"'(i/),
where (·, ¦> denotes the dual pairing for /\m(R") and /\'"(W), the spaces of w-vectors
and w-covectors of Ш", on which we use dual norms, the mass and the comass.
More generally, the representation of Г е %„(U) in this form is possible if we
only assume ??(?) < oo for all open sets Vei/. Any such current ? is said to be
representable by integration.
The last formula and the Lebesgue theorem allow us to extend ? to all m -forms
with ЦГЦ-summable coefficients (in particular, when the mass of ? is finite, to m-forms
whose coefficients are bounded Borel functions). For such ? we can define for any ЦГЦ-
measurable subset A of U (and, in particular, for any Borel set А С U) the restriction of ?
to A as a new current 7?_ A e 3>m{U) by
?\-?(?)= f[f,4>)d\\T\\, 4>e&"(U).
More generally, if /: U -> ? is any locally ЦГЦ-integrable function (in particular, any
locally bounded Borel function) we define
T\-fD>)= f[f,<l>)fd\\T\\, 4>e@m(U).
For any ? e $>m{U) the support of T, spt ?, is the smallest relatively closed subset С of
U such that ? (?) = 0 for all ? e S>m (U) with spt<? с U \ С The boundary operator
3:&m+i(U)^&m(U)
is defined to be dual to the operator of exterior derivative
d:9>m{U)->$>m+\u)

1030
?. Chlebik
by means of Stokes theorem. That means that for given ? e @>m+\ (U), dT is defined by
дТ(ф) = Т(аф), фе$'"(и).
A sequence of currents {7^} С S>„,{U) is said to converge weakly in U to ? if it
converges in the sense of distributions, i.e.,
Тк(ф) -> ? (?), for every ? e $m(U).
This weak convergence will be denoted by Tk —>¦ ? in U.
It is easy to prove that the mass is lower semicontinuous with respect to the weak
convergence. By the standard Banach-Alaoglu theorem we deduce that from a sequence
of currents {7jt} с $>m(U), with equibounded masses My(Tk) on each open set V <s U,
we can extract a subsequence convergent weakly to a current ? e @,„(U) representable by
integration.
Normal currents. We call a current ? e S>„,{U) locally normal if and only if ? is
representable by integration and either m = 0 or 9? is representable by integration. ?
is normal if it is locally normal and has compact support.
The set N,„(i/) = {T e ®m{U): ? is normal} is the union of the sets ?,„(??) = {? e
$>,„ (U): ? is normal and spt Г с К} corresponding to all compact sets К CU.
For Г е @m(U) we define
?(?) = ?(?) + ?(9?) incasem>0,
?(?) = ?(?) incasem=0.
For example, the и-dimensional locally normal currents in R" are the currents E" L/
corresponding to all functions / of locally finite variation in R".
Here E" e S>„(R") is the current defined by
Е"(ф)= 1[е\/\ег/\---Аеп,ф{х))а^"\х), 0e^"(R"),
e\,e2,...,e„ being the standard orthonormal basis in R" and e\ л ет А · · · л е„ the n-
vectorfield orienting R".
The current E" L A corresponding to .if"-measurable set А с R" is locally normal if
and only if A is a set of locally finite perimeter.
Real flat norm. Whenever Г is an w-dimensional current in U, and К с U is compact,
the FK-seminorm of ? (possibly +oo) is given by
FK(T) = inf{M(T -dS) + M(S): 5e^m+i(t/)with sptScK}.
The slightly weaker norm F, called the real flat norm, is defined by
F (?) = inf[FK (T): К с U is compact}.

Geometric measure theory: Selected concepts, results and problems
1031
The corresponding metrics using distances Fk(T\ - ??) (respectively F(T| - ??))
between T\ and T2 are defined in an obvious way.
Rectifiable currents. For a generic current ? representable by integration there is no
way of defining "tangent space". Even if spt ? is a smooth w -manifold, the w-vector ?
associated to ? has not to be related in any way to tangent space to spt T.
Federer and Fleming [21] introduced the subclass of rectifiable currents. This class has
good closure properties, and its elements enjoy, in an approximate sense, the differential
properties of smooth manifolds.
We recall that a subset S с К" is said to be m -rectifiable if, except for а Ж'"-пи\\ set
SO, it can be covered by the countable union of C1 smooth w-dimensional manifolds Sj,
7 = 1,2,....
For Ж'" almost all ? e S the approximate tangent space Tan ,5 ofS at ? is defined as the
tangent space to Sj at x. (One can show that, up to an Jf""-null subset of S, the definition
of Tan, S1 does not depend on the choice of smooth Sj's in the covering S С U^=o $j ·)
A current Г е 3>m(U) is said to be an integer multiplicity locally rectifiable current,
briefly a locally rectifiable current, if it can be expressed as
? (?) = /^(х),ф(х))в(х)аЖ'"(х), ф е ®"\U),
where the set S is an w-rectifiable Ж'"-measurable subset of U, the multiplicity function
?-.S -> Z+ is an Ж'"^^ locally summable positive integer valued function (i.e.,
fvnS \9\аЖ'" < oo for all open sets V <s U), and ? is an orientation on S. This means
that ? is an Ж'" -measurable w-vectorfield on 5" which for Ж'" almost every ? e 5" is
associated with Tanks' (i.e., ?(?) can be expressed in the form ?? л · · · л ?,„, where ??,...,
?,„ form an orthonormal basis for Tan, S). In this case we write ? in the form t(S, ?, ?).
If, moreover, spt ? is a compact subset of U, a current ? = t(S, ?, ?) is called rectifiable.
For example, if S is an m-dimensional C'-submanifold of U with Ж'"(У П S) < 00
for all open sets V <s U, and if ? is a tangent w-vectorfield on S orienting S, so that ? is
continuous and ||?|| = 1, then the classical operator integrating differential w-forms over
the oriented manifold S, is a locally rectifiable current.
Let^,„(i/) be the space of all w-dimensional rectifiable currents in U. (See Federer [19,
4.1.24 and 4.1.28] for a detailed definition and several alternate descriptions of rectifiable
currents.)
For any compact К с U we further set
&m(K) = {Te&m(U): sptTcK},
and when w > 0,
1т(К) = {ТеЯ,п(К): 9Ге^,„_|(К)},
l„(U) = {Te&m(U): ЭГе^-???/)}.
The elements of I,„ (U) are w-dimensional integral currents in U.

1032
?. Chlebik
The union of the groups
,?m(K) = \R + dS: R e &m(K) and S e^m+i(K)}
corresponding to all compact sets К с U is the group J?m(U) of m-dimensional integral
flat chains in U.
We largely use the notations and conventions established by Federer [19] in his book,
and listed there on pages 669-671. But it should be noted that spaces t%m(K), lm(K), and
^m(K) do not generally coincide with the corresponding spaces &т.к, 1т.к an(J &т.к
defined there except when К is a compact Lipschitz neighbourhood retract. Our exposition
is, in this respect, more closely related to the approach of Solomon [40].
Recall that К с К" is a Lipschitz neighbourhood retract in R" if there exists a
Lipschitzian map which retracts a neighborhood of К in Ш" onto К.
Integral flat metric. There is an integral analogue &ц of the real flat seminorm Fa:
(K cU being a fixed compact set), defined by
^(r) = inf{M(*) + M(S): T = R + dS, Re&n(K), Se&m+t(K)}.
The corresponding metric using distances ^к{Т\ - ??) between T\ and ?? induces so
called integral flat metric topology.
Deformation theory. One main object of study of general rectifiable or special (for
example, area minimizing) m-currents is to obtain strong and useful representations or
approximations of those currents by simple deformations of polyhedral chain of the m-
skeleton of the standard cubical subdivision of Ш".
The deformation theory which first appears in Federer and Fleming [21, 5.5] (see also
Federer [19, 4.2.1-4.2.9]) became a basic technique in the theory of normal currents.
Using multiple-valued function theory Almgren [1] was able to develop and apply the
deformation theory to the size bounded rectifiable currents.
A direct consequence of the deformation theorem of Federer and Fleming is the
following isoperimetric inequality.
ISOPERIMETRIC THEOREM. Suppose 0 < m < n, К С Ш" is convex and compact, and
S e I,„(K) with dS = 0. Then there is a T e I,„+i (K) with dT = S and
M(T) iC c(n, m)M(S)'J^1.
It has been improved by Almgren [2], who proved that the optimal value of the
isoperimetric constant does not depend on и.
Decomposition of integral currents. A current ? e I,n(IR") is called indecomposable
if and only if there exists no Se I»,(K") except 0 (the null w-current) and ? with
?(?) = ?E)+?(?-5).

Geometric measure theorx: Selected concepts, results and problems
1033
It may be shown that an integral current ? e I,„ (R") can always be written as a countable
sum of indecomposable currents Tt e I,„(E") such that
ОС ЗС
T = J2Ti and ?(?) = ? ?(?,).
i=\ i=l
The structure of 1-dimensional integral currents is relatively simple: every ? e 11 (?")
is a sum of finitely many oriented simple arcs and countably many oriented simple closed
curves, with finite total length.
However, in case 1 < m < ti, even indecomposable currents from I,„(R") can be
topologically very complicated; spt ? need not be w-rectifiable.
An important conclusion of the deformation theorem is the following:
Total boundedness lemma. Suppose 0 ^ с < oo and К с К" is a compact Lipschitz
neighbourhood retract. Then
A) N,„(/0 П {?: ЩТ) iC с} is ?K-compact,
B) \m{K) П {?: ЩТ) iC с} is &K-totally bounded.
Since N is lower semicontinuous, the Fa:-completeness of the set in A) is easy, so the
only issue in either conclusion is total boundedness.
The deformation theorem implies that given any ? > 0, a current as in A) (respectively
B)) is within ? of one of finitely many polyhedral chains in the appropriate flat metric.
5.2. Closure and compactness theorem
The following theorem and its corollaries convey the most essential information about
closure and compactness properties of integral and normal currents, and about the structure
of integral flat chains.
CLOSURE THEOREM. If К С К" is compact and m ^ 0 is an integer, then lm(K) is F-
closed in Nm(K).
Corollaries. (l)&m+](K)?Щ,+](К) = Im+i(К).
B) If ? e &m(K) andM(T) < oo, then ? e 3?,„{K).
Combining the closure theorem with conclusion 2 of the total boundedness lemma we
obtain the following:
Compactness theorem for integral currents. If К с Ш" is a compact Lipschitz
neighbourhood retract, and с < oo, then {T e lm(K). ЩТ) ^ с} is ^K-compact
in&m(K).
The closure and compactness theorems were proved by Federer and Fleming [21 ]. Their
proof relies on the measure-theoretic structure theory developed by Federer and discussed

1034
?. Chlebik
in Section 2. As its proof is quite difficult it has long been an obstacle to those seeking
an understanding of the closure theorem. Later other proofs have been given, independent
of the structure theory, by Solomon [40], Almgren [1], White [42,43], and Ambrosio and
Kirchheim [3].
The compactness theorem for integral currents leads directly to the existence of solutions
for a wide class of variational problems. In particular it allowed to establish the existence
theorem for the (measure-theoretic) Plateau problem
min{M(r): Telln(K), dT = S},
whenever К с К" is convex and compact and S e I,„_ ? (?) with dS = 0.
5.3. Regularity of minima! surfaces
Using the theory of integral currents we have obtained the existence of a solution ? e
I,n(IR") of the m-dimensional problem of Plateau. As it is a kind of weak solution, the
present research directs itself to the problem of regularity of m -dimensional "surface"
sptT \spt dT.
The first significant partial results on the problem of smoothness of area minimizing
currents have been obtained by De Giorgi, Reifenberg and Almgren:
(i) the regular points form a dense relatively open subset of sptT \spt9T,
(ii) in case m = ? — 1 the set of all singular points is Жт null.
We will discuss in some details hypersurfaces (i.e., w-surfaces in R",+l) only. In this
case we can use the formalism of De Giorgi [11]. A good account on this approach is the
bookofGiusti[22].
In this formalism, an w-surface in R'n+I is a boundary of a set ? of locally finite
perimeter. The main idea of De Giorgi is the following.
For ? e dE define an approximate normal vector
vr(x) = DXE(B(x,r))/\DXE\(B(x,r)).
It is possible to prove that if for some ro > 0 the vector vro(x) has length close enough
to 1, then H1TV40 IK(Jt)|| = 1· This is a crucial part of the proof from which it is easy to
show that dE is regular (even analytic) in a neighbourhood of x.
Once the generic regularity has been established, one can ask whether the singular set
can exist at all. It is natural to study behaviour of dE near a point ? e dE blowing up the
situation by sequences of expansive homotheties Txs.{ri I 0) around ? (as in Section 3,
Tx,r(y) = (y~ x)/r). Clearly, all the homothetic images of dE are minimal surfaces and
subsequence will converge in a measure to a set C, which is itself minimal. Moreover, С
is a cone (with vertex at 0), roughly speaking a tangent cone to 9 ? at ?.
One can conclude that dE is regular near ? if and only if С is a hyperplane. So the
existence of singularities of minimal hypersurfaces is reduced to the existence of singular
minimal cones. Almgren proved the non-existence of singular minimal cones in ? , and

Geometric measure theory: Selected concepts, results and problems
1035
Simons [38] extended this result up to R7 concluding that m-dimensional minimal surfaces
in R'n+1 are regular, for m ^ 6. This result is the best possible since the cone
C = {*eR8: ^2+JC2+JC2+JC2=JC2+JC2+JC2+JC2j
is a singular minimal cone. This was shown by Bombieri, De Giorgi and Giusti [4].
References
[1] F.J. Almgren Jr., Deformations and multiple valued functions. Geometric Measure Theory and Calculus of
Variations, Proc. Sympos. Pure Math., Vol. 44, Amer. Math. Soc., Providence, RI A986), 29-130.
[2] F.J. Almgren Jr., Optimal isope rime trie inequalities, Indiana Univ. Math. J. 35 A986), 451-547.
[3] L. Ambrosio and B. Kirchheim, Currents in metric spaces. Acta Mathematica 185 B000), 1-80.
[4] E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem, Inv. Math. 7 A969),
243-268.
[5] A.S. Besicovitch, On the fundamental geometric pmperties of linearly measurable plane sets of points (I),
Math. Ann. 98 A928), 422-484.
[6] A.S. Besicovitch, On the fundamental geometric properties of linearly measurable plane sets of points (II),
Math. Ann. 115 A938), 296-329.
[7] A.S. Besicovitch, On the fundamental geometric properties of linearly measurable plane sets of points (HI),
Math. Ann. 116 A939), 349-354.
[8] J.E. Brothers, Some open problems in geometric measure theory and its applications, Proc. Sympos. Pure
Math., Vol. 44, Amer. Math. Soc., Providence, RI A986), 441-464.
[91 E. De Giorgi, Su una teoria generate della misura (r — 1) -dimensionale in uno spazio ad r dimensioni, Ann.
Mat. Рига Appl. 36 D) A954), 191-213.
[10] E. De Giorgi, Nuovi teoremi relativi alle misure (r — \)-dimensionali in uno spazio ad r dimensioni,
Ricerche Mat. 4A955), 95-113.
[11] E. De Giorgi, Fmntiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa, Editrice Tecnico
Scientifica, Pisa A961).
[12] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 A931), 263-321.
[13] L.C. Evans and R.F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, Stud.
Adv. Math., CRC Press A992).
[14] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge A985).
[15] K.J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley A990).
[16] H.M. Farag, Unrectifiable I-sets with moderate essential flatness satisfy Besicovitch's 4 -conjecture. Adv.
Math. 149B000), 179-186.
[17] H. Federer, The Gauss-Green theorem. Trans. Amer. Math. Soc. 58 A945), 44-76.
[18] H. Federer, A note on the Gauss-Green theorem, Proc. Amer. Math. Soc. 9 A958), 447-451.
[19] H. Federer, Geometric Measure Theory, Springer, New York A969).
[20] H. Federer, Colloquium lectures on geometric measure theory, Bull. Amer. Math. Soc. 84 A978), 291-338.
[21] H. Federer and W.H. Fleming, Normal and integral currents, Ann. of Math. 72 A960), 458-520.
[22] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser A984).
[23] R. Hardt and L. Simon, Seminar on Geometric Measure Theory, Birkhauser, Boston A986).
[24] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc.
Amer. Math. Soc. 121 A994), 113-123.
[25] O. Kowalski and D. Preiss, Besicovitch-type pmperties of measures and submanifolds, J. Reine Angew.
Math. 379A987), 115-151.
[26] A. Lorent, Rectifiability of measures with locally uniform cube densities, Proc. London Math. Soc. B002),
in print.
[27] J.M. Marstrand, Hausdorff two dimensional measure in 3-space, Proc. London Math. Soc. 11 C) A961),
91-108.

1036
?. Chlebik
[28] J.?. Marstrand, The (<p,s) regular subsets of ? space. Trans. Amer. Math. Soc. 113 A964), 369-392.
[29] P. Mattila, ?ausdorff m-regular and rectifiable sets in ?-space, Trans. Amer. Math. Soc. 205 A975), 263-
274.
[30] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math. A995).
[31] E.F. Moore, Density ratios and(?, I) rectifiability in ?-space. Trans. Amer. Math. Soc. 69( 1950), 324-334.
[32] F. Morgan, Geometric Measure Theory -A Beginner's Guide, Academic Press A988).
[33] A.P Morse and J.F. Randolph, The ? rectifiable subsets of the plane. Trans. Amer. Math. Soc. 55 A944),
236-305.
[34] D. Preiss, Geometry of measures in K": Distribution, rectifiability, and densities, Ann. of Math. 125 A987),
537-643.
[35] D. Preiss and J. Tiser, On Besicovitch \ -problem, J. London Math. Soc. 45 A992), 279-287.
[36] T. Rado, On the Problem of Plateau, Springer, Berlin A933).
[37] С A. Rogers, Hausdorff Measures, Cambridge University Press A970).
[38] J. Simon, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 A968), 62-105.
[39] L. Simon, Lectures on Geometric Measure Theory, Proc. Centre for Math. Anal., Vol. 3, Australian Nat.
Univ., Canberra A983).
[40] B. Solomon, Л new proof of the closure theorem for integral currents, Indiana Univ. Math. J. 33 A984),
393-418.
[41] A.J. Volpert, Prostranstva BV i kvazilinejnyje uravnenija. Mat. Sb. 73 A967), 255-302.
[42] B. White, A new proof of the compactness theoremfor integral currents. Comment. Math. Helv. 64 A989),
207-220.
[43] B. White, Rectifiability of flat chains, Ann. of Math. 150 A999), 165-184.
[44] WP. Ziemer, Weakly Differentiable Functions, Springer, Berlin A989).

CHAPTER 25
Fractal Measures
Kenneth J. Falconer
Mathematical Institute. University of St Andrews, Fife KYI6 9SS, Scotland. UK
E-mail: kjf@st-andrews.ac.uk
Contents
1. Introduction 1039
2. Hausdorff and packing measures and dimensions 1039
2.1. Hausdorff measures 1039
2.2. Packing measures 1041
2.3. Calculation of Hausdorff and packing measures and dimensions 1042
2.4. Geometric measure theory 1044
2.5. Geometry of Hausdorff measures and dimensions 1046
2.6. Subsets of finite measure 1047
3. Measures with a fractal structure 1047
3.1. Local dimensions 1047
3.2. Dimension decomposition 1048
3.3. Multifractal measures 1049
3.4. Geometry of multifractal measures 1051
References 1051
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1037

This page intentionally left blank

Fractal measures
1039
1. Introduction
A fractal is a highly irregular set with a fine structure, that is with irregular detail at
arbitrary small scales. Often a fractal has some sort of self-similarity or self-affinity,
perhaps in a statistical or approximate sense. Many fractals of interest have a simple
recursive definition, and perhaps a 'natural' appearance.
Fractals are too irregular to be analysed using the traditional calculus or geometry of
smooth objects, and so were largely ignored by mathematicians until well into the 20th
century. Hausdorff measures provided the first major tool for studying such sets, but for
many years attracted only limited attention (largely by Besicovitch and his co-workers),
with individual fractals studied as pathological curiosities. The area received enormous
impetus in the late 1970s with the publication of Mandelbrot's essays [55,56] which
presented fractals as the rule, rather than the exception, in both mathematics and nature. At
the same time, computer technology had reached a level that made it possible to produce
good pictures of many and varied fractals. These developments led to a renaissance of
interest in Hausdorff measures and dimensions and in their applications to fractal analysis.
The first part of this chapter gives brief account of Hausdorff measures and their variants
and extensions as the natural tools for studying the geometry of fractals. The definition of
Hausdorff measure immediately leads to the notion of Hausdorff dimension, and many
properties are then conveniently expressed in terms of dimensions rather than measures. In
a sense, Hausdorff measures are measures that are distributed as uniformly as possible
across fractal sets. However, it is possible to construct measures that themselves have
highly irregular distributions across their support; indeed such measures arise naturally
in areas such as dynamical systems, number theory and turbulence. Measures with such a
fine structure are often thought of as 'fractal measures' or perhaps 'multifractal measures',
a term that reflects a single measure giving rise to a spectrum of fractal sets determined by
local densities. The second part of the chapter considers these fractal measures and their
analysis, which generally involves measures and dimensions of Hausdorff type.
There is now an enormous literature on fractal measures, and there is only space here
for a few selected references. Texts which give detailed treatments of fractal measures at
various levels of sophistication and which give many further references include [16,18,21,
32,62,83].
2. Hausdorff and packing measures and dimensions
This section gives a rapid treatment of Hausdorff and related measures. More details may
be found in the texts mentioned above.
2.1. Hausdorff measures
In his famous paper of 1919, Hausdorff [37] showed that a general construction of
Caratheodory could be used to construct 's-dimensional measures', where s is not
necessarily an integer. In particular, he showed that the middle-third Cantor set has positive

1040
K.J. Falconer
and finite log2/log3-dimensional measure. Ever since, such Hausdorff measures have
been a principal tool for studying fractal sets.
Let (X,d) be a metric space (often и-dimensional Euclidean space R" with the usual
metric). The diameter of a non-empty set U is
diami/ = sup(d(;t, у): х, yet/}.
For ? > 0 we term a finite or countable family of sets {i/,},^, a S-cover of F с X if
diam i/, ^ 5 for all / and F с (J^, i/,. For s ^ 0 we write
? * 1
?^(F) = inf{^(diaint/,)s: {t/,} is a<S-coverof F }. A)
Clearly, as S decreases, the allowable <5-covers for the infimum is reduced, so HSS(F) is
non-decreasing as S I 0. Thus the limit as S I 0 exists, and we define the s-dimensional
Hausdorff (outer) measure of F as
H\F) =lim HUF). B)
It is easily verified that 7is is a metric outer measure, that is
H*(EUF) = HS(E) + HS(F)
if ? and F are separated by a positive distance, and consequently that the Borel and Souslin
sets are W -measurable for s > 0. Moreover, W is Borel regular, that is for all F с X there
exists a Borel set F э F with Ws(?) = HS(F), see [18,62].
Hausdorff measures on Ш" extend the classical Lebesgue measures of length, area,
volume, etc. For example, if F is a subset of R or is a curve in R" for ? ^ 2, we have
W1 (F) = length (F)
and if F is a subset of R2 or a smooth surface in R" for ? > 3, we have
H2(F)= -area(F),
etc.
Hausdorff measures have many nice properties. For example, if F с R" and /: F -> R"
is a Lipschitz mapping, that is if
|/(*)-/00|<Ф-;у| for*,>-eF, C)
where о 0, then
H>(f(F))^csHs(F).
D)

Fractal measures
1041
In particular, this leads to the congruence property, that if /: R" -> R" is a congruence
or isometry then W{f{F)) = HS(F) and the scaling property; that if /: R" -> R" is a
similarity transformation with scaling ratio c, then
Hs(f{F))=csHs(F). E)
This generalizes the familiar scaling properties of length, area and volume, and is indicative
of the strong geometrical nature of Hausdorff measures.
A simple exercise using A) shows that if 0 ^ s < t then
H's(F)<:ni(F)S"\
On letting ? i 0 it follows that if HS{F) < oo, then H'(F) = 0 for all t > s. This fact
allows us to define the Hausdorff dimension of F с X as
dimw F = inf{s: Hs{F) < oo} = supji: Hs(F)>0}. F)
A number of important properties of Hausdorff dimension are immediate from the measure
properties of W. For example,
DC
dimw yj F,- = sup dim^ F,-. 0)
/=| 1ф'<зс
If F с R" and /: F -> R" is Lipschitz it follows from D) that
dimw /(F) <; dimw F (8)
with equality if / is a bi-Lipschitz mapping (with /"' : /(F) -> F also Lipschitz) such
as a (non-singular) affine transformation. A consequence of this is that if F is a smooth
m-dimensional submanifold of R" then dim/y F = m.
Sometimes more delicate Hausdorff measures are useful. A gauge function h : [0, oof ->
[0, oof is a strictly increasing function with /г@) = О and continuous on the right. The
Hausdorff measure construction goes through in just the same way on replacing (diam t/,-)s
by h (diam i/,-) in A) to give a generalized Hausdorff measure Hh. This can be useful for
analyzing sets F for which there is no real number s with 0 < HS(F) < oo. For example,
a Brownian path F in R", ? > 3, has (with probability 1) Hausdorff dimension 2, with
H2(F)=0, bm0<Hh(F) <ooif/2@ = f2loglogf-'.
2.2. Packing measures
Packing measures are in many ways dual to Hausdorff measures (which might be
termed 'covering measures'), and, rather surprisingly, were not introduced until the early
1980s [90,91,94].

1042 K.J. Falconer
Let (X, d) be a metric space. Given FCX.a (finite or countable) collection of disjoint
balls {?,·} is called a S-packing of F if each ball has centre in F and radius at most S. For
OOwe write
^(radius Bif: {B,} is a ^packing of F \. (9)
i=\ I
The limit
^(F) = lim^(F) A0)
exists, but Vq(F) need not be countably subadditive. To overcome this we set
EW):fiUF' A1)
i=\ i=l J
which is a Borel measure, and which is termed s-dimensional packing (outer) measure
onX.
An alternative approach is to work with the diameter based s-dimensional packing
measure. This is defined in a similar way, except that 'radius B,¦' is replaced by 'diamfi;'
in (9). The radius and diameter based packing measures are equivalent on E", but may be
very different for certain nonlinear metric spaces.
Packing dimension is then defined in the obvious way:
dimP F = M{s: Vs(F)<oo} =sup{s: Vs(F)>0}.
Basic properties of packing dimensions are easily derived from measure properties, for
example the analogs of G) and (8) hold for dim/>.
Not surprisingly, packing and Hausdorff measures are related by an inequality. Some
basic estimates together with the Vitali covering theorem establish that for all subsets F of
a metric space X and s ^ 0
HS(F)^VS(F)
and consequently
dim/y F ^ d'imp(F).
See, for example, [ 16,21,62] for further details of packing measures and dimensions.
2.3. Calculation of Hausdorff and packing measures and dimensions
Calculation of Hausdorff and packing measures and dimensions of a set F is often non-
trivial; the exact value of the Hausdorff measure of many simple fractals is still unknown.

Fractal measures
1043
Often, a direct covering estimate can give good upper bounds for Hausdorff measures.
Lower bounds may require indirect methods, and a frequent approach is to define a natural
measure supported by F and to examine its properties. In particular, if F с R" supports a
measure ? such that 0 < m(F) < oo, and 0 < с < oo:
if limsupM(B(;c, r))/rs <: с for all ? e F then W?(F) ^ m(F)/c, A2)
r—0
if ?\?\$???(?(?^))/?* ^c foralljceF then WS(F) ^ 2V(F)/c, A3)
r—>oo
if ???\??(?(?, r))lrs ^ с foralljceF then V\F) > 2'V(F)/c, A4)
if liminfM(B(jc,r))/rs>c foralljceF then -p?(F) ^ 2>(F)/c A5)
(see [13,26]). These properties illustrate the duality between Hausdorff and packing
measures.
To illustrate the basic approach, let F be the middle-third Cantor set, obtained from
the unit interval by repeatedly removing the middle-third of intervals, so at the kth
step of the construction we are left with a (closed) set Fk consisting of 2k intervals of
lengths 3~k, with F = Пь=о^· Tnen takmg tne intervals of Fk as а 3~к-со\ст of F
gives W3_k(F) ? 2k3~sk, so fti°g2/iog3(F) <; { and dimH F <; iog2/log3. On the other
hand, if ? is the natural measure supported by F which assigns each interval of Fk a
measure 2~k, a simple estimate gives ?(?(?, r)) <; 2rl0-2/|0=3 so that Ti}^2' |0?3(F) > |
anddimw F ^ log 2/log 3 by A2).
This type of calculation extends to a large class of 'self-similar' sets. For simplicity,
let /|,..., fm : M." -> Ш" be a family of (strict) contractions; such a family is called an
iterated function system (IFS). It may be shown [18,43] that there exists a unique
nonempty compact FCR", known as the attractor or invariant set of the IFS, satisfying
m
f = \Jmf). A6)
i = l
If the contractions f are similarities, that is if \f(x)- /?(>'I = ^?\?- y\ for all xjel",
the set F is termed self-similar. The type of calculation indicated for the middle-third
Cantor set extends to show that dimw F = dim/> F = s with 0 < W{F) < oo, and with
computable lower and upper bounds for HS(F), where s is the solution of
m
?'? = 1- A7)
i=l
provided that the /, satisfy the open set condition. This condition requires that the
components {//(F)}^., are 'reasonably separated' in that there exists a (non-empty) open
set U such that {/Hi/)}?!, are disjoint and (Jit ? /i(^) ? i/. This condition holds for
many familiar fractals, for example the von Koch curve and the Sierpinski triangle.

1044
K.J. Falconer
The theory of generalized capacities developed alongside Hausdorff measures, and there
are strong links between them. In particular, A2) may be used to show that, for F с ?",
dim/y F is the supremum value of s for which there exists a positive measure ? supported
by F with
? \?-?\-???{?)??{?)<??. A8)
This provides a further, now standard, technique for getting lower bounds for dimensions,
and is central in the study of the geometry of measures and dimensions.
Hausdorff and packing measures and dimensions have been calculated or estimated for
a wide range of fractals. Some of the main classes that have been considered are listed
below, with selected references.
• Self-similar sets (the /} in A6) are similarities) [18,43].
• Graph directed self-similar sets [3,66].
• Self-affine sets (the /, in A6) are affine maps) [20,22,68,76].
• Self-conformal sets (the ft are conformal maps) [6,85].
• Self-similar sets with infinitely many maps (/} are similarities with m = oo
in A6)) [58,63,64].
• Self-conformal sets with infinitely many maps (/, are conformal maps with m = oo
in A6)) [64].
• Random self-similar sets (ft are random similarities) [19,34,35,48,54].
• Sub/super self-similar sets ('=' is replaced by 'c' or "э' in A6)) [25].
• Sets defined by continued fractions [38,64].
• Sets defined by Diophantine approximation properties [4,15].
• Graphs and surfaces of self-affine functions [60,65].
• Paths and graphs of Brownian motion and its generalisations [1,46,92].
• Julia sets (invariant sets of analytic mappings) [85].
• Attractors or repellers of dynamical systems [23,49,52,67,78,88,93].
2.4. Geometric measure theory
One of the early uses of Hausdorff measures was the study the geometric structure of
subsets of E", a programme pioneered by Besicovitch through the middle part of the 20th
century. An early realisation [5], see [18,62], was that a compact subset F of E2 with
0 < ?) (F) < oo can be decomposed into a 'regular' or 'curvelike' part and an 'irregular'
or 'fractal' part which can be distinguished either measure-theoretically, in terms of the
existence of local densities, or geometrically, in terms of rectifiability or the existence of
tangents.
Higher dimensional results of this type require sophisticated machinery. A set F с E" is
called m-rectifiable if it is made up of countably many pieces which are Lipschitz images
of E", that is if there exist countable many Lipschitz maps /,: E'" -> E" such that
Wf\Q/;(e'"))=o.

Fractal measures
1045
A regular Borel measure ? on Ш" is m-rectiflable if there is an m-rectifiable set F with
?(?." \ F) = 0 and ? is absolutely continuous with respect to H'".
The fundamental result relating densities and rectifiability was proved in full generality
by Preiss in 1987 [81]. Fuller details are given by Chlebik [10] (Chapter 24 of this
Handbook), but briefly, if ? is a positive finite Borel measure on E" and the density
\??\?^?(?(?(?, r))/rm) exists and is positive and finite for ?-almost all x, then m is an
integer and ? is m-rectifiable. A consequence of this is that a Borel set Fwith H'"(F) < oo
is m-rectifiable (with m an integer) if and only if the density
. H'"(B(x,r)nF)
lim
r^o Br)'"
exists for W"-almost all ? e F, in which case this density equals 1 almost everywhere.
Moreover, there is a number 0 < c{m) < 1 such that if
Hm(B(x,r)(lF)
liminf > c(m)
/•^0 Br)'"
forW'-almostalljt e F then F is m-rectifiable. It has long been conjectured that c(l) = y,
but the best value to date is -^B + V46) ^ 0.732 [82].
Packing measures also provide characterizations of rectifiability. For example, if F is a
Borel subset of R" with 0 < V"\F) < oo, then V'"(F) = 7i"l(F) if and only if m is an
integer and the restriction of V" to F is m-rectifiable [62,86].
The modern approach to such geometric questions uses tangent measures, which are
limits of normalized enlargements of a given measure but which behave in a more regular
and tractable way than the original measure. More precisely, if a e M." and r > 0, let Ta.r
denote the transformation on measures defined by (??,-?)(?) = ?(? ? + a) for every set A.
A non-zero measure visa tangent measure of ? at ? if there are positive sequences r, -> 0
and c, such that c,Ta ;/? -> ? in the vague topology.
Tangent measures have been used to great effect in relating local features of a measure
to properties such as integral dimensionality and rectifiability. For example, the proof of
Preiss' theorem uses tangent measures to reduce density properties to more manageable
questions on the structure of m-uniform measures, that is measures ? with v(B(x, r)) =
const r'n for all ? e spt ? and r > 0.
Tangent measures have also been used to show that integral dimensionality and
rectifiability follow from even weaker density conditions, such as conditions relating to
average densities of the form
1 fTW(FnB(x,e-'))
liminf— I dt,
7-^oc ? J0 Bе-'У
see [69].

1046
K.J. Falconer
2.5. Geometry of Hausdorff measures and dimensions
Good geometrical properties are a key feature of Hausdorff measures and dimensions,
properties which in many way generalize those of classical length and area on curves and
surfaces. For example, conczmingproducts of sets ? с R" and F cRffl, there are numbers
q > 0 (depending on n, m, s, t) such that
c\Hs(E)H\F) ^ HS+'{E ? F) <: c2Hs(E)V'(F)
? ciVs+'(E ? F) <; c47>i(EO>'(F), A9)
so, in particular,
dim/y ? + dim/y F ^ dim/y (? ? F) ^ dim/y F + dim/> F
^ dim/>(F ? F) ^ dirnp F + dim/> F. B0)
More generally, there are integral inequalities of the form
Hs(Gx)dH'(x) iC cHs+\G), B1)
/
for G a Borel subset of R"+"' where G.v is the section {y e R": (jc, y) e G) for jc e R",
see for example [21,40,59,62].
For V a w-dimensional subspace of R", let projy : R" -> V be orthogonal projection.
Since projy is a Lipschitz mapping, for all F с R" and all subspaces V, we have
Hs(pro)v F) ^ W(F), and dim/^projy F) ^ min{w,dimw F). In fact this dimension
inequality is nearly an equality: for a Borel set F с R" and for almost all w-dimensional
subspaces V of R" (in the sense of the natural invariant measure on subspaces),
dim/y (projy F) = min{w, dim^ F).
Proofs of this type of identity nowadays use the potential theoretic approach A8), see [21,
50,62].
When F is a Borel set with dimw F = w and 0 < H'"(F) < oo the projection properties
onto an w-dimensional subset V are delicate. If F contains a rectifiable subset of positive
measure then ?.'"(???)? F) > 0 for almost all V, otherwise ?'"(???)? F) = 0 for almost
all V [18,62,81].
Packing dimensions behave less well under orthogonal projection. For example
d'imp ?
^ ?'??\?(???')? F) ^ min{w, dim/> F]
1 +(l/w- \/n)dimPE
for almost all w-dimensional subspaces V, with these inequalities best possible.
Nevertheless, dimp projy F is constant for almost all V, see [27,28].

Fractal measures
1047
These are also natural results on the dimensions of intersections. If ?, F are Borel
subsets of R" then
dimw(?n(? + Jt))^max{0,dimw(? ? F) - n\
for almost all ? e R", where F + x denotes F translated by ?. This follows using B1) with
G = ? ? ?, and is the best possible upper bound. The inequality in the other direction,
dimw(?na(F)) > dimw(?) + dimw(?) - ?
holds for a set of transformations ? e G of positive measure in a group of
transformations G in the following cases:
(a) G is the group of similarities;
(b) G is the group of congruences and either ? is a rectifiable set or dimw ? > \(n+\).
The proof uses potential theoretic methods [47,61,62]. Intersection properties involving
packing measures and dimensions are complicated [44].
2.6. Subsets of finite measure
For a measure to be of much use for studying a set F, the measure of F needs to be
positive and finite, or at least ?-finite. However, for many F, there is no value of s with
0 < W(F) < oo nor even any gauge function h such that 0 < Hk{F) < oo. One approach
then available is to seek a (topologically reasonable) ? с ? with positive finite measure.
Much technical work has been motivated by this aim. The basic result is that if ? с R" is
a Borel or Souslin set (that is a continuous image of a Borel set) with Hh{F) > 0, then there
exists a compact ? с ? with 0 < Hh (?) < oo. Early proofs of this [83,14] involve setting
up a 'net measure' on ? equivalent to the Hh and using the ultrametric structure to pick out
the subset ?. Recently, a completely different approach has been introduced [39,62] using
weighted Hausdorff measures to enable functional analytic techniques to be employed.
Analogous results are available for packing measures, see [45].
3. Measures with a fractal structure
We now turn to measures that are of such widely varying intensity that they may be thought
of as having a fine structure, that is as fractal measures in their own right. Indeed, a single
measure may give rise to many fractal sets.
3.1. Local dimensions
Given a Borel regular measure ? in R" with 0 < m(R") < oo, it is natural to define the
(lower) local dimension or local Holder exponent of ? at ? by
dimioc ?(?) = liminf
r^O logr
B2)

1048
K.J. Falconer
Thus, dimioc ?(?) = a if a is the greatest number such that, for all ? > 0, ?(?(?, г)) ^
const ra~? for all sufficiently small r.
Local dimensions are intimately related to Hausdorff and packing dimensions. For
example,
dim/y F = supjs: there exists ? with 0 < ?(?)< oo and
dimioc ?(-*) ^ s for ? almost all * e F},
see [26]. We say that ? has ejcaci (lower) dimension s if dim|0C ?(?) = s for ?-almost all x.
Surprisingly many measures have an exact dimension, for example, if ? is an invariant
ergodic measure of a Lipschitz mapping /: X -» X for X a closed subset of R", then ? is
exact dimensional [31].
For certain 'rich' measures ?, the sets
Fa = \x: dimioc ?(*) = <*} B3)
may be 'large' for a range of a. There are two natural ways to quantify Fa: in terms of the
measure ? itself, leading to the 'dimension decomposition' into components of different
exact dimensions, and in terms of Hausdorff dimension, leading to the 'multifractal
spectrum'.
3.2. Dimension decomposition
The dimension decomposition aims to split a measure up into exact dimensional
components, as far as this is possible. For ? a Borel measure on R" with 0 < ?(?") < oo,
and with Fa given by B3), the set
S={ae[0,n]: ?(?„)>0}
is finite or countable. For each aeSwe define a Borel measure ?" by
Mff(A)=M(AnF„) for А С R";
the measure ?" may be shown to be of exact dimension a for all a e S. We also define
md(A) = m[a\(JfA,
then ?° is a d/j^se measure, in that ?°(??) = 0 for all a. Then ? has a decomposition
into a countable number of exact dimensional components and a diffuse part:
? = ??"+?°,
aeS
see [ 12,26,84] for proofs of this.

Fractal measures
1049
3.3. Multifractal measures
An alternative, perhaps richer, approach is to look at the dimension of the sets at which
a measure takes a given local density. This approach stems from Mandelbrot's idea [54]
which was taken up in the physics literature [33,36].
Let ? be a finite Borel regular measure on R". It is convenient here to set
? & R . hm =a ,
r—o log r J
so that now Fa is the set of points ? at which the local dimension exists in a strong sense
and equals a, so that, roughly, ?(?(?, r)) ~ ra for small r. If ? is Lebesgue measure
restricted to [0, 1 ] then F\ =[0,1] and Fa = 0 for ? ? 1, a consequence of the uniformity
of Lebesgue measure. However, if ? is highly irregular, the set Fa can be substantial, with
dim/y Fa > 0, for a range of a. Such а ? is thought of as a multifractal measure, with its
(fine) multifractal spectrum defined as
/(a)=dimw Fa
for the range of ? for which Fa ? 0. Thus the multifractal spectrum of a measure attempts
to quantify the intensities of the singularities of the measure. There is an enormous
literature on multifractal analysis; general accounts may be found in [16,21,26,57,71,74,78].
Calculation of multifractal spectra is rarely easy; however there are several important
classes of measure for which the multifractal spectrum may be obtained as the Legendre
transformation of a straightforward 'auxiliary function'. This is best illustrated by the
standard example of a self-similar or multinomial measure.
Let /?,..., /„,: R" -> R" be an iterated function system of contractions satisfying the
open set condition, where f has scaling ratio c, < 1, and let F be the non-empty compact
attractor of this IFS, see A6). Let p\,..., p,„ be 'probabilities', so that p, > 0 for all /' and
Y%'-\ Pi = 1 · Then F supports a measure ? defined by the requirement that
? (У/, ° /ь ° · · · ° fh (F>) = Ph Ph ¦ ¦ ¦ Pik
for all к and all 1 $5 ?'?, b,. ¦ ¦, Ц· ^m. The measure ? satisfies
III
?(?) = ????(/?\?)). B4)
;=?
We now assume that ? is self-similar, that is that the f are similarities, which we also
assume satisfy the open set condition. We define the auxiliary function ?: R -> R by
1П
/ = l
Fa =

1050
K.J. Falconer
It is easy to see that ? is smooth, decreasing and strictly convex (provided that log pi/ log r,
is not constant for all /, a straightforward case we do not consider here). The multifractal
spectrum of ? turns out to be the Legendre transform of /3, that is
/(<*)= inf {P(q)+aq}. B5)
This may be established by defining an 'auxiliary measure' ? on F by i>(/,-, о Д ? · ¦ · о
fik(F)) = (pi,pi2 ¦¦¦piK)q(ritrb ¦¦¦??? and examining the local properties of ? using
A2)-A5).
The multifractal spectrum f(a) is defined and concave for amjn ^ a ^ «max, where
amin = mini ^(^m log ?; / logr/ and amax =
max|^,„logp,71ogr;, with /(a) attaining
a maximum of dim/y F.
There are many measures for which the multifractal spectrum is the Legendre transform
of an appropriately defined auxiliary function. Some of the main cases that have been
examined are listed below.
• Self-similar measures (/, are similarities in the IFS B4) defining the measure) [2,9,
71].
• Graph directed self-similar measures [17].
• Self-conformal maps (f are conformal maps in B4)) [8,79].
• Self-similar measures for infinite IFS (/) are similarities in B4) with m = oo) [58,63].
• Vector-valued self-similar measures [29].
• Quasi self-similar measures [75].
• Self-affine measures (/; are affine maps in B4)) [24,51,73],
• Random self-similar measures (/, are random similarities, p, are random
probabilities in B4)) [2,24,48,54,70].
• Random graph-directed self-similar measures [70].
• Equilibrium measures of hyperbolic dynamical systems [78,89].
• Invariant measures for complex maps [11,53].
• Occupation measures of Brownian motion and generalizations [41,77,87].
A very general approach to multifractal analysis utilizes measures of Hausdorff type,
see [7,71,78,80]. Given a finite regular measure ? on E" and q, ? e ? we may define
auxiliary measures Hq^ on E" using the following steps:
HfP(E) = infjJ3 ???,/'^???,/: ? с Ц ?, where
?, are balls centered in ? with diamfi, ^ S ,
7?·?(?)=????7^·?(?),
u a^o
?.?·?(?) sup ??0·?(?'). B6)
E'CE
(Using covers by balls centered in ? avoids difficulty when q < 0, and the final step is
needed to ensure monotonicity.)

Fractal measures
1051
Analogously to the definition of Hausdorff dimension, we define for q e R
/3(<7) = sup{/3: №р(тл) = 0}. B7)
For all measures ? there is an inequality with the Legendre transform of ?:
f(a) = dimw Ea ^ inf \fi(q) + qa\,
and there are many measures for which equality holds, including self-similar and self-
conformal measures, see [8,70].
Generalized packing measures V4^ may also defined along these lines [71].
3.4. Geometry of multifractal measures
It is natural to seek identities relating the multifractal spectra of measures to those of
their products, projections, intersections, etc., analogous to the geometric properties of
dimensions of sets mentioned in Section 2.5. Such properties are elusive, and though some
results are known, many open questions remain.
For products of measures there are good analogous of A9)—B1). Let ?, ? be finite
Borel measures on R" and R"\ respectively, and let ?\^ and V^ be the generalized
(q, /3)-Hausdorff and packing measures respectively, constructed from ?, see B6). Then
for?CR"andFcR'»,
cinlJip(E)ny(F) ? Hl^y(E xF)<: c2W/{E)Vy{F),
leading to inequalities between the fi(q) and f(a) functions for the measures
involved [72].
Results on projections of measures are more limited in scope, see [30,42,75]. For
slices of measures (that is intersections of measures with w-planes) and more general
intersections of measures, see [74] for a survey of the (often rather technical) results that
have been established.
References
[1] R.J. Adler. The Geometry of Random Fields, Wiley A981).
[2] M. Arbeiter and M. Patzschke, Random self-similar multifractals. Math. Nachr. 181 A996), 5^42.
[3] T. Bedford, Dimension and dynamics of fractal recurrent sets, J. London Math. Soc. B) 33 A986), 89-100.
[4] V.I. Bemik and M.M. Dodson. Metric Diophantine Approximation on Manifolds, Cambridge University
Press A999).
[5] A.S. Besicovitch. On the fundamental geometric properties of linearly measurable plane sets of points II,
Math. Ann. 115 A938), 296-329.
[6] R. Bowen, Equilibrium States and the Ergodic Theory ofAnosov Diffeomorphisms, Lecture Notes in Math.,
Vol. 470, Springer, Berlin A975).
[7] G. Brown, G. Michon and J. Peyriere, On the multifractal analysis of measures, J. Stat. Phys. 66 A992),
775-790.

1052
K.J. Falconer
[8] J. Cole, The multifractal geometry of graph directed self-conformal measures A999), to appear.
[9] R. Cawley and R.D. Mauldin, Multifractal decomposition of Moran fractals. Adv. Math. 92 A992), 196—
236.
[10] M. Chlebik, Geometric measure theory: Selected concepts, results and problems. Handbook of Measure
Theory, E. Pap, ed., Elsevier, Amsterdam B002), 1011-1036.
[11] P. Collet, R. Dobbertin and P. Moussa, Multifractal analysis of nearly circular Julia sets and the multifractal
formalism. Ann. Inst. H. Poincare Phys. Theor. 56A992), 91-122.
[12] CD. Cutler, The Hausdorff dimension distribution of finite measures in Euclidean space. Canad. J. Math.
38A986), 1459-1484.
[13] CD. Cutler, Strong and weak duality principles for fractal dimension in Euclidean space. Math. Proc.
Cambridge Philos. Soc. 118 A995), 393-410.
[14] R.O. Davies and С A. Rogers, The problem of subsets of finite positive measure. Bull. London Math. Soc. 1
A969), 47-54.
[15] M.M. Dodson, B.P. Rynne and J.A.G. Vickers, Diophantine approximation and a lower bound for Hausdorff
dimension. Mathematika 37 A990), 59-73.
[16] G.A. Edgar, Integral, Probability and Fractal Measures. Springer, Berlin A998).
[17] G.A. Edgar and R.D. Mauldin, Multifractal decompositions of digraph recursive fractals, Proc. London
Math. Soc. C) 65 A992), 604-628.
[18] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press A985).
[19] K.J. Falconer, Random fractals. Math. Proc. Cambridge Philos. Soc. 100 A988), 559-582.
[20] K.J. Falconer, The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103
A988), 339-350.
[21] K.J. Falconer, Fractal Geometry - Mathematical Foundations and Applications. Wiley A990).
[22] K.J. Falconer, The dimension of self-affine fractals 11, Math. Proc. Cambridge Philos. Soc. HI A992),
169-179.
[23] K.J. Falconer, Bounded distortion and dimension for non-conformal repellers. Math. Proc. Cambridge
Philos. Soc. 115 A994), 315-334.
[24] K.J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theor. Probab 7 A994),
681-702.
[25] K.J. Falconer, Sub-self-similar sets. Trans. Amer. Math. Soc. 347 A995), 3121-3129.
[26] K.J. Falconer, Techniques in Fractal Geometry. Wiley A997).
[27] K.J. Falconer and J.D. Howroyd, Projection theorems for box and packing dimensions. Math. Proc.
Cambridge Philos. Soc. 119 A996), 287-295.
[28] K.J. Falconer and J.D. Howroyd, Packing dimensions of projections and dimension profiles, Math. Proc.
Cambridge Philos. Soc. 121 A997), 269-286.
[29] K.J. Falconer and T.C. O'Neil, Vector-valued multifractal measures. Proc. Roy. Soc. London A 452 A996),
1-26.
[30] K.J. Falconer and T.C. O'Neil, Convolutions and the geometry of multifractal measures. Math. Nachr. 204
A999), 61-82.
[31] A.H. Fan, On ergodicity and unidimensionality, Kyushu J. Math. 48 A994), 249-255.
[32] H. Federer, Geometric Measure Theory, Springer, Berlin A969).
[33] U. Frisch and G. Parisi, Fully developed turbulence and intermittancy. Turbulence and Predictability
of Geophysical Flows and Climate Dynamics, M. Ghil, R. Benzi and G. Parissi, eds, North-Holland,
Amsterdam A988), 84-88.
[34] S. Graf, Statistically self-similar fractals, Probab. Theory Related Fields 74 A987), 357-394.
[35] S. Graf, R.D. Mauldin and S.C Williams, The exact Hausdorff dimension in random recursive
constructions. Mem. Amer. Math. Soc. 71 C81) A988).
[36] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.J. Shraiman, Fractal measures and their
singularities: the characterization of strange sets, Phys. Rev. A 33 A986), 1141-1151.
[37] F. Hausdorff, Dimension und ausseres Mass. Math. Ann. 79 A919), 157-179.
[38] D. Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets,
J. Number Theory 58 A996), 9-45.
[39] J.D. Howroyd, On dimension and on the existence of sets of finite positive Hausdorff measure, Proc. London
Math. Soc. 70 C) A995), 581-604.

Fractal measures
1053
J.D. Howroyd, On Hausdorff and packing dimension of product spaces. Math. Proc. Cambridge Philos. Soc.
119 A996), 715-727.
X. Hu and S.J. Taylor, The multifractal structure of stable occupation measure. Stochastic Process. Appl.
66A997), 283-299.
B.R. Hunt and V.Y. Kaloshin, How projections affect the dimension of fractal measures, Nonlinearity 10
A997), 1031-1046.
J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30A981), 713-747.
M. Jarvenpaa, Hausdorff and packing dimensions, intersection measures, and similarities, Ann. Acad. Sci.
Fenn. Math. 24 A999), 165-186.
H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika 42
A995), 15-24.
J.-P. Kahane, Some Random Series of Functions, Cambridge University Press A985).
J.-P. Kahane, Sur la dimension des intersections, Aspects of Mathematics and its Applications, North-
Holland Math. Library, Vol. 42, North-Holland, Amsterdam A986), 419-430.
J.-P. Kahane and J. Peyriere, Sur certaines martingales de Benoit Mandelbrot, Adv. Math. 22 A976), 131-
145.
A. Katok and B. Hasselblatt, Introduction to the Modem Theory of Dynamical Systems, Cambridge
University Press A995).
R. Kaufman, On the Hausdorff dimension of projections, Mathematika 15 A968), 153-155.
J. King, The singularity spectrum for general Sierpinski carpets. Adv. Math. 116 A995), 1-11.
O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press
A991).
A.O. Lopes, The dimension spectrum of the maximal measure, SIAM J. Math. Anal. 20 A989), 1243-1254.
B.B. Mandelbrot, Intermittent turbulence in self-similar cascades: divergence of high moments and
dimension of the carrier, J. Fluid Mech. 62 A974), 331-358.
B.B. Mandelbrot, Les Objects Fractals: Forme, Hasard et Dimension, Flammarion A975).
B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman A982).
B.B. Mandelbrot, Multifractals and \/f Noise, Springer A999).
B.B. Mandelbrot and R. Reidi, Multifractal formalism for infinite multinomial measures. Adv. Appl. Math.
16A995), 132-150.
J.M. Marstrand, The dimension of Cartesian product sets, Proc. London Math. Soc. 50 A954), 198-206.
PR. Massopust, Fractal Functions, Fractal Surfaces and Wavelets, Academic Press A994).
P. Mattila, Hausdorff dimensions and capacities of intersections of sets in ?-space. Acta Math. 152 A984),
77-105.
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press A995).
R.D. Mauldin and M. Urbanski, Dimensions and measures in infinite iterated function systems, Proc.
London Math. Soc. 73 C) A996), 105-154.
R.D. Mauldin and M. Urbaiiski, Confornwl iterated function systems with applications to the geometry of
continued fractions. Trans. Amer. Math. Soc. 351 A999), 4995-5025.
R.D. Mauldin and S.C. Williams, On the Hausdorff dimension of some graphs. Trans. Amer. Math. Soc.
298A988), 793-803.
R.D. Mauldin and S.C. Williams, Hausdorff dimension in graph directed constructions. Trans. Amer. Math.
Soc. 309A988), 811-829.
H. McCluskey and A. Manning, Hausdorff dimension for horseshoes, Ergodic Theory Dynam. Systems 3
A983), 251-260.
C.T. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 A984), 1-9.
P. Morters and D. Preiss, Tangent measure distributions of fractal measures, Math. Ann. 312 A998), 53-93.
L. Olsen, Random Geometrically Graph Directed Self-similar Multifractals, Longman A994).
L. Olsen, A multifractal formalism. Adv. Math. 116 A995), 82-196.
L. Olsen, Multifractal dimensions of product measures. Math. Proc. Cambridge Philos. Soc. 120 A996),
709-734.
L. Olsen, Self-affine multifractal Sierpinski sponges in R';. Pacific J. Math. 183 A998), 143-199.
L. Olsen, Multifractal geometry, Progr. Probab, Vol. 46, Birkhauser B000), 3-37.
T.C. O'Neil, The multifractal spectrum of quasi-self-similar measures, J. Math. Anal. 211 A997), 233-257.

1054
K.J. Falconer
[76] ?. Peres and B. Solomyak, Problems on self-similar sets and self-affine sets: an update. Progr. Probab.,
Vol. 46, Birkhauser B000), 95-106.
[77] E.A. Perkins and S.J. Taylor, The multifractal structure of super-Brownian motion, Ann. Inst. H. Poincare
Probab. Statist. 34 A998), 97-138.
[78] Y.B. Pesin, Dimension Theory in Dynamical Systems, University of Chicago Press A997).
[79] Y.B. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps
and Markov Moran geometric constructions, J. Stat. Phys. 86 A997), 233-275.
[80] J. Peyriere, in: Proceedings of the NATO ASI on Probabilistic and Stochastic Methods in Analysis with
Applications, NATO ASI Series, Vol. 372, Kluwerf 1992), 175-186.
[81] D. Preiss, Geometry of measures in R": distribution, rectifiability and densities, Ann, Math. 125 A987),
537-643.
[82] D. Preiss and J. Tiser, On Besicovitch xs \ -problem, J. London Math. Soc. 45 B) A992), 279-287.
[83] C.A. Rogers, Hausdorff Measures, 2nd edn, Cambridge University Press A998).
[84] C.A. Rogers and S.J. Taylor, Functions continuous and singular with respect to a Hausdorff measure,
Mathematika8A962), 1-31.
[85] D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 A982), 99-107.
[86] X. Saint Raymond and С Tricot, Packing regularity of sets in ?-space. Math. Proc. Cambridge Philos. Soc.
103A988), 133-145.
[87] N.-R. Shieh and S.J. Taylor, Logarithmic multifractal spectrum of stable occupation measure. Stochastic
Process 75 A998), 249-261.
[88] K. Simon, Hausdorff dimension for noninvertible maps, Ergodic Theory Dynam. Systems 13 A993), 199—
212.
[89] D. Simpelaere, Dimension spectrum of Axiom A diffeomorphisms I, II, J. Stat. Phys. 76 A994), 1329-1358,
1359-1375.
[90] D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups.
Acta Math. 153 A984), 259-277.
[91] S.J. Taylor and C. Tricot, Packing measure and its evaluation for a Brownian path. Trans. Amer. Math. Soc.
288 A985), 679-699.
[92] S.J. Taylor, The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 A986), 383-
406.
[93] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin A988).
[94] С Tricot, Two definitions of fractional dimension. Math. Proc. Cambridge Philos. Soc. 91 A982), 54-74.

CHAPTER 26
Positive and Complex Radon Measures
in Locally Compact Hausdorff Spaces
T.V. Panchapagesan*
Departamento de Matemdticas, Facultad de Ciencias, Universidad de los Andes, Merida, Venezuela
E-mail: panchapa@ciens.ula.ve
Contents
Introduction 1057
1. Preliminaries 1057
2. Regular extensions of (positive) measures 1059
3. Complex Radon measures and their properties 1062
4. Regular extensions of positive and complex measures 1068
5. Bounded complex Radon measures 1071
6. Characterizations of complex Radon measures 1073
7. Isomorphic representations of /CG")*,/CG". R)*,/CG")*, and /CG\R)*, 1077
8. Applications 1081
9. Generalization to Radon vector measures 1086
References 1090
*Supported by the C.D.C.H.T. project C-845-B-97 of the Universidad de los Andes, Merida, Venezuela.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1055

This page intentionally left blank

Positive and complex Radon measures
1057
Introduction
If ? is a right continuous complex function of finite variation in R" (i.e., ? is of bounded
variation in each bounded interval of R"), then ? induces a complex Lebesgue-Stieltjes
measure ??? on R", whose domain is a S -ring ? ? containing all the compact subsets
of R". If ft = МФ П ?(R"), then it is well known (see McShane A944) for the real
case and Panchapagesan A991a) for the complex case) that ? ? (respectively ???) is the
Lebesgue completion of ft (respectively ш*|тг) with respect to ?(???\??, ft) (respectively
toft).
Conversely, let ? be a complex measure on a <5-ring V containing the compact subsets
of R" and let ft = V ? ?(R") = {E e B(M."): |?|*(?) < oo), where |?| = ?(?, V) and
|?|* is the outer measure induced by |?|. If ? and V are the Lebesgue-completions of ?\?
and of ft, respectively, with respect to ft and ?\??, then there exists a right continuous
complex function ? of finite variation in R" such that ? ? = V and ? = ???. This result
is essentially the same as Theorem 54.2 of McShane A944), where McShane considers ?
to be real.
Since the positive and complex Lebesgue-Stieltjes measures on R" are precisely the
positive and complex Radon measures on R", respectively, in the sense of Bourbaki
A965a), the question arises whether one can generalize the above results to Radon
measures on a locally compact Hausdorff space ?. This question is answered in the
affirmative in Panchapagesan A992, 1993). The object of the present chapter is to give
a detailed exposition of these results.
Also a section is included to discuss some results obtained in Panchapagesan A998a,
1998b) as applications of the isometric isomorphism theorem, i.e., of Theorem 7.3. Finally,
the last section is devoted to state some results which generalize those in Sections 3,6 and 7
to Radon vector measures of Thomas A970).
By positive (respectively complex) measures we mean countably additive non negative
valued (respectively complex valued) set functions defined on a ring of sets. Unlike
Bourbaki A965a), the continuous linear functionals on K,(T) (see Section 1) are not called
real or complex measures.
1. Preliminaries
In this section we fix the notation and terminology. Also we give some definitions and
results from the literature.
? denotes a locally compact Hausdorff space. С is the family of all compact subsets
of T. CC(T) = {f:T -> C, / continuous, supp/ e C] is a normed space with the
supremum norm || ¦ Ц7- where ||/||r = supteT\f(t)\. For each С e C, CC(T,C) —
{/ e CAT): supp/ с С}. For С е С, let IC-CAT,C) ^ CAT) be the canonical
injection. Let ? be the inductive limit locally convex topology on CAT) induced by
the family {CAT, C), Ic)ceC where each CAT, C) is provided with the topology ?„ of
uniform convergence. Then we denote (CAT), ?) by K,(T). Clearly, ? is a Hausdorff
topology. fC(T, R) = {/ e K,(T), f real} and fC+(T) = {/ e K,(T): f > 0). K,(T, R)
is endowed with the relative topology induced by ?. ?,(?)* (respectively K,(T, R)*)

1058
Т. V. Panchapagesan
is the topological dual of the locally convex Hausdorff space K,(T) (respectively
/CG\R)).
A linear functional ? belongs to K,(T)* if and only if, for each С е С, there exists a
constant Kc such that |0(/)| sC Kc\\f\\? for all / e CC(T,C). Given ? e IC(T)*, we
define the conjugate ? of ? by 0(f) = 0(f) for / e /С(Г). 0 е K,(T)* is said to be real
if 0 = ?. Then it can be shown that 0 e K,(T)* is real if and only if 0(/) is real for all
/ e K,(T, R). ±@ + 0) and ^ @ - ?) for 0 e /С(Г)* are continuous real linear functionals
and are called the real part of 0 and the imaginary part of 0, respectively; are denoted by
Re 0 and Im 0, respectively. A linear functional 0 on /C (T) is said to be positive if ? (/) ^ 0
for / e /С+(Г). Then it can be shown that all positive linear functionals on K,(T) belong
to K,(T)*.
Given ? & ?,(?)*, there exists a unique positive linear functional |0|, called the absolute
value of ?, on K,(T) given by
|0|(/)= sup |0(g)| A.1)
|«|5?/. «e/C(D
for/e/C+(r).
/С(Г, R)* is a boundedly complete vector lattice under the partial ordering ?\ ^ 9j
given by 0|(/) ^ 6»2(/) for all / e /С+(Г). For 0 e /С(Г, R)*, 0+ = max@, 0) and
?~ = -min@,O) and ? has the canonical decomposition ? = ?+ - ?~. ? e ?,(?, R)*
has a unique extension to K,(T) as a continuous linear functional and we denote it also by
? and
\0(f)\<W\{\f\), fefC(T),
where \?\ is given by A.1). When ? e ?;(?)* is real, then \в\\/С(т.Ш) = Щка.Ш))+ +
@/CG\R))-- Given 0 e /С(Г)*, by an abuse of notation we denote the unique extensions
of ((Refl)|x;G-.K))+ and ((Re0)|/c(r.K))_ by (Re^)+ and (Re^)", respectively.
Similarly, are defined (Im0)+ and (Im0)_. Thus, ? = {(Re0)+ - (Re0)-} + /{(Im0)+ -
(Im0)-}.
For the above definitions and results the reader is referred to Bourbaki A965a,
Chapter III, § 1) and Panchapagesan A991 a, Chapter 6, §41).
Co (respectively U) is the family of all compact G& (respectively open) subsets of T. For
a non void class ? of subsets of ?, ? (?) (respectively ? (?)) denotes the <5-ring (respectively
?-ring) generated by S. B(T) = ?(??) (respectively BC(T) = a{C), BQ(T) = a(CQ)) is the
family of all Borel (respectively ?-Borel, Baire) subsets, of T. A subset ? of ? belongs to
BC{T) if and only if it is Borel and ?-bounded (in the sense, it is contained in a countable
union of compact sets) and hence the nomenclature for the members of BC(T).
Given a real measure ? on a ? -ring V, ? has the Jordan decomposition ? = ?+ — ?-
with ?+ and ?~ being finite positive measures so that ? is of finite variation on V.
Moreover, if ? = ? - ?, ?, ? positive measures, then ?+ ^ ? and ?~ ^ ? (see p. 50
of Dinculeanu A967)).

Positive and complex Radon measures
1059
The proof of Theorem III.5.17 of Dunford and Schwartz A958) can be adapted to prove
the following proposition which gives the concept of Lebesgue completion of a <5-ring.
PROPOSITION 1.1. Let ? be a complex measure on a S-ring V. Let V* = {E U ?: ? e V,
N CM eV with v(v, V)(M) = 0). For such ? U N eV*, let v(E U N) = v(E). Then
V* is a S-ring, V* D V, ? is well defined, v\p = ? and ? is a complex measure on V*. ?
(respectively V*) is said to be the Lebesgue completion of ? (respectively V) with respect
to V (respectively v).
By a measure (on a ring) we mean a positive measure.
From Chapter IV of Bourbaki A965a) we have the following theorem. The reader may
also refer to Hewitt and Stromberg A965), Naimark A959), Panchapagesan A991 a, 1991 b)
and Rudin A967).
THEOREM 1.2. Let ? be a positive linear functional on CC(T). Let M+(T) = {f:T ->
[0, oo], / lower semi-continuous], T+(T) = {f:T -> [0,oo]} and C+(T) = {/ e
CAT): f > 0). Let
?*(?)= inf sup{0(VO: t^g. феС+(Т)}
for ? С Т. Then:
(i) ?? is an outer measure on V(T).
(ii) Let ??* =(?сГ: ? is ?%-measurable]. Then ??* is ? ? -algebra and contains
B(T). We denote ?^\?t, by ??.
(Hi) д*(С) <оо, СеС.
(iv) ?*?(?) = ???{?*?(?): EcUeU},EcT.
(?) ?*(?) = sup^tC): С С Е, С е С] if ??*(?) is ?-finite and E e ??* or if
Ее U.
(vi) Given ? e ??* with ?*?(?) ?-finite, then there exist А, В е B(T) such that
А С ? С ?, ? ?-compact and ?%(?\?) = 0. Consequently, ?^(?) = ?^(?).
(vii) Let ?%* = ?*?\?{?? Then
?(?) = 1?<1?**, feCAT).
Moreover, if ? is a measure on B(T) satisfying (iii)-(v) above and if 9(f) =
fTfdv, f e CC(T), then ? = ?^0?. ?^0? is called the Borel-Radon measure
induced by ?.
2. Regular extensions of (positive) measures
We introduce several notions of regularity for measures defined on certain <5-rings or ?-
rings of subsets of ? and study the existence and uniqueness of regular extensions. These
results play a key role in the next section.

1060
Т. V. Panchapagesan
???????? 2.1. A measure ? on Bq(T) (respectively on BC(T), on B{T)) is a Baire
(respectively ?-Borel, Borel) measure if ?@ < oo for С e Co (respectively С е C).
Dehnition 2.2. Let Tl be a ring of subsets of ? with 5(Co) С U or 5(C) С U. A measure
? defined on 7? is said to be 7^-regular if
(i) ?@ <oo, CeCilft;
(ii) ?(?) = ?{?(?/): ? С ?/ e U ? 7г} for ? е 7г; and
(iii) ?(?) = sup^(C): С С ?, С е С ? 7г} for ? е 7г.
A complex measure ? on 7^ is said to be 7?-regular if, given ? e 1Z and ? > 0, there exist
С eTZnC and U e Ti f)U such that С С ? с ?/ and \v(G)\ < ? for all G e 7г with
GcU\C
Dehnition 2.3. A Borel measure ? on ?(?) is said to be Radon-regular if
(i) ?(?) = ???{?(?): ? С U е Щ for ? е ?(?); and
(ii) ?(?/) = 5??{?@: С С ?/, С е С) for U e W.
Using Proposition 17, Paragraph 15 of Dinculeanu A967), it is easy to prove the first
three statements of the following proposition while the fourth is due to Theorem 10.30 of
Hewitt and Stromberg A965).
Proposition 2.4.
(i) A Baire measure is Bo(T)-regular.
(ii) A measure ?? on S(Cq) is S(Co)-regular.
(iii) A complex measure щ on S(Cq) is S(Co)-regular.
(iv) If ? is a Radon-regular measure on B(T) and ? e B{T) with ?(?) ?-finite, then
?(?) = $\}?{?(?): С С ?, С еС}.
The following proposition is the same as Proposition 11 in Dinculeanu A967, § 14).
PROPOSITION 2.5. If С С U, С e С and U e U, then there exist Co e Co and Щ e
U П <5(Co) such that С С U0 с Со С U.
Using Propositions 2.4(iv) and 2.5 and the fact that each ? e BC(T) is ?-bounded, one
can easily prove the following
LEMMA 2.6. Let ? be a Radon-regular measure on B(T). Then:
(i) ifv = ?\^?· Men v is S(C)-regular;
(ii) if ? = ?|??G·), then ? is Bc (?)-regular.
The first part of the following lemma is due to Theorem 56.E of Halmos A950). For
С e Co, we can construct a sequence (/„) с C+(T) such that /„ \ ?? and by the
Lebesgue dominated convergence theorem ?@ = lim„ fT f„ ?? for a Baire measure ?.
Using this observation and Proposition 2.4(i) we have the following

Positive and complex Radon measures
1061
LEMMA 2.7. (i) Let ?? and ?? be Bc(T)-regular (respectively (ii) Baire) measures. If
f ???\ = ? /??2, feCAT) B.1)
Jt Jt
then ?? = ??-
THEOREM 2.8. Let ?? be a Baire measure on T. Then:
(i) ?? has a unique extension ?<- to BC(T) such that ?(- is a Bc(T)-regular measure;
(ii) ?? has a unique extension ? to B(T) such that ? is a Radon-regular measure;
(iii) If?(· and ? are as in (i) and (ii), then ??- = ?\?,(?)-
PROOF, (i) is due to Theorem 54.D of Halmos A950). For proving (ii), let
0(f) = ffdw, /еСДГ). B.2)
Then by Theorem 1.2 there exists the Borel-Radon measure ?°??? = ?(say) induced by
? and 9(f) is also given by B.2) with ?? replaced by ?. If ? = ?\?(^?), then by
Lemma 2.7(H) ? = ?? and, hence, ? extends ??. The uniqueness of ? follows from the
uniqueness part of (i), from Lemma 2.6(ii) and from (iv) and (v) of Theorem 1.2. D
Taking 9(f) = fT f ??, f e CC(T), where ? is a ??<(r)-regular measure and arguing
as in the proof of Theorem 2.8 applying Lemmas 2.6(ii) and 2.7(i) we obtain the following:
THEOREM 2.9. Every Bc(T)-regular measure has a unique extension ? to B(T) as a
Radon-regular measure.
THEOREM 2.10. Let ? be a S(C)-regular measure. Then:
(i) the unique extension ? of ? to BC(T) as a measure is Bc(T)-regular;
(ii) ? admits a unique extension ? to B(T) as a Radon-regular measure and ?\?? (j) =
?, where ? is as in (i).
PROOF, (i) As ? is finite on 8(C), ? admits a unique extension ? to a(C) = BC{T)
as a measure. Clearly, ?@ < oo for С e С If ? e BV(T) with ?(?) — oo, then
obviously Definition 2.2(ii) holds for ?(?) with TZ = BC(T). Let ?(?) < oo and let
? > 0. As ? is ?-bounded, there exists (C,,)^ С С such that ? С UГ с« ¦ By hypothesis,
there exists U„ e U ? 8(C) such that ? ? C„ С U„ and ?(?/„) - ?(? ? C„) < ?/B"),
? e N. Let U = U?° U„. Then ? cU eUn BC(T) and ?(t/) - ?(?) < ?. Thus ?(?)
satisfies Definition 2.2(H) with П = BC(T). With C„ as above, let E„ = \J" ? ? С*.
Then (?„)f С 5(C) and ?„ / ?. Then ?(?) = sup,, ?(?„) = sup,, sup^(C): С С ?„,
С eC}^ sup{M(C): С С ?, С е С) < Д(?), since ? is <S(C)-regular. Hence, ? is ?(·(?)-
regular.
(ii) Taking ? as in (i) and applying Theorem 2.9 to Д, we obtain a Radon-regular
extension ? of ? and, hence, of ?, to B(T). Moreover, ? is unique by the uniqueness
part of (i) and that of Theorem 2.9. ?

1062
?, V. Panchapagesan
If ? is a complex valued additive set function on a ring of sets TZ, then it is well known
that
sup |v(F)|<u(v, ¦?)(?)< 4 sup |v(F)|. B.3)
FcE.FeTZ FcE. Fell
Using B.3) one can prove the following
PROPOSITION 2.11. Let TZ be a S-ring of subsets ofX with 8(C) cTZor 5 (Co) С И and
let v, vj, i = 1,2, be complex measures on TZ. Then:
(i) If v\ and V2 are TZ-regular, then a\v\ + a2v2 is also TZ-regular, where a\ and a2
are scalars.
(ii) ? is TZ-regular if and only ifv(v, TZ) is TZ-regular.
(iii) Ifv(v,TZ) is TZ-regular, then Rev, Im v, (Rei>)+, (Rev)", (Imv)+ and (Imv)"
are TZ-regular.
(iv) If\v\ (E)\ ^ \v2(E)\for all ? еЯ and ifvi is TZ-regular, then v\ is TZ-regular.
3. Complex Radon measures and their properties
With each ? e fC(T)* we associate a unique complex measure ?? defined canonically on
a <5-ring Me containing 8(C) and study the properties of ?? and Mq.
The following result is the same as Proposition 15 in Bourbaki A965a, Chapter IV, §1).
PROPOSITION 3.1. For the positive linear junctionals ?\ and 02 on CC(T) we have:
@ ?*??+?2=?*??+??.
(ii) //0| ^??,????*?? <?*,.
For a positive linear functional ? on CC(T), let us recall from Theorem 1.2 that
?? = ?*\? t and ?^0? = ?*?\?(?)- ??°? is tne Borel-Radon measure induced by ?.
THEOREM 3.2. A measure ? on B(T) is Borel-Radon in the sense that there exists a
positive linear functional ? on CC{T) such that ? = ?^?? if and only if it is Radon-regular.
PROOF. Clearly, the condition is necessary by Theorem 1.2. Conversely, let 9(f) =
fT f ??, f e CC(T). Then ? is a positive linear functional on CC(T) and by the uniqueness
part of Theorem 1.2(vii), the condition is also sufficient. ?
From Theorem 1.2 we have the following:
PROPOSITION 3.3. Let ? eK,(T)* and let ?\ =Re# ande2 = lm0. Then:
(i) If ?? = [A CX: ??+(?) < ??, ??-(?) < oo, j = 1,2), then Me is a 8-ring and
contains 8(C).
(ii) For ? eMo, let ??(?) = {(??+ -??-) + '?{??+ -??-)}(?). Then ?? is a complex
measure on Mq.

Positive and complex Radon measures
1063
(iii) If ? is real, then ?? is real.
(iv) If ? is positive, then ?? is positive and finite.
Definition 3.4. Let ? e K,{T)*. The complex measure ?? in Proposition 3.3 is called
the complex Radon measure induced by ? and Me is called the domain of ??.
Notation 3.5. Let ? e K,{T)* and let ?? be the complex Radon measure induced by ?.
We denote ?(??, ??) by \??|. If ? is a real measure on a <5-ring ??? and 1Z\ is another
<5-ring contained in 7^2, then
(v,n\,n2)+(E) = %\^\v(F): FcE, F еП\}
and
(?,?],?2?(?) = -mf{v(F): ? С ?. F eTZ\}
for ? e IZi. If ? is a real or complex measure on 7^2, then
?{?,?\,??2){?)
In ? 1
J]| v(Ei)\: (?,)',' С 7г,, ?, П ?7¦ = 0. i ? j, {J ?, С ?
for ? e 7г2.
THEOREM 3.6. ?*>? # e /С(Г)*, # rea/. Then the following assertions hold:
(i) ?? is Mg-regular.
(ii) ?^ is of finite variation on Me-
(iii) ?^ ^ ?^+ a/id ?~ ^ ?^- /и ??^, where ?? = ?^ - ?^ " f^e Jordan
decomposition of ?? in the ?-ring Me.
(iv) \??\, ?^ and ?~ are Мд-regular.
(v) ?-tWo = (??\?(?)+· Мн U@ = (ЫаюГ ??^^(?«??(?·5(^))= Im«IIs<C)·
(vi) lM«IU(C). М<»Ъ(С)> M^li(C) and ??\?(? are S(C)-regular
(vii) (a) (???,{(?.??)+(?) = ^(?),?????.
(b) (??, 5(C), ??»?(?) = ?~(?), ? e ??«.
(c) и(д„,а(С),Л*„)=|Ы(Е). ЕеМв.
(viii) G/ve/г ? е Мв, there exist A, B eK = ?(?) ? Mff яис/г ?/??? А С ? С В and
??+(?\?) = ??-(?\?) = 0. Then ??~(?) = ??+(?), ??-(?) = ?„-(?) and
?? (respectively Me) is the Lebesgue completion of ??\?? {respectively of TV)
with respect to ?? {respectively to ?? \ц).
PROOF. By Theorem 1.2, ??+ and ??- are Me-regular and hence ?? and \??\ are also
???-regular by Proposition 2.11. Since Me is a <5-ring and ?? is a real measure, (ii) holds.
As ?? = ??+ - ??-, (iii) holds. As ?? + and ??~ are ??^-regular, (iv) holds by the
inequalities in (iii) and by Proposition 2.1 l(iv). Thus (i)—(iv) hold.

1064 T.V. Panchapagesan
(v) Let F e 8(C) and ? > 0. By (iv), given F e ?#, there exists Cf eC such that
CfcF and \^f,\(F\CF) < ?. Thus,
(VeWo)+(E)
= sup{Mff(F): Fc?. F e 5(C)} > sup^«@): FcF, FeMff}
> sup{Mff(F)-?: FcF, F e ??«} = ?+(?) - ?.
Hence, (??\?@))+ ^ M^U(C)- Since the reverse inequality is obvious, this proves the first
equality. Similarly, the remaining two equalities can be proved.
(vi) Clearly, it suffices to show that |Мб> ИгсС) is <5 (С)-regular. Let ? e 8(C) and ? > 0.
By (iv) there exist U e U ? ?? and С е С such that С С Е с i/_and \??\(U\C) < ?. As
F е С, by Proposition 2.5 there exists i/()eWn 8(C) such that F С t/(). If V = U ? ?/(),
then Fc V eWn<5(C)and |ms|(V\C) < ?.
(vii) (a) Let F e M#. By (iv) and (v) we have
= sup{M+(C): С СЕ, С eC}= sup sup^F): F С С, F e 5(C)}
Ccfc". CeC
^sup{Mfl(F): FCF, F e<5(C)} iC sup{M„(F): FcF, F e ??} = ?+(?).
Similarly, (b) is proved by using the ?? -regularity of ?~.
(vii) (c) For F e ??, given ? > 0, there exists a partition {Ej}" of F in ?? such that
" 1
]P|Mfl(?i)| > lMff|(F)- -?.
By (i), for each / there exists Cj e С such that С, с F, and |m«(G)| < ?/(In) for
GeMfl with G С Ei\d. Then ?',' 1д»(С/)| ^ ?'? 1д»(Е/I - ?/2 > \Чв\{Е) - ?. Thus
?(??, 5(C), My)(E) > |Mfl|(F). Since the reverse inequality is obvious, (c) holds,
(viii) This is immediate from (iii) and from Theorem 1.2(vi). ?
THEOREM 3.7. Let0e K,(T)*, ? real and let ?+ and ?? be as in Theorem 3.6. Then:
@ ME~ls(C) an^ Mfl la(C) a^w/f unique extensions ?^ ami ?,7> respectively, to B(T)
as Radon-regular measures and ?^~(?) = ?^(?), JLtff~(F) = ?# (?) /or F e
??(?)?????.
(ii) If ? = \??\\?(?> then ? has a unique extension ? to B(T) as a Radon-regular
measure and 0(F) = \??\(?)/?? ? e BC(T) ПМ«.
(iii) There exists a unique positive linear functional ? on CC(T) such that ? = ?^??,
where ? is as (ii).
(iv) \?\ ^ ?, where ? is as in (iii).
(?) ??+ = ?+ and ??~ = ?~ in ??.
(vi) Мв = Мщ and \??\ = ?\?\.

Positive and complex Radon measures
1065
PROOF, (i) The first part is due to Theorems 3.6(vi) and 2.10(ii). By Lemma 2.6(ii) and by
Theorem 3.6(iv) we have
? + (?) = sup{ji+(C): СсЕ, С eC}= sup^+(C): ССЕ, С еС)
= ?+(?)
for ? e BC(T) П ??. Similarly, the other result holds,
(ii) The proof is similar to that of (i).
(iii) This is due to Theorem 3.2.
(iv)Let / e C<(X) with suppf = C. Then
\0(f)\ = I f fd^e\B[C])\ < ? \f\dv{M\Bio.B(C))
\Jc I Jc
< j \f\dvfa\slC). 8(C)) < f \?\?\?„\ = f \f\dv
= f\f\dj$* = ,l,{\f\)
and, hence, by the definition \?\ (see A.1)), (iv) holds.
(v) By (iv) and by Proposition 3.1, ?*^ ^ ?*? and ц.щ = ?«+ + M«- in ? (?) П ??.
Consequently, by (ii) and (iii) and by Theorem 3.6(iii) we have
Ш(Е) = (?+ +?")(?) < (??++ ??-)(?) = ?„,(?) < ?^'?(?)
= \??\(?) for ? еВс(Т) П ??.
Hence, ?[?\(?) = \??\(?) for ? e ?,.(?) П ??.
Now let ? eMe. Then by Theorems 3.6(viii) and 1.2( vi) there exist А, В е ?((?)?)??
with A ?-compact such that А С ? С ? and ??+(?\?) = ??-(?\?) = 0. Hence, by
Proposition 3.1(i) we have pm(B\A) = 0. Consequently, ? = A U (E\A) e Мщ. Thus
Aifl с Мщ. Moreover, as A e BC(T) П M«, by Theorem 3.6(iii) we have ?\?\{?) =
????(?) = \??\(?) = ?+(?) + ?-(?) = ?+(?)+??(?) = \??\(?) and hence М|«||мн =
|?#|. Therefore, by Theorem 3.6(iii), we have |?«| = ?^ + ?^ ^ ?«+ + M^- = M|fl| = |М«1
in Mff. This shows that ??+ = ?+ and ??- = ?„ in M«. Hence (v) holds.
(vi) In the proof of (v) we showed that Мн С Мщ. If ? e Мщ, then by Theorem 1.2(vi)
there exists А, В еВ(Т)Г\Мщ such that А С ? С S, ?|«|(?\?) = 0 and ?|«|(?) < oo.
Then by Theorem 3.6(iii) and Proposition 3.1 (ii), it follows that A and E\A belong to ??
and hence ? e M«. Thus (vi) holds. ?
Now let us consider an arbitrary ? e K,(T)*.
THEOREM 3.8. Let ? e Х](Г)*. Г/геи the following assertions hold:
(i) ?^ /? ??-regular.
(ii) ?^ ? of finite variation in M#.

1066
Т. V. Panchapagesan
(iii) \??\ is Mo-regular.
(iv) u(Mfl|a(C),5(C)) = |Mfl||a(C).
(v) |Мй11г(С) is &(C)-regularand hence ??\&^) is S(C)-regular.
(vi) и(дй,5(С), Mfl) = |Mfl|.
(vii) G/ven ? e Mfl, ?/геге exist A,B eTL = B(T) ? Mff яис/г ?/??? ? С ? С В and
\??\(?\?) = 0. Consequently, ?$(?) = ?$(?) and ?? (respectively ??) is the
Lebesgue completion of ??\?? (respectively ??) with respect to TL (respectively
??\?)-
(viii) If ? = \??\\$(?· then ? has a unique Radon-regular extension ? to B(T) and
HE) = \??\(?)? ? e B,(T) П ??.
(ix) There exists a positive linear functional ? on CC(T) such that 0 = ?^??, where ?
is as in (viii). Moreover, \?\ ^ ?.
PROOF. Let ?\ = Re# and ?2 = Im#. As ?? = ??? ? ??2 and U\ ? U2 e W ? ?? for
Uj еИП ??[ for / = 1,2, (i) holds by Theorem 3.6(i). (ii) holds as ?? is a 5-ring. By (i)
and Proposition 2.11, (iii) holds. The proof of Theorem 3.6(vii)(c) holds here verbatim to
prove (vi). (iv) is immediate from (vi). The proof of (v) is similar to that of Theorem 3.6(vi)
but for the fact that we have to invoke (iii) of the present theorem instead of (iv) of the said
theorem. Thus (i)—(vi) hold.
(vii) Let ? e ??. Since ??+(?) < oo and ??-(?) < oo for j = 1,2, by
Theorem 1.2(vi) there exist A.fie BC(T) П Me such that А С ? С В, А ?-compact, and
??+(?\?) = ??7(?\?) = 0 for j = 1,2. Thus \??\(?\?) = 0. Also by (vi), |/ie|(F) =
?(?? 1тг, K)(F) for F e K. Thus (vii) holds.
(viii) The proof is similar to that of Theorem 3.7(ii).
(ix) The first part is due to Theorem 3.2. As in the proof of Theorem 3.7(iv), for
/ e CC(T) with supp / = C, we have
\0(f)\ = \S ?????€)+? f /???,\???\^ ? \?\??(??\?(.0,?@)
\Jc Jc I Jc
^J \f\dv = J \/\??^ = ?{\?\)
and, hence, by A.1) we have \?\ < ?. ?
Theorem 3.9. Let ? e fC(T)*. Then:
(i) |де(С)|^Д|в|(С)./ЬгСеС.
(ii) For A e Me, \??\(?) = 0 if and only if ?\?\(?) = 0.
(iii) ??? = ?|?| and \??\ = ?\?\.
PROOF. Let ?\ and вг be as in the proof of Theorem 3.8. Let С eC. Then by Theorem 1.2,
??+1Me and M#-1Ms are ??# -regular for ; = 1,2. Therefore, given ? e N, there exists t/„ e
W П ??„ such that С С ?/„, ??+(?„\0 < \/? and ??-(?„\0 < l/и for ;' = 1,2. Let
W„ = ?? Uk and W = ПГ W„. Then W e M« as ?? is a <S-ring. Moreover, С С W and

Positive and complex Radon measures 1067
??+(?\0 = ??? (W\C) = 0 for ; = 1,2. By Urysohn's lemma there exists /„ e C+(T)
with xc ^ /„ ^ xw„- If g„ = ?'? fi· tnen gn \ Xc a.e. with respect to ??+ and ?^- for
;' = 1,2. If С ? is the support of g \, then by the Lebesgue dominated convergence theorem
we have
lim f gn ??^ = ?+(? and lim [ g„ ??*°.? = ??- (С) for ; = 1, 2.
" Jt °J ? " JT ? ?
Then as gn -> xc Mjflj-a-e-, by the same theorem we have
|мй(С)| = lim / &???? = lim0(g,,) < lim |0|(g„) = lim / ?„<*?^?
Iй Ус, " " " Jt
= Ml«l(C).
Hence, (i) holds.
(ii)Let |де|(Л) = О. By Proposition 3.1 we have
Mjflj(A) < (m|S,| +M|«2l)(A) = u(M«,, Мй,)(А)+и(м«:.Мй:)(А)
= sup{u(Mff,,Mff,)(C): CCA, С еС)
+ sup{u(Mfl:,Mff2)(C): CCA, CeC}
= sup{u(MSl|i(C),a(C))(C): CCA, CeC}
+ sup{u(MS2|i(C),a(C))(C): CCA. С e С} ^ 2|м«|(А) = О
by (iv) and (v) of Theorem 3.6 applied to ?\ and ?7 and by Theorem 3.7(vi) and by (iii)
and (iv) of Theorem 3.8.
Conversely, let M|«|(A) = 0. Then ?\? |(A) = 0 for j = 1,2 and, hence, by
Theorem 3.7(vi) we have
iMfll(A) ^ ?(???,???)(?) + ?(??2,??:)(?) = ?\?,\(?) + ?\#2\(?) = 0.
Hence, (ii) holds.
(iii) Let ? еМв. Then, by Theorem 3.8(vii), there exist А, В e B(T) ? ?? with А с
? С S such that \??\(?\?) = 0. Then by (ii) ?]?\(?\?) = 0 and, hence, м^|(?\А) = 0.
Therefore, E\A e ??|#| and hence ? e ??^| since ??+(?) < oo and ??-(?) < oo for
; = 1,2. Thus ?? с ??]#|.
Conversely, let ? e M|fl|. Then by Theorem 1.2(vi) there exist А, В е ?(?) ? Aifl such
that А с ? С S and ???|E\?) = 0. Thus ?\?\(?) = ?\?\(?)- Then by Proposition 3.1,
??+(?\?) = ??-{?\?) = 0 for ; = 1,2. Consequently, E\A,E e ?? and thus
?? = ??\.
Let ? e ?? and let ? > 0. By Theorem 3.8(vi) there exists {f,}',1 С 8(C) with
?, П ?y = 0 for 1 ^ j and (J? Ei С ? such that ?? |м«(Е,-I > Ш(Е) - ?/2. As

1068
Т. V. Panchapagesan
Mflla(C) is <5(C)-regular by Theorem 3.8(v), there exists С, е С such that С, С ?, and
\??(??) - ?^@| < ?/B?) for/ = 1,2,..., п. Then by (i) we have
/1 /1
5^М1«|(с,-)>^|мй(С/)|
I I
1
- ? > |?«|(?) -?.
Hence, ?\?\(?) > |?^|(?). By (viii) and (ix) of Theorem 3.8, ?\?\(?) ^ ?°™(?) =
\??\(?) for ? e B( (?) ? Mff and, hence,
???? = |?»| ???,(?)??„. C.1)
Finally, if ? e Me, then by the proof of Theorem 3.8(vii) there exist А, В e ?(?) ? Mff
with Л С ? С S, A ?-compact, \??\(?\?) = 0 and \??\(?) = |?«|(?). Then by (ii)
and C.1) we have
\??\(?)= \??\(?) = ?]??(?) = ????(?) + ?1??(?\?) = ???\(?).
Hence, (iii) holds. ?
4. Regular extensions of positive and complex measures
Using the various notions of regularity given in Section 2 for positive and complex
measures defined on a <5-ring 7? containing 8(C) or ? (Co), we study the regular extensions
of positive and complex measures defined on ? (Co). The results of this section play a key
role in the sequel.
THEOREM 4.1. Let ?? be a finite (positive) measure on <5(Co). Then there exist unique
extensions ?, ? and ? of ?? to <5(C), BC(T) and B(T), respectively, such that ? is <5(C)-
regular, ? is Bc(T)-regular and ? is Radon-regular as measures. Moreover, ? = v\S{C) —
w\HC) and ? = w\BATy
PROOF. Let ?? also denote its unique extension to Bq(T) as measure. Then the existence
and uniqueness of the extensions ? and ? are due to Theorem 2.8. By Lemma 2.6,
? = v\&(C) an(J ?\?(? are <5(C)-regular extensions of ?? and hence by Theorems 2.8(i)
and 2.10(i), ? = v\S(C) = ?|«?- ВУ Lemma 2.6(ii) and by Theorems 2.8(i) and 2.10(i),
? = ?\???). ?
LEMMA 4.2. Let ?\ and ?? be finite (positive) measures on 8(Co) (respectively 8(C)-
regular measures on 5(C)). Then there exist unique extensions ?\, ?>2 and ац of ?\,
?? and ?? + ??, respectively, to 5(C) (respectively B(T)) as 8(C)-regular (respectively
Radon-regular) measures and ??, = ?>? + an.
?
> ^2\??(?:)\

Positive and complex Radon measures
1069
PROOF. Let ?3 = ? ? + ?? and let ?\, ?'-,, ?\ be the unique extensions of ? ?, ?? and ?^,
respectively, to Bo(T) (respectively ?, (?)) as measures. Clearly, ?', = ?, + ?'-,. Let
0j(f) = fjfdp'j, f e CC(T), for ; = 1, 2, 3. Then 03 = 0| + 0?. By Proposition 3.1,
M^ = M^°r + м?'г on B(T). Let ?, = ?^ if ?, is 5(C)-regular and Wj = ??><0 if M/
is defined on 5(Co). Then by Theorem 1.2, ?^? is Radon-regular and by Lemma 2.6(i),
Ме°Ъ(С) is <5(C)-regular. From the proofs of Theorems 2.8 and 2.9 it follows that Wj is a
unique extension of ? у with the desired properties of regularity, for j = 1, 2, 3. D
Lemma 4.3.
(i) If ? is a S(C)-regular real measure, let ц> = HatC»)- ^en vo = V+I^(C(,) an^
»0 = v'licC,,)-
(ii) Ifv\ and vi are S(C)-regular complex measures on 8(C) and if v\ \^c{)) — V-\&(C»)·
then v\ = vi.
PROOF, (i) Clearly, v? iC v+\S{Ci)) and v~ ^ v_|s<c„)- Let ?, = i^latC,,) and ?2 =
v~\s(C())- By Lemma 4.2 there exist unique <5(C)-regular extensions ?>(|, ?^, ?\ and ??
of v?, Vq , a)| and oil, respectively, such that ?\ + v^ = 2л + ^ on 5(C). By hypothesis
and by Proposition 2.11, v+ = lu\ and v~ = аи so that ? = a>T — ?? = i>(| — vj". Hence,
?+ ^ ?^ and ?~ ^ vj" and thus (i) holds.
(ii) Clearly, it suffices to prove the result for v\ and vi real. Let ?? = v\ \^co) = v2ls(C0)-
By Theorem 4.1 there exist unique <5(C)-regular extensions ?(| and ?~, of ?(| and ?^,
respectively, to 8(C). Now, by hypothesis and (i), v~[\&(C{)) = vtWc») = Mo so tnat ЬУ
the uniqueness part of Theorem 4.1 we conclude that vt = ?(| for j = 1,2. Similarly,
vj = ?^ for j = 1,2. Thus ?>? = vi. ?
THEOREM 4.4. Let ??, Ц) fee complex measures on 8(Cq), ?? bemg of bounded variation
on 8(Co). Г/геи:
(i) ?? ?<« ? unique extension ? to 8(C) as a 8(C)-regular complex measure, ? is real
(respectively positive) if vq is so.
(ii) The unique extension ?? of ?? to Bo(T) as a complex measure is B()(T)-regular.
(iii) ?? has a unique extension ? to BC(T) (respectively ? to B(T)) as a Bc(T)-regular
(respectively B(T)-regular) complex measure, ? and ? are real (respectively
positive) if ?? is so.
(iv) ? = ?|?(.(?) and ?5 = ?|?()(?) = ??„G>
(?) LetM =sup{u(W,,<5(C0))(?): ? e <S(C0)}. Then
sup{u(/j, TZ)(E): ЕеП} = М
for ? = ??\ ? or ? and К = B0(T), ВС(Т) or B(T), respectively.
PROOF, (i) Let ? ? = Re ?>? and v2 = Im ц>. By Theorem 4.1 there exist unique^ (C)-regular
extensions i>+ and vj of i>+ and v~, j = 1, 2 and let ? = (?* - vj") + /(?-? - ?-?)- By

1070
Т. V. Panchapagesan
Proposition 2.11, ? is <5(C)-regular, extends ?>? and is unique by Lemma 4.3. The rest of (i)
is obvious.
Let |??? = ?(??, ? (Co))- By Theorem 17.26 of Panchapagesan A991a) there exists a
unique extension ?? of ?? to Bq(T) as a complex measure of bounded variation. Moreover,
|??| = ?(??, ??(?)) extends |?0| to Bq(T) and
sup{|/75|(?): EeB0(T)} = M. D.1)
If ?\ = Re?o and ?? = ?????, then ?5 = (?+ -?~[) + '?(?? - ??) on ?()(?), where /? +
and /j" are the unique extensions of ?+ and ?~ to Bq(T) as measures for у = 1,2.
(ii) By Proposition 2.4(i), ^t and /j- are ??o(T)-regular for ; = 1, 2 and hence ?? is
SoG')-regular. _ _
(iii) By Theorem 2.8(H) there exist unique extensions a>; and yy of ^t and /j~,
respectively, to ?(?) as Radon-regular measures. For С е С, by Proposition 2.5 there
exists Co e C0 with С С Co so that w;(C) ^ ?, (Co) = /?y+(Co) ^ ?? by D.1) and
similarly, yy(C) ^ M. Consequently, wj(X) ?? ?? and Yj(X) ^ ? for ; = 1, 2. Then by
Propositions 2.4(iv) and 2.11, ? = (?\ — ?\) + /(?? - yi) is i3(T)-regular and extends
?? and ??- Moreover, (Reo))|^(C0) = ?\ and (ImoOlaiC,,) = >?2; ^?? and Ima> are B(T)-
regular. Let a and ? be also i3(T)-regular scalar extensions of ?\ and /ji, respectively.
Then, as Aта)|Во(Г) = 0, we have fTfd(lma)+ = fT f d(\ma)~, for / e CC(T) and
hence by Lemma 2.7(H), (Ima)+ = (Ima)" so that a is real. Similarly, ? is real. Clearly,
?+ +a~\S(c0) = ^7 +а+Ыс»)- Since a+ and a" are the unique Radon-regular extensions
of their respective restrictions to <5(Co), by Lemma 4.2 we have ?\ + a~ = ?\ + a+ and
hence a = ?\ — ?\ = Re ?. Similarly, ? = Ima> and hence ? is unique.
Taking ? = ?|?,G> by Lemma 2.6(H) we observe that ? is a Bt (T)-regular extension
of ??. As ?|^(? is <5(C)-regular by Lemma 2.6(i), the uniqueness of ? follows from
Lemma 4.3(H). From the above proof it is clear that ? and ? are real (respectively positive)
if ?? is so.
(iv) This follows from the uniqueness of ? and ? and from their definition.
(v) By applying the above extensions of |??| to BC(T) and B(T) we deduce the results
from D.1) from Proposition 2.5 and from Definition 2.2(iii).
The proof of the following theorem is similar to that of Theorem 4.4 and, hence,
omitted. ?
THEOREM 4.5. Let ? be a S(C)-regular complex measure of bounded variation with
sup{u(M, S(C))(E): ? e S(C)} = M. Then:
(i) The unique extension ? of ? to BC(T) as a complex measure is В'С(Т)-regularand
is real (respectively positive) if ? is so.
(?) ? has a unique extension ? to B(T) as a B(T)-regular complex measure and ? is
real (respectively positive) if ? is so. Moreover, ? = ?|?, G>
(iii) sup{v(?, BC(T))(E): ? e BC(T)} = sup{v(cu, B(T))(E): ? e B(T)} = ?.
COROLLARY 4.6. Every complex measure ?? on Bq(T) has a unique extension ? to
BC(T) (respectively ? to B(T)) as a Bc(T)-regular (respectively B(T)-regular) complex

Positive and complex Radon measures 1071
measure and ? (respectively ?) is real if ?? is real and ? (respectively ?) is positive if ??
is positive. Moreover,
sup{v(n,U)(E): ?eft}=sup{u(Mo,?o(O)(?): ? e B0(T)} < oo
where ? = ? or ? and ?? = BC(T) or B(T), respectively.
5. Bounded complex Radon measures
In this section we give several characterizations of a bounded linear functional ? e fC(T)*.
Definition 5.1. For ? e K,(T)*, let ||<9|| = sup{|<9(/)|: / e K,(T), \\f\\T ? 1}. ? e
K,(T)* is said to be bounded if ||#|| < oo.
Note that ? e K,(T)* is bounded if and only if ? e (CC(T), \\ ¦ \\T)* = (Со(Г), || ¦ ||r)*
where (Со(Г), || ¦ Ц7-) is the Banach space of all continuous complex functions vanishing
at infinity in T.
Notation 5.2. The linear subspace of all bounded ? e K,(T)* (respectively ? e
fC(T, R)*) is denoted by К,(Щ (respectively K,(T, Щ).
Definition 5.3. A complex Radon measure ?? on ? is said to be bounded if
sup{|/z»(E)|: EeS(C)}<oo.
Definition 5.4. For ? elC(T)* we define
\\??\\ = sup{v^e\S{C].8(C))(E): Ее 8(C)}.
Lemma 5.5. Let0efC(T)* and let E e 5(C()). Then
u(Mflla(C„), <HCo))(?) = ?(??,??)(?) = ?\?\(?).
In particular, ?(?? |5(C|)), &(Co))(E) = ?(??\?(?< <5(C))(?).
PROOF. Let ? = м^ад,) and let |v| = v(v, 5(C())). By Theorems 3.2 and 4.1 there exist
a unique Radon-regular extension |i>| of |i>| to B(T) and a positive linear functional ? on
Cc-CT) such that ? = ?^0?.
Let / e CC(T) with supp / с С e Co- Since / is integrable with respect to every
Baire measure (see p. 241 of Halmos A950)) and since ?()(?) ? С = B()(C), as in
the proof of Theorem 3.8(ix) we have |0(/)| = I fc /?(?«|?|)<?)? < fT \?\?(?^') =
iMI/l) and, hence, by A.1), \?\ sC ?. On the other hand, for ? e <S(C0), by
Theorem 3.9(iii) we have \v\(E) ^ ?(??, MH)(E) = ?\?\(?), so that д^ог|г(С„) ^Ml«lkc0)- If
? = (??^? - м^огIг(С(,)' then by Lemma 4.2 there exists a unique Radon-regular extension

1072 T.V. Panchapage.san
? of ? to B(T) such that ? + ?0^ = ?^? on B(T). Thus ?^?? «? ?${. Therefore, ? = \?\
and then by Theorem 3.9(iii) we have \v\{E) = ?^!(?) = ?\?\(?) = ?(??,??)(?) for
? e S(Co). The last part is due to Theorem 3.8(iv). ?
THEOREM 5.6. Let ?? be a complex Radon measure on T. Then the following assertions
are equivalent:
(i) ?? is bounded.
(ii) ? is bounded.
(iii) TZcMe, where Ti = B0(T) or BC(T) or B(T).
(iv) Mq is ? ?-algebra in T.
(v) sup{|Mfl(E)|: ? eTZ] < oo, where TZ= Bu(T) or BC(T) or B(T) or MH-
(vi) sup{u(Mff |тг, П)(Е): ? e ft} < oo, vWiere ?? = ?? or B(T) or BC(T) or B0(T)
or 8(C) or 8(C()).
(vii) sup{u(/iS|i(Co„a(C(,))(C): CeQ < эо-
(viii) ||?«|| < oo.
Moreover, \\?\\ = \\??\\ for ? e K,(T)*. The functional ? is bounded if and only if
Me = Мц* and when ? is bounded. \\??\\ is given by the supremum in (vi) with 11
being anyone of the the S-rings MH, B(T), BC(T), B0(T). 8(C) or ? ((X)) and also by the
supremum in (vii). In particular,
\\??\\ = ?{??\??),?(?))(?).
PROOF, ((i) =>· (ii)) Let feCc(T), with ||/||r ^ 1 and let supp/ = C. Then
\0(f)\ = I f /?(?„\???)\ < f \f\dvfaH\BlC),B(C))
\Jc I Jc
<f(M»li(C),i@)(C)<4sup{|/i«(?)|: ?e5(C)}
by B.3) and, hence, ? is bounded if ?? is bounded.
((ii) =f. (vii)) Let С e C(). By Proposition 2.5 there exists UQ eU Г) 5 (Co) such that
С С ?/0. Let / e С+(Г) with Xc ^ / ^ Xt;,,- Then ||/||r = 1 and by (ii) we have
v{fieWC),&(C))(C)< f |/??(?«|?(?,5@)
J и»
<f\f\d^) = \e\(f)^№\ E.1)
since ?(??\&??,?@)(?) = ?\?\(?) for ? e 5(C) by Theorems 3.8 and 3.9 and since
||0|| = PHI by Corollary 1 on p. 58 of Bourbaki A965a). If M() is the supremum in (vii),
thenbyE.1),Mo<||0||.
Since E.1) holds also for С еС, by (iv)and (v) of Theorem 3.8 we have
IIm«IKIH|.
E.2)

Positive and complex Radon measures
1073
Let ? = и(мй|а(С),5(С)) and let ? ? = $??{?(??\??· ??)(?): ? e К], where К is
one of the <5-rings in (vi). By Theorem 3.8(iv), ? = ?(??,??)\&<,?- For ? e Я(С). by
Proposition 2.5 there exists С e Co with ? С С and, hence, by Lemma 5.5 we have
?(?) ? Ш(С) = ?(??, Мв)(С) = u(/i«|i(Co„ S(C0))(C) whence it follows that MslC) =
Mq. By Corollary 4.6, MBo[T] - ??[?] and by Theorem 3.8(vii), MM„ = ??(?)- Since
?(??, ??) is ??^-regular by Theorem 3.8, we have Mm,, = M^C)-
The proof of the equivalence of the remaining assertions is easy and, hence, omitted.
From the proof of ((i)^(ii)) and from E.2) it follows that \\?\\ = ||?«|| forfl e ?,(?)*
(even thought ? is not bounded).
If ? is bounded, by (iv), ? e MH and by Theorem 3.9, Me = M\H\ = ??> . The condition
is also sufficient since Мц* is a ?-algebra in T. ?
6. Characterizations of complex Radon measures
Using the results of earlier sections we characterize the complex Radon measures in
terms of complex measures defined on S(Co) and in terms of those defined on 5(C)
which are further <5(C)-regular. Also we give another characterization which generalizes
Theorem 54.2 of McShane A944) to complex Radon measures. See introduction.
Lemma 6.1. Let ? е/С(Г)*, 0, = Re0 and fa. = Im0. Let ? = ?(?«|5,?,5@). Then:
(i) ? is S(C)-regular and if ? is the Radon-regular extension of ? to B(T), then
? — ubor
(n) B(T) f)M„ = {Ee B(T): v(E) < oo}.
(iii) Me = {E С ?: there exist А, В e B(T) ? Me such that А С ? С В, and
НВ\А) = 0]. _ _
(iv) If ? у = ??] \&ic) and ?+ and ?~ are the unique Radon-regular extensions of
?+ and ?~, respectively, to B(T) for j = 1,2, then ?«(?) = (?/" - ?~[)(?) +
;(?+ - M7)(?)/or ? е В(Т) П ??.
(?) If ? = {? e ?(?): 0(E) < oo), i/?e/j ?« ? ?/ге Lebesgue completion of ??\? with
respect to ??.
In short, ?? and ?? are uniquely determined by ?«|^(?-
PROOF, (i) By (iv) and (v) of Theorem 3.8, ? is <5(C)-regular. From the proof of
Theorem 3.9(iii) and from the Radon regularity of 0 and ?^4 it follows that 0 = ?^?.
(ii) Let TZ be as in (v). Then by (i) and by Theorem 3.9(iii), (ii) holds.
(iii) By Theorem 3.8(iv), v= \??\\&{?- ВУ the Me -regularity of \??\ and by the Radon-
regularity of 0 we have \??\{?) = v(E) for ? e B(T) ? ??- Then by Theorem 3.8(vii),
(iii) holds.
(iv) By (i) and (v) of Theorem 3.7, by the MH~-regularity of ??~ and by the MH—
I J J
regularity of ??-, we have ?| = ??~ and ?" = ??- in B(T) П Me, for j = 1, 2. Since

1074
Т. V. Panchapagesan
Mg = Me. П Me-,,
M+(?) = sup{Mt(C): С СЕ, С е С\ = sup^+(C): С С Е, С еС}
= ??+(?)
for ? е В(Т) П ??? and for ; = 1,2. Similarly, ??(?) = ?„-(?) for such ? and for
j = 1,2. Hence, (iv) holds.
(v) By (iv) and (vi) of Theorem 3.8, ?(??\?,?)(?) = ?(??,??)(?) for ? e К and
consequently, by (i) and Theorem 3.9, ?(??\??,??)(?) = 0(E) for ? ell. Now (v) is
immediate from (iii). ?
If ? = ?\ - 02, where ?\ and ?? are positive linear functionals on CC(T), then ?+ +?? =
?~+?\ and, hence, by Proposition 3.1 we have ?^? + ?^,? = ?^ + ?^ on ? (?). Hence,
we have the following
LEMMA 6.2. If 9\ and ?? are positive linear functionals on CC{T), then ?(?, _#2) =
??? - ??2 on 8(C).
LEMMA 6.3. Let ? be a 8(C)-regular real measure on 8(C). Let |?| = ?(?^(.0)). ?/\?\
is the unique Radon-regular extension of\?\ to B(T), let Tl= [E e B(T): |?|(?) < oo}.
Then:
(i) There exists a unique ? e K,(T)*, ? real, such that ?^? = ?- Moreover, ?+ =
??+Wc) and?~ = ?-н-Ыс)·
(ii) \?\ = ?\?\\&(?·
(in) \?\ = ?$.
(iv) ?? = ?(?)???. _
(?) Me is the Lebesgue completion ofTZ with respect to \?\\??-
(vi) If?+ and ?" are the unique Radon-regular extensions of?+ and ?~, respectively,
to B(T), then ??(?) = ?+(?) - ?-(?) for EjeTZ. _
Consequently, given F e ??, ??(?) = ??(?) = (?+ - ?~)(?) where А С F С В,
А, В е7г with |?|(?\?) = 0.
PROOF. Since ? is 3(C)-regular, by Proposition 2.11 the measures |?], ?^, andjjT are
«5(C)-regular. Then by Theorem 2.10(ii) the Radon-regular extensions |?|,?+ and ?" exist
uniquely on B(T).
(i) By Theorem^.2 there exist positive linear functionals ?\ and ?? on Cc (X) such that
?+ = дЬог and ?- = дЬог Let ? _ ?] __ ??_ Then by Lemma 6.2, ??\??? = ?- Then ЬУ
Theorems 3.6(v)and 3.7(v) we have ?+ = ?^+?^? an(J M~ = M«- Uo-
To prove the uniqueness of ?, if possible, let ? e /С(Г)* be such that ? = ??|^?·
Let ?? = Rew and ?? = Imw. Since ??,|5,? = 0, ??7\??? = ??-|«<0 Then ЬУ the

Positive and complex Radon measures
1075
uniqueness part of Theorem 2.10(H), we have ? °I = ? 1 so that
?2+(/> = f fdi^) = f fdi^) = ?2_(/) f0r / e Cc(T).
Thus W2 = 0. Hence, ? is real and ? = ??\&?? = (??- -MwOko = (?«- -Mfl-)ko·
Therefore, (??+ +Me-)b@ = (М^+ + ??-)??«7) and consequently, by Proposition 3.1 we
have ?\°++?~}\?^) = M^++ft,-)l5(C)- Thus, by the uniqueness part of Theorem 2.10(ii)
we conclude that ?!™' , = ?^ _, so that (?+ + 0~)(f) = (?+ + w~)(f) for
f e CC(T). Hence, ? = ? and thus ? is unique.
Since |?| = ?(??\&?), 8(C)) by (i), the assertions (ii) and (iii) hold by Lemma 6.1(i),
whereas (iv) follows from Lemma 6. l(ii). Similarly, (v) is immediate from Lemma 6. l(iii).
Finally, (vi) and the last part follow from (iv) and (v) of Lemma 6.1. ?
As a consequence of the above lemmas we give the following theorem, which among
other things, characterizes a complex Radon measure in terms of its restriction to 8(C).
THEOREM 6.4. (i) A complex measure ? on 8(C) is the restriction of a complex Radon
measure ?# if and only if ? is 8(C)-regular In such case, ? is unique and is called the
functional determined by ?.
(ii) Let ? be a 8(C)-regular complex measure on 8(C) and let \?\ = ?(?, 8(C)). Let \?\
be the unique Radon-regular extension ?[\?\ to B(T) and let 1Z = {E e B(T): \?\(?) <
oo}. Let ?? = Re ?, ?2 = ????, and ?+ and ?~ j be the Radon-regular extensions of
?| and ?^, respectively, to B(T) for j = 1,2. Moreover, by (i), let ? be the functional
determined by ?. Then:
(a) TZ = B(T)f)M„. _
(b) ? ? is the Lebesgue completion oflZ with respect to \?\\??- ^
(c) Given ? e ??, there exist А, В ell with А с ? с В such that |?|(?\?) = 0.
Moreover,
??(?) = (?+ - ?^)(?)+?(?? - ?!)(?).
Thus ?? is the Lebesgue completion of ?? |тг with respect to TZ.
(d) |?| = Д|0||з(С) and thus \?\ determines \?\.
(e) ? is real if and only if ? is real; ? is positive if and only if ? is positive. When ? is
real, ?+ and ?~ determine ?+ ????~, respectively.
PROOF, (i) The condition is necessary by Theorem 3.8(v). Conversely, let ? be 8(C)-
regular, with ?? = Re ? and ?? = ????. By hypothesis and by Proposition 2.11,?? and ??
are <5(C)-regular. Consequently, by Lemma 6.3 there exists 0,- e fC(T)*, 0,- real, such that
My = M0/ls(C) for; = 1,2. Let0 = 0| +???. Then ? eK,(T)* and ? = ?^?^?- Clearly, ?
is unique by the uniqueness part of Lemma 6.3(i).
(ii)(a) As ? = ??\&?), the result follows from Lemma 6. l(ii).
(b) This is the same as Lemma 6. l(iii).

1076
Т. V. Panchapagesaii
(c) This follows from (iv) and (v) of Lemma 6.1.
(d) By Theorems 3.8(iv) and 3.9(iii), |?| = ?(??, Me)\S{C] = MioikC) and< hence,
(d) holds.
(e) This is immediate from Lemma 6.3(i). ?
The following result generalizes Theorem 54.2 of McShane A944) to complex Radon
measures. The hypothesis that B(T) ? V = [? e ? (?): \v\(E) < oo} is the same as
B(T) П V = {? e B(T): \v\*{E) < oo} when ? = R" and in this case, the hypothesis
of P-regularity of ? in Theorem 6.5 is redundant.
THEOREM 6.5. Let ? be a 8-ring containing 8(C) and let ? be a V-regular complex
measure on V. Let ? = ?^? and let ? = ?(?· ? (С)). Then \v\ is S(C)-regular. Suppose
? = ?(?)?)? = {? eB(T): M(?)<oo},
where \v\ is the unique Radon-regular extension of \v\ to B(T). If ? and V are the
Lebesgue completions of ?\? and 1Z, respectively, with respect to ?? and ?\??, then there
exists a unique ? e fC(T)* such that ? = ?? and V = Me- Moreover, ? is real {respectively
? is positive) if ? is real {respectively if ? is positive).
PROOF. Let \?\ = ?{?, V). By hypothesis and by Proposition 2.11, \?\ is D-regular and
as 8(C) С Р, the argument given in the proof of Theorem 3.6(vi) holds here to show that
ЫкС) is «5(C)-regular.
Since ? is D-regular, as in Theorem 3.6(vii)(c) we have
?{?,&@,?)(?)=\?\(?)
for ? e V. Consequently, |i>| = ????^? and, hence, |?| and ? are <5(C)-regular. Then
M(?)=sup{M(C): CCE, С eC} = sup{|M|(C): СсЕ, СеС} = |д|(?)
for ? e П and hence М1тг = 1м11тг- Since 8(C) С К С V and since ?(?, 8(C), V)(E) =
\?\(?) for ? e ?, it follows that ?(?\?· K)(E) = \?\(?) for ? e TL. Hence, ?(?\?. Щ =
Since ? is 5(C)-regular, by Theorem 6.4(i) there exists a unique ? e K,(T)* such that
v = М«1г(С) and by (ii)(e) of the said theorem, the functional ? is real (respectively
positive) if ? is real (respectively positive). Moreover, by (ii)(d) of the said theorem,
? = M|fl|ls(C)- Then by hypothesis, by (a) and (b) of Theorem 6.4(ii) and by the fact that
М1тг= 1м11тг = ?(?\??,11), we conclude that V = MH-
Let ?? = Re ?, ?? = ????, ?\ = Re ? and v2 = Imv. Since ?? and ?? are P-regular,
by following an argument similar to that in the proof of Theorem 3.6(v) we note that
М/Ъ(С) = VJ and ?~\^? = vj for j = 1,2. Moreover, as 1/xllstC) is ^(C)-regular,
by Proposition 2.11 ?, and ?+ and v~ are 5(C)-regular for j = 1,2. Let vj~ and vj
be the unique Radon-regular extensions of v+ and vj to B(T) for j = 1,2. Then by

Positive and complex Radon measures
1077
Proposition 2.4(iv), by the hypothesis that ? is P-regular and by Proposition 2.11, we
have
it(?)= sup{vt(C): СсЕ, С eC}= sup{?j(C)¦. Cc?, €&0) = ?^{?)
for ? e ft and for ; = 1,2. Similarly, v~ (?) = mJ(?) for ? e ft and for j = 1.2.
Consequently, given ? e V = ??, by Theorem 6.4(ii)(c) there exist A, S_e ftjvith А с
? С S and M(B\A) = |?|(?\?) = 0. Then ??(?) = (vj - vf )(A) + /(?2+ - vJ)(A) =
(?+ -?~)(?) + ?(?? - ?7)(?) = ?(?) = ?(?) since by hypothesis ? is the Lebesgue
completion of ?\? with respect to ft. Hence, ?? = ?. ?
In the following theorem we characterize a complex Radon measure in terms of its
restriction to 8(Cq).
THEOREM 6.6. (i) A complex measure ? on ? (Co) is the restriction of a complex Radon
measure ?? and such ? e fC(T)* is unique. We say that ? is the functional determined by
vifv = Mfl|s(c0); ? is real {respectively positive) if ? is real {respectively positive).
(ii) If ? is a complex measure on S(Co) and ? is the unique extension of ? to 8(C) as
a S(C)-regular complex measure, then ? and ? determine the same functional ? e K,(T)*.
That is, v = ??\&@?)) and ? = ?.?\&??
In the following, let ?, ? and ? be as in (ii). Let v\ = Re ?, ?: = Im ?, ?\ = Re ? and
?? — ????. Then:
(iii) The unique S(C)-regular extensions v+ and v~ of v^ and vj, respectively, to S(C)
are given by v^ = ?+ and v~ = ?~, j = 1, 2.
(iv) // ? is real and \v\ = v(v, 8(Си)), then v+ = ?+?^,,), v~ = M~li(C„) and \v\ =
?(?, 8(C))\s[C„)- Consequently, v+, v~ and \v\ determine ?+, ?~ and \?\, respectively.
(?) If ? is complex, then \v\ = ?(?. 8(C))\S{c„) =M|«|l5(C0) sothat\v\ determines Щ.
PROOF. By Theorem 4.4(i) there exists a unique 8(C)-regular complex measure ? on 8(C)
such that ?\???) = ? and such ? is real (respectively positive) if ? is real (respectively
positive). Then by Theorem 6.4(i) there exists a unique ? e K,(T)* such that ? =
??\&(? and hence, ? = ??\&@»)- This functional is real (respectively positive) if v is
real (respectively ? is positive) by Theorems 4.4(i) and 6.4(ii)(e). Since ? determines ?
uniquely, it follows that ? determines ? uniquely. Thus we have proved (i) and (ii).
(iii) By Lemma 4.3(i), vj = m||5(C|)) and vj = ?]\&(?0) and as ?j and ?~ are 8(C)-
regular, by the uniqueness part of Theorem 4.1 we conclude that vj = ?j and vj = ?~
for; =1,2.
(iv) Let ? be real. Then by Theorem 4.4(i), ? is real and Lemma 4.3(i) implies
v+ = ?+?^,,) and v~ = ?" |г(С()). Consequently, by Theorem 6.4(ii)(e) the result holds.
(v) This is immediate from Lemma 5.5. ?
7. Isomorphic representations of fC(T)*, K,(T. R)% K(T)*7 and K,(T, Щ
Making use of the results of the last section we show that fC(T)* is isomorphic to the
complex vector space of all complex measures on 8(Co) and to that of all <5(C)-regular

1078
Т. V. Panchapagesan
complex measures on 8(C). The same isomorphism, when restricted to /CG\ R)*, is order
preserving and maps K,(T, R)* onto the real vector space of all real measures on 8(Cq) and
that of all <5(C)-regular real measures on 8(C). Using the results of Section 5 we also show
that IC(T)l (respectively /С(Г, R)?) (see Definition 5.1) is isometrically isomorphic to the
Banach space of all bounded complex (respectively real) measures on 8(Cq) and to that of
all bounded 8 (C)-regular complex (respectively real) measures on 8(C). Finally, the vector
space of all complex valued additive set functions of finite (respectively bounded) variation
on a ring of sets is shown to be isomorphic (respectively isometrically isomorphic) to
K,(T)* (respectively to K,(T)*) for a suitably chosen totally disconnected locally compact
Hausdorff space T.
Before giving the relevant theorems we fix the notation of various spaces of real and
complex measures.
Notation 7.1. In the following vector spaces, the operations of addition and scalar
multiplication are defined setwise. Mo(T) (respectively MAT)) denotes the vector space
of all complex (respectively <5(C)-regular complex) measures on 8 (Co) (respectively on
8(C)). M(T) (respectively MAT)) is the vector space of all ??(r)-regularjrespectively
i3c(r)-regular) complex measures on B(T) (respectively on BC(T)) and MAT) is the
vector space of all complex measures on BAT)- Mq(T)i, = {? e MAT): ? bounded) and
Mc(T)b = {? e MAT): ? bounded). The real vector spaces MrQ(T), M[.(T), M(T, R),
MAT,R), M0(T,R), Mr0(T)h, M[(T)h are the subspaces of the corresponding real
measures in MAT), MAT) etc., respectively.
THEOREM 7.2. Let ? :MAT) -> K,(T)* be given by ?(?) = ? if ?- ?.<*\&(? and
?0:???)-> K(T)* be given by ?0(?) = ? if ? = ?? |г(С()). Then:
(i) ? and ?? are well defined and are linear isomorphisms onto fC(T)*.
(ii) If ?(?) = ? (respectively ??(?) = ?), then ?(?(?,?@))) = \?\ (respectively
Фо(и(М(Со))) = |0|).
(iii) Let Ф(г) = Ф\м[\Т) and Ф^'1 = ??)\?\[?)· Then ??? (respectively ?^'}) is
an order preserving linear isomorphism of M[(T) (respectively of M'0(T))
onto /CG\R)*, where ?, < ?2 if ?, (?) iC ?2(?) for ? e 8(C) (respectively
for ? e 8(C0)). In particular, if ?"Vy = 0,-, ?7 e M[.(T), for 7 = 1,2, then
???(?\ ??2) = ?\ ??? where
??,????)= sup {?^,(?) + ?«,(?\?)}
FcE. F€S{C)
????(?)(?\ л ??) = ?\ ???, where
??]????)= inf \?? (F) + ????\?)\
FcE. Fe,5(CI ' '
for ? e 8(C). A similar result holds if ?\ and ?2 belong to MrQ(T).
(iv) MAT) (respectively MAT)) is the dual of)C(T) and
0{f) = f fdivLehiO). /е/С(Г), G.1)

Positive and complex Radon measures
1079
respectively
0(f) = j fd^e\b{Ct))\ /еЩ), G.2)
where ?(?) = ? (respectively ??(?) = ?).
(?) ?[.(?) (respectively M'Q(T)) is the dual offC(T. R) and an expression similar to
G.1) (respectively G.2)) holds ?/???(?) = ? (respectively ?^\?) = ?).
PROOF, (i) By Theorems 6.4(i) and 6.6(i), ? and ?0 are well defined. If ?(?\) =
?(?2) = ?, then ?? = ??\S(c) = ?3 and, hence, ? and similarly, ?? are injective. Making
use of Proposition 3.1 and Theorem 6.4(ii)(e) it can be shown that ? and ?? are linear. ?
is an onto mapping by Theorem 3.8(v), while ?? is evidently an onto mapping.
(ii) This follows from Theorem 6.4(ii)(d) for ? and from Theorem 6.6(v) for ??.
(iii) Ф(г) is order preserving by Theorem 6.4(ii)(e) and Proposition 3.1 and Ф0'' is
order preserving by Theorem 6.6(i) and Proposition 3.1. The rest of (iii) is an immediate
consequence of the order preserving property of these isomorphisms Ф"'1 and Ф() .
(iv) As in the proof of Theorem 3.7(iv) we have
0(f) = / ??(??]\?[€)) + ? / /?(?(,,\???)= / fd(jie\StC))
for / e CC(T) with supp/ = С where ?\ = Re# and ?? = 1тв. Thus G.1) holds. Since ?
is an onto isomorphism, the result holds for MC(T). Similarly, the result holds for Mq(T)
and ??.
(?) The proof is similar to that of (iv). ?
Theorem 7.3. Let S =J3c(T)Jjespectively B(T), B0(T), 8(C), S(C0)) and let M(S) =
MC(T) (respectively M(T), M0(T), Mc(T)b, M„(T)b). For ? e M(S), let \\?\\ =
sup{u^, ?)(?): ? eS]. Then:
(i) The mapping Ф^гЛК^) -> K,(T)l given by ?^(?) = ? if ? = ??\? 's weH
defined, and is an isomorphism onto ?.(?)^ and ||?5(?)|| = ||?|| for ? e M(S)
so that ?5 is an isometric isomorphism.
(ii) Each one of the spaces (M(S). || · ||) is the dualof(Cc(T), || · ||r) (or equivalently,
of(Co(T), || · ||7-)) and consequently, (M(S), || · ||) are Banach spaces.
(iii) Results similar to (i) and (ii) hold if K,(T)l and M(S) are replaced by K,(T. R)*?
and M'(S), respectively, where M! (S) = {? e M(S): ? real].
PROOF, (i) Let ??,?? e M(S) and let ?, ? e C. Clearly, ??? + ??? e M(S) and
(?? ? + ??2)\????) =?·?\ \Slc()) + ? ¦ М21г(С„)· Thus- by the uniqueness part of the various
assertions in Theorem 4.4 we conclude that Mo(T)i, is the image under a linear onto
isomorphism Г5: M(S) -> Mo(T)i, given by
rs(M) = MU(C„), ?^?(?).

1080
T.V. Panchapagesaii
Let ??(?) = (?? ° ^s)(m) for ? e A1(<S). where ?? is as in Theorem 7.2. Clearly,
?5 is a linear isomorphism of M(S) onto its image in fC(X)*. If ?^??) = ?, then
??(???(?0)) = ? and by hypothesis, sup{^(?)|: ? e <5(Co)} < 00. Consequently, by the
equivalence of (ii) and (vi) of Theorem 5.6, ? is bounded and hence <I>s(M(S)) С fC(T)fy
Conversely, if ? e K,(T)*} then by Theorem 5.6 ?? is bounded in ?? and ?? D B(T).
Consequently, ??\s belongs to M(S) and <Ps(Ph\s) = ?, so that <t>s(M(S)) = K,(T)*r
By the last part of Theorem 5.6, \\<Psiv)\\ = HmII for ? e M(S).
(ii) This is immediate from (i) and from the fact that K.(T)*} is the dual of (CC(T), || · ||r)·
(iii) The proof is similar to the earlier parts. ?
The following theorem is based on the Stone representation theorem for Boolean rings
and Theorems 7.2 and 7.3.
THEOREM 7.4. Let ? be a non void set and let TZ be a ring of subsets of ?.
Let M-jz (respectively (M-ji)b) be the vector space of all complex valued finitely
additive set functions of finite (respectively bounded) variation on TZ and let \\?\\ =
sup{u(M, "?)(?): ? eTZ] for ? e (Mn)b- Let M'n {respectively (Мхп)ь) be the
subspace of real valued set functions in M-r (respectively (M-jz)b)· Then there exists
a totally disconnected locally compact Hausdorff space ? such that M-jz is isomorphic
to K,(T)*; Mrn is order isomorphic to К,(Т.Ж)* and (Mn)b (respectively (M'n)b)
is isometrically isomorphic (respectively isometrically order isomorphic) to K,(T)*7
(respectively to К.(Т,Ш)*7). When TZ is an algebra in ?, the space ? can further be
assumed to be compact.
PROOF. By the Stone representation theorem for Boolean rings (see Dinculeanu A967,
§ 18)), there exists a totally disconnected locally compact Hausdorff space ? such that
TZ is ring-isomorphic to the ring TZ of all compact-open subsets of T. Let ? be such an
isomorphism from TZ onto TZ.
Let С e Co of T. Then by Proposition 2.5 there exist U„ e U П 5(Co) such that
С = П^° U„. Since the members of TZ form a base for the topology of Г, each U„ is of
the form Un = (J · ? (A,^), Anj eTZ. A.s С is compact, there exist A„y;, / = 1. 2,... ,k„,
in TZ such that С с U; = i ф^А>ч,)- If An = Ufl, Anji. then С = ПГ ф(А») and' hence'
С e ?(??). Since TZ С Co, it follows that S(C0) = S(TZ).
For ? e M-jz, let ?(?)(?) = ?(?-?(?)) for ? e TZ. Sincej?~'@) = 0 and since
each countable disjoint union {En}f in TZ with (J ? E„ = ? eTZ has E„ = 0 for all but
a finite number of и, it follows that ? = ? (?) is a complex measure on TZ and hence
admits a unique extension 0 to S(TZ) — <5(Cq) as a complex measure. Conversely, given a
complex measure ? on <5(Co), let ?(?~ '(?)) = v(E) for ? eTZ. Clearly, ? is well defined
on 7? and is a complex valued finitely additive set function on TZ. Since i>|^ is of finite
variation, ? e М-ц. Moreover, ? (?) = v\^. Also, the mapping ?: M-r -»· Mq(T) given
by ?(?) = {?(?)} is linear and bijective, where Mq(T) is as in Notation 7.1 and {?(?)}
is the unique countably additive extension of ? (?) to S(Co) = S(TZ). Consequently,
by Theorem 7.2(iv) M-r is isomorphic to K,(T)*. The other results follow on similar
lines. ?

Positive and complex Radon measures
1081
8. Applications
In this section we discuss some results obtained in Panchapagesan A998a, 1998b) as
applications of Theorem 7.3. Also we include a generalization of Theorem 4.4 to locally
convex Hausdorff space-valued (briefly, IcHs-valued) Baire measures, whose proof as
given in Panchapagesan A998b) is motivated by the proof of the former theorem.
In the sequel, ? (?) denotes the dual of (Со(Г). || · ||r) and is identified with the space
of all bounded complex Radon measures on Г - i.e., with the space K(T)*r In virtue of
Theorem 7.3 it is also identified with the Banach space of all (bounded) ??G>regular
complex measures ? on B(T) with ||?|| = ?(?. ?(?))(?).
Let К be a metrizable compact Hausdorff space. Proposition 8 of Dieudonne A951)
states that a sequence {?,,}^ in ? (?) is weakly convergent if and only if {?„(?/)} j* is
convergent in С for each open set U in K.
Later, for a bounded subset A of ? (?), Grothendieck A953, Theorem 2) gave several
characterizations for A to be relatively weakly compact. One of them is that, for each
disjoint sequence (i/,)^ of open sets in ?, lim, ?(?/,) = 0 uniformly in ? in A. As a
corollary of this characterization, he obtained the following generalization of Dieudonne's
result to (non-metrizable) locally compact Hausdorff spaces T.
THEOREM 8.1 (Grothendieck, 1953). A sequence {?,,}^ '" ? (?) is weakly convergent
in M(T) if and only if hm,, ?„(?/) e С for each open set U in T.
The hypothesis that lim„ ?„(?/) e С for each open set U in ? implies that the set {?,,} j*
is bounded in M(T).
The above generalization of Grothendieck cannot be considered optimum for the
following reason. In a metrizable compact space K, each open set is a Baire set and hence
an optimum generalization of Dieudonne's result would be the following:
A sequence {?„ }^ in M{T) is weakly convergent in M(T) if and only if lim„ ?„ (U) e С
for each open Baire set U in T.
Such a generalization with the additional hypothesis of boundedness of {?,,}^ is
obtained in Corollary 1 of Panchapagesan A998a). For this, Theorem 7.3 is used to give
a Baire version of Theorem 2 of Grothendieck A953). In fact, the following result is
obtained.
Theorem 8.2 (Theorems 1 and 2 of Panchapagesan A998a) - Baire and ?-Borel versions
of Theorem 2 of Grothendieck A953) and of Lemma VI.2.13 of Diestel and Uhl A977)).
Let ? be a locally compact Hausdorff space and let M(T) be the dual о/(Со(Г), || ¦ Цг)·
Let A be a bounded set in M(T). Then the following .statements are equivalent:
(i) A is relatively weakly compact.
(ii) For each disjoint sequence {i/,} of open Baire (respectively (ii)' ?-Borel) sets in T,
1????(?/,) = 0
uniformly in ? e A.

1082 T.V. Panchapagesan
(iii) For each disjoint sequence {i/,}^ of open Baire (respectively (iii)' ?-Borel) sets
inT,
limu(M|7e,ft)(t/,) = 0
I
uniformly in ? e A where H— Bq(T) (respectively BC(T)).
(iv) (a) Given ? > 0 and an open Baire (respectively (a)' ?-Borel) set U in T, there
exists a compact Gs С (respectively a compact C) such that С С Е and
supu(M|7e,ft)(t/\C)<f
?€?
where ??= Bq(T) (respectively BC(T)).
(b) For each ? > 0, there exists a compact Gs С (respectively (b)' compact C)
in ? such that
sup sup ?(?\??,??)(?) < ?
??? E€Tl.EcT\C
where 1Z= Bq(T) (respectively BC(T)).
(v) Given ? e B$(T) (respectively (\')\ Given ? e BC(T)) and ? > 0, there exists
a compact Gs С (respectively a compact C) in ? such that С С Е and
suPMeA ?(?\??, Tl)(E\C) < ? where K= B0(T) (respectively BC(T)). In other
words, ?\??(?) (respectively ?\?^?)) is uniformly Baire (respectively ?-Borel)
inner regular.
(vi) Given a sequence E„ \ 0 in Bq(T) (respectively (vi'): in BC(T)) and an ? > 0,
there exists no such that
sup ?(?\??, П)(Е„) <?
?€?
for ? ^ no, where ?? = Bq(T) (respectively BC(T)). In other words, ?\?{^?)
(respectively ?\???)) is uniformly countably additive.
(vii) Given ? e Bo(T) (respectively ? e BC(T)) and ? > 0, there exist a compact Gs
С and an open Baire set U in ? (respectively a compact С and an open ?-Borel
set U in T) such that С С Е С U and
supu(Ml7e,ft)(t/\C)<e
?€?
where ?? = Bo(T) (respectively BC(T)). In other words, ?\?{){?) (respectively
?\?,.(?)) is uniformly Baire (respectively ? -Borel) regular.
Using the equivalence of (i) and (ii) of Theorem 8.2 and following an argument similar
to that on p. 150 of Grothendieck A953) we have the following optimum generalization of
Dieudonne's proposition.

Positive and complex Radon measures
1083
COROLLARY 8.3 (Corollary 1 of Panchapagesan A998a) - Generalization of Proposition 8
ofDieudonneA951)). A bounded sequence {щ}^ in M(T) is weakly convergent in M(T)
if and only //lim, ?,(U) e С for each open Baire set U in T.
The above result is really an optimum generalization of the said result of Dieudonne and
is more natural and stronger than Theorem 8.1. Moreover, the boundedness of {?,}]'0 can
also be dropped in the above corollary. See Remark 9.18 below.
Using the strict Dunford-Pettis property (briefly, SDP-property) of Со(Г) and other
deep results, Grothendieck A953) characterized weakly compact operators on C(K)
whose range is contained in a complete IcHs. In fact, he obtained the following
THEOREM 8.4(Theorem6ofGrothendieckA953)). Let К be a compact Hausdorff space
and let X be a complete IcHs. Let u:C(K) ^> X be a continuous linear mapping. Then
the following statements are equivalent:
(i) и is weakly compact.
(ii) For each closed set F in К, и**(ху) e X, where u** is the biadjoint of и and the
dual X* ofX is endowed with the strong topology ?{?*, ?).
(iii) For each closed Gs С in К, н**(хс) e X-
(iv) For each increasing sequence (f,,)^ С C(K), with 0 < /„ < 1, (uf,)^0 converges
weakly in X.
Then Remark 2 on p. 161 of Grothendieck A953) says that without any difficulty the
techniques developed in earlier sections can be used to show that the enunciation of the
above theorem is textually valid for Со(Г), with ? locally compact and Hausdorff.
Later, Edwards A965) gave the details of the proof of the said remark, using the
techniques of Grothendieck A953). But, unfortunately his proof is incorrect and as
observed in Panchapagesan B000) Grothendieck's techniques hold if and only if ? is
further ?-compact.
On the other hand, Bartle-Dunford-Schwartz A955) developed a theory of integration
of scalar functions with respect to a countably additive Banach space-valued vector
measure and using it, they gave an alternative method to study weakly compact operators и
on C(K), К a compact Hausdorff space, when the range of и is contained in a Banach
space, and obtained an integral representation of the operators u. When и are weakly
compact, we call this the Bartle-Dunford-Schwartz representation theorem for u. They
also deduced the SDP-property of C(K) from the integral representation of и.
The Bartle-Dunford-Schwartz representation theorem is generalized in Panchapagesan
A998b) as follows.
Theorem 8.5 (Theorem 1 of Panchapagesan A998b) - Generalized Bartle-Dunford-
Schwartz representation theorem). Let X be an IcHs and let u:Co(T) —> X be a
continuous linear mapping. Then there exists an X**-valued vector measure (i.e., a vector
valued additive set function) m on B(T) with the following properties:
(i) x*m e M(T) for each x* e X*. and consequently m :B(T) -> X** is countably
additive in the ?(?**, X*)-topology.

1084
Т. V. Panchapage san
(ii) The mapping x* —* x*m of x* into M(T) is weak*-weak* continuous.
(Hi) x*uf= fTfdx*m for each f e CQ(T) and x* e X*.
(iv) {m{E): ? e B(T)} is ze-bounded in X**, when X** is endowed with the locally
convex topology ze of uniform convergence in equicontinuous subsets of X*.
Conversely, let m : B(T) -> X** be a vector measure satisfying (i) and (ii). Then there
exists a unique continuous linear map и: Co(T) -> X such that (iii) holds. Moreover,
m(E) = u**(xE) for ? e B(T) and m satisfies (iv).
Finally, the vector measure m satisfying (i)-(iii) is uniquely determined by the
continuous linear map и and is called the representing measure of и and m has ze -bounded
range in X**.
When X is quasicomplete, uf = fT f dm, for / e Со(Г), where the integral is defined
in the sense of Definition 3 of Panchapagesan A998b).
When X is quasicomplete and и is weakly compact, then uf = fT f dm for / e Co(T),
the vector measure m has range in X, and is countably additive and Borel-regular (in the
sense of Definition 5 of A998b)).
Remark 8.6. Clearly, the above theorem generalizes the Riesz representation theorem for
positive linear forms on Cq(T) to quasicomplete IcHs-valued weakly compact operators on
C0(T).
Using Theorems 8.2 and 8.5, 35 characterizations are given in Panchapagesan A998b)
for a quasicomplete IcHs-valued continuous linear map и on Со(Г) to be weakly compact.
These characterizations include those in Remark 2 of Grothendieck A953). In particular,
we have the following theorem, which subsumes the said remark.
THEOREM 8.7 (Theorem 3 of Panchapagesan A998b)). Let X be a quasicomplete IcHs.
Let и :Co(T) -> X be a continuous linear mapping. Then the following statements are
equivalent:
(i) и is weakly compact.
(ii) u**(xu) e Xfor all open sets U in T.
(iii) h**(xf) e X for all closed sets F in T.
(iv) u**(xu) & Xfor all open sets U belonging to BC(T).
(v) u**(xu) e Xfor all open Baire set U in T.
(vi) u**(xu) e X for all open sets U in ? which area-compact.
(vii) m**(xf) e X for all closed sets F in ? which are Gs-
(viii) u**(xu) e X for all open sets U in ? which are a countable union of closed sets
inT.
(ix) For each increasing sequence (f,)^ С Cq(T), with 0 < /„ < 1, (uf,) converges
weakly in X.
In Sections 2 and 4 we used the Riesz representation theorem for a positive linear
functional ? on CC(T), to prove the existence and uniqueness of ??(r)-regular(respectively
??f(r)-regular) extension of a complex measure on Bq(T) (see Theorem 4.4). In virtue of
Remark 8.6 the following question arises:

Positive and complex Radon measures
1085
As in the scalar case, can the integral representation of a weakly compact operator be
used to generalize Theorem 4.4 to X-valued countably additive vector measures ?? on
Bq(T), where X is a quasicomplete IcHs?
The question is answered in the affirmative in Theorem 10 of Panchapagesan A998b).
First the concept of 7?-regularity is generalized to vector measures, where TZ is a <5-ring
of sets in T, containing С or C0 (see Definition 5 of Panchapagesan A998b)). When 1Z is
Bq(T) (respectively BC(T), B{T)) we use the terminology of Baire regularity (respectively
?-Borel regularity, Borel regularity). Then using the integral representation theorem
(Theorem 8.5) and other theorems in Panchapagesan A998b), the following generalization
of Theorem 4.4 is proved.
Theorem 8.8 (Theorem 10 of Panchapagesan A998b)). Let ? be a locally compact
Hausdorff space and let X be an IcHs. Then each X-valued countably additive vector
measure ?? on Bq(T) is Baire-regular. If X is further quasicomplete, then there exists a
unique X-valued countably additive Borel-regular (respectively ?-Borel regular) vector
measure ? on B(T) (respectively ?, on B,(T)) such that ?|??)(?) = Mo (respectively
Мс-1в(,(Г) = Mo)· Moreover, ?? = ?\? (?).
Remark 8.9. The above result has already been proved by different methods. For
example, see Dinculeanu and Kluvanek A967). Kluvanek A967). Sion A969), Dinculeanu
and Lewis A970), etc. However, recently a new proof, which is simpler than all the known
proofs, has been given in Dobrakov and Panchapagesan B002).
Let us recall from the definition on p. 634 of Edwards A965) that a Banach space ?
is said to have the SDP-property if, for each weakly compact operator и : ? —> ?, X а
quasicomplete IcHs, и transforms each weak Cauchy sequence in ? into a convergent
sequence in X.
As remarked earlier, Grothendieck A953) showed that Cq(T) has SDP-property. Using
Theorems 8.5 and 8.7, an alternative proof is given in Panchapagesan A998b) (see
Theorem 11 there).
Pelczynski A960) showed that every continuous linear mapping u:C(K) —> ? is
weakly compact if ? is a Banach space with со <? E. This result was improved
in Pelczynski A962) where he proved that every unconditionally convergent operator
u:C(K) -> E,E an arbitrary Banach space, is weakly compact. (A continuous linear
map u: ? -> F, E, F, Banach spaces, is said to be unconditionally convergent if,
for every sequence (x,,)^ of vectors in ? with Y^ |**(л-„)| < oo for each x* e X*,
?^° u(x„) is unconditionally convergent in F.) Using the technique of reduction to
metrizable compact space and Theorem 8.4, Thomas A970) extended the former result
of Pelczynski to a continuous linear map и :Со(Г) -> X, X a quasicomplete IcHs with
со <t X- Using Theorems 8.5 and 8.7 and dispensing with the technique of reduction to
metrizable compact space, both the results of Pelczynski are generalized to continuous
linear mappings on Co(T) with range in a quasicomplete IcHs. See Theorems 12 and 13 of
Panchapagesan A998b). For an alternative simple proof see Remark 9.18 below.

1086
Т. V. Panchapagesan
9. Generalization to Radon vector measures
Thomas A970) developed a theory of Radon vector measures analogous to the theory of
essentially integrable scalar functions given in Chapter V of Bourbaki A965b). In this
section we briefly discuss the results announced in Panchapagesan A995/96) along with
others which generalize some of the results in Section 7 to these measures. A more detailed
complete exposition of these results including others will be published elsewhere. (See
Remark 9.18 below.)
Definition 9.1. Let X be a quasicomplete IcHs and let Г be the family of all continuous
seminorms on X. A linear mapping и : ?,(?) -> X is called an X-valued Radon operator
(or simply, a Radon operator when there is no ambiguity about the range space X) if, given
a compact set С in ? and ? e ?, there exists a (finite) constant Mc.p > 0 such that
р(иф)^Мс.р\\Ф\\т
for all ? e fC(T) with supp<? с С. (Though Thomas A970) developed his theory for
K,{T, E), it is equally valid for fC(T) also.)
Let T+(T) and M + (T) be as in Theorem 1.2. As ? is fixed, we shall denote them by
T+ and M + , respectively.
Following Thomas A970) we give the following definitions.
Definition 9.2. Let и : ?,(?) -^Ibea Radon operator. Let / e T+ and let ? e ?. If
/ e M+, we define
"*,(/)= sup р(иф).
\фК/.фе)С(Т)
If / has compact support, then we define
«·(/)= inf u'(g).
Finally, if / is arbitrary in T+, we define
"*(/) = sup{i/*(/2): 0< h ^ /, supp/г compact}.
For ? С Г, we define и* (Е) = и* (??·).
For / e K,(T), it is easy to show that и* (|/|) < oo for all ? e Г. This fact plays a key
role in the following:
Definition 9.3. Let u:K,(T) ->· X be a Radon operator. Let T°(u) = [f:T -> C,
"" (l/l) < oo for each pef). Then /С(Г) с Т°. T° is endowed with the locally convex
topology induced by the seminorms {и* ? e ?}. The closure of K,(T) in J"° for this
topology is denoted by C\(u) and all members of C\ (и) are said to be и-integrable.

Positive and complex Radon measures
1087
THEOREM 9.4. C\{u) is a vector space containing K,(T). C\(u) = {/ e T°(u): given
? > 0 and ? e ?, there exists ? e K,(T) such thatu'(\f — ?\) < ?].
Since р{иф) ^ и* A01) for ? e ?,(?) and ? e ?, it follows that и is continuous in
K,(T) also for the topology of C\ (u). Thus и admits a unique continuous linear extension
й to the whole of C\ (и) with values in X, the IcHs completion of X. However, as X is
quasicomplete, by Theorem 1.35 of Thomas A970) uf е X, for / eC\(u).
Definition 9.5. Given a Radon operator и : ?,(?) -н» X, for each / e ?| (и), the value
of the integral of / in ? is denoted by ii (/).
Remark 9.6. An X-valued Radon operator is referred to as Radon measure with values
in X in Thomas A970). For / e C\ (и), ?(/) is denoted by fT f du in Thomas A970).
Definition 9.7. Let и : ?,(?) -+ X be a Radon operator. We define Mu = {E cT: ?? e
?|(и)} andM„(?') = ii(x?).
THEOREM 9.8. For a Radon operator и on K.(T), Mu is a ring of sets and ?? is countably
additive on Mu.
Definition 9.9. In virtue of Theorem 9.8, the countably additive vector measure ?;,
is called the Radon vector measure induced by и and Mu is called the domain of ???.
(Compare Definition 3.4.)
Definition 9.10. A Radon operator и :K,{T) -> X is said to be bounded if, for each
реГ, there exists a (finite) constant Mp such that
р{иф)^Мр-\\ф\\т
forah>e/C(r).
Definition 9.11. A bounded Radon operator u:K,{T) -> X is said to be weakly
compact if its unique continuous linear extension to Cq(T) is a weakly compact operator
on (Со(Г), || · ||r). A Radon operator и : ?,(?) -+ X is said to be prolongable (in the sense
of Thomas A970)) if its restriction и\)сШ) is a weakly compact Radon operator for each
relatively compact open set U in T.
Using Remark 2 on p. 161 of Grothendieck A953), Thomas A970) proved the following
theorem, on which rests the major part of his theory.
Theorem 9.12 (Theorems 2.2 and 2.2 bis, Thomas A970)). Let u:K,{T) -> X be a
bounded Radon operator. Then the following conditions are equivalent:
(i) Every bounded Borel measurable function belongs to C\ (u).
(ii) For every open set ? in T, there exists a vector ?? e X such that ?*{??) =
fT Xwd{x*u) for each x* e X*. That is, the weak integral {in the sense of Thomas
A970)) of ?? belongs to X for each open set ? in T.
(iii) и is weakly compact on (Cq(T). \\ · \\т).

1088
Т. V. Panchapagesan
Thomas A970) argues that ((ii) =>· (iii)) by the results on p. 160 of Grothendieck
A953). Really, as Г is locally compact, he means Remark 2 on p. 161 of the said paper.
Unfortunately, till the publication of Panchapagesan A998b) the said remark remained
unproved as noted in the paragraphs following Theorem 8.4 above. The condition (ii)
of Theorem 9.12 says that ?*?? = (??**(??))(?*) for each x* e X*, where и** is the
biadjoint of и :Co(T) -> X. Consequently, ?**(??) = ?? e X. Hence, by Theorem 8.7
(which subsumes Remark 2 of Grothendieck A953)) и is weakly compact and, hence,
(iii) of Theorem 9.12 holds. Consequently, the theory of Radon vector measures developed
in Thomas A970) remains valid.
The following result generalizes a part of Theorem 5.6 to weakly compact Radon
operators.
THEOREM 9.13. Let u: K,(T) -> X be a Radon operator. Then the following statements
are equivalent:
(i) и is weakly compact.
(ii) Mu is ? ? -algebra in ? and С С М„.
(iii) KcMu, where П = В{Т) or ?,.(?) orB0(T).
(iv) Every bounded u-measurable scalar function (in the sense of Definition 1.18 of
Thomas A970)) belongs to C\(u) and и is prolongable.
(v) Every bounded IZ-measurable scalar function belongs to C\(u), where 1Z = B(T)
or BAT) or B0(T).
(vi) For each open set U e 1Z there exists a vector хц e X such that x*(xu) =
fT Xu d(jielt) for each x* e X*, where ??* = x*u e K,(J)*h and П = B(T) or
BC(T) or Bo(T).
If и is a weakly compact Radon operator on fC(T) and ifmu is the representing measure of
its continuous linear extension to (Со(Г), || · \\т) (see Theorem 8.5), then mu = /j.u\b(T)-
Moreover, for each ? e Mu,
??<(?)=\\?\???(??H
CtC
where С is directed by the partial ordering C\ ^ C2 ifC\ С С2 for С \, C: eC.
The following theorem characterizes prolongable Radon operators.
THEOREM 9.14. Let u: K,(T) -> X be a Radon operator. Then the following statements
are equivalent:
(i) и is prolongable.
(ii) г (Со) с ми.
(iii) 8(C) С Mu.
(iv) Mu is a ?-ring containing all relatively compact open sets in T.
(v) Mu is a S-ring containing С
(vi) ?,? is a ?-ring containing Co-
(vii) Every bounded scalar IZ-measurable function with compact support belongs to
C\(u), where П = B(T) orBC(T) orBq(T).

Positive and complex Radon measures
1089
Remark 9.15. In the light of the above theorem and Proposition 3.3, the members of
K,(T)* are precisely the scalar valued prolongable Radon operators on K,(T). Similarly, in
virtue of Theorems 5.6 and 9.13, the members of fC(T)*} are precisely the scalar valued
weakly compact Radon operators on K,(T).
Definition 9.16. Let и :K,(T) -> X be a Radon operator with the associated Radon
vector measure ?„. Then ?„ is called a weakly compact (respectively prolongable) Radon
vector measure if и is weakly compact (respectively prolongable).
Let ? be a <5-ring containing C() and let ?: V -> X be countably additive. Based on two
new concepts of Lebesgue-Radon completion and localized Lebesgue-Radon completion
of V (respectively of ?) with respect to ? (respectively to D), we have the following
theorem which generalizes Theorems 6.4 and 6.6 to prolongable Radon vector measures.
THEOREM 9.17. Let ?:8(C) -> X be countably additive. Then ? is the restriction of
an X-valued prolongable (respectively weakly compact) Radon vector measure ?? if and
only if ? is 8(C)-regular (see Definition 5 of Panchapagesan A998b)) (respectively and
if ? has relatively weakly compact range). In that case, и is unique and и is called the
prolongable (respectively weakly compact) Radon operator determined by ?. Moreover,
Mu is the localized Lebesgue-Radon (respectively Lebesgue-Radon) completion of 8(C)
with respect to ? and ?„ is the localized Lebesgue-Radon (respectively Lebesgue-Radon)
completion of ? with respect to 8(C).
Remark 9.18. An alternative simple proof of Theorems 12 and 13 of Panchapagesan
A998b) is given in Panchapagesan B002). The proposed generalizations in Theorems 9.13,
9.14 and 9.17 are closely related to the study of the Bartle-Dunford-Schwartz integral of
scalar functions with respect to a Banach space-valued and more generally, with respect
to a quasicomplete IcHs-valued countably additive measure defined on a ?-ring or on a
<5-ring of sets. Such a study is carried out in author's forthcoming papers [PI] (The Bartle-
Dunford-Schwartz integral, I. Basic properties); [P2] (The Bartle-Dunford-Schwartz
integral, II. ?/7-spaces, 1 ^ ? < oo); and [P3] (The Bartle-Dunford-Schwartz integral, III.
Integration with respect to IcHs-valued measures) which have been communicated for
publication. A measure-theoretic treatment of many of the results of Thomas A970) for
weakly compact and prolongable Radon operators on K,(T) with the improvement of
replacing open sets by open Baire sets is given in [P4] (The Bartle-Dunford-Schwartz
integral, IV. Applications to integration in locally compact Hausdorff spaces - under
preparation), where the hypothesis that {?,·}^ is bounded in Corollary 8.3 is shown to
be redundant by proving that, for each open set U in ?, there exists an open Baire set
V С U such that ?, (V) = ?,(?/) for all;'. [P5] (The Bartle-Dunford-Schwartz integral, V.
Complements to the Thomas theory of vectorial radon integration - under preparation)
obtains some of the results of Thomas A970) by simpler proofs, improves many of them
by replacing open sets by open Baire sets, gives a correct proof of Theorem 1.35 of
Thomas A970) whose original proof is incorrect and obtains Theorems 9.13,9.14 and 9.15.
Moreover, it is shown in [P5] that, for a prolongable (respectively weakly compact) Radon
operator и on K,(T) with values in a quasicomplete IcHs, C\ (u) given in Thomas A970)

1090
Т. V. Panchapagesan
is essentially the same as C\ (?„) given in [P3], where ?„ is the representing measure of и
as given in [P4] and for / e C\(u), Ct(f) = fT f du (see Remark 9.6)= fT f d\xu (the
Bartle-Dunford-Schwartz integral of / with respect to ?„ as given in [P3]). We would
also like to emphasize that [P4] and [P5] are not only based on [PI, P2, P3] but also on
Panchapagesan A998a, 1998b)
References
Bartle, R.G., Dunford, N. and Schwartz J.T. A955), Weak compactness and vector measures. Canad. J. Math. 7,
289-305.
Bourbaki, N. A965a), Integration, Chapitres I-IV, Hermann, Paris.
Bourbaki, N. A965b), Integration, Chapitre V, Hermann, Paris.
Diestel, J. and Uhl, J.J. A977), Vector Measures, Amer. Math. Soc., Providence, RI.
Dieudonne, J. A951), Sur la convergence des suites de mesures de Radon, Anais. Acad. Bras. Ciencias 23, 21-38.
Dinculeanu, N. A967), Vector Measures, Pergamon Press, New York.
Dinculeanu, N. and Kluvanek, 1. A967), On vector measures, Proc. London Math. Soc. 17, 505-512.
Dinculeanu, N. and Lewis, P.W. A970), Regularity ofBaire measures. Proc. Amer. Math. Soc. 26, 92-94.
Dobrakov, I. and Panchapagesan, T.V. B002), A simple proof of the Borel extension theorem and weak
compactness of operators, Czechoslovak Math. J., in print.
Dunford, N. and Schwartz, J.T. A958), Linear Operators, Part I. General Theory, lnterscience. New York.
Edwards, R.E. A965), Functional Analysis, Theory and Applications, Holt, Rinehart and Winston, New York.
Grothendieck, A. A953), Sur les applications lineares faiblement compactes d'espaces du type C( A"), Canad. J.
Math. 5, 129-173.
Halmos, PR. A950), Measure Theory, Van Nostrand, New York.
Hewitt, E. and Stromberg, K. A965), Real and Abstract Analysis. Springer, New York.
Kluvanek, I. A967), Characterization of Fourier-Stieltjes transforms of vector and operator valued measures,
Czechoslovak Math. J. 17, 261-277.
McShane, E.J. A944), Integration, Princeton University Press, Princeton.
Naimark, M.A. A959), Normed Rings, Noordhoff, Groningen.
Panchapagesan, T.V. A991a), Medida e Integracion, Pane I, Vols. 1 and 2, Universidad de los Andes, Facultad
de Ciencias, Merida, Venezuela (Spanish).
Panchapagesan, T.V. A991b), Integral de Radon en espacios localmente compactos у de Hausdorff, Cuarta
Escuela Venozolana de Mat., Universidad de los Andes, Facultad de Ciencias, Merida, Venezuela (Spanish).
Panchapagesan, T.V. A992), On complex Radon Measures I, Czechoslovak Math. J. 42, 599-612.
Panchapagesan, T.V. A993), On complex Radon measures II, Czechoslovak Math. J. 43, 65-82.
Panchapagesan, T.V. A995/96), On Radon vector measures. Real Analysis Exchange 21, 75-76.
Panchapagesan, T.V. A998a), Baire and ? -Borel characterizations of weakly compact sets in M( T), Trans. Amer.
Math. Soc. 350, 4839-4847.
Panchapagesan, T.V. A998b), Characterizations of weakly compact operators on Cq(T), Trans. Amer. Math.
Soc. 350, 4849-4867.
Panchapagesan, T.V B000), On the limitations of the Grothendieck techniques. Rev. Real Acad. Cienc. Exact.
Fis. Natur., Madrid 94, 437^440.
Panchapagesan, T.V. B002), Weak compactness of unconditionally convergent operators on Cq(T), Math.
Slovaca, in print.
Petczynski, A. A960), Projections in certain Banach spaces, Studia Math. 19, 209-228.
Petczynski, A. A962), Banach spaces on which every unconditionally converging operator is weakly compact.
Bull. Acad. Polon Sci. Ser. Sci. Math. Astronom. Phys. 10, 641-648.
Rudin, W. A967), Real and Complex Analysis, McGraw-Hill, New York.
Sion, M. A969), Outer measures with values in topological groups, Proc. London Math. Soc. 19, 89-106.
Thomas, E. A970), L'integration par rapport a line mesure de Radon vectorielle, Ann. Inst. Fourier (Grenoble)
20,55-191.

CHAPTER 27
Measures on Algebraic-Topological Structures
Piotr Zakrzewski
Institute of Mathematics, University of Warsaw, ul. Banacha 2. 02-097 Warsaw. Poland
E-mail: piotrzak@mimuw.edu.pl
Contents
Introduction 1093
1. Invariant measures on arbitrary G-spaces '094
2. Nonmeasurable sets for invariant measures ' ?2
3. Extensions of invariant measures · 107
4. Invariant measures on Polish groups 1109
5. Invariant measures on Polish G-spaces " И
6. Isometrically invariant measures on Euclidean spaces 3
References 1127
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1091

This page intentionally left blank

Measures on algebraic-topohgic/il structures
1093
Introduction
This article is entirely devoted to countably additive measures invariant under groups of
transformations. Our general framework is a G-space, i.e., a pair (X. G), where G is
a group acting on a set X. If, moreover, ? is a ? -algebra of subsets of X, we say that
? is G-invariant if gA e ? whenever g e G and A e ?. Then we also say that G acts by
measurable transformations on the measurable space (?, ?). We say that a (?-additive,
non-zero) measure ? : ?1 —» [0, oo] is G-invariant if /J.(gA) = ?(?) for every g e G and
Ael, i.e., if for each g e G the transformation (g,-):X->Xis measure-preserving.
Invariant measures are present in many parts of mathematics including harmonic
analysis, ergodic theory and topological dynamics.
A typical example of such a measure is the Lebesgue measure: one of its key properties
is that it assigns equal values to congruent sets, so that the measure of a set does not depend
on its position in the space.
The Lebesgue measure is a particular example of a Haar measure on a locally compact
topological group G: the only, up to a constant factor, regular Borel measure, positive on
nonempty open sets, finite on compact sets and invariant under the left shifts from G. Haar
measure plays an important role in abstract harmonic analysis and group representation
theory.
In ergodic theory one focuses on the case where a single measure-preserving
transformation ? of a probability measure space (?, ?. ?) is given, so that X becomes a Z-space
via the map (к, x) ь* Tkx for к e ? and ? e X. Invariant measures for more general group
actions are also of considerable interest.
But in this article we leave aside these aspects of the theory of invariant measures that
are specific for harmonic analysis or ergodic theory. Instead, given a group G acting by
measurable transformations on a measurable space (?, ?), we focus on the fundamental
problems concerning the interplay between algebraic, combinatorial and topological
properties of the action and the existence and properties of G-invariant measures defined
on ?. In particular, we are interested in the way in which in variance of a measure influences
the size and the structure of the underlying ?-algebra, and the existence and properties of
nonmeasurable sets.
General necessary and sufficient conditions for the existence of finite and ?-finite
invariant measures on arbitrary G-spaces are discussed in Section 1. We omit these
conditions obtained for the special case of Z-actions which do not have meaningful
extensions to more general classes of actions. Such conditions properly belong to classical
ergodic theory and are discussed in many standard textbooks on the subject (see, e.g.,
Friedman, A970)). We concentrate on results related to a characterization that was obtained
by Hopf A932) for the case of Z-spaces and was later generalized by Kawada A944),
Hajian and Ito A969) and others to arbitrary G-spaces. It is formulated in terms of the
notion of countable equidecomposability, first investigated by Banach and Tarski A924).
We admit after Hopf that such results are not fit for immediate applications but nevertheless
they throw a light on the intrinsic nature of G-spaces admitting invariant measures.
Section 2 is devoted to invariant measures defined for all subsets of a given set X. Results
are parallel to the work of Tarski A938) on the existence of finitely additive invariant
measures. Conditions imposed on a G-space X, which refute the existence of a G-invariant

1094
P. Zakrzewski
?-finite measure defined on V(X), at the same time guarantee that for an arbitrary G-
invariant ?-finite measure ?: ? -> [0, oo] on X, we have ? ? V{X), i.e., nonmeasurable
sets exist. These investigations are related to the well-known general measure problem,
whether there exists a ?-additive ?-finite measure defined on the power set of X and
vanishing on all singleton sets. Thus the answers, as is well-known, depend on set-theoretic
assumptions.
In Section 3 we deal with extensions of invariant measures, the matter closely connected
to the invariant version of the general measure problem. Given a G-invariant measure
?: ? -> [0, oo] on a G-space X we look for another G-invariant measure ?: ?' -> [0, oo]
such that ? ?. ?' and ?\ ? = ?. This reduces to finding a suitable ???-?-measurable set
that can be added to the domain of the measure ?.
In Section 4 topology enters into the picture, as we discuss the existence of Borel
measures on Polish (i.e., separable, completely metrizable) groups. In particular, standard
facts about Haar measure in locally compact groups are summarized. For the sake of
conceptual coherence and making the interplay between measure and topology more
transparent, we restrict ourselves to the case of second countable groups.
The discussion of invariant properties of Borel measures on Polish spaces is continued
in Section 5, where continuous or, more generally, Borel actions of Polish groups are
considered. Some parts of this section belong to descriptive dynamics, a branch of
descriptive set theory that has been recently attracting increasing interest. Our exposition
is based on the widely distributed set of lecture notes by A.S. Kechris A994).
In the last section we come back to extensions of invariant measure, this time
concentrating on isometrically invariant measures in Euclidean spaces R". We present
a survey of results connected with Sierpinski's problem, formulated in Szpilrajn A935),
concerning the existence of a maximal isometrically invariant extension of Lebesgue
measure.
1. Invariant measures on arbitrary G-spaces
We say that a group (G, ·, e) acts on a set X, if there is a map (g,x) ь* gx of G ? ?
into X, called an action of G on X, such that for every ? e X:
(i) ex = x;
(ii) for all gi, g2 e G, g](g2x)= (g\ ¦ gi)x.
We then say that (X, G), or just X, is a G-space.
It follows that for each g e G the function ? ь* gx is a permutation of X. If G itself
consists of permutations of X, then we tacitly assume that the action is given by gx = g(x)
and say that G is a group of permutations of X.
Given a G-space X a (G-)orbit is the set Gx = {gx: g e G). A subset А с X is (G-)
invariant if it is the union of orbits of its elements or, equivalently, it is closed under the
action of G. We say that the action is transitive, if X is the only orbit. The action is free, if
gx ? ? for any g ? e and ? e X.
If G is a subgroup of a group (X, ·), then G acts on X by left shifts, i.e., gx = g ¦ ?
for g & G, ? e X. Of course, this action is free and it is transitive if and only if G = X;
G-orbits coincide with the left-cosets of G in X.

Measures on algebraic-topological structures
1095
If ? is a ?-algebra of subsets of a set X, then we say that (?, ?) is a measurable (or
Borel) space. The elements of ? are called measurable sets.
By a measure on X we mean a ?-additive (unless explicitly stated otherwise), non-zero
measure defined on a ?-algebra ? of subsets of X. We always assume that {x} e ? for
any ? e X. A measure m : ? -> [0, oo] on X is diffused if it vanishes on all singleton sets,
i.e., m({jt})=0forall;t el
Given sets А, В e ? we write A c,„ ?, if m(A \ ?) = 0; if A c,„ ? and ? c,„ A, then
we write A =„, B.
Suppose a group G acts by measurable transformations on a measure space (?, ?, m).
The action of G is m-free if m*({.r e X: gx = x}) = 0 for any g e G \ {e}, where m*
stands for the outer measure of m.
A subset А С X is (G-)almost invariant (with respect to w) if m(gA ? A) = 0 for every
g e G, where ? denotes the symmetric difference. It is easy to check that the collection
?* = {A & ?: A is almost invariant}
is a ?-algebra of subsets of X contained in ?.
The measure m is (G-)quasi-invariant if m(A) = 0 implies m(gA) = 0, and (G-)
invariant if w(A) = m(gA), for every A e ? and g e G. It is (G)-ergodic, if w(A) = 0 or
m(X \ A) = 0 for any A e 2?*. Ergodicity of m is equivalent to the fact that the associated
measure algebra ?* /m is the two-element algebra.
Note that if the measure m is quasi-invariant, then the ?-algebra ?* is invariant. If
m is, moreover, ?-finite, then every A e ? is contained in a minimal almost invariant
set A* e ?*, where minimal means that A* c,„ ?, whenever AC, В and В е ?* (see,
e.g., Ramachandran and Misiurewicz A982)). If the group G is countable, then simply
A* = \Jg€GgA;'mgcncTal A* = \Jn€Ng„A for certain gn e G, и eN={0, 1,...}.
The problem of existence of invariant measures on an arbitrary G-space have been
approached from two different but closely related points of view.
(A) In classical ergodic theory the problem was formulated as follows (see Halmos
A956, p. 81)):
Given a group G acting by measurable transformations on a ?-finite measure
space (?, ?, m), find necessary and sufficient conditions for the existence of a G-
invariant, probability or ?-finite measure ? defined on ? and such that ??«?,
i.e., m is absolutely continuous with respect to ?. It is usually assumed that the
measure m is quasi-invariant (in this case elements of G are sometimes called
nonsingular transformations of the measure space (?, ?, m)). It is then required
that ? ~ m, i.e., the measures ? and m have the same null sets.
(B) Influenced by Tarski's work on finitely additive invariant measures (see Wagon
A986) or Laczkovich B002) in this Handbook) one searches for "purely
combinatorial" conditions for the existence of an arbitrary probability or ? -finite G-invariant
measure, when a group G acts by measurable transformations on a measurable
space (?, ?), with no measure defined on the ?-algebra ? given in advance.
The two approaches lead to closely related results. The reason is that if all G-orbits
are uncountable, then any ?-finite invariant measure defined on the ?-algebra ? has to
be diffused but even in relatively simple cases, when the ?-algebra is, say, countably

1096
P. Zakrzewski
generated, it may happen that it carries no ?-finite, diffused measure at all, even without the
additional requirement of G-invariance. So, natural necessary conditions for the existence
of an invariant measure have the form: "? carries a measure m satisfying some additional
properties."
In order to formulate the desired conditions, we need to recall the notion of equivalence
by countable decompositions, introduced by Banach and Tarski A924). Our terminology
is consistent with that from Wagon's monograph A986).
Given a group G acting by measurable transformations on a measurable space (?, ?)
we say that sets А, В & ? are countably (G-)equidecomposable (in ?), in symbols
А ~эо В in ?, if there is a partition of A into countably many sets An e ?, ? e N, and
elements gn e G such that the sets gn An form a partition of B.
A set A e ? is called countably (G-)paradoxical (in ?), if A contains disjoint subsets
A\, ?? e ? with A ~x A\ ~x At in ?.
A set A & ? is called G-negligible (in ?) if X contains pairwise disjoint subsets
A/, s ?, ? ? ?, with each A„ ~3C A in ?. Let Nq consist of all G-negligible elements
of ?. It can be proved (see, e.g., Chuaqui A977)) that the family No is closed under
countable unions and taking subsets in ?, i.e., it constitutes a ?-deal in ?. In fact, it is not
difficult to prove that the ? -ideal Nc is generated by countably paradoxical sets in ?, i.e.,
it is the smallest ?-ideal in ? containing all measurable subsets of X that are countably
paradoxical in ?.
Additionally suppose that m is a measure defined on ?. A set ? e ? is called (G-)
bounded (with respect to m), if ? ~эс A in ?1 implies m(Z \ A) = 0 for every A e ?,
Ac. Z, i.e., ? is not countably G-equidecomposable in ? with any measure-theoretically
proper measurable subset of itself.
It is not difficult to prove that ? is bounded if and only if the measure m vanishes on
all measurable subsets ? of ? with the property that ? contains pairwise disjoint sets
E„ e ?, ? e N, with each En ~x ? in ? (see, e.g., Kawada A944)).
In particular, we have the following:
PROPOSITION 1.1. Suppose a group G acts by measurable transformations on a measure
space (?, ?^). The following are equivalent:
(i) The set X is G-bounded with respect to m.
(ii) The measure m vanishes on all G-negligible sets from ?.
(iii) The measure m vanishes on all measurable sets that are countably G-paradoxical
in ?.
The following fundamental theorem on the existence of invariant probability measures
was first proved by E. Hopf A932) for the case of Z-actions and then generalized by
Y. Kawada A944) to arbitrary group actions.
THEOREM 1.2. Suppose a group G acts by measurable transformations on ? ?-finite
measure space (X, X\m) and assume that the measure m is quasi-invariant. Then the
following are equivalent:
(i) There exists an invariant probability measure ? on ? such that ? ~ m.
(ii) The set X is bounded with respect to m.

Measures on algebraic-topohgical structures
1097
Obviously, any invariant probability measure on ? vanishes on sets that are countably
paradoxical in ?, so the existence of such a measure ? with ? ~ m immediately implies
that X is bounded with respect to m. The nontrivial part of the Hopf-Kawada theorem
asserts that the converse is true.
Given sets А, В e ? we write A «; В if there isCel with С ~эс A in ?, С c„, В
and for every Q e ?*, if Q c„, ?* and m@) > 0, then m((B \ С) П 0 > 0, i.e., С ? ?
is a measure-theoretically proper subset of ? ? (?
Given sets ?,?,??? and « e N we write
A = nB + C
if A =„, ? ? U · · ¦ U A„ U С where ? ? A„, С are pairwise disjoint sets in ? with each
Ai ~oc ? in ?.
The essence of proof (which uses, as Kawada admits, some ideas taken from J. von
Neumann's 1936-37 classical lecture notes on "Continuous Geometry" - see von
Neumann A998)) are the following two lemmas.
Lemma 1.3. For every sets А, В e ? of positive measure there is a partition (up to a
measure zero set)
A* =„ (J Qn
of A* into pairwise disjoint sets Q„ e ?*, ? e N, such that for each ? e N m(Qn) > 0 and
AnQ„=n(BD Q„) + R„, where R„ « ? ? Q„.
Moreover, the partition is unique, up to a measure zero set.
We say that a set В e ? is a G-atom if m(B) > 0 and В is minimal with respect to <<C
in the sense that A <<C В implies m(A) = 0, for any A e ?. Aset Q e ?* is called discrete
if Q =m ?* for a certain G-atom B.
LEMMA 1.4. There are disjoint sets D,C e ?* such that X =„, DUC and the following
conditions are satisfied:
(i) Ifm(D) > 0, then D is discrete.
(ii) If m(C) > 0, ?/геи there is a decreasing sequence of sets C„ e ? with Co = C,
C,* =m С and C„ = 2C„+| for every n e N.
Moreover, such sets С and D are unique, up to measure zero sets.
Taking the lemmas above for granted, the proof is completed as follows.
Assume, without loss a generality, that m(X) = 1.
Let D and С be sets from Lemma 1.4. We shall define measures ?</, ?< on measurable
subsets of D and C, respectively (if m(D) = 0 or m(C) — 0, then we let ??? s 0 or ?( ? 0,
respectively).

1098
P. Zakrzewski
So assume that m(D) > 0 and let D= B* for a certain G-atom ?. Take A e ?, A c,„ D.
By Lemma 1.3 and the fact that ? is a G-atom, there exists a partition of D into sets
Qn e ?* such that m(Q0) = 0 and
Qn = n(B П Qn) for each ? > 0.
Fix и > 0 and additionally assume that A cw Q„. Then, using again Lemma 1.3, we
have
АПРк = к(ВПРк) for к = I,...,n,
where A* =,„ UILi ?k is the appropriate partition of A* into sets A e ?1*.
Set
For an arbitrary A & ? with A cm Z) define
00
??/(?) = ^?^",(???„)·
/f=l
Now assume that m(C) > 0 and take ???,? c,n C. Fix и e N. Note that С = 2"C„.
Using Lemma 1.3 we find a partition of A* into sets Qkn) e IT*, k = 0,...,2", such that
А П ?'"> = л(С* П ?'"') + R™ where ^'" « QkH).
Then let
/-w=E^'»(ei,,))
* = l
and define
Mc(A)= Hm /„(A).
n—>oc
It can be checked that this definition is correct.
Finally, for an arbitrary A e ? define
?(?) = ??(? П D) + ??(? П С).
For details see Kawada A944).
Kawada A944) noticed that this proof reveals the following uniqueness property of
invariant measures.

Measures on algebraic-topological structures
1099
THEOREM 1.5. Suppose a group G acts by measurable transformations on a measurable
space (?, ?). Then any invariant probability measure ? on ? is determined by its values
on ?*, the ? -algebra of measurable sets, almost invariant with respect to ?. Moreover,
if m is a probability quasi-invariant measure on ? such that X is bounded with respect
to m, then there exists a unique invariant probability measure ? on ? such that ? ~ m
and ?(?) = m(A)for any A e ?*. In particular, ifm is ergodic, then there exists a unique
invariant probability measure ? on ? with ? ~ m.
Shorter proofs of the Hopf-Kawada theorem were later found by Ramachandran and
Misiurewicz A982), and Ramachandran A989). The simplification was achieved by using
the semigroup of (countable equidecomposability) types (see Section 5 of Laczkovich
B002) with the obvious changes). Ramachandran's proof was based on a theorem of
Tarski (see Tarski A938, Theorem 1.55)) concerning measures on semigroups. In fact,
this theorem yields the existence of a finitely additive invariant measure equivalent to a
given quasi invariant measure m. The latter, however, suffices, due to the following result
of A.P. Calderon A955) (see also Friedman A970, Theorem 3.13)).
THEOREM 1.6. Suppose a group G acts by measurable transformations on ? ?-finite
measure space (?, ?, m) and assume that the measure m is quasi-invariant. If ? is a
finite, finitely additive, invariant measure defined on ? with ? ~ m, then the function ?
defined by
?(?) = infl ]T P(A„): A„ e ?, А с (J A„ forAe ?,
is a finite, ?-additive, invariant measure on ? with ? ~ m.
To formulate another necessary and sufficient condition for the existence of an invariant
probability measure we need the notion of a weakly wandering set.
Suppose a group G acts by measurable transformations on a measurable space (?, ?).
A set A e ? is called weakly wandering (under G) if there exist elements g„ e G, ? e N,
such that gn А П g„,A = 0 for ? ? т. Clearly, every weakly wandering set is G-negligible.
The following strengthening of the Hopf-Kawada theorem was first proved by Hajian
and Kakutani A964) for Z-actions and then generalized by Hajian and Ito A969) to
arbitrary actions (partial generalizations were obtained by several authors - see the
references in Hajian and Ito A969)).
THEOREM 1.7. Suppose a group G acts by measurable transformations on ? ?-finite
measure space (X, E,m) and assume that m is quasi-invariant. Then the following are
equivalent:
(i) There exists an invariant probability measure ? on ? such that ? ~ m.
(ii) The measure m vanishes on all weakly wandering sets in ?.
Under the hypotheses of Theorem 1.7 condition (ii) above is equivalent to each of the
following three assertions:

1100
P. Zakrzewski
(iii) m(A) > 0 implies \nii,€Gm(gA) > 0, for every A e ? - see Hajian and Kakutani
A964).
(iv) For every ? > 0 there is ? > 0 such that m(A) < ? implies m(gA) < ?, for every
Ae ?-seeSakaiA978).
(v) For every ? > 0 there is ? > 0 such that m(A) < ? and ? ~^ ? imply m(B) < ?,
for every А, В e ? - see Sakai A978).
Hajian and Ito used functional analytic methods in their proof, treating the existence of
an invariant measure as the problem of finding a fixed point under certain group of linear
operators on an appropriate Banach space.
More precisely, let us assume that the measure m is a probability measure and consider
the following action of the group G on the Hubert space L2(X, ?, m):
Ug(f) = f(g-]x)[w!,(x)]w2 forge G, /eL2(X,?,m),
where wg is the Radon-Nikodym derivative with respect to m of the measure mg~' (A) =
m(g~] A) for ??? Any common fixed point, i.e., a function /о е L2(X, ?, m) with
Ug(fo) — /o f°r all g e G, gives a desired invariant probability measure ? ~ m by the
formula
?(?)= / fl{x)dm{x) for A e ?.
Yet another proof of the Hopf-Kawada theorem was given by R. Chuaqui A977).
Chuaqui (quoted in Wagon A986, Question 9.13)) conjectured that the the existence of a ?-
finite measure m on ? which vanishes on all G-negligible sets is sufficient for the existence
of a G-invariant probability measure on ?, even without the additional requirement that m
is quasi-invariant. By a result of Halmos A947), this was known to be true for Z-actions.
Chuaqui's conjecture was confirmed in Zakrzewski A993a). More exactly, we have the
following result which also generalizes the Hajian-Ito theorem.
THEOREM 1.8. Suppose a group G acts by measurable transformations on ? ?-finite
measure space (?, ?, m). Then the following are equivalent:
(i) There exists an invariant probability measure ? on ? such that т«д.
(ii) The measure m vanishes on all G-negligible sets in ?.
(iii) The measure m vanishes on all weakly wandering sets in ?.
Let us now turn to the conditions for the existence of invariant ?-finite measures.
If a group G acts by measurable transformations on a measure space (?, ?, m), a set
? e ? is called (G-)a-bounded (with respect to m), if it is a countable union of bounded
sets.
In Kawada A944), quoted above, the following theorem is proved.
THEOREM 1.9. Suppose a group G acts by measurable transformations on a ?-finite
measure space (?, ?, m) and assume that m is quasi-invariant. Then the following are
equivalent:

Measures on algebraic-topological structures
1101
(i) There exists an invariant ? -finite measure ? on ? such that ? ~ m.
(ii) The set X is ?-bounded with respect to m.
(iii) The re is a bounded set A G ? with A* =w, X.
This result, for the special case of Z-actions, was later rediscovered by Halmos A947)
who also noticed that in this case the assumption that the measure m is G-quasi-invariant
may be omitted if we want the resulting invariant measure only to dominate m in the sense
of absolute continuity.
The following theorem from Zakrzewski A993b) generalizes the results of Kawada and
Halmos to arbitrary groups of measurable transformations of any ?-finite measure space
(?, ?, m). The new element that arises here is the necessity of adding to the condition
that X is ?-bounded another property, which guarantees the existence of a quasi-invariant
measure dominating m in the sense of absolute continuity (and is, therefore, superfluous if
m itself is assumed to be quasi-invariant).
Let Ic consist of all A e ? with m(gA) = 0 for every g e G. It is clear that /c is a
?-ideal in ?.
We say that a ?-ideal / in ? satisfies the countable chain condition in ?, if there is
no uncountable collection of pairwise disjoint sets in ? \ I. Clearly, m being ?-finite, the
?-ideal of m-measure zero sets satisfies the countable chain condition in ?.
THEOREM 1.10. Suppose a group G acts by measurable transformations on ? ?-finite
measure space (?, ?, m). Then the following are equivalent:
(i) There exists an invariant ?-finite measure ? on ? such that т«д.
(ii) The set X is ?-bounded with respect to m and the ?-ideal Ic satisfies the countable
chain condition in ?.
Moreover, the following are equivalent:
(i) The ?-ideal Ic satisfies the countable chain condition in ?.
(ii) There exists a quasi-invariant, ?-finite measure ? on ? such that Ic = {A e ?:
?(?) = 0}.
(iii) There exists a quasi-invariant, ?-finite measure ? on ? such that т«д.
In fact, the proof of the nontrivial part of Theorem 1.8, asserting the existence of an
invariant probability measure ? on ?, m «?, boils down to showing that ifm vanishes on
all weakly wandering sets in ?, then the ?-ideal Ic satisfies the countable chain condition
in ?.
In the special case when the group G is countable, the ?-ideal /c always satisfies the
countable chain condition in ?. But in fact in this case even an invariant ?-finite measure
always exists, the measure defined by ?(?) = \A П Gx\ for A e ?, i.e., the counting
measure concentrated on the orbit Gx of an arbitrary point ? e X, being a trivial example.
However, if we look for diffused invariant measures, then it follows that a quasi-invariant
?-finite diffused measure defined on ? exists, provided that the ? -algebra ? carries ? ? -
finite diffused measure at all. As is well known, the latter may not be true even if ? is
countably generated. On the other hand, even in the case of a Z-action, the existence of
a quasi-invariant ?-finite diffused measure does not imply that such an invariant measure
exists. An example is given in Zakrzewski A993b). It is based on the result of D.S. Ornstein

1102
P. Zakrzewski
A960) that there exists a Ъ-action by measurable transformations of the Lebesgue measure
space ([0, 1], C, ?) such that the Lebesgue measure ? on the ?-algebra С of Lebesgue
measurable subsets of [0, 1] is quasi-invariant and ergodic but there is no invariant, ?-
finite measure ? on С such that ? ~ ?.
Other conditions for the existence of an invariant ?-finite measure, equivalent with
a given quasi-invariant one, were obtained in Arnold A968) (see also Zakrzewski A993b,
Theorem 3.3), Rosenblatt A974) - it concerns the case when the group G is finitely
generated - and Rosenblatt A980) - it concerns the existence of an invariant measure
which is strongly equivalent with the given quasi-invariant measure in the sense that the
measures have the same sets of measure zero and the same sets of infinite measure).
Theorems 1.2 and 1.9 are in fact special cases of a more general theorem from Kawada
A944) on the existence of a strictly positive measure on a Boolean algebra A, invariant with
respect to a certain group of automorphisms of A (see Kawada A944, Satz 4)). Of course,
this is closely related to the problem of finding an invariant measure ? with ? ~ m, if a
measure m, quasi-invariant under an action by measurable transformations on a measurable
space (?, ?) of a group G is given. Indeed, the action of G on X canonically induces
the action of G on the associated measure algebra A= ?/m. This Boolean-algebraic
point of view is also present in Vladimirov A965), Chuaqui A977). Let us note that the
search for invariant measures on Boolean algebras is connected to the study of groups of
automorphisms and measure-preserving automorphisms of measure algebras (see Fremlin
B000) for an account of the subject and further references).
A remarkable feature of existence proofs from Kawada A944) is that they also
establish certain uniqueness properties of invariant measures. Let us mention the following
generalization of Theorem 1.5:
Suppose a group G acts by measurable transformations on a measurable space
(?, ?). Let ? be an invariant ? -finite measure on ? and suppose that A e ?
is a bounded set with ?* =? ?. Then ? is determined by its values on sets of
the form AC\Q, where Q e ?*.
In particular, we have the following corollary established earlier in von Neumann A940,
Lemma 3.3.4).
PROPOSITION 1.11. Suppose a group G acts by measurable transformations on
a measurable space (?, ?). If ? is an invariant ? -finite ergodic measure on ?, then ? is
unique, up to a constant factor, among invariant ? -finite measures ? on ? with ?«?.
An account of results connected to the Hopf-Kawada theorem on one hand and
a classification of G-spaces with a quasi-invariant measure, studied by H.A. Dye A959,
1963) and W. Krieger A969a, 1969b), on the other, may be found in Dang-Ngoc-Nghiem
A973).
2. Nonmeasurable sets for invariant measures
We say that a measure m is universal on a set X, if it is defined on the ? -algebra V(X) of
all subsets of X, i.e., if m-nonmeasurable sets do not exist.

Measures on algebraic-topological structures
1103
A particular case of the problem discussed in the preceding section concerns the
existence of invariant universal measures on G-spaces. As is well known, given a set X,
there might be fundamental set-theoretic obstacles preventing the existence of any ?-finite
diffused universal measure on X (see, e.g., D.H. Fremlin's article A993) for a thorough
discussion of this point). On the other hand, in many cases, the reason why a particular
measure cannot be defined on V(X) is precisely its G-invariance, regardless of any new
set-theoretic assumptions which might possibly be added to the usual Zermelo-Fraenkel
set theory with the axiom of choice (ZFC).
Let us call a set X large, if there exists a ?-finite or, equivalently, probability diffused
universal measure on X (in the commonly used terminology this means that there exists
a real-valued-measurable cardinal к ^ \X\ - see, e.g., Fremlin A993, 1C-D)); otherwise
call X small.
The prototype of arguments employing invariance properties of measures to show the
existence of nonmeasurable sets within ZFC is the Vitali construction.
By a selector or a transversal of a collection С of nonempty, pairwise disjoint subsets of
a set X, we mean a subset ? с X which intersects every element of С in exactly one point.
The classical Vitali subset ofR, constructed by G. Vitali A905), is a selector of the
family of all cosets of the rationals Q, treated as a subgroup of the additive group R. It
is easy to note that a Vitali subset ofR is nonmeasurable with respect to any translation-
invariant measure on R which assumes a finite non-zero value on [0, 1]. In particular, no
translation-invariant probability measure on R is universal.
In fact, Vitali's argument works equally well for, say, left action of a countable subgroup
of an arbitrary uncountable group G. Let Я be a subgroup of G and call a Vitali set of ? а
selector of the collection of all left-cosets of ? in G. Now, if ? is countably infinite and S
is a Vitali set for H, then S is nonmeasurable with respect to any finite invariant measure
on G. This follows directly from the fact that
G= [J hS,
i.e., G can be partitioned into countably many translates of S. The latter shows also that G
is countably paradoxical (in arbitrary pieces). (It is in fact not difficult to prove that in the
general case of an arbitrary G -space X, a subset ? of X is countably paradoxical (in
arbitrary pieces) if and only if there exists a countable subgroup ? of G such that the
intersection with ? of the ?-orbit of every point of ? is infinite - see, e.g., Zakrzewski
A991a, Lemma 2.2).)
Recall that according to a theorem of Tarski A938), given an action of a group G
on a set X, a finitely additive probability invariant measure on X exists if and only
if X is not paradoxical, i.e., X does not contain disjoint subsets X\, ??, each finitely
equidecomposable with X - see Laczkovich B002) for more details.
The exact analog of Tarski's theorem for countably additive measures is not true in
general, since there may be no ?-additive probability universal measure on X vanishing
on all countably paradoxical sets for purely set-theoretic reasons like, say, X being small,
which have nothing to do with the obvious obstacle that X is countably paradoxical. So
the following special case of Theorem 1.8, proved in Zakrzewski A991a), seems to be the

1104
P. Zakrzewski
optimal analog of Tarski's theorem in the countably additive case for arbitrary G-spaces
(see, however, Theorem 5.1).
THEOREM 2.1. For an arbitrary G-space (X, G) the following are equivalent:
(i) There exists a universal invariant probability measure on X.
(ii) There exists a universal probability measure on X vanishing on all countably
paradoxical subsets of X.
Let us now turn to ?-finite measures.
A.B. Khararazishvili A975) and, independently, P. Erdos and R.D. Mauldin A976)
clarified the situation for the case when a group acts on itself by (say) left shifts (see also
С Ryll-Nardzewski and R. Telgarski A978) for a generalization of this result).
THEOREM 2.2. Let G be an uncountable group acting on itself by left shifts. Then no
?-finite invariant measure on G is universal.
PROOF. Take a subgroup ? of G of cardinality ?? and let S с G be a Vitali set of H.
Suppose that ? is a ?-finite invariant universal measure on X. Then we cannot have
?(S) > 0, since this implies that the collection [hS: h e H] consists of uncountably many
sets of positive measure. But ?(?) = 0 is also impossible, since then G = [Jh€H hS has
measure zero (the union of any family of ? ?-many null sets with respect to a universal
?-finite diffused measure is a null set - see Fremlin A993, ID)). ?
The above argument proves also the nonexistence on G of a universal ?-finite measure,
quasi-invariant with respect to its uncountable subgroup H. On the other hand, for an
arbitrary countable group ? acting on a large set X, a universal ? -finite invariant diffused
measure always exists. (Proof. Assume that m is a universal probability diffused measure
on X. Fix a selector S of the collection of all Я-orbits such that m(S) > 0. Define the
measure ? by
?(?)= ]? km({x eS: \АГ)Нх\=к\) forACX, (*)
/teNU|Dc)
where oo · 0 = 0 and oo · t = oo, if t ? 0.)
The case of an arbitrary G-space X which admits a universal probability measure m
is covered by Theorem 1.10. However, it turns out that now the situation is simpler: the
existence of a quasi-invariant ?-finite universal measure on X implies the existence of
an equivalent invariant one. This is a consequence of the following structural theorem for
universal ?-finite quasi-invariant measures established in Zakrzewski A991 a) (for a shorter
proof, due to D.H. Fremlin, see Zakrzewski A997), Fremlin A993, U)).
THEOREM 2.3. Suppose G is a group of permutations of a set X. If m is a universal
quasi-invariant ?-finite measure on X, then there exists a countable subgroup ? of G
such that
m({x e X: gx ? Hx}) = 0 for every geG.

Measures on algebraic-topological structures
1105
Now, under the notation of Theorem 2.3, if S is a selector of the collection of all H-
orbits, then formula (*) above defines a G-invariant universal ?-finite measure ? with
? ~ m.
Combining this with Theorem 1.10 we have the following theorem (for more details see
Zakrzewski A991a, 1997)).
THEOREM 2.4. For an arbitrary G-space (X, G) the following are equivalent:
(i) There exists a universal invariant ?-finite measure on X.
(ii) There exists a universal quasi-invariant ?-finite measure on X.
(iii) There exists a universal ?-finite measure m on X such that there is no uncountable
collection ofpairwise disjoint sets outside the ?-ideal Iq = {AC X: m(gA) = 0
for every g e G).
In particular, Theorem 2.3 implies that // a group G acting on a set X contains an
uncountable subgroup G' whose action on X is free, then no universal ?-finite quasi-
invariant measure on X exists. This was earlier proved by Khararazishvili A983) (see also
Khararazishvili A996)) in the following stronger form: under the above assumptions, if
? : ? —> [0, oo] is ? ?-finite quasi-invariant measure on X, then every set ? e ? with
?(?) > 0 contains a nonmeasurable set. But in fact, Theorem 2.4 easily implies that,
in general, for an arbitrary G-space X, if there is no universal quasi-invariant ?-finite
measure on X, then any ?-finite quasi-invariant measure on X has the property that
every measurable subset of X of positive measure contains a nonmeasurable set. (Proof.
If ?: ? -> [0, oo] is a ?-finite quasi-invariant measure on X and ? e ? is such that
V(Y) с ?, then the universal measure m on X defined by m(A) = ?(? ? ?), for А с X,
satisfies condition (iii) above.)
More examples of natural G-spaces admitting no universal ?-finite invariant measures
can be found in Penconek and Zakrzewski A994), Zakrzewski A997).
While the results above concern the possibility of simultaneously making all subsets of
a given G-space X measurable with respect to an invariant measure, there still remains
the more modest problem, whether a particular subset of X can be "made measurable" by
an invariant measure. This question, aimed at getting a deeper insight in the properties of
nonmeasurable sets, is motivated by the following facts:
• Every Vitali subset ofR is nonmeasurable with respect to any translation-invariant
extension of the Lebesgue measure.
• There exists a Vitali subset of R which is measurable with respect to a certain
?-finite diffused translation-invariant measure defined on ? ? -algebra containing all
Lebesgue measurable subsets ofR (this is due to Khararazishvili A983, 1991, 1996)).
Along these lines the following theorem of Khararazishvili A980) generalizes
Theorem 2.2 in the case of Abelian groups.
THEOREM 2.5. Let G be an uncountable Abelian group acting on itself by left shifts.
Then there exists a subset A of G nonmeasurable with respect to any ?-finite invariant
measure on G.
It seems open whether the same holds for arbitrary groups. Let us call a subset A of an
arbitrary G-space X absolutely nonmeasurable if it does not belong to the domain of any
?-finite invariant measure on X. Khararazishvili A983, 1991, 1996) gave more examples

1106
P. Zakrzewski
of absolutely nonmeasurable sets in certain natural G-spaces but the problem, posed in
Khararazishvili A996), to find necessary and sufficient conditions for the existence of an
absolutely nonmeasurable subset of X, is still open.
We may also look at sets nonmeasurable with respect to extensions of a given invariant
measure. Answering a question of A. Pelc A986), S. Solecki A993a) obtained the
following general result, particular instances of which were earlier proved by
Khararazishvili A978) (the case of a translation invariant extension of the Lebesgue measure on E"
and Pelc A986) (the case of a left invariant extension of a (left) Haar measure on a locally
compact, ?-compact topological group; see also Hewitt and Ross A963, 16.13)).
THEOREM 2.6. Suppose G is an uncountable group acting on a set X and ? is ? ?-finite
invariant measure on X. If the action of G is ?-free, then each set of positive measure
contains a subset nonmeasurable with respect to any invariant extension of ?.
The above holds, in particular, for every uncountable group G acting on itself by left
shifts; this gives another strengthening of Theorem 2.2.
Let us emphasize a remarkable role played by Vitali sets in the constructions of sets
nonmeasurable with respect to invariant measures. If we define a Vitali set of a subgroup ?
of a group G acting on a set X as a selector of the collection of all Я-orbits, then it
is certainly true to say that playing with Vitali sets is the basic technique leading to the
results above.
Measurability properties of Vitali sets with respect to invariant measures were
investigated by Solecki A993b). He showed that under the hypotheses of Theorem 2.6 one can
always find a countable subgroup of G whose Vitali sets behave just like the original Vitali
sets of the rationals on the real line. More precisely, there exists a countable subgroup ?
of G such that each Vitali set of ? is nonmeasurable with respect to any invariant
extension of ?. On the other hand, under certain assumptions a subgroup of full cardinality
with properties like that cannot be constructed: If the cardinality of the group G has
uncountable cofinality, G acts freely on a set X and ? is ? ?-finite invariant ergodic measure
on X, then there exists an invariant extension ? of ? such that each subgroup ? of G
with \H\ = \G\ has a ?-measurable Vitali set. For the special case of additive subgroups
of ? the latter was strengthened by A. Nowik A996/97) who proved that there exists a
translation invariant extension of the Lebesgue measure on ? which measures at least one
Vitali set of each uncountable subgroup o/E. Recently, Solecki B001) showed the same
in the context of invariant extensions of the Haar measure on any locally compact, second
countable Abelian group.
Vitali sets of subgroups of ? were also studied by J. Cichon, A.B. Khararazishvili and
B. Weglorz A993). They proved, in particular, that // ? is an uncountable analytic (i.e.,
a continuous image of a Borel set) subgroup o/E then there exist a Vitali set of ? which
is Lebesgue measurable and all Vitali sets of ? are measurable with respect to a certain
isometrically invariant extension of the Lebesgue measure on E.
More information on various set-theoretic aspects of the theory of invariant measures
can be found in Khararazishvili A998).
We complete this section with a result which shows that the assumption that invariant
measures under consideration are ?-finite is crucial for all results concerning the existence
of nonmeasurable sets.

Measures on algebraic-topological structures
1107
We say that a measure m defined on a ? -algebra ? of subsets of a set X is semi-finite or
semiregular if every set of positive measure contains a set of positive finite measure.
Zakrzewski A987) and earlier in the Abelian case Pelc A982), solving a problem posed
by V. Kannan and S.R. Raju A980), proved the following theorem.
THEOREM 2.7. If G is an arbitrary large group, then there exists a universal semi-finite
invariant measure on G.
3. Extensions of invariant measures
We have seen in Section 2 that given a ?-finite invariant measure ?: ? -> [0, +oo]
on a G-space X satisfying some additional conditions, there are nonmeasurable sets
which cannot be "made measurable" with respect to any invariant extension of ?.
A complementary question is whether there is a nonmeasurable set which can be "made
measurable" by an invariant extension of ?. More precisely, whether a proper invariant
extension of ?, i.e., an invariant measure ? defined on an invariant ?-algebra ?' D ? and
such that ?| ? = ?, exists at all. This problem will be referred to as the invariant measure
extension problem. A particularly important instance of it, concerning isometrically
invariant extensions of the Lebesgue measure, whose investigations were initiated by
E. Szpilrajn (Marczewski) A935), will be discussed in Section 6.
The obvious necessary condition for the existence of a proper extension of a given
invariant ?-finite measure ?:?-> [0, oo] on a G-space X is that it is nonuniversal,
i.e., ? ? V(X). Moreover, without loss of generality we will always assume that the
measure ? is diffused. (For let С = {x e ?: ?({?}) > 0); С is a countable subset of X
and ? = ?, + ?? where ?\, ?2: ? -> [0, +??] are defined by ?? (A) = ?(? П С) and
?2(?) = ?(? \ C). If ?@ > 0, then ?? is a universal measure on X and it follows that
the measure ? has an invariant extension if and only if so does the measure ?2, which is
clearly diffused.)
In general, there are the following basic ways of obtaining an invariant extension
?:?'-> [0, +Oo]of ?:
(I) Find a set ? <? ? such that for any sequence (g„: ? e N) of elements of G, the
union UneN Sn В has inner measure zero. Let I consist of all subsets of the unions
of this form.
Then define:
?'= {? ? ?: A e ? and N el}.
?(???) = ?(?)
(see Szpilrajn (Marczewski) A935)).
(II) Find an almost invariant set ? ^ ?, then define:
?'= {(A, nZ)U(A2\Z): A\,A2eE\,
?((?| П ?) U (A2 \ ?)) = i · [?*(?, П ?) + ?*(?2 \ ?)
+ ?*(?|??) + ?*(?2\?)],

1108
P. Zakrzewski
where ?* and ?* stand for the inner and outer measure for ?, respectively - see
Los and Marczewski A949).
Recall that a set X is small, if there is no ?-finite diffused universal measure on X, i.e.,
there is no real-valued-measurable cardinal ^ \X\; otherwise X is large.
A. Hulanicki A962) used method II to prove the following general result.
THEOREM 3.1. Let G be a group of permutations of a set X such that \G\ ^ | X \ and X is
small. If ? is an invariant ? -finite diffused measure which is uniform, i.e., every subset of X
of cardinality smaller than \ X \ has measure zero, then ? has a proper invariant extension.
Pelc A986) applied method I to solve the invariant measure extension problem for the
case of an Abelian group acting on itself as follows.
THEOREM 3.2. Let G be an uncountable Abelian group acting on itself by left shifts.
Then every ?-finite invariant measure on G has a proper invariant extension.
Pelc conjectured that the same is true for any uncountable group G. A refinement of
Hulanicki's argument that lead to Theorem 3.1 above shows that it is indeed the case
when G is small. So one cannot falsify Pelc's conjecture on the basis of ZFC alone,
the nonexistence of large sets being relatively consistent with ZFC (see, e.g., Fremlin
A993, Ue). It is also known that the conjecture is true for some other algebraically or
topologically defined classes of groups (see Zakrzewski A996)) but the general question is
still open.
Thus the invariant measure extension problem for uncountable groups of permutations
is not solved even for the case of the left-shift action of a group on itself. On the other
hand, subsequent investigations started in Pelc A986) and then continued in Krawczyk
and Zakrzewski A991), brought a satisfactory solution for countable permutation groups.
So let ?: ? —> [0, +oo] be a ?-finite diffused measure on X, invariant with respect to a
countable group G of permutations of X.
A Hulanicki-style argument, based on method II above, shows that if X is small, then ?
always has a proper invariant extension.
On the other hand, if X is large, then ? can be universal (see the remarks following
Theorem 2.2). So assume that ? is defined on a proper ?-algebra ? of subsets of X. We
may additionally assume that ? is complete and every invariant subset of X is measurable,
since otherwise we can immediately extend ? using methods I or II, respectively.
The following result from Krawczyk and Zakrzewski A991) reveals the structure of
such measures. We say that a partition ? of a subset ? of X is a G-partition if A e ?
implies gA e ? for every g e G. We call ? a minimal G-partition of ? if there is no
G-partition ?' ? ? of ? with the property that every element of ? is the union of finitely
many elements of P'. It turns out that the structure of ? can be completely described in
terms of G-partitions of G-orbits.
THEOREM 3.3. Assume that G is a countable group of permutations of a set X and
? : ? —» [0, oo] is an invariant ?-finite complete diffused measure defined on X such that
every invariant subset А с X is in the ?-algebra ?. Let S be a selector of G-orbits.

Measures on algebraic-lopological structures
1109
Then there exists a function s ь* Ps, determined uniquely up to a measure zero set,
which assigns to each s e S a countable G-partition Ps of the orbit Gs, with the following
property:
(i) The ? -algebra ? consists of all sets Ac. X such that for ?-almost every s e S, the
set ADGs is the union of some pieces of the partition Ps.
Moreover:
(ii) For any A e ?,
?(?)= ]? к ¦ ?(? C\\J{Gs: s eSandAHGs ePf}).
JteNU[3c)
where ? is a selector of the family {Ps: s e 5") and Ps consists of sets which are
unions of к elements of the partition Ps (oo 0 = 0 and oo-t = oo if t ? 0).
(iii) The measure ? does not have a proper invariant extension if and only if ?-almost
every partition Ps is minimal.
Now the point is that properties of G -partitions of the orbit Gx of a point ? e X
are reflected by algebraic properties of G and the stabilizer of x, i.e., the group Gx —
{g e G: gx = g]. This is due to the fact that every G-partition ? of Gx has the form
? = {gHx: g e G), where ? is a subgroup of G containing G,- Moreover, such a P is
minimal if and only if every proper subgroup of ? containing G.( has infinite index in ?
(see, e.g., Kargapolov and Merzljakov A979)).
In view of this, we have the following corollary.
THEOREM 3.4. Suppose G is a countable group of permutations of a set X.
(i) If every nontrivial subgroup ? ofG has a proper subgroup of finite index in H, then
every ?-finite nonuniversal invariant measure ? on X such that the action of G is
?-free has a proper invariant extension.
(ii) If there exists a nontrivial subgroup ? of G, every proper subgmup of which has
infinite index in H, X is large and the action of G on X is free, then there exists a
? -finite nonuniversal invariant measure on X without proper invariant extension.
The additive group Q of the rationals is an example of a group every proper subgroup of
which has an infinite index. It follows that ifQ acts freely on a large set X, then there exists
a nonuniversal invariant ?-finite (or even a probability) measure on X without proper
invariant extension. Further applications to measures on Euclidean spaces R", invariant
with respect to groups of isometries of W, will be discussed in Section 6.
4. Invariant measures on Polish groups
In the previous sections we considered discrete G-spaces and discussed properties of
measures which were related only to the algebraic structure of underlying spaces. Now
we are going to let topology enter the scene. In order to make the interplay between

1110
P. Zakrzewski
measure and topology more transparent we decided to stay within the realm of standard
Borel spaces.
Let В(У) or ?(?) denote the ?-algebra of Borel subsets of a topological space (?, ?),
i.e., the ?-algebra of subsets of ? generated by the open sets of Y. A measurable space
(?, ?) is a standard Borel space if either X is countable and ? = V(X) or it is isomorphic
to (R, B(R)), i.e., if there is a bijection / between X and R such that A e ? if and only if
f[A] e B(R), for A^X. Equivalently, ? = ?(?) for a certain topology ? on X which is
Polish, i.e., admits a compatible separable complete metric (see Kechris A995, 12.5)).
By a Borel measure on a standard Borel space (?, ?) we mean a measure defined on ?.
A Borel measure on a Polish space X is a measure defined on the ?-algebra B(X). Note
that if m is a ?-finite Borel measure on a standard Borel space (?, ?), then m is diffused
if and only if it is nonatomic, i.e., any set of positive measure splits into two sets, each
having positive measure.
A Polish group is a topological group whose topology is Polish (a survey of some basic
facts from the theory of Polish groups may be found in the book of Becker and Kechris
A996)). Examples of Polish groups include (for more examples see Becker and Kechris
A996,1.3)):
(i) Hausdorff second countable locally compact groups: countable groups (with the
discrete topology), Lie groups, the Cantor group Z^;
(ii) (X, +), where X is a separable Banach space;
(iii) The group of homeomorphisms H(X) of a compact metric space (X,d), where
the group operation is composition, with the topology it inherits as a Gg subset
of C(X, X), the Polish space of all continuous functions / : X -> X with the sup
metric du(f, g) = supx€Xd(f(x), g(x)), for /. g e C(X, X);
(iv) The group of isometries lso(X,d) or just Iso(X) of a complete separable metric
space (X,d) with the topology of pointwise convergence. It can be proved that if
the space (X,d) is compact, then lso(X,d) is a compact subgroup of H(X). By
a result of J. van Dantzig and B.L. van der Waerden A928), if the space (X,d) is
locally compact and connected, then the group \so(X,d) is locally compact (for
recent extensions of the latter result to more general classes of spaces see Clemens
et al. B000)).
The problem of existence of a ?-finite invariant Borel measure on a Polish locally
compact group was solved by Haar A933), as is well known. The uniqueness part of the
following result is due to von Neumann A936).
THEOREM 4.1. Let G be a Polish locally compact group acting on itself by left shifts.
Then there exists a unique, up to a multiplicative positive constant, ?-finite invariant Borel
measure on G.
This measure is called the (left) Haar measure on G and we will denote it by ??-
Similarly, there is a right Haar measure ?'? defined by
?'?(?) = ??(?-*) forSeB(G).
In order to realize the essential difference of approach compared with the existence
arguments from Section 1, let us briefly sketch a more or less standard proof of existence

Measures on algebraic-topological structures
1111
of Haar measure (the "covering ratios" proof - for details see Parthasarathy A977,
Proposition 54.2) or Mycielski A974)).
Proof. Fix a compatible metric d in G and let Be denote the open neighborhood of the
identity of radius ?. Without loss of generality we assume that ? ? is compact. For any
compact set К с G and any ? ^ 1 let
?(?,?) = min{|C|: С is a finite covering of К with left shifts of Be}.
and then put
?(?)= lim _ ' ' ,
n^T E(B\, 1/n)
where ? is a nonprincipal ultrafilter of subsets of N.
Finally, for every open set U с G, let
?(?/) = sup{A(tf): К С f/, K compact}
and for any В е B(G) define
?(?) = inf(?(?/): В с U, U open}. ?
From this proof it may be seen that Haar measure is finite on compact sets. In particular,
we have the following proposition.
PROPOSITION 4.2. If G be a Polish locally compact group, then the following are
equivalent:
(i) Haar measure on G is finite.
(ii) The group G is compact.
For a compact group G it is customary to assume that y.c(G) = 1.
The uniqueness of Haar measure does not imply that the left and right Haar measures
on G coincide (even up to a multiplicative constant), since a (left) Haar measure is in
general not right-invariant. On the other hand, left Haar measure is always right-quasi-
invariant, i.e., quasi-invariant with respect to right shifts. This follows from the following
more general fact, which is also the first step towards von Neumann's proof of uniqueness
of Haar measure.
PROPOSITION 4.3. Let mi and m,- be arbitrary ?-finite measures on a Polish locally
compact group G. If mi is left quasi-invariant and m, is right quasi-invariant, then
m/ ~ mr. Consequently, m/ ~ mr ~ \xq.

1112
P. Zakrzewski
PROOF. Take an arbitrary set В е B(G). By Fubini's theorem and the appropriate quasi-
invariance of the measures under consideration we have
mi(B) = 0 <s> mi(g~] ?) = 0 for w/-almost all g e G
<?> mr(Bh~]) = 0 for mr-almost all h e G
&mr(B) = 0. ?
It is known that any Polish group admits a left-invariant compatible metric (the Birkhoff-
Kakutani theorem - for a proof see, e.g., Berberian A974, p. 28)). In view of a result of
S. Banach A937), if d is a left-invariant compatible metric on a Polish locally compact
group G, then (left) Haar measure on G is invariant under all isometries of the metric
space (G, d) (for various generalizations of this result see Segal A949), Segal and Kunze
A978, Corollary 7.5.1), Bandt A983)). If (left) Haar measure on G is right-invariant, G
is called unimodular. Since right shifts are isometries of the space (G,d) if and only if
the metric d is right-invariant, it follows that every Polish locally compact topological
group admitting a compatible (two-sided) invariant metric is unimodular. Examples of
such groups include Abelian groups and compact groups.
The next result reveals another important property of Haar measure and is the second
step towards the proof of its uniqueness.
PROPOSITION 4.4. A Haar measure on a locally compact Polish group is ergodic.
PROOF. Let A e B(G) be ?? -almost-invariant. It means that хд = хг1Д,дс-а.е., where
Хд is the characteristic function of A in G. By Fubini's theorem there is ho e G such
that хд(/?о) = хя-|д(Ло) for дс-а.е. g e G. Equivalently, either gho e A for ??-а.с.
g e G or gho <? A for ??-а.с. g e G. By the (right) quasi-invariance of ?? it follows
that ??(? \ A) = 0 or ??(?) — 0, respectively. ?
Now the uniqueness of Haar measure follows at once from Proposition 1.11.
A consequence of uniqueness is that unimodular groups may also be characterized as
follows: G is unimodular if and only if Haar measure on G is inversion-invariant, i.e.,
invariant under the transformation g ь* g_l. (Proof. If ?? is right-invariant, then ?'0 is
left-invariant. Then, by uniqueness, there is a positive constant с such that ??(?~]) =
?'?(?) - с ¦ ??(?) for any В e B(G). Now it suffices to take as В any symmetric set,
i.e., such that В = B~', of finite positive measure (for example the closure of a symmetric
neighborhood of the identity of a sufficiently small diameter) to conclude that с = 1. The
other implication is easy.)
Another consequence of uniqueness is the existence of a continuous homomorphism
A of G into the multiplicative group of positive real numbers satisfying the following
property:
??(?8) = ?(8)??(?) for all g e G, A e B(G).
The function ? is called the modular function of the group G. Clearly, G is unimodular if
and only if A(g) = 1 for all g e G. From this it becomes obvious why compact groups are

Measures on algebraie-topological structures
1113
unimodular. (Proof. If G is compact, then 1 = ??(?) = ??^?) = Mg)^c(G) = A(g)
for any g & G.) It is also easy to see that the function ? is the Radon-Nikodym derivative
???/??'?.
It should be underlined that the basics of Haar measure theory, surveyed above,
were appropriately generalized by A. Weil A940) to arbitrary locally compact groups.
Also, Weil's proof of uniqueness is entirely different from that of von Neumann (other
uniqueness arguments were given by H. Cartan A940) and D.A. Raikov A942)).
A complete account of Haar measure in general locally compact groups may be found
in many standard textbooks (for example, see Hewitt and Ross A963), Nachbin A965),
HalmosA974)).
Now let us turn to Polish groups which are not locally compact. S. Ulam (quoted by
J.C. Oxtoby A946)) proved that if G is a Polish, non-locally-compact gmup and m is any
left quasi-invariant Borel measure assuming at least one positive finite value, then every
open subset of G contains uncountable many pairwise disjoint left shifts of a compact
set of positive measure. Combining this with the theorem on existence of Haar measure
we immediately obtain a characterization of locally compact groups in terms of invariant
measures.
THEOREM 4.5. Assume that G is a Polish group. Then the following are equivalent:
(i) G admits a (left) quasi-invariant Borel ? -finite measure.
(ii) G is locally compact.
This characterization can be considerably strengthened. A standard Borel group is
a group G together with a ?-algebra ? such that (G. ?) is a standard Borel space and
the mapping (x,у) ь* xy~f is a Borel function from G ? G into G. If G is a standard
Borel group, then there is, of course, a Polish topology ? such that ? = ?(?) but it is not
necessarily true that there is one for which (G. ?) is a topological group. On the other hand,
if such a topology exists, then it is unique (see, e.g., Kechris A995, 12.24)).
A. Weil A940) (a weaker result for a broader class of groups) and G.W Mackey A957)
proved that // G is a standard Borel group and there exists a quasi-invariant Borel ?-
finite measure on G, then there is a topology ? giving its Borel structure and making G
a Polish locally compact group. The idea of a proof is to use a quasi-invariant Borel
probability measure ? on G to embed G into the unitary group U(H) of the Hubert space
? = L2(G, B(G), ?) (see, e.g., Parthasarathy A977, §56)).
The Mackey-Weil theorem shows that Polish locally compact groups can be
characterized in the following way.
THEOREM 4.6. Assume that G is a standard Borel group. Then the following are
equivalent:
(i) G admits a quasi-invariant Borel ? -finite measure.
(ii) G carries a Polish topology which gives its Borel structure and makes it a locally-
compact group.
Though, as we have just seen, in a non-locally-compact Polish group G, for any ? -finite
Borel measure on G, the ?-ideal in B(G) of measure zero subsets of G is neither left nor

1114
P. Zakrzewski
right invariant, it turns out that every Polish group G carries a certain natural (two-sided)
invariant ?-ideal in B(G). The following concept was introduced by J.PR. Christensen
A972) for the Abelian case and then extended by F. Tops0e and J. Hoffmann-J0rgensen
A980), and J. Mycielski to all Polish groups.
We say that a Borel (or, more generally, a universally measurable, i.e., measurable with
respect to the completion of every ?-finite Borel measure on G) subset A of G is Haar
null, if there exists a Borel probability measure m on G such that m(gAh) = 0 for all
g.heG.
It is not difficult to prove that in Polish locally compact groups Haar null Borel sets
coincide with the Borel sets of Haar measure zero. In an arbitrary Polish group G, Haar
null Borel sets form a ?-ideal in B(G) which is (two-sided) invariant, i.e., closed under
left and right shifts (the closure under countable unions requires proof- see Christensen
A972)). Another similarity of Haar null sets with the Haar measure zero sets, at least for
Abelian groups, is exhibited by the following result of Christensen A972) (for the case of a
(not necessarily Abelian) locally compact group this is the well known Steinhaus property
which is due to H. Steinhaus A920) in the case of G = Ш and to Weil A940) in general).
If G is an Abelian Polish group and A is a Borel (or universally measurable) set which is
not Haar null, then the set AA~' contains an open neighborhood of the identity of G. It
seems to be open, whether the same is true for a broader class of Polish groups.
Motivated by another property of the Haar measure zero sets in locally compact groups
Christensen asked, whether every collection of pairwise disjoint, universally measurable
sets which are not Haar null is countable. R. Dougherty A994) obtained negative answer
for many particular non-locally compact groups. Finally, Solecki A996) proved that // G
is a Polish non-locally-compact group admitting a (two-sided) invariant metric, then there
exists an uncountable collection of closed, pairwise disjoint sets which are not Haar null.
This leads to another characterization of the local compactness of a Polish group (admitting
an invariant metric) in measure theoretic terms.
THEOREM 4.7. Assume that G is a Polish group, admitting an invariant metric. Then the
following are equivalent:
(i) The ? -ideal of Haar null Borel subsets of G satisfies the countable chain condition
in B(G).
(ii) G is locally compact.
It again seems to be open, whether the same is true for a broader class of Polish groups.
Let us close this section by pointing out an important role played in the results above by
the assumption that invariant measures under consideration are ?-finite. Namely, Oxtoby
A946) proved the following theorem.
THEOREM 4.8. Every Polish group carries a Borel semi-finite invariant measure.
5. Invariant measures on Polish G-spaces
A G-space X is called a standard Borel G-space if G is a standard Borel group, X is
a standard Borel space and the action of G on X is a Borel function from G ? X into X.

Measures on algebraic-lopological structures
1115
If G is a Polish group, X is a Polish space and the action is continuous, then X is called a
Polish G-space. By a fundamental theorem of Becker and Kechris A996, Theorem 5.2.1)
// G is a Polish group, then any standard Borel G-space (?, ?) is Borel isomorphic to
a Polish G-space, i.e., ? = ?(?) for a certain Polish topology ? on X which makes the
action of G on X a continuous function from G ? ? into X.
Polish G-spaces have long been studied in several areas of mathematics including
ergodic theory, operator algebras and group representation theory. Of particular interest
are actions of Polish locally compact groups which cover the case of the group of integers
acting on X via the map (к, x) ь* Tkx, where Г is a homeomorphism of X, flaws, i.e.,
R-actions and actions of Lie groups. Other examples of Polish G-spaces include:
(i) The action of a Polish group G on itself by (say) left shifts.
(ii) The canonical action of a Polish group G on the quotient space G/H of all left-
cosets of a closed subgroup ? of G (it is a Polish space with the quotient topology
consisting of all U с G/H such that n~' [U] is open in G, where ?(?) = ? ? is
the canonical projection) given by (g, xH) ь* (gx)H, for g.x eG.
(iii) The action of the group of homeomorphisms G = H(X) on a compact metrizable
space X given by (g, x) ь* g(x) for g e H(X) and ? e X.
(iv) The action of the group of isometries G = Iso(X) on a Polish space X given by
(g,x)i-+ g(x) for g elso(X) and ? e X.
An excellent account of the topic (and more examples) is given in Becker and Kechris
A996).
The case when G is locally compact is the most important one as far as invariant
measures are concerned. There is, however, a general result valid for Borel actions
of arbitrary Polish groups, which in this case strengthens the existence theorems from
Section 1. Recall that Tarski's theorem asserts that the existence of a finitely additive
probability universal invariant measure on a G-space X follows already from the
assumption that the set X is not paradoxical. Becker and Kechris A996, Theorem 4.2.1)
and earlier M.G. Nadkarni A990) in the case G is countable, proved the following exact
analog of Tarski's theorem for countably additive measures on Polish G-spaces.
THEOREM 5.1. Let G be a Polish group and X a standard Borel G-space. Then the
following are equivalent:
(i) There exists an invariant Borel probability measure on X.
(ii) The space X is not countably paradoxical in Borel pieces.
Note that by the result of Becker and Kechris, quoted at the beginning of this section,
we may assume that X is actually a Polish G-space. Then the crux of the proof of the
nontrivial implication ((ii) => (i)) lies in the case G is countable. For assume that we have
already proved the theorem for the countable case. Let Go be a countable dense subgroup
of G and assume that X is not countably G-paradoxical in B(X). Then X is not countably
Go-paradoxical and by the countable case there exists a Go-invariant probability Borel
measure ? on X. It turns out that ? is also G-invariant, due to the following general
fact which may be established by a straightforward continuity argument. In a Polish
G-space any Borel probability measure invariant under a countable dense subgroup ofG
is G-invariant (it should be noted that the latter is not true for ? -finite measures).

1116
P. Zakrzewski
The countable case was proved by Nadkarni for Z-actions but, as noted by Becker and
Kechris, the argument works as well for any countable group with some easy modifications
in the proof. The following outline is extracted from Becker and Kechris A996, Section 4).
Let us assume that G is a countable group of homeomorphisms of a Polish space X and
that ? ? Nc, the ?-ideal in B(X) consisting of all G-negligible Borel subsets of X, i.e.,
sets A e B(X) such that X contains pairwise disjoint subsets A„ e B(X), ? e N, with each
A„ ~oc A in B(X). This is equivalent to the assertion that X is not countably paradoxical
in B(X). We assume, moreover, that all orbits are infinite, since otherwise the result holds
trivially.
Denote by B' (X) the ?-algebra of invariant Borel sets and recall that since the group G
is countable, A* = {Jg€G gA is the minimal invariant set containing a Borel set А с X.
The idea of the proof is similar to that of the proof of the Hopf-Kawada theorem
(Theorem 1.2), outlined in Section 1.
Given sets А, В e B(X) we write A < В if there is С е B(X) with С ~oc A in B(X),
С с ? and for every ? e В, С П Gx is a proper subset of В П Gx.
Given sets A, B,C e B(X) and ? e N we write
A = nB®C
if A = ? ? U · · · U An U С where ? ?,..., ?„, С are pairwise disjoint Borel sets with each
A, ~XB inB(X).
The following two lemmas are analogs of Lemmas 1.3 and 1.4.
LEMMA 5.2. For every Borel sets А, В с X there is a partition (up to a set from Nc)
a*=\Jq„
of A* into pairwise disjoint Borel sets Qn e ?' (?), ? e N, such that for each ? e N
А П Qn = n(B П Q„) ? tf„, wfcere tf„ < ? ? ?>„.
Moreover, the partition is unique, up to a set from Nc-
For every Borel sets ?, ? с X and .? e X, let
?, if л: е Q„,
[A/B](x)=ln .
1 0, otherwise,
in the notation of the preceding lemma.
LEMMA 5.3. There is a decreasing sequence ofBorel sets Cn with Co = X< C,* = X and
[C,,/C,,+ |](.x) ^ 2 modulo ?q, for every ? e N.
Taking the lemmas above for granted, the proof is carried out as follows.

Measures on algebraic-topological structures
1117
Fix a sequence (C„) from Lemma 5.3 and for any A e B(X) and ? e X let
[А/С„](дО
m(A,;t) = hm .
It can be proved that this definition is correct for all ? outside a set from Nc and the
function m(A, x) has the following properties:
(i) 0 ^ m(A, ?) ? 1 and m(X, x) = 1;
(ii) m(A, x) — 0 modulo Nc if and only if A e Nc;
(iii) m(gA,x) = m(A, gJt) = w(A, x) modulo Nc;
(iv) m(|J,ieN A„,;t) = HT=am(An<x) modulo Nc, whenever A„, ? e ?, arepairwise
disjoint Borel sets;
(v) if Q is an invariant Borel set and ? e Q, then
m(A, x) = m(A П Q,x) modulo Nq.
Now fix an invariant countable open base U for the topology of X and let
?(?) = infl ]? ?(?„): A„eW. А с (J A„ for A e B(X),
^;ieN /? eh' '
where ? :U ^> [0, oo] is defined by
T(A) = m(A,xQ),
for a suitably chosen *о е X, so that the function ? is an invariant probability Borel
measure on X (the point is that checking this requires only countably many conditions
of the form (iii) and (iv) above plus w@, .vq) = 0, hence the set ? of exceptional points is
in Nc and, since X & No, we may choose *? e X \ E).
For details see Nadkarni A990), Becker and Kechris A996).
As an immediate consequence of Theorems 4.1, 4.5 and 5.1, and Proposition 4.2 we
have the following characterization of compact Polish groups.
THEOREM 5.4. If G is a Polish group, then the following are equivalent:
(i) G is not countably paradoxical in Borel pieces.
(ii) G is compact.
Let us now consider actions of Polish locally compact groups. Classically, of particular
importance is the case when a locally compact group G acts continuously on a compact
metric space X, i.e., when X is a compact Polish G-space. N.N. Bogoliouboff and
N.M. Kryloff A937) proved that every compact Polish Ъ-space admits an invariant Borel
probability measure. This was later extended by S.V. Fomin A950) to actions by solvable
groups. More generally, we say that a Polish locally compact group G is topologically
amenable if there exists a finitely additive invariant Borel probability measure on G. Thus
every compact group is topologically amenable, the left Haar measure being a witness of

1118
P. Zakrzewski
its amenability. Other examples include Abelian and solvable groups (see Wagon A986,
p. 162)). (According to a theorem of A.L.T. Paterson A986) topologically amenable groups
are characterized among Polish locally compact groups as exactly these groups which are
not paradoxical in Borel pieces - this should be compared with Theorem 5.1 above.)
The following general theorem (see, e.g., Greenleaf A969) or Plante A983)) establishes
a necessary condition for the existence of invariant Borel probability measures on compact
Polish G-spaces.
THEOREM 5.5. Let G be a Polish locally compact topologically amenable group. Then
every compact Polish G-space X admits an invariant Borel probability measure.
A functional analytic proof of Theorem 5.5 is based on the fact that a locally compact
topologically amenable group G satisfies the following version of the Markov-Kakutani
fixed-point theorem: if G acts continuously on a compact convex subset К of a locally
convex topological space ? so that all the transformations ? ь* gx, g e G, are affine
(i.e., g(ax + A — a)y) = agx + A — a)gy for x,y e К and 0 ^ a ^ 1), then they have
a common fixed point (see, e.g., Wagon A986, p. 163)).
For an arbitrary Polish space X, let P(X) be the space of all probability Borel measures
on X, equipped with the topology generated by the maps m ь* f f dm, where/ varies over
the set of all bounded continuous real-valued functions on X. Then P(X) is a Polish space.
If, moreover, X is compact, then so is the space P(X) and, by the Riesz Representation
Theorem, it may be viewed as a convex, compact subspace of the dual space C(X, K)*
equipped with its weak*-topology (see Kechris A995, Section 17E).
Now the proof runs as follows: let ? = C(X, Ш)*, К = P(X) and consider the action
of G on К defined by g? = \xg~', where \xg~' (A) = ?(g~' A) for A e B(X). Since this
action is continuous, the Markov-Kakutani theorem, quoted above, gives a common fixed
point ? e P(X), which is an invariant probability Borel measure on X. (This should be
compared with the proof of Theorem 1.7 of Hajian and Ito, outlined in Section 1.)
Since the group Iso(X) of all isometries of a compact metric space (X, d) is compact,
hence topologically amenable, and X is a compact Polish Iso(X)-space, we have the
following corollary.
PROPOSITION 5.6. IfX is a compact metric space, then there exists an isometry-invariant
Borel probability measure on X.
Returning to general considerations, suppose that G is a Polish locally compact group
and X is an arbitrary Polish G-space. Denote by INVc the set of all invariant probability
Borel measures on X. Then INVc is a convex subset of the Polish space P(X). Let EINVc
denote the set of all ergodic measures ? e INVc ·
It follows from Theorem 5.1 that Nq is the collection of all Borel sets A^ X such that
?(?) = Ofor every ? e INVG. (Proof. Without loss of generality assume that the group G
is countable. If A ? Nc, then A* <? Nq and by Theorem 5.1 there is a measure ? e INVc
with ?(?*)= 1.)
A thorough study of INVG was undertaken by R.H. Farrel A962) and VS. Varadarajan
A963). Their main results may be summarized as follows.

Measures on algebraic-lopological structures
1119
THEOREM 5.7. Let G be Polish locally compact group and X a Polish G-space.
(i) Any measure ? e INVc is determined by its values on B' (X). Moreover, //INVc ?
0, then any probability measure defined on B'(X) which vanishes on every set
AeB1 (X) П No has the unique extension to a certain measure ? e INVc ·
(ii) If there exists an invariant pmbabilit}· Borel measure on X, then there also exists
an ergodic one.
(iii) The sets INVc and EINVG are Borel in the space P(X).
(iv) (A uniform ergodic decomposition theorem) If INVc ? 0, then there is a Borel
mapping ?, ? : ? ь* ?? from X onto EINVc such that
• the mapping ? is constant on G-orbits, i.e., figx = Д, for ? e X, g e G;
• for every ? e EINVG, ifXv = {x e ?: ?? = ?), then ?(?,) = 1;
• for any ? e INVc,
?(?)= / ??{?)??(?) for all A eB(X). (*)
Jx
Moreover, the decomposition map ? is uniquely determined modulo No-
Some remarks are in order concerning the above result.
• Assertion (i) is related to Theorem 1.5. The point is that when G is locally compact,
then for any Borel subset A of a Polish G-space X there exists a set A e B' (X) with
?(? ? A) = Ofor any ? e INVG such that A e B*(X) (where B*(X) is the ?-algebra
of all Borel subsets of X which are almost invariant with respect to ?). It follows that
the values of any measure ? e INVc on B*(X) are determined by its values on B'(X).
This also implies that a measure ? e INVG is ergodic if and only if?(?) = 0 or 1 for
any A eB'(X).
• In the "moreover" part of (i) the assumption that INVc ? 0 may be omitted. Indeed,
in view of Theorem 5.1, this already follows from the existence of any probability
measure defined on B' (X) and vanishing on each ? &?'(?)??? (since this implies
XiNG).
• Assertion (ii) is related to the fact that when G is locally compact, then EINVc is the
set of all extreme points of the convex set INVc (this is known to be false for arbitrary
Polish groups).
• In assertion (iv) we obtain a partition of X into invariant Borel sets Xv with the
property that (by (*)) ? e EINVc is the only invariant Borel probability measure
on the standard Borel G-space X,., i.e., Xv is uniquely ergodic. Note that this does
not imply that Xv is a single G-orbit. But it is so if, for instance, the group G
is compact (see remarks following Theorem 5.8 below); in this case the partition
{Xv: ? e EINVc} consists of all G-orbits and if Xv = Gxo, then the measure ? is
defined by
v(A) = Mc({?eG: gx0 e A)) forAeB(X),
where ? с is the Haar measure on G.

1120
P. Zakrzewski
Some generalizations of the Farrel-Varadarajan's results to quasi-invariant measures
were obtained by Y.I. Kifer and S.A. Pirogov A972), K. Schmidt A977), A. Ditzen A992)
and G. Greschoning and K. Schmidt B000).
So far in this section we have been looking at invariant probability measures only. The
case of ?-finite measures is less clear in general but it is best understood when the group G
is locally compact. Given a Polish G-space X let us say that X is (G-)homogeneous if the
action of G is transitive, i.e., if X coincides with the only G-orbit. Recall that ? is the
modular function of the group G and G.< = {g e G: gx = x} is the stabilizer of a point
? & X. Note that a transitive action of G on a Polish space X can be identified in a natural
way with the canonical action of G on the quotient space G/Gx; thus all homogeneous
G-spaces are essentially of the form G/H, where Я is a closed subgroup of G.
The study of invariant Borel measures on homogeneous G-spaces was undertaken by
A. Weil A940). The main results may be summarized as follows.
THEOREM 5.8. Suppose that G is a Polish locally compact group and X is a Polish
G-homogeneous G-space. Fix a point xq e X and denote by ? the modular function of the
group GX0.
(i) There exists a quasi-invariant probability Borel measure on X and any two such
measures are equivalent.
(ii) There exists an invariant ?-finite Borel measure on X if andonly ifS = A\GXo, the
restriction of ? to G ,<·„· Such a measure, if it exists, is unique, up to a constant factor,
and ergodic.
PROOF (Sketch). In order to prove (i), it suffices to take a probability Borel measure
? ~ ?? on G and then define
v(A) = ?({# e G: gx0 e A}) for A e B(X).
The equivalence of any two quasi-invariant ?-finite measures and the uniqueness of an
invariant ?-finite measure can be proved by a Fubini-type argument similar to that used for
the case of the left action of G on itself - see Propositions 4.3 and 4.4.
So the main part of the proof is to show that under the assumption that A\G.f0 = <5, there
is a non-negative Borel function ?: X —>¦ R such that the measure ? defined by
?(?)= / p(x)dv(x) forAeB(X).
JA
is G-invariant. For details see, e.g., Parthasarathy A977, Proposition 55.6). ?
In particular, a unique invariant Borel probability measure on a Polish homogeneous
G-space X exists, if the group G is a compact. On the other hand, X may possess such a
measure without G being compact - conditions for its existence were studied by S.P Wang
A976) and T.S. Wu and J.S. Yang A986).
If G is a unimodular locally compact Polish group but ? is its closed subgroup which is
not unimodular, then the quotient G-space G/H admits no invariant ?-finite Borel measure
(an example is given in Parthasarathy A977, Exercise 55.8)).

Measures on algebraic-topological structures
1121
A special case of a homogeneous G-space is a homogeneous Polish metric space (X,d),
i.e., an Iso(X)-homogeneous Polish Iso(X)-space. Recall that the group Iso(X) is Polish
but not necessarily locally compact, even if X is so. However, Banach A937) proved
that // X is a homogeneous Polish locally compact metric space, then there exists an
isometrically invariant Borel ? -finite measure ? on X. The existence of Borel measures on
metric spaces with various invariance properties was also studied by L.L. Loomis A945),
J. Mycielski A974, 1977), С Bandt A983), С Bandt and G. Baraki A986) and S. Gao and
A. S. Kechris B001, Theorem 6.10).
The more general problem of existence of a Haar-like Borel measure on a locally
compact topological space X, invariant with respect to a group G of homeomorphisms
of X, have also been considered by several authors. The following general result is due to
R.C. SteinlageA975).
THEOREM 5.9. Let X be a Polish locally compact space and G a group of
homeomorphisms of X onto itself Suppose that, for every disjoint pair of compact subsets of X, there
is a nonempty open set U С X such that no image ofU under any element ofG intersects
both of the compact sets. Then there exists an invariant ?-finite Borel measure ? on X.
In particular, the condition holds ifG is a group of isometries of a Polish locally compact
metric space (X, d). In this case, if at least one G-orbit is dense in X, then ? is positive on
open sets, finite on compact sets and is a unique, up to a constant factor, invariant Borel
measure on X with these properties.
As a matter of fact, Steinlage proved the theorem in a more general framework of
arbitrary locally compact Hausdorff spaces - it strengthened earlier results of Banach A937),
Segal A949), Segal and Kunze A978, Theorem 7.2), Y. Mibu A958) and J. Poncet A954)
(see also J.W. Roberts A974/75) where a similar result for compact spaces is proved).
Even if a given Polish G-space is not homogeneous, we can apply Theorem 5.8 to the
induced action of G on an arbitrary G-orbit (which is an invariant Borel subset of X -
see Becker and Kechris A996, 2.3.3)). In particular, it follows that if G is a Polish locally
compact group, then every Polish G-space X admits a quasi-invariant Borel probability
measure. But in fact the latter can also be proved by another method which gives some new
information (see, e.g., Kechris A994, Proposition 3.30)).
PROPOSITION 5.10. Let G be a Polish locally compact group, X a Polish G-space and
m a probability Borel measure on X. Then there exists a quasi-invariant probability Borel
measure m on X with the following properties:
(i) m'(A) = m(A)foranyAeBi(X);
(ii) m' <g ? for any quasi-invariant probability Borel measure ? on X with m«n;
(iii) m is ergodic if and only if m is ergodic.
PROOF. Let ? с be a (left) Haar measure on G and take a probability Borel measure
? = ?? on G. Put
m'(A)= J m(g-,A)d?(g) forAeB(X).
It is easy to see that this works.
D

1122
P. Zakrzewski
Let us finally briefly touch the case when G is a countable group and X is a standard
Borel G-space. In this case the existence of an invariant (or quasi-invariant) probability
(or ?-finite) Borel measure on X is really a property of the orbit equivalence relation Eg,
whose equivalence classes are the G-orbits.
Let us say that an equivalence relation ? on a set X is countable if every equivalence
class is countable. The equivalence relation ? с is Borel (as a subset of ? ? ?) and
countable. Conversely, J. Feldman and C.C. Moore A977) proved that if ? is a countable
Borel equivalence relation on a standard Borel space X, then there is a countable group G
and a Borel action of G on X such that ? = Eq. Now, given such an E, a ?-finite
Borel measure ? on X is called ?-invariant (E-quasi-invariant, respectively) if for any
Borel set А с X and a Borel 1-1 map f:A-+ X with f(x)Ex for all ? e A, we have
?(f[A]) = ?(?) (?(/[?]) = 0 <s> ?(?) = 0, respectively). It is easy to check that this is
equivalent to the assertion that ? is G-invariant (G-quasi-invariant, respectively) for any
countable group G and a Borel action of G on X with ? = Eq.
As we have already seen (Theorem 5.1), Nadkarni A990) proved that the existence
of an ?4nvariant probability Borel measure on X is equivalent to the fact that X is not
countably paradoxical in Borel pieces with respect to any Borel action of a countable group
inducing E. Let us say that ? is compressible if there is a Borel map /: X —>¦ X such that
for each ? e X, f[[x]E] is a proper subset of the equivalence class [x]e of x.
Now, Nadkarni's theorem may be stated in the following way, with no appeal to any
action that induces E.
THEOREM 5.11. Suppose ? is a countable Borel equivalence relation on a standard
Borel space X. Then the following are equivalent:
(i) There exists an ?-invariant Borel probability measure on X.
(ii) ? is not compressible.
Hence the existence of an f-invariant Borel probability measure on X tells us something
about the structure of E.
On the other hand, a ? -finite ?-invariant Borel measure on X trivially exists for any
countable Borel equivalence relation ? and it may even be proved that there always exists
a diffused one, provided that ? has uncountably many equivalence classes. Proof. By a
theorem of J. Silver A980), since ? is a Borel equivalence relation on a standard Borel
space X with uncountably many equivalence classes, there is a nonempty perfect set S с X
which meets every equivalence class in at most one point. Let m be a Borel probability
diffused measure on S and suppose G = {g„: ? e N}, go = id*, is an arbitrary countable
group of Borel automorphisms of X with Ec — E. Let S„ = g„S and define an f-invariant
?-finite Borel measure ? on X as follows:
?(A) = YJm(sng;l AnU,\\JSk
forAeB(X).
However, the existence of an ?-quasi-invariant ?-finite diffused Borel measure which
is additionally E-ergodic, again expresses an important property of the equivalence
relation E.

Measures on algebraic-topological structures
1123
We say that a ?-finite Borel measure ? on X is E-ergodic if ?(?) = 0 or ?(? \ A) = 0
for any invariant A e B(X), where a subset А с X is called invariant if it is the union of
all equivalence classes of its elements. This is equivalent to asserting that ? is G-ergodic
for any countable group G of Borel automorphisms of X with ? = Eg ¦ It follows from
Proposition 5.10 that if there exists a diffused E-ergodic probability Borel measure on X,
then there also exists an ?-quasi-invariant one.
A Borel equivalence relation ? on a standard Borel space X is called smooth if there is
a standard Borel space ? and a Borel map f.X^Y with xEy <s> f(x) = f(y) for any
x, у e X. It might be proved that// ? is countable, then ? is smooth if and only if ? has a
Borel transversal (see Kechris A995, 18.D)).
The following theorem is closely related to the so-called Glimm-Effros dichotomy - see
Becker and Kechris A996, Section 3.4).
THEOREM 5.12. Suppose ? is a countable Borel equivalence relation on a standard
Borel space X. Then the following are equivalent:
(i) There exists an ?-quasi-invariant, ergodic, diffused Borel probability measure
onX.
(ii) ? is not smooth.
It is not difficult to prove that if a countable aperiodic (i.e., with all equivalence classes
infinite) Borel equivalence relation ? is smooth, then it is compressible. But if ? is not
compressible, then by Theorems 5.11 and 5.7(ii), there even exists an ?-invariant, ergodic,
diffused Borel probability measure on X. An example of a nonsmooth compressible ? is
the Vitali equivalence relation on R (i.e., xEy «==> ? — у е Q).
6. Isometncally invariant measures on Euclidean spaces
The study of isometncally invariant measures on the Euclidean space W is particularly
appealing, since it is directly connected to the search for a "good" notion of a measure that
would generalize the notions of the length of an interval, the area of a region and the volume
of a solid. Though the Lebesgue measure ?„ : ?„ -> [0, oo] has been widely accepted as
the ultimate solution to the problem of finding a satisfactory way to "measure" complicated
subsets of E", we have to face the fact that some Lebesgue nonmeasurable sets still exist
(at least if we accept the axiom of choice - see Wagon A986, Chapter 13) for a discussion
of this point). One way to cope with this disadvantage is to consider isometrically invariant
extensions of ?„.
Let G be a subgroup of Iso(R") - the group of all isometries of R". Recall from Section 3
that given an invariant ?-finite measure ? defined on a ?-algebra ? of subsets of M." there
are two basic ways of obtaining an invariant extension Д : ?' —>¦ [0, +00] of ?.
The first is to find a nonmeasurable set such that the G-invariant ?-ideal I it generates
in V(R"), consists of sets of inner ?-measure zero. The extension ? is then defined on the
?-algebra generated by ? UI so that ?(?) = 0 for any N e I (see (Szpilrajn, 1935)).
The second boils down to finding a ?-almost invariant set A ? ? and then defining ?
on the ?-algebra generated by ? and A (see Los and Marczewski A949)).

1124
P. Zakrzewski
The existence of a proper isometrically invariant extension of ?„ was proved by Szpilrajn
(Marczewski) A935) using the first method. Szpilrajn constructed a generalized Bernstein
set ? с R" with the property that any countable union of isometric copies of Z, is neither
contained in nor disjoint from an arbitrary nonempty perfect subset ? of R". In the
same paper Szpilrajn stated the problem posed by Sierpinski, whether every isometrically
invariant extension of Lebesgue measure in R" has a proper isometrically invariant
extension. It is well known that if the continuum is large, then there exists an extension
of ?„ to a universal measure, i.e., a measure defined on V(R") (see, e.g., Fremlin A993,
1D)). However, no such extension can be isometrically invariant. Indeed, the classical Vitali
subset of R" is nonmeasurable with respect to any translation-invariant extension of ?«.
(In fact, by a theorem of Khararazishvili A978), every set of positive Lebesgue measure
contains a subset nonmeasurable with respect to any translation invariant extension of ? ?.)
Moreover, by Theorem 2.2, no universal, translation invariant, ?-finite measure on R"
exists at all. Thus no isometrically invariant extension of ?„ is universal and Sierpinski's
problem amounts to asking if a maximal one exists.
The negative answer, i.e., the theorem that no maximal isometrically invariant measure
on R" exists was first obtained independently by S.S. Phakadze A958) and A. Hulanicki
A962) under the additional assumption that the continuum is small (this is closely related to
Theorem 3.1). Then Khararazishvili A980) proved it without any additional set-theoretical
assumptions for the one-dimensional case. Finally, K. Ciesielski and A. Pelc A985) got
the negative answer in any dimension, showing even more than was required.
THEOREM 6.1. Suppose G is any subgroup o/Iso(R") containing all translations. Then
every G-invariant ?-finite measure on R" has a proper G-invariant extension.
The Ciesielski and Pelc's proof is also based on the first method of extending invariant
measures, described above. More precisely, it is showed that the space R" is a countable
union of G-absolutely negligible sets, where a set N с R" is called G-absolutely negligible
if the G-invariant ?-ideal, it generates in VCR"), consists of sets of inner measure zero
with respect to every G-invariant ?-finite measure on R". Now, given a ?-finite G-
invariant measure ?: ? —>¦ [0, oo] on R", if R" = IJisN ^' -wnere eacri ? 's G-absolutely
negligible, then N,0 ? ? for at least one /'о е N.
The idea of partitioning R" into countably many G-absolutely negligible sets is due to
Khararazashvili A983) who used it for the case when G is the group of all translations
of R". A more detailed account of the long history of Sierpinski's problem can be found in
Ciesielski A989, 1990).
In view of Theorem 6.1, Ciesielski and Pelc A985) stated the problem of characterizing
those subgroups G of Iso(R") for which the conclusion of their result remains true.
Using ideas of Hulanicki A962) they noticed that if the continuum is small, then no
restrictions on G are needed. (For the reasons, explained at the beginning of Section 3, we
will assume that all measures, we want to extend, are diffused.)
Ciesielski A990) proved, in particular, that it is enough to require that either the
action of G is transitive (this was earlier proved by Khararazishvili under the additional
assumption of the Continuum Hypothesis) or G contains uncountably many translations.
In fact, in the same paper Ciesielski studied a more general question: given a subgroup G of

Measures on algebraic-topological structures
1125
Iso(R") and a ?-finite G-invariant diffused measure ? on R" does there exist a proper G-
invariant extension of ?? He proved that it is so, provided that G contains an uncountable
subset ? with the property that every two different elements of ? coincide exactly on
a set of inner measure ? zero. Finally, Zakrzewski A990) noticed how to apply the key
technical tool from Ciesielski's paper to get the following strengthening.
THEOREM 6.2. Suppose G is a subgmup o/Iso(R") and let ? be a ?-finite G-invariant
diffused measure on R". If ? does not vanish on the set of points with uncountable
G-orbits, then ? has a proper G-invariant extension.
As a corollary we have the following solution to Ciesielski-Pelc's problem.
THEOREM 6.3. Let G be a subgroup o/Iso(R"). The following are equivalent:
(i) Every G-invariant ?-finite diffused measure on R" has a proper G-invariant
extension.
(ii) Either the continuum is small or exactly one of the following holds:
• all G-orbits are uncountable:
• there exists a common fixed point of all elements of G and all other G-orbits
are uncountable.
Theorem 6.3 reduces the problem of finding necessary and sufficient conditions for
extendabilityofagiven G-invariant ?-finite diffused measure on R" to the case in which G
is countable. This case was fully examined by Krawczyk and Zakrzewski A991) as a
particular instance of the general problem, whether a given ?-finite measure, invariant with
respect to a countable group of permutations of a set X, has a proper invariant extension.
The solution to the latter was presented in Section 3 (see Theorems 3.3 and 3.4). Combining
it with Theorem 6.3 leads, in particular, to the following characterization.
THEOREM 6.4. Let G be a subgroup o/Iso(R"). The following are equivalent:
(i) Every nonuniversal G-invariant ?-finite diffused measure on R" has a proper
G-invariant extension.
(ii) The set of all points ? e R" such that the G-orbit Gx is countable and there is
a nontrivial minimal G-partition ofGx is small.
On the other hand, Krawczyk and Zakrzewski A991) obtained the following theorem,
related to the original Sierpinski's problem.
THEOREM 6.5. Suppose G is any subgroup of Iso(R"). Then every nonuniversal
G-invariant ?-finite extension of the Lebesgue measure in R" has a proper G-invariant
extension.
Note that when we consider arbitrary groups of isometries, we have to assume that a
G-invariant extension of ?„, we want to extend, is nonuniversal. Indeed, as has already
been mentioned, if the continuum is large, then ?„ can be extended to a universal measure.
But as a matter of fact such an extension is rather poorly invariant with respect to
isometries, as another result from Krawczyk and Zakrzewski A991) shows.

1126
P. Zakrzewski
PROPOSITION 6.6. Assume that the continuum is large and let G be a subgroup of
Iso(E"). Then the following are equivalent:
(i) There is an extension of the Lebesgue measure to a G-invariant universal measure
onR".
(ii) The group G is discrete.
Here "discrete" refers to the topology inherited from Iso(E") and is equivalent to the fact
that each G-orbit consists of isolated points. For example, we can imagine an extension of
the Lebesgue measure to a universal measure invariant with respect to all translations by
the integers but this definitely is a very weak property.
To emphasize the fact that the non-discreteness of a group G implies that certain
extensions of ?„, even by countably many new sets, cannot be G-invariant, let us quote
an observation from Zakrzewski A991b): IfG is a non-discrete subgroup o/Iso(E"), then
every open set А С ?" is countably G-equidecomposable with R" (in arbitrary pieces).
Until now we discussed the basic question, whether a given invariant measure can
be properly extended in the invariant way. But there are also some interesting results
concerning particular properties of such extensions. For example, if we move from measure
spaces to the respective measure algebras, then we see that extending a measure ?:?—>
[0, +oo] to ?: ?' —>· [0, +oo] does not necessarily mean extending the associated measure
algebra. More precisely, it may happen that
WYeZ'BXeE ?(???) = 0. (*)
This means that the canonical embedding e: ?/? —>¦ ?'/? of measure algebras defined
by e([A]M) = [?]? is an isomorphism. Let us call such an extension weak, otherwise call
it strong.
Clearly, the first of the two methods of extension, recalled at the beginning of this
section, produces a weak extension, while the second gives a strong one. It was already
realized by Szpilrajn A935) that there exists a strong isometrically invariant extension of
the Lebesgue measure ?„ on R" .
Every Iso(E")-invariant weak extension of ?„ is Iso(E")-ergodic. Khararazishvili
A983) proved that there exists a strong Iso(E" yinvariant extension ?/?„ which is Iso(E")-
ergodic.
Another property of a weak extension is that it does not change the character of the
measure space (?", ?,?), i.e., the smallest cardinality к of a collection {Aa: a < к) с ?
such that for any set A e ? and every ? > 0 there is a < к with ?(? ? ??) < ?.
For the Lebesgue measure the character is Ко and the question, whether there exists
a translation invariant extension of the Lebesgue measure on ? of uncountable character,
was already stated by Szpilrajn A935). The affirmative answer was given by K. Kodaira
and S. Kakutani A950). In fact, Kakutani and Oxtoby A950) constructed a translation
invariant extension of ? ? of the maximal possible character (see also Hewitt and Ross
A963, 16.3), where a generalization to the case of Haar measure on a compact metric
group is proved).
THEOREM 6.7. There exists a translation invariant extension of the Lebesgue measure
on R of the character 22N|.

Measures on algebraic-lopological structures
1127
Using the same technique Khararazishvili A978) proved that there is an Iso(R")-
invariant extension ? : ?' —>¦ [0,+oo] of the Lebesgue measure on R" of the character
2? such that no restriction of ? to an lso(R")-almost invariant (with respect to ?) set
? e ?' is ergodic.
After realizing that the methods by which Sierpinski's problem was solved lead to weak
extensions, Ciesielski A990/91) posed the problem, whether every isometrically invariant
?-finite measure on R" has a strong isometrically invariant extension. The affirmative
answer is given in Zakrzewski A995/96), where it is also proved that ifG is any subgroup
o/Iso(R"), then every complete ?-finite G-invariant measure on R" which has a proper
G-invariant extension, has a strong G-invariant extension as well.
We conclude this section with another result from Ciesielski and Pelc A985): If the
continuum is large, then there exists a universal semi-finite isometrically invariant measure
onR".
References
Arnold, K. A968), On ?-finite invariant measures, Z. Wahrscheinlichkeitsth. verw. Geb. 9, 85-97.
Banach, S. A937), On Haar's measure. Appendix to Theory of the Integral by S. Saks, 2nd rev. edn, Monografie
Matematyczne, Vol. 7, Warszawa-Lwow.
Banach, S. and Tarski A. A924), Sur la decomposition des ensembles de poits en parties respectivement
congruents. Fund. Math. 6, 244-277.
Bandt, С A983), Metric invariance ofHaar measure, Proc. Amer. Math. Soc. 87, 65-69.
Bandt, C. and Baraki, G. A986), Metrically invariant measures on locally homogeneous spaces and hyperspaces.
Pacific J. Math. 121, 13-28.
Becker, H. and Kechris, A.S. A996), The Descriptive Set Theory of Polish Group Actions, London Math. Soc.
Lecture Note Series, Vol. 232.
Berberian, S.K. A974), Lectures in Functional Analysis and Operator Theory, Springer-Verlag, New York.
Bogoliouboff, N.N. and Kryloff, N.M. A937), La theorie generate de la mesure dans son application a I'etude
des systems dynamiques de la mecanique non-lineare, Ann. of Math. 38, 65-113.
Calderon, A.P. A955), Sur les mesures invariantes, С R. Acad. Sci. Paris 240, 1960-1962.
Cartan, H. A940), Sur la mesure de Haar, С R. Acad. Sci. Paris 211, 759-762.
Cichoii, J., Khararazishvili, A.B. and W?glorz, B. A993), On sets ofVitali's type, Proc. Amer. Math. Soc. 118,
1243-1250.
Christensen, J.PR. A972), On sets ofHaar measure zero in abelian Polish groups, Israel J. Math. 13, 255-260.
Chuaqui, R.B. A977), Measures invariant under a group of transformations. Pacific J. Math. 68, 313-329.
Ciesielski, K. and Pelc, A. A985), Extensions of invariant measures on Euclidean spaces. Fund. Math. 125. 1-10.
Ciesielski, K. A989), How good is Lebesgue measure?. Math. Intelligencer 11, 54-58.
Ciesielski, K. A990), Algebraically invariant extensions of ?-finite measures on Euclidean spaces. Trans. Amer.
Math. Soc. 318, 261-273.
Ciesielski, K. A990/91), Query 5, Real Anal. Exchange 16 A), 374.
Clemens, J.D., Gao, S. and Kechris, A.S. B000), Polish metric spaces: their classification and isometry groups.
Preprint.
Dang-Ngoc-Nghiem A973), On the classification of dynamical systems, Ann. Inst. H. Poincare 9 D), 397^425.
Ditzen, A. A992). Ergodic measures for Borel group actions. Preprint.
Dougherty, R. A994), Examples of non-shy sets. Fund. Math. 144, 73-88.
Dye, H.A. A959), On groups of measure preserving transformations 1, Amer. J. Math. 81, 119-159.
Dye, H.A. A963), On groups of measure preserving transformations 11, Amer. J. Math. 85, 551-576.
Erdos, P. and Mauldin, R.D. A976), The nonexistence of certain invariant measures. Proc. Amer. Math. Soc. 59,
321-322.
Farell. R.H. A962), Representation of invariant measures, Illinois J. Math. 6, 447^467.

1128
P. Zakrzewski
Feldman, J. and Moore, C.C. A977). Ergodic equivalence relations and von Neumann algebras. I, Trans. Amer.
Math. Soc. 234, 289-324.
Fomin, S.V. A950), On measures invariant under certain groups of transformations. Izv. Akad. Nauk SSSR 14.
261-274 (in Russian).
Fremlin, D.H. A993), Real-valued-measurable cardinals. Set Theory of the Reals, H. Judah ed., Israel Math.
Conf. Proa, Vol. 6, 151-304.
Fremlin, D.H. B000), Measure Theory. Vol. 3, Internet: ftp essex.ac.uk/pub/measuretheory.
Friedman, N.A. A970), Introduction to Ergodic Theorx. Van Nostrand-Reinhold. New York.
Gao, S. and Kechris, A.S. B001), On the classification of Polish metric spaces up to isometry. Mem. Amer. Math.
Soc., to appear.
Greenleaf, F-P. A969), Amenable actions of locally compact groups. J. Funct. Anal. 4, 295-315.
Greschonig, G. and Schmidt, K. B000), Ergodic decomposition of quasi-invariant probability measures
(Dedicated to the memory of Anzelm Iwanik), Colloq. Math. 84/85 B), 495-514.
Haar, A. A933), Der Massbegrif in der Theorie der kontinuierlichen Gruppen, Ann. of Math. 34. 147-169.
Hajian, A. and It6, Y. A969), Weakly wandering sets and invariant measures for a group of transformations.
J. Math. Mech. 18, 1203-1216.
Hajian, A. and Kakutani, S. A964), Weakly wandering sets and invariant measures. Trans. Amer, Math. Soc. 110.
136-151.
Halmos, PR. A947), Invariant measures, Ann. of Math. 48, 735-754.
Halmos, PR. A956), Lectures on Ergodic Theory. Publ. Math. Soc. Japan.
Halmos, PR. A974), Measure Theory. Springer. Berlin.
Hewitt, E. and Ross. K.A. A963). Abstract Harmonic Analysis, Vol. I. Grundl. Math. Wiss., B. 115. Springer,
Berlin.
Hopf, E. A932), Theory of measures and invariant integrals. Trans. Amer. Math. Soc. 34, 373-393.
Hulanicki, A. A962), Invariant extensions of the Lebesgue measure. Fund. Math. 51. 111-115.
Kakutani, S. and Oxtoby, J.C. A950), Construction of a non-separable extension of the Lebesgue measure space.
Ann. of Math. 52 B), 580-590.
Kannan. V and Raju, S.R. A980). The nonexistence of invariant universal measures on semigroups, Proc. Amer.
Math. Soc. 78, 482^484.
Kargapolov, M.I. and Merzljakov, Ju.I. A979), Fundamentals of the Theory of Groups. Graduate Texts in Math..
Springer-Verlag, Berlin.
Kawada, ? A944), Uber die Existenz der invarianten lntegrale. Japan. J. Math. 19, 81-95.
Kechris, A.S. A994), Lectures on definable group actions and equivalence relations. Preprint.
Kechris, A.S. A995), Classical Descriptive Set Theory. Graduate Texts in Math., Vol. 156, Springer-Verlag,
Berlin.
Khararazishvili, A.B. A975), On certain types of invariant measures. Dokl. Akad. Nauk SSSR 222, 538-540 (in
Russian).
Khararazishvili, A.B. A978), Some Questions of Set Theory and Measure Theory. Izd. Tbiliss. Gos. Univ., Tbilisi
(in Russian).
Khararazishvili, A.B. A980), Absolutely nonmeasurable sets in Abelian groups, Soobshch. Akad. Nauk Gruzin.
SSR 97, 537-540.
Khararazishvili, A.B. A983), Invariant Extensions of Lebesgue Measure. Izd. Tbiliss. Gos. Univ., Tbilisi (in
Russian).
Khararazishvili, A.B. A991), Vitali's Theorem and its Generalizations. Izd. Tbiliss. Gos. Univ.. Tbilisi (in
Russian).
Khararazishvili, A.B. A996), Selected Topics of Point Set Theory. Lodz University, Lodz.
Khararazishvili, A.B. A998), Transformation Groups and Invariant Measures. Set-Theoretic Aspects, World
Scientific, River Edge, NJ.
Kifer, Y.I. and Pirogov, S.A. A972), The decomposition of quasi-invariant measures into ergodic measures, Usp.
Mat. Nauk 27 E), 239-240.
Kodaira, K. and Kakutani, S. A950), A non-separable translation invariant extension of the Lebesgue measure
space, Ann. of Math. 52 B), 574-579.
Krawczyk, A. and Zakrzewski, P. A991), Extensions of measures invariant under countable groups of
transformations, Trans. Amer. Math. Soc. 326, 211-226.

Measures on algebraic-topological structures
1129
Krieger, W. A969a), On non-singular transformations of a measure space, I. Z. Wahrscheinlichkeitsth. verw.
Geb, 11, 83-97.
Krieger, W, A969b), On non-singular transformations of a measure space, II. Z. Wahrscheinlichkeitsth. verw.
Geb. 11,98-119.
Laczkovich, M. B002), Paradoxes in measure theory. Handbook of Measure Theory, E. Pap. ed., Elsevier,
Amsterdam, 83-123.
Loomis, L.L. A945), Abstract congruence and the uniqueness ofHaar measure, Ann. of Math. 46 B), 348-355.
Los, J. and Marczewski, E. A949). Extensions of measure. Fund. Math. 36, 267-276.
Mackey, G.W. A957), Borel structures in groups and their duals. Trans. Amer. Soc. 85, 134-165.
Mibu, Y. A958), On measures invariant under given homeomorphism group of a uniform space. Math. Soc.
Japan. 10, 405^429.
Mycielski, J. A974), Remarks on invariant measures in metric spaces, Colloq. Math. 32, 105-112.
Mycielski, J. A977), A conjecture of Ulam on the invariance of measure in Hubert's cube, Studia Math. 60, 1-10.
Nachbin, L. A965), The Haar Integral, Van Nostrand, Princeton, NJ.
Nadkarni, M.G. A990), On the existence of a finite invariant measure, Proc. Indian Acad. Sci. (Math. Sci.) 100,
203-220.
Nowik, A. A996/97), On a measure which measures at least one selector for every uncountable subgroup. Real
Anal. Exchange 22 B), 814-817.
Omstein, D.S. A960), On invariant measures. Bull. Amer. Math. Soc. 66, 297-300.
Oxtoby, J.C. A946), Invariant measures on groups which are not locally compact. Trans. Amer. Math. Soc. 60,
215-237.
Parthasarathy, K.R. A977), Introduction to Probability and Measure, Macmillan India, Madras.
Paterson, A.L.T A986), Nonamenability and Borel paradoxical decompositions for locally compact groups, Proc.
Amer. Math. Soc. 96, 89-90.
Pelc, A. A986), Invariant measures and ideals on discrete groups, Dissertationaes Math., Vol. 255, PWN,
Warszawa.
Pelc, A. A982), Semiregular invariant measures on Abelian groups, Proc. Amer. Math. Soc. 86, 423^426.
Penconek, M. and Zakrzewski, P. A994), The existence of nonmeasurable sets for invariant measures, Proc.
Amer. Math. Soc. 121, 579-584.
Phakadze, S.S. A958), The theory of Lebesgue measure, Trudy Tbiliss. Mat. Inst. 25 (in Russian).
Plante, J.E. A983), Solvable groups acting on the line. Trans Amer. Math. Soc. 278. 401^414.
Poncet, J. A954), Une classes d'espaces homogenespossedant line mesure invariante, C. R. Acad. Sci. Paris 238,
553-554.
Raikov, D.A. A942), A new proof of uniqueness ofHaar's measure, Dokl, Akad. Nauk SSSR 34, 211-212.
Ramachandran, D. A989), ? new proof of Hopf's theorem on invariant measures. Contemp. Math. 94, 263-271.
Ramachandran, D. and Misiurewicz, M. A982), Hopf's theorem on invariant measures for a group of
transformations, Studia Math. 74, 183-189.
Roberts, J.W. A974/75), Invariant measures in compact Hausdorff spaces. Indiana Univ, Math. J. 24, 691-718.
Rosenblatt, J.M. A974), Equivalent invariant measures. Israel J. Math. 17. 261-270.
Rosenblatt, J.M.( 1980), Strongly equivalent invariant measures. Math. Proc. Cambridge Philos. Soc. 88, 33^43.
Ryll-Nardzewski, С and Telgarsky, R. A978), The nonexistence of universal invariant measures, Proc. Amer.
Math. Soc. 69, 240-242.
Sakai, K. A978), Remarks on the existence of finite invariant measures for groups of measurable transformations,
Proc. Japan Acad. Ser. A Math. Sci. 54, 285-287.
Schmidt, K. A977), Lectures on Cocycles of Ergodic Transformation Groups, Lecture Notes in Math., Vol. 1,
MacMillan India, New Delhi.
Segal, I.E. A949), Invariant measures on locally compact spaces, J. Indian Math. Soc. 13, 105-130.
Segal, I.E. and Kunze, R.A. A978), Integrals and Operators, Springer-Verlag, Berlin.
Silver, J. A980), Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann.
Math. Logic 18, 1-28.
Solecki, S. A993a), On sets nonmeasurable with respect to invariant measures, Proc. Amer. Math. Soc. 119,
115-124.
Solecki, S. A993b), Measurabilitv properties of sets ofVitali's type, Proc. Amer. Math. Soc. 119, 897-902.

изо
P. Zakrzewski
Solecki, S. A996), On Haar null sets. Fund. Math. 149 C), 205-210.
Solecki, S. B001), Translation invariant ideals, Israel J. Math., to appear.
Steinhaus, H. A920), Sur les distances des points des ensembles de mesure positive. Fund. Math. 1, 93-104.
Steinlage, R.C. A975), On Haar measure in locally compact 7i spaces, Amer. J. Math. 97, 291-307.
Szpilrajn, E. A935), Sur /'extension de la mesure lebesguienne. Fund. Math. 25, 551-558.
Tarski, A. A938), Algebraische Fassungdes Massproblems, Fund. Math. 31, 47-66.
Tops0e, F. and Hoffmann-J0rgensen, J. A980), Analytic sets and their applications. Analytic Sets, Academic
Press, San Diego, pp. 317^401.
van Dantzig, J. and van der Waerden, B.L. A928), Uber metrisch homogene Raume, Abh. Math. Sem. Univ.
Hamburg 6, 374-376.
Varadarajan, VS. A963), Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109, 191-220.
von Neumann, J. A936), The uniqueness ofHaar's measure, Rec. Math. [Mat. Sb.] 1 D3). 721-734.
von Neumann, J. A940), On rings of operators III, Ann. of Math. 41.94-161.
von Neumann, J. A998), Continuous Geometry. Princeton Landmarks Math., Princeton Paperbacks, Princeton
University Press, Princeton, NJ.
Vitali, G. A905), Sul problema delta mesura dei gruppi di punti di una retta. Bologna.
Vladimirov, D.A. A965), Invariant measures on Boolean algebras. Mat. Sb. (N.S.) 67 A09), 440-460 (in
Russian).
Wagon, S. A986), The Banach-Tarski Paradox, Cambridge Univ. Press, Cambridge, UK.
Wang, S.P. A976), On density properties of certain subgroups of locally compact groups, Duke Math. J. 43,
561-578.
Weil, A. A940), [.'integration dans les groupes topologicues et ses applications, Actualites Sci. Indust., Publ.
Math. Inst. Strasbourg, Hermann, Paris.
Wu, T.S. and Yand, J.S. A986), Homogeneous space with finite invariant measure, Tamkang J. Math. 17, 153-
159.
Zakrzewski, P. A987), The existence of universal invariant semiregular measures on groups, Proc. Amer. Math.
Soc. 99, 507-508.
Zakrzewski, P. A988), On universal semiregular invariant measures, J. Symb. Logic 53, 1170-1176.
Zakrzewski, P. A989), The existence of universal invariant measures on large sets. Fund. Math. 133, 113-124.
Zakrzewski, P. A990), Extensions of isometrically invariant measures on Euclidean spaces, Proc. Amer. Math.
Soc. 110,325-331.
Zakrzewski, P. A991a), Paradoxical decompositions and invariant measures, Proc. Amer. Math. Soc. Ill, 533-
539.
Zakrzewski, P. A991b), When do equidecomposable sets have equal measures?, Proc. Amer. Math. Soc. 113,
831-837.
Zakrzewski, P. A993a), The existence of invariant probability measures for a group of transformations, Israel J.
Math. 83, 343-352.
Zakrzewski, P. A993b), The existence of invariant ?-finite measures for a group of transformations, Israel J.
Math. 83, 275-287.
Zakrzewski, P. A994), When do sets admit congruent partitions. Quart. J. Math. Oxford 45 B), 255-265.
Zakrzewski, P. A995/96), Extending isometrically invariant measures on R" -A solution to Ciesielski's query.
Real Anal. Exchange 21 B), 582-589.
Zakrzewski, P. A996), Extending invariant measures on topological groups. The Proceedings of the Tenth
Summer Conference on Topology and Applications, Ann. New York Acad. Sci., Vol. 788, pp. 218-222.
Zakrzewski, P. A997), The uniqueness of Haar measure and set theory, CoWoq. Math. 74, 109-121.

CHAPTER 28
Liftings
Werner Strauss
Mathematisches Institut A, Universitat Stuttgart, Postfach 80 11 40. D-70511 Stuttgart, Germany-
E-mail: strauss@mathematik.uni-stuttgart.de
Nikolaos D. Macheras
Department of Statistics. University of Piraeus, 80 Karaoli and Dimitriou str, 185 34 Piraeus, Greece
E-mail: macheras@unipi.gr
Kazimierz Musial
Institute of Mathematics, Wroclaw University, PI. Gruiidwaldzki 2/4, 50-384 Wroclaw, Poland
E-mail: musial@math.uni. wroc.pl
Contents
Introduction 1133
1. Terminology 1133
2. Existence of liftings and densities 1135
3. Liftings for functions 1140
4. Liftings on topological spaces 1147
5. Liftings on topological groups 1154
6. Permanence of liftings 1157
6.1. Liftings in products 1157
6.2. Strong liftings in products 1161
6.3. Liftings on projective limits 1163
6.4. Various Fubini products 1166
6.5. Applications of Fubini products to stochastic processes 1168
7. Liftings for abstract valued functions 0
8. Liftings and densities with respect to ideals of sets 1173
9. Beyond ?°°(,u) 1174
10. Further applications 1175
References 1178
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1131

This page intentionally left blank

Liftings
1133
Introduction
The classical monograph of A. and С Ionescu Tulcea A969a) provides a systematic
exposition of almost all results about liftings known at that time, in particular the basic
existence theorems of J. von Neumann and D. Maharam were given with a new and
more direct proof. The significance and power of the existence of liftings was illustrated
there by important applications to mathematical analysis, e.g., to the point realization
for automorphisms of spaces of measurable functions, to disintegration of measures,
representation of integral operators which is equivalent to the differentiation of vector
valued measures, and to the separable modification of stochastic processes. By a proper
use of liftings they succeeded to put many classical results in their final form. Practically
at the same time Kolzow A968) proved the equivalence of the existence of liftings, Vitali
differentiation, and Dunford Pettis theorem for locally convex spaces.
Nevertheless many interesting problems were left and formed the starting point for
further developments in this field, such as the existence proof for densities on arbitrary
finite measure spaces of Graf and von Weizsacker A975), the negative solution of the
so-called 'strong lifting problem' by Losert A979), the discussion of the existence of
(strong) Borel liftings by Mokobodzki A975), Fremlin A977) and that of the non-existence
of translation invariant Borel liftings for Haar measures by Johnson A980), Talagrand
A982), Kupka and Prikry A983), Losert A983), and Burke A993), the notion of lifting
compactness studied by Bellow A980) as well as Edgar and Talagrand A980), existence
results for strong Baire liftings by Grekas and Gryllakis A991), the application of forcing
methods for the non-existence of certain types of liftings (an example which demonstrates
that lifting theory provides challenging problems for other areas of mathematics) by
Shelah A983), Burke and Just A991) and Burke and Shelah A992), and the discussion
of permanence results mainly in products of probability spaces starting with a paper of
Talagrand A989) and subsequently developed by Burke A995), Fremlin B00?), and the
authors to mention probably the most important ones.
Because of the large number of contributions, in this article we can only give an overview
of these developments with short indications of methods and proofs. Concerning the
abounding number of applications we had to restrict ourselves drastically to either the
most spectacular ones or to the most recent ones, giving only references for all the others.
As far as we could single out, we have tried to incorporate any paper dealing with liftings
in the list of references at the end of this article. Sometimes we found it difficult to give
full credit to authors, since many results in that field are circulated unpublished, but on the
other hand they have become folklore.
1. Terminology
For a measure space {?, ?, ?) we denote by ? /? its measure algebra (a Boolean algebra
under its canonical Boolean operations), and r : ? -> ?/? is the canonical map (a Boolean
homomorphism). We assume throughout that ? is nontrivial, i.e., ?{?) > 0. _
К stands for one of the fields R of real numbers or С of complex numbers, R for the
extended real line, and N = {1, 2, 3,...} stands for the set of natural numbers.

1134
W. Strauss et a].
?^(?) is the space of all K-valued ? -measurable maps on ?. ?^ (?) is a K-algebra
underpointwise addition and multiplication together with multiplication by scalars from K.
?°(?) := ??(?) and ?^(?), the space of all R-valued ? -measurable functions are lattices
under pointwise order. The subspace ?^(?) of ?^(?) consisting of all strictly bounded
/ e ?^(?) (i.e., Il/ll := supwei2 |/(?)| < oo), is a Banach algebra and ?3€(?) := ?f (?)
is a Banach lattice.
A set N e ? with ?(?) = 0 is called a ?-null set and the ?-ideal of all ?-null sets is
denoted by ?? and ?/ is the ideal of all sets A e ? with ? (A) < oo. For А, В е Г we
write A = ? a.e. (?), or only ? ? ? for short if there arise no doubts about the measure,
if ???, the symmetric difference of A and B, is a ?-null set, and we write f = g a.e.
(?), or / = g for short, if {/ ? g} e 2?0 for f,ge ?^(?) U ?|(?)· The equivalence class
of all functions in ?^(?) U ?^(?) or of all sets in ?, that are ?-a.e. equal to / or to A
will be denoted by /* or by A*, respectively. Equivalent functions or sets will never be
identified.
The (Caratheodory) completion of (?,?,?) is written (?,?,?). The ?-algebra
generated by a family ? of subsets of ? is denoted by ?(?).
A measure space is locally determined if ? = {A<^ ?: ?? ? & ? for A e ??) and
(?, ?, ?) (or just ?) is semi-finite, i.e., ?(?) = sup^(fi): A э В e ? f) for any A e ?.
A measure space (?, ?, ?) is localizable if it is semi-finite and, for any S c. ? there exists
an ? e ? such that (i) ? \ Я e ?0 for any ? e ?, (ii) if G e ? and E\G e ?0 for every
? e ? then ? \G e ??. It will be convenient to call such a set Я an essential supremum
of ? in ? (see Fremlin A980, A6B)).
If (?, T, v) is a measure space and /: ? -> (9 is a measurable function such that
?(?) = ?(/~'(?)) for all ? e ? then ? is called the range of ? via / and we write
? = /(?).
A pretopological measure space is a quadruple (?,?, ?, ?) such that (?, ?) is a
topological space and (?, ?, ?) is a measure space. This notion is introduced only in
order to avoid repetitions of the form (?, ?, ?, ?) such that (?, ?) is a topological space
and (?, ?, ?) is a measure space.
A topological measure space is a quadruple (?,?,?,?) such that (?,?) is a
Hausdorff topological space and (?,?,?) is a measure space with ? с ?. Let
(?, ?, ?1, ?) be a topological measure space. The measure ? is ?-additive if for any
increasing family (G,)ie/ of open subsets of i? we have
G, ) =5???@.
For any Hausdorff topological space (?, ?) we denote by ?(?) its Borel ?-algebra, i.e.,
the ?-algebra generated by T. The Baire ?-algebra of (?, ?), i.e., the ?-algebra generated
by the system of all cozero subsets of ?, is written ??(?). We call (?, ?, ?, ?) a Baire
or Borel measure space, respectively if (?, ?, ?) is the completion of a finite measure
space (?, ?, v), where ? = ??(?) or A = ?(?), respectively. Let (?, ?) beaHausdorff
topological space. A measure on ??(?) is called completion regular, if for any В е ?(?)
there exist A\, Аг &??(?) such that A\ с ? с A3 and ?(?3 \ ??) =0. A measure ? on

Liftings
1135
?(?) is completion regular if its restriction to ??{?) is completion regular. A topological
measure space (?, ?, ?, ?) is called a category measure space if ? is the system of all
sets with the Baire property with respect to ? and ?? is equal to the system of all sets of
the 1st category in ?. We use the notion of a quasi-Radon measure space in the sense of
FremlinA974,72A).
For an arbitrary probability space (?, ?, ?) we call (R, ?, ??, v) its associated hyper-
stonian space if R is the Stone space of the measure algebra of (?, ?, ?), ? the topology
generated by [s(a): a e ?/?], where s(a) с R is the corresponding closed-open set of ?
according to the Stone duality, ?? denotes the ?-algebra of all subsets of R with the Baire
property (namely those sets А с R such that ???/ is a first category set for some open
subset U of R), and ? = ??? :TZ-> R where ?:??-* ?/? is the canonical epimorphism
and ? : ? /? -> R is unambiguously defined by ?(?) := ?(?) if a = A' for АеГ.
Throughout we assume the validity of the Axiom of Choice.
2. Existence of liftings and densities
A lifting for a given measure space (?, ?, ?) is a Boolean homomorphism ?: ? —> ?
with the additional properties
P(A) = A (LI)
and
p(A) = p(B) if ? ? ?, (L2)
i.e., besides (LI) and (L2) ? satisfies more explicitly the equations
p@) = 0 and ?(?) = ?, (Bl)
р(ЛПЯ)=р(Л)Пр(Я), (В2)
and
/)(AUS) = p(A)Up(B), (B3)
if A, Be ?. It follows
p(Ac) = [p(A)]c forAeT. (B4)
Conversely (B2) and (B4) imply (B3). We denote by ?(?) the space of all liftings for
(?, ?,?).
There is another way of looking at liftings. For ? e ? (?) we can define unambiguously
a Boolean homomorphism ?': ?/? —> ? by means of
?·(?'):=?(?) ifAeT

1136
W. Strauss et al.
with the property r ? ?* = idr/?, the identical map of ?/?. For this reason it is perhaps
more appropriate to call p* lifting (J. von Neumann's original definition), since this reveals
its algebraic character more precisely. But in applications the first given definition seems
to be more in common use. It is also clear that the measure ? enters only through the ?-
ideal ?? of its null sets into the lifting. Therefore measures on ? producing the same null
set ideal produce the same liftings, i.e., liftings depend in fact on the triple (?, ?, ??),
where ?? is an ideal in ? and there are generalizations of the notion of lifting along these
lines already in the paper of von Neumann and Stone A935).
In 1931 A. Haar raised the problem of the existence of a lifting for the Lebesgue measure
space on Rd, d e N. J. von Neumann gave a positive solution based on the classical
Lebesgue density. Clearly the same problem can be raised for arbitrary measure spaces
and there too, as we will see below, densities provide a useful step in the construction
of liftings. A lower density for the measure space (?, ?, ?) is a map <5 : ? -> ? with
the properties (LI), (L2), (Bl), and (B2) and for the notion of upper density we have to
replace (B2) by (B3) therein. If we define ?1'(A) := (8(AC))C for arbitrary maps ?: ? -> ?
then the operation ? -> <5C' is a bijection from the space ?{?) of all lower densities onto
the space ? (?) of all upper densities obeying the law (<$')' = <5 for all ? e ?(?) U ? (?),
where <$'' = ? if and only if ? e ? (?). For this reason we consider only lower densities and
call them "densities" for short.
The best known example of a (lower) density (being no lifting) is the Lebesgue density
D, defined by means of
DjA) :=LeR'; Urn ^n^U))!
[ s\0 ?(?8(?)) J
for Lebesgue measurable sets А с R'', ? the Lebesgue measure on R'', and B$(x) the ball
of center ? and radius ? > 0. Lebesgue's celebrated density theorem is just (LI) while the
other axioms of a lower density follow more or less by technicalities (see, e.g., Oxtoby
A971, Theorems 3.20 and 3.21)). Starting from the Lebesgue density J. von Neumann
A931) constructed liftings for the Lebesgue measure on R1' by a process which has been
generalized (see Graf and von Weizsacker A976), Traynor A974)) to arbitrary densities
on measure spaces (?, ?,?) and, at the same time, was made more transparent in the
following way. For ? e ?{?) and ? & ? define a filterbase
?(?):={???: ? e ?(?)}
and apply the axiom of choice to find an ultrafilter ??(?) finer than ?{?). Then put
? (?) := {???: A eU(w)} if A e ?.
It follows <$(A) с р(А) с <$' (A) for A e ? and this implies for complete measure spaces
(?, ?, ?) that p(A) e ? and ? satisfies (LI), while all other properties of a lifting are
immediate by construction.
THEOREM 2.1. If the measure space (?, ?, ?) is complete, then for any ? e ?(?) there
exists ape ?(?) such that ? (А) с р(А) с ?' (A) for all A e ?.

Liftings
1137
By an application of Theorem 2.1 to the Lebesgue density the first existence result of
von Neumann A931) is now immediate.
COROLLARY 2.2. There exists a lifting for the Lebesgue measure space on Ш , d e N.
The construction above leading to Theorem 2.1 makes it obvious that the liftings of
the corollary rely on a more or less arbitrary choice of an ultrafilter in a non-constructive
way with the consequence that any trace of uniqueness or "naturalness" is hopelessly lost.
Though starting from the, in a certain sense "natural" Lebesgue density, we cannot single
out some sort of "canonical" lifting. Taking any lifting ? for the Lebesgue measure on R
obtained by this process, we cannot without further information answer such a simple
question as, e.g., 0 e p(]—oo, 0]) or 0 e p(]0, oo[). On the other hand it can be easily seen
that the axiom of choice is necessary for producing a lifting for the Lebesgue measure
space (see, e.g., Burke A993)). For general measure spaces not even a "natural" density is
at hand as it was for the Lebesgue measure space. But for finite (even incomplete) measure
spaces (?, ?,?) with ?(?) > 0 a density can be constructed by transfinite induction
using the following two extension lemmata for densities. Clearly it is sufficient to consider
probability spaces for simplicity.
LEMMA 2.3. Let (?, ?, ?) be a probability space, ? a ?-subalgebra of ? with ?? с ?,
A e ?, and denote by ? the ?-subalgebra of ? generated by ? U {A}. Then for every
? e ?(? | ?) there exists ? ? & ?(? \ ?) extending ?. For ? e ?(? \ ?) may be chosen
?<??(? \?).
First note ? = {(D П A) U (E П A'): D, ? e ?] and choose elements B.C e /? such that
B=essinf{De/j: ACZ)} and С = essinf{D e ?: А' с D}.
Then
г((?>ПА)и(?ПАг))
:= [А П <5((Z) П S) U (? ? Bc))] U [Ac ? ?((? ? С) U (D ? С'))],
if D, ? ? ? is a solution given by Graf and von Weizsacker A976).
LEMMA 2.4. Let (?, ?,?) be a probability space, (??)„?? an increasing sequence of
?-subalgebras of ? with ?? с ?\, and let ?? be the ?-subalgebra of ? generated by
U,,eN *?и· Рог each n e N let be given ?„ ??(?\??) with <5„ | ?? = <5„, for all m < n. Then
exists ? ?? e ?(? \ ??) satisfying ?? | ?„ = ?„ for all n e N.
If ?,,(/) denotes a version of the conditional expectation of / e ?°°(?) with respect to
the ? -subalgebra ? of ? we may define (following Graf and von Weizsacker A976))
ОС ОС ОС
Soo(A) = ? U ? *т(\ЕчАхл) > 1 - 1/*}) if A e ??.
к=\п~\ т=п

1138
W. Strauss et al.
From Doob's martingale convergence theorem (see, e.g., Fremlin A989, Chapter 22, 1.6))
follows <$oo(A) = A a.e. (?).
A measure space (?, ?, ?) is called strictly localizable if there exists a family (A,-),-e/
in ? of pairwise disjoint sets A,- e ? with ? (A,) < oo for all /' e / such that
{^??=?,? = {????: ? ? A,- e ? for all / e/},
/e/
and
?(?) =^д(АП A,) for A e ?.
/e/
Such a family (A,-),-e/ is called a decomposition oi ?. We will derive existence theorems
from the following strictly more general extension theorem.
THEOREM 2.5. Let be given a strictly localizable measure space (?, ?, ?), a
?-subalgebra ? of ? with ?? с ?, and a density ? e ?(? \ ?). Then there exists a density 5 e ?(?)
with ? | ? = ?.
This and the next theorem are in fact theorems about probability spaces since the
generalization to strictly localizable spaces is obvious and purely technical. We give a short
indication of the proof for a probability space (?, ?,?) for later reference. Anyproofuses
induction in one form or another. We can apply induction taking the following steps.
(A) Choose the smallest cardinal d with the property that there exists a collection
? с ? of cardinality d such that the ? -algebra generated by ? U ? is dense in ?
for the pseudometric generated by ?. Let ? = {Ma)a<K be indexed by ordinals
less than к, where к is the first ordinal of cardinality d. For a ^ к denote by ??
the ?-subalgebra of ? generated by ? U {Mp: ? < a], where we may assume
Ma ? ?? for a < к. Inductively we can construct a family (&а)а^к of densities
Sa e ?(? | ??) with ?? | ?? = Sa for a ^ ? < ?.
(B) The induction starts with So = ?.
(C) The step from a to a + 1 is covered by Lemma 2.3.
(D) For a limit ordinal ? ? ? of an uncountable cofinality put Sa = U/3<a &?·
(?) For a limit ordinal ? ?. ? of countable cofinality apply Lemma 2.4.
(F) Finally put <$:= SK.
Densities constructed in this way will become important later on in Section 6. We
therefore turn their proof in a definition and call any density ? for a probability space
{?, ?,?) admissible if it is constructed inductively taking the steps (A) to (F) above
starting from the ?-subalgebra ? =?(?0); ??{?) denotes the family of all admissible
densities for ?. Clearly ??(?) ? 0.
The last theorem and Theorem 2.1 imply the next result.
THEOREM 2.6. Let be given a strictly localizable complete measure space (?, ?, ?), a
?-subalgebra ? of ? with ?? с ?, and a lifting ? e ?(? | ?). Then there exists a lifting
~p e ?(?) with ~?\? = p.

Liftings
1139
We call any ? e ?(?) admissibly generated if there exists a density S e ??{?) such
that <5(A) с p(A) for all A e ?; ???(?) is the (for complete probability spaces clearly
non-empty) class of all admissibly generated liftings for ?.
If in the last two theorems ? is the ?-subalgebraof ? generated by ?? with the obvious
lifting on it, we get the following existence theorems of Graf and von Weizsacker A976)
for densities and of von Neumann A931) and Maharam A958) for liftings, respectively.
DENSITY THEOREM. For every nontrivial strictly localizable measure space there exists
a density.
LIFTING THEOREM. For every nontrivial strictly localizable complete measure space
there exists a lifting.
Radon measure spaces are strictly localizable since they have a concassage (see, e.g.,
Schwartz A973)) and hence the lifting theorem applies.
COROLLARY. Each nontrivial Radon measure space has a lifting.
Around 1942 J. von Neumann gave an oral proof to S. Kakutani and D. Maharam for the
lifting theorem, but "the proof was unfortunately forgotten beyond hope of reconstruction"
according to Maharam A958). In 1958 D. Maharam gave a different proof based on her
structure theorem for measure algebras, reducing the general case to the product space
{0, 1}?, where к is any infinite ordinal, hence only needing a special instance of the
martingale convergence theorem. The proof indicated above using general martingale
convergence theorem in connection with induction was given by A. and C. Ionescu
Tulcea A961) and is today, in one form or another, standard in literature. It is an open
problem whether (even for probability spaces) the lifting theorem remains true without
the assumption of completeness. The most interesting case is when measure is defined on
the ?-algebra of Borel sets. A lifting ? e ?(?) is called a Borel lifting for a topological
measure space (?, ?, ?,?) if ? (A) e ?(?) for all A e ?, and a similar definition applies
for Baire liftings. According to Shelah A983) it is consistent with ZFC that there exists no
Borel lifting for Lebesgue measure on [0, 1] . On the other hand already von Neumann
and Stone A935) proved the existence of a lifting for the Borel measure space on [0. 1]
under the assumption of the continuum hypothesis. This has been later generalized to the
following result of Mokobodzki A975) andFremlin A977).
THEOREM 2.7 (CH). Subject to the continuum hypothesis any ?-finite measure space
with a measure algebra of cardinality less or equal to ?? has a lifting.
The assumption of the strict localizability implies that the basic measure space is locally
determined. Within the class of all complete locally determined (c.l.d. for short) measure
spaces, the strict localizability is in fact a necessary condition for the existence of a lifting.
To see it we need only to consider for a measure space weaker types of decompositions,
so called decompositions (ND) and (D), respectively, i.e., families (A, )ie/ in ? f such that
? (A) = ?,??? M(A n Ai) for a11 A e ?/ as well as ?(?, ? Ay) = 0 and ?, ? A, = 0,

1140
W. Strauss et al.
respectively for /' ? j in / (see Ellis and Snow A963)). Due to an application of the
axiom of choice decompositions (ND) exist for any measure space and a lifting converts
a decomposition (ND) into a decomposition (D). If (A, ),e/ is such a decomposition (D) it
follows supi6/ A* = ?' and by Fremlin A978, Theorem 2), the measure space (?, ?, ?)
is c.l.d. It is obvious that this argument remains true for a much weaker type of "lifting", the
so-called orthogonal lifting, i.e., for a map ? from ? into itself with the properties (LI),
(L2), as well as (?) ?(?) ? ?(?) = 0 for all А, В e ? with ? ? ? = 0. Another such
weaker notion has been considered by Kolzow A968), the monotonous lifting, i.e., a map
? from ? into itself with the properties (LI), (L2), and (?) ?(?) с ?(?) if А с ? for
?, ? e ?. Any density is a monotonous lifting, and from the existence of a monotonous
lifting ? follows that of an orthogonal lifting ? by taking ?(?) := ?(?) П ?'(?) if A & ?,
see Gapaillard A973) and Strauss A971). Bichteler A972) considers pre-densities, i.e.,
maps ? from ? into itself with the properties (LI), (L2), (Bl), and ?(??)?· · ·??(?*) =0
if ? ? П ¦ · ¦ П А^ = 0 for ? ?,..., Ak e ? and shows that the existence of a pre-density
implies the existence of a density in complete measure spaces. For this reason we get the
somewhat surprising result that within the class of all c.l.d. measure spaces the existence
of a lifting is equivalent to the existence of considerably weaker types of set functions.
THEOREM 2.8. For a c.l.d. measure space (?, ?, ?) the following conditions are
equivalent.
(i) There exists a lifting for (?, ?, ?).
(ii) There exists a density for (?, ?, ?).
(iii) There exists a pre-density for (?, ?, ?).
(iv) There exists a monotonous lifting for (?, ?, ?).
(?) There exists an orthogonal lifting for (?, ?, ?).
(vi) The measure space (?, ?, ?) is strictly localizable.
Any measure space has a c.l.d. version (see, e.g., Fremlin A978)) with the same
measure algebra if the measure space is localizable. At this point we should mention that
localizable c.l.d. measure spaces are precisely spaces with the Radon-Nikodym property,
equivalently the Riesz property (see Segal A951)) and that by Fremlin A978) there exist
c.l.d. localizable, non strictly localizable measure spaces, i.e., within the class of all c.l.d.
measure spaces the class of measure spaces with lifting is strictly smaller than the class
of all measure spaces with the Radon-Nikodym property. According to Halmos A950,
Section 31 (9), p. 131) there exist measure spaces without decomposition (D). For such
spaces even an orthogonal lifting cannot exist.
3. Liftings for functions
Given a measure space (?, ?,?) a (function-)lifting for C^(?) is a K-algebra homo-
morphism ?: ?^(?) -> ?^(?) (i.e., a K-linear, multiplicative map) with the additional
properties
v(f) = f, 00
<P(f) = <P(g) if/sgfor/,ge??0i). A2)

Liftings
1141
and
<p(l) = l. (n)
We denote by Л^(д) the space of all liftings for ?^(?). Any lifting ? for ?3€(?)
is a lattice homomorphism, i.e., <p(\f\) = \<p(f)\ and ?(/±) = ?(/)± for / e ?3€(?)
(see A. and С Ionescu Tulcea A969a)). The original question of A. Haar was about the
existence of a lifting ? for ?^(?), ? the Lebesgue measure on R'1, d e N satisfying
<p(f) = <p(f) for / e ?j^(?). This problem can be easily reduced to the existence
of liftings for sets in the following way. Given ? e Л (?) put for any simple function
/ = ?"=? aiXAi with ?,??,?,??(/ = 1 ?),
И
i=\
This defines unambiguously a K-algebra endomorphism of the algebra ??.(?) of all
K-simple functions over ? of norm < 1 having a unique extension to an algebra
endomorphism of ?^ (?) with px(f) = p~*-(f), since ??(?) is dense in ?^(?) by
the Lebesgue ladder theorem. Conversely for any ? e ?^(?) with <p(f) = <p(f) for
/ e ?jj? (?) we can define ape ?(?) by means of
p(A) := {?(??) = 1} for A e 27.
PROPOSITION 3.1. 77?e map ? e Л (?) -> ?00 e ???(?) « ? bijection which is completely
determined by the equation ?°°(??) = xpiA)for A e 27.
A different proof of Proposition 3.1 was given by von Neumann A931) in his first paper
on liftings based on the formula
p°(/)("):=inf{reQ: wep([f<r))}
ifp-.?^ ?, /??^(?). and ? e ?.
Then p° and p°° can be identified on ?3€(?) since ?°\?0€(?) e ?°°(?) and р°(хл) =
Xp(A)- As a corollary we get the next result.
THEOREM 3.2. For any c.l.d. measure space (?, ?, ?) the following conditions are all
equivalent.
(i) The measure space (?, ?, ?) is strictly localizable.
(ii) The measure space (?, 27, ?) /гая ? lifting.
(iii) 77геге ешм ? /////ng <р for ?? (?) iMC^ ^?? <?(/) = <P(/)/or ?" / е ^к (м)-
J. von Neumann noticed already in his first paper from 1931 that for the Lebesgue
measure on Ш the last theorem no longer holds true if the algebra ?^ (?) is replaced by
the algebra ? (?).

1142
W. Strauss et al.
As for liftings of sets there exist weakenings of the notion of lifting for functions
which arise naturally in de Possel derivation as well as in Fourier analysis in case of one
dimensional Lebesgue measure. Such a type is the linear lifting ? for ?°°(?), i.e., a map
? : ?°°(?) -> ?°°(?) which is a positive linear map with the additional properties A1),
A2) and (n) (hence ?(?) = a for all a e Ш). Clearly ?°°(?) с (?(?), where we write
G^) for the class of all linear liftings and indeed ??(?) := ?^(?). Note that a linear
lifting is already a lifting if it is a lattice homomorphism of ?3€(?) into itself. In fact
for / := хд - XA<, if A e ? follows ?(/+) л ?(/~) = 0. Since ?(/+) = 1 - ?(/~),
the inequality 0 < ?(/~) < 1 implies 0 < ?(/+) < 1, a contradiction. Hence ?(/) e
{0,1}?.
By a classical result of Lebesgue A910) we have
f(x)= lim —i- for a.a. ? e Rd
k^X ?(??(?))
for any sequence 0 < r^ -> 0 (к -> oo), and any Lebesgue integrable /, if ? is the
Lebesgue measure on Rd, d e N. For r^ := 1 /ifc, ? e Rrf and / e ?°°(?) writing
Uk(f,x) = —'
?(?„(*))
we get
|«*(/,Jt)| < II/IIoo for all к e N and all ? eW1.
Therefore, if we choose a free ultrafilter U on N, then there exists
\jj{(f){x):= Mm Uk{f,x) forall/e?эc(?)andл: eRd
keli
and ?\ e 0(?), since by Lebesgue's result ?/?(/) = / a.e. (?), hence ?? e ?°°(?) by
completeness of ? while all the other properties of a linear lifting follow for ?\ by
definition. This result can be generalized to more general measure spaces with suitable
derivation bases. For d = 1, ? the Lebesgue measure on ]—?,?] similar results are
obtained by Cesaro and Abel summability for the Fourier series of a Lebesgue integrable
function / on ]—?, ?]. In fact, if
1 Г
-?
is the nth Cesaro-mean,
1 /sin-
K„(x)=-l

Liftings
1143
the Fejer kernel, and
fr@):=h f f^P,@-t)dp(t), 0<r<l, -?<?<:?
is the harmonic of /, where ?,·(?) := A — r2)/(l — 2rcos# + r2) denotes the Poisson
kernel of /, then lim,,^^ ?„(?) = f(x) by a theorem of Lebesgue A905) (see Zygmund
A968, 3.9, p. 90)) and limr^i fr(x) = f(x) for a.a. ? е]-тг,?] by Fatou A906) (see
Hoffmann A965)). If /„ := /,_1/(I then again \?„(?)\, \f,,(x)\ ^ ll/lloc for ? e N,
? e]-?r, ?] for all / e ?°°(?) and if the ultrafilter ?/ is chosen as above, then by means
of
i/2(f)(x):=liman(x), fo(f)(x) :=\im f„(x), хе]-л,л],
neU neU
can be defined 1Д2, тДз е ?(?).
There are two procedures (both due to A. and С lonescu Tulcea A969a)) for converting
a linear lifting ? e (?(?) into a lifting in ?00(?) provided the basic measure space is
complete. First note that by means of
±(?):={?(??)=?}, ^(?):={??(??)>0} for A e ?
we can define ? e #(?), -^ e У (?) with ~? = (?I' (see A969a, p. 36)), so a solution
is given by choosing ape ?(?) with ?(?) с /э(А) с i/r(A) for A e Г, according
to Theorem 2.1. Then apply Proposition 3.1 of this section. But according to A. and
С lonescu Tulcea A969a, Chapter III, Section 2, Theorem 1), there is no need for resorting
to liftings for sets due to the following result paralleling Theorem 2.1.
THEOREM 3.3. Let be given a complete measure space (?, ?,?). For any ? e <?(м)
the set ?? := {и e <7(?): ??(?) ? ?(??) ? ??{?), A e ?} is a non-empty, convex and
compact subset of the locally convex space ?.^(?)??. An element ? e 0? is extremal in
?}? if and only if ? is a lifting for ?°°(?).
The existence of an extremal element in ??? follows now from the Krein-Milman
theorem.
In general measure spaces there are no "natural" linear liftings at hand, a situation very
similar to that for densities. But again there is an inductive construction which parallels
that one for densities in Section 2.
LEMMA 3.4. If (?, ?,?), ?, A and ? are as in Lemma 23 of Section 2 then for each
? e <?(? | ?) there exists ??<? <?(? \?) extending ?. For ? e ?0€(? | ?) may be chosen
?<???(? \?).

1144
W. Strauss et al.
First note that ?°°(? | ?) = {fXA + gXA<: fge ?^(? | ?)}, and if В, С е ? are
defined as in Lemma 2.3 of Section 2 then put
iA(/Xa +gXA') :=iA(/xb+^XB')Xa + VK/xo +gXc)XA·
for f,ge ?°°(? | ?), a formula completely analogous to the corresponding formula for
densities in Lemma 2.3 of Section 2 (see Graf and von Weizsacker A976, p. 156)).
LEMMA 3.5. If (?, ?,?) is a complete probability space, (?„),??? and ? эс are as m
Lemma 2.4 of Section 2 then for all ?,, e §{? \ ?„) with ?,, \ ?„, = ?,,, ifm ^ n e N there
exists ? тДоо e ?(? | ??) satisfying ?? \ ?„ = ?„ for и е N.
As in the above examples choose again a free ultrafilter WonN and put
????(?):=?™??(?„„(/))(?) for / e ??(? ? ??)< ? e ?.
(Here limne^ could be replaced by any Banach limit, see Dunford and Schwartz A958,
Chapter II, 4.22-23).) The existence of the limit is guaranteed by |?«(?,,,(/))(?)| ^
||/|| oo for и e N and ? e ?. Now Doob's martingale convergence theorem implies
VOo(/) = / a.e. (? | ??). Since (?,?,?) is assumed to be complete this implies
???(?) e ??(? I 'Jdc)· All other properties of a linear lifting are immediate by the
definition of ?? as a limit. Using Lemmata 3.4 and 3.5 for linear liftings in the same
way as for densities in Section 2 in an inductive proof taking exactly the same steps (A)
to (F) exhibited after Theorem 2.5 we get extension theorems for linear liftings and in the
same way as for densities the (non-empty) class AC?(?) of admissible linear liftings. But
note that we need completeness of the basic measure space.
THEOREM 3.6. If (?, ?,?) is a strictly localizable, complete measure space and ? a
?-subalgebra of ? with ?? с ? then for any ? e 0(? \ ?) there exists ? ? e Жц) with
?\? = ?.
If we take here ? as the ? -algebra generated by ?? then the existence of a linear
lifting follows. It should be however noted that according to Burke and Shelah A992)
it is consistent with ZFC, the Zermelo-Fraenkel set theory including the axiom of choice,
that ?°°(?) admits no linear lifting for many non-complete probability spaces including
Borel measure space on [0, 1]. For this reason there is no need for stating an existence
result for linear liftings on strictly localizable complete measure spaces, since there the
better result of the existence of liftings is available. In the context of extensions naturally
arises the problem of mapping of liftings and densities.
THEOREM 3.7. Let (?,?,?) and (?,?,?) be measure spaces together with a
measurable map /: ? -> ? such that ? = /(?) is the image measure of ? under f. If
(?, ?,?) is strictly localizable then for every ? &?(?) there exists a S e #(?) such that
a(/-'(/»)) = /-'(a*))
(C)

Liftings
1145
for all В е Г. If in addition (?, ?, ?) is complete then for given ? e ?(?) we may choose
? e ?(?) satisfying equation (C) and in that case we have
Sx(hof) = ix(h)of (Cx)
for all h eC°°(v).
In fact, if ? is the ?-subalgebra of ? generated by ?? and {f~[(B): В еТ},а density
<$o for ? | ? is defined by means of 5<)(A) := /~](?(?)) if A e ? and A = f~l(B) for
some ? e ?. By Theorem 2.5 we can extend <$() to a density <$ on ?. If ? is a lifting then <$o
is too and we can apply Theorem 2.6 to extend it to a lifting on ? because (?, ?, ?) is
assumed to be complete.
Since we have applied extension theorems for getting ? from ? it it obvious that ? is
not uniquely determined by ? via the equation (C). We call any ? e #(?) satisfying the
equation (C) an inverse density of ? and write /~' (?) for the class of all inverse densities
for ?. On the other hand for a surjective map / any ? e i?(v) satisfying (C) is uniquely
determined by <$ e #(?) since then ?(Z?) = /E(/_l (Я))) for S e ?. For this reason we
call ? ihe direct density of ? and write ? = /(<5). In the same way we can define the inverse
///f//ig and the direct lifiing. But note that for given ? e ?(?) in general no direct lifting
exists. Clearly Uretfd') /-'(f) is the class of all densities for ? having a direct density, and
similarly for liftings, but no inner characterization for the elements in this class is known
and as yet very poor partial results can be given, e.g., if ? := [В с ?: f~] (?)} and /
is injective then for any ? e ? (?) there exists a direct lifting, see Macheras and Strauss
A992, Lemma 2.2). We refer to Kupka A983) for his 'projection' Theorem 2.7 which gives
a positive result for the 'projection' of strong liftings in the presence of a disintegration.
Another way of projecting from products onto its factors is discussed by Macheras and
Strauss B000).
For linear liftings similar results as for liftings can be obtained by an application of the
extension Theorem 3.6.
THEOREM 3.8. If (?, ?, ?) is a complete, strictly localizable measure space and the
map f is surjective then for each ? e Q(v) exists ? ? e ?}(?) such that
*(hof) = (<p(h))of (C30)
for all h e ?°°(v). Again ? can be chosen as a lifting if ? is a lifting.
Starting from the last theorem the inverse and direct linear lifting can be defined in
analogy to the inverse and direct lifting. Note that the equation (C00) implies ? e f~](<p)
and ? e /_l(^) if ? is the lower density defined by ?(?):= {?(?,*,) = ? and ? is the
upper density defined by the formula ?(?) := {?(??) > 0} for A e ?.
The above indicated construction of linear liftings by means of de Possel derivation has
some sort of converse. As a preparation we need the following result in which (i), (ii),
(iii) were given by Maharam A958), Theorem 4 for lifted sets. It is remarkable in itself
since in general uncountable unions of measurable sets are no longer measurable, but it
has interesting consequences in addition.

1146
W. Strauss et al.
THEOREM 3.9. Let be given a c.l.d. measure space (?, ?, ?) with a density ? e #(?).
Then for every non-empty collection BQ ? with В с S(B)for В € В we have
(i) U?e?;
(") UB^UBeBS(B)?8(UB);
(iii) \J BeB B'= (UB)' (V denotes the upper bound of В in ? /?);
(iv) ?0 := LUexv ?(?) ^? ????? = ? a.e. (?);
(?) $\??)????{?) = ?({]?) if ? is directed upwards;
(vi) (?, ?, ?) is localizable.
There is a corresponding version for liftings of functions given by A. and C. lonescu
Tulcea A969a). J. Gapaillard asserts condition (i) of the last theorem for monotonous
liftings with a proof being convincing at least for finite measure spaces. As a first
consequence of Theorem 3.9 we get Vitali derivation bases (see Kolzow A968, Section 12)
for definition) in c.l.d. measure spaces with lifting by means of the next result of Kolzow
A968).
THEOREM 3.10. For given c.l.d. measure space (?, ?,?) with a lifting ? e ?(?) put
R :=LUer,- P(A) and define gp(w) := [A eEf. ? e ACp(A)| and a p (?) := {g: g is
a cofinal subset ofgp(cu)}for ? e R. Then (??(?))?(?/? is a strong Vitali derivation basis.
If conversely a weak Vitali derivation basis a = (?(?))?(?/?, Rc e ?? is given in a c.l.d.
measure space (?, ?,?) we define the lowerdePosselderivative with respect to a
Da(f)(w)= inf liminfiAJ-f
gea(u>) Aeg ?(?)
for all locally integrable functions, i.e., all ? -measurable functions such that //д is ?-
integrable for all A e ?/, and ? e ?. Put A,(A) := [D_ci(xa) = 1} for A e ?. Then Д,
is a density for ? which can be converted into a lifting by Theorem 2.1, i.e., we have the
converse of Theorem 3.10 which is also due to Kolzow A968), see also С. lonescu Tulcea
A971) and Sion A973).
COROLLARY 3.11. For each c.l.d. measure space (?, ?, ?) with ?(?) > 0 there exists
a lifting if and only if the measure space has a weak {strong) Vitali derivation basis.
In the situation of Theorem 3.10 the upper and lower de Possel derivatives with respect
to ap are given by
_ f. f ?? f.f ??
D(f)(w) = limsupJAJ , D(/)(^)=liminfi^f—
Аеар1ш) IJ-(A) Aea„Uu) ? (A)
for all locally integrable functions, i.e., for all ? -measurable functions such that /?? is ?-
integrable for all A e ?/ and all ? e R := LUer P(A). If in addition R = ? then they
satisfy D(f) = Z)(/) = p(f) for all / e ^(?), i.e., the lifting appears as a de Possel

Liftings
1147
derivation. The assumption LUer /°(^) = & ls trivially satisfied for every probability
space and can be achieved for any ? -finite measure space.
A linear lifting ? for ??(?) is a positive linear map from ?\?) into ?''(?) satisfying
the basic properties A1) and A2) of liftings, where Ср(ц) is the Banach space of all finitely
real-valued functions f e ?°(?) with ||/||p := /\/\?? < oo, for 1 ^ ? < oo. A. and
С Ionescu Tulcea A969a, Chapter IV, Section 4, Theorem 6) noticed that there can't
exist a linear lifting for ??(?), 1 ^ ? < oo, if there exists a non-negligible measurable
set A which is diffuse, i.e., whose class A' e ?/? does not contain any atom (a similar
conclusion holds true in case ? = 0 under an obvious definition of the linear lifting for
?°(?)).
With a similar proof one can see that under the same assumption there can't exist a lifting
? e ?(?) satisfying the additional condition ????^? ?«"> = ?^=? P(An) if A.„ e ?,
? e N, in fact only the properties (LI) and (L2) of a lifting are needed for this conclusion.
All these results are on the basis of ZFC, the Zermelo-Fraenkel set theory including the
axiom of choice. In Solovay's model of Zermelo-Fraenkel set theory (where the axiom of
choice fails) the above mentioned proof for the non-existence of a linear lifting for С (?)
in case 1 ^ ? < oo carries over to the case ? = oo since (/^(?))' = L1 (?) (a result of
Christensen A974)), hence <?(?) = 0 for Lebesgue measure ?, saying again that some
sort of non-constructive tool like the axiom of choice is needed for an existence proof of a
lifting even for Lebesgue measure (see Graf and von Weizsacker A976)).
4. Liftings on topological spaces
Throughout this section {?,?,?,?) is a pretopological measure space with (?,T)
being Hausdorff. Theorem 3.9 gives raise to two classical "lifting topologies" of A. and
С Ionescu Tulcea A969a), which according to A. and С Ionescu Tulcea A964b, p. 445)
are partially tracing back to J.C. Oxtoby and which convert lifted functions into functions
continuous with respect to the lifting topology. For given measure space (?,?,?) and
S e ?(?) the collection T>s := {8(A): Aeljisa basis for a topology tg on ?. We don't
assert ts <? ? at the moment, but it follows from Theorem 3.9 that tg П ? с ?^ := {A e
?: AC5(A)}.
The next theorem gives a collection of the basic properties of the density- and lifting
topologies ts, ?$, tp, and zp, taken from A. and С Ionescu Tulcea A969a) and A. Ionescu
Tulcea A967a). Before we state it, we need a definition. If (?, ?, ?, ?) is a pretopological
measure space, a density ? e ?(?) is called ?-strong, if G с 8(G) for all G e ? ? ?, and
topologies ? satisfying such a condition are called compatible with S by A. and С Ionescu
Tulcea A969a). It is well known that the Lebesgue density D on the Lebesgue measure
space on Rd is ? ''-strong for the euclidean topology Zd on E'7, and so is any lifting ?
for the Lebesgue measure satisfying D(A) с p(A) for all A e ? (such liftings exist by
Theorem 2.1).
THEOREM 4.1. Forgiven c.l.d. measure spaces (?, ?,?) with S e ?(?) and ? e ?(?)
we have the following results.
(i) ft с Ti с ?.

1148
W. Strauss et al.
(ii) Tg П ?? = 0 and for all A e ? there exists a G e z$ with A AG & ?^.
(iii) A subset К с ? is of the 1st category with respect to the topology ?& if and only
if К is closed and nowhere dense or equivalently if К е ??-
(iv) The topologies tp and ?? are extremally disconnected and tp с ??.
(?) The topology tp {hence ??) is Hausdorff if the set [p(A): A e ?) separates the
points of ?.
(vi) ??(?,??) = Cf(i2, ??) = {p°(/): / e ?°(?)? and p°(f) is the unique
continuous function with respect to tp (respectively, ??) in f.
(vii) Ch&jp) = Cb(n,Tp) = {px(f): feCx^)}.
(viii) Iftp is Hausdorff then the Stone space of the measure algebra ? /?? is the Stone
Cech compactification of^,tp).
If ? e {tp, ??) we can add the following conditions:
(ix) We have clT(A) = p(A)for all AeT and ? = ?(?, ??).
(?) ? is the unique T-strong lifting for the topological measure space {?, ?, ?,?).
(xi) The topological measure space (?, ?, ?, ?) is a quasi-Radon measure space.
Here (i) follows from Theorem 3.9 and (ii) and (iii) are from A. Ionescu Tulcea A967a).
A proof for (iv) to (viii) is available in A. and C. Ionescu Tulcea A969a, Chapter V,
Section 3), as well as the equation ?^(?, tp) = 0%(?, ??) in (?).
If S is the Lebesgue density (see Section 2) of the Lebesgue measure space (?, ?, ?)
on R and ? is a lifting with <5(A) с p(A) for A e ? then ?? is completely regular and
tp = ? ? by A. and С Ionescu Tulcea A969a, Chapter V, Section 4, Theorem 2). But there
exists a po e ?(?) such that ??) is finer than the euclidean topology on R and fA) ? ??).
For any ? & ?(?) with ?? finer than the euclidean topology on R the topology ?? is not
normal and every ? ? -compact К с R is finite by A. Ionescu Tulcea A967a).
It is now easy to see that generally the topological measure space (?,?, ?, ?)
from (vii) will not be a Radon measure space. Take for instance as (?, ?, ?) the Lebesgue
measure space with ? := zp for a lifting ? e ?(?) for which zp is finer than the euclidean
topology Ed on R. Since by the above remark any zp-compact subset К of R is finite the
inner regularity ?(?) = sup^(A"): К с А, К rp-compact} with respect to rp-compact
subsets К of R can't be true if ?(?) > 0.
If (?,?,?) and (?,?,?) are measure spaces, ? e #(?), and ? e ?(?) then a
?-?-measurable map f: ? -> ? is ts-t(-continuous if and only if /~'(f(S)) <?
5(/~'(?(#))) for all S e T. If moreover ? e ?(?), and ? e A(v) these conditions are
equivalent with f~[^(B))=6(f-[ (?)) for all В e ? as well as with <(/?) о / = 5(й о /)
for all/г ??°°(?).
PROPOSITION 4.2. If (?,?, ?, ?) a/id (<9,5, ?, ?) are topological measure spaces, the
map f is a T-S-continuous surjection, and f(8) existsforS e ?(?), then f (8) isT-strong
provided ? is ?-strong.
But note that under the assumptions of Proposition 4.2 for a strong lifting ? for ? an
inverse lifting ie/_1(i;) needs not to be strong, see the example before Theorem 6.20.
As we state below, the last theorem gives in fact a topological characterization for c.l.d.
measure spaces having a lifting and there are again characterizations by weaker types of

Liftings
1149
"liftings" in an abounding number, hence we can mention only the most spectacular ones.
We call a map ? from ?^{?) into itself satisfying the properties A1), A2) a monotonous
lifting if in addition <p(f) ^ q>{g) for/, g e ?°°(?) and / ^ g, it is called a bounded linear
lifting if ? is a linear map with \\?\\ : = sup{||<p(/)||/||/||oc: 0 < ||/||эс < oo} < oo, where
11/11 is the strict supremum and ||/||oc the essential supremum of a function / e ?^°(?),
it is called г. function lower respectively upper density if <p(f A g) = (p(f) Л (p(g) and
4>{fvg) =(p(f) v<p(g), respectively for/, g eCx^).
THEOREM 4.3. For given c.l.d. measure spaces (?. ?,?) the following conditions are
all equivalent with the existence of a lifting for ?°° (?).
(i) There exists a topology ? с ? such that the topological measure space
(?,?, ?,?) is a category measure space.
(ii) There exists a topology ? с ? such that ? ? ?? = 0 and a set К с, ? is of first
category if and only if it is closed and nowhere dense in ?.
(iii) There exists a topology ? с ? such that card(/* П Q,(i2)) = 1 for any f e
?°°(?), where C/,(i2) denotes the space of all ? -continuous, bounded real-valued
functions on ?.
(iv) There exists a monotonous lifting for ?°°(?).
(?) There exists a function lower respectively upper density for ?°°(?).
(vi) There exists a linear lifting for ?°° (?),
(vii) There exists a bounded linear lifting ? for ?°°(?) of norm \\?\\ < 3.
For (i) and (ii) compare Graf A973), for the sophisticated equivalence proof for (vii) see
Erben A983), where an example is given that the bound 3 cannot be improved.
We call ? e ?(?) (and ? e 0(?)) almost ?-strong, if there exists a set N e ?? such
that for all G e ? ? ? follows G \ N с <S(G) (respectively 1^(/) \ Nc = f \ Nc for all
/ e Сь№) П ?°°(?)). 5 and тД are called T-strong in case N = 0. It is obvious that
S e ??(?) is T-strong if and only if ? с ?^.
If ?d is the euclidean topology of Wl then it is well known that the Lebesgue density D
and any lifting ? for the Lebesgue measure with D(A) с p(A) for A e ? are ?f/-strong,
and so are the linear liftings ?/м, 1Д2 from Section 3 obtained by L. Fejer's theorem (see
Hoffmann A965, pages 20 and 33)). Any hyperstonian space has a uniquely determined
strong lifting, which is given by choosing the unique continuous function from each
equivalence class of /^(?). For any complete measure space (?, ?,?) any ? e ?(?)
is tp-strong as well as zp-strong by the definition of tp and zp.
PROPOSITION 4.4. If the topology ? is ?, 1 then for each ? e ?(?) the following
conditions are equivalent.
(i) ? is almost T-strong.
(ii) /0°° is almost T-strong.
(iii) There exists a N e ?0 such that p(F) CFU N for all closed Fel.
(iv) There exists a N e ?0 such that p°(f) \ Nc = f \ Nc for all f e C(i2Kl ?°(?),
where C(i2) denotes the space of all continuous functions from ? into E.
(v) There exists a N e ?0 such that poc(f)\ Nc = f \ Nc for all f e С 1^H^ (?),
where Сь№) denotes the space of all bounded continuous functions fют ? into R.

1150
W. Strauss et al.
Here we can choose the same set N e ?? in (i) to (iv), where in particular N = 0 for
T-stmng ? and remember that p, p°°, and p° are in biunique correspondence by means
of the equations ?°°{??)=?° (??) = ??(?) for A e ?.
The implication (i) =>· (iv) is quickly achieved by observing p° = po if po(f) =
sup{r e Q: ? e p({f > r})} since then po(/) I ^ > f I Nc for / e C(i2) and p0(-f) =
—po(f) hence po(f) \ Nc = f \ Nc if if ? is almost T-strong with universal set N e ??-
The implication (iv) =>· (ii) and the equivalence of (i) and (ii) are obvious.
Moreover (ii) =>· (i) works for almost T-strong ? e <?(?), since for a ?-,1 topology ?
follows xc = sup{h e Q,(i2): /i^xG)if GeT hence xc\yv ^ ?(??) I W therefore
G\N c^OforG e ? ? ? if ?_ is defined by i^(A) = [?(??) = 1} for A e ?, where
for yo e Л (?) follows ? = p. So we have in addition the following result.
PROPOSITION 4.5. If the topology ? is ?3 ? and the measure space (?, ?, ?) is complete
then the existence of a T-almost strong density for ? is equivalent with the existence of a
?-almost strong {linear) lifting. Here we may replace "almost strong" by "strong'".
If a topological measure space admits a T-strong density its measure has to be of full
support, i.e., 5ирр(д) = ?, since then for all G e ? follows G = 0 from G e ??- The
notion of the almost strong lifting allows us to cover in full generality the cases with
5ирр(д) ? ?, for which the notion of strong lifting is inappropriate.
PROPOSITION 4.6. If (?, ?,?) is a complete measure space with supp(M) = ?, then
fmm the existence of a T-almost strong lifting for ? follows the existence of a T-stmng
lifting.
This is easily achieved for ? e ?(?) with G\N с p(G) for G e ? ? ?, ? e ?? by
choosing an ultrafilter U(a>) finer than the filter basis {G e ?: ? e G) for ? e N and
putting
p(A) := (p(A) П Nc) U {? e ?: AeU(co)} for A e ?.
The space (?, ?, ?, ?) (or just ?) has the almost stmng lifting property (ASLP for short),
if there exists an almost T-strong lifting for ?. Since for general spaces (?, ?, ?, ?) there
is no "natural" candidate for an almost strong lifting, one possible issue is to check whether
arbitrary liftings are almost strong. This leads to the stronger notion of universal stmng
lifting property (USLP for short), which says that ? (?) ? 0 and every ? e ? (?) is almost
T-strong. Results sufficient for applications (polish spaces) rely on the next result which is
a generalization of a result from Maher A978), see alsoFremlin B00?), and Macherasand
Strauss A996a). Before we state it we need a modification of the purely topological notion
of network (see Gruenhage A984)). A family ? с ? is called a measurable network for
the pretopological measure space (?,?, ?,?) if for each G e ? there exists a subfamily
$cf such that G = UQ\ we denote by ?????(?) the least cardinal of a measurable
network for (?,?, ?,?).

Liftings
1151
LEMMA 4.7. If (?,?, ?,?) is a pretopological measure space with a measurable
network ? then ? e ?(?) is almost ?-strong if there exists N e ?0 such that G \ N с
? (G) for all G ef.
The last lemma remains true if we replace liftings by monotonous liftings.
It follows that a complete topological measure space (?.?, ?,?) has the USLP if there
exists a cardinal К with mnw^) < К and if for any family A<^ ? with card(^) < К and
?(?) = 0 for A e ? follows UAe ? and ?A?4) = 0. Indeed for given lifting ? e ?(?)
apply Lemma 4.7 for N := \^jFejr(F \ p(F)) if ? is a measurable network of cardinality
mnw^). For К = K| this gives the next result of Fremlin B00?, 453F).
COROLLARY 4.8. A complete strictly localizable topological measure space (?.?, ?,?)
with a countable measurable network has the USLP.
In particular a complete strictly localizable topological measure spaces (?,?, ?,?)
possesses the USLP if their topology ? is second countable. This is an improvement
of a result of Graf A975), see Maher A978). In particular, polish as well as locally
compact metrizable measure spaces have the USLP (see A. and С Ionescu Tulcea
A969a, Chapter VII, Theorem 8)). The next result is from Macheras, Musial and Strauss
B00?a).
PROPOSITION 4.9. Let (?,?, ?, ?) and (?, S, ?, ?) be Borel probability spaces and
f:: ? -> ?? measure preserving map. Suppose that ? admits a strong lifting ? which has
an almost strong inverse image lifting in ?(?). Then there exists a strong inverse lifting
? e ?(?) of ? under f if and only if f~\x(B)) ? G ? 0 implies ?(/~' (?) ? G) > Ofor
all В eS and all G e T.
To sketch the proof of the above proposition, let ? e ?(?) be an almost strong inverse
lifting of ? under /. Then there exists a null set N e ?? such that
G с V(G) U N for each G eT.
For each ? e N, let
?(?):={?: Ае/_|(Г), ???(?)},
S(w):={f-\G): GeS, f(w)eG].
and
?(?) := {А П G: A e ?(?), G e ? (?)}
and ?(?) с ? be the filter defined by
Jr(w):={E e ?: 3F eH(co) with Fc ? a.e. (?)}.

1152
W. Strauss et al.
Define a density ? e #(?) by means of
p(E) := [?(?) ?) Nc]u p*(E) for each ? e ?,
where p*(E) :={weN: Ее ? (?)).
It can be shown that ? e ?(?) П /_1 (?) and that ? is strong. By Theorem 2.1 there
exists ape ?(?) such that p(E) с p(?) for each ? e Г. It follows that ? is a strong
inverse lifting of ? under /.
A measure ? on a totally ordered space (?,?) is a measure defined on the Borel ?-
algebra ?(?) generated by the order topology, ? is the completion of ?(?) under ?,
where we denote the extension of ? to ? again by ?. Then the quadruple (?, ^, ?, ?)
is called a totally ordered measure space. For such spaces Sapounakis A983) proved the
existence of strong liftings.
THEOREM 4.10. Let (?,-?,?, ?) be a totally ordered measure space such that its
measure is of full support. Then there exists a strong lifting for ?.
For Baire and Borel measure spaces necessary conditions for the ASLP are given by
Babiker and Strauss A980a) as well as by K.P. Dalgas.
PROPOSITION 4.11. If (?, ?, ?, ?) is a Baire measure space with finite measure and a
completely regular topology ? then the ASLP implies that the measure ? is ?-additive and
completion regular. For any topological measure space (?, ?, ?, ?) with the ASLP the
measure ? is necessarily ?-additive.
It follows, e.g., that the Wiener measure restricted to the completion of the Baire ?-
algebra on R l°·'' does not have the ASLP. But the Wiener measure considered on the
completion of the Borel ? -algebra has the USLP (see, e.g., Macheras and Strauss A996b)).
Clearly the last proposition raises the problem whether the necessary conditions for the
ASLP given there are sufficient? By Babiker and Knowles A978), there exists a Baire
measure space (?,?, ?, ?) with compact ?, finite and non-atomic ? of full support,
that is ?-additive but not completion regular. This space is an example of a compact Baire
measure space without the ASLP.
Fremlin A979) has given an example of a Radon measure space on a compact set with
completion regular ? of full support but without the ASLP, improving a result of Losert
A979) which lacked completion regularity. The completion regularity is of interest here
because in important known cases where the ASLP holds true, the completion regularity is
also fulfilled. Mokobodzki A975) and Fremlin A977) gave a converse for Proposition 4.11.
It was noticed by Dalgas that this result is the main step towards a result on strong Borel
liftings improving the result of Mokobodzki and Fremlin. In the subsequent theorem of
Dalgas A99?) с stands for the cardinal of the set of reals. Different approaches have been
given by Musial A973) and Lloyd A974).
THEOREM 4.12 (CH). Let be given a topological measure space (?,?,?,?) with
finite measure ? and ? possessing a basis of the cardinality less or equal to с Then,

Liftings
1153
(?, ?, ?, ?) has a strong Borel lifting if and only if ? с ??, supp(M) = ?, and ? is
?-additive (?? is the completion ???(?) with respect to ?\?{?)). The theorem remains
true for ?-finite ? if ? is moderated or all finite Borel measures on (?,?, ?, ?) are
?-additive.
The proof of the last result relies on the following theorem of Fremlin A977).
THEOREM 4.13. Let be given a topological measure space (?,?, ?,?) with ? с ??
satisfying the following conditions.
(i) For any A e ? and any Q с ? with ?(? П G) = 0 for all G ? Q follows
?(????)=0.
(ii) If к is the cardinal of the measure algebra ? /? then the union of fewer than к sets
of measure zero is measurable and of measure zero.
(iii) 5ирр(д) = ?.
Then there exists a strong Borel lifting for (?,?. ?, ?).
Burke A993a, Proposition 3.6) gives an elementary proof that the existence of a Borel
lifting for the Borel-Lebesgue measure space on [0, 1 ] implies already the existence of a
strong Borel lifting for this measure space. It follows from the example of Losert A979)
that the Theorem 4.12 is no longer valid without the cardinality restriction. Dalgas gives
an example that for ?-finite measures spaces additional hypotheses must be imposed
for a corresponding characterization. Mauldin A978) gives related results, following the
original results of von Neumann A931). The first one states that subject to the continuum
hypothesis, any Borel measure space (?,?, ?, ?) has a Borel lifting if (?,?(?)) is Borel
isomorphic to a universally measurable subset of the unit interval [0, 1] of the reals, thus
generalizing a result of A. and С Ionescu Tulcea A969a, p. 182). The second one applies
Martin's axiom to construct liftings ranging in the c-algebra ??(?) of the topological
space (?, ?), i.e., in the the smallest algebra containing the Borel ?-algebra ?(?) and
being closed under unions of less than с sets. If the continuum hypothesis does not hold
true the c-algebra may be much larger than the Borel ? -algebra. This result is one of the
very few ones in lifting theory applying Martin's axiom. For more results on Borel liftings
we refer to Section 5 on translation invariant liftings.
Assuming the continuum hypothesis, first existence results for strong Baire liftings have
been given by Losert A980), see also Talagrand A978a), for a restricted class of measures,
i.e., for any product of less than or equal to Hi Radon probability measures of full support,
each on a compact metric space. Grekas and Gryllakis A992) improved this result as well
as a result of their own from 1991 by the following theorem.
THEOREM 4.14 (CH). Subject to the continuum hypothesis every product of less than or
equal to К 2 many completion regular probability measures, each supported on a product
of less than or equal to ? ? many compact metric spaces admits a strong Baire lifting.
The proof is based on a result for measures on product spaces satisfying a certain
condition which reduces in case of compact metric factor spaces to completion regularity.
The next theorem from Babiker and Strauss A980a) gives necessary and sufficient
conditions for the USLP.

1154
W. Strauss et al.
THEOREM 4.15. Let (?,?) be a locally metrizable space. Then a Borel measure space
(?, ?, ?, ?) with a finite measure ? has the USLP if and only if ? is ?-additive, and a
Baire measure space (?, ?, ?, ?) with finite measure ? and completely regular topology
? has the USLP if and only if ? is ?-additive and completion regular.
This theorem implies that under the mild set theoretic assumption of the non-existence
of measurable cardinals every metrizable space with finite Borel or Baire measure has the
USLP.
COROLLARY 4.16. Suppose that every closed discrete subspace of the metric space
(?, ?) has non-measurable cardinality. Then every Baire (respectively Borel) measure
space (? ,?,?,?) with a finite measure ? has the USLP.
For existence of strong liftings on products and projective limits, respectively we refer
to the Sections 6.2 and 6.3 below.
The interest in strong liftings comes from the following theorem of A. and С Ionescu
Tulcea A969a) on the existence of strict disintegrations (see the same book for
terminology).
THEOREM 4.17. A finite Radon measure space (?,?, ?, ?) over a compact set ? with
5ирр(д) = ? has the ASLP if and only if for each Radon measure space (X, S, ?, v) over
a compact set X and each continuous surjection ?: ? —* X with ? = ?(?) there exists a
strict disintegration of ? with respect to p.
The notion of a strong lifting has been generalized to the so-called W-lifting, for which
we refer to Levin A975) and for existence to Babiker and Strauss A980a, 1980c). Bichteler
A970, 1971) shows that the problem of the existence of strong liftings for Radon measures
on locally compact Hausdorff spaces can be reduced to the problem of the existence of
strong liftings for Radon measures on products of unit intervals. Bichteler A972) shows
that the set of all signed Radon measures ? on a locally compact Hausdorff space X such
that \?\ admits an almost strong lifting is a band in the Dedekind complete lattice of all
Radon measures on X, see also Bichteler A973) as well as С Ionescu Tulcea and Maher
A971).
For weakenings of the notion of strong lifting such as the so-called almost strong pre-
density, the idempotent lifting, and the almost ?-continuous lifting, respectively we refer to
Bichteler A972), Georgiou A974), Grekas A989), and Rinkewitz A997), respectively.
5. Liftings on topological groups
Throughout this section a topological group X carrying Haar measures together with its
?-algebra ? of Haar measurable sets (the common domain of all its left and right Haar
measures) and a left Haar measure ? on ? are given. Then (?, ?, ?) is called a Haar
measure space over X by Fremlin B00?). (The case of a right Haar measure can be reduced
to that of the left one.) The map ? e X -> jc_1 e X is the inversion operation. For s e X,

Liftings
1155
A<^X, and / : X -» К we consider sA := {sx: ? e A], the /e/f s-translate of A as well as
y(s)f, the /e/r s-translate of / defined by means of
(y(s)f)(x):=f(s-lx) for all * e X.
A density 5 e #(?) and a linear lifting ? e <?(?), respectively is called left-translation
invariant if
S(jA) = jS(A) for all A e ? and all s e X
and
\l/(y(s)f) = y(s)(i/(f)) for all / e ?^(?) and all s e X,
respectively. If again (see Section 3) p°° and p° are the liftings for functions in ?{^(?)
and ?|?(?) uniquely generated by a lifting ? e ? (?), then ? is left-translation invariant if
and only if ?00 and p° are such.
In the above definition we have fixed explicitly a Haar measure ? on X for easier
reference. But the definition of a left-translation invariant density and (linear) lifting is
completely independent of this choice, since all left (and right) Haar measures produce
the same domain ? and the same null sets ??. For this reason we can speak of Haar
densities and Haar (linear) liftings. In Abelian groups we simply speak on translation
invariant densities and (linear) liftings. For the next proposition compare A. and С Ionescu
TulceaA967, Section 3, Proposition 1) and Fremlin B00?,448C).
PROPOSITION 5.1. Any left-translation invariant linear lifting is strong. The same holds
true for translation invariant densities.
The Lebesgue density D defined in Section 2 for the Lebesgue measure space on Rd, d e
N, is translation invariant and similarly the examples 1Д2, ?3 quoted in Section 3 provide
translation invariant linear liftings on the circle group. A. and С Ionescu Tulcea A967,
p. 90), show that in contrast to the uniqueness of the Haar measure, Haar densities are by
no means uniquely determined even in case of the Lebesgue measure on the real line. By
the next two theorems (which are the analogues of the Theorems 2.1 and 3.3, respectively)
translation invariant densities and linear liftings can be converted into translation invariant
liftings.
THEOREM 5.2. For any left-translation invariant density 8 for ? there exists a left-
translation invariant lifting ? for ? with 8(A) с p(A)for all A e ?.
In fact by Theorem 2.1 we may first choose are ?(?) with 8(A) с ?(?) for all A e ?
and then take
p(A) := {x e X: ее ?-1 A)} for all A e ?,
where e denotes the identity of X.

1156
W. Strauss et ah
THEOREM 5.3. For any left-translation invariant linear lifting ? for ? there exists an
extremal element ? e ?}? (see Theorem 33 for the definition) which is a left-translation
invariant lifting for ?.
Compare A. and C. Ionescu Tulcea A967, Section 6, Corollary 3) for the last theorem.
The construction of left-translation invariant densities and linear liftings goes through
the induction steps (A) to (E) lined out for Theorem 2.5. But these steps become now more
complicated since the construction has to run through translation invariant ?-subalgebras.
In particular, in the step from an ordinal a to its successor ordinal a + 1 the extra difficulties
are a consequence of the fact that translation invariant ? -subalgebrafor a + 1 is much larger
than that one generated by the ?-subalgebra for a and the new element which enters.
A. and С Ionescu Tulcea A967) had overcome these difficulties by exploiting special
structural properties of locally compact groups, in particular of Lie groups, getting the
next fundamental result.
THEOREM 5.4. For each Haar measure space (?, ?, ?) on a locally compact group X
there exists a left-translation invariant lifting.
The last step from a locally compact group to a group X carrying Haar measures which
we are going to state now has been done by Fremlin B00?) using the fact that for a Haar
measure space (?, ?, ?) on X there exists a Haar measure space (?, ?, v) on a locally
compact group ? and a continuous map /: X -> ? such that ? = f^) and for all ? e ?
there exists a set F e ? with /_1 (F) с ? and ? \ /_1 (F) e ?0.
COROLLARY 5.5. For any topological group carrying a Haar measure there exists a left-
translation invariant lifting adequate for all its Haar measures.
For the special group X = {0, 1}' for a non-empty index set / with its "usual Haar
measure" the proof of the existence of a Haar density from Fremlin B00?, 345C), can be
modified to give a proof of the existence of an admissible translation invariant density and a
translation invariant admissible linear lifting being in particular a strong admissible density
and strong linear lifting, respectively.
By Theorem 5.2 there exists an admissibly generated translation invariant lifting, hence
by Proposition 5.1 an admissibly generated strong lifting.
Concerning translation invariant Borel liftings for Haar measure spaces Johnson A980)
proved that in ZFC no translation invariant lifting for the Haar measure on the circle group
R/Z can be a Borel lifting. This result was extended by Talagrand A982) to non discrete
compact Abelian groups. While R.A. Johnson used results from topological dynamics
on the circle group M. Talagrand's proof is simpler since it is based on the well known
construction of the Cantor set. Generalizing Johnson's procedure Kupka and Prikry A983)
succeeded to extend M. Talagrand's result to non-discrete (possibly non-Abelian) locally
compact groups replacing also Borel liftings by more general liftings ? with Baire property,
i.e., p(A) is a set with Baire property for any A e ? (note that every Borel set has this
property). M. Talagrand's simpler method was developed for the non-Abelian case first by
Losert A983) and subsequently by Burke A993) who succeeded in the next stated theorem
to improve the result of J. Kupka and K. Prikry.

Liftings
1157
THEOREM 5.6. In each non-discrete locally compact group X there exists a Borel set ?
such that for an arbitrary left-translation invariant lifting ? for the Haar measure space
(?, ?,?), theset p(E) is not universally measurable and does not have the Baire property.
Here a set А С X is called universally measurable if A is ?-measurable for every Radon
measure ? on X.
Fremlin A989) gives a list of open problems.
Questions 5.7. A) Is there a cardinal к such that the Haar measure on {0, 1}* has no
Baire lifting?
This problem is connected with the next unsolved one about the existence of product
liftings (see Section 6 for the definition) in incomplete probability spaces.
B) Do there exist product liftings for the Borel measure space on [0, l]2?
C) Does {0, l}*1 possess a Borel lifting?
By Mokobodzki's Theorem 2.7, subject to CH, the spaces {0, 1}^' for /=0,1,2 have
Baire liftings.
According to Burke and Shelah A992) it is consistently true with ZFC that there is no
Borel lifting on {0, \)K for any к.
D) Is the existence of a Borel lifting for {0, 1}'" consistent with 2K° > N3?
The problem of existence of translation invariant liftings has been generalized to the
problem of the existence of Q-invariant (linear) liftings for a given set Q of bi-measurable
maps s: ? -> ? over a measure space (?, ?, ?) by A. and С Ionescu Tulcea A969a,
p. 182) and Maher A974, p. 69). Besides the positive solution for the group of left
translations in topological groups carrying Haar measures from above (see Corollary 5.5)
another positive solution was given by A. Ionescu Tulcea A965) for the existence of a Q-
invariant linear lifting if (?, ?, ?) is strictly localizable and Q is a countable amenable
group. On the other hand von Weizsacker A977) proved on the basis of a lemma on
automorphisms for complete Boolean algebras criteria for the non-existence of ^-invariant
liftings. In particular he gives a maximality argument for the set of left translations
in the situation of Theorem 5.4, more precisely he showed in von Weizsacker A976),
Corollary A.3. on the basis of his fixed-point result the next theorem.
THEOREM 5.8. IfX isa connected locally compact group with left Haar measure then for
every set G of continuous bi-measurable and null-set preserving bijections on X which is
strictly larger than the set of all left translations there exists no lifting commuting with G.
6. Permanence of liftings
6.1. Liftings in products
We denote by (f]i6/ ??, <g)i6/ 27,-, (g),6/ ?/) or by <g)i6/ (?,, 27,-, ?,) or by (?,, ?,, ?,)
the product probability space of the probability spaces (??,, 27,-, ?,) (/ e /) and (??6/ ^"
<8>i6/ ??. <8>,6/ Mi) or <8>,6/ №. ??' ?/) denotes its (Caratheodory) completion. For each

1158
W. Strauss et at.
0 ? J с / we denote by (Qj, ?],??) the product measure space (?)iej(&i, IT,-, ?,-).
For any 0 ^ J с / the canonical projection of i2/ onto i2y is denoted by p./, where
Pi : = p(,¦} if / e / and the ?-algebra ? у' (Ej) с X1/ is written ITy. For a probability space
(<9, T, v) and a non-empty set / we write (?1, ?1 ,?1) for the product probability space
0/Ej(C/, ??,??) with all its factors (?,-, 2?,-, ?,-) equal to (<9, 7, v) for /' e /.
If / is a function defined on П/е/ ^ ап(^ (?<? · ¦ ¦¦ '?'«) e ??'=?^ are fixe(J. then
/(?,-, ?,„) is the function on ?/6/\|?, ?„] ?? obtained from / by fixing (?,-,,..., ?,-,,).
In a similar way the sets E(Wi ? ) being sections of a set ? С П/6/ ? are defined. In
case of the product of two spaces, we shall be using the notation /?, fe and ??, ?? rather.
During the last fifteen years a good deal of research in lifting theory concentrated on the
problem of the existence of liftings compatible with the product structure of probability
spaces leading to different types of compatibility of increasing complication, starting from
TalagrandA989).
Throughout what follows let be given a family ((??, 2?,-, ?,))/6/ of probability spaces
with index set / ? 0 and a probability space (?, ?, ?) with ?=??,?^??,?\?/ =
?/. If 0 ? J с /, then С) := {f о pj: f e ?^(?^)} is the set of functions, determined
by coordinates in J. Moreover,
0 A< := ? J1 (? A<) for A< e ?;, / e 7
ieV \e./ '
and
<S>fi ¦= (? /') °PJ for /' e ^(W), ' e У
ie./ \e./ '
are the cylinder sets and functions, respectively. We call S e #(?) a product-density for ?
if for all / e / exists 5,- e #(?,) such that
5 @ A,A = 0 5,- (A,-) for all A, e 27,-, i eF^I, F finite. (P)
The 5,-, / e / are uniquely determined by <5 via (P) but conversely S is only uniquely
determined on the cylinder sets by the family (<5,);6/ via (P). We therefore write S e
0i6/ <5, and call <5, the /th marginal of 5. ???(?) is the class of all product-densities, and
??(?) := ??,,(?) ? ?(?) is the class of all product-liftings for ?. Clearly the marginals of
any ? e ?,,(?) are in ? (?,) for / e /. In the same way we can define the class 0,,(?) of all
product linear liftings and the class ?^(?) : = 0?(?) ? ?30(?) of all product (function-)
liftings, where indeed ? e ?,,(?) if and only if px e ?^(?) and in that case the marginals
satisfy the equation (pi)°° = (p30)/ if / e /. The product-lifting was first investigated
in A. and С Ionescu Tulcea A969a, Chapter VIII). Then Talagrand A982) introduced
a consistent lifting. A lifting ? for a complete probability space (?, ?, v) is said to be
consistent if for each ? e N there exists a product lifting p" e A@ v) with all its /th

Liftings
1159
marginals equal to ? for / = 1,..., n. Consistent densities can be defined in the same
way. The existence problem for consistent liftings has been definitely solved by Talagrand
A989).
THEOREM 6.1. For each complete probability space (?. ?. v) there exists a consistent
lifting ? e A(v).
It is however not true that each lifting is consistent. Talagrand A988) assuming (CH)
constructed such a Borel lifting ? on [0, 1] with respect to Lebesgue measure that its
product p2 defined on Borel rectangles by p2(A ? ?) = ?(?) ? ?(?) cannot be extended
to a lifting on Lebesgue measurable subsets of the unit square (in particular it cannot be
also extended to a Borel lifting).
Question 6.2 (cf. Burke A993)). Can one produce in ZFC a lifting for Lebesgue measure
on [0, 1] such that p2 does not extend to a lifting on [0. I]2?
Question 6.3 (cf. Burke A993)). Is it a theorem of ZFC that there is no Borel lifting ?
on [0, 1] such that p2 extends to a Borel lifting on [0. I]2?
Question 6.4 (A.H. Stone). Is there consistently a Borel lifting on [0, 1] of bounded
Borel class?
The existence of a product lifting in case of a product of arbitrary complete probability
spaces was solved by Macheras and Strauss A996d). For an improvement of this result see
Theorem 6.11 below.
S & ?(?) and ? e 0(?) will be called respecting coordinates for ? if S(Ej) с E*j
and i/(C*j) с С* for all 0 ? J с /. The notion traces back to works of Burke A995)
and Fremlin B00?). For finite index set / all the above concepts make sense for ?-finite
measure spaces (?,-, IT,-, ?,-), /' e /, instead of probability spaces, since in that case there
exists a unique product measure. But again it is only a matter of technique to carry over
the results. So we are going to consider probability spaces only. The first essential result
concerning liftings respecting coordinates is due to Burke A995). It has been obtained by
an application of a theorem of Erd6s-Rado.
THEOREM 6.5. For any finite family ((?,-, ?,, ?,)),"=1 of complete probability spaces
there exists a lifting for (^)J = | ?, respecting coordinates.
Fremlin B00?, 345G), attempting to prove an infinite version of Theorem 6.5 took first
into account densities respecting coordinates. Applying D. Maharam's theorem on the
structure of measure algebras (cf. Fremlin A989,3.9)) Fremlin B00?) has got the following
result.
THEOREM 6.6. For any family ((?,. Ej, ?,)),6/ of complete probability spaces there
exists a separately additive density S for ? / respecting coordinates.

1160
W. Strauss et al.
Separate additivity above (D.H. Fremlin called it the (*) property) means that
8(AUB)=8(A)US(B) for any A e ?) and В е ?*? (*)
for all disjoint J, К с /.
It seems that the proof procedure applied above is restricted to complete probability
spaces and gives no information about the marginals. Probably without the completeness
assumptions one cannot obtain the separate additivity above. The admissible densities have
convenient properties from the product point of view and for this reason they are the
natural candidates for marginals in the existence results below. The next result of Macheras,
Musial and Strauss A999) gives not only a solution of the existence problem for densities
respecting coordinates and arbitrary probability spaces but at the same time it produces
a unifying approach to all former existence results for existence of product densities and
liftings as well as to those respecting coordinates. The conclusion of this result is on one
side weaker since it can not be shown that the density which is respecting coordinates
is separately additive in addition but on the other hand its proof is quite elementary (in
particular D. Maharam's theorem is not applied there) and the product probability needs
no completion.
THEOREM 6.7. Let ((?,-, 2?,-,?,-))?6/ be a family of probability spaces. If io e / is fixed,
then for each <5,0 e #(?,-0) and for arbitrary- <5, e ?#(?,) with i e / \ {io} there exists
? ? e ??(?) such that S respects coordinates and ? e <S>,6/ &· ^n Particular the theorem
holds true if all the densities <$,, i e /, are identical and admissible. As a consequence, it
follows that each admissible density is a consistent density.
Since ??(?,¦) ? 0 if /' e / the above result is a generalization of Macheras and Strauss
A995). The proof of Theorem 6.7 is based in principle on the inductive steps exhibited
before Theorem 2.6, but this time the induction is more complicated (see Macheras, Musial
and Strauss A999) for details). Applying Theorem 6.6 Fremlin B00?) proved the following
nice result.
THEOREM 6.8. For any family ((?,-, 2T;,ji;))ie/ of complete Maharam homogeneous
probability spaces there exists a lifting for <S),-6/Mi respecting coordinates.
The proof of Theorem 6.8 has been reduced to the next Theorem 6.9 of Fremlin B00?)
which is of an even more special nature by using the fact that the measure algebra of any
Maharam homogeneous probability space is isomorphic to some {0, 1}'' with its usual
measure v,- for i el.
THEOREM 6.9. For any set I any translation invariant lifting for the usual measure ? on
{0, 1}' respects coordinates.
Then Macheras, Musial and Strauss B00?c) applying Theorem 6.22 to the conclusion
of Theorem 6.8 were able to generalize Theorem 6.8 to the following form:

Liftings
1161
THEOREM 6.10. For an arbitrary family ((?,-, ?1,, ?,)),6/ of complete Maharam
homogeneous pmbability spaces and arbitrary finite collection (((У/, Tj, Vj))jej of complete
probability spaces there exists a lifting for «S>,6/ M/) § «8> /е./ vi) respecting coordinates.
In case when / = 0 one gets Burke's theorem with a completely new proof. It is as
yet an open problem whether for arbitrary families of complete probability spaces there
exist liftings respecting coordinates. Only partial results are known. The next result seem
to be the most far reaching for general complete probability spaces at the moment. (See
Macheras, Musial and Strauss A999) as well as Fremlin B00?) for this result.)
THEOREM 6.1 1. Let к be an ordinal and ({??, ??, ??))?<? be a family of complete
probability spaces. Then for each po e ?(??) and each collection {pa e AG ?{??): 0 <
a < к) there exists a lifting ? e ®a<K Pa respecting coordinates of each initial segment
ofK.
Considering linear liftings a general existence theorem was given by Macheras, Musial
and Strauss B000). The proof is based on the extension Lemmata 3.4 and 3.5.
THEOREM 6.12. ((?,-, ?{,?,)),-6/ be a family of complete probability spaces with
product pmbability space {?,?,?). If iq e / is fixed, then for each ?,0 e <?(?,0) and
for arbitrary ?, e ?<?(?,) with ie/\ {/<)} there exists ? ? e 0(?) such that ? respects
coordinates and ? e <S>,6/ ?'·
The above result gives also an alternative proof of Theorem 6.6.
It seems however that the extreme point method of Theorem 3.3 does not work while
trying to convert a linear lifting respecting coordinates found in Theorem 6.12 into a lifting
respecting coordinates.
6.2. Stmng liftings in products
Though weaker than liftings respecting coordinates, product liftings are useful for
transporting strong liftings from the factors onto the product as well as for attacking
a problem posed by Kupka A983), whether the product of two topological (especially
Radon) probability spaces has the ASLP if the factors have this property. In particular it is
natural to ask, whether the product of hyperstonian spaces has the ASLP. It is known from
Macheras and Strauss A996c) that the only possible candidate for a strong lifting on the
ordinary completed product (? ? ?, ? ? ?, ? §> ?, ? §> ?) of a hyperstonian probability
space (?,?, ?, ?) with itself is necessarily a product lifting of the canonical strong liftings
in the factors. If (?, ?, ?, v) is the hyperstonian space of, e.g., the Lebesgue probability
space on [0, 1 ], it follows from the next theorem that there exists no strong lifting for ? <g> ?
since ? ? ? g ? §> ?, the latter a result of Fremlin A976). But let us remark that subject
to the continuum-hypothesis the Radon product of (?, ?, ?, v) with itself has the ASLP
by Theorem 4.12 (see Macheras and Strauss A996c, Section 3, Remark 5) for details).
It remains an open problem, whether the Radon product of two Radon probability spaces

1162
W. Strauss et al.
with the ASLP has the ASLP, in particular it is unknown, whether the Radon product of
two hyperstonian measure spaces has the ASLP.
Let us remark that the existence result for product liftings, Theorem 6.11, can be stated in
an equivalent form in terms of products of lifting topologies which implies Theorem 6.13, a
permanence result for strong liftings on products. For the proofs of the next three theorems
see Macheras and Strauss A996d) as well as A996c). If ((i2,-,7/)),-6/ is a family of
topological spaces we write f],6/ % f°r the product topology on f],6/ ?-
THEOREM 6.13. Let be given a family ((?,, 7/, ?,, ?,)),6/ (/ a non-empty set) of
complete topological probability spaces, pj e Л (?,) for each i e I,_Jet (?, ?,^) be a
complete probability space such that ? = ?]/6/ ?,, ®2Г,- с ?, д|0F/2Г,- = <S>/6/Mi.
and let ? e ?(?) satisfy ? e ®,6/ Pi- Then we have that ?,6/ % <? ? and ? is strong
with respect to ?]|6/ % if and only if all p, are strong for i e /.
If (?, ?, H, v) is again the hyperstonian space of the Lebesgue probability space on
[0, 1 ] then it follows from Theorem 6.13 the surprising fact that the unique strong lifting
? for ? is neither consistent nor admissible and any consistent lifting ? for ? is not strong.
In particular, ? has the ASLP but not the USLP and the lifting topologies ta and ?? satisfy
ta ? t„, ?? ? ?? g ? <g> ?.
Under a mild set-theoretic assumption, i.e., assuming the non-existence of measurable
cardinals, a purely topological result of P.C. Curtis, M. Hendricksen, and J.R. Isbell
(see Gillman A960, p. 53)) says that a product of two topological spaces is extremally
disconnected if and only if one factor is extremally disconnected and the other one is
discrete. This implies the following result whose assumptions are satisfied, e.g., by the
Lebesgue measure space on [0, 1]. It answers the question, whether the product of lifting
topologies is a lifting topology to the negative.
THEOREM 6.14. If we assume the non-existence of measurable cardinals then for
two complete probability spaces (?(, ??, ?,) and pi e ? (?,) with non-discrete lifting
topologies tPl, ??? (/ = 1,2) the pmduct topologies tP] ? tp,, ??] ? ??2 are not extremally
disconnected. #(?\ ? ?2, ?, ?) is a complete probability space such that ?\ <g> ?2 с ?,
?\?\ ®?2=?\ <8> ??, tP] ? tP2, ??] ? ??2 с ?, then in particular we have
for each ? e ?(?), and if ? is stmng with respect to tP] ? tP2 respectively ??] ? ?? (for
example if ? e p\ <g) p2) then
tPl ? tp-, С tn and ??] хтР2Стп, but not equality.
The next result is an analogue of Theorem 6.11 for strong liftings.
THEOREM 6.15. Let к be an ordinal, ((??, %, ??, ??))?<? any family of complete
topological probability spaces with completed product space (?,?,?) and pmduct
topology T. Suppose that all measures ?? (? < ?) have full support and that ?? has

Liftings
1163
the ASLP while all other measures ?? for 0 < a < к have the USLP. Then for any stmng
lifting po & ?(??) there exist strong liftings pa e A(\ia)for 0 < a < к and a strong lifting
? e ?(?) such that ? e ®?<? ??, ? <? ? and
???? ?\ ??\ = ((?)??(?)? ?\ ??
?<^?<? ' ^?<? ' ?^?<?
for all A e &)?<??? and all a < к, that is ? is a strong lifting respecting individual
coordinates and initial segments of coordinates.
In particular, if ?? = B^a)for all a < к then ? = ?(?) and if all the measures ??
(a < к) are completion regular then ? is also completion regular.
The last theorem together with Corollary 4.8 imply the next classical result of Maharam
A958) and Kakutani A943).
COROLLARY 6.16. For every ordinal к there exists a strong lifting for the usual
measure ? on {0, 1}?.
Another consequence of the last theorem together with Corollary 4.8 is the next one
given by Fremlin B00?, 453H).
COROLLARY 6.17. Let ((??,??, ??,??))??? be a family of complete topological
probability spaces with completed product measure space (?, ?, ?) and product topology
T. Suppose that all measures ?? (a e /) have full support and that every Ta has a
countable measurable network. Then ? is ? ?-additive topological measure and has a
T-strong lifting.
6.3. Liftings on projective limits
Throughout what follows (??, ??,??, fap, I) denotes a projective system of probability
spaces which will be assumed complete if liftings are involved. If / = к, where к is an
infinite cardinal, we say that the projective system (??, ??,??, /??,?) of probability
spaces is continuous, if for every limit ordinal ? < к we have that (??, ??, ?^, {fa}a<^)
is the projective limit of the projective system (??, ??, ??, /??, ?}. For projective systems
of topological probability spaces a similar definition can be posed.
As for products and inductive limits suitable notions of compatibility of projective
systems and liftings or densities are crucial. A family (pa)aei °f densities pa e ?{??)
is called self-consistent, if
??(/??(?))=/-??(??(?)) (?)
for all A e ?? and all ?, ? e /with a ^ ?. For liftings pa e ?(??),? e / this is equivalent
to the ???-??? -continuity of the maps fap as well as with the equations
P?(hofaP) = (P?(h))ofaP
(pOC)

1164
W. Strauss et al.
for all h e ?°°(??) and all ?, ? e / with ? ? ? by Section 3. The latter equation can be
taken as a definition of a self-consistent family of linear liftings for the above projective
system. If (?, ?,?, (fa)aei) IS the projective limit of the above projective system a
density ? e ?(?) is called ? projective limit of the self-consistent family (pa)aei if
p{fal(A))=f-](pa(A))
for all A e ? and all a e /. Again for liftings (where we take the completed projective
limit) this is equivalent to the ???-??-continuity of the maps fa as well as with the validity
of the equations
px(b/„) = (p„xW)o/„
for all h e ?°°(?„) and all a e / and this equation can be taken as the definition of the
projective limit of linear liftings. In symbols we write ? e proja6/ lim pa for densities and
(linear) liftings. We assume throughout that all canonical projections fa of a projective
limit (?, ?, ?, (fa)aei) of probability spaces are surjections.
Suppose we have a projective system (?, ?,?, (fa)a<K)- In general, it is not obvious
whether there exists at all a self-consistent family of densities or liftings associated with
the system. By Macheras, Musial and Strauss B00?a) there is an answer to the positive in
the next result.
THEOREM 6.18. Let к be an infinite ordinal and (??, ??, ??, fap> к) a continuous
projective system of probability spaces with pmjective limit (?, ?,?, (/?)?<?)· Then
there exist a self-consistent family {??)?<? of densities Sa e ?(??) and a density ? e #(?)
such that S is a pmjective limit of the system (Sa)a<lc.
If all probability spaces of the pmjective system are complete and the pmjective limit is
taken completed, then we may in the above replace the word "density" by "linear lifting"
and "lifting", respectively, throughout.
We will give examples below showing the existence of self-consistent families over non
well ordered index set without projective limit. Necessary and sufficient conditions for the
existence of projective limit liftings are given by Macheras and Strauss A994).
THEOREM 6.19. Let be given a pmjective system (??, ??, ??, /??. ?) of complete
probability spaces with completed projective limit (?, ?,?, (fa)aei), ? self-consistent
family (pa)aei of strong liftings pa e ?(??) and a lifting ? e ?(?). Then the following
conditions are all equivalent:
(i) the pmjective limit topology ? of {???}aei 's contained in ? and ? is stmng w.r.t.
?;
(ii) the pmjective limit topology ? of(tPa)aei is contained in ? and ? is strong w.r.t.
T;
(iii) ? is a pmjective limit of (pa )ae/;
(iv) ? с ?p\
(?) ? с tp.

Liflings
1165
For projective systems of (complete) topological probability spaces we may ask for the
existence of self-consistent families of strong liftings and densities and of strong projective
liftings. In general self-consistent families of strong liftings do not have any projective limit
lifting as shown in the next example.
Let X := {0, 1}', where / is an index set with card(/) = N2, and let ? be the
probability measure on Bo(X) constructed by Fremlin A979). Denote by T(I) the
system of all finite subsets of /. For any a e JF(/) put Xa := {0, 1}" and ?? : =
fa(fJ-), where /„ is the canonical projection from X onto Xa, and denote by Ta the
discrete topology on Xa. For any ?, ? e ? (I) with a ^ ? denote by fap the canonical
projection from Xp onto Xa. If by ? is denoted the projective limit topology of the
family (Ta)aeyr{l), then (?,??(?),?, (f^aeFU)) 1S tne projective limit of the system
{Xa,Bo(Xa), ??, /??,?(?)). For any a e T(I) the map pa from B0(Xa) onto itself
defined by
pa(A) = A for any A eBo(Xa)
is a strong lifting in ?(??) and the family (pa)a€^(i) is self-consistent. Since ? does
not admit a strong lifting, in particular there cannot exist any lifting ? e ?(?), which is a
projective limit of (pff)ff6^-(/), because in such a case we should have G = p(G) for any
element G of the family
G-={fal(Ga): Gae%,aeT(I)}.
But since Q is a basis for the topology T, it follows from Lemma 4.7 that ? should be
strong; this yields a contradiction. The same example shows also that an inverse lifting of
a strong one under a continuous and measure preserving map is not in general strong. In
fact, for each a e T{I) there exists by Theorem 3.7 a pa e ?(?) which is an inverse lifting
of pa under /„. But according to Fremlin A979) pa cannot be strong. But for well ordered
index set there is always a positive solution due to Macheras, Musial and Strauss B00?a).
THEOREM 6.20. Let к bean infinite ordinal and (Xa,Ta, ??, ??, /??, ?) be a projective
system of complete topological probability spaces. Suppose that (?, ?, ?, (/?)?<?) " 'he
projective limit of the above projective system and {??)?<? is a self-consistent family of
strong densities ?? e ?{??). Then for the projective limit topology ? of (Ta)a<K follows
ТсГ and there exists a strong density ? e ?( ? ) such that ? is a projective limit of the
system (??)?<?. ^ ^ _
In particular, if for every a < к we have ?? = B(Xa) then ? = B(X) and, if in addition
? = Bq(X) and all the measures ?? are completion regular, then ? is also completion
regular. The same is true if we replace "densities" by "liftings" throughout.
The last result raises the problem of the existence of a self-consistent family of strong
liftings for projective systems of complete topological probability spaces. Even for well
ordered index set the answer is to the negative in general as witnessed by the following
example. Take Fremlin's Radon probability measure ? on X := {0, l}^2 which has no
strong lifting and is supported by X (cf. Fremlin A979)). If к is the smallest ordinal of

1166
W. Strauss et al.
cardinality N2, then (?, ?(?), ?, (fa)a<K) is the projective limit of the projective system
{Xa,B(Xa), ??, /??,?) of probability spaces, where Xa := {0, ?}", /?? (respectively /„)
is the canonical projection from ?? onto Xa (respectively from X onto Xa), and ?? is the
image measure /?(?) on B(Xa) for a ^ ? < к. Assume that there exists a self-consistent
family (pa}a<K of strong liftings pa e ?(??). Then by Theorem 6.20 there exists a strong
lifting ? e Л (?); this yields a contradiction.
On the other hand we know two classes of spaces with projective systems admitting
a self-consistent family of strong liftings even for arbitrary index set. These are the
hyperstonian spaces with measure of full support and the extremally disconnected Baire
spaces where each set of the first category is closed, endowed with a category measure
(cf. Macheras and Strauss A994, Remark 2.2(iii) (b) and (c)), which includes also the
definitions of the above notions).
6.4. Various Fubini products
The most difficult type of lifting in products arises if we ask for lifted functions with
measurable respectively lifted sections or even for Fubini-like formulas. There is still a
comparatively satisfactory answer for densities in the next theorem. All results in this
subsection are taken from the paper Musial, Strauss and Macheras B001) if not otherwise
indicated.
THEOREM 6.21. Let (?,?,?) be an arbitrary probability space. If ? e ??(?) then
for each (?, ?, ?) and each S e #(?) there exists ? e ?{? <g) v) with the following
properties:
(i) ? ?<5<8>?;
(ii) [?{?)]? = ?{[?(?)]?)^?????? and ? e ?® ?;
(iii) [?(?)]? is ?-measurable for all ? e ? and ? e ? ®T.
Admitting the completeness of both measure spaces one can find ? e ?(? <g> v) satisfying
the properties (ii), (iii), and the following two properties:
(iv) ?(? ? S) с S(A) ? ? (?) for all A e ? and В еТ:
(?) ?([?(?)]? U [<р(Ес)]ш) = 1 /or ?// ? е ? and ? е ? % ?.
The condition (iii) cannot be improved essentially. More precisely, if S e ?(?), ? e
??(?) and ? e ?(? <g> ?) are such that (ii) is satisfied and
[?(?)]? = ?([?(?)]?) for all ? e ? and ? e ? § ?
then either ? or ? is purely atomic.
For liftings a corresponding result looks as follows:
THEOREM 6.22. Let (?, ?, ?) and (?, ?, v) be complete probability spaces. For each
? e AGA{v) and each ? e ?(?), there exists ? e ?(? ® v) such that the following
conditions are satisfied:
(a) ? e ? <8>?;
(b) [?(?)]? = ff([7r(E)L)/or ?????? and ? &?&?.

Liftings
1167
Equivalently,
n~epx®a~ and [???)]? = ??([*??)]?)
for each f e ?°°(? <§> ?) and each ? e ?.
The above result is in a sense the best possible, as it follows from the next theorem.
THEOREM 6.23. Let (?, ?, ?) be a complete non-atomic and perfect probability space
and let (<9, T, v) be a complete non-atomic probability space. There exist no liftings
? e ?(?) and ? e ?(? ® ?) satisfying the following two conditions:
(j) there exists ? e ? such that for each ? e ?®?
[<p(E)f e ?-
(jj) for each ? e ? ® ? there exists a set Ne e ?? such that
[?(?)]? = ?([?(?)]?) for each ? ???.
In particular, no ? e ?(?) is an admissible density.
So we are left with the following problem for given complete probability spaces
(?,?,?.)???(?,?,?).
Question 6.24.
(i) Does there exist a lifting ?? e ?°°(? § ?) such that [<?^(/)]? e ?°°(?) and
[??(/)]? eCoc(?)fo?аnfeC^x(?®v)c?ndfo?гLn(?,?)e? ? ??
(ii) Can we choose in addition ?? e ?^(? § ?) in (i) for some marginal liftings?
In case both these complete probability spaces coincide with the Lebesgue probability
space over [0, 1] subject to the continuum hypothesis we get an answer to the positive
for Question 6.24(i) just by choosing a Borel lifting according to Theorem 4.12 for
the 2-dimensional Lebesgue probability on [0, l]2. But what is the position without the
continuum hypothesis?
In case of linear liftings Macheras, Musial and Strauss B00?d) proved the following:
THEOREM 6.25. Let (?, ?, v) be a complete separable probability space and (?, ?, ?)
a complete probability space. Then for each ? e <?(?) and each ? e AQ(v) there exists a
? e 0(? <8> v) such that
(i) (p(g ® h) = p(g) <g> r(h) for all g e ?зс(м) and h ??^(?);
(ii) for each f e ?°°(? ® v) and each we ?
M/)L = r(H/)]J·

1168
W. Strauss et al.
It remains an open question whether the separability assumption is necessary. However
again there is no hope for a linear product lifting with all sections being lifting invariant.
The subsequent theorem of Macheras, Musial and Strauss B00?d) is a direct consequence
of the appropriate part of Theorem 6.21.
THEOREM 6.26. Let (?,?,?) be a complete probability space and (?,?,?) be an
arbitrary complete probability' space. Assume that there are ? e 0(?), ? e Q(v) and
? e 0(? <8> v) such that the inequalities
И/)]ш@) ="([*>(/)] J@) and [<p(f)]\o>) = p([<p(f)f)(o)
hold true for all (?, ?) and all f e ?^ (? <§> ?). Then at least one of the measures is purely
atomic.
Question 6.27. Does there exist ? e 0(? ? ?), ? ? ^?) and ? e Q(v) such that
? ep®a, [?°°(/)]? e Cx(v) and [<px(f)f e ?^(?) for all / e ?^(? ® v) and for all
(?, ?) &? ? <9?
Question 6.28. Does there exist a ? e <?(? § ?) and a ? e ^(v) such that for all
/ e ?°°(? § ?) exists tf/ e ?0 with ?([<?(/)]?) = [?(/)]? e ??(?) for all ? ? tf/
and [<p(f)f e ?°°(?) for all ? e 6>?
For general complete probability spaces these problems are open as far as we know.
The next two results are related to the topic of this section since they also concern
some measurability properties of liftings from a different point of view. They are due to
Christensen A974).
THEOREM 6.29 (CH). Let ? be a separable ?-algebra on ? (i.e., generated by countably
many sets and containing all points) and let ? be a probability on ?. Let ? be the
weak*-topology on the space Loc(?). Then there exists ? e 0(?) such that for each
probability measure ? defined on the product ?-algebra ??(?,^(?)) <g> ? the function
if, ?) -> p(f)(w) is measurable with respect to the completion of v. In particular each
functional f* —> p(f)(a>) is universally measurable on ?,^(?).
THEOREM 6.30. We keep the notation of Theorem 6.29. Assume that ? is non-atomic and
let Ba (?) be the ? -algebra of sets possessing the Baire property with respect to the weak* -
topology of ?.°°(?). If ? e Л (?), then for almost every ? e ? there exists a measure ??
on ??(?) such that the functional f -> p(f)(a>) is not measurable with respect to the
completion of ??.
6.5. Applications of Fubini products to stochastic processes
Theorem 6.22 has applications to functions of two variables and stochastic processes.
All results in this subsection are taken from Musial, Macheras and Strauss B001). Let

Liftings
1169
(?,?,?), (?,?, ?) be complete probability spaces and let {Хв}неп be an arbitrary
real-valued stochastic process on (?,?,?). If {??}??(-) is another stochastic process
then it is called to be equivalent to {Хн}неп if for each ? e ? we have XH = ? ? a.e.
(?) (the exceptional set may depend on ?) according to Talagrand A987). [Хв}не(-> is
said to be measurable if the map (?, ?) -> ?«(?) is measurable with respect to the
product ?-algebra ? ® Т. {Хн}не(~> is bounded if the family {XH: ? e ?] is bounded
in ?,°°(?). There are several papers concerning the existence of measurable (or separable)
processes that are equivalent to a given process (cf. Cohn A978), Talagrand A987, 1988)).
Sometimes a measurable process equivalent to a bounded {Хн}ве(-) can be defined by
setting ?? = р(Хв), where ? e ?(?) and the initial process X or (?, T, v) satisfy some
additional conditions. In particular, the lifting ? in Cohn A978) is assumed to be strong
and ? is taken to be an interval. In general however, a strong lifting might not exist on
an investigated topological measure space. With the help of Theorem 6.22 we get, - by
a different method-, the following two results stated in Cohn A978) under additional
topological assumptions.
THEOREM 6.31. For each ? & AGA(v) and each bounded measurable stochastic
process {XeJflew on a space (?,?,?) there is a collection of measurable functions
{УеЬем °n (?, ?, ?) satisfying the following conditions:
(i) ?.(?) = ?(?.(?)) for each ???.
(ii) There is ? ? e To such that for every- ? ? ? ? we have XH = Ye a.e. (?) and
{??\???? is a measurable stochastic process on (?,?,?).
(iii) There is ?? e ?? such that for every ? ? ??
? .(?) = ? .(?) a.e. (?).
(iv) If ? is a separable metric space and (Хн)н$М\ '·* continuous in probability, then
(??)???? is separable. Furthermore, every countable dense subset of&\ ? ? is a
separating set for (Yh)h$mx-
In the terminology of Cohn A978) the process {J«}«6w is called ?-canonical. Another
application of Theorem 6.22 is the next result.
THEOREM 6.32. For each ? e ???(?) and each bounded measurable stochastic
process {??)??? on a space (?,?.?) there is a measurable process {Yh}h€(-> on
(?, ?, ?) that is equivalent to {Хн}не(-> and satisfies the following conditions:
(i) ??=?(??) for each ?;
(ii) There exists a set ?? e ?? such that for each ? <? ?? we have
?(?) = ???) a.e. (?).
Notice that in Theorem 6.32 we had to use properties of ? (X) in order to assure the plane
measurability of Y, where ? is given by Theorem 6.22. Direct defining of the process ?
by setting ?? = p(Xe) for quite an arbitrary lifting might destroy the plane measurability
properties of the process X (cf. Cohn A978)).

1170
W. Strauss et al.
One of the important problems in the theory of functions of two variables is pointing out
of conditions guaranteeing the plane measurability of a separately measurable function.
A notion of a stable set investigated by D.H. Fremlin and M. Talagrand (see Talagrand
A984)) turned out to be very fruitful in this field. In particular, Talagrand A984, 10-2-1)
proved that if ? is a compact set, ? is a Radon measure on ?, /: ? ? ? ^ R is
measurable as a function of the first variable and continuous as a function of the second
variable, then / is measurable, provided the family {fH: ? e ?] is a stable set. Applying a
result of Fremlin A983) on stable sets together with Theorem 6.22, we get a similar result
for functions that have lifting invariant sections. It may be considered as a strengthening of
M. Talagrand's method from Talagrand A988) of modification of a stochastic process with
the help of consistent liftings.
THEOREM 6.33. Let (?,?,?), (?, ?, ?) be complete probability spaces and, and
{Хв)ве&) be an arbitrary real-valued bounded stochastic process on (?,?,?) with
measurable paths (i.e., all functions ?? are ?-measurable). If the set {X«: ? e ?] is
stable then for each ? e ???(?) the process {YH}HeH given for each ? by Yh '¦= ?(??) ls
? <g> ? -measurable and equivalent to {Хн}не(->-
COROLLARY 6.34. Let (?, ?,?), (?, ?, ?) be complete probability spaces and let
/:йх e^Rteo separately measurable function such that the set {fH: ? e ?] is
stable. If there exists ? e ???(?) such that for each ? e ? the equality p(fH) — fH
holds true, then f is ? ® ?-measurable.
7. Liftings for abstract valued functions
Throughout let be given complete probability spaces (?. ?, ?) and a completely regular
Hausdorff space T. For any lifting ? e ?(?) a Baire-measurable function f-.?^?
induces a Borel-measurable map pj (/) from ? into ??, the Stone-Cech compactification
of T, defined via the formula
hoPT(f) = p(hof) {0TheCh(T).
According to Bellow A980) the map / is called lifting compact if for every ? e ?(?)
we have pr(f)(M) e ? for ?-a.a. ? e ?. This implies h о pr(f) = h ° f a.e. (?)
for every h e C/,(T), where the null-set involved depends on h e C/,(T). If for lifting
compact / the stronger equivalence pr(f) = f ?-a.e. holds true for any ? e ?(?),
then / is called strongly lifting compact by Babiker et al. A986), and a completely
regular Hausdorff topological space ? is (strongly) lifting compact by definition, if for
any complete probability space (?, ?,?) every Baire-measurable map f-? -> ? is
(strongly) lifting compact. Lifting compactness of completely regular Hausdorff spaces is a
property lying strictly in between strong measure compactness and measure compactness
by Bellow A980, 6.1) as well as Edgar and Talagrand A980, Theorem 1). By Bellow
A980) lifting compactness has good stability properties.

Liftings
1171
Theorem 7.1.
(i) An arbitrary subspace of a compact metrizable space is lifting compact.
(ii) Every Baire set in a lifting compact space is lifting compact.
(iii) Any continuous bijection between completely regular spaces with Baire measurable
inverse transforms a lifting compact space into a lifting compact one.
(iv) A countable product of lifting compact spaces is lifting compact.
The results (ii) to (iv) are analogs of results proved by Moran A969) for (strongly)
measure compact spaces. A Banach space ? under its weak topology is lifting compact if
and only if every ?-valued scalarly measurable function is scalarly equivalent to a Bochner
measurable function by Bellow A980, Section 6, Remark 2). Every subspace of a compact
metric space is strongly lifting compact. Moreover, this holds true for every strongly lifting
compact space. The next three theorems are taken from Babiker et al. A986).
Theorem 7.2.
(i) An arbitrary subset of a strongly lifting compact space is strongly lifting compact.
(ii) Countable unions of strongly lifting compact Baire subsets of a completely regular
Hausdorff space are strongly lifting compact.
(iii) A countable product of strongly lifting compact spaces is strongly lifting compact.
(iv) The image of a strongly lifting compact space under a continuous surjection with
Baire measurable section is strongly lifting compact.
(v) A measure compact space is strongly lifting compact if every point has a strongly
measure compact neighborhood.
The next result states a relation between strong lifting compactness and the USLP. Its
proof relies on Theorem 3.7.
THEOREM 7.3. If the map f-.?—* ? is strongly lifting compact and ? is the image
measure of ? under f on the ?-algebra В := [В с ?: f~' (?) e ?], then the topological
measure space (T, T, B, v) has the USLP, if ? denotes the completely regular Hausdorff
topology of T.
From the last theorem together with the existence of a compact Radon probability
space without strong lifting (see Section 4) follows that neither lifting compactness nor
strong measure compactness imply strong lifting compactness, nor does strong lifting
compactness imply strong measure compactness, as witnessed by the standard Lebesgue
non-measurable subset of [0, 1]. For general metric spaces strong lifting compactness is
equivalent with measure compactness, which means in that case that every closed discrete
subspace has non-measurable cardinal.
It is an open problem, whether the converse of Theorem 7.3 is true, compare Macheras
and Strauss A992). The next result gives among other equivalent conditions an answer to
the positive for (E, weak), a metrizable locally convex spaces ? under its weak topology.
An essential tool for its proof is A. Tortrat's Theorem 8, from Tortrat A975).
THEOREM 7.4. For a metrizable locally convex space ? the following conditions are all
equivalent.

1172
W. Strauss et al.
(i) (E, weak) is strongly lifting compact.
(ii) Every Baire probability measure space based on (E, weak) has the USLP.
(iii) Every Baire probability measure space based on (E, weak) has the ASLP.
(iv) (?, weak) is completion regular and measure compact.
(v) Every Baire probability measure ? on (E, weak) is supported by a ?-measurable
closed linear subspace of ? which is separable with respect to the metric of E.
(vi) (?, weak) is measure compact and every Borel subset of (E, metric) is
measurable with respect to any Baire probability measure on (E, weak).
(vii) Every scalarly measurable function from a complete probability space into ?
agrees a.e. with a Bochner measurable function.
(viii) (E, weak) is measure compact and {0} is a Baire subset of ? with respect to
(?, weak).
(ix) (E, weak) is measure compact and there exists a sequence in ?", the topological
conjugate of E, which separates the points of E.
(?) (?, weak) is measure compact and there is a continuous linear injection from
(E, metric) into RN.
(xi) (?, weak) is measure compact and submetrizable.
If the locally convex space ? is normable, we may add the following condition.
(xii) (E,weak) is measure compact and there is a continuous linear injection from
(E,norm) ?????°°(?).
It can be seen from the last theorem that within the class of all metrizable locally convex
spaces strongly lifting compact spaces can be characterized in a purely topological way.
In Strauss A992) the strong lifting compactness of conjugate Banach spaces under their
weak* topology has been discussed. In this class the equivalence of the condition (vii) in the
last theorem with strong lifting compactness as well as with the conditions corresponding
to conditions (ii) respectively (iii) of the last theorem breaks down. There are similar
characterizations for strongly lifting compact functions in Babiker et al. A986). The
conjugate ??([0, 1]) of C/,([0, 1]) is a non-separable Banach space which is submetrizable
under its weak topology. The mild set-theoretic assumption that the continuum is measure
compact implies that (??([0, 1]), weak) is measure compact. Therefore by the last theorem
?([0, 1 ]) is an example of a non-separable strongly lifting compact space.
In connection with the definition at the beginning of this section one should mention the
following construction of a lifting of Banach space valued functions, which is commonly
used in the context of differentiation of vector measures and integration of vector functions.
If X is a Banach space and X' is the space of continuous functionals on X then for
given function /: ? -> X', satisfying (x, f) e L^ip), where by definition (*, /)(?) : =
(f(co))(x) for every ? e ? and for every ? e X, we can for each ? e ?(?) define a
function ?(/):? -> X' by setting (x,p(f)(w)) := p((x,f})(a>) where ? e X for all
? & ?. Von Weizsacker A978) proved that p(f) is measurable when X' is equipped with
its weak* -topology. This function is an essential tool in investigating several aspects of
integration. More details and abandon references can be found in Musial B002).
In fact we have the following result equivalent to the existence of a lifting (compare
A. and С Ionescu Tulcea A962, 1969a) and Kolzow A968)):

Liftings
1173
THEOREM 7.5. For given c.l.d. measure spaces (?, ?,?) the following conditions are
all equivalent:
(i) There exists a lifting on ?°°(?);
(ii) For any normed space X and any bounded linear operator и from L1 (?) into
X' (the topological dual space of X) there exists a map f from ? into X' with
(x,f) е?°°(м)/ога//л:еХ and\\f\\ ? N1 such that J{x, f)gd^ = (x,u(g'))
for all ? e X and all g e ?' (?).
8. Liftings and densities with respect to ideals of sets
The notion of measure-theoretical lifting and density is a particular case of more general
notions of lifting and density with respect to an ideal of sets. One may call them
respectively by ^-lifting and j7-density. Given a measurable space (?, ?) and an ideal
J С ? one says that a Boolean homomorphism ?: ? -> ? is an J7-Iifting if p(A) = A
for all A e ? and p(A) = p(B) whenever A = В in the sense of J. Similarly j7-density
is introduced.
As in the case of measure spaces there is also an equivalent formulation for the Boolean
algebra ? /J. In fact the original paper of von Neumann and Stone A935) has been written
in the language of ideals. In spite of this, the theory of J'-liftings is much less developed
than the theory of measure liftings. Here are some known facts. Maharam A977) noticed
that the following result follows easily from Graf A973):
THEOREM 8.1. If ? is a Baire space (i.e., no non-empty subset of ? isofthe first category
in itself or, equivalently, in ?), and ? is the ? -algebra generated by Borel subsets and the
?-ideal С of sets of the first category, then there is a strong lifting on ?/C.
For some time there has been an open question whether each Boolean algebra of the
form ?IJ which additionally satisfies the countable chain condition has a lifting. Shelah
A998) answered it negatively.
THEOREM 8.2. There is a cardinal к (к = 2*°++ suffices) a ?-ideal J С V(k) and
a ?-algebra ? С V(k) containing J and such that ?/J satisfies the countable chain
condition but ?/J has no lifting.
The next theorem of von Neumann and Stone A935) is a classical tool for converting a
density into a lifting in the situation of this section. Let Ka be a fixed infinite cardinal. Let
us remember that a lattice V is called (conditionally) ?-complete for some cardinal к, if
every subset W of V having cardinality ^ к (with lower and upper bounds in V) has an
infimum and a supremum in V.
THEOREM 8.3. Let J be an ideal in a Boolean algebra В which is conditionally к-
complete for all к < Ha. Ifcard(B/J) ? Ha, then for any J-density ? there exists a
J-lifting ? with <p(B)^p(B)for all В е В.

1174
W. Strauss el ai
Von Weizsacker A976, Theorem B.3) shows that the condition card(??/j7) ^ K„ in the
last theorem cannot be weakened to card(B/J) ^ ??+?, even if one is interested only in
the existence of linear liftings.
THEOREM 8.4. Assume sup ? < ?? for all families ? of cardinals with card(T) < Ka
and ? < N„/or all ? e ?. Then there exist set algebras Bi, ideals Ji in Bi, Ji-densities
?? for / = 1,2 such that the following conditions hold true.
(i) Ji is к-complete for all к < Ka/or i = 1,2.
(ii) B\/J\ and Bi/Ji are isomorphic with card(??,/J/) = ??+? for / = 1,2.
(iii) There exists no linear lifting p\ :?^(?|) -> C^(B\) satisfying р\(Хв) > ??^?)
for all В еВ{.
(iv) There exists a linear lifting pi: ?^(??) -> C^(Bi) satisfying рг(Хв) ^ XK(fl)/or
?// ? e Й2, but there is no Ji-lifiing ~p~, for Bi satisfying ?? (В) с ~р2(В) for all
BeB2-
At last we formulate a few facts concerning liftings on the set of natural numbers.
It is well known that if J is the ideal of all finite subsets of N, then there is no J7-Iifting
on V(N).
To formulate the next result denote by #A the cardinality of А с N. Then, define a
density measure ? : V(N) -> [0, 1] by setting for each А с N
#(АП{1,2,...,»})
?(?) := Lim ,
where Lim is a Banach limit on l^.
With this concept, we have the following result of D. Maharam and P. Erd6s (see
MaharamA976)):
THEOREM 8.5. Let ? be a densit}- measure on N and let J := {А С ?: ?(?) = 0). Then
there is no J-lifting on V(H).
9. Beyond Сж(?)
As we have mentioned at the end of Section 3 it has been proved already by A. and
С Ionescu Tulcea A969a) that there is no lifting on ?''(?) if 1 < ? < oo and ? is non-
atomic. It turns out however that each lifting can be extended far beyond ?^(?), as the
following result of Monakov-Rogozkin A974) shows
THEOREM 9.1. Given ? e ?(?) there exists an ideal K,(p) in the lattice ?'^(?) with the
following properties:
(i) Cx^)C>C(p). ^
(ii) There is a lifting ? on K,(p) which is an extension of p.
(iii) IC(p) is the largest ideal in ?^(?) possessing the above two properties.
(iv) The extension of ? onto K,(p) is unique.

Liftings
1175
More precisely, / e K,(p) if and only if f/g e ?^(?) for a function g e ?^(?)
such that g ^ 1 and /0(l/g) = 1/g. One can also say that K,(p) = ?(?(?, T)) where
? identifies functions ?-equivalent, ? is one of the lifting topologies and C(i2, T) is the
space of T-continuous real-valued functions.
Monakov-Rogozkin A974) proves also that if ? e 0(?) \ ?(?) then a counterpart of
fC(p) for yo does not exist.
10. Further applications
Lifting theory has so many applications in mathematical analysis that it is impossible to
give a detailed account for all within this article since this would afford too much additional
terminology. But we want to give references now for some of the applications which could
not be mentioned as yet.
Jirina A959) proved the existence of disintegrations for measures under separability
assumptions. The most common procedure taken now is to eliminate separability by an
application of liftings, see С lonescu Tulcea A965b), A. and С lonescu Tulcea A969a),
Hoffmann-Jorgensen A971), Pellaumail A972), Chatterji A973), Valadier A973), Saint-
Pierre A975), Pachl A978), Heller A983), Babiker and Strauss A982), Choksi and Duncan
(unpublished), and Rinkewitz A997). In A. and С lonescu Tulcea A969a) and Babiker
and Strauss A982), the application of a lifting comes indirectly in via the Dunford
Pettis theorem. Heller A983) determined the class of all Baire measures which can be
disintegrated with the help of liftings. Corresponding results are given by Rinkewitz
A997).
We are going to recall two particular applications of liftings to disintegration. Let (?,?)
be a measurable space and let ?: ? -> ? be a measurable map (i.e., p~] (T) с ?).
? : ? ? ? -> [0, 1] is a regular conditional probability if
(cp 1) Pg is a probability measure on ? for all ? e (9;
(cp2) ? -> ??(?) is ? := ? (?)-measurable for all A e ?;
(cp3) ? (А П p'[(B)) = fB ??(?)??(?) for allA e ? and all В е Т.
The next result was given by Hoffmann-J0rgensen A971).
THEOREM 10.1. Let ? be a Hausdorff space, ? is a regular probability on ?(?) (i.e., ?
is inner regular with respect to compact sets), (?,?) is a measurable space, ?:?—*?
is a measurable function and ? = ?(?). Then there exists a regular conditional probability
? : ?(?) ? ? -> [0, 1] such that for every ? e ? the measure PH is regular.
PROOF (Sketch). We assume for the simplicity that ? is compact. If / e C(i2) then we
define a ?-continuous finite signed measure vt on ? by setting for every В e ?
?/(?)= ? ???.
Jp~hB)
Let ?(?, f) be a Radon-Nikodym derivative of vf with respect to v. One easily sees that
?(?, f) is bounded in L°°(v) by ||/||эс· Take an arbitrary ? e ?(?) and set for every

1176
W. Strauss el al.
/?C(C)
P(-,f)-=p(p(-,f))-
It follows from the properties of 17 that ~?(?, ·) is for each ? a positive continuous linear
functional on С (i2). Moreover, for every ? and / e C(i2) we have
?@,??) = 1 and |^,/)|«;||/||зс.
Hence, for every ? there is a regular probability P# on ?(?) such that for all / e C(i2)
P@,/) = f fdPe.
The rest of the proof is a matter of calculations. ?
The second application of liftings we are going to present here is an application to
disintegration of compact measures. We follow the terminology of Pachl A978).
Definition 10.2. Let (?, ?) be a measurable space and let F>, T, v) be a probability
space. Let к be a probability on ? ® ? such that ?(? ? S) = v(B) for every ? e T.
Suppose that for every ? e ? there is a ? -algebra ?? on ? and a probability /># on 2^
such that the following two conditions are satisfied:
(a) for each Ae ? there exists N e ??(?) such that A e ?? for all ? e ? \ ? and the
function ? \ N э ? -> ??(?) is ?|(<9 \ ^V)-measurable;
(b) if A e Г and В е Г then
/c(A ? ?)= ??{?)??{?).
Jb
The collection {(.?^, Ро)}о6б> is called a v-disintegration of к.
Here is the main result of Pachl A978).
THEOREM 10.3. Let (?,?,?) and (?,?,?) be probability spaces and let к be a
probability on ? <g> ? with marginals ? and v. Assume that ? is complete and ?
is approximated by an ?-compact lattice К С ? which is closed under countable
intersections. Then there is ? ?-disintegration {(??, Рв)}вев °fK suc^ ?^?? ?? D К, and
К, approximates Pg for every ?.
Basic idea OF the proof. Let ? be a lifting on L^iv). By the Radon-Nikodym
theorem, for each A e ? there exists a ?-measurable function hA such that for every
В е Г, we have
?(? ? B)= / hAdv.
JB

Liftings
1177
For each ? e ? define a function ?# on K, by ??(?) := р(Ик(в)). It needs some work to
show that for each ? there exists a probability ?? ???(?,) such that ?? ^ ?? on /C and its
completion Pfi gives a ?-algebra ?? satisfying the requirements of the theorem (see Pachl
A978) for details). D
It should be mentioned here that using the above result J.K. Pachl proved that a
restriction of a compact measure (in the sense of Marczewski A953)) to a sub-a-algebra
is also a compact measure, thus answering a question of E. Marczewski.
For applications to the realization of homomorphisms see A. and С Ionescu Tulcea
A969a, Chapter X), Graf A980a), Babiker and Graf A983), Vesterstr0m and Wils A969),
Fremlin A989, 4.12 and 4.14), and Fremlin B00?).
Strongly connected with the above are applications of liftings to the existence of
measurable selections and sections discussed by Edgar A976), Talagrand A978), Losert
A980), Graf A980, 1982), and Kupka A983). The results are too technical to be presented
here so we mention only two of them, taken from Graf A980a).
THEOREM 10.4. Let ? фЪ be a Hausdorff space. Let ? be a non-trivial complete strictly
localizable measure which is locally determined and semifinite. Then, let ? : ?(?) —> ?/?
be a Boolean ?-homomorphism such that ? ? ? is a Radon measure on ?. Then there is
a E-B@)-measurable map f : ? -> ? with f'[(B)e ? (?) for all В e ?(?), i.e., ? is
induced by f.
Proof (Sketch). If ? e ?(?) then ? ? ? :?(?) -> ? is a homomorphism. Let ?,(?) be
the collection of all compact subsets of ?. Then, for each ? let ?,? := {К e ?,(?): ? e
? ? ?(?)}. One can easily see that П/С<„ ? 0. In fact for almost all ? the set ?\?,? is a
singleton. Consequently we can define /: ? -> ? by taking /(?) e ?\??. f satisfies the
conclusion of the theorem. ?
The next result is related to the extension of measures, ? ? 0 is Hausdorff, (?, ?, ?)
is a finite measure space and ?: ? -> ? is a measurable map. Denote by ? (?, ?) the
collection of all positive measures ? on ?(?) such that ? = p{v).
THEOREM 10.5. If ? e ? (?, ?) is Radon, then ? is an extreme point of ?{?, ?) if and
only if there exists ? ?-?(?) measurable weak section f : ? —* ? for ? with ? = /(?)
(i.e., ?(??/-'/>(?))=0/???//?? ?).
The separable modifications of stochastic processes and the domination of measures
are treated by A. and С Ionescu Tulcea A969b; 1969a, Chapter VII, Section 7 as
well as Chapter IX, Section 7). Here again the lifting eliminates restrictive hypotheses
(metrizability, for example) needed previously. For separable measurable modifications of
empirical processes and regularizations of stochastic processes see in addition Talagrand
A987, 1988), respectively. Schreiber et al. A971), use liftings for the construction of
probability measures corresponding to stochastic processes. This construction applies
algebraic models for probability spaces introduced before by Dinculeanu and Foias A968).
In these models liftings apply as well.

1178
W. Strauss et al.
A. and С Ionescu Tulcea (cf. A969a)) applied lifting in order to describe the space of
functionals on the space of Bochner integrable functions. Further applications of that type
can be found in Dinculeanu A967) and several papers concerning vectorial integration. For
applications of the consistent lifting to the Pettis integral see Talagrand A984).
С Ionescu Tulcea A965b) applies the strong lifting property and linear liftings for the
decomposition of measures into its ergodic parts and in A965) he uses the translation
invariant lifting (see Theorem 5.4) to solve a problem on almost stable sets in locally
compact groups posed by A.B. Simon.
Marechal A969) gives a definition of a measurable field of Hubert spaces over a
Hausdorff space based on the almost strong lifting property and shows that isomorphisms
between measurable fields can be induced by pointwise isomorphisms. She gives
decompositions of operators on Hubert space by means of liftings in Marechal A968,
1969).
Von Weizsacker A978) applies liftings to the regularization of functions with Radon
image measures and arbitrary completely regular range. Applications to subdifferentials
and convex functions are given by Levin A975).
Graf A995) gives an application of the lifting to self-similar measures. A similar
procedure was taken by Schief in an unpublished note.
Another application of the lifting theory is the existence and uniqueness of preimages of
a given measure (see, e.g., Edgar A976) as well Graf A980a, 1982)). Also Lipecki A998)
used liftings for the extension of measures.
References
Babiker, A.G. A981), Lifting properties and uniform regularity of Lebesgue measures on topological spaces,
Mathematika 28, 198-205.
Babiker, A.G. and Graf, S. A983), Homomorphism compact spaces, Canad. J. Math. 35, 476-558.
Babiker, A.G. and Knowles, J.D. A978), An example concerning completion regular measures, images of
measurable sets and measurable selections, Mathematika 25, 120-126.
Babiker, A.G. and Strauss, W. A980a), Measure spaces in which every lifting is an almost ?-lifting, Arab J. Math.
1, 11-21.
Babiker, A.G. and Strauss, W. A980b), Almost strong liftings and ?-additivity. Measure Theory Proc.
Oberwolfach 1979, D. Kolzow, ed., Lecture Notes in Math., Vol. 754, Springer, Berlin, 221-226.
Babiker, A.G. and Strauss, W. A980c), Measure spaces in which every lifting is an almost strong H-lifting,
Measure Theory Proc. Oberwolfach 1979, D. Kolzow, ed.. Lecture Notes in Math., Vol. 754, Springer, Berlin,
228-232.
Babiker, A.G. and Strauss, W. A982), The pseudostrict topology on function spaces. Rend, lstit. Mat. Univ.
Trieste 14, 99-105.
Babiker, A.G., Heller, G. and Strauss, W. A984), On a lifting invariance problem, Measure Theory Proc.
Oberwolfach 1983, D. Kolzow and D. Maharam-Stone, eds, Lecture Notes in Math., Vol. 1089, Springer,
Berlin, 79-85.
Babiker, A.G., Heller, G. and Strauss, W. A986), On strong lifting compactness, with applications to topological
vector spaces, J. Austral. Math. Soc. Ser. A 41, 211-223.
Bellow, A. A980), Lifting compact spaces. Measure Theory Oberwolfach 1979, Proceedings, Lecture Notes in
Math., Vol. 794, Springer, Berlin, 233-253.
Bhaskara Rao, K.P.S. and Bhaskara Rao, M. A983), Theory of Charges. Academic Press.
Bichteler, K. A970), A reduction of the strong lifting problem. Invent. Math. 11, 159-162.
Bichteler, K. A971), An existence theorem for strong liftings, J. Math. Anal. Appl. 33, 20-22.

Liftings
1179
Bichteler, К. A972), On the strong lifting property, Illinois J. Math. 16, 370-380.
Bichteler, K. A973), A weak existence theorem and weak permanence properties for strong liftings, Manuscripta
Math. 8, 1-10.
Bliedtner, J. and Loeb, P.A. B000), The optimal differentiation basis and liftings of L*-, Trans. Amer. Math. Soc.
352,4693^4710.
Burke, M.R. A993), Liftings and the property of Baire in locally compact groups. Proc. Amer. Math. Soc. 117,
1075-1082.
Burke, M.R. A993a), Liftings for Lebesgue measure, Israel Math. Conf. Proc. 6, 119-150.
Burke, M.R. A995), Consistent liftings, Unpublished notes of 1995/01/23.
Burke, M.R. and Just, W. A991), Liftings for Haar measure on @, 1 }k. Israel J. Math. 73, 33^44.
Burke, M.R. and Shelah, S. A992), Linear liftings for non-complete probability spaces, Israel J. Math. 79, 289-
296.
Carlson, Т., Frankiewicz, R. and Zbierski, P. A994). Borel liftings of the measure algebra and the failure of the
continuum hypothesis, Proc. Amer. Math. Soc. 120, 1247-1250.
Carothers, D.C. A990), Order continuous Borel liftings, Rocky Mountain J. Math. 20, 51-57.
Carothers, D.C. A992), Liftings into countably complete Banach lattices, Houston J. Math. 18. 467^472.
Chatterji, S.D. A973). Disintegration of measures and lifting. Vector and Operator Valued Measures and
Applications, Academic Press.
Choksi, J.R. and Duncan, R. (unpublished). Disintegration of measures and measure-valued conditional
expectation, unpublished and undated notes.
Christensen. J.PR. A974). Topology and Borel Structure. Math. Studies. Vol. 10, North-Holland. Amsterdam.
Cohn, D.L. A978), Liftings and the construction of stochastic processes. Trans. Amer. Math. Soc. 246. 429^438.
Dalgas, K.P A99?), On the existence of strong Borel liftings. Preprint.
Dieudonne. J. A948), Sur le theoreme de Lebesgue-Nikodym (III). Ann. Univ. Grenoble 25. 25-53.
Dieudonne, J. A951), Sur le theoreme de Lebesgue-Nikodym (IV). J. Indian Math. Soc. 15. 77-86.
Dinculeanu, N. A967), Vector Measures, Pergamon Press and VEB Deutscher Verlag der Wissenschaften.
Dinculeanu, N. and Foia§, С A968), Algebraic models for measures, Illinois J. Math. 12, 340-351.
Donoghue, W.F. A965), On the lifting property, Proc. Amer. Math. Soc. 16, 913-914.
Dunford, N. and Schwartz, J.T. A958), Linear Operators Part I, lnterscience, New York.
Edgar, G.A. A976), Measurable weak sections, Illinois J. Math. 20, 630-646.
Edgar, G.A. and Talagrand, M. A980), Liftings of functions with values in a completely regular space, Proc.
Amer. Math. Soc. 78, 345-349.
Eifrig, B. A972), Ein nicht-standard Beweis fiir die Existenz eines Liftings, Arch. Math. 23, 425^427.
Eifrig, B. A972a), Ein nicht-standard Beweis fiir die Existenz eines starken Liftings in C^ @. 1 ], Contributions
to Non-Standard-Analysis, W.A.J. Luxemburg and A. Robinson, eds, North-Holland, Amsterdam, 81-83.
Eifrig, B. A975). Ein Nicht-Standard-Beweis fiir die Existenz, eines Liftings, Measure Theory, Proc. Conf.,
Oberwolfach, 1975, Lecture Notes in Math., Vol. 541, Springer. Berlin, 133-135.
Ellis, H.W. and Snow. D.0.A963). On (L1)* for general measure spaces, Canad. Math. Bull. 6,211-230.
Erben, W. A983), Topologische und masstheoretische Liftings, Ph.D. thesis. University of Stuttgart.
Farah, 1. A998), Completely additive liftings. Bull. Symbolic Logic 4, 37-54.
Fatou, P. A906), Series trigonometriques et series de Taylor. Acta Math. 30, 335^400.
Fillmore, PA. A966), On topology induced by measure. Proc. Amer. Math. Soc. 17, 854-857.
Fremlin, D.H. A974), Topological Riesz Spaces and Measure Theory, Cambridge University Press, Cambridge.
Fremlin, D.H. A976), Products of Radon measures: A counter-example, Canad. Math. Bull. 19, 285-289.
Fremlin, D.H. A977), On two theorems of Mokobodzki, Note of 1977.
Fremlin, D.H. A978), Decomposable measure spaces, Z. Wahrscheinlichkeitsth. verw. Geb. 45, 159-167.
Fremlin, D.H. A979), Losert's example. Note of 18/9/79. University of Essex, Mathematics Department.
Fremlin, D.H. A980), Consequences of Martin's Axiom, Cambridge University Press, Cambridge.
Fremlin, D.H. A983), Stable sets of measurable functions. Note of 17 May 1983.
Fremlin, D.H. A989), Measure algebras. Handbook of Boolean Algebras, J.D. Monk and R. Bonnet, eds,
Elsevier.
Fremlin, D.H. B000a), Measure Theory, Vol. 1, The Irreducible Minimum, published by Torres Fremlin.
Fremlin, D.H. B00?), Measure Theory, Vols. 2-5.

1180
W. Strauss et al.
Fuglede, B. A971), The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier
(Grenoble) 21, 123-169.
Gapaillard, J. A970), Sur un theoreme de Maharam, С R. Acad. Sci. Paris 271A, 39^41.
Gapaillard, J. A973), Relevements monotones. Arch. Math. 24, 169-178.
Gapaillard, J. A975), Relevements sur une algebre d ensembles. Measure Theory Proc. Oberwolfach 1975,
Lecture Notes in Math., Vol. 541, 137-153,
Gardner, R.J. A975), The regularity of Borel measures and Borel measure compactness, Proc. London Math. Soc.
30,95-113.
Georgiou, P. A973), Liftings and disintegration. Bull. Soc. Math. Grece (N.S.) 14, part A. 56-74.
Georgiou, P. A974), A semigroup structure in the space of liftings, Math. Ann. 208, 195-202.
Georgiou, P. A980), On "idempotent" liftings. Measure Theory Oberwolfach 1979, Proceedings, Lecture Notes
in Math., Vol. 794, Springer, Berlin, 254-260.
Gillman, L. A960), A P-space and an extremally disconnected space whose product is not an F-space, Arch.
Math. Basel 11, 53-55.
Goldman, A. A977), Mesures cylindriques. mesures vectorielles et questions de concentration cylindrique.
Pacific! Math. 69, 385^413.
Graf, S. A973), Schnitte Boolescher Korrespondenzen und ihre Dualisierungen, Ph.D. thesis, University of
Erlangen-Niimberg.
Graf, S. A974), On a disintegration theorem of Dorothv Maharam, Note of 1974.
Graf, S. A975), On the existence of strong liftings in second countable topological spaces. Pacific J. Math. 58,
419^426.
Graf, S. A980), Measurable weak selections. Measure Theory, Proc. Oberwolfach 1979, D. Koelzow, ed., Lecture
Notes in Math., Vol. 794, Springer, Berlin, 117-140.
Graf, S. A980a), Induced ?-homomorphisms and a parametrization of measurable sections via extremal
preimage measures, Math. Ann. 247, 67-80.
Graf, S. A982), Selected results on measurable selections, Rend. Circ. Mat. Palermo B) Suppl. 2, 87-122.
Graf, S. A995), On Bandt's tangential distribution for self-similar measure, Monatsh. Math. 120, 223-246.
Graf, S. and von Weizsacker, H. A976), On the existence of lower densities in noncomplete measure spaces,
Measure Theory Proc. Oberwolfach 1975, Lecture Notes in Math., Vol. 541, 155-158,
Grekas, S. A985), On the existence of idempotent liftings. Bull. Soc. Math. Grece (N.S.) 26, part A, 47-52.
Grekas, S. A987), On the strong lifting property for products, Bull. Soc. Math. Grece (N.S.) 28. part A, 63-70.
Grekas, S. A989), On the existence of idempotent liftings, Proc. Amer. Math. Soc. 107, 367-371.
Grekas, S. and Gryllakis, C. A991), Completion regular measures on product spaces with application to the
existence of Baire strong liftings, Illinois J. Math. 35. 260-268.
Grekas, S. and Gryllakis, С A992), Measures on product spaces and the existence of strong Baire liftings.
Monatsh. Math. 114, 63-76.
Gruenhage, G. A984), Generalized metric spaces. Handbook of Set-Theoretic Topology, K. Kunen and
J.E. Vaughan, eds, Elsevier, Amsterdam.
Halmos, P. A950), Measure Theory. Van Nostrand, Princeton.
Hansel, G. A972), Theoreme de relevement et measures bivalentes, Ann. Inst. Poincare 8,395—401.
Hebert, D.J. A973), A general theorem for decomposition of linear random processes, Proc. Amer. Math. Soc.
38, 331-336.
Heller, G. A983), Zur Desintegration topologischer Masse, Ph.D. thesis. University of Stuttgart.
Hoffmann, K. A965), Banach Spaces of Analytic Functions, 2nd edn, Prentice Hall, Englewood Cliffs, NJ.
Hoffmann-j0rgensen, J. A971), Existence of conditional probabilities. Math. Scand. 28, 257-264.
Ionescu Tulcea, A. A965), On the lifting property (V0, Ann. Math. Stat. 36, 819-828.
Ionescu Tulcea, A. A966a), Sur le relevement fort et la desintegration des mesures, С R. Acad. Sci. Paris 262A,
617-618.
Ionescu Tulcea, A. A966b), Sur la domination et la desintegration des mesures. C. R. Acad. Sci. Paris 262A,
1142-1445.
Ionescu Tulcea, A. A967a), Liftings compatible with topologies. Bull. Soc. Math. Grece 8, 116-126.
Ionescu Tulcea, A. A967b), On the lifting property. Proceedings Symposium in Analysis. Queen's University,
Kingston Ontario (June, 1967).

Liftings
1181
Ionescu Tulcea, A. A973), On pointwise convergence, compactness, and equicontinuity in the lifting topology
(/), Z. Wahrscheinlichkeitsth. verw. Geb. 26, 197-205.
Ionescu Tulcea, A. A974), On pointwise convergence, compactness, and equicontinuity II, Adv. Math. 12, 171-
177.
Ionescu Tulcea, A. and Ionescu Tulcea, С A961), On the lifting property (/), J· Math. Anal. Appl. 3, 537-546.
Ionescu Tulcea, A. and Ionescu Tulcea, С A962), On the lifting property (//), representation of linear operators
on spaces LPE, 1 < ? < oo, J. Math. Mech. 11, 773-769.
Ionescu Tulcea, A. and Ionescu Tulcea, С A963), Problems and remarks concerning the lifting property,
Technical report, US Army Research Office, Durham, NC. September 1963.
Ionescu Tulcea, A. and Ionescu Tulcea, С. A964а), On the lifting property (III). Bull. Amer. Math. Soc. 70,
193-197.
Ionescu Tulcea, A. and Ionescu Tulcea, С A964b), On the lifting property (IV). Disintegration of measures, Ann.
Inst. Fourier (Grenoble) 14, 445^472.
Ionescu Tulcea, A. and Ionescu Tulcea, С A967), On the existence of a lifting commuting with the left translations
of an arbitrary locally compact group, Proc. of the Fifth Berkeley Symposium on Mathematics, Statistics, and
Probability, Berkeley, California, 1967, Vol. II, 63-67.
Ionescu Tulcea, A. and Ionescu Tulcea, C. A969a), Topics in the Theory of Lifting, Springer, Berlin.
Ionescu Tulcea, A. and Ionescu Tulcea, С A969b), Liftings for abstract valued functions and separable stochastic
processes, Z. Wahrscheinlichkeitsth. verw. Geb. 13, 114-118.
Ionescu Tulcea, C. A955), Deux theoremes concernant certains espaces de champs de vecteurs. Bull. Sci. Math.
79, 106-111.
Ionescu Tulcea, С A965a), On the lifting property and disintegration of measures. Bull. Amer. Math. Soc. 71,
829-842. (Invited address presented before the Chicage Meeting of the AMS on April 9, 1965.)
Ionescu Tulcea, С A965b), Remarks on the lifting property and the desintegration of measures. Technical Report,
US Army Research Offices, Durham, NC.
Ionescu Tulcea, С A966), Surcertains endomorphismes de L?(Z. ?), С. R. Acad. Sci. Paris 261A, 4961^4963.
Ionescu Tulcea, С A971), On liftings and derivation bases. J. Math. Anal. Appl. 35, 449^466.
Ionescu Tulcea, С A972), Liftings commuting with translations. Proceedings of the Sixth Berkeley Symposium
of Math. Stat, and Probability, Vol. 2, Univ. of California Press, Berkeley. 97-100.
Ionescu Tulcea, С and Maher, R.J. A971), A note on almost strong liftings. Ann. Inst. Fourier (Grenoble) 21,
35^41.
Jirina, M- A959), On regular conditional probabilities. Czechoslovak Math. J. 9, 445^450.
Johnson, R.A. A978), Existence of strong liftings commuting with a compact group of transformations. Pacific
J. Math. 76, 69-81.
Johnson, R.A. A979), Existence of strong liftings commuting with a compact group of transformations II, Pacific
J. Math. 82, 457^461.
Johnson, R.A. A980), Strong liftings which are not Borel liftings. Proc. Amer. Math. Soc. 80, 234-236.
Kakutani, S. A943), Notes on infinite product measures, II, Proc. Imperial Acad. Tokyo 19, 184-188.
Kolzow, D. A968), Differentiation von Massen, Lecture Notes in Math., Vol. 65, Springer, Berlin.
Kupka, J. A983), Strong liftings with applications to measurable cross sections in locally compact groups, Israel
J. Math. 44, 243-261.
Kupka, J. and Prikry, K. A983), Translation invariant Borel liftings hardly ever exist. Indiana Univ. Math. J. 32,
717-731.
Kupka, J. and Prikry, K. A984), The measurabilitv of uncountable unions. Amer. Math. Monthly 91 B), 85-97.
Lahiri, B.K. and Chakrabarti, S. A991), Density topology, Math. Student 59, 89-108.
Lebesgue, H. A905), Recherches sur la convergence des series de Fourier. Math. Ann. 61, 251-287.
Lebesgue, H. A910), Sur I' integration des functions discontinues, Ann. Sci. Ecole Norm. Sup. 27, 361^450.
Levin, V. A975), Convex integral functions and the theory of lifting, Russian Math. Surveys 30, 119-184 (Uspekhi
Mat. Nauk 30).
Lipecki, Z. A998), Quasi-measures with finitely or countably many extreme extensions, Manuscripta Math. 97,
469-481.
Lloyd, S.P. A974), Two lifting theorems. Proc. Amer. Math. Soc. 42, 128-134.
Losert, VL. A979), A measure space without the strong lifting property. Math. Ann. 239, 119-128.

1182
W. Strauss et al.
Losert, V.L. A980), A counterexample on measurable selections and strong lifling. Measure Theory Proc.
Oberwolfach 1979, D. Kolzow, ed.. Lecture Notes in Math., Vol. 794, Springer, Berlin, 153-159.
Losert, V.L. A983), Some remarks on invariant liftings. Measure Theory Proc. Oberwolfach 1981, Lecture Notes
in Math., Vol. 1080, Springer, Berlin.
Macheras, N.D. A984), Permanenzprobleme fiir induktive Limiten, Dissertation, Universitat Erlangen-Niimberg.
Macheras, N.D. A989), On inductive limits of measure spaces and projective limits of LI'-spaces, Mathematika
36, 116-130.
Macheras, N.D. A998), On the Lx-spaces of inductive limits of measure spaces, Atti Sem. Mat. Fis. Univ.
Modena 44, 513-523.
Macheras, N.D., Musiat, K. and Strauss, W. A999), On products of admissible liftings and densities, Z. Anal.
Anwendungen 18, 651-667.
Macheras, N.D., Musiat, K. and Strauss, W. B000), Linear liftings respecting coordinates. Adv. Math. 153, 403-
416.
Macheras, N.D., Musial, K. and Strauss, W. B00?a), On strong liftings on projective limits. Preprint.
Macheras, N.D., Musial, K. and Strauss, W. B00?b). A filter approach to product densities and liftings.
Manuscript.
Macheras, N.D., Musial, K. and Strauss, W. B00?c), On the existence of liftings respecting coordinates. Preprint.
Macheras, N.D., Musiat, K. and Strauss, W. B00?d), Linear Fubini liftings. Preprint.
Macheras, N.D. and Strauss, W. A992), On various strong lifting properties for topological measure spaces.
Rend. Circ. Mat. Palermo B) Suppl. 28, 149-162.
Macheras, N.D. and Strauss, W. A993), On the permanence of almost strong liftings, J. Math. Anal. Appl. 174,
566-572.
Macheras, N.D. and Strauss, W. A994), On strong liftings for projective limits. Fund. Math. 144, 209-229.
Macheras, N.D. and Strauss, W. A995), Products of lower densities, Z. Anal. Anwendungen 14, 25-32.
Macheras, N.D. and Strauss, W. A996a), On self-consistent families of almost strong liftings in projective systems,
Atti Sem. Mat. Fis. Univ. Modena 44, 105-111.
Macheras, N.D. and Strauss, W. A996b), Products and projective limits of almost strong liftings, Atti Sem. Mat.
Fis. Univ. Modena 44, 119-133.
Macheras, N.D. and Strauss, W. A996c), On products of almost strong liftings, J. Austral. Math. Soc. Ser. A 60,
311-333.
Macheras, N.D. and Strauss, W. A996d), The product lifting for arbitrary products of complete probability
spaces, Atti Sem. Mat. Fis. Univ. Modena 44, 485^496.
Macheras, N.D. and Strauss, W. A999a), On Products of Hvperstonian spaces, Atti Sem. Mat. Fis. Univ. Modena
47, 369-382.
Macheras, N.D. and Strauss, W. A999b), On consistent lower densities, Atti Sem. Mat. Fis. Univ. Modena 47.
Macheras, N.D. and Strauss, W. B000), Mazur-Orlicz theorem in lifting theon; J. Math. Anal. Appl. 241, 122—
133.
Macheras, N.D. and Strauss, W. B00?), On strong product liftings. Preprint.
Maharam, D. A958), On a theorem of von Neumann, Proc. Amer. Math. Soc. 9, 987-994.
Maharam, D. A976), Finitely additive measures on the integers, Sankhya, Ser. A 38, 44-59.
Maharam, D. A977), Category. Boolean algebras and measures. Lecture Notes in Math., Vol. 609, Springer,
Berlin, 1243-135.
Maher, R.J. A974), Strong liftings and Borel liftings. Adv. Math. 13, 55-72.
Maher, R.J. A978), Strong liftings on topological measured spaces. Studies in probability and ergodic theory.
Adv. Math. 2, 155-166.
Maitland Wright, J.D. A969), A lifting theorem for Boolean ?-algebras. Math. ? 112, 326-334.
Marczewski, E. A953), On compact measures. Fund. Math. 40, 113-124.
Marechal, O. A968), Operateurs decomposable dans le champs mesurables d'espaces de Hilbert, С R. Acad.
Sci. Paris 266A, 710-713.
Marechal, O. A968a), Decomposition des ope'rateures dans le champs mesurables d'espaces de Hilbert, C. R.
Acad. Sci. Paris 267A, 636-639.
Marechal, O. A969), Champs mesurables d'espaces Hilbertiens, Bull. Sci. Math. 93, 113-143.
Mauldin, R.D. A978), Some effects of set-theoretical assumptions in measure theory. Adv. Math. 27, 45-62.
McMinn, T.J. A975), Commuting and topological densities and liftings. Trans. Amer. Math. Soc. 211, 1-22.

Liftings
1183
Mokobodzki, G. A975), Relevement borelien compatible avec une classe d ensembles ne'gligables. Application
a la desintegration des mesures, Seminaire des probabilites IX, 1974/5, Lecture Notes in Math., Vol. 465,
Springer, Berlin, 437^*42.
Monakov-Rogozkin, A. A974), Spaces admitting a lifting. Works in Mathematics and Physics, Tallin, 5-14 (in
Russian).
Monakov-Rogozkin, A. A978), The structure of spaces of measurable functions that admit lifting. Acta Comment.
Univ. Tartu 448, 12-20 (in Russian).
Monakov-Rogozkin, A. A981), Spaces which do not have the linear lifting property. Acta Comment. Univ. Tartu
504, 10-16 (in Russian).
Monakov-Rogozkin, A. A991), A description of measure spaces with liftings. Acta Comment. Univ. Tartu 928,
73-88.
Moran, W. A969), Measures and mappings on topological spaces, Proc. London Math. Soc. 19, 493-508.
Musial, K. A973), Existence of Borel liftings, Colloq. Math. 27. 315-317.
Musiat, K. A980), Projective limits of perfect measures. Fund. Math. CX, 163-189.
Musial, K. B000), Lifting and some of its applications to the theory of Pettis integral. Rend. Istit. Mat. Univ.
Trieste 31, Suppl.l, 1-35.
Musial, K. B002), Pettis integral. Handbook on Measure Theory, E. Pap, ed., Elsevier, Amsterdam, 531-586.
Musial, K. and Macheras, N.D. B000), Liftings of Pettis integrable functions. Hiroshima Math. J. 30, 205-213.
Musiat, K., Strauss, W. and Macheras, N.D. B001), Product liftings and densities with lifting and density
invariant sections. Fund. Math,
von Neumann, J. A931), Algebraische Reprasentanten der Funkionen "bis auf eine Menge von Masze Nulf,
Crelle's J. Math. 165, 109-115.
von Neumann, J. and Stone, M.H. A935), 77k· determination of representative elements in the residual classes of
a Boolean algebra. Fund. Math. 25, 353-378.
Oxtoby, J.C. A971), Measure and Category, Springer, Berlin.
Pachl, J.K. A978), Disintegration and compact measures. Math. Scand. 43, 157-168.
Pellaumail, J. A970), Sur la derivation d'une mesure vectorielle. Bull. Soc. Math. France 98, 305-318.
Pellaumail, J. A972), Application de /'existence d'un relevement a un theoreme sur la desintegration des mesures,
Ann. Inst. H. Poincare 8, 211-215.
Rinkewitz, W. A997), Zur Existenz. Desintegration und Choquet-Darstellung von Urbildmassen - Ein Zugang
iiber Stonesche Darstellungsraume, Ph.D. thesis. Ludwig-Maximilians-Universitat, Miinchen.
Rodriguez-Salinas, B. A978), ?-spaces of Suslin and Lusin, The strong •'lifting" property. Rev. R. Acad. Cienc.
Exact. Fis. Natur. Madrid 72, 541-557 (Spanish).
Rodriguez-Salinas, B. A978a), Radon measures of type ? and measures with ''lifting". Rev. R. Acad. Cienc.
Exact. Fis. Natur. Madrid 72, 605-610 (Spanish).
Rodriguez-Salinas, B. A979), Suslin and Luzin semi ?-spaces. Strong lifting property. Rev. R. Acad. Cienc.
Exact. Fis. Natur. Madrid 73, 33^40 (Spanish).
Ross, D.A. A990), Lifting theorems in nonstandard measure theory, Proc. Amer. Math. Soc. 109, 809-822.
Ryan, R. A962), The lifting property and direct sums, MRC, Technical Report No. 328, US Army, University of
Wisconsin.
Ryan, R. A964), Representative sets and direct sums, Proc. Amer. Math. Soc. 15, 387-390.
Sapounakis, A. A983), The existence of strong liftings for totally ordered measure spaces. Pacific J. Math. 106,
145-151.
Saint-Pierre, J. A975), Desintegration dune mesure поп borne. Ann. Inst. H. Poincare 8. 275-286.
Schief, A. A99?), SOSC and OSC are equivalent. Unpublished and undated notes, Ludwig-Maximilians-
University, Munich.
Schreiber, B.M., Sun, T.C. and Bharucha-Reid. A.T A971), Algebraic models for probability measures
associated with stochastic processes. Trans. Amer. Math. Soc. 158, 93-105.
Schwartz, L. A973), Radon measures on arbitrary topological spaces and cylindrical measures, Oxford Univ.
Press.
Segal, I.E. A951), Equivalences of measure spaces, Amer. J. Math. 73. 275-313.
Shelah, S. A983), Lifting problem of the measure algebra, Israel J. Math. 45, 90-96.
Shelah, S. A989), Baire irresolvable spaces and lifting for layered ideal. Topology Appl. 33, 217-221.
Shelah, S. A998), Lifting problem with the full ideal, J. Appl. Anal. 4, 1-17.

1184
W. Strauss et al.
Sion, M. A973), A Theory of Semigroup Valued Measures, Lecture Notes in Math., Vol. 355, Springer, Berlin.
Strauss, W. A971), Funktionalanalytische Fassungdes Satzes von Radon-Nikodvm I, J. Reine Angew. Math. 249,
92-132.
Strauss, W. A974), Die Obstruktion zur strengen Lokalisierbarkeit eines Maszraumes, Manuscripta Math. 12,
1-10.
Strauss, W. A975), Retraction numbers, liftings and the decomposability of a measure space. Bull. Acad. Polon.
Sci. Ser. Math. Astronom. Phys. 23, 27-33.
Strauss, W. A992), On strong lifting compactness for the weak* topology. Rocky Mountain J. Math. 22, 1057—
1081.
Talagrand, M. A978), Non existence de certaines sections et applications ? la theorie du relevement, C. R. Acad.
Sci. Paris 286A, 1183-1185.
Talagrand, M. A978a), En general il n'existe pas de relevement line'aire borelien fort, C. R. Acad. Sci. Paris
287A, 633-634
Talagrand, M. A981), Non existence de relevement pour certaines mesures finiement additives et retracte de /8N,
Math. Ann. 256, 63-66.
Talagrand, M. A982), La pathologic des relevements invariants, Proc. Amer. Math. Soc. 84, 379-382.
Talagrand, M. A984), Pettis Integral and Measure Theory, Mem. Amer. Math. Soc., No. 307.
Talagrand, M. A987), Measurability problems for empirical processes, Ann. Probab. 15, 204—212.
Talagrand, M. A988), On Liftings and the Regularization of Stochastic Processes, Probab. Theory Related Fields
78, 127-134.
Talagrand, M. A989), Closed convex hull of set of measurable functions. Riemann-measurable functions and
measurability of translations, Ann. Inst. Fourier (Grenoble) 32, 39-69.
Tortrat, A. A975), Prolongements ?-reguliers, applications aux probabilite's Gaussiennes, Symposia Mathemat-
ica21 (convegno sulle Misure su Gruppe e su spazi Vettoriali ), INDAM, Rome 1975, 117-138.
Traynor, T. A974), An elementary proof of lifting theorem. Pacific J. Math. 53, 267-272.
Valadier, M. A973), Desintegration d'une mesure sur un produit, С R. Acad. Sci. Paris 276A, 33-35.
Vesterstr0m, J. and Wils, J. A969), On point realizations of L^-endomorphisms, Math. Scand. 25, 178-180.
Volcic, A. A982), Liftings and Daniel! integral. Measure Theory Oberwolfach 1981, Proceedings, Lecture Notes
in Math., Vol. 945, Springer, Berlin, 180-186.
von Weizsacker, H. A976), Some negative results in the theory of liftings. Measure Theory, Proc. Conf.,
Oberwolfach, 1975, Lecture Notes in Math., Vol. 541, Springer, Berlin, 159-172.
von Weizsacker, H. A977), Eine notwendige Bedingung fur die Existenz invarianter mass-theoretischer Liftings,
Arch. Math. 28,91-97.
von Weizsacker, H. A978), Strong measurability. liftings and the Choquet-Edgar theorem. Vector Space
Measures and Applications II, Proc. Conf. Univ. Dublin, 1977, Lecture Notes in Math., Vol. 645, Springer,
Berlin,
von Weizsacker, H. A982), The nonexistence of "liftings" for arithmetic density, Proc. Amer. Math. Soc. 86,
692-693.
Zygmund, A. A968), Fourier Series, Vol. I, 2nd edn, Cambridge Univ. Press, Cambridge.

CHAPTER 29
Ergodic Theory
Frank Blume
Department of Mathematics, John Brown University, Siloam Springs. AR 72761, USA
E-mail: fblume @acc.jbu. edu
Contents
Introduction 1187
1. Basic examples 1189
2. Ergodic theorems 1192
3. Ergodicity 1196
4. Recurrence 1200
5. Mixing 1203
6. More about convergence in ergodic theory 1210
7. Entropy and information 1214
8. Constructions in ergodic theory 1226
References 1231
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1185

This page intentionally left blank

Ergodic theory
1187
Introduction
Historically, ergodic theory emerged from the study of dynamical systems which in
turn originated in Newton's formulation of the laws of motion and the universal law of
gravitation. In his theory Newton described a dynamical system consisting of ? point
masses by a set of differential equations that completely determined the motion of the point
masses given certain initial conditions, However, this approach to the study of dynamical
systems was beset with grave difficulties, because even in the case of a three-body system
an explicit solution to the equations of motion was impossible to obtain. Facing this
dilemma, Henri Poincare initiated at the end of the 19th century a qualitative analysis
of dynamical systems which shifted the focus away from differential equations to global
geometric properties of the phase space and transformations inherent in the system. His
famous recurrence theorem (Theorem 4.1) which establishes the arbitrarily close future
return of a system to states that existed in the past can be regarded as the prototype of an
ergodic-theoretic argument (see Furstenberg A981)).
The modern formulation of ergodic theory dates back to the theorems of von
Neumann and Birkhoff (Theorems 2.2, 2.4 and 3.2) concerning the equality of time and
space means of physical quantities in dynamical systems. This question, known as the
ergodic hypothesis, was first introduced by Boltzmann in connection with the theoretical
foundations of statistical mechanics. For intuitive reasons, we will frequently make
references to physical interpretations of ergodic-theoretic results (see Petersen A996)),
but it also needs to be mentioned, that following the work of von Neumann and Birkhoff,
ergodic theory soon developed a life of its own and lost to a certain extent its ties to physics
and in particular to statistical mechanics. On the one hand the assumption of ergodicity
(Definition 3.1) used to demonstrate the equality of time and space means was difficult to
verify even for simple systems such as ideal gases, and on the other hand the discovery
of rich connections to other fields such as probability (Section 1), information theory
(Section 7) and number theory (Example 3.4 and Section 4) generated interest in ergodic
theory as an independent area of research.
The theory of dynamical systems is divided into three major branches: differentiable
dynamics - the study of diffeomorphisms on differentiable manifolds, topological
dynamics - the study of homeomorphisms on topological spaces and ergodic theory which
is best described as the study of statistical properties of measure-preserving systems.
Here we say that (?, ?, ?, ?) is a measure-preserving system, if (?, ?, ?) is a complete
probability space and ?: X -> X is a measure-preserving transformation in the sense that
? satisfies the following conditions:
(a) T is bijective,
(b) ТА, T-'AeSforall A e Sand
(c) ?(?~[?) = ?(?) for all AeB.
The assumption of bijectivity is not always included in the definition of measure-preserving
transformations, but it is convenient and does not represent a major restriction.
For a given measure-preserving transformation, we may define a group action of the
integers on X via the map (n, x) н> T"x where ? e ? and ? e X. It is of considerable
interest to consider more general group or semigroup actions on X, and in particular the
case of measure-preserving flows, that is actions of R on X, is very important, but for the

1188
F. Blu me
sake of notational and conceptual coherence we will restrict ourselves in this exposition to
the main case that the group acting on X is Z.
In modeling dynamical systems via measure-preserving transformations on a probability
space, we think of each point ? in the state space X as representing perfect knowledge
about the system at the level of abstraction that is considered relevant. For example if we
consider a system to be completely described by the positions and momenta of its parts,
then ? represents perfect knowledge of these two properties - position and momentum.
Measure theory naturally enters into the picture as we observe that for practical and also
principal reasons (uncertainty in quantum mechanics), perfect knowledge of a system's
state is unattainable so that instead of identifying single points ? e X we may only assume
knowledge of a set of states A e В of which ? is a member (see Petersen A996)). Given
this measure-theoretic approach, it is natural to represent measurable, that is observable,
physical quantities by measurable functions /: X -> К (or C). In order to explore the
evolution of the system over (discrete) time, we examine the values of a measurable
quantity / along the orbit of a state ? e X under the action of T, that is we examine
the sequence [f(Tkx)}^=Q. In such a mathematical model the ergodic hypothesis, as
introduced by Boltzmann, is equivalent to the assertion that the time mean
exists and is equal to the space mean fx f ?? for almost all ? e X with respect to ?
(Theorem 3.2). In other words, the claim is here that the space mean of an observable
quantity can be derived almost surely from discrete measurements along the time evolution
of a single state ? e X. This idea of deriving global information from a single state is
very important and will resurface in Section 7 in Theorems 7.23 and 7.31 (see also the
concluding remark in Section 6).
Besides the quantitative problem of establishing the existence of longterm averages (in
the mean or pointwise along orbits under the action of T) we may also study on a more
qualitative level the recurrence behavior of sets. The earliest result in this context was the
above mentioned theorem by Poincare which, in its simplest form, states that for any set
A & В with ? (A) > 0 do not only there exists an ? e N such that ?(? П T~n A) > 0.
Recurrence statements of this type do not only have profound philosophical implications
for our understanding of dynamical systems, but also provide an important link to the field
of combinatorial number theory. Several major results are listed in Section 4.
Another important problem in ergodic theory is the classification problem. We say that
two measure-preserving systems (?, ?, ?. ?) and (?, С, ?>, S) are (metrically) isomorphic
if there are sets Xo С X and YqGY and a map ?: X \ Xo -> ? \Yo such that
(a) ? is bijective,
(b) ? and ?~] are measurable,
(c) м(Х0) = у(У0)=0,
(d) ??(?) = ??(?) for all ? e X \ X0 and
(e) ?(?~[ A) = v(A) for all measurable sets А с ? \ Yo-
The problem of distinguishing measure-preserving systems at the level of metric
isomorphisms remained essentially unsolved until Kolmogorov A958) introduced entropy as an

Ergodic theory
1189
isomorphism invariant and thereby created an entirely new field of research. The entropy
of a system is a single number (invariant under isomorphisms) that in a sense measures the
average degree of randomness in the orbits of points under the action of ? (Section 7).
Prior to the work of Kolmogorov, measure-preserving systems could only be classified
at the level of spectral properties, which can be defined in sole reference to the induced
unitary operator
UT:L2(X,B,v)-> ???,?,?), / н> / о Г,
and do not take fully into account the action of ? on individual points. We say that two
measure-preserving systems (?,?, ?. ?) and (?, С, ?, S) are spectrally isomorphic if there
exists a unitary operator V : L2(X,B, ?) -> L2(Y,C, v) such that VUT = UsV. Examples
of dynamical properties that serve as spectral invariants are ergodicity (Section 3) and
mixing properties (Section 5). These invariants are completely characterized by Uj and are
intuitively best described as quantitative recurrence properties. For example, a measure-
preserving system (?, ?, ?, ?) is strongly mixing if and only if for all A e В we have
Hindoo ?(? П Т~"А) = ?(?J (Theorem 5.4). We see that here the general qualitative
statement of the Poincare recurrence theorem is replaced by the stronger quantitative
requirement that A and T~" A are approximately independent for all sufficiently large n.
Finally, we also need to mention the representation problem as an important question
in ergodic theory. We may for example ask which measure-preserving transformations
can be represented by (that is, are isomorphic to) diffeomorphisms on differentiable
manifolds or homeomorphisms on topological spaces. One such result is stated in Section 3
(Theorem 3.10), but for a more detailed discussion of this and many other important
subjects the reader is referred to standard expositions of ergodic theory and dynamical
systems such as Petersen A983), Cornfeld et al. A982), Friedman A970), Halmos and
von Neumann A958), Katok and Hasseblatt A995) and Rudolph A990). The very well
researched account in Petersen A983) served as the main reference for the standard results
and examples of ergodic theory in the present exposition.
1. Basic examples
In this section we list a number of the most important examples of measure-preserving
systems. Several of these highlight the close relationship between ergodic theory and
probability.
Stationary random processes. Assume that С is a ? -algebra of subsets of a given set Y.
For ? := Yz we denote by Bo the ?-algebra of subsets of ? generated by cylinder sets of
the form
[y = {y,>}„ez e ?: y,u eB\,..., y„k e Bk},
where к e ?, ? ?,..., щ е ? and B\, ?* е С. For a probability measure ? on Bo, we
denote by В the completion of Bo with respect to ?. The triple (?,?, ?) is said to be a

1190
F. Blume
{discrete time) random process with state space (Y, C). If the random process (?, ?, ?) is
stationary in the sense that for all m e ? and all cylinder sets as above we have
?({?<??: у,,, еВи...,у,ч е Вк})
= м({у е ?: y„l+„, e B\,..., Ущ+т е ?*}),
then the left shift ? :? ->¦ ? defined via the equation
ipy)n -=Уи+\
is easily seen to be measure-preserving with respect to ? (because ? preserves the measure
of cylinder sets). Therefore, (?,?, ?,?) is a measure-preserving system. Conversely,
every measure-preserving system gives rise to stationary random processes via finite or
countable partitions. To see this, let us assume that (?, ?, ?. ?) is a measure-preserving
system and or = {Av: у e Y] is a finite or countable measurable partition of X, in the sense
that the index set ? is finite or countable, the sets in a are pairwise disjoint and measurable
and Uvey Ay = X. For ? := Yz we define a map ?: X -> ? via the equivalence
(фх)„ = у <S> ?"? e Av.
Denoting by Co the ? -algebra generated by cylinder sets of the form
{y = {yn}nez e ?: y„,=z\,---,y,,k = Zk} with zi, ¦ ¦ ¦, ?* e У,
the map 0 is clearly measurable with respect to Co and В (i.e., 0~'Co С В), and the
equation v(A) := ?@-1 A) for A e Co defines a probability measure on Co- If С is the
completion of Co with respect to v, then the fact that Г is measure-preserving implies that
(?, C, v) is a stationary random process.
Bernoulli shifts. A Bernoulli shift is a special case of a measure-preserving system
generated by a stationary random process. Here the state space (Y, C) consists of a finite
set ? = {1,..., m), and the ?-algebra С consists simply of all subsets of Y. The set ?
is frequently also referred to as a finite alphabet. In order to define a measure ? on the
?-algebra Bo generated by the cylinder sets in ? = ?", we assume that p\,..., p,„ are
strictly positive real numbers such that ?'"=, ?,¦ = 1. Then for any elements г ?,..., ц е ?
we define
и{{у={Уп}пе%?Я: y„t =i\ y„k = /*}) = p,, ¦ ¦ ¦ Av
Extending ? to a probability measure on the completion В of Bo with respect to ?, the
system (?, ?, ?, ?) is measure-preserving where, as above, ? is the left shift on ?.
Markov shifts. Markov shifts form another class of measure-preserving systems
generated by stationary random processes. The finite state space (Y, C) and the ?-algebra Bo of

Ergodic theory
1191
subsets of ? are given as in the case of Bernoulli shifts. To define the measure ?, we
assume that A = (fl,y) is an m ? m stochastic matrix (i.e., a,y ^ 0 for all i, j, and the sum of
the components in each row of A is equal to one). Then according to the Perron-Frobenius
theorem (see Varga A962)) there exists a row probability vector ? = {p\,..., p,„) (i.e.,
Pi ^ 0 for all / and ??=, ?, = 1) such that
pA = p.
Given the vector p, we define
?({???: уП[ = i\,...,y„k = ik}) = р^щ^2- ¦ ¦ aik_{ik.
Extending ? again to a probability measure on the completion В of Bo, it follows that
(?, ?, ?, ?) is a measure-preserving system.
Gaussian systems. The importance of Gaussian measure-preserving systems is due to the
fact that Gaussian random processes are frequently encountered in probability theory and
also in statistical mechanics and other branches of the physical sciences. We set ? := Rz
and denote by С the ?-algebra of Borel subsets of R so that the state space is (R,C).
A measure ? on the ?-algebra Bo generated by the cylinder sets in ? is said to be a Gauss
measure (see Shirayayev A984)) if for any integers n\ < ¦ ¦ ¦ < nk there is a row vector
(m|,..., mk) e K* and a symmetric positive-definite к ? к matrix В = {b\j) with inverse
B~' = A = (ajj) such that for any Borel set ? с R* we have
?({?&?: (уП1,...,у„к)еЕ})
VdetA
/ eXP\ 2 ? a'j(x' ~ m'^xJ
2?
Extending ? to a probability measure on the completion В of Bo, the Gaussian random
process (?, ?, ?) is known to be stationary (see Cornfeld et al. A982)) if for all /, j eZ
the projections ?,¦: ? -> R and ?;¦: ? -> R onto the /th and ;'th factors of ? respectively
(i.e., ni(y) := yi and тгДу) := у,) satisfy the following conditions:
(a) fn ?,- ??? = /? ???^? ='. w and
(b) fn (тг, — m)(jij — m)d[i depends only on the difference i — j.
In this case (?,?, ?,?) is a Gaussian measure-preserving system, and the vector and
matrix components ш, and bjj are given by the equations w, = m and biy = /? (?,,, —m) ?
(Knj -™)??.
Rotations on the circle and on compact Abelian groups. For a e R we define a map
7^ : [0, 1 [ -> [0, 1 [ via the equation
Га;с := ? + a mod 1 = ? + ? — [? + ?],

1192
F. Blume
where we denote by [t] as usual the greatest integer function, that is [t] = max{n e Z:
? ^ t]. Identifying [0, 1[ with the unit circle К in the complex plane, Ta can also be
represented by the equation
? е2л'х '= g27r'<-v+0?'
Since Ta obviously preserves Lebesgue measure ? on the ?-algebra В of Lebesgue
measurable subsets of [0, 1 [, we see that ([0, 1 [, ?, ?, ??) is a measure-preserving system.
In the more general situation that G is a compact Abelian group and g e G, the map
Tg:G -> G, Гял· := gjt is measure-preserving with respect to Haar measure on G, because
Haar measure is translation invariant.
Translations on the torus. As a generalization of rotations on the circle and as another
special case of a rotation on a compact Abelian group, we define a map Ta : [0, l[m ->
[0, l['n on the m-dimensional torus via the equation
Tax := (jc? +ot\ mod 1,.. .,x„, +a„, mod 1),
where a = (a\,... ,am) e R'". Then Ta is a measure-preserving transformation with
respect to Lebesgue product measure on [0, 1 ['".
Finite interval exchanges. Assume that I\,..., /,„ are pairwise disjoint half open
subintervals of [0, 1 [ whose union is [0, 1 [. A map ?: [0, 1 [ -> [0, 1 [ is said to be a finite
interval exchange transformation if there are real numbers a\,...,a„, such that
(a) Tx = ? + аь for all ? e h and
(b) TIjHTIj =0foralliV;.
Given these conditions, it is clear that ? is bijective and measure-preserving with respect
to Lebesgue measure on [0, 1 [. We also note that a rotation Ta on the circle is a special
case of a finite interval exchange with m = 2, I\ = [0, 1 - [<*][, h = [1 - [a], 1[, a\ = [a]
and «2 = [a] — 1.
2. Ergodic theorems
In this section we give an overview of the principal convergence theorems of ergodic
theory with particular emphasis on von Neumann's mean ergodic theorem and Birkhoff's
pointwise ergodic theorem. The proof of von Neumann's theorem follows the exposition
in Parry A981).
THEOREM 2.1 (Mean ergodic theorem in Hilbert space). Assume that ? is a Hilbert
space, U : ? -> Ha unitary operator and let ? := {/ e H. Uf = /}. If ? : ? -> ? is
the projection of ? onto M, then
lim
n—>oc
I"
Y,Ukf-Pf
?
k=0
= 0.

Ergodic theory
1193
PROOF. Let N := {Uf - / | / e #}. If g A. N then (i//, g) = (/, g) for all / e #. This
shows that (t/-'g - g) J. / for all / e #. Therefore, i/"'g = g = Ug and ~N С М.
Since t// = / implies (/, Ug - g) = 0 for all g e Я, we also have ? с N . Hence
H = M®~N. Furthermore, if g:= f- Pf, then g e N and therefore, j-t ?"?? Ukg -* 0
andiE"lii/V^JP/. ?
THEOREM 2.2 (Mean ergodic theorem (von Neumann, 1932a)). Let (?,?,?, ?) be a
measure-preserving system and assume that f 6 L-(X, ?, ?). Then we have
lim
11—»OC
/i-l
^foTk-E(f\M)
k=0
= 0,
where ? = {A e ? \ ? ? = ? а.е.), and E{f\M) is the conditional expectation of f
with respect to Л4.
PROOF. The mean ergodic theorem is an immediate consequence of Theorem 2.1 applied
to ? := L2(X, ?, ?) and the induced operator Uj, where Urf = f°T. D
Remark 2.3. The mean ergodic theorem in Hubert space is also valid for linear
contractions, that is linear operators U: ? -> ? for which \\Uf\\ ^ ||/|| for all / e ?
(see Petersen A983)).
THEOREM 2.4 (Ergodic theorem (Birkhoff, 1931)).
preserving system and let f e ? (?, ?, ?). Then
Let (?,?,?,?) be a measure-
/i-l
1
lim -Yf(Tkx) = E(f\M)(x) a.e.
k=0
and
lim
и—>oo
/i-l
-У>оГ*-?(/|.М)
k=0
= 0.
The question of whether Birkhoff's pointwise approach or von Neumann's mean
convergence approach is more adequate for modeling physical reality has been the subject
of considerable philosophical debate. At first sight it appears that taking pointwise averages
along the orbit of a single state is a true reflection of scientific practice because each value
f(Tkx) represents the outcome of a specific measurement. On the other hand, according
to quantum theory each measurement changes the state of the system, so that pointwise
convergence statements may not only be unnecessary, but actually impossible to match in
the laboratory. Even if quantum effects are not taken into account, the measuring process
itself (such as the reading of a needle) may represent an average due to the inertia of the

1194
F. Blume
measuring device so that von Neumann's mean convergence approach may indeed be the
more plausible one. (For further discussion along these lines see Petersen A996).)
Of central importance for the proof of the ergodic theorem is the so-called maximal
function:
I"
/*(*):= sup-?/(Г*дг)
%=0
The following theorem is due to Wiener A939) and Yosida and Kakutani A939).
THEOREM 2.5 (Maximal ergodic theorem). If (?,?,?,?) is a measure-preserving
system and f e L\X,B, ?), then
f fa-?^?.
J[f*>0]
COROLLARY 2.6. For all a eR we have
fd?>a?({f*>a}).
L
PROOF. Apply Theorem 2.5 tog := f-a. ?
To see how the maximal function is instrumental in the proof of the ergodic theorem,
we will now show how pointwise convergence of the ergodic averages follows from the
maximal ergodic theorem. We follow the exposition in Petersen A983) where the reader
may also find a complete proof of the ergodic theorem.
Proof of pointwise convergence in Theorem 2.4. For ?, ? e ? with a < ?, we
define
? 1 ""' 1 ""' )
???:= \xe X: liminf- У" f(Tkx) < ? <? < limsup- Y^ f(Tkx) \.
I "^°° " to "— л ыо J
It is sufficient to show that ?(???) = 0, because then the union over all rational a and ?
has measure zero as well, so that the limit exists a.e. Since ??.? is a ?-invariant subset of
{/* > ?], we may apply Corollary 2.6 to the restriction of ? to ??.? to conclude that
f f??> ??(??.?).
J ??.?
Applying essentially the same argument to — / it is not difficult to see that
fd?^a?(Ea.?),
JEa,

Ergodic theory 1195
and therefore.
??(??.?)!? /?????(??.?).
J??.?
Since a < ?, this is possible only if ?(??.?) = 0. ?
Just as von Neumann's mean ergodic theorem, the maximal ergodic theorem can be
derived from a more general operator version: A map U: L1 (?, ?, ?) -> L1 (?, ?, ?) is
called positive contraction if
(a) ||?//||? ? \\f\U for all f ? L^X.B, ?),
(b) ?/6?.'(?.?.?) / > ° a-e- => i// > 0 a.e.
Theorem 2.7 (Maximal ergodic theorem for operators (Hopf, 1954)). IfU is ? positive
contraction on L1 (?, ?, ?) and /ei,1 (?, ?, ?), then
? ???^?,
Jlf*>0]
where f*(x) := sup„6N I Щ !/*/(*)·
A very simple proof of this theorem was given by Garsia A965).
Using the maximal inequality for operators, it is possible to prove the following very
general convergence theorem due to Chacon and Ornstein A960).
THEOREM 2.8 (Chacon-Ornstein theorem). If {?,?,?) is ? ?-finite measure space,
0 ^ /, g e V (?, ?, ?) and U: L' (?, ?, ?) -> L' (?, ?, ?) is a positive contraction, then
?-1 /?-1
?"*//?"**
*=0 *=0
converges to a finite limit a.e. on {x e ?: ??=0 Uk g{x) > 0}.
Another important strengthening of the ergodic theorem is due to Kingman A968). We
say that a sequence {F„ }„6pj of measurable real-valued functions on X is subadditive with
respect to a measure-preserving transformation ? if
Fm+n(x) iC F„(x) + F,„(T"x) a.e. for all n, m e N.
THEOREM 2.9 (Subadditive ergodic theorem). Assume that (?, ?,?, ?) is a measure-
preserving system. If{ F„ }„6n is a subadditive sequence ofintegrable real-valued functions
on X such that
inf - [ F„
? eN П Jx
?? > — oo,

1196
F. Blume
then there exists a T-invariantfunction F e V (?, ?, ?) such that
F„(x)
lim = F(x) a.e.
a—>oo ?
and
lim
77^00
?
= 0.
For more about subadditive ergodic theory see Kingman A973).
Wiener considered the problem of finding a criterion for the ergodic averages
7, ?2"=? f ° Tk t0 ^ dominated by an integrable function. To formulate his result, we
define
log+W :=
0
if;te@, l[,
log(jc) if ? e [1, 00 [.
Theorem 2.10 (Dominated ergodic theorem (Wiener, 1939)). Let (?,?,?,?) be a
measure-preserving system. Iff \f\ log+ |/| ?? < oo, then
sup
716N
71-1
k=0
eLl(X,B,p).
Remark 2.11. Petersen A979) extended this result to ergodic measure-preserving flows
and showed that the condition / |/|log+|/|??? < oo is not only sufficient, but also
necessary if / ^ 0.
3. Ergodicity
Historically, the notion of ergodicity was introduced by Ludwig Boltzmann in connection
with the foundations of statistical mechanics. Here it was desirable that the time mean
of a physical quantity is equal to its space mean (see Theorem 3.2(d)). Later, the role
of ergodicity for the description of systems in statistical mechanics such as ideal gases
proved to be less significant than the presence of a large number of particles (atoms or
molecules). However, from a purely mathematical point of view the importance of the
concept of ergodicity cannot be overestimated, because it serves as a central assumption
in the formulation of a wide variety of theorems and plays an important role as a spectral
invariant.
Definition 3.1. A measure-preserving system (?, ?, ?, ?) is said to be ergodic if all
measurable ?-invariant subsets of X have measure 0 or 1.
THEOREM 3.2 (Equivalent characterizations of ergodicity). We assume that (?, ?, ?, ?)
is a measure-preserving system. Then the following statements are equivalent:

Ergodic theory
1197
(a) (?, ?, ?, ?) is ergodic.
(b) Every measurable ? -invariant function (i.e. f = f ? ? a. e.) is constant a.e.
(c) The induced transformation Ut '¦ L2(X, ?, ?) -> L2(X, ?, ?) has a
one-dimensional eigenspace corresponding to the eigenvalue 1.
(d) For each f ??,'(?,?,?) we have
1 ""' Г
lim -Yf(Tkx)= / ??? a.e.
jt=0 J ?
(e) For all А, В е В we have
1 ""'
lim -??(?~????\?)=?(?)?{?).
и—» ос п *—*
к=0
(f) For ?// f,geL2(X, ?, ?) we have
1 ""'
lim -Y\№f,g)=(f,D(i,g).
*=0
(g) For a// / e L2(X, ?, ?) with (f,l)=0we have
1 ""'
lim -?(?/?/,/) = 0.
PROOF of (a) =>· (b) =>· (d) =>· (a). If (?,?,?, ?) is ergodic and / is ?-invariant, then
Er := {x e X: f(x) > r} is ?-invariant for all rel. Thus ?(??) = 0 or 1. This proves
that / is constant a.e., because otherwise there would exist anreR with 0 < ?(??) < 1.
The implication (b) =>· (d) follows from Theorem 2.4 by observing that E(f\M) is ?-
invariant and fx f ?? = fx ?(/\?)??. Finally, to prove ((d) =>· (a)), we assume that
? e Bis ?-invariant. Then ? ?"?? XE(Tkx) = xe for all ? e N and therefore, d) implies
that xe is constant a.e. This proves that ?(?) = 0 or 1. ?
For a complete proof of Theorem 3.2 see Petersen A983) and Parry A981).
Remark 3.3. In analogy to Theorem 3.2(d), the limit in the Chacon-Ornstein theorem
can be shown to be a constant if we impose on a positive contraction the additional
assumption of conservative ergodicity. If a positive contraction U:L](X,B,?) ->
L1 (?, ?, ?) is conservative ergodic in the sense that for all 0 ^ / e L1 (?, ?, ?) with
?({/ > 0}) > 0 we have
oo
^Ukf(x) = oo a.e..

1198
F. Blume
then the limiting function in the Chacon-Ornstein theorem has the constant value
Examples 3.4. (a) Bernoulli shifts. For Bernoulli shifts condition (e) of Theorem 3.2
is easily verified for cylinder sets, and using standard approximation arguments it can be
shown to hold for all measurable sets A and B. Thus, Bernoulli shifts are ergodic.
(b) Markov shifts. A stochastic matrix A is said to be irreducible, if for every pair of
indices (/, j) there exists an ? e N such that the element at position (/,j) in A" is strictly
positive. If (?, ?, ?, ?) is a Markov shift on a finite alphabet, then (?, ?, ?, ?) is ergodic
if and only if the stochastic matrix associated with A is irreducible. For a proof see Petersen
A983).
(c) Rank one transformations. Assume that ?: [0, 1] -> [0, 1] is a measure-preserving
transformation with respect to Lebesgue measure ? on [0, 1]. We say that ? has rank one
if for every ? > 0 and for every NeN there exist pairwise disjoint intervals I\,..., /„ for
some ? > N such that ?(/| U ¦ ¦ ¦ U /„) > 1 - ? and ?G7*?/*+|) = 0 for 1 ^ к ^ ? - 1.
Using the Lebesgue density theorem (see Rudin A987)) it is not difficult to see that a rank
one system is ergodic. For more information concerning rank one systems see Ferenczi
A990).
(d) Irrational rotations on the circle. If Tax := ? + amodl is an irrational rotation
on [0, 1[ (i.e., ? e R \ Q), then the measure-preserving system ([0, 1[, ?, ?, ??) (where
? denotes Lebesgue measure) is ergodic. To prove this statement we assume that / e
L2([0, 1[) is ?-invariant with Fourier expansion f(x) = Y^L-^ а„е2л'"х. Then we have
27Г//1Л
? ane2*""=f(x) = f(Tx)=f(x+a)= ? а„е2™юе
and therefore, a„ = а„е2л'"а for all ? e Z. This shows that a„ = 0 for all ? ? 0, because
the irrationality of ? implies that ?2?'"? ? 1 for ? ? 0. Hence / = ?? a.e., and with / = ? a
for some ?-invariant set A e B, the result follows directly from Definition 3.1.
Using the ergodicity of Ta it is not difficult to prove Hermann Weyl's A910) famous
equidistribution theorem. If / is a subinterval of [0, 1 [, then
1'·?? . ,. . card({0^ к ^n - 1: ka mod 1 e /})
л^оо ? ?—' п-»зс п
Weyl A916) improved this result using Fourier analysis to show that the equation above
is still valid if ?/ is replaced by a Riemann integrable function. Due to this early work of
Weyl and more recently, the work of Bourgain( 1988a, 1988b, 1989), a very active field of
research has developed at the interface of ergodic theory and Fourier analysis. A few results
are listed in Section 6, but for more information on this subject and further references the
reader is referred to Petersen and Salama A995).
(e) Translations of the torus. If a = (a \,..., <*,„) e E'" and Ta is a translation on the
torus [0, 1 ['" (see Section 1), then Ta is ergodic if and only if a is rationally independent
in the sense that VkeZ'"k -a e ? ? ?: = 0 (for a proof see Petersen A983)).

Ergodic theory
1199
Another important question concerns the existence and uniqueness of ergodic measures
for homeomorphisms on compact metric spaces.
Dehnition 3.5. We say that a system (?,?) is a cascade if X is a compact metric
space and ?: X -> X is a homeomorphism. (X, T) is said to be uniquely ergodic if there
is exactly one ?-invariant Borel probability measure on X.
THEOREM 3.6. If ? is the unique ?-invariant Borel probability measure of a uniquely
ergodic cascade (X, T), then (?, ?, ?, ?) is ergodic.
PROOF. For any set A e В we define Мл(б) := ?(?? ?)/?(?). If there were a T-
invariant set A e В with 0 < ?(?) < 1, then ?^ and ??\? would be distinct ?-invariant
measures on X (because ?^(?) = 1 and мх\л(А) = 0) contradicting the assumption of
unique ergodicity (see Katok and Hasselblatt A995)). ?
THEOREM 3.7. (X, T) is uniquely ergodic if and only if for every continuous function
f: X —> X the averages
1 ""'
converge uniformly to a constant function on X. In case that (?, ?) ? uniquely ergodic,
with invariant measure ?, the constant is equal to fx f ??.
Remark 3.8. According to the Kryloff-Bogoliouboff Theorem A937) any continuous
map on a compact metric space X has an invariant Borel probability measure. The proof
of this result is based on the Riesz representation theorem (see Rudin A987)). Details
are provided in Katok and Hasselblatt A995) where the reader will also find a proof of
Theorem 3.7.
As explained in Furstenberg A981), Theorem 3.7 can be applied to obtain the following
interesting generalization of the equidistribution theorem (Example 3.4(d)):
THEOREM 3.9. If p(x) is a real polynomial with at least one coefficient other than the
constant term irrational, then for every subinterval I o/[0, 1 [ we have
card({0 ^k^n-\: p{k) mod 1 e /})
lim =?(/).
и—>зс п
This result was first proved by Weyl A916). Dynamical proofs were given by
Furstenberg A960), Hahn A965) and Postnikov A966).
We conclude this section with a very general representation theorem which asserts that
every ergodic measure-preserving transformation on a Lebesgue space is isomorphic to
a minimal, uniquely ergodic homeomorphism on a compact metric space. Here we say
that a homeomorphism ?: X -> X on a compact metric space X is minimal if for every

1200
F. Blume
x e X the orbit {T"x: ? e Z) is dense in X. This result was first proved for weakly
mixing transformations (see Section 5) by Jewett A970) and then extended to ergodic
transformations by Krieger A972) using the Krieger generator theorem (Theorem 7.16).
THEOREM 3.10. IfT.X -> X is an ergodic measure-preserving transformation on a
Lebesgue probability space (?, ?, ?), then there is a minimal homeomorphism S on the
Cantor set С such that (S, C) is uniquely ergodic and isomorphic to (?, ?, ?, ?).
4. Recurrence
One of the earliest results in the study of measure-preserving systems is due to Poincare
A899). In contrast to the ergodic theorems of von Neumann and Birkhoff the Poincare
recurrence theorem gives us only a qualitative rather than quantitative description of the
longterm behavior of a measure-preserving system.
THEOREM 4.1 (Poincare recurrence theorem). Assume that (?, ?, ?, ?) is a measure-
preserving system. Then for every A € В we have
(ЭС ОС \
А П p| [Jr-"A = ?(?).
,n=0"='« /
In other words, almost all points of A are infinitely recurrent with respect to A.
PROOF. Following the exposition in Parry A981), we note that the sets
ЭС
Am := (J T-"A
n—m
form a decreasing sequence in the sense that A„, D Am+\ for all m e No. Since Г-'" Aq =
A,„ for all m e N, it follows that for all m\,mi ^ 0 we have ?(?„,,??,„2) = 0 and
therefore, ?(??? ?,^=? ?>?) = 0- The result follows by observing that А С Ао- ?
A stronger, quantitative statement concerning the recurrence of sets has been established
by Khintchine A934). For this we need the following definition:
Dehnition 4.2. A set ? с ? is said to be relatively dense if there exists an integer К
such that
? П {;',...,;' + ?] ? 0 for all ; e ?.
The proof of Khintchine's recurrence Theorem uses the Hilbert space theory of
L2(X, ?, ?) and the mean ergodic theorem (see Parry A981)).

Ergodic theory
1201
THEOREM 4.3. //(?, ?, ?, ?) is a measure-preserving system, then for every A e В and
every ? > 0 the set
?? := [? e ?: ?(? ? ?~" A) > ?(?J - ?}
? relatively dense.
We will now turn our attention to theorems concerning multiple recurrence, which is
a subject that has found very interesting applications in combinatorial number theory.
We begin by stating two famous number-theoretic results that have been shown to be
consequences of a multiple recurrence theorem by Furstenberg and Katznelson.
THEOREM 4.4 (Van der Waerden's theorem A927)). If {C\,..., C,} is a partition of
the positive integers, then there is a j e {1, r] such that Cj contains arbitrarily
long arithmetic progressions, i.e. for every к &N there exist integers a, n e N with
[a,a + n,...,a + (k~ \)n] CC,.
Van der Waerden's theorem is a corollary of a more general result obtained by Szemeredi
A975). To formulate this theorem we need the following definition: A set S С If is
said to have positive upper Banach density if for some sequence of products П„ : =
[a„. ?, bn. ?] ? ¦ · ¦ ? [a„,r, b„.r] С If with \im„-+x{b„.j - a„j) = oo (for all ;') we have
card^nn,,)
lim > 0.
н-юо card(n„)
For Szemeredi's theorem we only need to consider sets of positive upper Banach density
in Z, but for later reference in Theorem 4.8 we gave here already the general definition for
the r-dimensional case.
THEOREM 4.5 (Szemeredi's theorem A975)). Any set of positive integers with positive
upper Banach density contains arbitrarily long arithmetic progressions.
THEOREM 4.6 (Furstenberg and Katznelson A978)). Let Т\,Ъ, ¦ ..,Tm be commuting
measure-preserving transformations (i.e. T, oTj = TjoTi) on a probability space (X.B, ?)
and let A e В with ? (A) > 0. Then
1 "
lim sup - ^?(?~?? ? ¦ ¦ · ? T~"A) > 0.
»—>эс П
In particular, there exists an ? e N such that
?G7"??---??„7"?) > 0.
As a corollary we obtain the following multiple recurrence theorem:

1202
F. Blume
Theorem 4.7 (Ergodic Szemeredi theorem (Furstenberg, 1977)). Let (?, ?, ?, ?) be a
measure-preserving system, m e N and A e В with ? (A) > 0. Then there exists an ? e N
such that
?(? П Т~"А П · ¦ ¦ П Т-('"~])"А) > 0.
It is important to point out that Furstenberg's proof of the ergodic Szemeredi theorem
(which preceded the more general Theorem 4.6) is based on ergodic theory—it does not use
combinatorial number theory. Using the ergodic Szemeredi theorem, Furstenberg was thus
able to provide purely ergodic-theoretic proofs of the Theorems of van der Waerden and
Szemeredi. Furthermore, using Theorem 4.6, Furstenberg and Katznelson A978) obtained
the following multidimensional extension of Szemeredi's theorem:
Theorem 4.8 (Multidimensional Szemeredi theorem). We assume that S С 27 has
positive upper Banach density. If F С 27 is a finite set, then there is an integer ? e N
and a vector a e 27 such that a + nF С S.
An interesting polynomial version of Theorem 4.5 with its corresponding generalization
of Szemeredi's theorem (Theorem 4.10) has been obtained by Bergelson and Leibman
A996):
THEOREM 4.9. If T\, T2,..., Tm are commuting measure-preserving transformations
on a probability space (?,?,?) and p\(n),..., p„,(n) are polynomials with rational
coefficients taking integer values on the integers such that p,@) = 0 for i = 1,..., m,
then for all A e В with ? (A) > 0, we have
limsup - ??(?-?,?)? П · · · П Г„7''"'(А)А) > 0.
?^ сю П
k=\
THEOREM 4.10. Assume that S С 27 is a set of positive upper Banach density and let
P\ (n),..., p,„(n) be polynomials with rational coefficients taking integer values on the
integers such that pi @) = 0 for i = 1,..., m, then for any vectors b \,..., b„, e 27 there
exist an integer ? and a vector a e 27 such that a + pi {n)b[ e S for all i = 1,..., w.
For further improvements and refinements of Theorems 4.3 and 4.4 the reader is referred
to Furstenberg and Katznelson A985,1986, 1991), Bergelson et al. A996) and in particular
also Furstenberg A981) which gives an excellent account of the connections between
ergodic theory, topological dynamics and combinatorial number theory.
Finally, we wish to make a remark on the philosophical significance of Theorem 4.9.
As a trivial consequence of the Poincare recurrence theorem, we see that for every
measure-preserving system (?,?,?, ?) and every A & В there is an ? e N such that
?(? П Т~"А) > 0. Interpreting this statement as a prediction of arbitrarily close future
return of a system to states that existed in the past, it is interesting and surprising to notice
that such arbitrarily close returns also happen along polynomial sequences, because if p(n)

Ergodic theory
1203
is a polynomial with the properties stated in Theorem 4.9, then for any A e В there is an
? e N such that ?(? ? Т~р{п) А) > 0 (apply Theorem 4.9 to Т\=Ъ = ?, p\ (?) = ? and
P2(n) = p(n) + n). So in a certain sense the world looks the same in polynomial time as
in standard linear time.
5. Mixing
Definition 5.1. A measure-preserving system (?, ?, ?, ?) is said to be weakly mixing
if
1 ""'
lim -???(?"*???)-?(?)?(?)|=0 forallA,fie?,
*=0
and we say that (?, ?, ?, ?) is strongly mixing if
lim ?(?~"? ?) ?) = ?(?)?(?) for all A,B eB.
Given these definitions and Theorem 3.2, it is easy to see that
strong mixing =>· weak mixing =>· ergodicity.
Furthermore, strong mixing, weak mixing and ergodicity are isomorphism invariants.
THEOREM 5.2 (Equivalent characterizations of weak mixing). For any
measure-preserving system (?, ?, ?, ?) the following statements are equivalent:
(a) (?,?, ?, ?) is weakly mixing.
(b) Ш^оо^'Ц^иУ^) ~ {f,\){\,g)\=0forall f,geL\X,B^).
(c) For all f e L2(X, ?, ?) with (/, 1) = 0 we have
1 ""'
lim -?(^/./>=?·
A=0
(d) There is a set ? С N of density zero (i.e., lim,,^^ card(Ai П {1,...,«})/« = 0)
such that
lim ?(?-'???) =?(?)?(?) for all A, Be В.
(e) ? ? ? is weakly mixing on ? ? X with respect to the product measure ? ? ? (see
also Section 8 for a definition of products).
(f) ? ? S is ergodic on ? ? ? for each ergodic measure-preserving system (Y, C, v, S).
(g) ? ? ? is ergodic.
(h) All measurable eigenfunctions of Ut are constant a.e.

1204
F. Blume
(i) (Krengel, 1972) For all A e В there exists a strictly increasing sequence {«a}„6n
С N such that the intersection of the ?-algebras Cm (m ^ 0) generated by the sets
{T~""'+k А: к eN] is trivial (in that it contains only sets of measure zero or one).
We wish to give a partial proof of Theorem 5.2 following the exposition in Parry A981)
and Petersen A983). For this we need the following analytic fact (see Walters A975)):
LEMMA 5.3. If {а„}„ец is a bounded sequence of non-negative integers then we have
???,,^?? - J2"kZ0ak = 0 if and only if there is a set ? С N of density zero such that
???,?^??. п<?м an = 0.
Proof of (а) <s> (d) and (a) => (e) =>· (g) => (h) =>· (a) in Theorem 5.2. The
equivalence (а) о (d) is a trivial consequence of Lemma 5.3. To prove (a) =>· (e) we
assume that (?, ?, ?, ?) is weakly mixing. For ? ? ? to be weakly mixing on ? ? ?
it is sufficient to show (using standard approximation arguments) that
1 ""'
lim - V \? ? ?((? ? T)~kA, ? ?? ? ?, ? ??)
?—>эс ? '?-~' ~
— ? ? ?(?\ ? ??)? ? ?(?\ ? ??)\ =0
for all ? ?, Аг, В], ?2 е ?. According to (d) there are sets ? ?, Mi с ? of density zero,
such that
(lim ?(?~"???]??)=?(??)?(??) for/e {1,2}.
Hence
lim ? ? ?((? ? ?)~"?? ? ?^ П ?, ? ??)
= ? ? ?(?\ ? ??)? ? ?(?\ ? ??).
Applying the equivalence (a) o (d) to ? ? ?, we conclude that (a) implies (e). The
implication (e) => (g) is trivial. To see that (g) implies (h), we assume that Ut has a
nonconstant eigenfunction / e L2(X, ?, ?) corresponding to the eigenvalue ? e C. Since
Ut is unitary, we have |?| = 1. Thus, for g(x, y) := f(x)f(y) we obtain UTxTg(x,y) =
Ut/(x)Ut f(y) = g(x,y) and therefore, g is a nonconstant ? ? ?-invariant function so
that ? ? ? is not ergodic according to Theorem 3.2(b). Finally, we will provide a brief
outline for the proof of (h) =>· (a) that illustrates the use of the spectral theorem. We
denote by V the closed linear span of the eigenfunctions of Ut and for / e V1- and
g e L (?, ?, ?) we define a measure ? on the spectrum o(Ut) of Ut via the equation
v(A) := {E(A)f g), where ? is the projector-valued measure generating Ut according to

Ergodic theory
1205
the spectral theorem. Then it follows that
" *=0 " k=0UalUT>
I ""' Г
kklkdv{k)dv^)
k=QJ'r(t>T)xat,UT)
=1
a(Uj)y,a(Uj) n(\ — ??)
( ?-^??(?)??{?).
A)
The last step is permissible, because 1 — ?? = 0 if and only if {?, ?) is a point on
the diagonal ? of a(Ur) ? ?(???) and /i has measure zero, because using the spectral
theorem, it is not difficult to show that ?([?}) = 0 for all ? e ?(???)- Since the integrand
in A) converges to zero a.e., the bounded convergence theorem and Lemma 5.3 imply that
Hindoo + ?*=? \(Ujf,g)\ = 0. If now / is an arbitrary function in L2{X,B, ?), then (h)
implies that / - (/, 1) e V1 and therefore,
, n— I | «- I
0= Hm -??(?/?/-</,??|= lim -???-?-?/??1·*)!-
*=0 it=0
Replacing / and g by хл and ?^, it follows that ? is weakly mixing.
D
For a complete proof of Theorem 5.2 the reader is referred to Petersen A983), Parry
A981) and Krengel A972) (an interesting improvement of Krengel's characterization of
weak mixing in (i) has been given by Bergelson A993)).
THEOREM 5.4 (Equivalent characterizations of strong mixing). For a measure-preserving
system (?, ?, ?, ?) the following statements are equivalent:
(a) (?,?, ?,?) is strongly mixing.
(b) For all f,ge L2(X, ?, ?) we have
lim {UT f,g) = (f, I)(I,g).
17—>00
(c) (Renyi, 1958) For each A e В we have
lim ?(?""?? ?)=?(?J.
II—>00 V '
(d) (Blum and Hanson, 1960) For all ? e [l,oo[, all strictly increasing sequences
{ki}ieN С ? and all f e U7(X, ?, ?) we have
lim
II—>3C
№'-f.
???
:0.

1206
F. Blume
(e) (Ornstein, 1972) (?, ?, ?, ?) is completely ergodic (i.e., (?, ?,?, ?") is ergodic
for all ? e Z) and there is а с e R such that for all А, В е В we have
limsupM(r_"A ? ?) ? ??(?)?(?).
?—>??
Examples 5.5. (a) Bernoulli shifts. For Bernoulli shifts it is easy to see that
lim ?(?~"????)=?(?)?(?)
for all cylinder sets A and B. Standard approximation arguments show that this equation
actually holds for all measurable sets A and B. Thus, Bernoulli shifts are strongly mixing.
(b) Markov shifts. An m ? m stochastic matrix A is said to be aperiodic, if there is no
i e {1,..., m) and no integer к > 1 such that the condition a" > 0 (where a" is the /th
element on the diagonal of A") implies that к divides n. If (?,?, ?, ?) is a Markov shift
on a finite alphabet, then (?, ?, ?, ?) is strongly mixing if and only if the stochastic matrix
associated with A is irreducible and aperiodic (see Petersen A983)).
(c) Finite interval exchanges. If ?: [0, 1[ -> [0, 1[ is a finite interval exchange
transformation and ? is Lebesgue measure on [0, 1 [, then ([0, 1 [, ?, ?, ?) is not strongly
mixing (see Cornfeld et al. A982)), but possibly weakly mixing (see Choe A993)).
(d) Lebesgue spectrum. We say that a measure-preserving system (?,?,?,?) has
Lebesgue spectrum of multiplicity N, if there is a set ? of cardinality N (possibly infinite)
and a set of functions {f\j: ? e ?, jeZ) which, together with the constant function
1, form an orthonormal basis for L2(X, ?, ?) such that Urfx.j = fx.j+\ for all ? e ?
and j e Z. It can be shown that a measure-preserving system is strongly mixing if it has
Lebesgue spectrum (see Petersen A983)).
(e) A weakly mixing system that is not strongly mixing. We have already mentioned
that strong mixing implies weak mixing, but it is actually not easy to see that the reverse
implication is false. We will now give a brief outline of a construction due to Chacon
A969) which not only yields a weakly mixing system that is not strongly mixing, but also
illustrates well how measure-preserving transformations can be constructed using cutting-
and-stacking procedures (see also Section 8). We begin by assuming that we are given a
finite tower (or stack) ? =(I\,..., I„) of disjoint intervals I\ ,...,/„ of equal length. The
intervals I\,..., /„ are also referred to as the levels of ?. We define a measure-preserving
transformation
и—I n
7|:U''-U'i
? = | ?=2
such that the restriction of T\ to Д for к e {1,..., ? — 1} is an affine map onto h+\ ¦ We
then 'cut' the tower ? vertically into three substacks of equal width and stack up these
three subtowers with a spacer (that is an interval equal in length to the width of the three
subtowers) inserted between the second and the third subtower. The resulting tower Г2
consists of Ъп + 1 levels (see the diagram below). We proceed to define ?? in analogy to
T\ in such a way that the restriction of Ti to one of the first Ъп levels in Г2 is an affine

Ergodic theory
1207
map onto the next higher level. Now we repeat this cutting-and-stacking procedure with ??
in place of ? to produce a tower гз of height 3Cи + 1) + 1 and continuing this way we
obtain a sequence of towers {?„} where each tower is associated with a transformation T„.
The pointwise limit ? of the transformations T„ is a measure-preserving transformation
(with respect to Lebesgue measure) on the union X of the levels of the towers ?„. Since X
has clearly finite measure, we have constructed a measure-preserving system (?, ?, ?, ?),
where we denote by ? Lebesgue measure divided by the Lebesgue measure of X.
79
?
To see that ? is not strongly mixing, we choose a set A e В such that ? (A) < \,. Since
the levels of the towers ?„ generate B, it follows that for large ? most of the levels of
?„ are almost entirely contained in A or in X \ A. Denoting the length of ?„ by /„ and
observing that the first /„ levels in ?„+| are mapped onto the second /„ levels, it follows
that ?{?'" А П A) > ^?(?) > ? (AJ contradicting strong mixing. To see that ? is weakly
mixing, we assume that / is an eigenfunction of Ut corresponding to the eigenvalue ?.
Then for large ? the function / is almost constant on almost all levels of ?„. If / were
actually constant on each level of ?„ then the construction of ?,,+ ? from ?„, as illustrated
above, would imply that ?'" = ?'" + ? and therefore ? = 1. Since ? has rank one we know
that ? is ergodic (see Example 3.4(c)), and it would follow that / is constant a.e. on X.
Thus according to Theorem 5.2(h) we would be able to conclude that ? is weakly mixing.
For a detailed discussion that makes these arguments precise see Petersen A983).
It is interesting to notice that in a certain sense almost every measure-preserving system
is weakly mixing but not strongly mixing. To formulate this result, we assume that
(?,?,?) is a Lebesgue space. We say that two measure-preserving transformations on
X are equivalent if they differ only on a set of measure zero, and we define the weak
topology on the set of equivalence classes of measure-preserving transformations on X via
the statement
lim T„ = T ? ??6? lim ?(?,,????) = 0.
?—>oo /?—>эс

1208
F. Blume
It can be shown that with respect to the weak topology as defined above, the set of
weakly mixing transformations on X is residual (Halmos, 1944), while the set of strongly
mixing transformations is of first category (Rokhlin, 1948). Along these lines Glasner and
King A998) established a very general zero-one law that classifies sets of transformations
defined via certain isomorphism invariants as either residual or meagre.
We now turn our attention to the subject of higher orders of Mixing. Our first theorem in
this context says that a weakly mixing measure-preserving system is weakly mixing of all
orders. For a proof see Furstenberg A981).
THEOREM 5.6. Assume that (?,?,?,?) is a weakly mixing measure-preserving system.
IfAo, A\,...,Aj e В and p\ < ¦ ¦ ¦ < pj e N then for all ? > 0 the set
{keN: |?(?0??-^?| ? ¦ ¦ · ? T^'J Ay) - ?(?0)?(?|) ¦ ¦ -M(Ay-)| >?}
has density zero (see Theorem 5.2(d)).
Given the statement of Theorem 5.6, it is obvious that for all ? > 0 we have
1 ""'
limsup- ^|?(?0? ?-^'?, ? ¦¦ ¦ ? ?'^??
"^°° n k=o
-?(?0)?(?|)···?(?7)| ?? ?.
Therefore, using Lemma 5.3, we obtain the following corollary:
COROLLARY 5.7. Assume that (?,?,?,?) is a weakly mixing measure-preserving
system. IfAo, A |,..., Aj e В and p\ < ¦ ¦ ¦ < p; e N then
1 "_l
lim - ???(????-^?| ? · ¦ · ? T~kpJ А Л - ?(?0)?(?,) ¦ ¦ -?(?,-)| = 0,
and there is a set ? С N of density zero, such that
lim ?(?0??-"',|?|?-??-"^?.)=?(??)?(?|)-?(?7).
A result similar to Theorem 5.6 can also be obtained for mildly mixing systems (see
Furstenberg A981)):
DEnNlTlON 5.8. Assume that (?, ?, ?, ?) is a measure-preserving system and let / e
L2(X, ?, ?). We say that / is rigid if there is a sequence {/ц }„6n С N such that
lim ||/-t/"l/IU=0.

Ergodic theory
1209
Definition 5.9. A measure-preserving system (?,?,?, ?) is said to be mildly mixing
if there are no nonconstant rigid functions in L2(X, ?, ?).
Note: Using Theorems 5.2(h) and 5.4(b) it is not difficult to show that
strong mixing =>· mild mixing =>· weak mixing.
THEOREM 5.10. Assume that (?, ?, ?. ?) is a mildly mixing measure-preserving system.
IfAo, ? ?,..., Aj eB and p\ < ¦ ¦ ¦ < pse N then for all ? > 0 the set
[k e ?: \?(?0 П Г-**" А, П ¦ ¦ ¦ П T'ki''Aj) - ?(?,)?(?2) · ¦ ¦ ?(?*)| < ?}
is relatively dense (Definition 4.2). For stronger versions of this statement see Furstenberg
A981).
Now we will discuss higher order strong mixing. A measure-preserving system
(?, ?, ?,?) is said to be strongly mixing of order m, if for all Ao, A,„ e В we have
lim ?(?0 П ?""' ? ? ? · ¦ · ? ?-("?+ ¦+"'")Am)
Л| И,,,-» 00 '
= ?(?0)?(?|)-?(?„,).
Kalikow A984) showed that strong mixing implies strong mixing of order three (and
by the same argument strong mixing of all orders) for rank one transformations, and
Ryzhikov A993) extended this result to finite rank transformations. Furthermore, Host
A991) demonstrated that strong mixing implies strong mixing of all orders for systems
with singular spectral measure and Thouvenot showed that if it is true that two-fold mixing
implies three-fold mixing for zero entropy systems (see Section 7), then this implication
is true for all systems. However, in general the problem of whether strong mixing implies
higher order strong mixing is still open.
An important class of measure-preserving transformations that are strongly mixing of
all orders is formed by the so-called К-automorphisms.
Definition 5.11. A measure-preserving transformation Г on a probability space
(?, ?, ?) is said to be а К-automorphism ('K' stands for 'Kolmogorov'), if for all m e N
and all sets Aq, ..., A,„ e В we have
lim sup |?(?0? ?)-?(??)?(?)| =0,
"^°°B6C,f(/l, A,„)
where C™(A \,..., A,„) is the smallest ? -algebra containing the sets TkAi ioxk^n and
ie(l,...,m).
Given this definition it is easy to show that every /^-automorphism is strongly mixing
of all orders, but the reverse conclusion does not hold (see Rudolph A990) and Cornfeld
etal. A982)).

1210
F. Blume
Theorem 5.12 (Equivalent characterization of ^-automorphisms). IfT:X-*Xisa
measure-preserving transformation on a probability space (?, ?, ?), then ? is a
K-automorphism if and only if there is a sub-? -algebra Л С В such that
(a) Г'ЛсА
(b) U^-oo T"A generates B, and
(c) nL-oo Т"Л " the trivial ?-algebra {0, X).
In Section 7 (Example 7.18(c)) we will give other characterizations of К -automorphisms
that establish an important link to the theory of entropy. For a proof of Theorem 5.12
see Cornfeld et al. A982). Furthermore, it can be shown that every ?-automorphism has
countable Lebesgue spectrum (see Parry A981) or Petersen A983)).
Finally, we wish to discuss briefly the concept of topological mixing for cascades
(Definition 3.5) and its relation to measure-theoretic mixing.
Dehnition 5.13. A cascade (X, T) is said to be:
(a) topological^ ergodic if all proper closed Г-invariant subsets of X are nowhere dense
inX;
(b) topological^ weakly mixing if for all nonempty open sets ?, ? С X the set {n e N:
T"A П ? ? 0} contains arbitrarily long blocks of consecutive integers;
(c) topologically strongly mixing if for all nonempty open sets ?, ? С X there exists an
N eN such that T"A ? ? ? 0 for all ? > ?.
If (?, ?) is a cascade and ? a ?-invariant Borel probability measure with support X,
then it is easily verified that measure-theoretic ergodicity, weak mixing and strong mixing
(Definitions 3.1 and 5.1) imply topological ergodicity, weak mixing and strong mixing
respectively. However, the converses of these implications are not true, even if (X, T) is
uniquely ergodic (see Kolmogorov A953) and Petersen A970)).
6. More about convergence in ergodic theory
In this section we list several non-standard ergodic theorems. We begin with a discussion
of averages along subsequences.
THEOREM 6.1 (Bourgain, 1989). Let (?,?,?, ?) be a measure-preserving system and
assume that p(n) is a polynomial such that p(n) e Ifor all ? e Z. Then for any q > 1 and
any f e L4(X, ?, ?) the averages
l-Zf(TP*>x)
converge pointwise a.e.

Ergodic theory 1211
THEOREM 6.2 (Furstenberg and Weiss, 1996). If (?,?,?,?) is a measure-preserving
system and f,g? L°°(X, ?, ?), then the averages
are convergent in L2(X, ?, ?).
THEOREM 6.3. Assume that (?, ?, ?, ?) is a measure-preserving system and let /, g,h e
L°°(X, ?, ?) and a,b,c e N. Then the averages
1 ""'
-~TuaT"fUbT"gU^h
? '—'
k=0
are convergent in L2(X, ?, ?).
For a proof and further discussion of this result see Conze and Lesigne A984, 1988)
and Furstenberg and Katznelson A990). A very general result for convergence along
polynomial sequences in weakly mixing systems was obtained by Bergelson A987).
THEOREM 6.4. Assume that p\ (и),..., p,„(n) are polynomials with integer coefficients
such that each pj is nonconstant and each pi — pj with i ? j is nonconstant as
well. If (?,?, ?,?) is a weakly mixing measure-preserving system and f\,...,f„ e
L°°(X,B,v),then
= 0.
In general, for a sequence A = {?„bisN С N and a given measure-preserving system we
may examine the convergence behavior of the averages
M,(A,f)(x):= J ? /(?^)'
where A(t) := {к е A: 1 ^ к ^ t] for all t > 1. Using Fourier analytical methods
developed by Bourgain, it can for example be shown (see Rosenblatt and Wierdl A995))
that if A = {?„}„?? is the (strictly increasing) sequence of prime numbers, then for
every measure-preserving system (?. ?,?,?) and every / e L°°{X,B. ?) the limit
lim,^oo M, (A, /) exists in L2(X, ?, ?). This approach is also easily extended to randomly
generated sequences. To see this, we assume that (?,C. P) and {УлЬегг is an i-i-d.
sequence of 01-valued random variables on ? with P(Y„ = 1) = ? and P(Y„ = 0) = 1 - p.
For each wefiwe define a sequence ?? via the equivalence ? e ?? о· У» (?) = 1. Then
lim
'f^U^k)fr..U?'»(k)fm- f /,??..· [ /„??
,„ Jx Jx

1212
F. Blume
for almost every ? e ? the sequence ?? is ergodic in the sense that for any measure-
preserving system (?, ?, ?, ?) and any / e ?,°°(?, ?, ?) the limit linv^oc ?,(??, f)
exists in L2(X,B, ?) and is equal to fx f ??. A more general result concerning pointwise
convergence along random subsequences is due to Lacey A994):
THEOREM 6.5. For an ergodic measure-preserving system (?, С, P, S) andan integrable
function ? : ? —> Ъ we set
k-\
Tk(w):=J2s(siv) for all к е N.
i=0
If fnSdP ф0, then there is a set ?? e С with ?(??) = 1 such that for every ? e ??,
every measure-preserving system (?, ?, ?, ?) and every integrable function f on X the
averages
*=l
converge a.e. with respect to ?.
Intuitively, Theorem 6.5 addresses the question of the influence of time fluctuations on
experimentally collected data. For example, we may ask what happens if instead of making
a measurement every 10 hours, we make small random errors around the designated target
times. In Theorem 6.5 the possibility of such errors is taken into account via the sequence
of measuring times Tk(w) generated by the transformation S and the function S. As stated
in Theorem 6.5, the averages of measurements taken along such perturbed time sequences
will almost surely converge so that in a sense we have a wide latitude for error. For further
discussion of time fluctuations see Petersen A996).
In the context of random ergodic theorems we also wish to mention the original random
ergodic theorem by Ulam and von Neumann A945):
Theorem 6.6 (Random ergodic theorem). Let (?,?,?) and (?,?, ?) be probability
spaces and assume that for every ? e ? we are given a measure-preserving transformation
??: X —>¦ X such that for every measurable function f on X the map (?, ?) ь* f(T0)x)
is measurable on ? ? ? with respect to the product measure ? ? P. Then for all
f e L] (?,?,?) and almost all sequences {?„}„6? in the product measure space
??^^?, С, ?) the averages
1 "
k=\
converge a.e. with respect to ?.
Note: The proof of this theorem is based on a so-called skew product construction which
is defined in Section 8.

Ergodic theory
1213
An important strengthening of Birkhoff's pointwise ergodic theorem was obtained by
Wiener and Wintner A941).
THEOREM 6.7 (Wiener-Wintner theorem). //(?, ?, ?, ?) is a measure-preserving system
and f & L1 (?, ?, ?), then there is a set Xq e В with ?(??) = 1 such that
??
lim -У/(Ткх)е'к
*=0
exists for all ? e К and all ? e Xo.
The significance of this theorem lies in the fact that convergence of the ergodic averages
occurs on a set Xo of full measure that is common to all ? еШ. For a single ? еШ,
the existence of the ergodic averages on a set of full measure is a trivial consequence
of the ergodic theorem, but it is not at all obvious that such a set exists for all ? e К
simultaneously, because ? is not countable. As a generalization of the Wiener-Wintner
theorem, Bourgain, Furstenberg, Katznelson and Ornstein obtained the following result
(see Bourgain A989)):
THEOREM 6.8 (Return-times theorem). If (?,?,?,?) is a measure-preserving system
and f e L°°(X, ?, ?), then there is a set Xq&B with ?(??) = 1 such that
I"
lim -^f(Tkx)g(Sky)
k=0
exists for all ? e Xo, every measure-preserving system (Y, C, v, S), every g e L1 (Y,C, v)
and a.e. у e ? with respect to v.
There is an interesting connection between the Wiener-Wintner theorem and the
ergodic Hilbert transform (Theorem 6.10) via a double maximal inequality for the helical
transform. For a measure-preserving system (?, ?, ?, ?) and a function /: X -> С we
define the helical transform as
H**f(x):= sup
neN.flelf
A f(Tkx)eike - f(T-kx)e-,kt>
k = \
THEOREM 6.9 (Campbell and Petersen, 1989). If'(?,?,?,?) is a measure-preserving
system and f e L2(X,B, ?), then for all ? > 0 we have
II fll2
?{???: ?** f(x) > ?} ^ ^.

1214
F. Blume
THEOREM 6.10 (Ergodic Hubert transform (Cotlar, 1955)). We assume that (?, ?, ?, ?)
is a measure-preserving system. If f e L2(X, ?, ?), then the sums
" f(Tkx)-f(T~kx)
converge pointwise a.e. as ? —> oo.
The theorem by Campbell and Petersen is a profound result which is closely related
to Carleson's theorem concerning the pointwise convergence of Fourier series (see also
Assani and Petersen A992) and Assani A993)). Furthermore, efforts have been made to use
the helical transform to describe the full spectral measure of the induced unitary operator
corresponding to a measure-preserving transformation via individual sample paths, that is
via orbits of points under the action of T. For a more detailed discussion of this subject the
reader is referred to Petersen A996).
7. Entropy and information
One of the central problems in ergodic theory is the classification problem: How can
we decide whether two given measure-preserving systems are isomorphic or not? One
option is to distinguish systems via dynamical properties such as ergodicity or mixing.
For example, a Bernoulli shift is not isomorphic to a finite interval exchange, because
the former is strongly mixing while the latter is not (see Examples 5.5). Another option
is to distinguish measure-preserving systems according to the spectrum of the induced
unitary operator. For the special case of discrete spectra, a very satisfactory answer to the
classification problem is given by the following theorem (see von Neumann A932b) and
Halmos and von Neumann A942)):
THEOREM 7.1 (Discrete spectrum classification theorem). We assume that S and ?
are measure-preserving transformations on a non-atomic Lebesgue probability space
(?, ?, ?) such that the induced operators U$ and Ut both have discrete spectrum (that
is the eigenfunctions of Us and Ut span L2(X,B, ?) respectively). Then the measure-
preserving systems (?,?,?, S) and (?,?,?,?) are isomorphic if and only if the set of
eigenvalues of Us is equal to the set of eigenvalues ofUj.
The case of transformations with continuous spectrum remained intractable until
Kolmogorov A958) introduced entropy as a new isomorphism invariant to show that there
are non-isomorphic measure-preserving systems with identical continuous spectra. We will
now begin our discussion of entropy by defining the information function for finite (or
countable) partitions. For a finite (or countable) index set / we say that a = {A,¦: i e /} is
? finite (or countable) measurable partition of a probability space (?, ?, ?) if
(a) A, e В for all г el,
(b) А, П Aj¦ = 0 for all i, j e / with i ? j, and
(c) m(U,6,A,)=1-
?

Ergodic theory
1215
If a = {A,-: i e /} and ? = {?}: j e J] are two finite (or countable) partitions, then the
refinement of a and ? is defined as
??? = {?,?)?] |/e/, j ej].
For a given point ? in a probability space (?, ?, ?) and a given finite (or countable)
partition a = {A,: / e /} we can gain some information about the location of ? in X
by identifying the set A e a of which ? is a member, and the smaller the measure of
A, the larger is the gain in information. Thus it is reasonable to define the information
function associated with a by an equation of the form /„(jc) = J^Aea ??(?)?(?(?)),
where ? : [0, 1] -> R is a positive decreasing function. We say that two finite (or countable)
partitions a and ? are independent if
?(? ? ?) = ?(?)?(?) for all A e ? and all S e /3.
It is natural to require that for independent partitions a and /3 the information function
???? associated with the refinement of a and ? is equal to the sum of the information
functions Ia and ??. This implies that ? should satisfy the equation ?(?(?)?(?)) =
?(?(?)) + ?(?(?)). Imposing the additional requirement that ? is continuous, we are
led to define ? as a negative logarithmic function, and by convention, we choose 2 as the
base of the logarithm.
Dehnition 7.2. If ? is afinite (or countable) partition of the probability space (?,?,?),
then the information function associated with a is
/ff(jc):=-^XA(jc)log2M(A) for all ? e X
Aea
and the entropy of a is
H{a):= ? ??{?)??.
Jx
Remark 7.3. If ? is afinite (or countable) partition and
i0 ifjt=0,
J(x)- |-;clog2(jc) if* e @,1],
then Definition 7.2 implies that
H(a)=Yjf^{A)).
Aea
(Note: If ? is not finite, then it is possible that ? (a) = oo.)

1216
F. Blume
In order to work efficiently with entropy and information, it is necessary to introduce also
the concepts of conditional entropy and conditional information. For a probability space
(?, ?, ?) and a sub-a-algebra ? С В we denote by ?(?\?) the conditional expectation
of xa with respect to T, i.e., ?{?\?) := ?(?? \?) for all AeB.
Definition 7.4. For a finite (or countable) partition ? of a probability space (?, ?, ?)
and a sub-a-algebra ? С В we define the conditional information function with respect to
? and .Fas
Ia\r{x) ·= - J] ХлОО log2 ?(?\?){?) for all jc e X,
and the conditional entropy of a with respect to F is
Я(а|Л:= f ??^(?)??.
Jx
Remark 7.5. If ? is a finite (or countable) partition, then the conditional information
function and the conditional entropy with respect to the ?-algebra ?{?) generated by the
sets in/3 are denoted by ??\? and ?(?\?), and the conditional expectation of ? a is denoted
by ?{?\?). It is easy to see that
?(?\?)(?)=???(?? /рч
tfl ?(?)
and
therefore,
??\?(?) =
?(?\?) =
¦¦ ??^?(.)??&
Лба v
fie/3
fie/3
Furthermore, if ? is trivial so that ?{?) = {0, X) is trivial, then ??\? = Ia and ?{?\?) =
? (a).
A few basic, easily verifiable properties of entropy and information are listed in the
following proposition:
PROPOSITION 7.6. Assume that a and ? are finite (or countable) partitions of a
probability space (?, ?, ?) and let ? and Q be sub-?-algebras ofB.
(a) ????\? = Ia\T + IfiiavF· where we denote by ay ? the ?-algebra generated by ?
and the sets in a.
(b) H(av ?\?) = ?(?\?) + ?(?\?? ?).
(c) If ? С д, then Н(а\П ? H(a\Q).

Ergodic theory
1217
(d) If ? is a refinement of a (i.e., ? ? ? = ?), then ?(?\?) ^ ?(?\?).
(e) ?(???\?)^?(?\?) + ?(?\?).
(f) If ? (?), ? (?) < oo, then a and ? are independent if and only if ? (? ? ?) =
?(?) + ?(?).
For a sequence {.7>}„6n of sub-a-algebras, we denote by V(?L| ?n tne ?-algebra
generated by the sets in the union UhLi ·?>· The next theorem is a consequence of the
martingale convergence theorem.
THEOREM 7.7. Assume that (?, ?, ?) is a probability space and {Tn}„eN is an increasing
sequence of sub-?-algebras of В (i.e., T\ С ?? С · · ¦)¦ Ua 's a finite partition ofX, then
lim H(a\Tn) = H\a\\l T„
\ /i = l /
If (?, ?, ?, ?) is a measure-preserving system and ? is a finite (or countable) partition
of X, then we denote by a"n the refinements of a under T, that is
n
< := T~ma ? · · · ? T~"a = \/ T~ka
к =tn
is the partition consisting of all the possible intersections of sets of the form T~k A with
A e a and m ^ к ^ и. Furthermore, we denote by a^ and a"^ the ?-algebras generated
by the sets in the unions \JfL,„ T~ka and U"=-oc T~ka respectively.
THEOREM 7.8. If (?,?,?,?) is a measure-preserving system and a is a finite (or
countable) partition ofX with ? (a) < oo, then
n—*oo ?
exists and is finite.
PROOF. Following the exposition in Petersen A983), we begin by showing that the
sequence {#„}„6pj with H„ := Н(а'^^) is subadditive. For и, m e N we have
Hn+m = H(al~] ? ?-?'-') ^ H(a''-i) + H(T~"a'^) = H„ + Hm.
Therefore, Hj„ ^ jH„ and Hj ^ jH\, so that {#,7;'},6? is bounded and L : = liminf/_3c
HjH < oo. Now let ? > 0 and choose a large j with Hj/j < L + ?. Then for ? > j and
к := [n/j] we have
L-^iktZfiini^ + Mj <!i±liL+eh
? kj kj к
and this clearly shows that L = lim„^oc H„/n. ?

1218
F. Blume
We are now ready to define the entropy of a measure-preserving transformation:
Definition 7.9. Assume that (?, ?, ?, ?) is a measure-preserving system.
(a) If ? is a finite (or countable) partition of X, then the entropy of ? with respect to a
is
h(a,T):= lim ° .
n—>oo n
(b) The entropy of ? is defined as the supremum of h(a, T) over all finite partitions of
X, i.e.,
h(T):= sup h(a,T).
[a\a finite]
Remark 7.10. It is easy to see that if (?,?,?,?) and (Y,C,v,S) are isomorphic
measure-preserving systems, then h(T) = h{S). In other words, entropy is an isomorphism
invariant.
THEOREM 7.11. Assume that (?, ?, ?, ?) is a measure-preserving system.
(a) If a is a finite partition ofX, then
h(a,T)= lim ?(a\a"\ = ?(a\al°).
(b) If a and ? are finite (or countable) partitions ofX with ? (?), ? (?) < oo, then
htf,T)^h(a.T) + H(P\a).
(c) For allkeZ we have h(Tk) = \k\h(T).
Now we will discuss an alternative approach to the definition of entropy for ergodic
transformations via counting of names (see Rudolph A990)). For this it is useful to
consider a finite partition a = {A\,..., A,,,} of a probability space (?,?,?) as a function
from X into a finite set {a\,...,am\ of symbols or 'names'. More precisely, if a : X ->
{??,..., a,„} is a measurable function, then the sets A, = a~' {a,} are the elements of the
associated partition. If Г is a measure-preserving transformation on X, then for all ? e N
and all ? e X the T, a, ?-name of ? is
a„(x) := (a(x), a(Tx),..., ?(?,,_ '*)).
Given this definition, it is easy to see that
<-'={<*,;'{*}·. *е{а|,....а,„}"}.

Ergodic theory
1219
In other words, the names a„ (x) provide a coding for the sets in a'^ . For ? > 0 we say that
a collection of names S С {? ?,.. .,am}" is ?-good if the measure of the union Uses'*/?'!5)
has measure greater than 1 — ?, and we define
N(T, ?, ?, ?) := min{cardE'): S is ?-good}.
In words: ?(?,?,?,?) is the smallest number of sets in a'^'] that cover a set greater than
1 — ? in measure.
Dehnition 7.12. Assume that (?,?,?,?) is a measure-preserving system and let
a: X -> {a\,.. .,am] be a finite partition of X. Then
ut-r ч log2 NG\ or, ?, ?)
h(T, ?,?,?) := —f and
?
h(T, ?, ?) := lim inf h(T, ?, ?, ?).
?—>эс
THEOREM 7.13. If (X, В, ?, ?) is an ergodic measure-preserving system, then
lim sup/г (?, ?, ?, ??) ^ ????^?,?,?,??)
?—>?? ?—» ??
for all О < ?? < ?2 < 1 and a// ^im'fe partitions a of X. Furthermore, the limit
lim?^oh(T, ?, ?) exists and
h{T)= sup lim h(?, ?, ?).
[a\a finite] ?—>0
(For a proof see Rudolph A990).)
Entropy as an isomorphism invariant is of course only useful, if it can be easily
calculated. For the computability of entropy, the concept of generating partitions is of
crucial importance:
Dehnition 7.14. If (?, ?, ?, ?) is a measure-preserving system, then a finite partition
? of X is said to be a generator with respect to ? if orf^ = B.
The following theorem is due to Kolmogorov A958) and Sinai A959).
THEOREM 7.15 (Kolmogorov-Sinai theorem). //(?, ?, ?, ?) is a measure-preserving
system and a is a generator with respect to T, then
h(T) = h(a,T).
(For a proof of this theorem see Parry A981).) For finite entropy ergodic measure-
preserving systems on Lebesgue probability spaces generators with respect to ? do always
exist (see Theorem 7.16). So in principle, entropy can be calculated via the Kolmogorov-
Sinai theorem for a very large class of measure-preserving systems.

1220
F. Blume
THEOREM 7.16 (Krieger generator theorem A970)). // ? is an ergodic measure-
preserving transformation on a Lebesgue probability space (?, ?, ?) such that h(T) < oo,
then there exists a finite partition aofX which is a generator with respect to T.
If it is not possible to explicitly determine a generator, then the following theorem
can still be useful in calculating the entropy of a measure-preserving transformation (see
RokhlinA967)).
THEOREM 7.17. //(?, ?,?,?) is a measure-preserving system and {a,,} is an increasing
generating sequence of finite partitions of X (i.e., a„+\ ? a„ = an+\ for all ? e N and
\I^L\an = B), then
h(T)= lim h(a„,T).
n—»oc
EXAMPLES 7.18. (a) Bernoulli shifts. Assume that (?,?,?,?) is a Bernoulli shift on
the finite alphabet {1,..., m] with weights p\ pm. If a is the time-zero partition of
?, i.e., a = {A\,..., Am] with A, := [x e ?: xq = i], then ?"*?, = {? e ?: Xk = i}.
Therefore, a^ contains all cylinder sets and it follows that a^x = B. Since the partitions
?, ?~'?,?~2?,... are pairwise independent. Proposition 7.6(f) and the Kolmogorov-
Sinai theorem imply that
#(?"~'? ? "
h(a) = h(a,a)= lim V ° ; = lim - У^Н(а~ка) = ? (a)
/I—>ЭС Ц II—>ЭС П *—'
к=0
in
i=\
In words: The entropy of a Bernoulli shift is equal to the entropy of the time-zero partition.
(b) Markov shifts. Assume that (?, ?, ?, ?) is a Markov shift determined by an m ? m
stochastic matrix A and a corresponding probability vector p. Using similar techniques as
in a), it can be shown (see Petersen A983)) that
m
h(p) = ~ X) PiOij log2 ajj.
i.j = \
(c) К -automorphisms. If (?, ?, ?, ?) is a measure-preserving system, then ? is a K-
automorphism if and only if ? has completely positive entropy, that is if and only if
h(a,T) >0
for all non-trivial finite partitions ? of X. Furthermore, it can be shown that ? is a A"-
automorphism if and only if for any finite partition a we have
oo
f>r = {0,x).
/7=0

Ergo die theory
1221
For proofs of these results see Rokhlin and Sinai A961) and Cornfeldet al. A982).
(d) Finite interval exchanges. If ?: [0, 1[ -> [0, 1[ is a finite interval exchange
transformation, then h(T) = 0. This result follows from the more general fact that every
measure-preserving transformation of finite rank has entropy zero (see Ferenczi A990)).
(e) Discrete spectrum. If (?, ?, ?, ?) is a measure-preserving system such that Ut has
discrete spectrum, then h(T) = 0 (for a proof see Petersen A983) or Parry A981)).
(f) Product transformations. Let (X\,B\,?\.?\) and (Xi.Bj, ??, ?>) be measure-
preserving systems and let ? := ?| ? Ъ be the product transformation on the product space
(?? ? Xi,B\ <8>??2,М| XM2)-If<*i and ?? are finite partitions of ? ? and X2 respectively,
then
? = <*? x<*2:={A| ? Ai: A| ea\, Ai e ??}
is a finite partition of ? ? ? ??. It is easy to see that ?^-1 = (??)'?_? ? (??)^-1 and
^KH) = %i)S"')+^(KH)·
Thus, /?(?, ?) = h{ot\, ?\) + h(ai, ?>). Choosing increasing generating sequences {ot\„}
and {<*2.п} and setting a„ :=а\ш„ х ??.„, it is not difficult to show that {a,,} is an increasing
generating sequence in the product space and therefore, Theorem 7.17 implies that
h(T\ хГ2) = й(Г|)+й(Г2).
A very important new area of research in the theory of entropy developed when Ornstein
A970a, 1970c) showed that two Bernoulli shifts are isomorphic if they have the same
entropy. In other words, for Bernoulli shifts entropy is a complete isomorphism invariant.
We will summarize here briefly the main results of Ornstein's theory.
Dehnition 7.19. Let ? > 0 and assume that a and ? are finite partitions of a probability
space (?, ?, ?). We say that a and ? are ?-independent, if there is a family Q С ? such
that the following conditions are satisfied:
(a) M(UB6gS)> 1-е;
(b) Елеа ЫА\В) -?(?)\ < ? for all B eQ.
Dehnition 7.20. Assume that (?,?,?,?) is a measure-preserving system. A finite
partition ? of X is said to be weakly Bernoulli if for every ? > 0 there exists an m e N such
that for all ? e N the partitions aJJ and <*!,"'_„ are ?-independent.
THEOREM 7.21 (Friedman and Ornstein, 1970). Assume that (?,?,?,?) is an er-
godic measure-preserving system with a weakly Bernoulli generator. Then the system
(?,?,?,?) is isomorphic to any Bernoulli shift of the same entropy.
Remark 7.22. If (?, ?, ?, ?) is a Bernoulli shift, then the time zero partition a of ? is a
generator such that org and <*!,"_„ are independent and in particular also ?-independent for
all m, ? e N. Thus, Bernoulli shifts are weakly Bernoulli, and therefore, Bernoulli shifts

1222
F. Blume
of equal entropy are isomorphic. It can also be shown that the time zero partition of a
strongly mixing Markov shift (see Example 5.5(b)) is weakly Bernoulli so that a strongly
mixing Markov shift is isomorphic to any Bernoulli shift of the same entropy (see Petersen
A983)). Ornstein A970b) also proved a strengthening of Theorem 7.21 by replacing the
assumption of 'weakly Bernoulli' successively by the weaker assumptions of 'very weakly
Bernoulli' and 'finitely determined'. An interesting new proof of Theorem 7.21 (and
its extension to the very weakly Bernoulli case) constructing an isomorphism explicitly
(rather than just establishing the existence of an isomorphism as in the original proof by
Ornstein) was given by Keane and Smorodinsky A979a) (see also Keane and Smorodinsky
A977, 1979b, 1979c)).
Given that the entropy of a partition a is defined as the integral of the information
function /„, it is interesting to examine the convergence behavior of Ia„-\. If (?, ?,?,?)
is a measure-preserving system, then / ,,-iU) can be regarded as a measure for the amount
of information encoded in the и-name a„(x). The next theorem says that in the ergodic case
this information converges to the entropy h(T) a.e. We encounter here again the important
theme of deriving global information about a measure-preserving system from a single
sample path. More precisely, the information encoded in the orbit of a point ? under the
action of ? is almost surely equal to the entropy of T.
THEOREM 7.23 (Shannon-McMillan-Breiman theorem). // (?, ?, ?, ?) is an ergodic
measure-preserving system and a is a countable partition ofX with ? (a) < oo, then
/„-.(*)
lim —2 =h{T) a.e.
n^oc ?
For the original proof of this theorem see Shannon A948), McMillan A953) and
Breiman A957). A proof using the Martingale convergence theorem is given in Petersen
A983) and for a proof in the spirit of Definition 7.12 and Theorem 7.13 see Rudolph A990)
or Shields A996).
We will now discuss briefly some facts related to zero entropy systems. If (?, ?, ?, ?)
is a measure-preserving system and ? is a finite (or countable) partition of X, then the tail
field of a is
-?
j=-oo
A proof of the following theorem is given in Rudolph A990) and Smorodinsky A971).
THEOREM 7.24. Assume that (?, ?, ?, ?) is a measure-preserving system.
(a) If a is a finite partition ofX, then h(a, T) = 0 if and only if а С ??-?,.
(b) If a and ? are finite partitions ofX such that а С ??, then h(T, a) = 0.
Given that there are numerous interesting examples of transformations with entropy
zero (in fact, the set of entropy zero transformations is residual with respect to the weak

Ergodic theory
1223
topology), we may ask whether the concept of entropy can be successfully employed
to study such systems. The most natural approach is here to replace the rate и by a
4-'
slower rate a„ and to examine the behavior of the sequence {?(?^ ')/an}n6pj instead
of {#(orQ-l)/n}„6pj. Unfortunately, it can be shown (see Blume A997)) that for any
aperiodic measure-preserving system (that is for any system for which the set of periodic
points has measure zero) and any positive monotone increasing sequence {a„}„6N with
Hindoo ?,,/и = 0 the supremum over all finite partitions of the limit inferior of the
sequence {#(??~')/?„}„6? is always 00,so that a meaningful invariant directly analogous to
h(T) cannot be obtained in this way. However, if instead of taking the supremum, we look
for lower bounds of entropy convergence rates, then we have the following nontrivial result:
THEOREM 7.25 (Blume, 1998). Assume that ? is a completely ergodic (i.e., Tk is
ergodic for all к e Z) measure-preserving transformation on a Lebesgue probability space
(?,?,?).
(a) There exists a positive concave function g:[l,oo[—» Ш (depending on T) with
/1 g(x)/x dx = 00 such that for all nontrivial finite partitions a of X we have
lim sup ———- > 1.
«^00 g(\og2n)
Note: A slightly weaker, but more general version of this statement for aperiodic
systems was given by Blume A997).
(b) There exists a positive monotone increasing sequence {ап}„ещ (depending on T)
with lim„^oo an = 00, such that for all nontrivial finite partitions a of X we have
liminf v ° ; > 1.
«->¦ oo a„
Another possibility for generalizing the concept of entropy is to consider refinements of
partitions along subsequences. This leads us to the notion of sequence entropy:
Dehnition 7.26. Assume that (?,?,?,?) is a measure-preserving system and let
A = [кп}пещ be a sequence of integers. Then, for all countable partitions ? of X we set
hA(a, r):=limsup vv'~' >-,
n—>зс П
and the sequence entropy of ? is defined as
hA(T):= sup hA(a,T).
[a\H(a)<oo]
Dekking A980) successfully used sequence entropy as an isomorphism invariant to
distinguish between an ergodic transformation and its inverse. The following theorem lists
a number of interesting properties of sequence entropy (see Kushnirenko A967), Newton
A970) and Saleski A977)):

1224
F. Blume
THEOREM 7.27. Assume that (?, ?, ?, ?) is a measure-preserving system.
(a) If {a„}„6pj is an increasing generating sequence of finite partitions ofX, then for all
sequences А С N we have
йд(Г) = lim йд(а„, Г).
и—>oo
(b) If A = {к„]„ем is relatively dense (see Definition 4.2), then
Ид(Т) = h(T)\imsup—.
I!-» ОС П
(c) (X, ?, ?, ?) is strongly mixing if and only if supAcB йл(Г, а) = H(a) for all
positive increasing sequences of integers В and all a with ? (a) < oo.
(d) (?, ?, ?, ?) is weakly mixing if and only if supA йл(Г, a) = ? (a) for all a with
H(a) < oo.
We will now turn to the concept of topological entropy and the variational principle.
We assume that ? is a homeomorphism on a compact metric space X, so that (X,T) is a
cascade. A collection U of nonempty subsets of X is said to be an open cover of X if each
set in U is open and U(y6W^ = %¦ An open cover V is a refinement of U if every element
of V is a subset of an element in U. In this case we write V > U. By N(U) we denote the
minimal cardinality of subcovers of U (which is finite, because X is compact), that is
N{U) :=min{card(W): WcWis an open cover of X},
and we define the entropy oiU as
H(U) := log2 N(U).
The refinement of two open covers U and V is
WvV:=(№: W = U ? V ? 0 for some i/eWandVeV)
As in the case of partitions of a probability space, we denote by U"n the refinements of an
open cover under ?, i.e.,
?
U'^x := T~mU ? · · · ? T~"U = \J T~kU.
k—m
Using the easily verifiable inequality H(UvV)^ H(U) + H(V), it can be shown that the
limit
h(U,T):= lim v ° ;
exists and that h(U, ?) ?, H(U).

Ergodic theory
1225
Dehnition 7.28. If (X, T) is a cascade, then the topological entropy of ? is defined to
be
htop(T):= sup h(U,T).
и
An alternative but equivalent definition of topological entropy that makes explicit use
of the metric structure of X was given by Bowen A971). By contrast, the definition of
topological entropy in Definition 7.28, due to Adler et al. A965), applies also to the case
that X is a compact Hausdorff space. In analogy to Theorem 7.17 the following result is
useful in computing topological entropy:
THEOREM 7.29. If {Ып}п(,ц is an increasing generating sequence of open covers (i.e.,
U„ ^ U„+\ and for every open cover V there exists an ? e N such that V $5 U„), then
htop(T) = lim h(U„,T).
The variational principle as stated below establishes a very important link between
topological entropy and measure-theoretic entropy (see Goodwyn A969, 1972), Dinaburg
A970) and Goodman A971)). It allows us to deduce many properties of topological
entropy from the corresponding properties of measure-theoretic entropy.
THEOREM 7.30 (Variational principle). Assume that (X, T) is a cascade. For a T-
invariant Borel probability measure ? on X (see Remark 3.8), we denote by h?(T) the
measure-theoretic entropy of the measure-preserving transformation ? with respect to ?.
Then
hiop(U) = sup{/?M(r): ? is a T-invariant Borel probability measure on X}.
The last subject in this section is a discussion of the relation between entropy and
Kolmogorov complexity. The definition of the algorithmic complexity of a finite 01-name,
that is a finite sequence in the 'letters' 0 and 1, is based on the concept of Turing machines
(for a definition of Turing machines see Jones A973) or White A991)). For a finite
alphabet A := {1,..., m] we define A* : = U,?L| л" ¦ With the appropriate coding (see
White A991)) a Turing machine ? can be regarded as a function that outputs elements
s e A* on an input r e {0, 1}* via a certain algorithm. The complexity K\j(s) of a name
s e A* with respect to ? is defined to be the minimal length of all 01-names r e {0, 1}*
which produce s when used as input for M. If there is no such input r, then Km(s) := oo.
More precisely, if we denote by l(r) the length of a 01-name s e {0, 1 }*, then
KM(s)=\m{n{l{r)- геГ'М1 ifM-'{i}^0,
loo ifM-'{5}=0.
For any ? e Аг we denote by ?"? the name w(m)w(m + 1)· ·?(?). If (X, T) is a cascade
and U = {U\,..., Um] is a finite open cover of X, then for ? e X we define
<pu(x) := {? e Az: T"x e ?/?(,„ for all neZ]

1226
F. Blume
and we set
Km (?'? )
sup-?"?(jc,6/, T) := limsup min —^—-,
п-юо и>ефц(х) П
in{-KM(x,U,T):=hminf min ?^° '.
п-юс 0;ефи(х) П
Since any universal Turing machine N is asymptotically optimal in the sense that for
every Turing machine ? there exists a constant См such that K^(s) ^ Km(s) + См for
all s e Л* (see White A991)), it follows that for any pair of universal Turing machines
N\ and N2 we have sup-A";v,(*,?/, T) = sup-KN^(x,U,T) and inf-AJv,(-*,?</, T) =
inf-A"yv2 (x ,14, T). This observation allows us to define
sup-K(x,U,T) :=sup-Kn(x,U,T) and
mf-K(x,U, T) := sup-KN(x,U, T),
where N is an arbitrary universal Turing machine. Now we set
sup-K(x,T) :=sup[sup~K(x,U,T): 14 is an open cover of X} and
inf-A"(;c, T) := sup{inf-A"(;c, 14, T): 14 is an open cover of X}.
The following theorem shows that the algorithmic complexity of trajectories in ergodic
measure-preserving systems is almost surely equal to the entropy of the system, so that
once again we are able to derive global information about a system from a single sample
path.
THEOREM 7.31 (Brudno A987), White A991, 1993)). If (?, ?) is a cascade and ? is a
?-invariant Borel probability measure on X, then
sup K(x,T) = inf K(x,T) = hu(T) a.e.
8. Constructions in ergodic theory
Products. If (?,?,?,?) and (Y,C,v,S) are measure-preserving systems, then the
product map ? ? S defined via the equation
? xS(x,y):=(Tx,Sy)
is measure-preserving on the product space (? ? ?, ? <%> С, ? ? ?).

Ergodic theory
1227
Skew products. Assume that (?, ?, ?, ?) is a measure-preserving system and {(Sx, Y,
C, ?): ? e X} is a family of measure-preserving systems such that the map (x,y)h+ Sxy
is measurable from ? ? У to У (with respect to the product measure ? ? ? on ? ? ?).
Then the map
г:ХхУ^ХхУ, (x,y)^(Tx,Sxy),
is a measure-preserving transformation on (? ? ?, В <g> С, ? ? ?). Note: The ergodicity
of skew products has been discussed by Anzai A951) for the special case that ? = G is
a topological group with Haar measure v, and Sx is right-multiplication with an element
f(x) e G. A formula for the entropy of skew products was found by Abramov and Rokhlin
A962): For a finite partition ? of ?, ? e N and ? e X we define
?'1(?) := S-'?? S-'S-'J ? ··· ? S-*S^ ¦ ¦ ¦ S.,^,
oo
?\°(?)·-=\/?"(?) and
M0,S):= f Я(^|^(дс))^.
./?
With these definitions it can be shown that
h(r) = h(T) + sup{hT(fi, S): ? is a finite partition of ?}.
Cutting-and-stacking and infinite interval exchanges. We already encountered one
example of a cutting-and-stacking procedure in Example 5.5(e), but we will now discuss
how in fact almost every measure-preserving system can be constructed in a similar
way. To give an exact definition of a cutting-and-stacking procedure (see Arnoux et al.
A985)) is somewhat cumbersome and we will therefore be content to give an informal
description. At any stage in a cutting-and-stacking procedure, we are given finitely many
towers of finitely many intervals (in the special case of Example 5.5(e) the number of
towers was equal to one at every stage in the process). The intervals in a given tower
have equal length, but the length of the intervals may vary between different towers.
We proceed to cut each of these towers into finitely many subtowers (the number of
subtowers may be different for different towers), add possibly a finite number of spacers
(see Example 5.5(e)) on top of each subtower and stack up subtowers of equal width to
obtain a new collection of finitely many towers. The measure-preserving transformation
corresponding to such a cutting-and-stacking process maps each level in a given tower
onto the next higher level. Furthermore, a cutting and stacking process can always be
arranged in such a way that the union over all stages in the process of all the levels in
the towers is equal to [0, 1[. It can be shown that every aperiodic measure-preserving
system (i.e., the measure of periodic points under ? is equal to zero) is isomorphic to a
measure-preserving transformation on [0, 1 [ (with respect to Lebesgue measure) obtained
from a cutting-and-stacking process as described above. Moreover, it is not difficult to

1228 F.BInme
see that every measure-preserving transformation obtained from a cutting-and-stacking
process can be represented as an infinite interval exchange. Thus we obtain the following
theorem:
THEOREM 8.1 (Arnoux et al., 1985). Any aperiodic measure-preserving transformation
on a probability space is isomorphic to an interval exchange transformation ? on [0, 1 [,
that is a map ? : [0, 1 [ -> [0, 1 [ which satisfies the following properties:
(a) There exists a strictly increasing sequence {t„}„ещ С [0, 1 [ and a sequence {а,,} С К
such that to = 0, lim,,^^ t„ = 1 and T(x) =x + a„ for all ? e /„ := [?„_ ?, t„ [.
(b) Г(/„)с[0, \[forallneN.
(c) ? is bijective.
Adic transformations. An interesting combinatorial approach for representing cutting-
and-stacking procedures (that is aperiodic measure-preserving transformations) has been
introduced by Vershik A989, 1991). The so-called adic transformations are defined on the
set X of all infinite paths in a graph which has the following characteristics:
(a) The graph consists of infinitely many levels with a finite number of vertices in each
level.
(b) The vertices in each level are connected only to vertices in the levels directly above
and below.
(c) In each level the vertices are ordered from left to right.
A ?-algebra on X is naturally generated by the cylinder sets of infinite paths. Furthermore,
we introduce a partial order on X such that two paths
x = (x\,x2,X7,,...) and у = (у\,у2,уз,..-)
are comparable if there is an ? e N such that ? к = у к for all к > п. For any pair
of comparable paths ? and у we say that ? < у if xn < y„ for the largest integer
? for which xn ? yn. For a given nonmaximal path ? we define ? ? to the smallest
path у which is larger than x. In order for ? to represent a cutting and stacking
process, it is necessary to remove from X a countable set of minimal and maximal
elements (for details see also Vershik and Livshits A992)). To illustrate how a cutting-
and-stacking procedure can be represented by an adic transformation, we consider an
example:
Cutting-and-stacking procedure:
~J
~1
t
о
t

Ergodic theory
1229
Equivalent adic transformation:
The measure of a cylinder set С = {x e X \ x\ = a\,...,x„ = an} is here 4r·
Furthermore, the 2" cylinder sets of the form {x e ? | ?\ = a\,..., x„ = a,,} represent
the 2" levels obtained in the «th step of the cutting-and-stacking procedure.
Rokhlin towers. The usefulness of the so-called Rokhlin tower decomposition for the
efficient formulation of proofs and as a basis for various constructions in ergodic theory
depends on the following theorem:
THEOREM 8.2 (Rokhlin lemma (Rokhlin, 1948)). If (?, ?, ?, ?) is an aperiodic measure-
preserving system, then for any ? > 0 and any ? e N there is a set A e В such that the sets
А, ТА,..., Г"-1 A are pairwise disjoint and the measure of their union is greater than
1-?.
Remark 8.3. The Rokhlin Lemma is a consequence of the more general Backward Vitali
Lemma (see Rudolph A990)) which is useful for example in the proof of the pointwise
ergodic theorem and in the proof of Theorem 7.13.
Induced transformations. If (?,?, ?, ?) is a measure-preserving system and A e В with
? (A) > 0, then according to the Poincare Recurrence Theorem, for almost all ? e A there
exists an ? e N such that T"x e A. Thus, идС*) := ???{? e ?: ?"? e A] is well defined
for almost all ? e A and this allows us to define a map Та on A (up to a set of measure
zero) via the equation
ТАх:=Т"л(х)х.
It is easy to see that with дд := ?/?(?) and Ba ¦= {В П А \ В е В} the system
(А, Вл, мл, Та) is measure-preserving. The entropy of an induced transformation is given
by the formula
h(T)
h(TA)= -^ (Abramov, 1959).
?(?)
For a given measure-preserving system, it is possible to topologize the space ?4? '¦=
{Та: A e ?, ?{?) > 0} of all induced transformations via the metric d(TA,Te) : =
?(???). Del Junco and Rudolph A996) showed that for an ergodic measure-preserving

1230
F. Blume
system (?,?,?, ?) the set of all weakly mixing induced transformations is residual in
Mt and if (?, ?, ?, ?) has positive entropy, then the set of induced A'-automorphisms is
residual in Mt as well.
Factors. If (?,?,?,?) and (Y, C, v, S) are measure-preserving systems, then a map
0 : X —» У is called a homomorphism or factor map if
(a) ф-^СсВ,
(b) фТ = Бф а.е., and
(c) ?@-'?) = v(A)forall AeC.
If ? is a factor map, then (Y,C, v, S) is said to be a factor of (?, ?, ?, ?). Factor maps
play a prominent role in the theory of joinings (see below) and are therefore also useful
in Ornstein theory, because as shown in Rudolph A990) joinings can be used to prove
Ornstein's isomorphism theorem.
Joinings and disjointness. If X = (?,?,?,?) and ? = (Y,C,v,S) are measure-
preserving systems, then a ? ? S'-invariant complete probability measure ? on ? ? У
is said to be a joining of X and ? if for all A e В and В е С we have
(a) A x В is measurable with respect to ?, and
(b) ?(? ? ?) = ?(?) and ?(? ? ?) = ?(?).
To discuss the connection between joinings and factor maps, let us assume that X and ?
are factors of a measure-preserving system (?,?,?,??) with factor maps ?: ? -> X and
? : ? -> У. If we restrict ? to the smallest ?-algebra containing ?~] ?? ?~]0, then the
countably additive set function
?(? ? ?):=?(?-^??)?-^?)
has a unique extension to a joining of X and ? (see Rudolph A990)).
On the convex space J(X, ?) of all joinings of X and ? (convexity means here that
pk\ + A - ?)?2 e J(X, ?) for all ?,, ?2 e J(X, ?) and all pe[0,l])a metric d can be
defined (see Rudolph A990)) such that
(a) (J(X, Y), d) is compact, and
(b) for any sequence {?,,} с J(X, ?) and any ? e J(X, Y) we have
lim <*(?„,?) = 0
n—>oo
if and only if \??\„^??„(? ? ?)=?(? ? ?) for all A e В and В е С.
THEOREM 8.4. IfX = (?, ?, ?, ?) and ? = (?, С, ?, 5") are ergodic measure-preserving
systems, then ? is an ergodic joining of X and ? if and only if ? is an extreme point of
J(X,Y) (see Rudolph A990)).
The space J(X, Y) is never empty, because we always have ? ? ? e J (?, ?). If ? ? ?
is the only joining of X and ?, then we say that X and ? are disjoint. The concept
of disjointness allows for the following interesting characterizations of ergodicity, weak
mixing and the A"-property:

Ergodic theory 1231
THEOREM 8.5. Assume that X = (?, ?, ?, ?) is a measure-preserving system.
(a) X is ergodic if and only if X is disjoint from any identity map on a Lebesgue space.
(b) X is weakly mixing if and only if X is disjoint from all isometries on compact metric
spaces.
(c) ? is а К-automorphism if and only if X is disjoint from any ergodic system
(Y, C, v, 5") for which h(S) = 0.
For the case of self-joinings, that is joinings of a system with itself, the space J(X, X)
contains always (in addition to the product measure ? ? ?) the diagonal joining
?0(? ? ?):=?(???)
and the off-diagonal joinings
?„(?? ?):=?(???"?).
(Note: The study of self-joinings was initiated by Rudolph A979).) We say that X has
two-fold minimal self-joinings if У (X, X) is the convex hull of {? ? ?) U {?„}„6?.
THEOREM 8.6. If ? = (?,?,?,?) is completely ergodic (i.e., T" is ergodic for all
? e Z) and has two-fold minimal self-joinings then ? commutes only with its powers T"
and X has no nontrivial factors.
In generalizing the concept of self-joinings we may also consider multiple self-joinings
and in particular off-diagonal joinings supported on graphs of the form {(T"]x,..., T"kx) \
? & X]. Partitioning the index set {1,..., k] into disjoint subsets, each set in the partition
determines itself an off-diagonal measure, and the directproduct of the resulting measures
is called a product of off-diagonal joinings. We say that X has к-fold minimal self-joinings
if all k-fo\d multiple self-joinings of X are in the convex hull of the products of off-
diagonal joinings. Del Junco et al. A980) showed that the Chacon transformation as
defined in Example 5.5(e) has minimal self-joinings of all orders. Given the statement
of Theorem 8.6, this implies that the Chacon transformation has no nontrivial factors.
Furthermore, Glasner et al. A992) showed that any system with 3-fold minimal self
joinings has minimal self-joinings of all orders. For proofs of the results in this section,
applications of joinings to Ornstein theory and further references, the reader is referred to
Rudolph A990) and Thouvenot A995).
References
Abramov. L.M. A959). Entropy of induced automorphisms. Dokl. Akad. Nauk SSSR 128. 647-50.
Abramov, L.M. and Rokhlin, V.A. A962), The entropy of a skew product of measure-preserving transformations.
Vestnik Leningrad Univ., No. 17, 5-13. Amer. Math. Soc. Transl. Ser. 2. Vol. 48 A965), 225-65.
Adler, R.L., Kohnheim, A.G. and McAndrew. M.H. A965). Topological entropy. Trans. Amer. Math. Soc. 114,
309-19.
Anzai, H. A951), Ergodic skew product transformations on the torus. Osaka Math. J. 3. 83-99.

1232
F. Blume
Arnoux, P., Omstein, D.S. and Weiss. B. A985). Cutting and stacking, interval exchanges and geometric models,
Israel J. Math. 50, 160-168.
Assani, I. A993), The helical transform and the a.e. convergence of Fourier series, Illinois J. Math. 37, 123-146.
Assani, I. and Petersen, K. A992), The helical transform as a connection between ergodic theory and harmonic
analysis. Trans. Amer. Math. Soc. 331, 131-142.
Bergelson, V. A987), Polynomial ergodic theorems, Ergodic Theory Dynam. Systems 7. 337-349.
Bergelson, V., Furstenberg, H. and McCutcheon, R. A996). IP-sets and polynomial recurrence, Ergodic Theory
Dynam. Systems 16, 963-974.
Bergelson, V., Komfeld, I. and Mityagin, B. A993), Unitary I}1-actions with continuous spectrum. Proc. Amer.
Math. Soc. 119, 1127-1134.
Bergelson. V. and Leibman, A. A996). Polynomial extensions of van der Waerden's and Szemeredi's theorems.
J. Amer. Math. Soc. 9. 725-753.
Birkhoff. G.D. A931). Proof of the ergodic theorem, Proc. Nat. Acad. Sci. USA 17. 656-660.
Blum. J.R. and Hanson, D.L. A960). On the mean ergodic theorem for subsequences. Bull. Amer. Math. Soc. 66.
308-311.
Blume. F. A997), Possible rates of entropy convergence. Ergodic Theory Dynam. Systems. 17. 45-70.
Blume, F. A998), Minimal Rates of entropy convergence for completely ergodic systems. Israel J. Math. 108,
1-12.
Bourgain. J. A988a). On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61.
39-72.
Bourgain, J. A988b). On the pointwise ergodic theorem on L1' for arithmetic sets. Israel J. Math. 61. 73-84.
Bourgain. J. A989). Pointwise ergodic theorems for arithmetic sets (Appendix: The return time theorem). Publ.
Math. IHES 69. 5^45.
Bowen, R. A971), Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153,
401^414.
Breiman, L. A957). The individual ergodic theorem of information theory, Ann. Math. Stat. 28. 809-11.
Correction, ibid. 31 A960). 809-810.
Brudno. A. A. A987), Entropy and the complexity of the trajectories of a dynamical system. Trans. Moscow Math.
Soc. 2, 127-151.
Campbell, J.T. and Petersen. K. A989). The spectral measure and Hilbert transform of a measure-preserving
transformation. Trans. Amer. Math. Soc. 313. 121-130.
Chacon. R.V. A969). Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22,
559-562.
Chacon, R.V. and Omstein. D.S. A960). A general ergodic theorem, Illinois J. Math. 4. 153-160.
Choe, G.H. A993), Weakly mixing interval exchange transformations. Math. Japon. 38 D). 727-734.
Conze, J.-P. and Lesigne, E. A984), Theoremes ergodique pour des mesures diagonales, Bull. Soc. Math. France
112, 143-175.
Conze, J.-P. and Lesigne, E. A988), Sur un theoreme ergodique pour des mesures diagonales. С R. Acad. Sci.
Paris, Ser. 1306, 491^493.
Comfeld, I.P., Fomin, S.V. and Sinai. Y.G. A982), Ergodic Theory. Springer, New York.
Cotlar. M. A955). Л unified theory of Hilbert transforms and ergodic theorems. Rev. Mat. Cuyana 1. 105-167.
Dekking F.M. A980), Some examples of sequence entropy as an isomorphism invariant. Trans. Amer. Math. Soc.
259A). 167-183.
Dinaburg. E.I. A970). The relation between topological entropy and metric entropy. Dokl. Akad. Nauk SSSR
190. 19-22. Soviet Math. Dokl. 11. 13-16.
Ferenczi. S. A990), Systemes de rangfini. These de doctorat. Universite d'Aix-Marseille.
Friedman. N.A. A970). Introduction to Ergodic Theory. Van Nostrand Reinhold Studies. No. 29. New York.
Friedman, N.A. and Omstein. D.S. A970). On isomorphisms of weak Bernoulli transformations. Adv. Math. 5.
365-394.
Furstenberg. H. A960). Stationary Processes and Prediction Theory. Ann. Math. Studies, Vol. 44. Princeton
University Press, Princeton. NJ.
Furstenberg. H. A977), Ergodic behavior of diagonal measures and a theorem of Szemere'di on arithmetic
progressions. J. Anal. Math. 31. 204-256.

Ergodic theory
1233
Furstenberg. H. A981). Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University
Press. Princeton, NJ.
Furstenberg. H. and Katznelson. Y. A978). An ergodic Szemeredi theorem for commuting transformations,
J. Anal. Math. 34. 275-291.
Furstenberg. H. and Katznelson. Y. A985). An ergodic Szemeredi theorem for IP-systems and combinatorial
theory, J. Anal. Math. 45. 117-168.
Furstenberg. H. and Katznelson, ? A986). ??,-sets, Szemeredi's theorem and Ramsey theory. Bull. Amer. Math.
Soc. 14 B), 275-278.
Furstenberg. H. and Katznelson. ? A990), Nonconventional ergodic averages. The Legacy of John von
Neumann. Proc. Sympos. Pure Math.. Vol. 50. Amer. Math. Soc.. Providence. Rl. 43-56.
Furstenberg. H. and Katznelson, ? A991), ? density version of the Hales-Jewett theorem, J. Anal. Math. 57 B),
64-119.
Furstenberg, H. and Weiss, B. A996). A mean ergodic theorem for jj ?^=? f(T"x)g(T"~ ?). Convergence in
Ergodic Theory and Probability, Ohio State Univ. Math. Res. Inst. Publ.. Vol. 5. de Gruyter, Berlin, 193-227.
Garsia, A.M. A965), A simple proof of E. Hopf's maximal ergodic theorem, J. Math. Mech. 14. 381-382.
Glasner, E., Host, B. and Rudolph, D.J. A992). Simple systems and their higher order self-joinings. Israel J.
Math. 78, 131-142.
Glasner, E. and King, J.L. A998), A zero-one law for dynamical properties. Contemp. Math. 215, 231-242.
Goodman, T.N.T. A971), Relating topological entmpy and measure entmpy. Bull. London Math. Soc. 3, 176—
180.
Goodwyn, L.W. A969), Topological entmpy bounds measure-theoretic entmpy, Proc. Amer. Math. Soc. 23. 679-
688.
Goodwyn, L.W. A972), Comparing topological entmps with measure-theoretic entmpy, Amer. J. Math. 94, 366-
388.
Hahn, F.J. A965), Skew pmduct transformations and the algebras generated by exp(p(»)). Illinois J. Math. 9,
178-189.
Halmos, PR. A944), In general a measure-preserving transformation is mixing, Ann. Math. 45, 786-792.
Halmos, PR. and von Neumann, J. A942), Operator methods in classical mechanics II, Ann. Math. 43, 332-350.
Halmos, PR. and von Neumann, J. A958), Lectures on Ergodic Theory, Chelsea, New York.
Hopf, E. A954), The general temporally discrete Markov pmcess, J. Rat. Mech. Anal. 3, 13-45.
Host, B. A991), Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel
J. Math. 76, 289-298.
Jewett, R.I. A970), The prevalence of uniquely ergodic systems, J. Math. Mech. 19, 717-729.
Jones, N.D. A973), Computability Theory: An Intmduction, Academic Press,
del Junco, ?., Rahe, A.M. and Swanson, L. A980), Chacon's automorphism has minimal self-joinings, J. Anal.
Math. 37, 276-284.
del Junco, A. and Rudolph, D. A996). Residual behavior of induced maps. Israel J. Math. 93, 387-398.
Kalikov, S.A. A984), Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory
Dynam. Systems 4, 237-259.
Katok, A. and Hasselblatt, B. A995), Intmduction to the Modern Theory of Dynamical Systems. Cambridge
University Press, Cambridge.
Keane, M. and Smorodinsky. M. A977). A class offinitary codes, Israel J. Math. 26, 352-371.
Keane, M. and Smorodinsky, M. A979a), Bernoulli schemes of the same entmpy arefinitarily isomorphic, Ann.
Math. 109, 397^406.
Keane, M. and Smorodinsky, M. A979b). Finitary isomorphism of irreducible Markov shifts, Israel J. Math. 34,
281-286.
Keane, M. and Smorodinsky, M. A979c), The finitary isomorphism theorem for Markov shifts. Bull. Amer. Math.
Soc. 1, 436-438.
Khintchine, A.I. A934), Eine Verscharfung des Poincareschen "Wiederkehrsatzes". Сотр. Math. 1, 177-179.
Kingman, J.F.C. A968), The ergodic theory of subadditive stochastic pmcesses, J. R. Stat. Soc. Ser. В 30. 499-
510.
Kingman, J.F.C. A973), Subadditive ergodic theory, Ann. Probab. 1, 883-909.
Kolmogorov, A.N. A953), On dynamical systems with an integral invariant on the torus, Dokl. Akad. Nauk SSSR
93, 763-766.

1234
F. Blume
Kolmogorov, A.N. A958), New metric invariants of transitive dynamical systems and automorphisms ofLebesgue
spaces, Dokl. Akad. Nauk SSSR 119, 861-864.
Krengel, U. A972), Weakly wandering vectors and weakly independent partitions, Trans. Amer. Math. Soc. 164,
199-206.
Krieger, W. A970), On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc.
149, 453^464.
Krieger, W. A972), On unique ergodicity, Proc. Sixth Berkeley Symp. A970), Vol. 1, University of California
Press, Berkeley and Los Angeles, 327-346.
Kryloff, N. and Bogoliouboff, N. A937), La theorie generate de la mesure dans son application a I'etude des
systemes dynamiques de la mecanique поп Imeaire, Ann. Math. 38 A), 65-113.
Kushnirenko, A.G. A967), On metric invariants of entropy type. Uspekhi Mat. Nauk 22. 57-65, Russian Math.
Surveys 22 A967), 53-61.
Lacey, M., Petersen, K., Rudolph, D. and Wierdl, M. A994), Random ergodic theorems with universally
representative sequences, Ann. Inst. H. Poincare Probab. Statist. 30 C), 353-395.
McMillan, B. A953), The basic theorems of information theory, Ann. Math. Stat. 24, 196-219.
von Neumann, J. A932a), Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18, 70-82.
von Neumann, J. A932b), Zur Operatorenmethode in der klassischen Mechanik, Ann. Math. 33, 587-642.
Newton, D. A970), On sequence entropy 1. Math. Systems Theory 4 B), 119-125.
Ornstein, D.S. A970a), Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4, 337-352.
Ornstein, D.S. A970b), Imbedding Bernoulli shifts inflows. Contributions to Ergodic Theory and Probability,
Lecture Notes in Math., Vol. 160. Springer. New York, 178-218.
Ornstein, D.S. A970c). Two Bernoulli shifts with infinite entropy are isomorphic, Adv. Math. 5, 339-348.
Ornstein, D.S. A972). On the root problem in ergodic theory, Proc. Sixth Berkeley Symposium A970), University
of California Press, Berkeley and Los Angeles. 345-56.
Parry, W. A981), Topics in Ergodic Theory, Cambridge University Press, Cambridge.
Petersen, K.E. A970), A topologically strongh mixing symbolic minimal set, Trans. Amer. Math. Soc. 148, 603-
612.
Petersen, K.E. A979), The converse of the dominated ergodic theorem. J. Math. Anal. Appl. 67, 431^436.
Petersen, K.E. A983), Ergodic Theory, Cambridge University Press, Cambridge.
Petersen, K.E. A996), Ergodic theory and the basis of science, Synthese 108, 171-183.
Petersen, K.E. and Salama, I. A995), Ergodic Theory and its Connections with Harmonic Analysis, Cambridge
University Press, Cambridge.
Poincare, H. A899), Les methodes nouvelles de la mecanique celeste, I A892), II A893) and III A899), Gauthier-
Villars, Paris and Dover, New York, 1957.
Postnikov, A.G. A966), Ergodic Problems in the Theory of Congruences and of Diophantine Approximations, Tr.
Mat. Instit. Steklova 82, 3-112. Proc. Steklov Institute of Math. 82, Amer. Math. Soc., Providence, RI, 1967.
Renyi, A. A958), On mixing sequences of sets. Acta Math. Acad. Sci. Hungar. 9, 215-228.
Rokhlin, V.A. A948), A 'general' measure-preserving transformation is not mixing, Dokl. Akad. Nauk SSSR 60,
349-351.
Rokhlin, V.A. A967), Lectures on the theory of entropy of transformations with invariant measure, Uspekhi Mat.
Nauk 22, 3-56, Russ. Math. Surveys 22, 1-52.
Rokhlin, V.A. and Sinai, Ya.G. A961), Construction and properties of invariant measurable partitions, Dokl.
Akad. Nauk SSSR 141, 1038-1041, Sov. Math. 2A966), 1611-1614.
Rosenblatt, J. and Wierdl, M. A995), Pointwise ergodic theorems via harmonic analysis, Ergodic Theory and its
Connections with Harmonic Analysis, London Math. Soc. Lecture Note Ser., Vol. 205, Cambridge University
Press, Cambridge, 3-151.
Rudin, W. A987), Real & Complex Analysis, McGraw-Hill, New York.
Rudolph, DJ. A979), An example of a measure-preserving map with minimal self-joinings and applications,
J. Anal. Math. 35,97-122.
Rudolph, D.J. A990), Fundamentals of Measurable Dynamics, Oxford University Press, New York.
Ryzhikov, V.V. A993), Joinings and multiple mixing of finite rank actions, Funktsional. Anal, i Prilozhen. 27 B),
63-78.
Saleski, A. A977), Sequence entropy and mixing, J, Math. Anal. Appl. 60, 58-66.
Shannon, C.E. A948), A mathematical theory of communication, Bell System Tech. J. 27, 379^423, 623-656.

Ergodic theory
1235
Shields, P.C. A996), The Ergodic Theory of Discrete Sample Paths, Graduate Studies in Math., Vol. 13, Amer.
Math. Soc.
Shiryayev, A.N. A984), Probability, Springer, New York.
Sinai, YG. A959), Flows with finite entropy, Dokl. Akad. Nauk SSSR 125, 1200-1202.
Smorodinsky, M. A971), Ergodic Theory; Entropy, Lecture Notes in Math., Vol. 214, Springer, New York.
Szemeredi, E. A975), On sets of integers containing no к elements in arithmetic progression. Acta Anth. 27,
199-245.
Thouvenot, J.-P. A995), Some properties and applications of joinings in ergodic theory, Ergodic Theory and its
Connections with Harmonic Analysis, Cambridge University Press, Cambridge.
Ulam, S.M. and von Neumann, J. A945), Random ergodic theorems. Bull. Amer. Math. Soc. 51, 660.
Varga, R.S. A962), Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Vershik A. A989), A new model for ergodic transformations, Banach Center Publ., Vol. 23, 381-384.
Vershik A. A991), The Fibadic expansion of real numbers and adic transformation. Report no. 4, 1991/92, Inst.
Mittag-Lefler.
Vershik, A.M. and Livshits, A.N. A992), Adic models of ergodic transformations, spectral theory, substitutions
and related topics, Adv. Soviet Math. 9, 185-204.
van der Waerden, B.L. A927), Beweis einer Baudet'schen Vermutung, Nieuw. Arch. Wisk. 15, 212-216.
Walters, P. A975), Ergodic Theory - Introductory Lectures, Lecture Notes in Math., Vol. 458, Springer, Berlin.
Weyl, H. A910), Uber die Gibbs'sche Erscheinung und verwandte Konvergenzphanomene, Rend. Circ. Mat.
Palermo 330, 337^407.
Weyl, H. A916), Uber die Gleichverteilung von Zahlen mod. eins. Math. Ann. 77, 313-352.
White, H.S. A991), On the Algorithmic Complexity of the Trajectories of Points in Dynamical Systems, Doctoral
Dissertation, University of North Carolina at Chapel Hill.
White, H.S. A993), Algorithmic complexity of points in dynamical systems, Ergodic Theory Dynam. Systems 13,
807-830.
Wiener, N. A939), The ergodic theorem, Duke Math. J. 5, 1-18.
Wiener, N. and Wintner, A. A941), Harmonic analysis and ergodic theory, Amer. J. Math. 63, 415^426.
Yosida, K. and Kakutani, S. A939), Birkhoff's ergodic theorem and the maximal ergodic theorem, Proc. Japan
Acad. 15, 165-168.

CHAPTER 30
Generalized Derivatives*
Endre Pap
Institute of Mathematics, University of Novi Sad, Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia
E-mail: pape@eunet.yu, pap@im.ns.ac.yu
Arpad Takaci
Institute of Mathematics, University of Novi Sad, Trg D. Obradovica 4,21000 Novi Sad, Yugoslavia
E-mail: takacsi@eunet.yu, takaci@im.ns.ac.yu
Contents
Introduction 1239
1. Sobolev spaces and derivatives 1239
1.1. Introduction 1239
1.2. Sobolev spaces 1242
1.3. Imbedding theorems and traces of functions '244
2. Distributional derivatives 1246
2.1. Introduction 1246
2.2. Distribution spaces 1248
2.3. Fundamental solutions '251
3. The Mikusinski operational calculus '254
3.1. Introduction 1254
3.2. Operators 1254
References 1259
*The authors want to thank for the partial financial support of the Project in the Fields of Basic Research
"Mathematical models of nonlinearity, uncertainty and decision" A866) supported by the Ministry of Science,
Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1237

This page intentionally left blank

Generalized derivatives
1239
Introduction
Many problems with partial differential equations (PDEs) appearing in applications
cannot have classical solutions, and this fact forced research throughout the twentieth century
on new concepts of the solutions and their differentiability. At the same time, the
classical spaces of functions became insufficient and inadequate for such new, "generalized"
solutions. These new theories developed in the thirties of the last century, mainly in the
works of S.L. Sobolev and K. Friedrichs, and one can say that they have been intensively
developing ever since.
In this chapter we shall expose three types of generalized derivatives, namely that of
Sobolev type, distributional one, and the one appearing in the Mikusinski operational
calculus. Of course, some elements from the theory of Sobolev and distribution spaces, as
well as the construction of the Mikusinski operator field, will also be given. Characteristic
applications to PDEs will be demonstrated.
1. Sobolev spaces and derivatives
1.1. Introduction
The notion of generalized derivative, closely connected to the notion of generalized
solution of differential equations, was introduced in the thirties by Sobolev A935, 1936),
Friedrichs A934), though some ideas in that direction appeared much earlier, e.g., in
B. Levi's paper from 1906, In order to give an idea how to introduce a generalized
derivative of a (say) locally integrable function, let us start with a partial integration. If
/ and ? are continuously differentiable complex valued functions on E, the latter with
compact support, i.e., ?{?) = 0 outside some closed bounded interval [—?, ?], then it holds
f f'(x)^(x~)dx = ? Г(хЩх~)ах = /(?)??\"_?- ? f{x)V<S)dx
JR J-a J-a
= - f f{x)V<7)dx. A)
Recall that the support of a continuous function ?:?->?, where ? с К" is a region,
denoted by supp<p, is the closed set defined by
supp<p = {x ? ?: ?(?) ^0}.
In view of the usual scalar product (·, ·) of complex valued functions from i--(R), given
by
(/?, /2) := f /? (x)Mx)dx, /?, /2 e L2(R),
JR
the left- and the right-hand side of A) can be written as
(/'.*>) = -(/,*>'). B)

1240
?. Pap and A. Takaci
Now if / belongs to the Hubert space L2(R), then it might not have a derivative (in the
usual sense) at each point ? e R. However, there might exist a function g e L~(R), such
that for each ? e C0' (R) (the vector space of continuously differentiable functions with
compact support) it holds
(*.*>) = -(/,*>')· C)
Then g e L2(R) from C) is the generalized first derivative (in the Sobolev sense) of
/eL2(R).
Example 1.1. The function /, given by,
0 for ? < 0,
? for ? ^ 0,
has no (classical) derivative at the point ? — 0. However, the Heaviside function #, given
by
? 0 for ? < 0,
A for jc > 0,
is the generalized derivative on R of the function /.
In order to show this, note that for an arbitrary but fixed ? e Cq (R), there exists an R > 0
such that ?(?) = 0 for ? ^ R, implying ?'(?) = 0 for every ? ^ R. Therefore
/¦+OC rR „ /¦«
(/>')=/ f(x)<p'(x)dx= ??'(?)??=??(?)\0- \-?(?)??
J-oc J0 JO
(R
= - / ?{?)??.
Jo
On the other hand, we have
/+oc ^ К
?{?)?{?)?? = ? ?(?)??.
-oo ^0
The preceding two equalities imply that for every ? e C,
1
(/(jt)i!>'W + i/W*>W)^ = 0,
which means that the generalized derivative of the function / exists on R, and is equal
to H.
Further on, ? will always denote a region, i.e., a connected open set in R", and a will
denote an и-tuple (ot\, or,,) e NQ!; by definition, we put |a| := a\ +aj-\ +<*„.

Generalized derivatives
1241
Recall that C*(i2), к e No, is the vector space of functions ?: ? -> С such that ?(?),
0 ^ |?| ^ ?, is a continuous function on ?, while ?^(?) is a proper subset of &{?),
whose elements ? have their supports contained in a compact set К с ? (depending on ?).
Analogously to B), the ath generalized derivative (in the Sobolev sense) of / e L2(i2),
for a = (??,..., an) e NQ! and ? a region in R", is defined as follows.
????????? 1.2. The function f(a) e L2(i2) is the ath generalized derivative (of order
|or|) on the region ? of a function / e L2(i2) if for every function ? e C(, (?) it holds
f /WD°^j^ = (-l)^ f /«"(*M^.
Together with the notation /@,) for generalized derivative, one can use also the usual
notations for classical derivatives, such as Da f or Э^/ДЭ^ ¦ ¦ ¦ dxna"), provided one
distinguishes the two derivatives. Of course, if / e С1 (?), then the generalized derivative
/(o?) and the corresponding classical one, Da f, are equal.
Whenever the ath generalized derivative fia) of a function / e L2(i2) exists, then it is
unique and is independent of the order of differentiation. It also holds that if all first order
generalized derivatives of a function / e L2(i2) exist and are zero, then this function is
equal to a constant almost everywhere. Note that not every function from L-(i2) has a
generalized derivative in the sense of Definition 1.2. For instance, the function f{x\, xi) =
sgn*| does not have the first order generalized derivative /"*' for a = A,0), i.e., the
generalized partial derivative in jc ?, on the unit disc 6@, l) = (j: = (xi,j:i)el2; |*| < 1}.
Next we give another "counterexample".
Example 1.3. The function /, defined on the unit disc ?@, 1) by
Ia for ? ? + jc2 < 1, ?-> > 0,
b for JC]" + JC, < 1, ?2 < 0,
has first order generalized derivatives in jci both on the upper and on the lower open
semidisc (i.e., for X2 > 0 and for xi < 0), but for ? ? b it does not have a generalized
derivative, with respect to jc2, on the whole unit disc.
It is a well known fact that whenever a function / e L2(i2) has a classical derivative
Da f on ?, ? = (??,..., a„), then it certainly has every lower order derivative D^ f,
where ? = (?\,..., ??) and /S/ ^ aj, 1 ^ j ^ ?. However, the following example shows
that things are different with the generalized derivative in the sense of Definition 1.2, giving
us an essential difference between the classical and generalized derivatives.
Example 1.4. The function /: ?@, 1) -> ?, defined by
/Cx) = sgn.xi + sgn*2,

1242
?. Pap and A. Takaci
does not have first order generalized derivatives df/dx\ and df/dx2 on 6@, 1). However,
it does have the second order mixed generalized derivative 9 f /{dx\dxi) on 6@, 1); in
fact, we prove next that the last generalized derivative on ?@, 1) exists and is equal to 0.
To that end, we have for every ? e Cq(S@, 1))
\ /(¦*)„ „ dx
JB@.\) дХ\дХ2
dx
= I sgn*i dx + I sgn*->-—-—dx
JB@,l) дх\дх2 JB@,\) дх\дх2
JB@.l)n|.r|.r,<0| дХ[дХ2 7?@.?)?|.?|.?|>0| дх\дХ2
-[ ^dx+( ^dx
J В@.\)П[.х\х2<0\ дх\дХ2 JB@. |)П|.г|.г><0| дх\дХ2
= 0 = / 0-?(?)??.
?@.|)
A necessary and sufficient condition for a function / e L2]a, b[ to have a generalized
derivative is that it is absolutely continuous on ]a, b[ and, moreover, that f'eL ]a, b[. The
last supposition of the existence of the derivative /' does not imply the absolute continuity
of the function /. For example, the function /, given by
ijt2sin;t-2 for0<;t<l,
10 for ? = 0,
has a derivative on the interval [0, 1], but /'? L']0, 1[ and so /' i L2]0, 1[. Note that the
fact that a function / has a generalized derivative /' e L' ]0, 1 [, still does not always imply
the absolute continuity of /.
In the upper analysis, instead of L2(Q), one can take the (larger) space L|-oc(i2), whose
elements are square integrable on every compact set contained in ?.
1.2. Sobolev spaces
As before, ? will always denote a region in Ш", i.e., a connected open set in Ш".
Dehnition 1.5. Let ? be a non-negative integer. The Sobolev space Wk№) consists
of all functions / e L2(i2) having a generalized derivative /@° e L2(Q) for every a,
0<:\a\<:k.
Note that for к = 0 we simply have \?°(?) = L2(Q).

Generalized derivatives 1243
The scalar product on ??<(?), defined by
(f\g)w4v)--= ? (/""? = ? / f'w'g^Wdx,
induces the norm
1/2
jJfa\x)\Jdx]
\/\\*4?) = (? Jjfia)w\2dx)
which turns W*(i2) into a separable Hubert space.
Next we define another class of spaces; historically, it was an alternative (though
equivalent!) approach to Sobolev spaces.
Dehnition 1.6. Let A: be a non-negative integer. The space ??<(?) is the completion of
'(?)·
the space C*(i2) in the norm || · || wk,ny
In other words, ??<(?) is the set of all limit points ofCauchy sequences in C*(i2) in the
norm || · ?1??*(?)· Surprisingly enough, only in 1964 it was proved by Meyers and Serrin,
see Adams A975),
??<(?) = ??<(?).
This important result can be stated also as follows.
THEOREM 1.7. If ? is a bounded region, then
C*(i2) = W*(i2), ik=0, 1,2,...,
where the closure ?/??<(?) is taken with respect to the norm \\ ¦ || Wk,?).
Slightly changing Definition 1.6 by replacing the space C*(i2) with its subspace
C?°(i2), whose elements are infinitely differentiable functions on ? having compact
supports contained in ?, we come to the following definition.
Dehnition 1.8. Let ? be a non-negative integer. The Sobolev space Wk№) is defined
by
??(?) = ^(?) (* = 0,1,2,...),
where the closure of C^°(i2) is taken with respect to the norm || · II iv*tr?)·

1244
?. Pap and A. Takaci
One can prove that the space Wk(Rn) is equal to Wk(R"). In general, if R" is replaced
with an arbitrary region, then this is not always true. It turns out that the segment property
of a region is a sufficient condition for the equality
Wk(Q) = Wk(Q).
Recall that a region ? has the segment property if for every ? from the boundary of ?
there exists a neighborhood Ux of ? and (another) element yv e ? such that for every
? e ? П Ux the element ? + tyx belongs to ?.
Intuitively, if a continuously differentiable function / on ? equals zero on the boundary
of a bounded region ?, then it should belong to the space W1 (?). In fact, this is true if the
region ? is a starshaped locally quadratic bounded region. Recall, a bounded region ? is
starshaped if there exists jco e ? such that for every ? > 1 the set {л:: xq + (x — xo)/k e ?)
is a subset of ?.
An important generalization of the Hubert space Wk (?) is the Banach space ??<·?(?),
for ? ^ 1, obtained by replacing the space L2(i2) and its usual norm in Definition 1.5 by
Lp(i2) and its norm. If, moreover, ? = [?] + q, for ? e N and q e ]0, 1[, then \??<?(?) is
the so-called Sobolev-Slobodeckij space. Using the Fourier transformation, one can extend
the space ??<(?) for negative values of k. For such generalizations, and its application to
PDEs, one can consult, e.g., Adams A975), Nikol'skii A975), Triebel A978), Hormander
A983-85), Wloka A987).
1.3. Imbedding theorems and traces of functions
When seeking the solutions of PDEs, one often firstly finds generalized solutions; of
course, whenever possible it is desirable to replace them by classical solutions, i.e., to
have a certain regularity of solutions. The statements giving relations between Sobolev
spaces and (more classical) spaces of continuously differentiable functions are usually
called imbedding theorems. In fact, from the very first theorems of this type, obtained by
S.L. Sobolev himself in the thirties, it became clear that they are well suited for the study
of PDEs and applications to problems in mathematical physics.
THEOREM 1.9 (The Sobolev lemma). Let ? С К" be a bounded region with the boundary
9? belonging to the class Ck. If к > s + и/2, s e No, and f e W*(i2), then there is a
function g eCs(i?) such that f = g almost everywhere.
One of the consequences of Theorem 1.9 is that, if ? = 1, then Wk+i ]a, b[ с Ck[a,b]
for every к е No, and, moreover, the imbedding is continuous.
The mutual relation between Sobolev spaces of different order is given in the following,
Rellich's, theorem, see Pap et al. A997).
THEOREM 1.10 (The Rellich lemma). // ? is a bounded region of R" and m > к
(m,k e No), then the imbedding map of the space ?'"(?) into the space ??<(?) is a
compact operator.

Generalized derivatives
1245
In general, compact mappings, in particular compact imbeddings, have numerous
important applications in analysis, e.g., in the analysis of the spectra of linear elliptic partial
differential operators over bounded domains.
Let / e W*(#) and let ? be a subregion of a region ?'. Then the extension of the
function / on ?', given by f(x) = 0 for ? e ?' \ ?, belongs to WkW).
Many PDEs arising in applications can not be divided from appropriate boundary
conditions, e.g., the Dirichlet problem. If a solution / of such a PDE is a sufficiently
smooth function on a region ? with a sufficiently smooth boundary 8?, there are no
problems about assigning values of / on 9?. However, if / belongs to a Sobolev space,
then / is, in general, defined only almost everywhere. In particular, the boundary 9? is
of measure zero, meaning that the value of / at the boundary is not uniquely determined.
This implies that one has to precise what is the value of a generalized solution of PDE on
the boundary - this important and not at all easy question leads to the notion of the trace
of a function.
Clearly, the trace of a continuous function /, defined on ?, on a surface S с ? should
be a continuous function on S from C(i2), which coincides almost everywhere with /. The
general case has to be handled with care, as will be exposed next.
Let ? с Ш" (? ^ 2) be a bounded region and S с ? a locally quadratic (n — 1)-
dimensional surface with the property that for every xq e S there exist regions U(xq) and
V and С' -diffeomorphism ? from U onto V such that some subset of ? П U is mapped
on some и-dimensional parallelepiped ? с V, and S П U is mapped on a side or union
of sides of P. Then there exist constants С ? > 0 and Ci > 0 such that for every function
/ e С' (?) and ?, 0 < ? < 1, we have
[ \f\2dS<:^- f \f(x)\2dx+C2sT f \Dif(x)\2dx. D)
Js ? ?? ^\'?
The preceding inequality D) can be written also in the following form
{Jslfl2dS) ^Cnfllw'^
for some constant C3 > 0.
Now if we define the operator Г : С' (?) -> L2(S) by
Г(/) = /1эя,
then one can show that there exists a unique linear and bounded extension ?: W' -> L (S)
of T. The obtained value of T(f) (a function from L2(S)) is the trace of the function
/ e W1 (?) on the surface S, which will be denoted by /Is·
One can interpret the trace of a function / e W1 (?) also in the following way. From
the fact that С' (?_)_is dense in the space ?\?), it follows that there exists a sequence
(//)./€N from ?](?) such that Нт^зс \\fj - f\\wi(n) =0. Then by D)
\\fk~ fm\\C-(S) ^C\\\fk -/??????'(?), k'm eN·

1246
?. Pap and A. Takaci
Hence (//)y6N is a Cauchy sequence in the complete space L~(S), which gives the
existence of an element /5 e L2(S) satisfying
lim \\fj-fs\\L2lS)=0.
Now the obtained function /5 is the trace of / e W' (?) on the surface S.
2. Distributional derivatives
2.1. Introduction
The approach that led to to the derivative in the Sobolev sense was exposed in the
previous section. Next, we shall see further generalization of the derivative, namely in
the distributional sense. For a motivation, we shall start with two problems.
Example 2.1. The problem of measurement of a physical value at a point xq; in fact, one
prefers measuring its average value in a sufficiently small neighborhood of xq-
For an example, let S be the density of a point located at the origin, with initial mass
m = 1. If one spreads this mass throughout a central ball ?@, r) in R3 with a radius equal
to r > 0, then its average density becomes
fr(x) =
\x\\ < r,
4лт3'
0, ?лгИ >r.
Clearly for all r > 0 it holds fB(Q r) f,-(x)dx = 1. However, what about the density for
r = 0? Denoting it by S — S(x), then it is natural to ask
I.
S(x)dx=\. E)
On the other hand, this density <$(-*) is equal to
+00, ? = 0,
5(jc) = ,
@, jc^O,
which contradicts E), and, in particular, it holds
fr(x) ¦/*· S(x), asr^0 + .
Still, for every continuous function ? on R3 it holds
/ fr(x)<p(x)dx^><p@)= &{?)?(?)??, r -> 0+, F)
Jb@.i) Jmy

Generalized derivatives
1247
meaning that fr -> S as r -> 0+ holds in a "weak" sense. In fact, the function ? has been
replaced by afunctional denoted again by ?:
<5:C(K3)^C, defined by ?(?):=?@). G)
Actually, this functional is exactly the Dirac's delta function, introduced by Paul Dirac
in the late twenties of the last century.
Example 2.2. Let the hyperbolic second order PDE
дхду
be given, where и is the unknown and / a given continuous function on R2.
If we multiply (8) by ? e C2(R2), then after two integrations by parts, we get
/ f(x,y)<p(x,y)dxdy = / —-<p(x,y)dxdy
JR2 Jr2 охду
f
JR
d-<p(x,y)
u(x,y)—r~z dxdy.
: дхду
This means that (8) holds in the "weak" sense, provided that the linear functional on
C2(K2), given by
<рь* / и(х,у)—— dxdy, (9)
УК2 дхду
equals to the functional
?"-* / f(x,y)<p(x,y)dxdy.
JRi
Thus the functional (9) is a weak solution of Equation (8), but it can also be taken as a
weak second order mixed derivative of the solution u.
Let us note that the two functionals, namely ? from G) and the one from (9) are not
functions in the classical sense, but rather some new elements - distributions. In his
famous book from 1951, L. Schwartz, using the somewhat earlier developed theory of
locally convex spaces, gave the mathematical foundation of the theory of distributions.
Other classical references on the distribution theory and PDEs are, e.g., Gelfand and Shilov
A958), Vladimirov A976), Hormander( 1983-85).

1248
?. Pap and A. Takaci
2.2. Distribution spaces
If ? is an open set in R", then an infinitely differentiable function ?: ? -> С is in the set
Co°(i2) if it has a compact support. The space ?>(?) is the set C^(i2) endowed with the
convergence defined as follows:
A sequence (<p/)y-6pj/rom ?>{?) converges to a function ? {from ?>(?)) if there exists
a compact set К С ? such that for every j eN it holds supply С К, and, moreover, for
every multiindex a e Щ the sequence (da/dxa)(pj uniformly converges to (da/3??)?, as
j -> oo.
Dehnition 2.3. Adistribution ? on ? is a linear continuous functional on ?>(?), where
the continuity of ? means that for every sequence (<py)y6pj which converges to zero in
?>(?) it holds
lim T(<pj) = 0.
The set of distributions on all ? will be denoted by ?>'(?)\ clearly it is a vector space.
The value ? (?) of a distribution ? e ?'(?) at a function ? e ?>(?) is also denoted by
(?,?).
The space of locally integrable functions on ?, L|OC(i2) is a proper subspace of the
space of distributions ?>'(?). Namely, if / is a locally integrable functions on ?, then the
functional Tf on ?>(?), defined by the relation
(Tf,<p):=jf(x)<p(x)dx, ?€Р(Й), A0)
is, in fact, a distribution from ?>'(?). The mapping i-|OC(K) -> V, given by / ь> Г/, is
injective, i.e., if /, g e i-|'oc(IR), then Tf = Г? if and only if / a= g. The distributions of
the type Tf, i.e., defined with a function / e i,,1 (R) by A0), are called regular. A typical
example of a non-regular (or: singular) distribution is the delta distribution given by G),
and also its generalization, the functional Sa, a e ?, given by
(??,?)=?(?), ?&?{?). A1)
A characterization of distributions is given in the following theorem.
THEOREM 2.4. A necessary and sufficient condition for the continuity of a linear
functional ? on ?>(?) is that for every compact set К С ? there exist numbers С > 0
and m e No such that
\(T,<p)\^C-pK,m(<p), ???(?), A2>

Generalized derivatives
1249
where the norms рк.т (·) are given by
in
??,??)= ? sup|D°V(*)|. A3)
м=оле*
In general, the number m in A2) depends on the compact set K. If, however, m can be
chosen independently of K, then ? from A2) is said to be a distribution of finite order;
the smallest m with this property is then called the order of ? in ?. It is easy to see that
every regular distribution, i.e., the distribution Tf defined by an / e /^(.?), see A0), is
of order 0. The characterization of distributions of order 0 gives the following theorem; in
short, it states that such a distribution is a (Radon) measure, and vice versa.
THEOREM 2.5. A distribution ? is of order 0 on E" if and only if it is a continuous linear
functional on the set C(R").
The delta distribution Sa, a e R, is clearly of order 0, which, in view of Theorem 2.5,
means that it can be observed as a measure on a Borel subset ? of the open set ?:
( il, if ?
Sa(E):= / Sa(dx)=\
J ? 10, if ?
We saw in Definition 1.2, how the generalized derivative in Sobolev sense of a function
from L2(i2) was defined. Next, in a similar manner, one introduces the distributional
derivative of a distribution.
Let a = (? ?, <*2, · · ·, <*„) ф @,0, 0) be a multiindex from Nq and / a function from
Cl0,|(K")· Then, for every ? e ЩШ), it holds (compare A))
f Daf{x)-<p(x)dx = (-\)^ [ f(x).EP<p(x)dx.
JR" JW<
Since both / and its partial derivative Da f define unique regular distributions Tf and
???), respectively, the last equality can be written as
№'·'>-<-?'·???^
Thus for an arbitrary element ? e ?>'(?) the ath distributional derivative of T, T(a\ is
defined by
(r<°\v):=(-l)^r,??), ?*?{?). A4)
The obtained functional ?@?) is also a distribution from ?>'(?). It is important to note the
fact that the space ?>(?) is closed under differentiation of arbitrary order. This implies an

1250
?. Pap and A. Takaci
essential property of the space ?>'(?), namely that every distribution has a distributional
derivative of arbitrary order, and, moreover, that the order of differentiation is irrelevant.
Perhaps this is one of the main reasons why distributions turned out to be so useful in the
theory of PDEs.
The distributional derivative is a linear operation, a continuous mapping from ?>'(?)
into itself, and, moreover, the Leibniz formula holds.
Let us see few examples.
Example 2.6. The kth derivative of the delta distribution is given by
(a(*\v) = (-!)<*) ?-v@), „eP.
Example 2.7. We have
/•+3C
(?', ?) = -[?(?), ??(?)\ = - / H(x)D(p(x)d)
Jo
f+OC
- / ??(?)??=?@)
Jo
Thus
(?,?), ?&?\
?' = ?. A5)
where ? is the delta distribution given in G) (with R3 replaced by E).
Next, if the continuous function / is given by
\x, if-OO,
??, if jc <c 0,
then /' = H, compare with Example 1.1, which together with A5) implies /" = ?.
Moreover, in view of Example 2.6, it holds
/(*+2)=i(*)i keNo. A6)
The derivative almost everywhere of H, DH, is the locally integrable function 0, since
the point ? = 0, where ? has no derivative, has measure 0. Thus from A5) it follows
?' = ? ф 0 = DH, meaning that the two derivatives: the distributional and the classical
one, are not always the same. The relation between these two derivatives in a rather general
case is given in the following theorem, see, e.g., Vladimirov A976).
THEOREM 2.8. Let f be a continuously dijferentiable function on the set R\J, where
J is a discrete set J = {x\, X2,. ¦.} (i.e. J has no finite accumulation point). At every xj,

Generalized derivatives
1251
j e J, f is assumed to have a discontinuity of the first kind, i.e., both the right- and the
left-hand side limits f(Xj ± 0) exist (but are not equal). Put sj = f(xj + 0) — f(xj — 0).
Then it holds
{Tf)' = TDf+Y^SjSX),
where (Tj)' is the distributional derivative of the regular distribution Tf, while To/ is the
regular distribution defined by the locally integrable function Df.
In Example 2.7, we saw that the S distribution can be obtained as a finite order derivative
of a continuous function. Actually, the following theorem gives an essential property of
distributions.
THEOREM 2.9. Let a distribution ? e ?>'(?) and an open set ?' such that ?' С ?, be
given. Then there exist an r e No and a continuous function f on ?', with support in ?',
such that
(T,<p)=[f(r\<p), „??(?').
In short, every distribution is locally a finite order (distributional) derivative of a
continuous function. Note that, in general, the order of the derivative, r, depends on ?'.
2.3. Fundamental solutions
One of the main results on linear PDE of order m
P(D)u = f, A7)
where
P(x,D)= ? aa(x)Da,
aa are given functions in some region ?, and / is a given function, in the context of
distributions, is the representation of the solution of A7) in the form of convolution of
/ and a special solution, ?, of Equation A7) with right-hand side equal to the delta
distribution.
DEHNITION 2.10. A distribution ? is the fundamental solution of a differential
operator P(x, D) from A7) if it satisfies the equation
P(x,D)E = S.

1252 ?. Pap and A. Takaci
Example 2.11. The function /: R2 -> R given for (x, y) = x +<)' ЬУ
f(x+iy)=-.—L- A8)
? x+iy
is the fundamental solution for the Cauchy-Riemann operator
-1/9 9
9 = - — + / —
2\9jc 9y
Note that / is a locally integrable function on R2, since
\f(z)\ = - ¦— for ? =x+iy,
? \?\
and thus it defines a distribution on R2. Let us prove that the function / from A8) is the
fundamental solution of the Cauchy-Riemann operator, i.e., that we have
?/,?) = ?@). A9)
We start from the equality
\ J-?? V' ?? 2? Jr\dx ^ dy)x+iy
Changing the variables ? — rcost, у = rsini, for 0 < r < oo, 0 ^ t ^ 2?, and taking
<Pi(r, t) = <p(rcosf, rsini), we obtain
^---1Сш-т^-
Therefore we have
C/,*>>=-^- f (?(??, t) - <pdO, t))dt + ^- f (<р(г,Ъг)-<р\(г,0))аг.
2? Jo ?? J0
By the compactness of the support of the function ?, and hence also of ?\, and the
periodicity of ?\, we obtain A9).
One of the great successes of the distribution theory is the following theorem, proved by
Malgrange A955-56) and Ehrenpreiss A954) in the mid fifties (see Pap et al. A997)).
THEOREM 2.12. Every linear partial differential operator with constant coefficients P(D)
has a fundamental solution E, i.e., there exists a solution of the equation
P(D)E = S.

Generalized derivatives 1253
The existence of the fundamental solution ? from Theorem 2.12 enabled Nirenberg
A955) to obtain the following result.
Theorem 2.13. Let
P(D)= ? ?"°? B°)
be a linear partial differential operator with constant coefficients. Then for any given
function f from C^° there exists a solution и in Cx of the equation
P(u) = f
The solution и is given by the convolution
u = E*f, B1)
where ? is the fundamental solution of the partial differential operator P(D) from
Theorem 2.12.
In all three main equations of mathematical physics, the fundamental solutions are
known.
Example 2.14. The function ?:RxR^ R given by
1 ? 1/2 ift>\x\,
E(x,t)=-H(t-\x\)=\' ' B2)
2 [0 if t < \x\,
is the fundamental solution of the one-dimensional wave equation. In other words, it holds
д2Е д2Е _
W~^x2=&'
The fundamental solution of the two-dimensional wave equation is the function
Ег{х, t) =
2?^?2-\?\2
where for ? — (x\,X2) e K2, we put \x\2 = x^ +x2.
Example 2.15. The locally integrable function ? given by
E(x)= = for и ^ 3,
??B-?)\?\"-2

1254 ?. Pap and A. Takaci
where ?„ = 2??"/2/ Г(и/2), is the fundamental solution for the Laplace operator
92 92
? = —= +^.
Example 2.16. The function
ж a H(t) ( m2\
E(x, t) = , exp — ,
is the fundamental solution for the heat equation, i.e., it holds
Эй ->
— ~azAu=S(x,t).
at
Further generalizations one can find in Colombeau A985), and in Pap B002), Chapter 34
of this Handbook.
3. The Mikusinski operational calculus
3.1. Introduction
Simultaneously with the introduction and the fast development of the theory of
distributions, in the early fifties the Polish mathematician J. Mikusinski with his collaborators
gave the foundations and systematic presentation of his operational calculus in a book first
published in 1959, see Mikusinski A983). Essentially, he introduced a field ? of
convolution quotients, much in the way the field Q of rational numbers is obtained from the
ring ? of integers. The elements of M, called (Mikusinski) operators, can be taken both as
a generalization of complex numbers, but also as a kind of generalized functions. Due to
its deep connection to other mathematical theories (in particular, with that of generalized
functions and the Laplace transformation), convenient algebraic properties, but also its
relative simplicity, the Mikusinski operator calculus had impact on numerous authors both in
theoretical and applied mathematics. Perhaps, the field ? is best suited for (both ordinary
and partial) differential equations with initial conditions.
The continuation of Mikusinski's book A983), written by J. Mikusinski and his
collaborator T. Boehme A987), presented the advanced topics and results obtained in a period
of thirty years till the mid eighties, in particular on the problems connected with the
topological and convergence structure of the Mikusinski operator field ?.
3.2. Operators
Next we present the main notions from the Mikusinski operational calculus; the complete
exposition is given in the mentioned monographs by Mikusinski A983) and by Mikusinski
and Boehme A987).

Generalized derivatives
1255
Throughout this section, we denote by C[0, oof (respectively ?) the additive group of
continuous (respectively locally integrable) functions on the interval [0, oof.
Let / and g be two continuous functions on the interval [0, oof, i.e., /, g e C[0, oof. The
convolution of the functions / and g is the continuous function /*gon [0, oof, defined
by the integral
(f*g)(t)=[f(T)g(t-T)dz, f>0. B3)
Jo
It is easy to check that C[0, oof is a commutative ring, when equipped with the
multiplication given by the convolution; in other words, the convolution * in C[0, oof
is a commutative and associative binary operation which distributes over the addition
of functions. From now on, we shall write simply fg for the continuous function on
[0, oof from B3). (One must distinguish the convolution product fg = f*g from the
multiplicative product (/ · g)(t) — f(t) ¦ g(t), t > 0.)
It is important to note that C[0, oof has no unit element for the convolution, since there
is no continuous function /' = i(t) such that (/ * /)(f) = /(f) for each / e C[0, oof.
For the construction of the Mikusinski operator calculus, essential is the following
Titchmarsh theorem.
THEOREM 3.1. If the functions f and g of class C[0, oof are not identically equal to zero,
then neither is their convolution f * g identically equal to гею.
In other words, C[0, oof has no divisors of zero, which implies that its quotient field can
be defined. This field shall be denoted by ? and called, by its inventor, the Mikusinski
operator field. Its elements, the (Mikusinski) operators, are quotients of the form
f/g, /eC[0,oo[, 0#seC[0,oo[.
The last division is observed in the sense of convolution from B3), meaning that two
operators
/i/g, and f2/g2 B4)
are equal iff (/| *g2)(t) = (h *gi)@ for each t e [0, oof.
The sum and the product of the operators f\ jg\ and frjgi from B4) are defined by
/l h _ f\g2+f2g\ f\ fl __ f\ fl
g\ 82 g\g2 ' g\ gl g\gl'
One easily sees that the right-hand sides of the upper two equalities are invariant under the
replacement, say, of the quotient f\ /g\ with the quotient /з/#з, as long as (f\ * #з)@ =
(/з *g\)U) for each t^0.
In view of the obvious analogy with the arithmetic of fractions (since the usual addition
and multiplication of operators can be treated in the same way as with real or complex
numbers), it is no surprise that the field ? has very good algebraic properties.

1256
?. Pap and A. Takaci
Let us remark that the set ?, equipped with the usual addition and the multiplication
considered in the sense of convolution, is also a ring without divisors of zero provided that
the usual equality of functions is replaced with the equality almost everywhere. Clearly, С
is a subring of ?; however, the set of the convolution quotients from С coincides with the
field ? obtained from С
Each continuous function / = f{t). t ^ 0, defines a unique operator (fg)/g, where
g is an arbitrary function from C[0, oof which is not identically equal to zero. It is
convenient simply to denote this operator by /; we then write / = {/@1 and say that
/ is defined by the continuous function /. The subset of M, consisting of the operators
representing continuous functions, is denoted by M(; clearly C[0, oof can be bijectively
mapped onto Mc.
Note that the identity operator / = ///, / e C[0, oo[ and / # 0, does not belong to
Melt is convenient to write simply 1 for the identity operator / in M, which allows the equality
1 / — / for any operator f &M- However, one must distinguish it from another important
element from M, namely the integral operator ? = {{}, thus defined by the function
identically equal to 1 on [0, oof. Clearly I e Mc\ its name comes from the equality
[Jo J
The inverse operator to I, 1 /? is the differential operator denoted by s, which, however,
does not belong to M.c (otherwise, the identity operator would also be in M(·). Note that
s is not exactly the usual derivation operator, namely for a function / with a continuous
derivative /' on [0, oof it holds
sf={f'(t)}+f@). B5)
Since any two operators can be multiplied, the expression sf in B5) has sense in ?
even if / is a continuous (or just a locally integrable) but not necessarily differentiable
function on R. In this way, one can observe the operator sf as the generalized derivative
of/.
If ? e N, then the «th power of ?, the operator ?", is defined by the continuous function
t"~] /(n - 1)!, t >0. Clearly, it can be observed as the operator of repeated integration; its
inverse, the operator s", is the nth power of the differential operator s. If / is an и-times
continuously differentiable function on [0, oof, then the following generalization of B5)
holds:
{/(,,)W} = snf - f@)s'-] /'"""(?)/.
Moreover, for ? e R and a > 0 it holds ?a = {ta~]/?(?)}, and these operators
are exactly the well known Riemann-Liouville (convolution) operators of fractional
integration. Thus the operator sa, the inverse of the operator la, is the operator of fractional
differentiation, provided that a > 0.

Generalized derivatives
1257
The step function HT(t), ? > 0, is defined by
0, t<r,
A, t > ?.
Then the operator
e-5T:=i{WT@}, ?>0, B6)
is the i/ii// operator (or: translation operator) and the last term is justified by the following.
If / is a locally integrable function, then
«-"{/('>} = {
0, 0^?<?,
f(t-?), ? ?.
Let us remark that the identity operator 1, the powers of the operator s", ? e N, and the
shift operator e~ST, ? > 0, correspond to the delta distribution, its «th derivative and the
delta distribution supported by the single point ?, respectively. In general, the connection
between the field ?4 and the space of distributions is complicated; there are operators (e.g.,
arising from the heat equation) that are not distributions, and vice versa. In particular, any
distribution with unbounded support both to the left and to the right cannot be identified
with any operator.
In applications, of great importance are the polynomials in operator s given by
P„(s) = a„s"+a„_is"-' + · · · +a\s +a(),
where a„ ? 0, ?„_?,..., ao are numerical constants. Operations on these polynomials are
treated in the same way as in ordinary algebra. A rational operator (in s) is a quotient of
two polynomials in s:
a„s" +?„_|?"_? ? \-a\s+a0
Rn.m(s)
fomJm +fo„I_|S-1 +---+b]S+b0'
where a„ ? 0, ?„_?,..., ao, bm ? 0, fo,„_|, bo, are complex numbers and m, ? e N. If
these numerical constants are real, then the rational operator Rn.m(s) can be decomposed
into a linear combination of the powers of the following simple fractions
1 1 s-D
s-A' (s-BJ+C2' (s-EJ+F2'
for appropriate real numbers A, B,..., F, and, if ? ^ m, of a polynomial of order ? — m
in s.
A rational operator /?„.,„ (s) represents a continuous function only if the power of the
denominator, w, is greater then the power of the numerator, n, i.e., if m > n. If, however,
m = и, then the rational operator /?„.,„(s) can be written as
Rn.m(s)=^I + Rt,
bm

1258
?. Pap and A. Takaci
where Rc is an operator representing a continuous function. The solving and analysis
of regularity of solutions of linear differential equations with appropriate initial value
conditions is rather simple, since one works with polynomials and rational operators in
s as in arithmetic.
There is a close connection between a set of (one sided) Laplace transformable functions
and a subset of the field ? of Mikusinski operators, established and analyzed by Ditkin
and Prudnikov A975). We give only some examples in order to represent this connection;
they show that the Mikusinski calculus can be taken as a generalization of the classical
right-sided Laplace transformation. To that end, let us note first that the functions
r, , neN, ? ?, B7)
s s-l (s-a)" (s-aJ+b2
where s stands for the complex variable, are the Laplace transforms (in appropriate half
planes of the complex plane) of the functions
м-\ eat ?
1, e', , -ea'sinbt, B8)
(n-1)! b
respectively. However, if in B7) we treat s as the differential operator from Л4, then these
operators are exactly represented by the functions from B8) respectively - the inverse
Laplace transforms of the functions from B7). Of course, there are continuous functions
on [0, oof, e.g., Bi — l)e'~, which are not Laplace transformable, since the integral
/ e-stBt-l)e'~dt
JO
does not converge for any seC, Thus the following problem
x'(t)-x(t) = 2t + Bt-l)e'2, Jt@) = 0, B9)
can not be solved using the Laplace transform. The differential equation in the field ?
s ? — ? = 2 + /,
where / = {Bt — \)e'~] corresponds to the problem B9). It, however, has a solution in the
field ?:
2 + /
+ ^—=\2е'+ f е'-т{2т-\)ет'ат\ ={е' + еГ'}.
s - 1 [ Jo J
An operational function и (х) is a function that maps a set of real numbers into the set of
operators (see Mikusinski A983, Part 3, Chapter I)). An operational function u(x) is said
to be continuous in a finite open interval У с К if it can be represented in the form
u(x) = q{u(x,t)\,
C0)

Generalized derivatives
1259
for some nonzero operator q from M, and some continuous function (in the usual sense)
u(x, t) on the domain {{x, t), ? e J, t ^ 0}. Moreover, an operational function u{x) is
said to have a continuous first derivative u'{x) on an open interval У с К if it is given
by C0) and the function u(x,t) has a continuous partial derivative j^u(x, t) in the upper
mentioned domain. Then, by definition, it holds
u'(x) = q\—u(x,t)\.
The derivatives of higher order of an operational function are also defined in this manner.
Let as note that the properties of continuous operator derivatives are similar to the usual
properties of the derivatives of numerical functions.
In the field M, one mostly uses the type I convergence (shortly: convergence). By
definition, a sequence of operators (an)„6pj converges to an operator a iff there exists an
operator b ? 0, such that (ban)neu is a sequence of continuous functions on f 0, oo [ which
converges uniformly on every finite interval to the continuous function ba.
Unfortunately, the type I convergence is not topological, i.e., there is no topology in
? such that the type I convergent sequences are exactly those that are convergent in this
topology. For that reason, another convergence was introduced by J. Mikusinski, namely
type I' convergence. A sequence of operators (a„ )пещ converges type Г to an operator a iff
each subsequence of (a„)„6pj has a subsequence which type I converges to a. The type Г
convergence is the smallest extension of type I convergence which is topological.
One can obtain the field of Mikusinski operators also if one starts from the ring С of
locally integrable functions on [0, oof. The space С is usually endowed with the topology
defined by the family of seminorms || · || ?, ? > 0, where
||/||7·= ? \f@\dt, T>0.
Jo
The subspace of С consisting of all the functions / such that || · || 7- > 0, for any ? > 0,
is denoted by ?0· The algebra of all operators of the form f/g, f e C, g e Co, is
denoted by .Mo· The convergence in С is by definition the convergence in all seminorms
|| · || 7-, ? > 0. Thus the sequence (xn)neN, of the elements from M, converges type / to
? e M, if there exist representations x„ = f„/g, ? = f/g, /«, f,g e ?, g # 0, such that
/„ -> / in C.
The problems connected with the convergence, topology and approximations in ? were
analyzed, besides Mikusinski's books A983) and (with Boehme) A987), e.g., in Burzyk
A983), Pap and Takaci A990), Takaci and Takaci A997).
References
Adams, R.A. A975), Sobolev Spaces, Academic Press.
Burzyk, J. A983), On the convergence in the Mikusinski operational calculus, Studia Math. 75, 313-333.
Colombeau, J.F. A985) Elementary introduction to new generalized functions. North-Holland Math. Stud.,
Vol. 113, North-Holland, Amsterdam.

1260
?. Pap and A. Takaci
Ditkin, V.A. and Prudnikov, A.P. A975), Operational Calculus, Visa Skola, Moscow (in Russian).
Ehrenpreiss, L. A954), Solutions of some problems of division, I, Amer. J. Math. 76, 883-903.
Friedrichs, K. A934), Spektralzerlegung halbbeschrankter Operatoren und Anwendung zu Spektraizerlegung von
Differenzialoperatoren, Math. Ann. 109, H. 4-5, 465^487.
Gelfand, I.M. and Shilov G.E. A958), Generalized Functions I. II, III Fizmatgiz (in Russian).
Hormander, L. A983-85), The Analysis of Linear Partial Differential Operators I-IV, Springer, Berlin.
Levi, B. A906), Sul principio di Dirichlet, Rend. Circ. Mat. Palermo 22, 293-359.
Mikusinski, J. A983), Operational Calculus, Vol. I, PWN - Polish Scientific Publishers, Warszawa and Pergamon
Press, Oxford.
Mikusinski, J. and Boehme, Т.К. A987), Operational Calculus, Vol. II. PWN - Polish Scientific Publishers,
Warszawa and Pergamon Press, Oxford.
Malgrange, B. A955-56), Existence et approximation des solutions des equations ma derivees partielles et des
equations de convolution, Ann. Inst. Fourier (Grenoble) 6, 271-355.
Nikol'skii, S.M. A975), Approxinmtion of Functions of Several Variables and Imbedding Theorems, Springer,
Berlin.
Nirenberg, L. A955), Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math. 8,
648-674.
Pap, E. B002), Pseudo-additive measures and their applications. Handbook of Measure Theory, E. Pap, ed.,
Elsevier, Amsterdam, 1403-1468.
Pap, E., Takaci, A, and Takaci, Dj. A997), Partial Differential Equations through Examples and Exercises,
Kluwer Texts Math. Sci., Vol. 18, Kluwer, Dordrecht.
Pap, E. and Takaci, Dj. A990), On linear differential equations in spaces of locally integrable functions and of
Mikusinski operators. Math. Nachr. 148, 197-208.
Schwartz, L. A950-51), Theorie des Distributions 1,11, Hermann. Paris.
Sobolev, S.L. A935), Le probleme de Cauchy dans I'espace des fonctionnelles. Dokl. Akad. Nauk SSSR 3 G),
291-294.
Sobolev, S.L. A936), Methode nouvelle ? resoudre le probleme de Cauchy pour les equations lineaires
hyperboliques normales. Mat. Sb. No. 1, 39-72.
Takaci, Dj. and Takaci, A. A997), Numerical methods for operator differential equations, PAMM Monograph
Booklets, Technical University of Budapest.
Thomson, B.S. B002), Differentiation, Handbook of Measure Theory. E. Pap, ed., Elsevier, Amsterdam, 179—
247.
Triebel, H. A978), Interpolation Theory, Function Spaces, Differential Operators. North-Holland Math. Library,
Vol. 18, North-Holland, Amsterdam.
Vladimirov, VS. A976), Equations of Mathematical Physics, Nauka. Moscow.
Wloka, J. A987), Partial Differential Equations, Cambridge University Press, Cambridge.

CHAPTER 31
Real Valued Measurability, Some Set-Theoretic
Aspects*
Aleksandar Jovanovic
School of Mathematics, Universiry of Belgrade, Stiidentski trg 16. 11000 Belgarde, Yugoslavia
Entail: aljosha@infosky.net
Contents
Introduction 1263
0. Notation 1264
1. RVM - first equiconsistency 1268
2. Cardinal monotony 1274
3. Nonregular ultrafilter 1277
4. Rudin-Keisler order 1279
5. Measures and normal Moore spaces 1281
6. Counting remarks 1283
7. Some complementary results 1286
8. Set-theoretic measure theory 1288
References 1290
*This work was partly supported by project "Mathematical models of nonlineanty, uncertainty and decision"
A866) financed by Ministry of Science, Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1261

This page intentionally left blank

Real valued measurability, some set-theoretic aspects
1263
Introduction
Besides mostly used (totally atomic) probability measures on finite elementary event
spaces which easily posses nice properties of totality and maximal additivity, nontrivial
measures in the case of infinite sets usually tend towards one of the two ancient roots:
arithmetics and geometry.
The former-counting measures, accumulating measures, tend to measure the whole and
bigger by measuring elementary and smaller.
The later, initiating in dichotomy: zero measure dots versus length measuring intervals,
with a desire to measure everything (totality) tend to satisfy a natural demand (well
established much earlier in integral calculations) that the addition and subtraction of a
lot of points does not change the result, and that the measure of union of disjoint intervals
equals their lengths sum (additivity), well established much earlier in summation of infinite
series.
In working mathematics, the expressions beginning like: "let ? be ... probability on
space X..." are common and widely needed. The axiom of choice AC, besides supplying
all sorts of measures, was essential in Vitali's construction of Lebesgue nonmeasurable
set, which crashed the totality of Lebesgue measure. And having an atotal geometric
probability means that there are (quite big) events without probability at all (while
obviously their small parts have probability zero).
These issues attracted serious attention early in the century. An early solution to this kind
of discomfort was discovered by Banach A923) who proved that there are total extensions
of Lebesgue measure in R1 and R2 which are finitely additive and isometry invariant. Thus,
infinite additivity is broken on some (quite big) events.
On the countable infinite event spaces situation is simplified by the fact that measure
additivity is limited by the space cardinality, which for the countable spaces means finite
additions only.
As shown in Solovay A970), weakening of AC to dependent choice DC leaves room to
kill Vitali sets and allow Lebesgue measure to be total.
On the other hand reducing axiom of choice to DC might leave us with no measure at
all. Such model of ZF + DC is given in Pincus and Solovay A977), thus leading to a nice
Universe of Mathematics with no probability at all except finite probabilities.
Variety of related notions and properties were studied in fine details in twentieth century.
This study revealed that the measure problem is tightly related to the well ordering
principles and "cardinal geometry of universe of mathematics" - displacement function-
solution of the continuum problem, compactness properties of the infinitary languages
and their combinatorial equivalents, with specific positions in the hierarchy of the strong
axioms of infinity, elementary embeddings of models of set theory. It's solutions are closely
related to other important problems in other areas of mathematics. Curiously, measure
problem became a kind of periodic: it surfaces as a big issue every ten to twenty years,
either by it self or involved in some other context. The content of this chapter intended to
present some of the major components in this dance of properties.
Somewhat complementary exposition is given in Laczkovich B002) in this Handbook.

1264
A. Jovanovic
0. Notation
Axioms for set theory emerged between hundred and eighty years ago and are nicely
stabilized since then. The axioms were probably selected as fundamental, simple, practical
and sufficiency general for the need - development of all Mathematics in uniform way
and set theory represents a kind of "fundamental geometry" of the Universe in which
Mathematics happens. Set theory is the first order theory with e as the only nonlogical
symbol and following axioms.
Empty set: C.x) ("x is empty") C;t)(Vy)(->y e x).
Extensionality: (V*)(Vy)(Vz)[(z e ? ++ ? e у) -> ? = у)].
Foundation: (V;t)[Cy)(y e ?) -> Cy)(y e ? ? (Vz)--(c е Jt л ; е у))]·
Pair set: (У*)(Уу)(Зг)(Уи)(и ег^к=ху« = у).
Union set: (V;c)Cy)(Vz)[z e у «-> (Зи)(г еилке *)].
Power set: (Vjt)Cy)(Vz)[z ey« (Уи)(и e:^«e *)].
Replacement: If <p(;t, y) is a formula with ? and у free, not involving ? then
(V;t)(Vy)(Vz)M;t, у) л ?(?, ?) -* у = г) -»·
(VM)Cu)(Vy)[y ev^ Cjc)(jc e и л <pU, у)].
Infinity: Cjc) ("? is ?"). (We denote the set of natural numbers, the first initial ordinal and
its cardinality by ?.)
AC/DC
Axiom of choice: "every set has a choice function".
Dependent choice: Let ? ? 0 and R a binary relation on x. Assume that
(Vye*)Cze*)(y/?z).
Then there is a map h: ? -> jc such that for all и е ?
й(и)Кй(и + 1).
First 8 axioms constitute Zermelo Fraenkel set theory ZF.
jc is an ordinal, Ord(jc) iff ? is e transitive and well ordered by e. Thus Ord is a unary
predicate, but it is nicely organized with e as well (well ordered) so we prefer to think
about this predicate as one object like sets, that is somewhat bigger than usual sets (with
| Ord | = oo) and call such objects proper classes. Thus instead of ???{?) we write ? e Ord.
Ord is related to the sets similarly to the proportion: ? and the finite sets of natural numbers.
In arithmetics finite objects are treated, no one mentioning infinity but all these objects
dwell in ? = ?. Expanding to set theory was initiated by the basic need of Mathematics -
to investigate infinity. If somehow Cantor diagonal argument did not exist, then we would
have integral infinity, in one piece. However, maybe then mathematics and especially set
theory would not be so exciting. In either case we (would) have Cantor-Russell paradox
or Russell paradox (with Cantor's diagonal proof) showing that infinity always leaks out:

Real valued measuiability, some set-theoretic aspects
1265
"the world is bigger than the objects in it". The universe of all sets is usually denoted by V.
Thus, ? e V -?· ? = ? or V = {? \ ? = ?]. On the other hand we have cumulative hierarchy
(Von Neumann):
VO = 0,
V„+i = P(Va) (all subsets of Va),
Va = ? V/j when ? is a limit ordinal, and
?<?
V= (J Va.
aeOrd
It is nice that V = V, i.e., that V of cumulative hierarchy is equal to the universe of all sets.
V is transitive (e transitive).
Thus, everything takes place in (predicate, "big thing", proper class, universe) V. Instead
of collecting all subsets Godel's hierarchy of constructible sets Lis defined using definable
subsets only: a subset X с Л is definable in A (with parameters) if there is a formula
<?(??>?,..., u„) of the language of set theory and elements ? ? atl & A such that
X = \a e A | (A, e) \=?[??\ ...?,,]}.
Thus,
La+\ = {x I x <? La л ? is definable in La),
La = ? Lfi for limit a, and
?<?
aeOrd
V = L stays for "all sets are constructible". Godel A938) proved equiconsistency of ZF
and ZFL (ZF + V = L) and proved that AC and GCH are theorems of ZFL, whence AC
and GCH are consistent with ZF.
If у = {? ? ?[?,?\,.. .,<*„]} for ? ?,.. .,<*„ e Ord, we say that у is ordinal definable and
write OD(y) or у e OD. If transitive closure of ? TC(jc) с OD we say that ? is hereditarily
ordinal definable and write HOD(jc) or ? e HOD. Similarly ODR means definable from
ordinals and reals and HODR for hereditarily ODR. All combinations of consistency
results equalizing some (none) elements of the chain L с HOD с V is established by
Levy and McAloon together with the relationship between OD and HOD.
A first order structure (M,E) for the language {e} is a model of set theory if the
axioms of set theory hold in (M,E). The model is transitive if ? is transitive. If ZF is
consistent then its models are infinite. By Skolem-Lowenheim theorem, if consistent ZF
has a countable model. Let 3?? and У1, be structures of a first order language С and for

1266
A. Jovanovic
f: ? -* N we say that it is an elementary embedding, in symbols /: 9Л < 9? iff for every
formula ?(?\ ...xn)oiC and all ? ? a„ e ?
mt=<p[al...a„] iff 4l\=<p[fai...fa„].
Transitive e models of ZFC containing all ordinals are called inner models. The nontrivial
elementary embeddings (different from identity) of V are important in set theory and model
theory. If j is such an embedding, then j(a) ^ a and there is a first ordinal moved by j -
its critical point.
Vopenka PRINCIPLE. If A: is a class of models of language C, then there are ОТ, У1 e К
such that m<<Jl.
Generic extensions (full details are well presented in, e.g., Jech A971a, 1971b, 1978)
or Kunen A980)). For a first order formula ? and a complete Boolean algebra В е 9Jt,
where 9Я is a model of set theory, the Boolean valued satisfaction \\<р\\в е В is defined by
induction on formula complexity, so that \\?\\ = 1 when ? is a logical axiom. Usually
the same is for axioms of ZF. Let h be an iW-complete homomorphism of ? to 2
(corresponding to a generic ultrafilter G). Composition of the Boolean valued satisfaction
with h represents the standard satisfaction, leading to the minimal extension Wl[h](=
9Jt[G]) of the initial model Ш of ZF, containing h (and G) which satisfies/violates certain
property, depending on the selection of В and h.
Compactness. An important property of the first order logic is given in the
COMPACTNESS THEOREM. A set of sentences ? has a model if and only if every finite
subset of ? has a model.
Its generalizations to infinitary languages attracted some attention as well. Infinitary
language Ck has к variables, and besides usual ways of formula formation has infinitary
conjunctions and quantifications: if Г is a set of formulas of Ck and \Г\ < к then лГ is a
formula of Ck; if ? e Ck and X is a set of variables, | X | < k, then (VX)<p is a formula of
Ck- Some generalizations of compactness:
Definition. A cardinal к > ? is weakly compact iff for every set ? of sentences of Ck
with \?\ < к, ? has a model if every subset of ? of cardinality < к has a model.
A cardinal к > ? is strongly compact iff for every set ? of sentences of Ck, ? has a
model if every subset of ? of cardinality < к has a model.
A cardinal к > ? is medium compact if the following is true.
Let ??, ? < к, be sentences of Ck and let U/j<a ?? nas a model for each a < k. Then
U/3<* ?? has a model.
None of the increased compactness can be proved to exist from original axioms of set
theory. They are introduced with axioms that guarantee the existence of a weakly compact
cardinal or a strongly compact cardinal or a medium compact cardinal.

Real valued measurability, some set-theoretic aspects
1267
Ultrafilters - binary measures. By ultrafilter theorem, AC supplies plenty of nonprincipal
ultrafilters over each cardinal. List some definitions. Let ? be a [0, 1] valued finitely
additive measure on P{k) (this includes ultrafilters). Cardinal norm or simply norm is
defined as
||?||=????{?|?(*)>0}.
Measure additivity is given with
add (?) = min||jc| ? ? с dom(M) л (Vy e ?)(?(?) = 0) ? ?([_Jjc) > ?!.
A cardinal к is RVM (real valued measurable) if it has a measure of additivity k. A measure
? on P(k) is (?, ?) regular iff there is ? с dom(M), |?| = ? and if X с ? and |X| > a
then ?(??) = 0. ? is regular if it is (?, A:)-regular.
There is plenty of regular ultrafilters. Nonpricipal ultrafilters whose all elements are
of the same cardinality are called uniform. There are plenty of uniform ultrafilters on
each cardinal. The existence of nonregular ultrafilters, known as a problem of Gillman
and Keisler and related problems of powers of ultrapowers were of major importance
in the development of theory of ultrafilters with variety of solutions, dependable on set
theory axioms contributed by Prikry, Jensen, Magidor, Woodin, Laver. But additivity of
an ultrafilter can be either ? or a measurable cardinal. Similarly for nonbinary (total)
real valued measures, their additivity is either ? or a real valued measurable cardinal.
Thus: a cardinal к is measurable iff there is a nonprincipal ultrafilter ? on к whose
additivity is k. Scott proved that the existence of measurable cardinals implies that there
are nonconstructible sets, i.e., V ? L. If к is measurable then there is a normal measure ?
over A:: for / e kk if f{x) < ? ??(?) (?({? | f(x) < ?}) — 1) then there is a constant b
such that f(x) = b AEi?) (?({.? | fix) = b)) = 1).
Weakening normality we get weakly normal ultrafilter: ultrafilter ? over к is weakly
normal iff for / e kk if fix) < ? AEi?) then there is a constant b such that fix) < b
??(?).
Strengthening normality we get compact cardinal: к such that for every ? > к there is
an ultrafilter ?? over ?*(?) of additivity к (?(?) = [? с ? | |jc| < к}).
Coincidentally, compact are just strongly compact, measurable are just medium compact
and cardinals with weakly normal ultrafilters are tied to the weakly compact cardinals.
Mention the supercompact cardinals. For ? < ? let Q = {x e Pa-(?) | ? e x]. Let ? be a
binary measure over Pk (?) of additivity к, containing all Q.
Let fix) e ? for all ? in ? (?), ? is normal over ?-(?) if for some a < ?, ?({* e
? (?) | fix) = <*}) = 1. Then к is supercompact iff for all ? there is a normal measure on
Pk U).
All compactness properties are properties of high concentration and of the increasing
strength and consistency strength. Detailed treatment one can find in Drake A974), Chang
and Keisler A973), Jech A978).
Mention that if j : V -> V is a nontrivial elementary embedding then the first ordinal
moved by j is a measurable cardinal. Strong compactness and super compactness are

1268
A. }ovano\4c
expressed as further closure properties of nontrivial j 's. If Vopenka principle holds then
there are many supercompact cardinals (see Jech A978), Solovay et al. A978)).
1. RVM - first equiconsistency
In early thirties Ulam A930) set the field with must-had-been surprising discoveries
collected in the following two theorems.
THEOREM 1.1. if there is ? ?-additive nontrivial measure on S, then either there exists a
two valued measure on S and \S\^ the least strongly inaccessible cardinal, or there exists
an atomless measure on 2K|) and 2K() ^ the least weakly inaccessible cardinal.
THEOREM 1.2. If there is a real valued measurable cardinal < 2K(I then there is a
total extension of Lebesgue measure. If к is real valued measurable then к is weakly-
inaccessible. The additivity of a {total) measure is real valued measurable cardinal.
One can see instantly that CH simplifies the situation: if CH is true then there is no
real valued measurable cardinal and no ? additive total extension of Lebesgue measure.
Thus the first consistency result of Godel provides a model (L) without total ? additive
extensions of Lebesgue measure. Scott A961) proved inconsistency of existence of any
total nontrivial ? additive binary measure with Godel's axiom of constructibility V = L.
Thus in L there are no total ? additive measures. The minimal universe L(?) containing a
binary ? additive measure ? must be larger than L. Opposite to disagreement of CH and
total ? additive measures on R, Silver proved consistency of GCH with the existence of a
binary ? additive total measure.
In 1966 Solovay proved this remarkable theorem.
THEOREM (Solovay A971)). The following theories are equiconsistent.
A) ZFC + "there is a total ? additive extension of Lebesgue measure".
B) ZFC + "there is a measurable cardinal".
That the consistency of B) implies the consistency of A) is proved with Cohen method
of forcing with nicely chosen Boolean algebra and impressively nice harmony of measures.
Let 9Л be a model of B) and let к be a measurable cardinal.
The proof- building a model - generic extension 9Jt[G] of ground model ШТ, has two
components:
(a) make 2H° ^ к in the generic extension;
(b) keep к real valued measurable.
Then, the rest follows from theorems of Ulam.
All details of the proof which is partly shortened here, can be found in Jech A978) or
Solovay A971).
Let ? ^ к be such that ??|> = ?. The simple measure space (S,-, ?[, ?') for i e / = ? ? ?
given with: S,= {0, 1), Tx¦ = P(St), ?'@) = 0, ?'({0}) = ?'({?) = з, м'({0, 1}) = 1.

Real valued measurabilirv, some set-theoretic aspects 1269
Let (S,F, ?) be a product measure space for / e /; S = П;е/ ?, ? be the least ?-
complete field of subsets of S containing the sets {t e '{0, 1} 11(/') = 0} for /' e /. Let В
be the measure algebra T/lhe, ideal of sets of ?-measure 0. В satisfies countable chain
condition c.c.c. and is a complete Boolean algebra. Define a ?-additive measure ?? on В
with M2([-*]) = ?(?)- Let G be an 9Л-generic ultrafilter on B. Then in 9Jt[G] the cardinals
are preserved because В satisfies c.c.c. In Wl[G], 2K" = ?. First, BK<»)OTlGl < (|B|K»)OT =
?. On the other hand in 9Jt[G] there are ? subsets of ?. For (a, n) e /, let UaM e ? be
such that
?/?.? = {?€/{0, 1} |/(?,?)= ?}.
Let «„,„ = [?/„,„] (e ?). For a < ? let x„ be the ?-valued subset of ? such that
\\h eXa\\= ua„ (? < ?).
Let xa be the G-interpretation of x«. It can be checked that ? ? ? implies xa ? ??,
confirming that in 9Ji[G], 2^" = ? > к.
Proof that к remains real valued measurable in 9Jt[G]. So, let к be a measurable with
a binary measure ?\ in 9Л and let ?, ?, ?2, ? be as above. We shall define a ?-valued
name ?3 and show that for any generic ultrafilter G, ?3 the G-interpretation of ?? is a
к -additive measure on k. Thus we have a nice interference of measures ??, ?, ? 2 and G
resulting in ?3 with desired properties.
Let a e ? \ {0} and let A e 9JtB be a ?-valued name such that a ll· А с ?. For each
? < A; let
,.. . ?2(?· ||?? ?||)
/a (A, a) = —
M2(a)
?? is A;-additive, hence /?(?,?) = ???{?\) for some real r. Let ??(?) = r. It has the
following properties
(i) all· A] =A2 implies ?„(?|) = ??(?2),
(ii) all· А] С A2 implies ??(?|)<?„(?2),
(iii) for X С к in Ш ? ? (?) = 1 implies ?? (?) = 1, ?? (?) = 0 implies ??(?) = 0,
(iv) let A^, ? < ? < к be such that a ll· Af с ? and ?7^/7 implies ? ll· A^ П A,, = 0. Let
A be such that a ll· A = Uf <y ^?· Then ?-additivity of ?? implies
Ma(A) = ^??,^??),
(?) ? ^ b implies ??(?) ^ ?/,(?).
Letr be a real number from [0, 1] and let {an \n < ?] be a partition of ? e B. If ?«„(?) < r
for all и, then for almost all ?, ?2(?„ · ||? e A||) < r · ??(?„), and thus for almost all a,
??(? ¦ ||? e A||) < r ¦ ?2(?). Hence ??(?) < r. As a consequence we have
if for every nonzero b ^ a there is a nonzero с ^ b such
that ?<-(?) < r then ??(?) < r. (*)

1270
A. Jovanovic
Similar holds for ^, >, ^. If b ll· А с к, define
?^(?)= inf ?«(?).
?? satisfies (i), (ii), (iii) but it is not additive and instead of (iv), we only have
М?(А)>]ГМ*(А?) for ? < к.
Let G be a generic ultrafilter. In 9Jt[G] we define дз: P(k) -» [0, 1] with
M3(A) = supM*(A),
b€G
where A is a name for A. Let ? 3 be the canonical name for ??,, defined in fflB using the
canonical G. дз does not depend on the name A for A, А] сд2 implies ??,(?\) ^ дз(Л2)·
ForXe9^3(X) = 1 ifM,(X) = 1 and ?3(?) = 0 if ?, (?) = 0.
The proof that ??, is к additive. Start with the meaning of ??. For a real r e Wl,
0 < r ^ 1, we claim that
??(?)^? iff fclh/?3(A)>r.
If ??(?) ^ r then for every generic G for which be G, мз(Д) > г, hence foil· ?., (A) >r.
Opposite if b ll· ? 3(A) > r then
foil· V? < г ЗЛ e G ?^(?)^?
that is
V<7 <r Vc<fo3diCc ?*|(A)^q.
Let 4 < r; we claim that ??(?) ^ 4. If ? ^ b then Vc < ? 3d ^ с such that ?} (A) > 4 and
hence ?^?) > ^. Thus ??(?) ^ 4. Since this holds for all q < r, we have ??(?) > r.
Proof that ??, is finitely additive. Let A, A| and At be such that every condition
forces that A is the disjoint union of A| and A2. If ? and гг are real numbers and
b ll· (?3(?|) > f\ and /?з(Аг) ^ r2) then (by first claim) b ll· ?3(A) > ? + r2; hence
?3(?) > ?3(?|) +?3(?2)· Conversely, assume that ?3(?) > ?3(?|) + ??(?2). There
are reals r\, гг е 9Л and kG such that
fo ll· /? 3 (? ?) < f\, ?3(??)<?2 and ? 3(A) ^ ? + ?.
Since b ll· ? з(А|) < r?, there is for each с ^ fo some ii < с such that ?</(?|) < r\; hence
?;,(?|) < ? and ?b{Aг) < ?, and so ?,*(?) ^ ?/,(?) < r\ + r2. Contradiction.

Real valued measurability, some set-theoretic aspects
1271
Now since дз is finitely additive it suffices to show that
?<? ?<?
for any family {?? | ? < ?] of less than к subsets of A:. So let у < к and let Af, ? < у and
A be such that ||A = \J^<Y Af || = 1 and let
?3(?)> ]Гмз(А?).
Then there is г е 9Л and beG such that
b\l· 2_]?3(??) < f and дз(А)>г.
Let ? be an arbitrary finite set and let Ag = Uf e? ^i ¦
Since
M3(A?)iC ^дз(А^)
= 1
we have b lh дз(А^) < r. By (*) we get ?/,(??-) < r.
Since Mfc(Af) < r for all finite ? с у, it follows that ?/,(?) < r, whence ?,*(?) ^ r, a
contradiction.
This completes the proof that in 9Jt[G] дз is (a nontrivial) A:-additive measure on k.
Proof that consistency of
ZFC + "there is a total extension of Lebesgue measure"
implies the consistency of
ZFC + "there is a measurable cardinal".
Certain combinatorial properties of measures and their zero measure ideals are essential:
saturation of ideal and normality. For a Boolean algebra В we say that it is ? saturated
if whenever {b$: ? < ?] is a pairwise disjoint family of elements then b$() = 0 for some
?? < ?. Let sat(S) be the least ? such that ? is ? saturated. For a nontrivial ideal / С Р(к)
we say that it is ? saturated if В = P(k)/I is ? saturated and denote sat(/) = sat(B).
Useful information on sat is given in the following propositions.
PROPOSITION (Tarski). Let В a Boolean algebra, let sat(S) ^ K0. Then sat(B) > K0 and
sat(Z?) is regular.

1272
A. Jovanovic
PROPOSITION (Ulam). Let к be a real valued measurable with nontrivial measure ? on k.
Let I be the ideal of sets of ? measure zero. Then I ist<\ saturated.
With respect to an ideal / on к we say that А с к is of measure zero if A e /. Otherwise
we say that A is of positive measure.
Let / be а к -complete ideal on к > ?. The following definition of normal ideal is
tightly related to normal (ultra) filters. Incompressible function corresponds to the minimal
unbounded function in the case of normal ultrafilter.
We say that / is normal if whenever Kt has positive measure, and /: В -> A: is such
that /(?) < ?, ? e В, then there is ?' с ? of positive measure, and a ? < к such that
f(t) = i, ???'.
THEOREM 1.3. Let к be an uncountable cardinal, I a k-saturated nontrivial ideal on
P(k). Then there is a normal nontrivial ideal, J on P{k), with
satG)$sat(/).
If I is a measure ideal, so is J.
PROOF. Begin with some definitions. Let /: A -> k. We say that / is almost bounded
if there is a ? < к such that {? e A | /(?) > ?) has measure zero. We say that / is
nowhere bounded if for each ? < к, {? e A | F(?) ^ ?) has measure zero. Finally, / is
incompressible if
A) /is nowhere bounded.
B) If ? с л has positive measure, g: В -> к and g(?) < f(%) for all ? e B, then g is
almost bounded.
These concepts are only important if A has positive measure. If A has measure
zero, every map /': A -> к is simultaneously almost bounded, nowhere bounded, and
incompressible.
LEMMA 1.1. Let f': A -> к be nowhere bounded. Then we can write A as the disjoint
union of sets В and С such that
A) f \ В is incompressible.
B) There is a g:C -> к with g(?) < /(?). ? e C, such that g is nowhere bounded.
LEMMA 1.2. There exists an incompressible function f:k —> k.
We can now prove Theorem 1.3. Let h:k -> к be incompressible. Define a subset
J с ? (?) by
J = {А с jt | /г (Л) е/}.
It is straightforward to check that У is a A:-complete ideal, and that к <? J. Since h is
nowhere bounded, h~[ ({?}) has /-measure zero. Hence {?} has ./-measure zero. An easy
check shows that satG) 5i sat(/).

Real valued measurability, some set-theoretic aspects
1273
We check that У is normal. Suppose A has positive У-measure, and g : A -> к is such
that g(?) < ? for ? e A. Put В = h~[(A). Then ? has positive /-measure (since A has
positive У-measure). Let f :B —> kbe, the composition
Then if ? e ?, /(у) = g(h(y)) < h(y). But /г is incompressible. It follows that there
is a ? < к such that {y e В \ f(y) ^ ?) has positive /-measure. Since / is к -complete and
? < к, there is a ?' ^ ? such that D = {y e В \ f(y) = ?'} has positive /-measure. Let
? = {? e A | g(y) = ?'}. Then ? e D+> g(h(y)) = ?' <-> h(y) e E. Thus D = h~](E).
It follows that ? has positive У-measure. That is, given g, A as above, there is an ? с А
of positive У-measure on which g is constant. Thus У is normal.
Finally, let к be a real-valued measurable cardinal. Let ? be a nontrivial A:-additive real-
valued measure on к. Let / be the ideal of sets of measure zero of ?. Let h: к —> к be
/-incompressible. Define v. P(k) -> [0, 1] by
?(?) = ?(/2-'(?)). (+)
Let У be as above. Then ? is easily checked to be A:-additive and nontrivial, and У is the
ideal of sets of ?-measure zero. Thus У is a measure ideal. The proof of Theorem 1.3 is
complete. ?
We now assume that к is as above, but that У is a nontrivial ^-saturated normal ideal.
LEMMA 1.3. Let A have positive measure. Let h : A -> к be such that /?(?) < ?, for all
? e A. Then h is almost bounded.
The inner model construction in the form in which it is used in the proof. Let A be a set.
We are going to construct a class, L[A], with the following properties:
A) L[A] is an e-model of ZFC.
B) L[A] is transitive and contains all ordinals.
C) А П L[A] is in L[A].
D) L[A] is the smallest transitive class satisfying (l)-C).
Suppose that ? is a set. Let e, be the e relation on x. Let Ал be А П x. Let ? be a
first order language with one two place predicate, e, and one one place predicate A. We
interpret С in the relational system 2lv = (x; е.,, Ал) in the obvious way. A set В с ?
is 2l.r-definable if there are elements yi, v„ of ? (possibly ? = 0) and a formula
? (у, у\,..., y„) of С such that
В = {уех\Ъх\=ф(у,У1 у»)}.
We put D(x, A) = {? с ? \ ? is 21 v-definable).
We now define L„[A] for ? e OR by induction on a:
Lo[A] = Q,
a = ? + 1:LJA] = D{Lp[Al A),
a is a limit ordinal: L„[A] = U/j<ff Lfl[A].

1274
A. Jovanovic
There is a canonical bijection
Fa : OR ^ L[A].
Fa is definable in set-theory. I.e. there is a set-theoretical formula \j/{x,y,z) which
expresses
"y is an ordinal and Fa(y) = z".
Let ? be the conjunction of a large finite number of axioms of ZFC.
_Let N be a transitive model of ?. Let ? be a set. Suppose that ? ?? ? e N. Put
A = ADN.
Then for ? an ordinal of N, La[A] = La[A] = L%[A] and FA(a) = Fj(a) = F^(a).
Let <*o be the least ordinal not in N. Then <*0 = OR^, and i-fA]^ = Lao[A] = LaQ[A].
Now let к be an uncountable cardinal. Let У be a normal nontrivial ?-saturated ideal in
P(k). Let_L[7] be defined as before. Put J = L[J] П J. It is straightforward to check that
in L[J], У is a normal nontrivial ?-saturated ideal in P(k).
THEOREM 1.4. Suppose that J, k, ? are as above and that ? < к. Then in L[J], к is
measurable.
The proof of this theorem uses the adaptation of argument in Silver's theorem:
THEOREM (Silver A971)). If к is measurable with normal measure ? and it's measure
zero ideal J then in L[J], GCH holds.
Using method of this theorem, Solovay proved that
"? < ? -> 2* < ?" is true in L[J].
This was sufficient in showing that к is measurable in L[J] with the use of the following
result of Tarski A930).
THEOREM 1.5. Let I be a nontrivial ideal in P(k). Suppose I is ? saturated for some
? < к. Suppose further that ? < ? -> 2K < к. Then к is measurable, and the quotient
algebra В = P(k)/I is totally atomic.
This completes the proof of Solovay's theorem on equiconsistency of total extension of
Lebesgue measure and measurable cardinals.
2. Cardinal monotony
Condition of cardinal monotony which is satisfied for e.g. finite probabilites with uniform
distribution:
\x\^\y\ => mW^W

Real valued measurability, some set-theoretic aspects
1275
is trivial in the presence of cardinal bisection, e.g., for Lebesgue measure, if CH holds,
? additivity prevents that a set of size less than continuum has a positive measure. The
same is true for any measure over continuum with CH.
In the case of the total ? additive measures observe first their indices.
The theorem of Ulam does not leave too much room: if a cardinal is RVM then it has
to be ^ 2? or measurable, seemingly there are no RV measures in the interval B?, 1st
measurable).
This restriction seems to eliminate the possibility of other RV large cardinals, the RV
counter parts of large cardinals (RV Strongly compact, RV super compact), obtained from
large cardinals by substitution of binary with RV measures in their defining properties
(JovanovicA980, 1981)).
There is a small leak however: Ulam's theorem states in fact that for every measure ?
add(M)?B<", 1st measurable),
with no restrictions on cardinal measure norm ||?||; besides the obvious add(M) < ||?|| ^
|index (?)| what variations are allowed here? Strait forward we get the following questions.
Is there any RV measure ? such that add^) < ||?||?
Is there any RV measure ? over continuum such that add^) < ||?||? The same for
extensions of Lebesgue measure.
Search for cardinal invariants in Solovay's model results in the following.
For the measure ?? which makes к RVM and ??, constructed from ?? which extends
Lebesgue measure to all sets of reals we have
add^3) = add^ \) — k,
11дз|1 = 11Д||| = *,
|??<1(??)|=? and |ind(/Z3)| = 2?.
In the construction of the model one can choose to have either к < 2? or к — 2?. The first
case gives sets of reals of positive ? з measure but of cardinality smaller than 2? (unusual),
thus answering the starting question of cardinal monotony.
To answer the last questions the procedure is similar as in Solovay's construction with
stronger hypothesis. Let к be ? strongly compact with ? > к, in the ground model 9Jt. Let
U be an uniform к complete ultrafilter over ?. Let (x, ?, ??) be product measure space
???{0,1} and let В be the corresponding measure algebra JF/the ideal of sets of measure 0.
The algebra В and 9Л-complete homomorphism h are used to obtain a Cohen type model
which preserves cardinals since В has countable chain condition and in which 2? — ? or
2? > ? as desired. From the ultrafilter U a uniform real valued measure ? is constructed
with the following properties
???(?) = ?,
???=?,
add^) = к < ?.

1276
A. Jovanovic
RV-large cardinals. Introduce other real valued large cardinals on the example of real
valued compact cardinals. We say that к is real valued strongly compact or real valued
compact cardinal iff for every ? such that cfk > к (other ? are obviously excluded) there is
a measure ?: ?(?) -> [0, 1 ] such that \\? || = ? and add^) = k. Obviously if к is strongly
compact cardinal then к is real valued compact. The analog of Ulam's theorem can be
proved similarly as the original theorem, thus
PROPOSITION 2.1. If к is real valued compact then either к ^ 2? or к is strongly compact.
THEOREM 2.1. Consistency of the theory
ZFC + "there is a strongly compact cardinal"
implies that the theory
ZFC -f "there is a real valued compact cardinal к ^ 2? " is consistent.
PROOF. First we note that in Solovay's method the measure ?? which makes к
measurable in the ground model is independent of measure ?? on measure algebra used to force
2? = ? > к. On the other hand it is the additivity of ?? what counts not merely its index.
So if ?? is ?: additive binary measure over <5 such that ||?|| — ? and if ?? is as before we can
apply Solovay's method and with sequence of measures as in Solovay's proof, we obtain
in 9Jt[G] a measure ? which maps ?{?) into [0, 1] which is к additive and properly real
valued or non atomic and uniform, i.e., ||?|| = ?.
All the construction took place in 9Jt[G]. It can be checked that ? and ? above are
independent. Thus, let к be strongly compact cardinal in the ground model 3??. Let К be
the class of binary measures witnessing compactness of k:
K = {?s\c?.?^k, ??:?(?)^{0,1], \\?*\\=&, add(Ma) = *}.
Do the forcing as before. In 9Jt[G] let us form a class A" of real valued measures: for
?$ e К as noted before make ?'& which is non atomic and such that
add(/4)=add^,5) and \\?'?\\ = ||??||.
By known facts it is clear that
?' = {?'&\????}^<?[0].
A" witnesses that к is real valued compact cardinal < 2?.
How about the other consistency implication? The model L[K], the inner model
constructed from a class of real valued measures К is not a solution, its worst problem
being appearance of a lot of new cardinals. Thus we can only state the following conjecture:
consistency of ZFC + "there is a real valued compact cardinal ^ 2'"" implies consistency

Real valued measurability, some set-theoretic aspects
1277
of the theory ZFC + "there is a strongly compact cardinal". Similarly for other RV large
cardinals (e.g., weakly compact, supercompact,...). ?
Real valued measurable cardinals, with Solovay's discovery pulled to the continuum
of real numbers or it's part, the (consistency) power of binary measurable cardinals. The
fruitful investigation of saturated ideals, done by Jech, Prikry, Magidor and others (see Jech
A978)) was a kind of main generalization of Solovay's arguments.
In the times when full zoology starting with measurable cardinals up to the  = 1"
was well mapped, it looked quite odd that the real valued measurables are so powerful
and so lonely, which pushed the search for the RV large. Kunen did similar thing with
PMEA which is discussed in Section 5. Prikry A975) defines real valued supercompact
mentioning that it's consistency is derived from super compact by the method of Solovay.
All the three sources, quite similar and at the similar time - seventies, use Solovay's
forcing for the downward relative consistency and upward constructibility relative classes
of measures for the partial back consistency. The applications like in Section 5 made these
kind of axioms interesting and useful in problem solving and classifications.
3. Nonregular ultrafilter
Let us now mention the first example of a nonregular countably incomplete ultrafilter due
to Jack Silver in the following theorem
THEOREM 3.1. If к is a real valued measurable then there is an ultrafilter over к which
is not (?, ?) regular for all ? < к such that c/? > ?.
PROOF. Let ? be a A: additive measure over k. Let D be any ultrafilter extending the filter
{x <? к ? ?(?) = 1). Thus ? e D implies ?(?) > 0. The following remark is useful:
for any sets xa, a e ?, there is a countable set С с ? such that
(+)
aeX c/€C
Proof of (+). Let
У a — ?? ? У а
disjoint and {ja )'<* = {Jaxa> s0
Ц UXa)=? (Uya)= Л >*&<*)¦
aeX
Choose С so that
^2^(Уа) = ^2^(уа).
Ct?C

1278
A. Jovanovic
Let c/? > ?, xa e D, a < ?; put
There is some ? such that ?, ?' ^ ? implies ? ? = ? a (otherwise, since c/? > ?, we would
have ? ? descending chain of reals). For each ? > ?, let С ? с ? - /3, С/з countable, be such
that
?( U Xa) = ?? (which = b- Use (+))·
Since c/? > ?, there is set ? с ? with \Z\ = ? such that /3 < /3' in ? implies that every
element of С ?', is greater than any element of С ?. For each ? e Z, let
w = U ¦*«¦
Since
?( U ¦*<* ~^) =0
one has: ???????) = 0, for /3, /3' e ?; also ??'/з) = ?? > 0. So
N/3eZ
hence
?(?^) = ^>0'
/3eZ
But
?^= ? ?^'<«>·
/JeZ f-.?^? ???
(V/3)/(/3)eC,)
So for some /: ? -> ? such that for all /3 e Z, /(/3) e С/з we have
P| xfw фЪ.
aeZ
Thus we have ? distinct xa whose intersection is nonempty.

Real valued measurability, some set-theoretic aspects
1279
Using more sophisticated arguments, Magidor proved consistency of existence of
nonregular ultrafilters over <«2 and а>з relative consistency of huge cardinals. His
nonregular ultrafilter F over 013 has cardinality jump
? ? ^ 013, while (obviously) I I 013
F
_ 2??
F
Nonregular ultrafilters are hard to obtain, jumping ultrafilters even harder. D
4. Rudin-Keisler order
For /: ? -> ? and ultrafilters D and ? (over ? or its subset),
E^RKD iff E={y\f-l(y)eD\,
E^RKD iff ?sCRKD and D sCRK E.
Complexity classification of nonprincipal ultrafilters due to Rudin and Keisler is related
to combinatorial properties of ultrafilters, hence the structure of Rudin-Keisler order (RK
order) of types of ultrafilters depends on the axioms of set theory (see Comfort and
Negrepontis A974)). The existence of minimal elements is characterized by normality
properties which are connected with a sort of completeness degree of ultrafilters.
If there is a minimal unbounded function / for D it generates a <rk minimal ultrafilter
? in the set of all uniform ultrafilters over k. The minimal unbounded function for ? is the
identity function (hence ? is weakly normal). Thus we can say that D above contains the
information on a minimal element which is type unique, below it in ^rk for set of uniform
ultrafilters over k. If the function / above is such that every function g:k —* k, such that
{/ I g(i) < /(/)} e D, is equal to a constant modulo D, then the ultrafilter ? is normal and
it is the minimal element in ^rk of all (nonprincipal) ultrafilters over k.
Type equivalence in ^rk preserves norms, thus the set of all <rk equivalence types
over some index can be divided into layers of ultrafilter norms which are cardinals from ?
up to the cardinality of a given index /, position of uniform ultrafilters over /.
What information can one level contain about the others? The existence of normal
ultrafilters implies that there are no types below them, and weakly normal ultrafilters are
minimal within the top layer. If there are minimal elements at lower levels those are type
equivalent to some weakly normal ultrafilter within the same layer. A generalization of
weak normality could be useful in the above consideration.
Let ? be a cardinal ^ k. An ultrafilter D over к is ?-weakly normal iff there is a minimal
unbounded function /: к -> a modulo D, i.e., iff for every g <d f there is a constant
? < a such that g <dY-
Weak normality trace of ultrafilter D over к is then defined as WNtr(D) = {a \ D is
?-weakly normal).
Some elementary facts we give in the following lemma.

1280
A. Jovanovic
Lemma 4.1.
A) If D is "only" normal or "only" weakly normal over k, then
WNti(D) = {k};
if D is ordinary ultrafilter then WN tr(D) = 0;
B) if a e WNtr(D) then in ^rk there is a minimal element below D in the layer of
ultrafilters of norm a.
As a consequence we have that ultrafilters with |WNtr| > 1 are essentially multiply
minimal in ^rk-
The existence of ?-weakly normal ultrafilters for a smaller than the cardinality of the
index are interesting. The first examples come from properties like an instance of strong
compactness. Namely if к is ?-strongly compact then there is an к -complete uniform
ultrafilter ? over ? which is (k, X)-regular. It is simple to check that WNtr(?) = {a < ? |
cf. a ^ k], which can be scaled down using the following theorem.
THEOREM 4.1. Weakly normal trace is preserved by c.c.c. extensions.
Consequently, starting from an instance of strong compactness, (ultrafilter E) tuning a
little Solovay forcing and Silver's construction, one can get an ultrafilter F over ? < 2?,
with WNtr equal to WNtr(?).
RK extension to RV measures. Let fi(k) denote the space of all ultrafilters over k. Let
M(k) be the space of all [0, 1] total measures on k. Clearly fi(k) с М(к). We can naturally
extend ^rk from fi(k) to ? (к) using (+) from Solovay equiconsistency proof: for v,
? e M(k) put
i'^rksM iff ? (?) = ?(/?_1(?)) for some h e kk.
Similarly to RK put %RKs for ? <RKs ? and ? sSrks v. Put rRKs^) = {v\ ? %RKs M)· In
this way we obtain a combined classification for binary and real valued measures.
List some facts for RKS.
Lemma 4.2.
(i) For every ? e ? (к), | rRKs (?) I ^2*;
(ii) \M(k)/^RKS\=22k-
(iii) ? e P(k) implies ?^?(?) = trks^). thus ultrafilters and measures do not mix.
(iv) P(k)/^RK с M(k)/^RKS; in fact P(k)/^RK is an initial part in <rks·
(?) ? ^rks ? implies that add(i>) < add^) and \\v\\ ^ ||?||,
(vi) normal, weakly normal and ?-weakly normal ultrafilters keep their positions in
^rks; normal measures are minimal among measures.
It would be interesting to see if other normality properties (weakly, a -weakly) can nicely
be translated to measures and learn more about possible complexity of ^rks·

Real valued measurability, some set-theoretic aspects
1281
5. Measures and normal Moore spaces
Moore spaces are regular spaces, which have a development. The problem if every normal
Moore space is metrizable, known as Normal Moore Space Conjecture (NMSC) was
unresolved for decades, attracting exclusive attention. Treatment of number of related
and restricted properties preceded its solution by Nyikos and Fleissner. A lot of work
contributed to the fact that this problem is nicely studied and investigated in a variety
of details (Tall A984)). Essential in problem solution was use of the following strong
property.
PMEA. For every k, the product measure on 2k can be extended to a total 2H° additive
measure.
Using Solovay forcing Kunen proved (Fleissner A984)) that consistency of ZFC + 
strongly compact cardinal" implies consistency of ZFC + PMEA.
First half of the normal Moore space problem was solved by Nyikos A980):
THEOREM (PMEA). Let X be a space of character < с Every normalized collection of
subsets of X is well separated.
PROOF. Let X be a space of character < с and let [Ca I a < ?) be a normalized family
of subsets of X. Thus for each Л с ?, there exist disjoint open subsets UA and VA of X
containing \^j[Ca I a e A] and \J{Ca \ ? ? A] respectively.
Let ? be a c-additive measure, extending the usual product measure on 2A, such that
?(?) exists for all А с 2?. We identify ? with the set of all ordinals whose cardinal is < ?.
For each a e ?, let
Ba = {fe2x\f(a)=l].
NotethatMB^= 1 and ?(?? - ??) = 1/4 for all ?, ? e ?, ? ? ?.
Each / e 2? is the characteristic function of some subset A/ of ?: Af = {a e ? | /(a) =
1}·
For a given a and a given ? e Ca, let {UY(p)\y < kp] be a base of open sets for the
neighborhoods of p, with kp < с Let
MP, Y] = {/ e 2? | Uy(p) с UAf or UY(p) С VAj}.
For a fixed p, \J{A[p, ?]\ ? < kp] = 2?. This is because, for any given / e 2?, either
Ua{ or VAf is an open set containing Ca, hence there is а у such that UY(p) is contained
in the one which contains Ca.
Because ? is c-additive and {A[p, ?] | ? < kp] is a collection of fewer than с subsets
that fill up 2?, the transfinite sequence
\jA[p,
? <?
o^s^kp

1282
A. Jovanovic
converges monotonically to 1. By the principle of Archimedean order, there exists a finite
set of indices {?\ (?),..., ?„?(?)} such that
d\jA[p,Yi(p)]\>T/*.
Let ?? be such that
tip
(P)(P)·
1 = 1
Then ?(?[?,?,,])> 7/8.
If we do this for all a and ? e Ca, we then have
?(?[?,??]??[?,?4])>3/4
for all p, q. Suppose peCa,qeCfi,a^p. Because ?(?-?) = 1/4 there exists
/ e A[p, Yp] П A[q, ?4] П (Ba - Bp).
From /(a) = 1, /(/3) = 0 it follows that ? e UAr q ? VAf- And from / e A[p, y,,] it
then follows that UYp(p) CUAf. Similarly, UY<l(q) CVAr Therefore,
Uyp(p)nUY4(q) = e>.
For each a, let
Ua=\J{Uyp(p)\peCa}.
Each Ua is an open set containing Ca and by the preceding paragraph,
????? = ? ????^?.
Hence {Ca I a < ?} is well separated. ?
COROLLARY (PMEA). Every normal Moore space is metrizable.
After theorem of Nyikos the obvious question remained: could PMEA be replaced with
some weaker property in the theorem of Nyikos? More precisely: what is the consistency
strength of NMSC? This enigma was resolved by Fleissner who proved the following
theorem (Fleissner A984)).
THEOREM 5.1. Consistency o/ZFC + NMSC implies consistency o/ZFC +  a
measurable cardinal".

Real valued measurability, some set-theoretic aspects
1283
The complete characterization of NMSC consistency strength meets similar difficulties
like RV strongly compact. Mitchell made it more precise by proving consistency of
hypermeasurable from NMSC. Together with original work of Solovay the above leads
to a sort of more old style mathematical and less set-theoretic conclusion.
ZFC +  total ? additive measure extending Lebesgue measure" is equiconsistent with
ZFC + "a large portion of NMSC".
Formulation "a large portion of NMSC" should mean that there must be some rather
exotic nonmetrizable example of NMS consistent with ZFC +  total ? additive extension
of Lebesgue measure".
In this way "existence of total ? additive extension of LM" is closely related to
"metrizability of all NMS". It is interesting that these properties remain related in
weakened form.
Concerning LM extensions first mention that Banach and Kuratowski proved:
CH implies that LM can not be extended by even countable many sets.
The following result of Carlson and Prikry solves the middle case with respect to
measure extensions.
Theorem 5.2.
(a) If set theory is consistent, it is consistent to assume in addition that 2 " is anything
reasonable and that given any < 2K(I sets of reals LM may be extended to measure
all of them.
(b) If the consistency of к weakly compact is assumed in addition then LM may be
extended to measure 2K° sets.
Beside PMEA people are using some weaker forms.
CMEA (c Measure Extension Axiom). Given с subsets of c2 there is a c-additive measure
extending the usual product measure and defined on those sets.
WMEA (Weak Measure Extension Axiom). For any к < с, for any collection of к subsets
of k2, the usual product measure on k2 can be extended to a A;-additive measure measuring
these sets.
Consistency of WMEA is proved from the consistency of ZFC, consistency of cMEA
from weak compactness.
THEOREM (Tall A984)). PMEA (cMEA) (WMEA) implies normal Moore spaces (of
weight < c) (of weight < c) are metrizable.
6. Counting remarks
What is so nice about a countable set? The fact that we can count it! Set counting is a
sort of simplest measurement = determining the size of a set. Since Cantor discovered
that infinity is not unique and that there is a vast array of sets (Va, a e Ord), each more
numerous than its predecessors, one basic need remains: try to maintain size counting as

1284
A. Jovaiwvic
simple as needed. This is usually accomplished with the axiom of choice (AC) implying
that all sets are bijective to (initial) ordinals (alephs). In fact well orderability of arbitrary
set means that it can be counted step by step, one element at each step, through transfinitely
many steps if necessary, but this procedure "ends" certainly, in the similar way the counting
of finite or countable sets is done. Thus
2 " = Nff+/(ff).
The index function / is nondecreasing and satisfies cofinality condition cf(Kff+/-(ff)) >
Cf(Kc).
(From Cantor Theorem) f(a) ^ l,aeOrd.(CHo)/@)= 1 and (GCH <s>) f(a) = 1,
a & Ord hold in L. /@) = 2 (for example) in Cohen's original model. Easton proved
that for regular cardinals / can take arbitrary values (obviously respecting cofinality and
nondecreasing conditions). So, for regular Ha, f(a) can be: 1,2, 3,..., ? + 1,..., ?\,...
it can be ? + 1,..., ??+\,.... Generally f(a) ^ 2Ku. When /(?) = 2?? we could say that
it is reasonably unbounded at a and then 2Ku = N2n„ .
Quite surprisingly, Silver A974) proved that for singular cardinals displacement /
depends on its values on predecessors:
THEOREM 6.1. If к is a singular cardinal of uncountable cofinality then:
2"=?+??(?:) implies 2k = k+.
(Here AE(k) stays for "stationary in k'\) Thus, if GCH holds AE below к it is true at к
as well. More generally, the way f is bounded AE bellow к will determine the way it is
bounded at k.
Regular cardinals satisfying equation K„ = a are weakly inaccessible. Regular к is
strongly inaccessible if ? < к implies 2Л < к. GCH implies that weakly inaccessible
cardinals (wine) and strongly inaccessible cardinals (sine) coincide. Existence of inaccessible
cardinal can not be proved in ZFC. Even, nicer, in ZFC + 1 sine" the consistency of
ZFC is proved, hence by Godel theorem ZFC + 1 sine" is stronger theory. If к is the 1st
sine, it is followed by 2nd, 3rd, ..., kth, ... then by hyper inac, hyper hyper, etc., quite
exhaustive sequence of properties one stronger than the predecessors and with the stronger
consistency strength.
The inaccessibility properties have been iterated to postulate bigger, stronger and
consistency-stronger cardinals, all the way up through the ordinal line. Nice journey
through their world is given in Drake A974), Jech A978).
Ulam A930) proved that measurable cardinal is sine. The results of Hanf, Scott, Keisler
and Tarski in 1960s established that inaccessible bearing weakly compact are very large,
but that measurable cardinals are even much larger. In fact if к is measurable with normal
measure ? then {a < к | a e sine) has measure 1. The same with all of inaccessible
properties, the same for weakly compact: {a < к \ a is WC) has measure 1.
Similar concentration properties show strongly compact, supercompact, extendible
cardinals, Vopenka principle and huge cardinals, scratching the ceiling of cardinal skies,

Real valued measurability, some set-theoretic aspects
1285
i.e., where there is no more room for iteration of cardinal properties, since they start
producing inconsistency. Thus, position in the hierarchy of large cardinals, practically the
coordinate on the Ord line corresponds to the consistency strength of a given property. In
fact the Ord line serves as simplest space (line) where relative position of sets reflects
their relative consistency strength. It might seem that there are uselessly many strong
cardinal properties, that those things cannot exist or that the cost/benefit ratio is weakly
promising when using them in "mathematics". However, if they serve in solving harder
exercises or problems, or even more, e.g., allow us to characterize properties or problem
solutions, then no doubt we will make no faster progress if we ignore them when they are
involved!? Equalization or weighing the properties on the Ord scale tightly connects sets of
mathematical properties with syntactic-language properties, with elementary embeddings
of models of set theory, with combinatorial properties of binary measures or their
ultrapowers incompressible (normalizing) elements. Strong axioms of infinity are nicely
mapped in Drake A974), Jech A978), usually divided in small large, medium large (weakly
compact) and large large cardinals (> measurable).
Consistency of GCH with existence of measurable cardinal was proved by Silver
(?,[?] ? GCH). In fact similar theorem to the Silver's for singular cardinals holds for
measurable cardinals (the first restriction of Eastons freedom):
THEOREM 6.2. If к is measurable (with normal measure ?) then
2V = v+ AE(k) implies 2k = k+
(f(v)=\ AE implies f(k)=\).
Thus if GCH holds AE below к then it is true at к as well. More generally, the way f is
bounded AE below к will determine the way it is bounded at k:
f(k)Kot(j\(f(a),e)\
? '
Kunen proved that 2k > k+ for measurable к is stronger property than measurability,
i.e., allowing models with many measurable cardinals.
That the size of continuum with total measure is big proved by Ulam A930):
THEOREM 6.3. [f к is a real valued measurable (< 2K(I) then к is weakly inaccessible.
Solovay proved that if к is real valued measurable then it is the kth inaccessible and
Mahlo cardinal.
Consequently if с is real valued measurable then:
(a) с is cth cardinal;
(b) 2K° = Кп+дО) = K/@) = N2tv or /(°) = 2*">thus / is unbounded at 0.
In other words if 2K° is real valued measurable then 2K(I makes extraordinary jump in
Ord. How peculiar it is, probably the best illustration is provided by the following theorem
of Prikry.

1286
A. Jovanovic
THEOREM 6.4. //2K° is real valued measurable then 2K = 2*° for all infinite ? < 2*4
Hence, the continuum displacement function /, first makes enormous jump at 0, then it
is constant for all indexes smaller than its value at 0.
2«"=Ka+f{ah /@) = 2K° and
2K| = 2*2 = · · · = 2Ka = · · · = 2K» for a < 2*°,
thus /@) = /A) = ··· = /(?) = ··· = 2?? fora<2K°,
quite nice diversity from that / = 1 at GCH to this exceptional - strongly hyperbolic
geometry of the universe in the presence of the total measure on reals. Thus geometrically
we could say that the total measure on reals strongly (locally) curves the universe of
mathematics.
Here we list some examples of continuum displacement behavior influenced by
nonregular ultrafilter.
Let D be a jumping ultrafilter over K|5 and K^4 = K|5. If ? ???? ^ K>5 then 2*'5 <
Кзо. If 2K° = ??+? then there is no jumping ultrafilter over N15. For displacement / in
2K" = На+ца), in circumstances similar to above for Ha, we have /(?) ^ a, thus / is
well bounded.
Remark. The measurement | |: V -> Cards is in fact a nice universal, totaly additive
measure. Its restriction to Va+\ (a e Ord) ranges within the well ordered N^+,(a) and
measures all subsets of Va. For regular | Va |, it is essentialy | Va \ = Д,-additive: X с Va,
such that |X| = | Va\, is X = U/j<y УА an^ ? <^a tnen for some ? \??\ — |X|.
7. Some complementary results
Reducing axiom of choice AC eliminates Vi tali's examples of Lebesgue nonmeasurable
sets. Back in 1964 Solovay proved the following theorem, published in Solovay A970).
THEOREM 7.1. Suppose that there is a transitive model o/ZFC + "There is a strongly
inaccessible cardinal" (ZFC + /).
Then there is a transitive model of ZF in which the following propositions are valid.
A) The principle of dependent choice (= DC).
B) Every set of reals is Lebesgue measurable (LM).
C) Every set of reals has the property ofBaire.
D) Every uncountable set of reals contains a perfect subset (P).
E) Let {Ax I x e R) be an indexed family of nonempty sets of reals with index set the
reals. Then there are Borelfunctions, h\, hi mapping К into R such that
(a) {x I h\ (jc) ^ Ax] has Lebesgue measure zero.
(b) {x I h2(x) ? ??] is of first category.

Real valued measurabllity, some set-theoretic aspects
1287
Mycielski and Swierczkowski A964) proved that if ZF + DC + ? has a transitive
model, then ZFC + "There is a strongly inaccessible cardinal" has a transitive model. This
shows that the inaccessibility is indeed needed in this theorem's assumption. Concerning
ZF + DC + LM B), Solovay anticipated that the inaccessibility is not needed.
The Theorem 7.1 is related and complemented with the following two theorems.
THEOREM 7.2. Suppose that ZFC + / has a transitive model. Then so does the theory
ZFC + GCH plus analogs of B) to E) of Theorem 7.1. Analogs are obtained by substitution
of "set of reals " with "set of reals definable from a countable sequence of ordinals ". Thus
the analog of B):
B') Every set of reals definable from a countable sequence of ordinals is Lebesgue
measurable.
THEOREM 7.3. Assume the hypotheses of Theorem 7.1. Then there is a transitive model
of ZFC in which 2H° = tt> and the analogs of B)-E) of Theorem 7.1 referred to in
Theorem 7.2 are valid.
Instead of 2K° = Щ one can have a similar theorem with 2s" = anything reasonable.
With stronger assumption:
there is a transitive model of ZFC + "there is a strongly inaccessible weakly
compact cardinal",
Solovay made a model of Theorem 7.3 in which Martin's axiom is true.
Theorem 7.2 is proved in the model of Azriel Levy obtained in the following way. Let
9Л be the initial model:
9Л N ZFC + "There is a strongly inaccessible cardinal".
Let ? be strongly inaccessible in 3??. For an ordinal ? define Px to be the set of
functions / whose domain is a finite subset of ? ? ?, range (/) с ? and f({a,n)) < a if
{a,n) edom(f). Order ?? with с. If ? e ЭК then (?*K* = ??.
Let G be an OT-generic filter on ?? and let <Jt = 9K[G].
With very reach contrapunctus in cofinal set of lemmas developed on the techniques of
Levy, Solovay proves the fuge.
LEMMA 7.1. Let U be a set of reals in VI which is ЧЖ-R-definable. Then 9t N U is
Lebesgue measurable.
The other properties listed in Theorem 7.2 are proved to be true in 9? as well.
Further, let 9?? = (HOD)''1. Solovay proves that reals and sequences of ordinals in УХ
and 9?| are the same. DC holds in 9?| and if A e "Я\ then in 9?, A is iW-R-definable. In
9?? every set of reals has the property of Baire and every uncountable set of reals contains
a perfect subset.
Once again reconsidering Lebesgue measure (and its extensions) the closely related are
the properties and their intensities: Axiom of choice, additivity, translation invariance and
measure domain (i.e., total/partial on R).

1288
A. Jovanovic
Some more subtle property combinations treated in Pincus and Solovay A977), led to
the theorem of Pincus and Solovay.
THEOREM 7.4. It is consistent with ZFC that there is an I* such that:
(a) /* is a finitely additive translation invariant extension of Lebesgue measure to a
measure on [0, 1].
(b) No ultrafilter on the set of reals is definable from ordinals, reals and I*.
In the same paper it is shown that generally it is harder to obtain nonprincipal ultrafilters
than nonprincipal measures in ZFC, ZF + DC. The following very important theorems are
proved there.
THEOREM 7.5. It is consistent with ZFC that no nonprincipal ultrafilter on ? is ODR but
there exists a definable (in fact projective) nonprincipal measure on ?.
THEOREM 7.6. It is consistent with ZF + DC + Hahn-Banach theorem that every ultra-
filter on any set is principal.
The last theorem, nicely complements the theorem of Luxemburg A969) which proved
that in ZF Hahn-Banach theorem is equivalent to the existence of a finitely additive real-
valued measure on every Boolean algebra.
The model for Theorems 7.3 and 7.4 is the same, 9Л = L(r) where r is a random element
of2<4>.
9Л is similar to the models of Solovay A971). Continue with the model У1 of sets
hereditarily definable from ordinals, reals and F (the enumeration of orbits of countable
restrictions of r). The measures used to prove these theorems are of the kind introduced in
Solovay A971) but on different indexes.
8. Set-theoretic measure theory
The research of so far mentioned problems expanded to the set theoretic measure theory.
It is not easy to list all mathematicians with major contributions. The gigantic paper of
Fremlin A993) integrated impressive collection of results since Solovay set the base,
including comprehensive references and historic remarks. There has been general diffusion
of set theory into measure theory with exhaustive discussion of fine details.
The following two theorems from Gitik and Shelah A989) represented a major advance
in measure algebra complexity description.
THEOREM 8.1. Let к bean atomlessly measurable cardinal with witnessing probability v.
Then the Maharam type of(k, Pk, v) is at least min(k{+w\ 2k).
THEOREM 8.2. Let (X, PX, ?) be an atomless probability space with к = add(M). Then
the Maharam type of (X, PX, ?) is at least min(J((+"'), 2k), and in particular is greater
than k.

Real valued measurability, some set-theoretic aspects
1289
COROLLARY 8.1. With the assumptions of Theorem 8.2. Let (Z,3, ?,?) be a Radon
probability space ofMaharam type ? ^ min(A:(+";), 2k). Then there is an inverse-measure-
preserving function f:x—*Z.
Corollary 8.2. With the assumptions of preceding corollary, there is an extension of ?
to a k-additive measure on PZ.
Fremlin further developed measure and integral calculus in the presence of real valued
measurable cardinals. He gave a number of theorems with detailed characterizations
involving all aspects of real valued measurability.
The classification of indescribability hierarchy, combinatorics and partition properties
related to the real valued measurability is given. A number of forcing absoluteness, and the
Scott type reflection arguments known for binary measures are presented.
Let us mention the following consequences.
(a) Let к be an atomlessly-measurable cardinal and / a set of cardinal less than k, ? a
second order formula, Ci,..., Q relations on / and ?\, ?,„ e /. Then for any
random real p.o. set P,
(/,Ci,..., ?,„)>=*> iff ll-p(/.C| \m)t<p.
(b) Let Ci,..., C„ be relations and ??, ..., ?,? ordinals, all with definitions which are
absolute for random real forcing. Let ? be a second order formula such that
ZFC I- "for every atomlessly measurable cardinal к > max, <?„, ??,
^?\,...?„;?\,...,???)?=?".
Then for every atomlessly-measurable cardinal к > max,^„, ?,,
[a \a < k, (a; C\, ...,?,„)?=?}
belongs to the RVM filter of k.
Some examples of nice measure theoretic implications are given in the following
theorems.
THEOREM 8.3. Let к be a real valued measurable cardinal and ? a normal witnessing
probability on k. Let (?, ?) be a Radon probability space and f: ? ? к -> a bounded
function. Then
J(j' ?(?,????)\?(??)^ J (j'/(?,?)?(??)\?(??).
THEOREM 8.4. Let к be an atomlessly measurable cardinal, let Xq Xm-\ be Radon
probability spaces ofMaharam type less than к and let f: f]KM *i -+ R be a bounded

1290
A. Jovanovic
function. Let ? :m —» m be a permutation such that the two repeated integrals
f(xo,...,xm-.l)dxm-i J... \dxQ,
Г= / ( '"( / /(·?°'···'·?'"?)^·?^('"-1))··· )dx<r@)
/(-(/¦
both exist. Then I = /'.
THEOREM 8.5. Let (x, p) be a metric space.
(a) X is Bore! measure-complete iff there is no real valued measurable cardinals.
(b) If X is complete (as metric space) then it is Radon iff there is no real valued
measurable cardinal ^ d(X).
Further, impact of real valued measurable cardinals to the properties of p.o. sets is
presented. Finally, Fremlin A993) presents discussion of quasi-measurable cardinals.
References
Banach, S. A923), Surle probleme de la measure. Fund. Math. 4. 7-33.
Bing, R.H. A951), Metrization of topological spaces. Canad. J. Math. 3, 175-186.
Bing, R.H. A965), A translation of the normal Moore space conjecture. Proc. Amer. Math. Soc. 16, 612-619.
Chang, C.C. A967), Descendingly incomplete ultrafilters. Trans. Amer. Math. Soc. 126, 108-118.
Chang, C.C. and Keisler, H.J. A973), Model Theory. North-Holland, Amsterdam.
Cohen, P.J. A964), The independence of the continuum hypothesis I, II, Proc. Nat. Acad. Sci. USA 50, A963)
1143-1148,51A964) 105-110.
Cohen, P.J. A965), Independence results in set theory. The Theory of Models, J.W. Addison, L. Henkin and
A. Tarski, eds, North-Holland, Amsterdam, 39-54.
Comfort, W.W. and Negrepontis, S. A974), The Theory of Ultrafilters, Springer, Berlin.
Devlin, K.J. A973), Aspects of Constructibility, Lecture Notes in Math., Vol. 354, Springer, Berlin.
Dickman, M. A. A974), Large Infinitary Languages - Model Theory, North-Holland, Amsterdam.
Drake, F.R. A974), Set Theory, North-Holland, Amsterdam.
Easton, W.B. A964), Powers of regular cardinals, Ph.D. Dissertation, Princeton University A964); Ann. Math.
Logic 1A970), 139-178.
Fodor, G. A966), On stationary sets and regressive functions. Acta Sci. Math. (Szeged) 27, 105-110.
Fremlin, D.H. A993), Real-valued-measurable cardinals. Israel Math. Conf. Proc., Vol. 6, 151-304.
Fleissner, W.G. A974), Normal Moore spaces in the constructible universe, Proc. Amer. Math. Soc. 46, 294-298.
Fleissner, W.G. A975), When is Jones' space normal?. Proc. Amer. Math. Soc. 50, 375-378.
Fleissner, W.G. A976), A normal collectionwise Hausdorff not collectionwise normal space. Gen. Topology Appl.
6, 57-64.
Fleissner, W.G. A982a), Normal nonmetrizable Moore space from continuum hypothesis or nonexistence of inner
models with measurable cardinals. Proc. Nat. Acad. Sci. USA 79, 1371-1372.
Fleissner, W.G. A982b), If all normal Moore spaces are metrizable, then there is an inner model with a
measurable cardinal. Trans. Amer. Math. Soc. 273, 365-373.
Fleissner, W.G. A984), The normal Moore space conjecture and large cardinals. Handbook of Set-Theoretic
Topology, K. Kunen and J.E. Vaughan, eds, North-Holland, Amsterdam, 733-760.
Fleissner, W.G., Hansell, W. and Junnila, H.J.K. A982), PMEA implies Proposition P, Topology Appl. 13, 255-
262.

Real valued measurability, some set-theoretic aspects
1291
Godel, K. A938), The consistency of the axiom of choice and of the generalized continuum hypothesis. Proc. Nat.
Acad. Sci. USA 24, 556-557.
Godel, K. A939), Consistency-proof for the generalized continuum hypothesis. Proc. Nat. Acad. Sci. USA 25,
220-224.
Godel, K. A940), The Consistency of the Continuum Hypothesis, Ann. Math. Studies, Vol. 3, Princeton Univ.
Press, Princeton, NJ; printing 1966.
Gitik, K. and Shelah, S. A989), Forcing with ideals and simple forcing notions, Israel J. Math. 68, 129-160.
Hanf, W.P. A960), Models of languages with infinitely long expressions. Intern. Cong. Logic, Methodology and
Philosophy of Science, Stanford University, Abstracts of contributed papers (mimeographed).
Hanf, W.P. A964), lncompactness in languages with infinitely long expressions. Fund. Math. 53, 309-324.
Hanf, W.P. and Scott, D.S. A961), Classifying inaccessible cardinals. Notices Amer. Math. Soc. 8, 455.
Jech, T.J. A967), Non-provability of Souslin 's hypothesis. Comment. Math. Univ. Carolin. 8, 291-305.
Jech, T.J. A968), ?, can be measurable. Israel J. Math. 6. 363-367.
Jech, Т., Magidor, M., Mitchell, W. and Prikry, K. (unpublished). Precipitous ideals.
Jech, T.J. A971a), Trees, J. Symb. Logic 36, 1-14.
Jech, T.J. A971b), Lectures in Set Theory with Particular Emphasis on the Method of Forcing. Lecture Notes in
Math., Vol. 217, Springer, Berlin.
Jech, T.J. A978), Set Theory, Addison-Wesley, New York 1978.
Jensen, R.B. A972), The fine structure of the constructible hierarchy, Ann. Math. Logic 4, 229-308.
Jones, F.B. A937), Concerning normal and completely normal spaces. Bull. Amer. Math. Soc. 43, 671-677.
Jones, F.B. A966), Remarks on the normal Moore space metrization problem. Ann. Math. Stud. 60, 115-120.
Jovanovic, A. A980), Uniform measures over continuum. Teoria della misura e sue applicazioni, atti del
convegno, Trieste, 127-129.
Jovanovic, A. A981), On real valued measures. Open Days in Model Theory and Set Theory, Proc. of a
Conference held at Jadwisin, Poland, W. Guzicki, W. Marek, A. Pelc and С Rauszer, eds, 145-152.
Jovanovic, A. and Mijajlovic, Z. A993), On weak normality and selectivity ofultrafilters. Bull. Greek Math. Soc.
35, 3-9.
Keisler, H.J. A971), Model Theory for Infinitan Logic, North-Holland, Amsterdam.
Keisler, H.J. and Tarski, A. A964), From accessible to inaccessible cardinals. Fund. Math. 53, 225-308;
corrections: ibid. 57, 119.
Ketonen, J. A971), Everything you wanted to know about ultrafilters. but were afraid to ask. Dissertation, Univ.
of Wisconsin, Madison, WI.
Ketonen, J. A972a), On non-regular ultrafilters, J. Symb. Logic, 37, 71-74.
Ketonen, J. A972b), Strong compactness and other cardinal sins, Ann. Math. Logic 5, 47-76.
Ketonen, J. A973), Ultrafilters over measurable cardinals. Fund. Math. 77, 257-269.
Ketonen, J. A974), On the existence of p-points.
Ketonen, J. A976), Non-regular ultrafilters and large cardinals. Trans. Amer. Math. Soc. 2241.
Ketonen, J. A976), Open problems in the theory of ultrafilters.
Kunen, K. A968), Inaccessibility properties of cardinals, Ph.D. Dissertation, Stanford Univ., Stanford.
Kunen, K. A970), Some applications of iterated ultrapowers in set theory, Ann. Math. Logic 1, 179-227.
Kunen, K. A971), On the GCH at measurable cardinals. Logic Colloquium '69, R.O. Gandy and C.E.M. Yates,
eds, North-Holland, Amsterdam, 107-110.
Kunen, K. A980), Set Theory, An Introduction to Independence Proofs, North-Holland. Amsterdam.
Kunen, K. A984), Random and Cohen reals. Handbook of Set-Theoretic Topology, K. Kunen and J.E. Vaughan,
eds, North-Holland, Amsterdam, 887-911.
Kunen, K. and Prikry, K.L. A971), On descendingly incomplete ultrafilters. J. Symb. Logic 36, 650-652.
Kuratowski, K. and Mostowski, A. A968), Set Theory. North-Holland, Amsterdam.
Kurepa, D. A935), Ensembles ordonnes et ramifies, Publ. Math. Univ. Belgrade 4, 1-138.
Laczkovich, M. B002), Paradoxes in measure theory. Handbook of Measure Theory, E. Pap ed., Elsevier,
Amsterdam, 83-123.
Levy, A. A960), A generalization ofGodel's notion of constructibiliry, J. Symb. Logic 25, 147-155.
Luxemburg, W.A.J. A969), Reduced products of the real number system. Applications of Model Theory to
Algebra, Analysis and Probability, Holt Reinbout and Winston, New York.

1292
A. Jovaitovic
Magidor, M. A974a), How large is the first strongly compact cardinal?
Magidor, M. A974b), Combinatorial characterization of supercompact cardinals. Proc. Amer. Math. Soc. 42,
279-285.
Magidor, M. A976), How large is the first strongly compact cardinal?. Ann. Math. Logic 10, 33-57.
Magidor, M. A977), On the existence of nonregular ultrafilters. Technical report.
Menas, Т.К. A973a), On strong compactness and supercompactness. Dissertation. Univ. of California, Berkeley.
Menas, Т.К. A973b), On strong compactness and supercompactness. Thesis, University of California, Berkeley.
Mostowski, A. A948), On the principle of dependent choices. Fund. Math. 35. 127-130.
Mycielski. J. and Swierczkowski. S. A964). On the Lebesgne measurabiliry and the axiom of determinateness.
Fund. Math. 54, 67-71.
McAloon, K. A971), Consistency results about ordinal definability, Ann. Math. Logic 2, 449^467.
Nyikos, PJ. A980), A provisional solution to the normal Moore space problem, Proc. Amer. Math. Soc. 78,
429^35.
Prikry, K.L. A970), Changing measurable into accessible cardinals. Dissertationes Math. (= Rozprawy Mat.)
68.
Prikry, K.L. A971), On a problem of Gillman and Keisler. Ann. Math. Logic 2. 179-188.
Prikry, K.L. A975), Ideals and powers of cardinals. Bull. Amer. Math. Soc. 81, 907-909.
Pincus, D. A974), The strength of the Hahn-Banach Theorem. Victoria Symposium on Nonstandard Analysis.
Lecture Notes in Math., Vol. 369, 203-248.
Pincus, D. and Solovay, R. A977), Definability of measures and ultrafilters. J. Symb. Logic 42 B), 179-190.
Przymusinski, T. A975), A note on collectionwise normality and product spaces. Collect. Math. 33, 65-70.
Przymusinski, T. and Tall, F.D. A974), The undecidability of the existence of a non-separable normal Moore
space satisfying the countable chain condition. Fund. Math. 85. 291-297.
Rowbottom, F. A964), Doctoral dissertation. University of Wisconsin.
Sacks, G.E. A969), Measure-theoretic uniformity in recursion theory and set theory. Trans. Amer. Math. Soc.
142,381^420.
Scott, D.S. A961), Measurable cardinals and constructible sets. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron.
Phys. 7, 145-149.
Shelah, S, A972), Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math.
Shoenfield. J.R. A967), Mathematical Logic, Addison-Wesley. Reading. MA.
Silver, J.H. A966), Some applications of model theory in set theory, Ph.D. Dissertation. Univ. of California,
Berkeley.
Silver, J.H. A971), The consistency of the GCH with the existence of a measurable cardinal. Axiomatic Set
Theory, D.S. Scott, ed., Proc. Symp. Pure Math., Vol. 13, Part 1, Amer. Math. Soc., Providence, RI, 391-396.
Silver, J.H. A974), On the singular cardinals problem, Proc. of Intern. Congress of Mathematicians, Vancouver
1974, 265-268.
Solovay, R.M. A970), A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92,
1-56.
Solovay, R.M. A971), Real-valued measurable cardinals. Axiomatic Set Theory, D.S. Scott, ed., Proc. Symp.
Pure Math., Vol. 13, Part 1, Amer. Math. Soc.. Providence. RI. 397^428.
Solovay, R.M. A972), Strongly compact cardinals and the G.C.H., Proc. 1971 Tarski Symp.
Solovay, R.M., Reinhardt, W.N. and Kanamori, A. A978), Strong axioms of infinity and elementary embeddings.
Ann. Math. Logic 13, 73-116.
Sierpiiiski, W. A938), Functions additives non-completement additives et functions non-measurables. Fund.
Math. 30, 96-99.
Tarski, A. A930), Une contribution a la theorie de la mesure. Fund. Math. 15, 42-50.
Tarski, A. A945), ldeale in vollstandige Mengenkorpen 11. Fund. Math. 33, 51-65.
Tall, F.D. A969), Set-theoretic consistency results and topological theorems concerning the normal Moore space
conjecture and related problems. Thesis, University of Wisconsin, Madison, 1969; Dissert. Math. 148, 1-53.
Tall, F.D. A974a), On the existence of normal metacompact Moore spaces which are not metrizable. Canad. J.
Math. 26, 1-6.
Tall, F.D. A974b), ?-points in ?? - ?, normal nonmetrizable Moore spaces and other problems of Hausdorff,
TOPO-72, General Topology and its Applications. Proc. Univ. of Pittsburgh Topology Conf., 1972. R.A. Lo
et al., eds. Springer, New York, 501-512.

Real valued measurability, some set-theoretic aspects
1293
Tall, F.D. A979), The normal Moore space problem. Topological Structures II, P.C. Baayen and J. van Mill, eds.
Math. Centre, Amsterdam, 243-261.
Tall, F.D. A982), Collectionwise normality without large cardinals, Proc. Amer. Math. Soc. 85, 100-102.
Tall, F.D. A984), Normality versus collectionwise normalin. Handbook of Set-Theoretic Topology, K. Kunen
and J.E. Vaughan, eds, North-Holland, Amsterdam, 685-732.
Vopenka, P. and Hrbacek, K. A966), On strongly measurable cardinals. Bull. Acad. Polon. Sci. Ser. Sci. Math.
Astron. Phys. 14, 587-591.
Ulam, S. A930), Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16, 140-150.

CHAPTER 32
Nonstandard Analysis and Measure Theory
Peter A. Loeb
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL6I80I, USA
E-mail: loeb@math.uiuc.edu
Contents
1. Introduction 1297
2. Extending the real number system , 1297
3. Calculus 1300
4. Further principles 1303
5. Set-theoretic measure theory 1303
6. A lattice approach to measure theory 1305
7. Internal functionals on continuous functions 1312
8. Lebesgue measure 1314
9. Representing measures in potential theory 1314
10. Poisson process 1316
11. Anderson's Brownian motion . . , 1317
12. The martingale convergence theorem , 1319
13. On an infinite number of independent random variables , 1322
14. Exact law of large numbers and independence 1326
References . . . , 1327
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1295

This page intentionally left blank

Nonstandard anahsis and measure theory
1297
1. Introduction
Infinitesimal numbers have been used by mathematicians for thousands of years as a means
of expressing intuitive mathematical ideas. Such numbers were always considered to be a
useful fiction since for a positive number to be infinitesimal, it would have to be greater
than 0 and yet smaller than any positive number you might write using a decimal expansion.
This fiction has always had its critics. In 1734, Bishop Berkeley called the infinitesimals
of Newton's calculus foundations "ghosts of departed quantities." He was questioning the
intellectual consistency of atheists who believed in infinitesimals.
It was not until 1960, that the "fiction" of infinitesimals became a reality with the
work of Abraham Robinson [28], Now, once Robinson's discovery is understood, one
can use it to obtain new results in many areas of current mathematical research. The key
to these applications is found in the strong intuitive feeling we have for the behavior of
infinite mathematical structures based on an understanding of related finite structures.
Using Robinson's work, we can transform finite constructions into infinite ones without
losing the initial combinatorial properties.
Here is an example: Brownian motion is the random motion of microscopic particles
suspended in a liquid or gas. An important part of mathematical probability is the
construction of mathematical models for this motion. An intuitive model for Brownian
motion works with a particle performing a random walk with steps of infinitesimal length.
For this model, one divides time into infinitesimal intervals; in each time interval, the
particle moves in a straight line over an infinitesimal distance. At the end of each time
interval, the particle chooses at random a new direction for its motion. Until recently,
probabilistdid not know how to make a mathematically correct version of this construction,
but nevertheless, they thought of Brownian motion in these terms. For them, it was a useful
fiction. Now, as we shall see, it is a correct model for Brownian motion.
2. Extending the real number system
Before saying more about this aspect of probability theory, as well as measure theory in
general, we first discuss Robinson's extension of the real number system. A more detailed
discussion can be found in the first two chapters of [12,24]. Just as the real numbers can be
constructed from Cauchy sequences of rational numbers, a number system with infinitely
small and infinitely large numbers can be constructed from sequences of reals. Sequences
tending to 0 represent infinitesimal numbers, and sequences tending to +oo represent
infinitely large numbers. Of course, one never works with equivalence classes of Cauchy
sequences of rational numbers when doing real analysis. Similarly, the construction using
sequences to extend the real numbers is for the most part irrelevant in applications. Since it
is important to convey how one actually works with infinitesimals, we concentrate on what
is relevant for applications.
Any particular project in which mathematics is applied employs a universe built from
a set X of individuals; perhaps X is the set of real numbers or the set of real numbers
together with points in some topological space or some Banach space. These individuals
may in fact be sets, but this is ignored in all subsequent considerations. The superstructure

1298
PA. Loeb
V(X) based on X consists of all of the set theoretic objects one can form from X in a finite
number of steps. All subsets of X are in V(X)\ also the set of all functions from X into the
set of all subsets of X is an element of V(X), etc. For now, we assume that we are working
with the set of real numbers R and the superstructure V(R); the general case is essentially
the same. Each Borel subset of R is an element of V(R); the collection of Borel subsets
of R is an element of V(R); every Borel measure on R17 is an element of V(R); etc. The
point is, V (R) contains everything one needs to consider in doing ordinary real analysis.
One augments R with additional points when working with some larger structure such as
a topological space.
What do we really know about V(R)? With the few exceptions of folklore and some
theorems hidden in desks, our knowledge is contained in the books of any superb library.
If we assume a common notation, that knowledge can be exhibited by a list of theorems.
We think of V (R) as "reality", but our only hold on that reality is a collection of theorems.
In this chapter, we need to make a distinction between reality and the strings of symbols
that we use to describe reality.
Let us start with the unrealistic assumption that every object in V(R) has a name. Of
course, there is a difference between a name and an object. The symbol 5 is different
from the number it names; for example, the Roman numeral V names the same object.
The symbol sin is not a function; it is a symbol we use to name a function. Using the
names of objects in V (R) together with quantifiers, brackets and other logical symbols, we
form theorems about V(R). We assume that all possible theorems about V(R) are gathered
together in a collection T. This is an idealized collection that contains every statement we
actually know to be true about V(R).
Robinson showed that the set R (or in general X) can be imbedded in a larger set *R
(or *X) so that the following properties hold:
A) Every name of an object in V (R) is associated as the name of a similarly constructed
object in V(*R). (For example, the name "sin" names a function from *R to *R.)
Names of elements of R name the same element when R is imbedded in *R.
B) (Transfer Principle) The scope of the universal quantifier V can be limited so that
when a theorem in ? is interpreted in V(*R) using the new meaning of the names
that appear in that theorem, the theorem is again true.
C) *R contains positive infinitesimal numbers, i.e., numbers that are larger than 0 but
smaller than 1 /n for any ordinary natural number ?. The reciprocal of such elements
are infinite or "unlimited" in that they are greater than ? for any ordinary natural
number ?.
To understand the Transfer Principle, we need to understand that "all" is a primitive
notion. There is no way we can specify what we mean even by "all subsets of the natural
numbers"; in terms of the language we actually use, we can only describe a countable
number of such sets. It is possible, therefore, to cheat, and in a consistent way limit the
meaning of "all" so that the Transfer Principle holds.
Although *R is not unique, we will assume that we have fixed such a set so that all
three properties are satisfied. Now we have two "realities" described by the theorems in T.
Working just with T, we cannot distinguish between V(R) and V(*R). For this reason,
intuition about the real number system can be transferred to the larger number system. At
the same time, we can take note of the differences between the two systems and obtain

Nonstandard analysis and measure theory
1299
new information not codified in T. This means that we have two modes of viewing and
working with V(*R). On the one hand, we see it as the usual system in which we do real
analysis, and on the other hand, it is a larger number system.
To say more, we need some terminology. Each object A in V(R) is called standard. The
object with the same name and same formal properties in V(*R) is denoted by *A; *A is
called the nonstandard extension of A. We do not use a star in denoting the elements of R
when imbedded in *R. For example, 5 is the same number regardless of whether it is in R
or*R.
An object in the superstructure V(*R) is called internal if it is an element of the
nonstandard extension of a standard object. This means, for example, that each element of
*R is internal, but otherwise, this is not a helpful definition. The following facts are helpful.
First, the nonstandard extension of a standard object is internal. Second, any object that can
be described just in terms of objects already known to be internal is itself internal. We call
such a description "an internal description." Every element of an internal set is internal.
Noninternal objects are called external. In interpreting the sentences of ? in V(*R), we
interpret "for all" as meaning "for all internal." For example, we interpret "for all subsets
of R" in V (*R) as meaning for all internal subsets of *R.
Let N denote the set of natural numbers. We call *N the nonstandard natural numbers.
Where are the new elements? Using ? to denote "or", a theorem of ? states that
V/ieN, /isC3 =>· n = lvn=2vn = 3.
We interpret this sentence as saying
V« e *N, ? <c 3 =>· ? = 1 ? ? = 2 ? ? = 3.
This means there are no new elements in *N below 3. With similar reasoning, we see that it
is necessary to go through all of the standard elements before we come to any new members
of *N. How do we know there are any new elements? A theorem of ? states that
VreR, 3/ieN with |r-«K 1.
Since there are elements of *R greater than any ordinary natural number, there must also
be elements of *N greater than any ordinary natural number.
We call the usual natural numbers limited, and we call the new elements of *N unlimited.
The set of unlimited elements of *N is denoted here by *NX\ it is an external set. To
see that this is so, we note that if *N;x were internal, then the theorem of ? that says
that every nonempty subset of N has a first element would force the existence of a first
unlimited element m. In this case, N would contain a last element, namely m — 1, but this
is impossible. It now follows that the set N of limited natural numbers is also an external
subset of *N; if it were internal, we could describe *N;x with the internal description
"*Noo := *N \ N". It is convenient that N is external, since the formal name for N is already
used for *N.
We call *R the nonstandard real numbers. By transfer of the properties of R, *R is an
ordered field extension of R. What does *R look like? First, there are elements greater

1300
P.A. Loeb
than any ordinary natural number; we call these the positive unlimited elements of *R.
Multiplying each of these by — 1, we have the negative unlimited elements of *R. Any
other element of *R is limited; this means that it is bounded in absolute value by some
ordinary natural number.
Among the limited elements are the infinitesimal elements. A number ? in *R is
infinitesimal if |?| < 1 /? for each ordinary natural number n. Zero is the only real number
that is infinitesimal. When two numbers a and ? differ by an infinitesimal, we write a ~ ?.
Of course, a ~ 0 means that a is infinitesimal.
If ? is limited in *R, it determines a Dedekind cut in R: (seR:Op).N R-" s < p].
This Dedekind cut, determines a unique real number r such that p — r. Thus, every limited
element ? e *R is infinitely close to a unique real number r called the standard part of p.
We write r = °porr = stp. We will write °p = +oo if ? is positive and unlimited, and
°p = — oo if ? is negative and unlimited.
Every real valued function / with domain А с R, has an extension */ mapping * A into
*R. All rules for / apply to */. For example,
V/oe*R, \?\ = ?J.
It follows that for each a e A, *f(a) = f(a). That is, */ is a true extension of /. We
will omit the star when */ is applied to points in A. We next give some examples due to
Robinson of how the calculus works using these extended functions.
3. Calculus
A function /defined on А с R is continuous at a point r e A, if and only if for each ? e *A
with ? — r we have *f(p) — f(r). That is. if you change from r by an infinitesimal,
you change the value of the function by an infinitesimal. If. for example, f(x) = .v, and
? ? ~ 0, then for each real number ?,
(? + AxJ = x2 +2-x ¦ ?? + (??J ~ ?2.
To see why this criterion for continuity holds, we assume that / is continuous in the usual
sense and fix r e A. Fix ? > 0 and a corresponding ? > 0 in R so that the following is a
theorem in T:
Vie A, \s-r\<& => |/(i)-/(r)]<e.
We interpret this theorem in V(*R) in terms of *A and */. If ? — r, then of course
|p — r| < <5, and so \*f(p) - f(r)\ < ?. Since ? is an arbitrary positive real number, it
follows that *f(p) — f(r).
To see why the converse implication holds, we use a principle called "Downward
Transfer." It says that if a sentence using only the names of standard objects is true for
V(*R), then it must be true for V(R) since its negation cannot be in T. Here, we assume
that for each ? e * A with ? ~ r we have *f(p) — f(r), and we fix an ? > 0 in R. Now the

Nonstandard analysis and measure theory
1301
following sentence is true when the symbols R, A, and / are interpreted as *R, * A and */,
respectively.
3<S > 0 in R such that Vs e A, |s - r| <S =>· |/(i) -/(r)| < ?.
The existence requirement is satisfied by any positive <5 ~ 0. It follows that what we have
written is a theorem in T. A similar proof shows that a function / is uniformly continuous
on a set А с R iff for each pair a, /3 in * A, a ~ /3 =>· */(<*) - */(?).
Here is an important theorem due to Robinson; it generalizes to arbitrary topological
spaces.
THEOREM 3.1 (Robinson). A set А С R is compact if and only if for each ? e * A, f/?ere
/j ana e A with ? — a.
PROOF. Assume A is compact but 3p e*A not in the monad of any standard point of A.
Then Va e A, |p - a\ is a limited, non-infinitesimal number, so 3<5„ > 0 in R such that
\p — a\ ^ <5„. Since A is compact, there is a finite set {??,..., an] с A and numbers
? ? <5„ in R+ with <5, = <$<,, so that using л for "and", the following sentence holds
forR:
(Va-) [x e А л \x -a\\ ^?\ л · · · л \х -дя-|| ^5я-1 =>¦ ^ -?«? < ^«]·
By transfer, |p - a„ \ < Sn, which contradicts the choice of <5„ — 5a„. It follows that if A is
compact, then each ? e *A is infinitely close to some element of A.
Conversely, if A is not compact, then it is easy to see that there is a sequences (On) of
open sets forming a covering of A with no finite subcovering. For each /; e N, let ? (?) be
an element of A not in the first ? sets Ot. Then we have
(Vn)(Vit) [??????????^?(?)(???].
By transfer, for /j e *?^, ? = *?(?) i*Ok for any standard к eN. Since for each a e A,
ЗА: e N with ? e *(?*, ? is not infinitely close to any standard point of A. D
COROLLARY 3.2. A continuous function f on a compact set A is uniformly continuous
on A.
PROOF. If we have a ~ ? in *A, then there is an r e A with ? ~ r ~ /3, and so
*f(a)~f(r)~*ftf). О
Robinson's compactness criterion is used in his proof that a continuous function /
defined on [0, 1] assumes its maximum value. First, fix ? e *Ncc, and consider the internal
subset S of *[0, 1] described as follows:
Vpe'R peS <==> 3k e *N with к sC ? and ? = —

1302
RA. Loeb
That is,
*=<1Д...,^,1
? ? ?
The set S is in internal one-to-one correspondence with an initial segment {1,2,..., H]
of the nonstandard natural numbers *N. A set with this property is called a hyperfinite set,
and we think of it as a finite set since it satisfies the formal definition of a finite set. That
is, let Vf denote the finite subsets of R. A subset of *R is hyperfinite if and only if it is
in *Vf- Any theorem about all elements of Vf is interpreted in V(*R) as a theorem about
all hyperfinite subsets of *R. In particular, every hyperfinite set has a maximum element.
To return to our problem, consider the hyperfinite set
*/? = |·/(^).·/(|) v(^).V(D
Let */ (k/H) be the maximum element of this set, and let r = st(k/H). Then / takes its
maximum value at r, since if s e [0, 1 ] and j/H is the next larger or equal point of S, then
/<·>=·/(?)<·/(?)=/(?.
Since the first and last terms are real numbers, f(s) ? f(r).
The notion of a hyperfinite subset is applicable for the extension of any standard set,
not just R. Hyperfinite sets are extremely useful since combinatorial results continue to be
valid even when the external cardinality is infinite.
The derivative of a function / at a point ? is not the ratio [f(x + ?*) — f(x)]/Ax, it is
the standard part of this ratio. For example, if f(x) — x2, then the ratio is
(x+AxJ-x2 2xAx + (AxJ
= = 2x + Ax — 2x.
Ax Ax
The integral of a continuous function / on [a,b], is given as follows after fixing
? e *Noo and setting Ax = (b - a)/H:
L
b
f(x)dx=st
T,*f(a + l-tr{b-a))Ax
Here, we have extended the summation operator to *Vf- In general, in applying the
integral, one partitions the interval [a, b] over which one is integrating into subintervals
[xi-\, Xi], where Axj =x,- -x,-\ ~0. One then approximates each part <2, of the quantity
one wants with the /th term of the hyperfinite Riemann sum. Let us express the error in
this approximation as the product e, - ?*, where e, e *R. As in [14], we want to have each
ej ~ 0, so that
\У2е' ' Ах'\ ^ (maxleil) · ? — ?) ~
0.

Nonstandard analysis and measure theory
1303
4. Further principles
We will from now on assume that we are working with a set X of individuals that
contains R but may be larger than R. A principle of nonstandard analysis that generalizes
the existence of infinitesimals in *R is the following: If a set S is an element of V(X),
then there is a hyperfinite set 5" с *S such that for each a e S,*a e 5". One can say that
S c. S' ?*S, and all of the containments are strict if S is infinite.
To see that this principle forces the existence of infinitesimals, one can take S equal to
the strictly positive real numbers, and let ? be the minimum element in 5". Here is a proof,
which generalizes, showing that if А с R is not compact, then there is an element ? e*A
that is not infinitely close to any element of A: Let {Oa} be an open covering of A with
no finite subcovering. Let {?/?} be a hyperfinite set extending the collection {*Oa}. By
assumption, there is a p e *A that is not in any ??? and thus is not in any *Oa. It follows
that ? is not infinitely close to any element of A.
Finally, to make nonstandard measure theory work, we need an additional property for
our nonstandard models, called ? ? -saturation. This means that any ordinary sequence
(A„: ? e N) of elements from an internal set ? is just the initial part of an internal
sequence {A„: ? e *N) of elements from E. Every "ultrapower" model, for example, has
this property.
5. Set-theoretic measure theory
We now turn to the nonstandard measure theory initiated by the author in [ 16]. Working in
an ? ? -saturated enlargement, we can construct a hyperfinite set Л as the set of elementary
outcomes in a conceptual experiment in the "nonstandard world". For coin tossing, for
example, Л can be the set of internal sequences of 0 and 1 of length ? e *Noc. Given
such a hyperfinite set A, we can let С consist of all internal subsets of A. The collection
С is an internal ?-algebra, but it is also an algebra in the ordinary sense. Suppose ? is
an internal probability measure on {A,C). For the coin tossing experiment, for example,
each internal set A would have internal cardinality \A\ and would be given the probability
?(?) = \?\/2? in *[0, 1]. In the general case, we can form a finitely additive real-valued
measure ? on (A, C) with values in the real interval [0, 1] by setting ?(?) = ??(?(?)). We
need to extend ? to a countably additive measure on the ?-algebra generated by C.
To consider this extension, let's start with an arbitrary internal measure space (A, C, ?).
We will assume that the internal measure ? takes only limited values, although a more
general theory is also available. When A is a hyperfinite set, one usually sets С equal to the
collection of all internal subsets of A, and ? is usually an internal probability measure. In
general, С is an internal ?-algebra as well as an algebra in the usual sense in the set A. Let
? be the finitely additive, real valued measure on (A, C) defined at each A e С by setting
?(?) = ??(?(?)). Let a(C) denote the ?-algebra in the ordinary sense (using the ordinary
natural numbers) generated by С The collection ?@ is, in general, an external collection
of subsets of A. We can extend ? to a measure ?/, defined on the measure completion
LM(C) of ?(?), and thus obtain a standard measure space (A, Ltl(C), ??.) on A, by using
the Caratheodory Extension Theorem in the following way.

1304
RA. Loeb
First note that when a sequence (A,-: /' e N), indexed by the ordinary natural numbers,
consists of pairwise disjoint elements of С and the union A is also in С then A is actually a
finite union since all but a finite number of the A, are empty. Here is the simple proof: Using
? ?-saturation, we extend the sequence (A,-: /' e N) to an internal sequence (A,: /' e *N);
the set
S:=|ine*N: AC [J A, |
is internal and contains *?^, so, since *N;x is external, S must contain some standard
natural number. It now follows that ?(?) = ???? М(Л)- By the Caratheodory Extension
Theorem, the finitely additive measure ? has a unique ?-additive extension ?/, defined
on the completion L?(C) of the ?-algebra a(C). Now (A,Lfl(C), ?^ is an ordinary
finite measure space formed on the internal set A. We note that in much of the literature,
(A, L?(C), ??.) is called a Loeb measure space and ?/, is called a Loeb measure.
Given ? in ??(?), it is well known that for any ? > 0 in R, there are sets A eCs and
В eC„ with А с ? с ? such that ??.(? \ ?) < ?. By saturation, we may assume that A
and В are actually sets in C. To see this, we suppose that (Bn: ? e N) is an increasing
sequence in С with limit B. We may extend the sequence to an internal increasing sequence
indexed by *N, and choose an unlimited integer ? such that ??.(?/) = 11гпиеК^^(В„).
A similar proof works for A. It also follows from saturation that there is a set С е С
such that the symmetric difference ? AC is a null set in LM(C). These set approximation
properties characterize L/((C). They have also been used by several authors to define
LM (C). (See, for example, [ 1,29].)
Given an *R-valued function / on A, we set °f(x) = st(f(x)) if f(x) is limited in *R.
We set °f(x) = +oo if f(x) is positive and unlimited in *R, and we set °f(x) = -oo
if f(x) is negative and unlimited in *R. It follows from the set approximation properties
that a function ^:X->KU {+oo, —do) is measurable with respect to L/( (C) if and only if
there is an internally C-measurable function / such that °f = g ??,-a.e. The function / is
called a lifting of g. If such an / takes only limited values, then g is integrable, and by the
set approximation properties
0 J /?? = J°fdvLL = jgdvLL. A)
If / ^ 0, then for each ? e N, / л /; ^ /, so
f"fdiiL ¦= \™? f{°f) ?????. = \imx f °(f ??)???. ? ° f f ??.
It follows that Equation A) holds when
|[?/|-(?/|?/?)]??~0 ?/,?*?^ B)
Condition B) is a simple expression of the condition called S'-integrability in the literature.
(See [3]; we have taken a different approach to arrive at this notion.) If, on the other hand,

Nonstandard analysis and measure theory
1305
we know that g is integrable, then it is easy to see that for a sufficiently small ? e *NX,
replacing / with the function / - Хц/1^,,) gives an S-integrable lifting of g. That is, with
this replacement, the internal integral of / is limited and Equation A) holds.
Let X be the nonstandard extension of a compact Hausdorff space Y. For each у е Y,
the monad of у is the set
m{y):= ? *0.
О open
ye О
Clearly, this definition generalizes the definition of a monad in the real line. Again,
Robinson's criterion for compactness holds: A set A is compact if and only if each point
in *A is in the monad of a point in A.
For each Borel set В с У, set
B:=\J{m(y): ye В}.
Let ? be an internal Baire probability measure on X. As in [5,19], a Borel measure ?
on ? can be defined by setting ?(?) = vl(B) for each Borel set В с ?. The measure
? restricted to the Baire sets is the standard part of ? with respect to the weak*
topology on Baire measures. By Robinson's criterion for convergence and clustering, the
correspondence between ? and ? yields a nonstandard approach to the weak convergence
of measures. We shall take a lattice theory approach for the development of these facts.
The above measure construction for hyperfinite coin tossing was used by the author
in [16] as the underlying space for standard coin tossing. Similar constructions were used
for Poisson processes in [ 16] and to produce representing measures for harmonic functions
in [17].
The coin tossing space was then used by Anderson [3] to make rigorous the probabilist's
intuitive treatment of Brownian motion as an infinitesimal random walk and the Ito integral
as a pathwise integral. We will discuss these constructions below.
First, in the next two sections, we will use a lattice approach to the integral to combine
both set theoretic measure theory and the construction of measures on topological spaces.
6. A lattice approach to measure theory
In classical analysis, the approach to general integration is either through measure theory
or through functional analysis with a lattice formulation of the integral. Starting from
the notion of "length", for example, one constructs Lebesgue measure and the class
of Lebesgue measurable sets. Lebesgue measurable functions form an extension of the
class of simple functions (linear combinations of characteristic functions of sets of finite
measure), and the Lebesgue integral extends the obvious calculation for simple functions.
The Riemann integral can also be viewed as a positive linear functional on the space of
continuous functions with compact support; as such, it is represented by a measure. That
measure is, of course, Lebesgue measure; it is obtained by extending the Riemann integral

1306
P.A. Loeb
from the continuous functions with compact support to the class of measurable functions
and then noting the action on the characteristic functions of measurable sets. With this
approach the Lebesgue integral is constructed before Lebesgue measure.
In the development to follow, we will extend a positive linear functional from an internal
lattice of *R-valued functions to an external space of "measurable" real-valued functions.
Moreover, ? ? -saturation will allow a development from first principles here with no use of
the Riesz Representation Theorem. Indeed, the Riesz theorem will follow as a consequence
of our construction.
We assume throughout that the constant function 1 is a member of the internal lattice and
has a finite integral. This corresponds to starting with a finite measure space or a compact
topological space. The original development of these results for more arbitrary measure
spaces and locally compact rather than just compact topological spaces can be found in
[21,22]. A recent modification by Bianconi et al. in [7] produces a nonstandard approach
to the Henstock-Kurzweil integral on a finite, closed interval of the real line.
Recall that a vector lattice of functions on a set X is a vector space with a pointwise
ordering. That is, (/ +g)(x) = f(x) +g(x), (af)(x) = a(f(x)) and / iC g if f(x) ? g(x)
for all ? & X. A lattice is closed under the operations ? and л; i.e., the function / ? g
defined by setting (/ ? g)(x) — f(x) ? g(x) at each ? e X is in the lattice; so is the
function / л g = — (—/ ? — g). A linear mapping ? from one lattice to another is called
positive if T{f) ^ 0 whenever / ^ 0; ? is called a functional if the range is in the scalar
field.
In our development here we will use the following notation:
(a) X will denote an internal set in an ? ? -saturated enlargement of a structure containing
the real numbers R.
(b) L will denote an internal vector lattice of *R-valued functions on X.
(c) / will denote an internal positive linear functional on L.
We will assume that 1 e L and /A) is limited in *R.
Example 6.1. The set X can be a hyperfinite set and L the space of all *R-valued internal
functions on X. The functional / can be determined by uniform counting measure; i.e.,
1(f) = A/\?\)??€? f(x) where |X| denotes the internal cardinality of X.
Example 6.2. A more general construction starts with an arbitrary internal probability
space (?, ?, v). (I.e., ? is an internal ?-algebra and ? an internal measure with v(X) = 1.)
Here, L can be the space of internal Д-measurable simple functions and /(/) = f f dv
for all / e L.
Example 6.3. A third example is constructed on the internal compact set X = *[0, 1]
with L the set of continuous functions on X. Here, / can be any positive linear functional
on L; a prime example is the nonstandard extension of the Riemann integral: /(/) =
*fofWdx-
From the internal lattice L, we construct two external vector spaces Lq and L over the
real numbers R.

Nonstandard analysis and measure theory
1307
Definition 6.4. The class of null functions Lo is the set of all internal and external *R-
valued functions h on X such that for any ? > 0 in R there is a ? e L with | h | < ? and
? (?) ^ ?. The class L is the set of real valued functions / on X having representation
/ = ? + h for some ? e L and h e Lo- Given such a representation of an / e L, we set
I(f) = °I(<p).
First we need some preliminary results showing, among other things, that L is a lattice
and / is a well-defined positive linear functional on L. Note that when ? e L П Lo,
?(?) ~ 0. Also note that because of our assumption that 1 e L and L(l) is limited, 1 e L.
PROPOSITION 6.5. The sets Lo and L are vector lattices over R. /// is in L with f =
<p+hfor<p e L andh e Lo, then 1{\?\) is limited in *R. Ifg is also in L with g = ? + к for
\j/ e L and к e Lq, then (f ? g) - (? ? ?) e Lo and (/ л g) - (? л тД) e Lq. Moreover, if
f =g (that is f = (p + h = \l/+k) then ?-? e Lf)L0, whence 1(f) = °?(?) = °?(?).
It follows that I is a well-defined positive linear functional on L.
PROOF. It is easy to see that Lq is a vector lattice and L a vector space over R. To show
that ?(\?\) is limited, fix ? in L with \h\ ^ ? and ? (?) ^ 1. Then <р-у^/^<р + У-
Since / is real valued, the internal set {« e ?: ? — ? ? ?] contains every unlimited element
of *N and thus some limited element. Similarly, ? + ? ^ — ? for some ? e N. It follows,
since /A) is limited, that ?(\?\) = ? (? ? 0) + ?(-? ? 0) is limited.
Now fix ? > 0 in R, and ? in L with \h\ + \k\ ^ ? and ?(?) < ?. From the arbitrary
choice of ? and the inequality
(? V ?) - ? = (? - ?) ? (? - ?) ?? (? + h) V (? +k) = f V g <: (? V ?) + ?
it follows that (/ ? g) - (? ? ?) e Lq. The rest is clear. ?
The reader should note that we have assumed no continuity properties for the internal
functional /. For standard Daniell or Radon lattice integration theory, one assumes at least
that if /„ \ 0, then / (/„) \ 0. This corresponds to the assumption in measure theory that
the original finitely additive measure is countably additive on the initial algebra. With K|-
saturation, however, we can establish continuity for the external functional / in the form
of a monotone convergence property; we need no continuity assumption on the internal
functional /.
THEOREM 6.6 (Monotone convergence). If {f„: ? e N) is an increasing sequence in L
with real upper envelope F and sup /(/„) < +oo, then F e L and 1(F) = lim /(/«)-
PROOF. By replacing /„ with /„ - f\, we may assume that each /„ ^ 0. By
Proposition 6.5, we may fix ?? e L and hn e Lo for each ? e N so that /„ = ?„ + h„ and
0 ^ ?? ^ ?,?+1 -By the ? ? -saturation of our enlargement, there is a ?? e L with ?? ^?,,???
each ? e N and °?(?,?) = lim„epj °?(?„). We need only show that F - ?? e L0. Fix ? > 0
in R. Choose for each ? e Nai/r„ e L with \h„\ ^ ?„ and ?(?„) ^ ?/2". By ? ?-saturation,
we may extend the sequence {?,,: « e ?) to an internal sequence [?,,: ? e *N} с L. We

1308
P.A. Loeb
may also choose к е *Мзс so that 0 ^ ?„ and ? (?,,) < ?/2" when 1 ^ ? ^ ? in *N. Setting
? = ?*?=\ ??, we have ? (?) < ?. Now for each ? e N,
?,? - ? < <P„ - V» < <Ри + «» ^ F ^ A + ?)(?? + ?),
so
(<Pn -<??)-? ^ F -??????+A + ?)?.
The rest is clear. D
The next two results exhibit the close relationship between the internal lattice L and the
external lattice L. This corresponds to the approximation of external but measurable sets
by internal measurable sets in the set-theoretic approach to nonstandard theory in Section 5
and in [16].
Theorem 6.7 (Internal approximation theorem). A real valued function f on X is in L
if and only if for each ? > 0 in R there exist functions ?\ and ?? in L with ?\ ^ / ^ ??
andI(\//2~ ?\) < ?, in which case, °?(?\)? 1(f) iC °?(?\) +?.
PROOF. First assume / = ? + h e L with ? e L and h e Lq. Fix ? > 0 in R. For
each ? e N, choose ?„ e L with \h\ ^ ?„ and ?(?„) < ?/?. Setting ?\ = ? - ?2 and
?2 =? +?2, we have ?\ ? / ^ ?? and /(?/? — ?\) < ?. Moreover, given any ?\ and ??
in L satisfying these conditions, we have for each ? e ?, ?\ — ?„ ^ ? ^ ?? + <р,г, whence
°/?) - ?/? ^ °/(<p) = /(/) ^ °/(iA2) + ?/? ^ °/(???) + ? + ?/?.
It follows that °/(i/f|)<c /(/) sC °/(?/?) +?.
Assume now that / is an arbitrary real valued function on X for which there exists
an increasing sequence {?„: ? e ?} с L and a decreasing sequence {?,,: ? e ?} с L with
?? ? f ? ?? and ?(?? —?„) < \/n for each// e N. By ? ?-saturation, we may extend both
sequences to *N and choose a \j/n e L such that for each ? e ?, ?„ ^ ???^. ??, whence
?? — ?? ^ f — ?? ^ ?? — ??- It follows that f — ?? e Lq and thus / e L. D
Given an *R-valued function g on X, we will let °g denote the extended-real valued
function on X defined by setting °g(x) — st(g(x)) for each ? in X. Here, st(g(x)) equals
+00 or —00 if g(x) is unlimited in *R. For any function g taking values in R or *R, we
set g+ — g ? 0 and g~ = — g ? 0.
PROPOSITION 6.8. If ? e L and ?(?) is limited in *R for each ? e X, then °? e L and
?(°?) = ??(?).
PROOF. By assumption, °/(l) < +00. The proposition follows from the fact that for each
? > 0 in R, \°? - ?\ < ?, so °? - ? e Lq. ?

Nonstandard analysis and measure theory
1309
Definition 6.9. Let M+ denote the set of nonnegative, extended-real valued functions g
on X such that for each и е N, g An е L, and set 7(g) = sup/(g ли) for all g in M+. Let
?? = {g- g+ e ??+ andg~ e ??+). For each g in M, set 7(g) = -/(g+)- У (g^) if at least
one of the right hand values is finite. Let
B={A?X: XAeM+) = {A<zX: ?? e L\.
For each A in ? let ?(?) = J(xA).
Note that in going from "integrable functions" to "measurable functions", we only need
to worry about infinite values. If we do not make our initial assumption that 1 e L and / A)
is limited, then this definition must be replaced with a more complicated one (see [21 ]).
THEOREM 6.10. The collection В is ? ?-algebra in X and ? is a complete, countably
additive, finite measure on (X, B).
PROOF. The fact that ? is a ?-algebra and ? is countably additive follows from
the monotone convergence theorem. Of course, for each A e ?, ?(?) ^ /A). The
completeness of ? follows from the internal approximation theorem and the fact that a
real valued function / = / + 0 in Lq is also in L. The rest is clear. ?
The next proposition contains preliminary results concerning the space M; from these
results it will follow that J(g) = §xgd^ for each g e M+. Note that in the above
definition, we may replace the truncations g л ? with truncations g л / ??? arbitrary
elements / ^ 0 in L since g л / = sup„g a f An; therefore, J(g) = sup{/(/): / e L,
f^gl
PROPOSITION 6.11. Fix g e ?+, ? e M+ and a ^ 0 in R. Then g + p, ag, gv ? and
g ? ? are in M+. Moreover, J(g + p) = J(g) + J(p), J (ag) =aJ(g), and if g iC ? then
J(g) ^ J(p)- Ifign- ? e N} is an increasing sequence in M+ with upper envelope G, then
GeM+ and J(G) = supJ(g„).
PROOF. For и eN, (g + p) An = [(g An) +(p An)] An e L,and for ? > 0, (ag) ли =
a(g A n/a) e L. It is easy to seejhat J(g) + J(p) ^ J(g + p). The reverse inequality
follows from the fact that if / e L and / sC g + ? then / л g ^ g and / - (/ л g) ^ p.
The rest is clear. ?
THEOREM 6.12. A nonnegative extended-real valued function g on X is B-measurable if
and only if g e M+ in which case
J(g)= f gdii.
PROOF. Fix g in ? + and let A = {g > 1}; we will show that ?? e L. Let / = (g л 2) -
(g A 1). Then / e L and ?? = lim(l л nf) e L\. Now for any positive a in R, ?? e L

1310
P.A. Loeb
when A = {g > a] = {g/a > 1}, and by the monotone convergence theorem, the same is
true for ? = 0. Therefore, if g e M+ then g is ?-measurable. The converse and the equality
J(g) = f gdp are obtained from the corresponding facts for ?-simple functions. D
Remark 6.13. In light of the above theorem, we may now call J an integral.
If ? e L then for any ? e ?, °(—? ? ? л п) = -? ? °? л ? e L, so °? e M. We say
that ? is ? "lifting" of °?. It may not be the case, however, that J(°\(p\) = °/(|<p|); ? may
take a large infinite value on an internal set A with ? (? a) — 0, i.e., ? (A) = 0. Functions
for which the standard part of the internal integral equals the integral of the standard part
of the function form an important class in this integration theory.
DEFINITION 6.14 (Variation of Anderson [3]). We say that ? e L is S-integrable ???(\?\)
is limited and J(°<p) = °?(?).
Proposition 6.11 shows that a limited valued (and therefore bounded) function ? is S-
integrable; the proof is much simpler than that for the measure-theoretic case ([16]). The
following criterion for the S-integrability of unbounded functions is an application of the
usual procedure for extending integrals from bounded functions to unbounded ones, with
the finiteness of the internal integral following as a bonus.
PROPOSITION 6.15. A function ? in L is S-integrable if and only if for each ? e *N3C,
?(\?\-\?\??)~0.
PROOF. We may assume ? > 0. By the definition of the integral and the equality of the
standard part of the integral with the integral of the standard part for bounded functions,
J С ?) = sup J (°? An) =sup°/(<p An) =lim°/(<p An).
,i я "
Since °?(?) = °?(? - ? л п) + °?(? л п) for each n e *N, if for each ? e *Noc we
have ?(\?\ - \?\ л ?) ~ 0 then for some finite /; e ?, °?(?) ^ 1 + °?(? An) < +oo,
and moreover, lim„ °?(? An) = °I(?). The converse is clear. ?
If / = / dv, then the condition given in some of the literature for ? to be S-integrable is
that ?(\?\) is finite and fA \?\ dv^O for any internal set A with v(A) ~ 0. Note that we do
not need to check the finiteness, and only the integral over sets of the form {\?\ ^ ?) need
be checked. It then follows that for any A with v(A) ~ 0, fA \?\ dv ~ 0. This definition
shows that there is a close connection between what is called "uniform integrability" of a
family of functions and the S-integrability of the functions in the extension of the family.
Uniform integrability for a family of measurable functions is the condition that ./]/,>„ I/I
goes to zero uniformly as / ranges over the family.
We next address the following question: Given g ^ 0 in ?, is there a lifting in L?

Nonstandard analysis and measure theory
1311
PROPOSITION 6.16. Given g ^ 0 in M, there is ? ?^? in L such that for each ? e N,
(g л ?) — (? л п) е Lo, whence for some ? e *?;?,
J(g) =sup7(g ли) = sup°/(<p ли) = °?(? A /j).
71 И
If A is the ?-null set \J„€N{°(g A n) - °(<p л n) ф 0} аи<* ? = {g = +00} ? {> = +oo),
f/геи g = °? < +oo ои X — (A U ?), /fg /ms a finite integral, then ? ? ? is S-integrable
and ?(? U ?) = 0. (/и this case, ? ? ? is called an S-integrable lifting of g.)
PROOF. We may choose sequences [?„: n e N} and [\j/„: n e N) in L so that 0 ^ ?„ ^
g An ^?,,,?,, ^ ?„+\ and /(i/r„ — <p„) < 1 /и for each n e N. Given к ^ w ^ ? in N,
i/fm л ? ^ g л и ^ ?^ a n ^ <pm л ?.
By ? ?-saturation, we may choose a <p e L so that for every m and и with m ^ ? in N,
??? A n ^ ? A n ^ <pm A n. Thus,
VneN (g ли) -(? ли) e Lo. ?
We now return to the case of Example 6.2.
Example 6.17. Let X be an internal set, ? be an internal algebra on X, and ? an
internal finitely additive measure on (?, ?) with v(X) finite in *R; for example, (?, ?, v)
may be an internal probability space. Let / be the ?-integral on the class L of internal
?-simple functions. For each A in ?, let v\(A) = st(v(A)). Then ? is an algebra in
the ordinary sense, and v\ is a finitely additive, real valued measure on A. The lattice
construction produces a standard measure space {?, ?, ?) that extends the finitely additive
space (?, A, v\); i.e., Л с ? and ?(?) = v\ (A) for each A e ? If ? e ? and ? > 0 in R,
then from the existence of functions ? and ? in L with ? ^ ?? ^ ? and ? (? — ?) < ?,
we obtain sets A\ = {? > 0} and A2 = {? ^ 1} in A. Clearly
A|CfiCA2 and v(A2 - A,) iC ? (? - ?) < ?.
Using ? ? -saturation, we may extend an increasing sequence A'\ and a decreasing sequence
A'2 corresponding to ? = \/n and find а С е Л such that ?(??? = 0. These are the
internal approximation results that characterize externally measurable sets in the measure-
theoretic approach to this integration theory in [16]. Now, g on X is ?-integrable if and
only if g has an S-integrable lifting, and for an S-integrable ? e L,
? °??? = ° ? ???.

1312
PA. Loeb
7. Internal functionals on continuous functions
The topological example considered in Example 6.3 is based on the set X = *[0, 1]. The
lattice L is the set of internal, continuous functions on X, and / is a positive linear
functional on L. An example of / is the nonstandard extension of the Riemann integral.
Instead of just the interval [0, 1 ], this section deals with an arbitrary standard set ? supplied
with a compact Hausdorff topology ? in an enlargement of a structure containing ? and R.
We assume that the enlargement is /c-saturated with ? ^ ? ? and к ^ Card(T), so that in
particular, the enlargement is ? ?-saturated. We set X =*Y.
Since the topology ? is Hausdorff, each point ? in X is in the monad m(y) of a unique
standard point у in У; i.e., у = st(x). With each extended-real valued function g on Y, we
associate the extended-real valued function g on X, where g(x) = g(st(x)). That is, we
spread the value of g at a standard point over the monad of that point. The standard part
map has played an important role in nonstandard measure theory; it's inverse appears here
in the form of the mapping #—>¦?. For each set А с У, we set A equal to the union of the
monads of the points in A. The set A = if-1 [A]. Moreover,
Хл =Xa-
We now fix an internal positive linear functional / on *C(Y), with /A) limited in *R
(recall that C(Y) denotes the space of continuous, real valued functions on Y). We apply
our general results and notation with L = *C(Y). In interpreting the following result, the
reader should keep in mind that a subset of ? is compact if and only if it is closed. The
result says that the inverse image with respect to the standard part map of each compact set
in К is measurable.
PROPOSITION 7.1. For each compact set К с. ?, К еВ and if
aK=M{°l{*f): feC(Y), ??^/???},
then ?(?) — ??.
PROOF. By saturation, there is a function ? e *C(Y) — L with ?*?: ^ ? ^ X? and
°?(?) —aK. Given any / e C(Y) with ?? ^ f ^ 1, and given ? > 0 in R, we have
V^Xjf <(!+?) ¦*/-
It follows that ?? — ? e Lq, ?? e L, and
?{?) = ^??) = 0?{?) = ??. ?
Since everything works nicely for compact sets in ?, they also work for the ?-algebra
generated by the compact sets, i.e., the Borel sets. (In the more general theory in [21] for a
non-compact space, one must work harder.)

Nonstandard anahsis and measure theory
1313
THEOREM 7.2. Let ? = {S с ?: Be В], and let ??(?) = ?(?) for each В e М.
Then ? is ? ?-algebra in ? containing the Borel ?-algebra, and ?? is a complete,
regular measure on (Y, Л4). A function g on ? is M-measurable if and only ifg is B-
measurable on *Y, in which case, ifg ^ 0 then fY g ??? — / ^??. For each f e C(Y),
PROOF. It is easy to see that ?4 is a ?-algebra, and we have just shown that it contains
the closed subsets of Y. Therefore, it contains the Borel sets. To show that ? у is regular,
choose В e ? and ? > 0 in R. Fix ? e L with 0 sC ? sC ?$ and °?(?) ^ J(Xb) - ?. Let
К = {st(x): ?(?) > 0). This is the standard part of an internal set of near-standard points,
so by a result of Luxemburg (see [12,25]), К is compact. Of course, К с В. Since
??(?) = ?(?)^°?(?),
it follows that ??{?) is the supremum of the measure of compact sets contained in B.
The same is true for ? \ B. This means that the measure ?? is regular. The relationships
between the measures and the integrals on ? and *Y follow from the corresponding
relationships for simple functions. If / e C(Y), then */ e L and / = °(*/) e L.
Moreover, /- */ e L0. Therefore, fY f ??? = f^, J?? = °I(*f). О
Example 7.3. If ? = [0, 1] and / is the extension of the standard Riemann integral on Y,
then a real valued g on ? is Lebesgue integrable if and only if 'g = ? + h where ? e *C(Y)
and h e Lq. In this case, the Lebesgue integral of g equals the standard part of the internal
Riemann integral of ?. A bounded g is Riemann integrable if and only if for any ? > 0,
there are continuous functions ? and q on [0, 1] with ? ^ g ^ q and fQ q — pdx < ?.
It follows that g is Riemann integrable if and only if °(*g) e L (result of the author and
A. Cornea.)
If ? is a positive linear functional on C(Y), and / = *T, Then for all / e C(Y),
fY f ??? = °I(*f) = T(F). Thus ?? represents T. This is the Riesz Representation
Theorem for the Dual space of C(Y) when ? is compact. A similar but more difficult
proof in [21 ] works for the continuous functions with compact support on a locally compact
space.
Numerous applications of nonstandard measure theory to problems in standard analysis
have used the standard part map to convert an internal measure to a standard one. (See, for
example, the articles of Anderson and Rashid [5] and Loeb [19], as well as the examples
below on Lebesgue measure and representing measures in potential theory.) The standard
measure on ? is the standard part of the internal measure with respect to the weak*
topology.
The above correspondence gives weak convergence results since if v„ —> v, that is, for
all continuous /, f f dv„ —>¦ f f dv, then ? =st(vtl) for ? e *Мзс. Here the standard part
of the internal measure ?? is taken in the weak* topology.

1314
P.A. Loeb
8. Lebesgue measure
Another way to construct Lebesgue measure on [0, 1], is to use a hyperfinite partition
{0, l/H, 2/H, ...,(#- \)/H} with equal mass \/{H + 1) at each point. One then forms
the standard measure from this measure, and projects down by the standard part map. This
is the construction of Lebesgue measure used by Anderson in his paper [3]. This measure
generates a hyperfinite Riemann sum, so it is clear from the fact that the resulting standard
measure on [0, 1 ] is the standard part of the internal one in the weak* topology, that the
resulting measure is Lebesgue measure. Anderson later generalized the construction to
Radon measures in [4].
9. Representing measures in potential theory
For this example, we work with harmonic functions on the unit disk in the complex plane
С Let Dr denote the open disk [z e C: |z| < r], and let D = D\. Let C,- be the circle
(геС: |г| = r), and let С = С ?. All measures we consider will be Borel measures. Let
Р(г, x) be the Poisson kernel (|г|2— И2)/|г — х\г, and let xq denote the origin. The space
?) consisting of all positive harmonic functions on D taking the value 1 at xo is convex
and compact with respect to the topology of uniform convergence on compact subsets of D,
i.e., the ucc topology.
It is well known that every continuous function on С has a harmonic extension on D and
that not every harmonic function on D is obtained in this way. On the other hand, by the
Riesz-Herglotz theorem there is for each h e ?' a probability measure v/, on С such that
h = j P(z,-)vh(dz).
The mapping ? м> Р(г, -) from С into W1 (with the ucc topology) is a homeomorphism.
We may think of ?/, as a measure either on С or on the collection of harmonic functions
{Р(г, -): z e С). The latter point of view is that of Martin boundary theory and Choquet
theory. The simplest realization of Choquet theory deals with a triangle. Each point inside
and on a triangle is represented by a unique affine weight on the extreme points of the
triangle, i.e., on the vertices. For the compact, convex set ??, the extreme points are the
functions {Р(г, -): ? e С), and each h eH] is represented by a unique probability measure
vh on this set. While the usual construction of vh is simple for the disk, it does not
generalize without going to an ideal boundary. The theory we have discussed in the previous
sections does yield a generalizable construction of i>/, by extracting a measure from the
function h that would otherwise be lost at the boundary of D. We give a brief description
of that construction. More details can be found in the original work [ 17,18,20] from which
this example is taken.
First, we recall that for each point ? e Dr, there is a Borel measure ??, called harmonic
measure, on the circle C,- that gives the value at ? of the harmonic extension of any
continuous function on Cr - Moreover, normalized Lebesgue measure on Cr is harmonic
measure ?[0 with respect to the origin jco- Given h e ?', the measures h - ???(), 0 < r < 1,
are probability measures, and ?/, is the weak* limit as the radius r tends to 1. This

Nonstandard analysis and measure theory
1315
construction of vf, does not work for more general domains and potential theories, but
a modification is valid in these more general settings. First we modify the construction of
vh for the unit disk.
Fix ZieW1. For each r < 1, let {AJ} form an interval partition of Cr, and choose y\ e
AJ". Let 8Yr denote unit mass at the point y\. The net of measures ?? Ь(.у\)Ц-гХй(.А\) - <V
converges in the weak* topology to the measure v/, on С The directed set for this net is
constructed so that the direction is given by letting r tend to 1 and refining the partitions
{AJ}. To see that v/, is in fact the weak* limit of this net, note that the integral of any
continuous function / with respect to one of these measures with support in Cr is a
Riemann sum approximation to the integral of / with respect to the measure h - ??
To obtain a construction that works in general, we do not use finite combinations of
measures concentrated on the points of D, instead, we want combinations of point masses on
the function space [0, +oo]D. Given r and a partition {AJ} of Cr, the function ? н» ??(??{)
is a harmonic function on Dr. It is the solution of the Dirichlet problem for the function that
is 1 on AJ and 0 on the rest of C,-. When we divide by ?\ (Ар, the new function is equal
to 1 at the origin xr,- Let <5;r be unit mass on the function that is equal to ?\(AJ)^.A(AJ)
in Dr and is identically 0 on and outside C,; the point mass <5;r is a measure on the function
space [0, +oo]D supplied with the product topology. (The restriction of the product
topology is the ucc topology on the set of positive harmonic functions on Dr taking the value 1
at xr,.) Again, the measures J^i Ь(у\)ц.гх (AJ) - <5[ have vh as a weak* limit as r approaches
1 and the partitions {AJ} are refined. The limit measure is on the set
{P(z,-): zeCjcH1 C[0,+oo]D.
This construction of 17, continues to work in quite general potential theoretic settings; it
does not use the Martin boundary. The proof that this construction works was obtained
in [18] by interpreting a construction of representing measures given in [17]; that
construction was the first application after coin tossing and Poisson processes of the general
measure theory from [16] described above. In [17], the measurability of the standard part
map was used to replace the standardizations of internal measures with standard measures
on the standard compact set ?). Here, for the special case of the disk D, is the construction
from [17] of vn. We start with a circle Cr С *D with r ~ 1, and an interval partition {A;r}
so fine that every standard harmonic function has infinitesimal variation on each set AJ.
We choose a point y;r in AJ for each /. Suppressing the superscript r, we have
V*eD, h(x)= f ^(?)???(?)-?^(??)???(?,)^~^-.
JCr ??()(??)
The family of weights */г(>,;)мЛA(А,) is made into an ordinary probability measure ??
using the general measure theory discussed above. The measure ?? is supported by the
set of nonstandard harmonic functions ??{?\)/???(??). This is a set of positive, internal
harmonic functions on Dr, with each function taking the value 1 at xr,. The mapping S on
this set of functions given by the formula
S(g)(x) = °(g(x)), V*eZ),

1316
PA. Loeb
is essentially the standard part mapping with respect to ?' supplied with the ucc topology.
The measurability of S, established for this special case by Loeb in [17] and generalized
by Anderson and Rashid in [5], allows one to project the measure ??, onto ?'. The process
preserves affine combinations of harmonic functions and yields representing measures, so
by a corollary of a result by Cartier, Fell, and Meyer (see [17]), the final measure is the
unique representing measure v\, on the extreme points {P(z, -): ? e C] of ?0.
10. Poisson process
The Poisson process is a model for events that occur with frequency ? in time. Here is
a way of using an intuitive model for this process to construct an appropriate probability
space. It is taken from [16].
Let ? be an infinite factorial in *N. Note that for any standard rational number q, q - ? e
*N. Divide the interval [0, ?[ into ?2 equal intervals, [0. \/?[, [?/?, 2/?[,..., [(?' — 1)/
?, ?[. For simplicity, we fix a standard rational number ?, and we let ? = к - ? e *N. We
let ? denote all internal ways that ? distinguishable points can be put into the ?-
intervals [k/?, (k + \)/?[. That is, ? consists of internal sequences ? = (?,·: 1 ^ / < ?) with
each ?, a nonstandard integer between 1 and ?-. Each point in ? has internal
probability \/?'?. We now have an internal probability space (?, ?, ?) from which we obtain a
standard probability space (?, Llt (?), ??.)- For any point ? = k/?, let ??(?) be the
number of points that the outcome ? puts in the interval [0. kjr}\. It is relatively easy to show
that for ?/.-almost all ?, ??(?) is finite for all finite ? and increases at most once in any
monad. If we set
?,{?) = sup/VT(a>)
for t e R and ? e ?, we get a standard Poisson process with parameter ?. Let us confine
our calculation to showing that if t\ and ti are standard rational numbers with t \ < ь, and
s = ti — 11, then the ??.-probability of the internal event A that there are exactly к points
inside [t\, t2[ is given by the Poisson distribution.
There are exactly s - ? subintervals inside [t\, fi[. and the ?-probability of any one of
the ? points being put into [t\, ь[ is i/?//?" = s/? = ks/?. Therefore, using the binomial
distribution to compute ?(?), we have
?(?) =
У·'
(у -k)\kl
к
.? J
-?* г
(ksy
(?*)
к I
У
- ?-Vv*
к\ (?-ky.Y
к Г
ks
?
1 -
ks
??-k
? J
ks
?
?
1-
ks
-?-к
(???
к\
? J
¦IJLl(A).
That is, ??^(?) is given by the Poisson distribution.

Nonstandard analysis and measure theory
1317
Each outcome of the Poisson process gives an increasing function that increases by 1 at
each event. One can show that ?/.-almost surly, only a finite number of events occur in a
finite time, and only one event occurs in a monad. (See [16].)
Another model for the Poisson process tosses a biased coin at each time change j/?.
The coin has very small probability of coming up heads; if it does, one considers that an
event has occurred at that time.
11. Anderson's Brownian motion
Here is a summary of Robert Anderson's construction of Brownian motion in [3]. Recall
that Random variables X\,...,X„ are independent if
It
Vai,...,a„, P(X| <<*, A---AX„<a„) = Y\P(X, <<*,),
?
where ? denotes the probability measure.
A Brownian motion is a family of random variables [Xt: t e [0, 1]} such that
A) X0 = 0,
B) if s ? < t\ ^ S2 < t2 ^ ¦ ¦ -^ s„ < t„ are points in [0, 1], then the random variables
Xt] — XSl,..., Xt„ — Xs„ are independent.
C) for t > s in [0, 1], the probability that X, - Xs ^ a is ?(?/^/? - s), where ?
denotes the standard normal distribution given by ?(?) = 1/V2tt · jtx e~' I dt.
The distribution -?{?/?) is the normal distribution with mean 0 and variance ?2.
Starting with an infinite coin toss {— 1, 1}'' where ? is an infinite factorial, ? denotes all
2'' internal outcomes of a coin toss, ? is the collection of internal subsets, ? is the internal
probability measure, and (?, L?(A), ??.) is the complete standard probability space they
generate.
We divide *[0, 1] into ? intervals of length At = \/?. At each division point, we toss a
coin and move right (if the coin is heads) or left (if the coin is tails) by Ax = s/At. Let
Xi(a>) = wi for I ^ i ?. ?. Then for a given coin toss ?, our position at nonstandard time
t e*[0, 1] is
j Ui'l
?{?,?)=—???{?).
Here, [-] denotes the greatest element of *N less than or equal to the argument. For
any pair (t,w) e [0, 1] ? ?, let ?(?,?) = °?(?,?). Anderson shows in [3] that ? is a
standard Brownian motion built on the nonstandard space ?. Here is a central result in
his development; it uses the well-known Reflection Principle. We write C([0, 1]) for the
standard space of continuous, real valued functions on [0, 1].
THEOREM 11.1. For iit-almost all ? e ?, ?(?,?) is limited for all t e *[0, 1], and
0(-.")еС([О,1])-

1318
PA. Loeb
PROOF. Given m, ? e N, let ?„?? be the internal set defined using internal sups and infs
as follows:
?????
!o>e ?: 3/ <n with sup ?(?,?)- inf ?(?,?)> — >.
reU/«.(/ + l)/"l ie[i/n.(i + l)/n| wj
Then for ? = ? ? ?,
?.(?,„?)^??( sup ?(?,?)- inf ?{?,?)>\/??\
\e[0.l/n| re[0.l/n] /
?
^ ? - ? I max
?-
2m
? ? ¦ u\ max У^?, > ) + ? - ?( min > <
Ыцл^ 2m/ \|<*<х^
^«-????,¦> ^\+2«-??[>,<-^
(this is the Reflection Principle)
2m
= 4„.? ]?>,
/чД
??
4и · 1 - *?
??
2т
= *-(?-*(??
by the transfer of the central limit theorem and the continuity of i/r. Therefore,
An
For ?? I{2m) > 1,
/2? J^i/ilm)
MZ.(^m«)^2«
p+oa
e 4Ldt =4n ¦ e *«·.
•fill Cm)
Let ?' = ? - Um (?„ ?««- Then ??,(?') = 1 - sup,„ inf„ ?/.№,«) = 1-
Fix ? ? ?. If for some t e *[0, 1], ?(?,?) is unlimited, then ? e ???? for all
standard m and и, whence ? g ?'. If for some s and r e *[0, 1] with s_/ we have
°|?(?,?) - ?(?,?)| = ? > 0, then for w > 2/a, we have ? e i2m„ for all ? e N, whence
?? ?'.
Now suppose ? e ?'. Then ?(?,?) is limited Vf e *[0, 1]. The variation of ?(-,?)
on monads in infinitesimal; one says that ?(-,?) is "S'-continuous". Moreover, ?(-,?)

Nonstandard analysis and measure theory
1319
on [0, 1] is the standard part with respect to the uniform topology on C([0, 1]) of this
internal S-continuous function on *[0, 1], so ?(-,?) is continuous. ?
Anderson also shows in [3] that the projection of the coin tossing measure from ?
onto the space of continuous functions is Wiener measure. Anderson continued his work
with an application to the stochastic (or Ito) integral. The idea, initiated by Ito, is to
form for each path an integral with respect to the changes ??(-,?) in the corresponding
Brownian motion. Such a pathwise definition of the integral can not be obtained directly.
However, there is no problem with the internal integral with respect to the random walk
? (·, ?). Anderson showed that the corresponding stochastic process equals the Ito integral.
Extensions by many authors of Anderson's work on Brownian motion and the stochastic
integral continues to this day. See for example [1] and related references.
12. The martingale convergence theorem
We next describe the application of a general limit theorem of Jiirgen Bliedtner and the
author to the martingale convergence theorem. The original work can be found in [8].
The general limit theorem established in [8] is an approach to the many theorems
in analysis and probability theory that say that for each Borel measure ? and for
some reference measure ?, the Radon-Nikodym derivative ??/?? is equal ?-almost
everywhere to a limit along a directed set I of ratios /?,(?,?), /' e I, defined in terms
of ? and ?. In [8], such limit theorems are reduced to an easier form. The essential idea is
that the desired limit result is established for all ? once it is shown that for any measurable
set ? and any ? with v{E) = 0, lim; /?,(?,?) = 0 ?-almost everywhere on E. Indeed,
since the set of Borel measures is closed with respect to scaling, it is enough to show that
limsup, Rj(v,a) ^ 1 ?-almost everywhere on E.
The formulation of this result is simple and yet general enough so that it is easily applied
in quite diverse settings. The technique is applied in [8] to several areas: Boundary limit
theorems in potential theory, measure differentiation theorems, the martingale convergence
theorem in probability theory. The last, is the topic of this section, but we will present a
somewhat simpler form of the full result.
We start with a standard probability space (?, Z, P) and an increasing sequence of ?-
algebras (?„: ? e N) with ? = °<Х]„€щ?>,)- That is, ? is the smallest ?-algebra containing
U„eN ?"· We let ^" and $ denote, respectively, the collections of all finite signed measures
on (?,?„) and (?, ?). We will use the notation v\Zn for the restriction to Zn of a measure ?
defined on a larger ?-algebra. Recall that an L1 -bounded martingale (/?„: ? e N) is a
sequence of functions on ? such that for each ? e N,
(i) h„ is Z„-measurable,
(ii) ?„ e Sn where VA e Zn, ?„(?) := fA hndP,
(iii) Vw > и, ???\?„ =?„,
(iv) sup,, fA\h„\dP < +oo.
The general technique established in [8] yields an easy proof of the following
convergence theorem of Andersen and Jessen [2].

1320
PA. Loeb
THEOREM 12.1 (Andersen-Jessen). Fix ? e S. For each ? e N, let ?? = ?\??, and let h„
be the Radon-Nikodym derivative of the nonsingular part of ? with respect to ? and
?n. Then the L] -bounded martingale (hn: ? e N) converges ?-almost everywhere to the
Radon-Nikodym derivative of the nonsingular part of ? with respect to P. That is,
?? ???
— (?) = hm h„ (?) = hm —— (?)
dP ? ? dP
for ?-almost every ? e ?.
For the proof, we need some notation and a result in the form of a "maximal inequality."
Definition 12.2. For each ? e S, h'- denotes a fixed nonnegative version of ??{/??
(?,-measurable). Also,
VMe<S, VAef, VikeN, Ak=\cueA: sup/if (?) > ll.
PROPOSITION 12.3. For each ? eS,if AeEk,then ?(?*)> P(Akfl).
PROOF. This result follows by taking a finite subset
{к <к+ 1 < ··· <Hm)cN,
setting ?-1 = 0, Bn = {? e A: h^+n > 1} - U"=-1 Bi forO^n^m, and noting that
m . in I in \
n=0JB" n=0 \я=0 /
The Anderson-Jessen result follows immediately from the general limit result from [8]
discussed above and the following proposition. ?
PROPOSITION 12.4. Fix ? e ? with P(E) > 0 and ? e S with v(E) = 0. Then
limsupftj'(w) ^ 1 for P-a.e. ? e E.
/eN
PROOF. We will show that P(f\k=\ El) = °- Fix ? > 0. Since ? is generated by the
?,-, there is а к e N and an A e ?* such that (P + v)(AAE) < ?. By Proposition 12.3,
P{Akv) ^ v{Akv), and by definition, ?(?) = 0, so,
P(Ek) ^ P(E - A) + P(Ak) ? P(E - A) + v(A - ?) < ?.
It follows that Р(Г(к=\ ?,*) = 0- D

Nonstandard analvsis and measure theory
1321
Not all martingales are generated by a measure. An example is "double or nothing": You
start with one dollar and at each play you either double your money or lose everything. On
the other hand, nonstandard measure theory provides a way to use Andersen and Jessen's
result to establish Doob's [11] convergence theorem for L1 -bounded martingales. Fix an
L'-bounded martingale (/?„: ? e N). We let (*?,?^, Pl) denote the standard, complete
probability space formed from the internal space (*?. *?. *P) using our general measure
construction. Similarly, for each ? e *N, we let (*?,??, ?„) denote the standard space
formed from (*?,*?„,*?„). Fix an index ? e *Noc. The ?-algebra В = a(\JneNB„)
generated by the collection {??„: /; e N) is a subset of ??. That is, for each n e N,
?„????„???.
Let ? = ??\?. Given n e N, we have v„ = v\B„ since for each A in the generating
algebra *?„,
v„(A) = °{*?„(?)) = °{*??(?)) = ?(?).
Moreover, for each A e *?„,
?„(?) = °(*?„(?)) = ° [ *hnd*P= ( °{*hn)dPL
J A J A
since
lim f [\hn\-(\h„\Am)]dP=0.
in-* oc J ?
That is, *h„ is an S'-integrable lifting of °(*hlt). Since the extension of v„ from *?„
to B„ is unique, vn <<C Pl, and the Radon-Nikodym derivative dv„/dPL = °(*h„). By
the Andersen-Jessen theorem, °(*/г„) —> dv/dPi ??,-a.e. on *?. Using Proposition 12.5
below, one can "push down" this result and show that the original sequence h„ converges
P-a.e. on ?. Moreover, the limit is a function / such that °(*/) = dv/dPi on *? \ U,
where U is a set of Pl -measure 0 in ??.
PROPOSITION 12.5. Let (?, ?, ?) be a finite measure space and {/?,: /' e 1} a family of
finite, measurable functions indexed by a countable, ordered set I с R+ with the ordering
inherited from E+ and N С I. Let {* Y, LA (?). ?^.) denote the standardization of the space
(*Y, *?, *?) in an ? ? -saturated nonstandard extension of a standard structure containing R
and Y. For each i e I, let g( = °(*/г,). Then the functions hi, i e I, converge k-almost
everywhere on ? to afunction h ifandonly if the functions gj, i el, converge ??-almost
everywhere on *Y to afunction g, in which case g = °(*h) ki-a.e. on *Y.
PROOF. If the /?, 's converge almost everywhere to h. then by Egoroff 's theorem, the g, 's
converge almost everywhere to °(*h). For the converse, fix ? > 0 in R, m e N, and any
finite set w = г ? < г'2 < · · · < i„ in I. Set
A = {yeY: 3j, l^j^n. with |/г,„(у) - й,-;0>)| > ?}.

1322
PA. Loeb
Set
B = {xe°Y: 3j, 1 <;<;«, with \glj(x)\= +00 от \g,„(x) -gij(x)\ > ?}.
Then *A с ?, and so ?(?) = ??.(*?) ^ kL(B). By Egoroff's theorem, hi(y), i e I,
satisfies a Cauchy condition for ?-a.e. у eY. ?
13. On an infinite number of independent random variables
For our last example of the application of nonstandard measure theory, we will discuss
the recent work of Yeneng Sun in [31-33] dealing with an infinite number of equally
weighted independent random variables. We begin by recalling the difficulties in the
classical framework. Rather than speak of one individual tossing a coin over and over
again, one would like to speak of an infinite number of equally weighted individuals,
acting independently, who each toss a coin once. There are measure theoretic problems.
One cannot put a uniform probability measure on a countable set, therefore, the set
of individuals must be uncountable. Examples are Lebesgue measure on [0, 1], or the
standardization, as discussed above, of an internal uniform measure on a hyperfinite set.
Once one gets beyond a countable set, there are classical measurability problems. Here is
a sketch of those problems. First a particular case.
Let ? = R10·'1 be the space of real-valued functions onT= [0, 1]. Let ? be Lebesgue
measure on T, and let ? be a product measure on ? constructed from a non-Dirac
probability distribution on R. An example of the measure on R for coin tossing would
be weights 1 /2 at 1 and -1. Note that the existence of such a continuum product measure
is guaranteed by the Kolmogorov existence theorem. For t e ?, let f, be the tth coordinate
function, i.e., ft((o) = w{t), where ? e ? is a function on T. Then, / is a real-valued
process with a continuum of independent and identically distributed (iid) random variables.
For a fixed ?, the function /(·, ?) on ? is denoted by /?; it is called a sample function.
Here is Doob's 1937 result [10] for this case.
PROPOSITION 13.1. Let h be any real-valued function on [0, 1]. Let
Mh = [?: /? (r) = ?(?) = h(t) except for countably many t e ?}.
Then Mf, has ?-outer measure one.
PROOF. Let В be any measurable set in the continuum product ?-algebra on ?. The set
? is in the ? -algebra generated by a countable number of cylinder sets. Therefore, В is
determined on a countable index set С in the sense that for any ? and ? in ?, ifa(t) = fi(t)
for all t e C, then a e В if and only if ? e B.
Now suppose that В contains ??/,. Take any ? e ?. Define ? so that it agrees with ?
on С and with h on ? \ C. Since w'(t) = h(t) except for countably many t e ?, we have
?' e Mh, and thus ?' e B. On the other hand, since ? and ? agree on C, the property of
С implies that ? e В also. This means that ? = ?, and hence the outer measure P*(Mh)
of Mh is one. ?

Nonstandard analysis and measure theory
1323
Remark 13.2. If one takes h to be non-Lebesgue measurable, then so is every function
in Mh- Thus, the outer measure of all non-Lebesgue measurable samples must be one. On
the other hand, if one wants the sample means to be some number a almost surely, then
one can take h to be the constant function taking the value a. Since P*{Mh) = 1, one can
trivially extend ? to a measure ? with ? (Mi,) = 1, and then claim that almost all sample
means are equal to a. Sun notes that the arbitrary choice of a simply makes the statement
meaningless. The claim is actually based on a more absurd underlying statement, namely,
that almost all sample functions take an arbitrary constant value a.
The following proposition of Sun shows that no matter what kind of measure spaces
are taken as the parameter and sample spaces of a process, independence and joint
measurability with respect to the usual measure-theoretic product are never compatible
with each other except for the trivial case. This also means that even if one chooses a very
large ? -algebra on the parameter space of a nontrivial independent process so that all the
sample functions are measurable, the process itself is still not jointly measurable. The proof
can be found in Sun's original work, e.g., [33].
PROPOSITION 13.3. Let (?,?, ?) and (?,?,?) be any two probability spaces with a
complete product probability space (I x X,I<g> ?,? <g> v), and let f be a function from
? ? ? to a separable metric space. If f is jointly measurable on the product probability
space, and for ? <g> ?-almost all (z'|, b) e / x /, /;, and /,2 are independent, then, for
?-almost all i e /, f is a constant random variable, where f is the function on X defined
by /(/, ·)·
Yeneng Sun's solution to the measurability problems that occur with ordinary products is
to work with the internal product of an internal index set and an internal probability space.
He then takes the standardization of that internal product space. That is, he applies to
the internal product space the general construction of a standard measure space described
in the previous sections of this chapter. It follows from Keisler's Fubini theorem, given
below in Theorem 13.6, that this space has the product properties one needs. Now one can
speak of each element of the index set representing an independent random variable, or in
applications, an independent individual. Also interesting is the fact, as we shall see, that
notions of independence that are distinct in the finite case become essentially the same. To
expand on these ideas, we must first discuss facts about the internal and external product
spaces.
Fix two internal probability spaces (?,?,?) and (Y, B, v). A set of the form ? ? ?
with A e A and В e Bis called a measurable rectangle and A <g> В denotes the *? -algebra
generated by the internal algebra of all * finite disjoint unions of measurable rectangles.
The internal product measure of ? and ? on A <g> В is denoted by ? <g> v. Let A <g> В be the
completion of A <S> В with respect to ? <g> v; the corresponding extension of the measure
? <g> ? is also denoted by ? <g> v. Analogous notation will be used for the standard product.
The next result shows that the usual product ??(?) <g> LV(B) is contained in the product
??®?(?®?). The proof is taken from [26].
Proposition 13.4 (R. Anderson [3]). If ? e ??(?) ® LV(B), then
? e ??®?(?®?) and (? ® ?)?.(?) = ??. ® ??.(?).

1324
PA. Loeb
PROOF. We use the notation ??? for the null sets of a measure ?. If N e NVL, then for
any standard ? > 0, there is an internal measurable set В з N with the internal measure of
? < ?. It follows that ? ? ? e ?/"(?®,.);. Similarly, if N e ?/"?;, then ? ? ? e ?/"(?®„)?.
Suppose that A e LM (Л) and ? e L,,(??). Then there exist sets U e A and V e В such that
UAA e7VMt and VAfi e7V,7 . Since
(?/ ? V)A(Ax fi)c((f/AA)x У)и (X ? (VAfi)) еЛА(м®,,);.
we have ? ? ? e LM®,,(.4 <g> B) and,
(?®?)?.(?? ?)~?®?(?/ ? V) = ?(?/) · ?( V)
~ ??.(?) ¦ vL(B) = Mi.® vL(A ? ?).
It follows that LM(^)(g) L„(??) С ?,?®,,(? ®?) and (?®?)?. coincides with ?^. ®vl on
LM(-4) <8) L^S). Finally, let N с ?? e ?,ДД) ® ?,,,(?) with ?^ ® ?>?.(??) = 0. We have
seen that ? e Л/[М®„^. It follows that N is in the complete measure space Llt®v (A <g> B),
and^®v)L(N)=0. ?
Yeneng Sun has shown that the containment of LM (A)<§Lv (B) in the space LM®„ (A <S> B)
is strict if and only if both ?^. and vi have non atomic parts. (See, for example, [33].) The
original example showing strict containment is due to D. Hoover. We next give a
modification of that example from [26]; it is due to D. Norman.
Example 13.5. Fix ? e *?^. Let X be the set {1,2,.... H), and let ? be the internal
power set of X. Let ? and ? be the respective counting measures on X and ?. Here, A and
В are the internal subsets of their respective sets. Now, LM®,,(.4 <g> B) is formed from the
internal counting measure on the internal product. The set
F = {(i,S)eX ? Y: i e S}
is internal, and by symmetry, the internal cardinality \F\ = ^|Х||У|. It follows that
F e L^.iA® B),and ^®v)L(F) = \. We will show that F i LM(*4)®L„ (B) .Assume
the contrary; then ? ? ®vl(F) = \. Now, there is a sequence of rectangles ?, ? ?, e.with
/' e N and each A,- e L^A) and each ?, e Ltl(B), such that F is contained in the union,
and the sum of the product measures ???\ ? ?. (A,)· vl(Bi) < 1. We may even assume that
for each /, A, e A, and ?, e ?. As in Section 5, we will only need the rectangles indexed
from 1 to some limited integer m to cover the internal set F. The complement of the union
of these rectangles is also a finite union of internal rectangles, and for one of these, С ? D,
we must have ??(?@) > 0 and st(v(D)) > 0. Now,
\{yeY: VjceC, ? i y}\ =2W_|C| = |У| · 2_|C|.
Since F П (С ? D) = 0, we have D с {>> e Y\ Vx eC,x <? y], so
?(?))<: ?-! = 2~|C1.
\Y\

Nonstandard anah'sis and measure theory
1325
Since 5?(?@) > О, С is not a finite set. But then v(D) ~ 0, so st(v(D)) = 0. This is a
contradiction.
Even though in general we have strict containment of Lfl(A) <g> LV(B) in the space
??®?(?®?), we do have Keisler's Fubini theorem from [13] for ??®?(?? ?).
THEOREM 13.6 (Keisler-Fubini). Let f : ? ? ? -> R be (? ® v)L-integrable. Then
(i) For [iL-almost all ? e X, f(x, ·) is vi-integrable.
(ii) ? ы>- fY f(x,y)dvL(y) is \xL-integrable.
Ciu) fx fY f(x,y)dvL dpL(x) = /??? f ?(? ® v)L.
PROOF. First we show that if А с ? ? ? has (? <g>i>)z. -measure 0, then for ??,-a.e. ? e X,
the section A(x) = {y e Y: (x, у) е A) has vL-measure zero. Pick a decreasing sequence
of internal sets E„ e A <g> В containing A with (? <g> ?) ? -measure going to zero. Let ? be
the intersection. For each set En, the internal Fubini theorem holds, so
(? ? ?)?. (Ел) = st(ji ® ?)(?„) = ° ? ?(?„(·?)??(.?)
= J °?(??(?)) ????) = (' vL {?„(?)) ????).
For each ? e ?, the sequence VL{En(x)) is decreasing, and the integrals go to 0. Moreover,
for each n, 0 ^ vL{E{x)) ^ у^(?„(л:)). Therefore, vL{E(x)) = 0 ??,-a.e. Since А с Е,
A(jc)c ?(jc)forall;c e X, and when vL (?(*)) = 0, A(x) e ?,,,(?) and МА(*)) = 0.
Now pick an /S-integrable lifting F of /, and set G() = / F(-, y)dv(y). We need to
show that for ??,-a.e. x, the function F(x, ·) is an /S-integrable lifting of f(x, ·) and that
G is an /S-integrable lifting of g := //(·, y)dvi(y). This will imply the first two parts of
the theorem, and the last part follows from the calculation
I f(x,y)d(ji®v)L=0 ? ???®? = ° ? ??)??(?)= /g(*)d/iz.W
= ft /"/(*, y)dvz.()')Wz.(*).
It follows from the first part of the proof that F(x, ·) is a lifting of/(jc, ·) for ??,-a.e. x. For
the S-integrability, we let Fm = —m ? F A m for each m e N. Then, as in the discussion in
Section 5,
0 ii / l· J I F - F„, \dv\ ???. ? ° ? [( J | F - Fm \??\??\
= ° /*|F-Fm|d/i®v->-0

1326
PA. Loeb
as m —>¦ oo. This means that for ? t-a.e. x, ° f \F — F„,\dv —>¦ 0, and for such an ?, F(x, ·)
is S-integrable. A consequence of this is that for ??. -a.e. jc,
°G(x) = ° ? F(x,y)dv(y) = ? f(x, y)dvL(y) = g(x),
whence G is a lifting of g. If G,„ — f F„,(x, y)dv(y), then \G\ ? m, and
0^° j\G-G,„\d??a j(j\F-F),\dv\d?^O
as w —>¦ oo. It follows that G is S-integrable, and the theorem is proved. D
14. Exact law of large numbers and independence
Recall that for if random variables X and ? are independent, then the covariance
cov(X, Y) = E((X - EX)(Y - ??)) = ?(??) - EX ?? = 0
since E{XY) = EXEY. Also recall the law of large numbers deals with a sequence
of random variables X, on a probability space. If the X, are independent with the
same distribution and have finite mean ?, then the new sequence of random variables
(?','=\ ??)/? tends to the constant random variable ? a.e. This means that for almost all
points у of the probability space, the value of the sequence at у tends to the constant ?.
For our discussion here, (?,?,?) and (?, ?, ?) are internal probability spaces, and
(? ? ?, ? <g> ?, ? ® ?) is the relevant internal product space. Their associated standard
measure spaces are respectively (?, L-,??), ??.). (*2. Lp(A), Pi.), and (? ??, L-A®p(T®
?), (? <g> P)z.). All processes on the latter space are assumed to be measurable with respect
to the ?-algebra Lk®P{T <& A).
Definition 14.1. A real-valued process / on the space (? ? ?, L-A®p(T ® ?), (? ®
P)z.) is said to satisfy the law of large numbers if for almost all sample realizations ? e ?,
the mean of the sample function f@ on the parameter space ? is essentially a constant.
Remark 14.2. When / satisfies the law of large numbers, the relevant constant must be
equal to the mean of / viewed as a random variable on the joint space of parameters and
samples. This means that for Pi.-almost all ? e ?, fTfC0(t)dkL(t) = ffTxn fd(k®P)L-
Note that there is no requirement of identical distributions in the following result due to
Sun (e.g., see [33]).
THEOREM 14.3. Let f be a real-valued square integrable process on the product
space (? ? ?,??®?(? <g> ?), (? <g> P)l)- If the random variables ft are almost
surely uncorrelated, then f satisfies the law of large numbers, where the almost sure
uncorrelatedness condition means that for (? <g> k)L-almost all (t\,ti) ?? ? ?, ftl and ft2
are uncorrelated.

Nonstandard analvsis and measure theorx
1327
PROOF. This is a consequence of Keisler's Fubini theorem, Theorem 13.6, and the
condition of uncorrelatedness. That is, writing f, for f(t, ·), we have
(/?@-?/,)??? dPL
( (/(??,?)- Ef,,)dkL{t\) J (/(h,")- Eft2)dkL(t2)\dPL
f {f(t\,a>)- Eftl)(f(t2,o>)- Eft2)dkL(ti)dkL(t2)\dPL
( (/(f,, ?) - ?/„ )(/(f2, ?) - Eft2) d(k ® k)LjdPL
= jj (J (ft, - Eft] )(/„_ - Efh)dPL\ d(k ® ?)?. = 0.
Thus, for PL-almost all ией, /?(/?@ - Ef,)dkL = 0, so ?/? = ?/. ?
A classical example of Bernstein shows that there are three events that are pairwise
independent but not mutually independent. Similarly, pairwise independence is also strictly
weaker than mutual independence for a finite collection of random variables. We finish with
statement of Sun's striking result (e.g., see [33]) showing that pairwise independence is, in
fact, almost identical to mutual independence in an ideal setting.
THEOREM 14.4. Let f be a process from (? ? Q,LA®PG ® A),(k ® P)l) to
a separable metric space X. The random variables f, are almost surely pairwise
independent if and only if they are almost mutually independent, i.e., for any ? ^ 2,
ft,, ft2, ¦ ¦ ·> ft,, ore mutually independent for (k")L-almost all (t\, ti, ¦ ¦ ·, t„) e T".
References
[ 1 ] S. Albeverio, J.E. Fenstad, R. H0egh-Krohn and T. Lindstr0m, Nonstandard Methods in Stochastic Analysis
and Mathematical Physics, Academic Press Series on Pure and Applied Mathematics, Vol. 122, Orlando
A986).
[2] E.S. Andersen and B. Jessen, Some limit theorems on set-fimctions, Danske Vid. Selsk. Mat. Medd. 25 E)
A948), 1-8.
[3] R.M. Anderson, A non-standard representation for Brownian motion and ltd integration, Israel J. Math. 25
A976), 15^46.
[4] R.M. Anderson, Star-finite representations of measure spaces. Trans. Amer. Math. Soc. 271 A982), 667-
687.
[5] R.M. Anderson and S. Rashid, A nonstandard characterization of weak convergence, Proc. Amer. Math.
Soc. 69A978), 327-332.
[6] L. Arkeryd, Loeb solutions of the Boltzmann equation. Arch. Rational Mech. Anal. 86 A984), 85-97.
[7] R. Bianconi, J. Prandini and С Possani. A Daniell integral approach to nonstandard Kurzweil-Henstock
integral, Czechoslovak Math. J. 49 A24) A999), no. 4, 817-823.

1328 P.A.Uieb
[8] J. Bliedtner and P.A. Loeb, A reduction technique for limit theorems in analysis and probability theory. Ark.
Mat. 30A) A992), 25-43.
[9] N.J. Cutland and Siu-Ah Ng, The Wiener sphere and Wiener measure. Ann. Probab. 21 A) A993), 1-13.
[10] J.L. Doob, Stochastic processes depending on a continuous parameter. Trans. Amer. Math. Soc. 42 A937),
107-140.
[11] J.L. Doob, Stochastic Processes. Wiley, New York A953).
[12] A.E. Hurd and P.A. Loeb, An Introduction to Nonstandard Real Analysis. Academic Press, Orlando A985).
[13] H.J. Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 B97) A984).
[14] H.J. Keisler, Elementary Calculus, An Infinitesimal Approach. 1st edn A976): 2nd edn, PWS-Kent A986).
[15] H.J. Keisler, Infinitesimals in probability theory. Nonstandard Analysis and its Applications. N.J. Cutland,
ed., Cambridge University Press, Cambridge A988). 106-139.
[16] P.A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory.
Trans. Amer. Math. Soc. 211 A975), 113-122.
[17] P.A. Loeb, Applications of nonstandard analysis to ideal boundaries in potential theory. Israel J. Math. 25
A976), 154-187.
[18] P.A. Loeb. A generalization of the Riesz-Herglotz theorem on representing measures. Proc. Amer. Math.
Soc. 71A) A978), 65-68.
[19] P.A. Loeb, Weak limits of measures and the standard part map. Proc. Amer. Math. Soc. 77 A) A979),
128-135.
[20] P.A. Loeb, A construction of representing measures for elliptic and parabolic differential equations. Math.
Ann. 260A982), 51-56.
[21] P.A. Loeb, A functional approach to nonstandard measure theory. Proceedings of the S. Kakutani
Retirement Conference, Contemp. Math. 26A984), 251-261.
[22] P.A. Loeb, Measure spaces in nonstandard models underlying standard stochastic processes. Proceedings
1983 International Congress of Mathematicians in Warsaw, PWN, Warsaw A984), 323-335.
[23] P.A. Loeb and H. Osswald, Nonstandard integration theory in topological vector lattices. Monatsh. Math.
124A997), 53-82.
[24] P.A. Loeb and M.P.H. Wolff, Nonstandard Analvsis for the Working Mathematician, Kluwer. Amsterdam
B000).
[25] W.A.J. Luxemburg. A general theory of monads. Applications of Model Theory to Algebra, Analysis, and
Probability, W.A.J. Luxemburg, ed.. Holt, Rinehart, and Winston, New York A969).
[26] H. Osswald. Measure and probability theory and applications. Nonstandard Analysis for the Working
Mathematician, P.A. Loeb and M. Wolff, eds, Kluwer, Amsterdam B000).
[27] E.A. Perkins, A global intrinsic characterization of Brownian local time, Ann. Probab. 9 A981), 800-817.
[28] A. Robinson, Non-Standard Analysis, North-Holland, Amsterdam, 1966.
[29] K.D. Stroyan and J.M. Bayod, Foundations of Infinitesimal Stochastic Analysis. North-Holland Studies in
Logic. Vol. 119. Amsterdam A986).
[30] K. Stroyan and W.A.J. Luxemburg, Introduction to the Theory of Infinitesimals. Academic Press, New York
A976).
[31] Y.N. Sun, Integration of correspondences on Loeb spaces. Trans. Amer. Math. Soc. 349 A997), 129-153.
[32] YN. Sun, A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN.
J. Math. Econom. 29 A998), 419-503.
[33] Y.N. Sun, Conventional operations on nonstandard constructions. Nonstandard Analysis for the Working
Mathematician. P.A. Loeb and M. Wolff, eds, Kluwer, Amsterdam B000).

CHAPTER 33
Monotone Set Functions-Based Integrals
Pietro Benvenutia, Radko Mesiarb, Doretta Vivonaa
aDipartimento di Metodi e Modelli Matematici, per le Scienze Applicate. Universita degli Studi "La Sapienza ",
Via A. Scarpa. 16. 00161 Roma. Italy-
E-mail: benvenut@dmmm.imironial.it. vivona@dmmm.uinroinol.it
Slovak Technical University, Radlinskeho 11. 81368 Bratislava. Slovakia and
Systems Research Institute. PAN, ul. Newelska 6. 01-447 Warszawa. Poland
E-mail: mesiar@leo.svf.stuba.sk
Contents
1. Introduction 1331
2. Choquet and Sugeno integrals 1333
2.1. Preliminaries 1333
2.2. Basic and simple functions 1334
2.3. Choquet integral 1336
2.4. Sugeno integral 1340
2.5. Similarity of Choquet and Sugeno integrals 1342
3. Pseudo-additions and pseudo-multiplications 1343
3.1. Pseudo-addition ® 1343
3.2. Pseudo-difference ? 1345
3.3. Pseudo-multiplication О 1345
3.4. Additional conditions with pseudo-ring structures 1348
4. General fuzzy integral 1352
4.1. Integral of simple functions 1354
4.2. Integral of measurable functions 1358
4.3. Additional properties of integral with pseudo-ring operations 1361
4.4. Extension of the integral to [-F, F] 1363
5. Examples 1364
6. Conclusions 1376
Acknowledgement 1377
References 1377
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1329

This page intentionally left blank

Monotone set functions-based integrals
1331
1. Introduction
The classical probability theory works with ?-additive measures and one of its most
important tools is the Lebesgue integral. The Lebesgue integral is a monotone non-negative
linear functional on the space of bounded measurable functions; its construction is strongly
linked with the additivity of the underlying measure.
However, the additivity of a measure may be rather restrictive when modeling several
real situations, e.g., the weights of criterion groups in multi-criteria decision making.
The interaction phenomena put them out of the framework of additive measures. Further,
additive measures are closed under (non-negative) linear combination. However, the
application of some other aggregations of measures, e.g., min or max value of two measures
of every measurable set, may violate the additivity.
To avoid the above mentioned problems, several generalizations of classical probability
measures have been proposed so far, such as pseudo-additive measures, possibility
measures, belief functions, A:-monotone capacities, к -order additive measures, etc. All
these generalizations are covered by fuzzy measures, i.e., monotone set functions vanishing
at the empty set (with some continuity properties when the universe is infinite). These set
functions are sometimes called also premeasures.
When dealing with a pseudo-additive measure, constructions similar to the one of
Lebesgue integral were used (Riecanova A982), Weber A984, 1986), Marinova A986),
Sugeno and Murofushi A987), Ishihachi, Tanakaand Asai A988), Pap A990), Kolesarova
A993)).
However, for a general monotone set function (fuzzy measure) we cannot get an
acceptable integral applying the above mentioned construction methods. Therefore, some
modified integral construction has to be applied, all adapted to the most expressive property
of a general fuzzy measure: its monotonicity. There have been a number of contributions
dealing with integrals based on a general monotone set function (fuzzy measure), which
for the sake of brevity will be called fuzzy integrals through this chapter. The fundamental
one and the most important of them is known as "Choquet integral" after its introduction
in the fuzzy literature by Schmeidler A986). Really this integral has been defined first
by Vitali A925) using an outer measure and successively by Choquet A953-54) in the
field of capacities. De Giorgi and Letta A977) and independently Greco A977) have given
the definition of the integral with respect to a general fuzzy measure and pointed out his
fundamental property: the horizontal additivity.
The Choquet integral is defined for non-negative functions; it is comonotone additive
(i.e., additive for comonotone functions) and homogeneous with respect to the
multiplication by non-negative reals. For any event A, the Choquet integral of the characteristic
function 1 a with respect to any fuzzy measure ? is just the measure ? (A) of the event. Among
several works devoted to the Choquet integral, we recall, e.g., Wang and Klir A992), Den-
neberg A994), Pap A995). For the applications of the Choquet integral, we mention On-
isawa et al. A986), Tanaka and Sugeno A991), Murofushi and Sugeno A993), Sugeno
and Kwon A995), Grabisch A996). A recent overview of state-of-art in this domain is
contained in Grabisch, Murofushi and Sugeno B000).
An alternative approach to the Choquet integral was developed by Sipos A979) who
considered all real functions, not only the non-negative ones. There are two possible

1332
P. Benvenuti el a].
extensions of the original Choquet integral to the class of all measurable real functions,
namely the asymmetric and the symmetric integral: see Sipos A979), Denneberg A994),
Pap A995). The asymmetric one is comonotone additive and positively homogeneous; the
other is completely homogeneous (with respect to the multiplication by a real number,
positive or negative) but it is neither additive nor comonotone additive.
Another fuzzy integral important for the applications of fuzzy set theory in many
domains was introduced by Sugeno A974). The Sugeno integral (in its original form)
deals with functions whose range is contained in [0. 1], i.e., with fuzzy sets, and with
normed fuzzy measures. In several publications, this integral is simply called "fuzzy
integral". It is comonotone maxitive, i.e., pseudo-additive for comonotone functions with
respect to the operation ?: ? ? у = sup{.v, у}. It is л-homogeneous, too, i.e., pseudo-
homogeneous with respect to the operation л: х л у = inf{.v. у]. Similarly as in the case of
the Choquet integral, also the Sugeno integral gives exactly the value ? (A) when applied
to the characteristic function 1 д.
Some generalized integrals have been introduced with respect to a general fuzzy measure
with properties similar to those of Choquet or Sugeno integrals. We recall the Ralescu and
Adams A980) generalization of the original Sugeno integral, the Murofushi and Sugeno
integral A991) based on r-conorm, the Choquet-like integrals of Mesiar A995).
The similarity between the Choquet and the Sugeno integrals was observed by several
authors (Suarez and Gil A986), de Campos, Lamata and Moral A991), Murofushi and
Sugeno A991), de Campos and Bolanos A993), Mesiar A995), Benvenuti and Vivona
A996a, 1996b)). Taking into account this similarity, Benvenuti and Vivona built a general
fuzzy integral based on two binary operations ? (pseudo-addition) and © (pseudo-
multiplication), similar to those introduced by de Campos and Bolanos in the finite spaces.
We recall furthermore the Pan-integral of Yang A985) (see also Wang and Klir A992)),
the multi-linear integral of Matsushita and Kambara A996), but we will not consider them
in this chapter since they do not have the characteristic properties of Choquet and Sugeno
integrals; namely, no type of additivity can be expected for these and the output for the
characteristic function \A is not ? (A), indeed it depends on the set A, not merely by
its measure but by the values of the measure on the family of the subsets of A. Another
interesting class of fuzzy integrals was introduced by Imaoka A997). Based on a ?-additive
atom-less measure on the Borel subsets of the unit square and constructed by means of a
copula (Schweizer and Sklar A983), Nelsen A999)), they include the Choquet and the
Sugeno integrals, are monotone and, when applied to a characteristic function 1д, they
give ? (A). A generalization of Imaoka's approach was recently proposed by Klement,
Mesiar and Pap B001). However, Imaoka's integrals do not possess in general any kind of
additivity, so we will not consider them, too.
In this chapter, using the procedure given by Benvenuti and Vivona, we will introduce a
class of general fuzzy integrals with respect to a general fuzzy measure. A fuzzy integral
is associated to any couple of fitting operations ? and ©. We request the properties which
characterize both the Choquet and the Sugeno integrals. These integrals, the integral of
Ralescu and Adams and the one of Weber A986) are particular cases in the class introduced
in this chapter. As a matter of fact, in the case when ? = + and © — · our approach results
just the Choquet integral (on the interval [0, oo]); in the case ? = ? and © = л working
on the unit interval [0, 1 ] we get the original Sugeno integral, while in the case of the

Monotone set functions-based integrals
1333
interval [0, +00] the fuzzy integral of Ralescu and Adams is obtained. Similarly, the case
in which ? = ? and © is any strict r-norm on [0, 1] leads to the integral proposed by
Weber.
Note that if the considered fuzzy measure is finitely ©-additive then our integral is a
special case of the Sion integration method introduced in A973). However, the possible
lack of ?-additivity by a general fuzzy measure puts our integral out of the Sion concept.
In general, fuzzy integrals can be viewed as specific aggregation operators: see Bouchon-
Meunier A998) and Benvenuti and Vivona B000). Some aggregation operators (with finite
or infinite number of inputs) are represented by integrals on a suitable fuzzy measure space;
the fuzzy measure is a tool in this representation; its values come from an interval [0, ?]
which need not be linked with the interval [0, F] related to the inputs. To keep this
generality, we even abandon the request of reconstruction of the original fuzzy measure
from the respective integral.
The exposition is organized as follows. In Section 2, the properties of the Choquet and
the Sugeno integrals are recalled, discussed and compared. Section 3 is devoted to an
appropriate choice of the binary operations ? and 0. These operations strongly influence
the properties of the resulting integrals. We assume for the pseudo-multiplication О only
the essential properties. Indeed, the additional properties of ? result in the additional
properties of the corresponding fuzzy integral, e.g., the associativity of ? results in the
©-homogeneity of the integral and the reconstruction of the original fuzzy measure from
its related integral is linked with the existence of a left-neutral element for the pseudo-
multiplication which we should work with. The construction of the general fuzzy integral
is done in Section 4. Finally, in Section 5 we give several examples of operations ? and ©,
with the corresponding fuzzy integrals.
As an informative overview of recent results in the fuzzy measures and fuzzy integrals
theory and application we recommend to the interested readers the volume of Grabisch,
Murofushi and Sugeno B000).
2. Choquet and Sugeno integrals
2.1. Preliminaries
Let ? be an abstract space, A a ?-algebra of subsets of ?. Let ? : А -> Ш = [0, oo] be
a fuzzy measure, i.e., a non-decreasing set function with ?@) = 0. The triple (?, ?, ?) is
called a fuzzy measure space. No assumption is made in general about the continuity of ?;
we say that ? is continuous from below if:
C„ e A C„ /С => м(С„) / м(С).
We recall that a fuzzy measure is additive (is a valuation on A) if
?(???)+?(??\?) = ?(?)+?(?) VA,SeA

1334
P. Benvenuti et al.
Moreover, we say that ? is maxitive (v-additive) if
?(???)=?(?)??(?) ??,???
To any function f :? -> R = [-00,+00], the family Cf =: {Cf(x): ? e R) U
{CUjc): jc e R) is associated, with C/(jc) = {? e ?: /(?) > jc) (= {/ > *} in short)
and С'Лх) = {? e ? | /(?) ^ ?] (=[f ^ ?] in short). The family C/ is a chain, i.e., it is
completely ordered with respect to the set inclusion. We call Cf the chain associated to f.
In order to examine the additivity or the maxitivity of the integrals it is useful to
consider the concept of comonotone (common monotone) functions (see Denneberg
A994)). Every function/ : ? -> R introduces on ? a semi-order relation: ?\ <f coi «=>
f{cu\) < f{o>i). The comparison between the two semi-order relations, introduced by two
functions / and g, suggests a binary relation ~ in Rn which is symmetric and reflexive
but not transitive.
Definition 2.1. / and g are comonotone (/ ~ g) iff the two corresponding semi-
orders are non-contradictory, i.e., there exists no pair ?\, ??_ \? ? such that ?>? </ <??
and a>2 <g <W|.
PROPOSITION 2.2. Given two functions f and g in Rn, let Cf and Cg be the two chains
associated to f and g, respectively, and C/, the chain associated to the function h = f +g;
the following conditions are equivalent (see Dennenberg A994)):
(i) f~g\
(ii) CfUCg=Ch;
(iii) there exist two continuous non-decreasing functions ?/ and ?? from R tо R such
that:
<Pf(z) + <pg(z) = z, f(w) = <pf{h(w)), ?(?) = (^(/?(?)).
2.2. Basic and simple functions
We shall denote with ?(?), or shortly T, the set of all measurable (Д-measurable)
functions from ? to R+: that means that the chain Cf associated to / e ?(?) is a subset
of ?. This definition of measurability coincides with the classical one, because Л is a
? -algebra.
For any a e R+ and any A e ?, the function b(a, A) defined by:
fa ifa>eA,
0 if ? g A,
is called basic (simple) function.
A measurable mapping s: ? -> R+ is called a simple function if its range is finite.
We shall denote by ?(?), or shortly S, the set of all simple functions from ? to R+.
The simple functions have several representations by means of basic functions. The most

Monotone set functions-based integrals
1335
known representation from the classical measure theory is based on the notion of the finite
partition of ?. To any simple function s, with range {a\, ai, ¦ ¦ ¦, a,,}, a finite partition
{A|, ?2,..., A,,} is associated, where A,¦ = {? e ? | /(?) — ?,}. The function s admits
the representations:
? «
i = J]fo(flf,A/)s\/fo(fli,Ai). B.1)
1=1 1=1
The possible null term in B.1) can be omitted, and by assuming:
0 < ? ? < 02 < ¦ ¦ ¦ < a„, B.2)
so we obtain univocally the classical standard representation, with minimal number of
used basic functions.
The family S of simple functions is fundamental in the construction of an integral
because it is dense in the class ? of measurable functions. As a matter of fact, any
measurable function is the limit of a non-decreasing sequence of simple functions;
therefore any functional defined on ?, which is continuous from below, is completely
determined by its values on S.
The Lebesgue integral of a simple function s is defined taking its classical
representation B.1) and putting:
I sdm— II y^fo(a,-, A,) J dm = y~]q,w(A,·).
The structure of the classical measure space (the ?-additivity of the measure) allows
also the use of any additive representation by means of basic functions, in order to
express univocally the integral as similar sum. The lack of additivity in the case of fuzzy
measures requires another type of representation of simple functions. In order to express
the comonotone additivity of the Choquet integral and the comonotone maxitivity of the
Sugeno integral, it is convenient to introduce a representation by means of a system of
comonotone basic functions.
PROPOSITION 2.3. Any simple function s admits a comonotone additive step
representation:
s = Yib(ci,Ci) B.3)
1 = 1
and a comonotone maxitive step representation
n
s = \/b(a,,C,) B.4)
1 = 1
with C\ DC2D -DC,, D 0 (strictly).

1336
P. Bemenuti et al.
PROOF. We assume: C, = U"=/A/ and from B1^ and B·2) we obtain directly the
expression B.4). We obtain easily the expression B.3) from B.1) taking: c\ = a\, ст =
a2-a\,...,c„=a,,-a„-\. ?
The representations B.3) and B.4) (again omitting the possible null term) are minimal
for the number of basic functions. They are characterized by common conditions С ? D
C2 D ¦ ¦ О C„ D 0, and {d} С С,. Note that Cs = {?, C,. C2, ¦ ¦ ¦, C„, 0}.
2.3. Choquet integral
We recall the definition of the Choquet integral and its fundamental properties. For more
details see Denneberg( 1994) and Pap A995).
Definition 2.4. The Choquet integral is the functional ?(?) -> M+ defined in this way:
S/??=: f p(Cf(x))dx. B.5)
The function ? -+ ?(? f(x)) is non-increasing, so the integral is always meaningful in the
sense of Cauchy-Riemann.
The fundamental properties of the Choquet integral are the following:
(CH. 1) BASIC VALUES. For any basic simple function it is:
b{c,C)d? = c¦?{C) VceE+. VCeA B.6)
with convention 0 · oo = oo · 0 = 0.
/
PROOF. Equation B.6) is immediate from B.5) because if / = b(c, C), then we have
Cf(x) = С for O^x <candCf(x) = tf, for ? ^ с ?
In particular, for с = 1, the basic function coincides with the characteristic function 1 с
of the set С and the Choquet integral reconstructs the given fuzzy measure:
/
\€??=?@ VCeA
(CH. 2) MONOTONICITY.
fUg => ? /??^? 8??.

Monotone set functions-based integrals
1337
PROOF. The property is immediate consequence of the monotonicity of ?. Indeed, if
/ ^ g, then it is Cf(x) с Cg(x) and so m(C/(*)) ^ ц.(Сх(х)) for all xeR+. ?
(Ch. 3) Homogeneity.
? (с-/Lд = с· ? f ?? VceE*.
PROOF. For с = 0 the relation is obvious. Suppose с > 0; from C-/U) = C/(x/c), we
obtain:
?(?/)??=? ii{Cf{xlc))dx=c- ? ?(€ f(y))dy=c ¦ j /??.
(CH. 4) COMONOTONE ADDITIVITY.
f~g=* f (/+8)?? = { fdp + f 8??. B.7)
PROOF. First we consider the function h = f + g and the two functions ?/ and ?? given
by (iii) of Proposition 2.2. We define a pseudo-inverse of ?( putting:
?)(?) = sup{f: <pf(t) ^ x] = inf{f: <p/@ > x].
So we have:
(Pf(y)>x<=*yxp*(x) and f(cu) = (pf(h(w))>xt=$h(co)>(pf(x).
From these equivalences and the relation /(?) = <p/ (h(co)) we obtain
? /??= [ vi{Cf(x))dx = ? ?{?„{?}(?)))??
Starting from the Lebesgue measure /, we introduce on the Borel sets of E+ the
measure mf defined by mf(B) = 1({?: ?*(?) e B}). This is exactly the Lebesgue-
Stieltjes measure induced by the function ?/ because:
mf([a, b[) = ?([?/(?), ?/?)[) = ?,?) - ?,(?).
By means of the substitution у = ?* (?) (see, for instance, Halmos A950, Sec. 39)) we
obtain:
/ ???= ? ?(?,(>0)^

1338
P. Beiwenuti el al.
Equally, using the function <pg we have:
where mg is the Lebesgue-Stieltjes measure induced by ??. Now, by Proposition 2.2(iii)
we get <pf(y) +<pg(y) = y- So we obtain mf+m4=l and finally
/¦00 /-X /-ЭС /·
/ ?(?,?(?))???/+ n(Ch(y))dmg= / ^{Ch(y))dy = it hdii.
D
(Ch. 5) Horizontal additivity. Given a measurable function f and с е R+, we
obtain a horizontal additive decomposition by cutting the function f at the level с and
putting
0 iff(?) <Cc,
/(?) - с if /(?) > с.
/ = (/лс) + /+, withft+(co) = V
For any horizontal additive decomposition the Choquet integral is additive:
j /?? = {(/ ??)??+? f+??. B.8)
PROOF. The functions (/ л с) and /c+ are comonotone; B.8) is a special case of como-
notone additivity. ?
(CH. 6) CONTINUITY FROM BELOW. If the fuzzy measure ? is continuous from below,
then the Choquet integral is also continuous from below, i.e., for any non-decreasing
sequence {/„}„ем it is:
f (sup f„J ?? = sup ? f„ ??.
PROOF. If we assume that the fuzzy measure ? is continuous from below, then we
obtain the continuity from below of the Choquet integral from the well known theorem
of monotone convergence for the Cauchy-Riemann integral. In fact, if /,, / f pointwise,
then Cf„{x) / Cf(x) and by continuity of ? we obtain ?(?/?(?)) /д(С/(х)), too. D
THEOREM 2.5. If ? is ? ?-additive measure, the Choquet integral coincides with the
Lebesgue integral:
?????
fan.
?

Monotone set functions-based integrals
1339
PROOF. Let s be a simple function, from the comonotone step representation B.3) we see
that
С,(дс):
С ? if 0 ^ ? < a\,
C, if a,¦ _ ? ^ ? < at,
0 if a„ ^ jc.
So, by the definition B.5) we obtain, for any simple function:
/
sd? =а|Д(С|) + (а2 - ai)M(Ci) + (аз -а?)м(Сз)Н
+ (а„ -а„_|)м(С„).
Now, С„ = А„ and, as the fuzzy measure ? is ?-additive, then ?@ — m(C,-+i ) = ?(?,).
So we obtain
f ,?<*? = ]?4?(?,)
^ i=l
This is as well the value of the Lebesgue integral of the same function s. The identity of the
Choquet integral and Lebesgue integrals on ?(?) implies the identity on the entire family
?(?) because both integrals are continuous from below. ?
Note also that the additivity of ? is enough to ensure the additivity of the corresponding
Choquet integral, see Denneberg A994).
Moreover, if the fuzzy measure ? is continuous from below, then the Choquet integral
of a given function / can be represented by means of the Lebesgue integral of the same
function in a measure space (?,?/,??)-), which is however depending on /.
THEOREM 2.6. In a fuzzy measure space (?,?,?), with ? continuous from below, the
Choquet integral of any function f coincides with the Lebesgue integral in the measure
space (?,A/,m f):
????=?
fdmf,
?
where A/ is the ? -algebra generated by С / and m / is the ? -additive measure induced on
A / from the restriction of ? to the chain Cf.
PROOF. If ? is continuous from below, its restriction of Cf really admits a unique
extension to A/ (see Halmos A950, Sec. 13)). The Choquet integral of the function /
with respect to ? is equal to the Choquet integral of the same function with respect to
this extension m/ because ? and m f are equal on Cf. By Theorem 2.5 the latter integral
coincides on the family of all Af -measurable functions with the Lebesgue integral in the
measure space ^,A/,mf). Obviously, / is A/ -measurable. ?

1340
P. Benvenuti el al.
2.4. Sugeno integral
We recall the definition of the Sugeno integral and its fundamental properties. For more
details see Pap A995).
Let (?,?,?) be a fuzzy measure space with ? : A —>¦ [0, 1]; we consider the family
T\ (?) of all measurable functions from ? to [0, 1].
Definition 2.7. The Sugeno integral is the functional ?\(?) -> [0, 1] defined in this
way:
? ???=: \J (* ? m(C, (*))). B.9)
velO.ll
The fundamental properties of the Sugeno integral are the following:
(Su. 1) Basic values.
/
b(c,C)d? = cл?(C) Vc e [0. 1]. VC e A.
PROOF. If / = b(c, C), then we have C/(x) = С for 0 ^ ? < с and Cf(x) = ttforx^c.
Therefore from definition we obtain:
/
b(c,C)d?= \J (jc ??(?) =слд(С). ?
velO.cl
The value 1 is the unit element for the operation л in the interval [0, 1]. The Sugeno
integral of the characteristic functions gives back the fuzzy measure:
/
\???=?@ VCeA.
(SU. 2) MONOTONICITY.
/^?=> f /??? $ gd?.
i /??? i
PROOF. If/<g,thenitisC/(Jc)CCs(Jc)andsoM(C/-(Jc))^M(C1?(Jc))foralU e [0, 1[.
This inequality involves by means of B.9) the inequality between the two integrals. D
(SU. 3) Л-HOMOGENEITY.
i (??/)??=??<& f ?? Vc e]0, 1].

Monotone set functions-based integrals
1341
PROOF. We have: Ссл/(х) = С/(?) for* < с and Ссл1(х) = 0 for ? ^ с. So we obtain:
У дслд(С?.л/(дс))= V дслд(С/(дс)) = сл( V дг л ? (С/Ч*)) V
.telO.I] леЮ.<1 VvelO.ll '
D
(Su. 4) Comonotone maxitivity.
/~g => i (fvg)dv = f /???? 8??. B.10)
PROOF. If / ~ g, by Proposition 2.2 the two sets C/(x) and Cg(x) belong to the same
chain and therefore it is C/(x) => Ся(х) or Ск(л) 2 С/Ч*) for every * e [0, 1[. In both
cases
/i(C/VJf(*)) = M(C/(*) U Q(*)) = д(С/(дс)) ? м((Сч(*)).
The equality B.10) follows easily from the definition B.9) and from the algebraic
properties of operations ? and л. ?
(Su. 5) Horizontal maxitivity. Given a measurable function f e ?\(?) and с е
]0, 1[, we obtain a horizontal maxitive decomposition by cutting the function f at the
level с and putting:
? ...,, ,V/ . .0 iffW^c,
/ = (/Ac)v/(v, withf?(co) =
/(?) iff(w)>c
For any horizontal maxitive decomposition the Sugeno integral is maxitive:
|/^ = |(/ac№v|/>. B.11)
PROOF. The functions (/ л с) and fv are comonotone; B.11) is a special case of
comonotone maxitivity. ?
(Su. 6) CONTINUITY FROM BELOW. If the fuzzy measure ? is continuous from below,
then the Sugeno integral is continuous from below, too, i.e., for any monotone non-
decreasing sequence {/„}«eN:
i isup/„j<iM = supi ?„??.
PROOF. If /„ / / pointwise, then it is С/¦„(*) / Cf(x). If we assume that the fuzzy
measure is continuous from below, then we have ? л д(С/„(*)) У х л д(С/(*)). ?

1342
P. Benvenuti et al.
2.5. Similarity ofChoquet and Sugeno integrals
The properties (Ch.) characterize univocally the Choquet integral. The properties (Su.)
are exactly the same as (Ch.) except for replacement of operations + and · with the
operations ? and л, respectively. Likewise, they characterize univocally the Sugeno
integral. Many characterization theorems are known in the literature. We expose here two
theorems, which are formally identical under replacement of the operations + and · in R+
with the operations ? and л in [0, 1 ].
We assume that a fuzzy measure space (?, ?, ?) is given, with
? : ? -> Ш+ continuous from below.
THEOREM 2.8. The Choquet integral is the only functional Lch'-F(H) -> E+ which
verifies the properties (Ch. 1, 5, 6).
PROOF. Let Lch'-F(H) -> K+ be a functional which satisfies the properties (Ch. 1,
5, 6). First we consider a simple function s e <S(i2) and its comonotone additive
representation B.3); this representation is obtained by means of ? - 1 repeated horizontal
additive decompositions. From the properties (Ch. 1, 5), which are supposed to be true for
the functional Lch and are true for the Choquet integral, we obtain for both the same value
ons:
Lch(s) = ?^ ·?@= ? .?^?.
?=? J
The functional Lch and the Choquet integral are identical on the class <S(i2) of simple
functions; both are monotone in this class, which is dense in ?(?). By the property
(Ch. 6) of continuity from below, which is assumed for the functional Lch and holds for
the Choquet integral, the functional Lch and the Choquet integral are identical on the
whole ?(?). As a matter of fact, any function / e ?(?) is the limit of a non-decreasing
sequence of simple functions: s„ / /; from the property (Ch. 6) we obtain:
Lch(f)= Hm LCh(s„)= lim f s„d?= ? f ??.
?—>эс ?—юс J J
a
If the fuzzy measure is not continuous from below, then the Choquet integral results into
a functional in general not continuous from below. Nevertheless the remaining properties
(Ch. 1, 2, 3, 4, 5) need not characterize univocally the Choquet integral.1 To keep the
universality of properties characterizing the Choquet integral we will deal with fuzzy
measures continuous from below.
'Take, e.g., ? defined by: ?@) = 0, ?(?) = ос, ?(?) = 1 if ? ? 0. ?. Then the Choquet integral gives:
???? = sup/, if inf/ = 0, $f ?? = ос otherwise. Let L be the functional defined by: L(f) = sup/, if
/"' @) / 0, L(f) = oc else. The functional L fulfills, as well, the properties (Ch. 1. 2. 3. 4. 5). but it is different
from the Choquet integral if the space ? is not finite.

Monotone set functions-based integrals
1343
THEOREM 2.9. The Sugeno integral is the only functional Lsu'-^\(^) -> [0. ? which
verifies the properties (Su. 1, 5, 6).
PROOF. The proof is exactly the same as in the previous theorem, except for replacement
of the operations +, · with the operations v, л and the use of the comonotone maxitive
step representation B.4) instead of the comonotone additive step representation B.3). D
The property of monotonicity for both Choquet and Sugeno integrals is involved by
(Ch. 1, 4) or (Su. 1, 4), respectively. The stronger properties of comonotone additivity or
of comonotone maxitivity are involved by the weaker properties of horizontal additivity or
of horizontal maxitivity in the context of the functionals which verify the axioms (Ch. 1,
5) or (Su. 1, 5), respectively.
The above theorem shows the way in order to give a common definition of Choquet
and Sugeno integrals. Moreover, it suggests a unique procedure for building up a
general integral by means of two pseudo-operations ? (pseudo-addition) and ? (pseudo-
multiplication). This integral uses a general fuzzy measure and includes the Choquet and
the Sugeno integrals as particular cases.
3. Pseudo-additions and pseudo-multiplications
In the previous section, we have seen that the Choquet integral is strongly related to the
usual arithmetic operations of addition + and multiplication · on the interval [0, oo].
Similarly, the Sugeno integral depends on the operations ? and л on the interval [0, 1].
In order to find a common framework for both Choquet and Sugeno integrals, we have to
deal with a general pseudo-addition ? and a general pseudo-multiplication ? which must
be fitting to each other. The algebraic properties of functionals taken into account in the
integration procedures result to the corresponding properties of the operations ? and ?
which we have to work with. In general, ? is supposed to be a continuous generalized
triangular conorm (see Sugeno and Murofushi A987), Pap A990, 1995), Wang and Klir
A992), Grabisch, Murofushi and Sugeno A992), Mesiar A995), Benvenuti and Vivona
A996a, 1996b), Benvenuti and Mesiar B000a, 2000b)).
3.1. Pseudo-addition ?
Definition 3.1. Let F e]0, oo]; a binary operation ? : [0, F]2 -> [0, F] is called a
pseudo-addition on [0, F] if the following properties are satisfied:
(Al) a® b = b®a (commutativity)
(A2) a ^ a', fo < b' =>· ? ? b ^ ?' ? b' (monotonicity)
(A3) (? ? b) ? с = ? ? (b ? с) (associativity)
(A4) ??0 = 0?? = ? (neutralelement)
(A5) a„ -> a, bn -> b =>¦ a„ ? b„ -> ? ? b (continuity).
The axioms (A2)-(A5) ensure that a pseudo-addition ? on [0, F] forms an /-semigroup
of Mostert and Shields A957). The general form of the pseudo-addition is commutative.

1344
P. Benvenuti el al.
So, the commutativity follows from the remaining axioms; we have included also it only
for the sake of completeness.
In the following proposition we recall the general structure of the operation ? in an
/-semigroup; for more details the reader can see Mostert and Shields A957), and the
volume of Klement, Mesiar and Pap B000).
We note preliminarily that, by virtue of the continuity of the operation ?, the set of
©-idempotent elements C® = {a e [0, F]: a® a = a] is closed. It is never empty because
0, F e C®. So, the complement Ce = {a e [0, F]: ? © a > a} is the union of a discrete
system (countable, finite or, in case, empty) of disjoint open intervals:
Ce=(J]a*-&[.
keK
PROPOSITION 3.2. Let ? be a pseudo-addition on the interval [0, F]. There exists a
system of disjoint intervals ]<**, Д: [, k e К, К finite or countable, and an associatedfamily
of continuous strictly increasing mappings gk : [otk,Pk] -* [0, oo] with gk(ak) = 0 such
that:
a®b= (**~'((**(?)+**(&)) л g*(ft)) if]a,b[e]ak,Pk[2, (ЗЛ)
lavi) otherwise.
Each function gk is determined uniquely up to a multiplicative positive constant.
For the closed set C® the two extreme cases are possible: C® = {0, F) C$ = ]0, F[ and
C® = [0, F], C^ = 0. In the first case the pseudo-addition assumes one of the following
forms:
Example 3.1.
a(Bb = g-](g(a)+g(b)), C.2)
being g : [0, F] -> [0, oo] an increasing bijection.
Example 3.2.
a®b = g-]((g(a)+g(b))Al), C.3)
being g : [0, F] -» [0,1 ] an increasing bijection.
The form C.2) shows an isomorphism with the usual addition on [0, oo]; this is obtained
in particular when g is the identity map. The form C.3) shows an isomorphism with the
truncated addition on [0, 1].
In the second case the pseudo-addition is the idempotent operation v:
Example 3.3.
a®b = avb. C.4)

Monotone set functions-based integrals
1345
3.2. Pseudo-difference ©
For a given b e [0, F] the function ? н> fo ? ? need not to be strictly monotone. The
equation b®x = a, with a^ b has always a solution, but the solution need not be unique.
So, the pseudo-difference is not always univocally defined. We shall use, together with a
given pseudo-addition ?, the following (minimal) pseudo-difference, compare also Weber
A983).
Dehnition 3.3. The pseudo-difference is the binary operation ©:[0, F] ? [0, F] ->
[0, F] defined by:
aQb = inf{x e[0, F]: b®x ^ b}. C.5)
The pseudo-difference is defined on the whole square [0. F] ? [0, F], not only if a > b.
When a < b, then ? ? b = 0, and when a > b, then ? ? b > 0. More, when a ^ b, then
fo© (fl9i>) = fl.
3.3. Pseudo-multiplication ©
We will build an integral for the Д-measurable functions (with values in the interval [0, F])
on a fuzzy measure space (?,?, ?) with ?(?) — ? @ < ? < ??). The integral will
be a functional with some basic properties connected, as well, with the given pseudo-
addition ©. Any integration procedure uses another binary operation ©, which is called
pseudo-multiplication. The expected properties of the integral determine the next minimal
properties of © which we have to require.
Our approach assumes that the integral of any basic simple function b(a, A) depends on
the event A only by means of its measure ?(?):
r®
/ b(a,A)?d? = I(a,?(A)).
This formula should be valid for any a e [0, F], A e ? and for any fuzzy measure
? : ? -> [0, ?], with ? = ?(?). Therefore the function / defines the so called pseudo-
multiplication ©: [0, F] ? [0, M] -> [0, F] as
aQb=:I(a,b). C.6)
We expect that the integration of the zero function results to zero. The zero function can
be expressed either as fo@, A) for arbitrary A e ? or as b(a, 0) for arbitrary a e [0, F].
Recall that ?@) = 0 and thus it should be
0©?(?) = ?©0 = 0. C.7)
The integral is supposed to be monotone, comonotone ?-additive and continuous
from below whenever the underlying fuzzy measure ? is continuous from below. All

1346
P. Benvenuti el al.
above mentioned expected properties, when applied merely to the basic simple functions,
determine the minimal assumptions on ? which we have to require and we are going to
state.
Let ?? < a2 and A\ С Ai\ it is ?{?\) < ?(??) and b(a\. A\) ^ b(ai, Ai). For the
monotonicity of the integral we must have
????(?|)<;?2??(?2). C.8)
For any ??, 02, A, the two functions b(a\, A)andfo(a2, A) arecomonotoneandfo(fii|, ?) ?
b(a2, A) = b{a\ ? аг, A). For the comonotone ©-additivity of the integral we must have:
(?? ? ?2) ? ?(?) = (?, ? ?(?)) ? (?2 ? ?(?)). C.9)
Finally, let ?, / a and ?, / A, then fo(a,. A,) / b(a, A), and, if the fuzzy measure is
continuous from below, it is also ?(?,) / ?(?). In order to have the continuity from
below of the integral, the following must hold:
?,??(?,)/???(?). C.10)
The formulae C.7)—C.10) should be valid for any ?,, for any A, and for any fuzzy measure,
i.e., for a, e [0, F] and ?{?,) e [0, M], as ? =?(?). Therefore for a correct building up
of a general fuzzy integral the monotonicity, the left distributivity, the left continuity of the
pseudo-multiplication © are unavoidable.
Dehnition 3.4. Let ? be a given pseudo-addition on [0, F] and let ? e]0, oo]. A binary
operation ©: [0, F] ? [0, ?] -» [0, F] is called a ?-fitting pseudo-multiplication if the
following properties are satisfied:
(Ml) aQ0 = 0Qb = 0 (zero element)
(M2) a^a\b^b' =>· a © b ^ a © b' (monotonicity)
(M3) (a®b)Oc = aOc®bQc (left distributivity)
(M4) (sup,, an) © (sup,„ b,„) = sup,, ,„ a„ © b„, (left continuity).
Note that we will follow the usual convention to the priority of operations, i.e., a © с ? b © с
means (a © c) ? (b © c).
The left distributivity (M3) corresponds to the Cauchy equation in the I-semigroup
([0, F], ?) for all functions fh(x) = xQb,be[0, M\.
fb(x®y) = fb(x)®fb(y), x,ye[0,F]. C.11)
Consequently we have the following characterization of all ?-fitting
pseudo-multiplications.
THEOREM 3.5. The binary operation ©: [0, F] ? [0, ?] -> [0, F], defined by
cQb = fh(c)

Monotone set functions-based integrals
1347
is a ®-fitting pseudo-multiplication if and only if (f, | b e [О, М}) is a non-decreasing
system of left-continuous solutions of the Cauchy equation C.11) such that
/b@)=0, /o(c) = 0, Mc) = sup f,(c)
a<b
for all с e [0, F] and all b e ]0, ?].
The class of solutions of the generalized Cauchy equation is closed with respect to the
concerned pseudo-addition (see Benvenuti, Vivona and Divari B002), Proposition 2.1).
So the class of ?-fitting pseudo-multiplications is, as well, closed with respect to the
operation ?.
COROLLARY 3.6. Let Q\ and ©2 be two pseudo-multiplications: [0, F] ? [0, ?] ->
[0, F]. If ? ? and ©2 are (B-fitting, then also the operation ?* = (??) ? (©2) given by:
a ©* b = (a ©| b) ? (a ©2 b) is a ?-fitting pseudo-multiplication.
PROOF. From the two systems (/1./, | b e [0, ?]) and (fu, I b e [0, ?]), associated to the
two operations ©| and ©2, we obtain the system (/,* = f\j, ? fi.b I b e [0, M]) which is
a non-decreasing system of left-continuous solutions of the Cauchy equation. The pseudo-
multiplication ©* is associated to this system. ?
However, the structure of all solutions of the generalized Cauchy equation was recently
characterized in Benvenuti, Vivona and Divari B002). So, the above theorem allows to
describe explicitly the ?-fitting pseudo-multiplications in several specific cases.
Example 3.4. If the pseudo-addition has the form C.2), then each left-continuous
solution of the Cauchy equation C.11) has the form:
/(*) = ?"'(? ·?(*)), ??[0,??], C.12)
with the convention 0 · 00 = 00 · 0 = 0. The null solution of the Cauchy equation is obtained
in particular for ? = 0. The system (//,) defines a left-continuous non-decreasing map
?: [0, ? ] -> [0, oo] with ?@) = 0. So, in this first example the general form of the pseudo-
multiplication is
aOb = g-](g(a)-i(b)). C.13)
Example 3.5. If the pseudo-addition has the form C.3), then each left-continuous
solution of the Cauchy equation C.11) has the form:
/W = «"'((^«W)a1), ??{0}?[1,??]. C.14)
The system (//,) defines a left-continuous non-decreasing map ?: [0, ?] -> {0} U [ 1, oo].
So, there is a constant bo e [0, M] such that k(b) = 0 (and then //, = 0) whenever b < bo

1348
P. Benvenuti el al.
and k(b) ^ 1 whenever b > Ь$. Hence, in this second example the general form of the
pseudo-multiplication is
??& = *_?((*(?)·?(&))??). C.15)
Example 3.6. For the pseudo-addition ? = ? the left distributivity and the right one
follow from the monotonicity of 0. Therefore, any binary operation which satisfies (Ml),
(M2) and (M4) is fitting with the pseudo-addition v.
Right unit. If the integral is assumed to represent an aggregation operator (Benvenuti
and Vivona B000)) then the property of idempotence is requested: the integral of any
function constant on the whole ? must give the value of the function. From C.3) we
obtain
/? /-?
с ? ?? = / b(c, ?)??? = ?? ?(?).
So, ? = ?{?) must be a right unit for the pseudo-multiplication:
(M5) aQ ? = a (right unit element).
Nevertheless the idempotence is not necessary in other applications of the integral; so
in general we do not require the condition (M5) in the definition of ?-fitting pseudo-
multiplication.
Example 3.7. In the form C.13) the condition (M5) is equivalent to ?(?) = 1. So
the form of the pseudo-multiplication verifies (M5) with any non-decreasing and left-
continuous function ?: [0, ?] -> [0, 1 ] with ?@) = 0 and ?(?) = 1. In the form C.15) the
condition (M5) is still equivalent to ?(?) = 1 but in this case the condition is very strong
because the function ? can assume only the values 0 and 1. The pseudo-multiplication is
determined up to the value of the parameter bo e [0, ? [. It is ? ? b = 0 for b ^ bo and
a © b = a for b > bo-
3.4. Additional conditions with pseudo-ring structures
We assume from now on that the interval [0, M] (values of the fuzzy measure) coincides
with the interval [0, F] (values of the functions). This is frequent in the applications except
when the integrals represent special aggregation operators (Benvenuti and Vivona B000)).
If the binary operation © is defined on the square [0, F]2, as well as the operation ?, the
two operations give to the interval [0, F] the structure of a pseudo-ring. In this case, some
other properties are meaningful for the integral and some additional conditions must be
requested to the pseudo-multiplication for their fulfillment.

Monotone set functions-based integrals
1349
Left unit. Let ?: [0, F] ? [0, F] -> [0, F] be a pseudo-multiplication with left unit u:
и ? b = b for all b e [0, F]. The basic simple function b(u. A) = LI? is called the pseudo-
characteristic function of the event A. Now, from C.3) we have
/
иАОац. = иО ?(?) = ?(?).
If we expect that integration of a pseudo-characteristic function will result in the underlying
fuzzy measure ?, then the existence of a left unit element и for the pseudo-multiplication
? is needed.
(M6) There exists и e ]0, F] such that uQ b = b (left unit element).
PROPOSITION 3.7. The only pseudo-addition ?, which admits a ®-fitting pseudo-
multiplication with left unit и which is ®-idempotent, is ? = v.
PROOF. Let и be ?-idempotent and as well left unit for a ?-fitting pseudo multiplication.
Then we have: a®a = uQa®uQa = {u®u)Qa = uQa = a for any a e [0, F].
Consequently ? = v. ?
Any binary operation non-decreasing and left continuous in both variables, with ? ? 0 =
0 ? b = 0, is fitting with the pseudo-addition v. The equation uQb = b may be true for all
b e [0, F] and и running in some interval. Therefore, in a v-fitting pseudo-multiplication
the uniqueness of the left unit can be missing.
PROPOSITION 3.8. If ? ? ? and ? is a (B-fitting pseudo-multiplication with left unit u,
then (with notations of Proposition 3.2) there exists к so that: и e ]<**, Д-[, gkWk) = oo,
and the pseudo-multiplication is univocally determined on the strip ]ak, ??<] x [0, F] by:
aeb=\ 8^((8k(a)gi,(b)/gk(u)) л gh(M) ifb e ]<*„, ?„[, he К,
\b ifbeC®.
C.16)
PROOF. From Proposition 3.7 the left unit и is not ?-idempotent; so, there exists к such
that и e ]?*,&[. If fo e Ce, from и О b = b and Lemma 4.6 of Benvenuti, Vivona and
Divari B002), we deduce a Q b = f,,(a) = f,,(u) = b for all a e ]ak, pk[.lf b<?C®, it is
b e ]<*/,, /3/,[ for some h e K. Theorem 4.7 of the same note ensures that we have for all
a e ]ak, ?? ¦
aQb= fh{a) = g^({k„ ¦ gk(a)) ? g„(fi,))
with some ?& e ]0, oo] and with limitation ?/, · #НД0 > gh(Pn) if gk(fik) < oo. Moreover,
the condition uQb = b request that ?* · gk (и) = gi, (b), so we obtain C.16) for a e ]ak ,pk[.
But, if gk(Pk) is finite, the limitation
. ,a , gh(b)-gk(fik)
gk(u)

1350
P. Benvenuti et al.
cannot be fulfilled for all b e ]<**, Д [. Therefore gk(fik) must be necessarily oo.
Finally, C.16) is true for a = ??-, as well, because of the left continuity of the pseudo-
multiplication. D
COROLLARY 3.9. 1/(&??, any (B-fitting pseudo-multiplication possesses not more that
one left unit.
PROOF. Suppose и е]ог*,Д-[ is a left unit of a ?-fitting pseudo-multiplication. Evidently,
from C.16), no other different left unit exists in the interval ]at, ? к¦.]¦ Moreover, from C.16)
we see that:
inf a © b = sup{c eCe: с ^ b}, aQft = ? ?) = : inf{c e Ce: с > b].
a>ak
Further, from monotonicity and left continuity (in both variables) of the
pseudo-multiplication ? we obtain:
ak Ob ? sup( inf a Ob') =??) =: sup{c eC®\c<b].
So, for a < otk, b g Ce, we have aQK <*(b) < & and, for a > &., fo ^ Ce. we have
а О b > Д ? b = ??) > b. Therefore no other left unit different from и exist, neither in
[0,a*]norin[&,F]. D
Remark 3.8. The existence of a left unit element for the pseudo-multiplication ? is
really a strong property. Not all pseudo-additions ? admit a fitting pseudo-multiplication
with a left unit element. This is, e.g., the case of the pseudo-addition C.3).
Example 3.9. The only pseudo-multiplication with left unit и fitting with the pseudo-
addition C.2) is defined by:
aOb = g-x{{g(a)g(b))lg(u)). C.17)
The existence of the left unit element и forces the commutativity of the
pseudo-multiplication given by C.13).
The pseudo-multiplication C.17) doesn't verify the condition (M5). For the pseudo-
multiplications fitting with the pseudo-addition C.2), the conditions (M5) and (M6) are
incompatible.
Example 3.10. A pseudo-multiplication with left unit, fitting with a general
pseudo-addition C.1) with at least а к such that gk(fik) = oo, is obtained assuming the
expression C.16) in the strip ]<**, Д] ? [0, F] and, e.g.,
aOb = 0 if0<^a^oik, aOb = F if/3*<a<F.
This is the minimal pseudo-multiplication with left unit и e ]«;·, /3*[.

Monotone set functions-based integrals
1351
An other example is the maximal form of the ?-fitting pseudo-multiplication with left
unit u: in association of the expression C.16), Corollary 3.9 and ?0? = ?0?) = ?, we
assume for b > 0:
aQb = a(b) if0<a^ak, aQb=F if ft <a^F.
Associativity. The associativity of the pseudo-multiplication is not required in general
but we must assume this property if we want the ©-homogeneity of the integral.
r® r®
/ (cOf) ??? = ?? f Q ??, с е [0, F].
As a matter of fact, the ©-homogeneity restricted only on basic simple functions implies
the following equality:
(M7) (a © b) © с = a © (b © c) (associativity).
Right distributivity. As well the right distributivity of the pseudo-multiplication
is not required in general. Nevertheless, it must be required if we want the integral to
be ?-additive when the measure is ?-additive. In fact, for А П В = 0, if ?(? U ?) =
?(?)??(?) we get:
/? p®
(b(a,A)®b(a,B))Od?= / b(a, A U ?) ??? = a © (? (?) ®?{?)).
If we request the ?-additivity of the integral, then it must be:
/? /-? r®
(?>(?,?)??>(?,?))?<*?= / b(a,A)?d?® b(a,B)Od?.
So the ?-additivity of integral is possible only if the following equality holds:
(M8) aO(b®c) = aOb®aQc (right distributivity).
Commutativity. Commutativity of © is not required, in general; furthermore this
property does not correspond to any meaningful property of the integral. However, in some
special cases, the commutativity may be required, or may be forced by other required
properties:
(M9) ? ? b = b © a (commutativity).
Some authors - Pap A990, 1995), Wang and Klir A992), Mesiar A995), Mesiar and
Rybarik A995), Benvenuti and Vivona A996a, 1996b) - have considered the sometimes
so-called pan-operations. These operations are a general pseudo-addition ? satisfying
the properties (A1-A5) and a ?-fitting pseudo-multiplication satisfying all properties
(M1-M9), i.e., all properties seen above.
Further, commutative, associative, monotone binary operations © with neutral element
и е ]0, F[ are proper generalizations of the uninorms recently introduced by Klement,

1352
P. Bemenuli el al.
Mesiar and Pap A996), Yager and Rybalov A996). Therefore, left-continuous uninorms
(see De Baets A998, 1999)) are appropriate candidates for pseudo-multiplication when
? = v. On the other hand, if и = F and ? is commutative and associative, ? should be
a left-continuous triangular norm (see Schweizer and Sklar A983), Klement, Mesiar and
Pap B000)).
Remark 3.11. To keep the maximum of generality we assume for the pseudo-
multiplication, in the next section, only the properties (M1-M4). These are sufficient to
define and build univocally an integral for the Д-measurable functions in a general fuzzy
measure space with properties similar to those of Choquet and Sugeno integrals.
4. General fuzzy integral
The aim of this section is to define a general fuzzy integral on an arbitrary fuzzy measure
space (?, ?, ?) by means of any given pair of fitting pseudo-operations ? and 0. This
integral shall be endowed with properties which are corresponding to the characteristic
properties of Choquet and Sugeno integrals. It coincides with the former or the later in
special cases.
The instruments which we use in order to construct the general fuzzy integral are the
following:
A) A fuzzy measure space (?, ?, ?) with ? :A^> [0. ?], ? = ?(?) e ]0, oo].
B) A pseudo-addition ?: [0, F] ? [0, F] -> [0, F], which gives to the interval [0, F]
the structure of I-semigroup.
C) ? ?-fitting pseudo-multiplication О: [0, F] ? [0, ?] -> [0, F].
We do not request any property of continuity for the fuzzy measure ?, nevertheless we
shall recognize in this section that the best background for the general fuzzy integral is a
measurable space equipped with a fuzzy measure continuous from below.
The general fuzzy integral is a functional defined on the family ? of Д-measurable
functions /: ? -> [0, F]:
j /Od/i:.F-*[0,F].
We request for the general fuzzy integral the following properties.
(Ge. 1) Basic values.
/ Ь(с,СH<*м = сОд(С).
In particular, if the pseudo-multiplication admits a left unit element и, the integral of the
pseudo-characteristic functions Uc = b(u, C) reconstructs the given fuzzy measure:
/ Uc ???=?@ VCeA

Monotone set functions-based integrals
1353
(GE. 2) MONOTONICITY.
/? /-?
f?d?? ????.
(GE. 3) COMONOTONE ?-ADDITIVITY.
/? /-? /-?
(/?,?)???=/ /????/ #???.
(Ge. 4) Horizontal ?-additivity. Given a measurable function / and a number
с e [0, F], we obtain a horizontal ®-additive decomposition by cutting the function / at
the level с and putting:
/ = (/лс)ф(/©с).
We request that the general fuzzy integral is ?-additive for any horizontal ?-additive
decomposition:
/? /-? /¦
f?d? = (/?^???®
f?d?= (/лсH<*мф/ (/??)???.
Horizontal ?-additivity is a weaker form of comonotone ?-additivity because the two
functions (/ л с) and (/ ? с) are comonotone.
(Ge. 5) Continuity from below. If ? is continuous from below we request that the
integral is continuous from below, i.e., for any non-decreasing sequence {/„}„ен:
/ (sup/„j ©<*M = sup / /,,???.
We define the general fuzzy integral as a functional ? -> [0, F], which satisfies the
properties (Ge. 1,2, 3, 4, 5). In reality, this functional is univocally determined by the
properties (Ge. 1, 4,5), as well as the Choquet and Sugeno integrals, which are determined
by properties (Ch. 1, 5, 6) and (Su. 1, 5, 6), respectively. We begin building the functional
on the family S of the (measurable) simple functions from ? to [0, F]. Afterwards, we
recognize that the functional so defined in S verifies all properties requested. Finally, by
means of the continuity from below, we extend the functional on the whole family ? of all
Д-measurable functions and we prove that the so obtained functional verifies as well all
properties (Ge. 1,2,3,4,5).

1354
P. Benvenuti et al.
4.1. Integral of simple functions
Any simple function s admits the standard classical representation:
11
s = \Jb(aj,Ai), D.1)
i=\
with 0 < ? ? < 02 < ¦ ¦ ¦ < a„ and А, П Ay = 0 for i ? j ¦ More, it admits many comonotone
?-additive step representations (shortly ?-step representation)
m
s = 0fo(c,,C,), withCi 2C2 2---2C,,,. D.2)
i=\
The standard ?-step representation
?
s=®b(c*,C*) D.3)
i=l
is the minimal one for the number of steps and for the height of the single steps; it is
obtained from the classical one taking in D.2)
It
m=n, C* = \^JAj and c* = a\, c\ = аг ??\, ..., с* = ?,, ? ?„-? ·
j='
In order to define the integral of a simple function s we can use its standard ?-step
representation because it is obtained by means of ? — 1 repeated horizontal ?-additive
decompositions. For any functional L® :S -> [0, F] which satisfies (Ge. 1) and (Ge. 4),
by repeated applications of these properties, we obtain necessarily:
m
^%) = фс*©М(С*). D.4)
Now, we associate to any ?-step representation D.2) of a simple function s the
expression:
m
? = фс,Ом(С,). D.5)
7=1
LEMMA 4.1. The expression D.5) gives the same result for all ®-step representations of
any given simple function s. Therefore
m m
? = 0 c; ? ?(?) = 0c,* ? м(С,*) = L®(s) D.6)
?'=? ? = ?
for any ?-step representation of the simple function s.

Monotone setfiinctioiis-based integrals 1355
PROOF. 1. First we consider the presence in the representation D.2) of a step subdivided
horizontally in two steps. Let CJ+\ = Cj, the two steps b{cj,Cj) and b(cJ+\, C]+\) form
a horizontal cut of the single step b(Cj ? cJ+\, Cj). From the left distributivity of ? it
is: Cj ? ? (С/) ? с,-+| 0 д(С^) = (с,- ? с,-+|) ? ? (С;)· Therefore the two steps can
be replaced by the single one fo(cy ? c,-+i, C/) without modifying the result ? of the
expression D.5).
2. Secondly, we consider the possible presence of an inessential step. If c\ ?· · -®Cj-\ =
a, and ?,- ? Cj = a,-, the step b(Cj,Cj) is inessential in the representation D.2). We
prove that the corresponding term cj ? ? (Су) in D.5) is also negligible. Let Sj =
c\ ? ?((?|) ? · · · ? Cj-\ ? ?^-?), from the monotonicity of the two operations and
from the left distributivity, it is
Sj > ^\®---®?]^)??^]-\) = ????^?-\).
As the pseudo-addition ? is continuous, there exists w e [0, F] such that Sj¦ = w ? ?, ?
?^}-\). Again from the monotonicity it holds:
Sj ^ Sj+\ = Sj ? Cj ? ?(G7) ^ ?? ? ?, ? ?(?>-?) ? с, 0д(Суч)
= ?? ? (?,- ? с,-) ? [i(Cj- \) = w ? ?, ? ?(G7-1) = 5;.
So it is S; = Sj+\ and the inessential step can be eliminated without modifying the result ?
of the expression D.5).
3. Then we consider the possible presence in D.2) of a non-minimal step. Let Cj be
not minimal, the step b(cj,Cj) can be replaced by b(c'-,Cj), with c' < cy, because the
difference is the inessential step b(cj ? c',Cj).
4. Finally, starting from any ?-step representation, by elimination of all inessential steps
and all subdivision of a single step in two or more steps, we attain the standard one, without
modifying the result ? of the expression D.5). ?
The functional L® is univocally defined by the expressions D.4) or D.6): it can be
evaluated for any given simple function s starting from the standard ?-step representation
or from any ?-step representation.
LEMMA 4.2. The functional Le, defined on S by D.6), is non-decreasing.
PROOF. We consider the representations of two simple function by means of a common
partition {A,: / = 1,2,...,«}
II И
i = \/fo(a,,A,), ?' = \/??\,??)
i=\ i=\

1356
P. Bemvnuti et al.
with ?| ^ Д2 < · · · ^ a„ and a\ ^ a'-, < · · · ^ a'ir The relation s < s' involves a; < a- V/.
For evaluating the integrals we consider two ?-step representations:
II II II
* = ф/>(с,,С,), i' = 0fo(C;,C,) withC,=[jAy. D.7)
i = l , = l j=\
It is sufficient to prove that
In m
фс,Ом(С,)<фс;Ом(С,) D.8)
7=1 7=1
for a, = a,' for / ? к and а к < a'k.
If к = ?, we assume in D.7): c; = c;' for / < ? and hence necessarily c„ < c,',. The
inequality D.8) is a direct consequence of the monotonicity of the operations ? and ©.
\f к < ?, we assume c-, = c\ for / ? к к + 1 and putting d = a'k Q a* we take
Ck =ak ??*-?, 4 =ck®d, ck+\ = d®c'k+r c'k+] =ak+\ Qa'k.
The two sides of D.7) have the same terms except Ck О ?(Ck)®Ck+\ ? ?(Ck+\) in the
left side and c'k ? ? (С к) ? с'к+, ? ?(Ck+1) in the right one. Now for these terms we have:
Ck ? М(С*) ? Cjt+i ??(Ck+\) = «0 д(С*) 0d ?m(C*+i) ? q.+ | ? ?(<7*+?)
<с*Од(С*)ФйОд(С*)фс;+|Од(С*+|)
= q©M(CjtHq.+ | OM(Cjt+i).
The inequality D.7) is now a direct consequence of the associativity and monotonicity of
the operation ?. D
LEMMA 4.3. The functional L®, defined in S on D.6), is comonotone (B-additive.
PROOF. Let s, s' e Sm- If s ~ s' the corresponding chains C, and Cs· belong to the chain
CS0j' =: {i2, С ?, Сг, ¦ ¦ ¦, С„, 0). The functions s, s', s ? s' admit the following ©-step
representations, in which there are present, possibly, some inessential or null steps:
II II II
i = 0fo(c,,C,), i' = 0fo(c;,Ci). s®s' = Qb(ci®c'„Ct).
1=1 i=l i=l
By means of the left distributivity of ? and the associativity of ? we obtain from D.6):
L®(f®g) = L®(f)®L®(g). ?
LEMMA 4.4. If ? is ®-additive and © is distributive on both sides, then the functional
L®, defined in S on D.6), is ®-additive.

Monotone set functions-based integrals
1357
PROOF. For a given simple function we consider the classical representation and any ©-
additive representation, with basic functions not necessarily comonotone:
in
s = ®Hdj,Dj). D.9)
i=\
We express the sets by means of the atoms {P\, Рг, ¦ ¦ ¦, Pn] of the partition generated by
the families [Dj} and {A,}:
Ai=\JPh, DJ=\Jph
heH, neK,
where tf, = {h: Ph С A;) and K; = {h: Pn с Dj] are subsets of {1,2, ...,N].
{H\, Нг, · · ·, H„) is a partition of {1, 2,..., N) and D.9) is equivalent to
a, = ? dj for all /г such that Ph с А,.
Oj2/>,,
So we have
/1
0а,Ом(А,) = 0фа,Ом(/7,) = 0ф( ф ^JOM(ft)
?=? 1 = 1 he Hi i=\ heH,^ Dj^p,,
? ?
= ? ? ^??(^) = ? ? ^??(?,)
/?=? Dj2P,, /?=? ?,,??^
= ?^??(??).
7=1
For a given simple function s we obtain the same result when we use the classical
representation or any ?-additive representation, comonotone or quite arbitrary. But, if we
consider the standard ©-step representation or any ?-step one, the result gives the value
of the functional L®; therefore we have for any ?-additive representation of s:
0^0M(D;) = L®(s) = ??, ? ?(?,).
j=\ i=\
This formula shows clearly the asserted ?-additivity of the functional Le and the
possibility to evaluate it starting, as well, from the classical representation. D

1358
P. Benveiiuti el al.
4.2. Integral of measurable functions
In order to define the general fuzzy integral, the functional L® should be extended to
the whole family ? of all measurable functions. The procedure is similar to those used
sometimes for the Lebesgue integral. Given a function / e f, we consider the class S/ of
all simple functions s such that s < /.
Dehnition 4.5. We define for any / e T, in agreement with the requested
property (Ge. 5):
/
f(ddii=:%\^{L®{s):s^Ss]. D.10)
THEOREM 4.6. The integral defined above verifies the requested properties (Ge. 1, 2, 3, 4).
PROOF. We first observe that the integral D.10) is an extension to ? of the functional Le;
this is a consequence of Lemma 4.2:
f
s(ddii = l®(s) VseS. D.11)
The property (Ge. 1) is true for Le by virtue of D.6) and for the integral, as well, by virtue
of D.11). The monotonicity (Ge. 2) of the integral is evident directly by definition. Now
we give the proof of the comonotone ?-additivity (Ge. 3), so (Ge. 4) is proved, too.
Given two functions / and g in T, we take
/? r®
fOd? and ?' < / gOd?^,
then there exist two simple functions s eSf and s' e Sg such that
L®(s) > ? and L®(s') > ?'.
We consider the standard ?-step representations
? ?
?=| ?=?
and we construct two modified functions ? and J' by replacing the sets C, and C\ with:
C,¦ = ????: /(<й)>фсЛ and С-= \? e ?: g(w) ^ фс'Л.

Monotone set functions-based integrals
1359
It is easy to see that
11 И
s <: s = ф/>(с,, Ci) ? f, s' <: Г = 0fo(c,', C\) < g.
i=l i=l
Now, if / ~ g, then s ~ J' and from Lemma 4.3 we obtain:
L®(s®s') = L®(s) ? Le(s') > Le(s) ? Le(s') > ? ? ?'.
As (? ? ?') e «S/e^, and ?, ?' are arbitrary we obtain:
/? /-? /-?
(f?g)Od?^ fQd?®^ #???.
We prove now the opposite inequality by using the functions <pj and <ря of
Proposition 2.2(iii). To any given s e S/®g we associate two comonotone simple functions
Sf = iff о s and Sg =ifsos. We note that Sf eS/ and s g e S,,, and Sf®sg =s. Therefore,
from Lemma 4.3, the definition D.10) and the monotonicity of the operation ?, we obtain:
/? /-?
fOd?? / g?d?.
For arbitrariness of s e Sf®g we conclude
/? /-? /-?
(f®g)Od?^ fOd?®j 8???. ?
THEOREM 4.7. If the fuzzy measure ? is continuous from below, then the integral defined
by D.10) is continuous from below.
PROOF. First, we take a sequence {/itheN of measurable functions which converges
pointwise to a simple function s e S and we consider this function given by its classical
representation and its standard ©-step one:
II It
fk/s = \fb(a,,Ai) = ($b(c*,C*).
i=\ i=l
Let с,,..., c,', be real numbers such that:
I
c'j < c* V/ and ?,_? < ??c- = a\ < a\ for /' = 2, 3,..., n.

1360
P. Bemenuti el al.
We construct the sequence
sk =Q)b(c';, ? with C* = {? e ? | /t(<u) > a,'}·
We get:
• Sk < fk because sk (?) = a] =>· ? e cf =>· /t (?) > ?,',
• Cf с С/ because ? e С* =>· 5 (?) ^ ? (?) > ?- =>· 5 (?) > ?;,
• С* ?' C, because ? е С,- =>· /* (?) ?' s (?) ^ ?, > ? ¦.
Therefore:
/¦? " "
lim / fk?d?^ lim ? с,' ©м(С*) =0с,' O^Cf).
к^+ос J k^ + oc ^^
/=1 ' = 1
From arbitrariness of с' we obtain / /;· © ?? / L®(s).
Now we assume /* /* /, with f e J7. Let s e <S/\ then fk As / s and so
/? /·? /·?
fkOd?^ lim / (fk?s)Od?= ?©???.
As s is arbitrary in S/ and because of monotonicity (Ge. 2), we obtain:
/? /-?
fkOd?/J /???(/). ?
The following theorem generalize Theorem 2.8 and Theorem 2.9 and shows that really a
measurable space, with a fuzzy measure continuous from below, is suitable for construction
of a general fuzzy integral.
THEOREM 4.8. In a fuzzy measure space (?,?,?), with ? continuous from below,
the general fuzzy integral is the only functional Lce-F -> [0, F] which verifies the
properties (Ge. 1, 4, 5).
PROOF. The proof is exactly the same as in Theorem 2.8 after replacement of the
operations + and · by any couple of fitting pseudo-addition ? and pseudo-multiplication ©. D
Remark 4.1. Given any pseudo-addition ? : [0, F] ? [0. F] -> [0, F] and any ©-fitting
pseudo-multiplication О: [0, F] ? [0, ?] -> [0, F], the construction of the general fuzzy
integral is applicable in any fuzzy measure space (?, ?, ?) with ?(?) ^ ?. The general
fuzzy integral possesses all the properties (Ge. 1-5); it is idempotent if ?{?) is a right unit
for the pseudo-multiplication. Moreover, the construction leading to the definitions D.4)
and D.10) is the only possible in agreement with (Ge. 1,4,5). So the general fuzzy integral
defined above is characterized by these properties.

Monotone set functions-based integrals 1361
From the construction of the general fuzzy integral its pseudo-additivity with respect to
pseudo-multiplication follows.
COROLLARY 4.9. Under the notations of Corollary 3.6,
J ?((?\)®(??))??=([ /????\®(? f ?? ??\ D.12)
PROOF. For any ?-step representation D.2) of a given simple function s we obtain, by
virtue of the associativity of the pseudo-addition ?:
in . m \ ? in
0c,-((Oi) ? @2))м(С,) = ( 0c,¦ ?? м(С,)) ? ? 0c, ?2 ? (С/)
i=\ 4=1 ' 4=1
This equation shows the relation D.12) for the integrals of all simple functions. The
validity of the same relation for any measurable function follows immediately from
definition D.10). ?
4.3. Additional properties of integral with pseudo-ring operations
In this subsection we consider the structure of pseudo-ring on the interval [0, F]. We
assume that the two operations ? and © are both defined on the square [0, F] ? [0, F].
We deal with some properties of the integral, depending on conditions requested to the
pseudo-multiplication in addition to the fundamental ones (M1-M4).
THEOREM 4.10. If the pseudo-multiplication is associative and right distributive, then
the integral is Q-homogeneous, i.e.:
/® /-?
(??/)??? = ?© / f ???.
PROOF. By virtue of the left continuity of ? it is sufficient to prove the equality only for
the simple functions. We consider any ?-step representation of a simple function; from the
right distributivity we have:
It It
i = 0fe(C;,Ci) => flOi = 0fe(fl©Ci,Ci).
i = l f=l
From Lemma 4.1 and by means of associativity and right distributivity we obtain:
r® " / " \
/ (a?f)?d? = 0(fl ? с,) ? д(С) = ? ? I 0с, ? д(С,) 1
J i=\ 4=? '
г®
= aQ /???. ?

1362
P. Benvenuli el al.
THEOREM 4.11. If the fuzzy measure ? is ®-additive and continuous from below, and
if the pseudo-multiplication ? is distributive on both sides, the general fuzzy integral is
(&-additive and coincides with the integral of Sugeno-Murofushi (see A987)):
/? /·?
/???=? fdii = SM(f,ii,®,Q).
PROOF. The integral of Sugeno-Murofushi is ?-additive and continuous from below. It is
defined in S by means of the ?-additivity, starting from the classical decomposition. The
coincidence of the two integrals on S is given by Lemma 4.4 and on ? by continuity from
below. D
For the general fuzzy integral a representation theorem similar to Theorem 2.6 is true.
THEOREM 4.12. Let ?/ be the restriction of ? to the chain C/, let A/ be the ?-
algebra generated by C/ and m? the (B-additive continuous from below measure induced
on A/ from ??. If the pseudo-multiplication is distributive on both sides, the general
fuzzy integral of the function f coincides with the Sugeno-Murofushi integral on the fuzzy
measure space (?,?/, /я?):
/? ?-л
fOd?= fdmJ=SM(f,mJ,®,0).
??
PROOF. The function / is A/-measurable. ?
THEOREM 4.13. In the case of pan-multiplication ? the general fuzzy integral coincides
with the so called Choquet-like integral ofMesiar:
/? /·?
fQd?= / hdk® = SM(h, ??, ?, ?).
??
The function h(x) = ?@/(?)) is the same as in the Choquet integral. The ?-additive
measure ?? is defined on the Borel class of [0, +oo], it is continuous from below and
determined by the condition ?([0, ?]) = ?, Vjc e [0, +oo] (see Mesiar A995)).
The integral of a given measurable function /: ? -> [0, F] is a functional on the set ?
of all fuzzy measures ? : A -> [0, F]. The set ? is partially ordered by the relation:
Ml -<M2 «=> ??(?)^ ?2{?L? e A.
THEOREM 4.14. The general fuzzy integral is a monotone functional on M'-
J fOd?^?:J
fQd?^ fQd?

Monotone set functions-based integrals 1363
PROOF. The inequality is evident for the simple functions (from monotonicity of the
operations ? and ©). For the measurable functions it follows from the continuity from
below of the integral. D
The set ? is closed with respect to the pseudo-addition ? :? ? ? -> ? defined by:
M = Ml0M2 «=> м(А) = Д|(А)фд2(А)УАе Л.
THEOREM 4.15. If the pseudo-multiplication is as well right distributive, then the integral
is ^-additive with respect to the pseudo-addition of measures:
/? /·? /-?
/Оф|Фд2)=/ fOd?^(B f ???2. D.13)
PROOF. It is sufficient to prove the equality for the simple functions. As a matter of fact,
from the right distributivity of О and from the associativity of ? we have:
фс,0(М|(С;)Фм2(С,))=ф(с,Ом|(С,)фс,Ом2(С,))
f = l /=1
= (фс,ОМ|(С,))©@с,Ом2(С,)),
i.e., by definition D.6) the equality D.13) holds. ?
4.4. Extension of the integral to [— F, F]
The Choquet integral, defined for functions whose range is in [0, oo], admits two different
extensions to the whole real line [—oo, oo]: the so called asymmetric Choquet integral and
the symmetric one (see, e.g., Denneberg A994) and Pap A995)).
The asymmetric Choquet integral is defined for finite fuzzy measure and measurable
function / : ? -» [—oo, oo] by
(a)
J fdix = ^ f+?? - j f-??'1
where /+ = / ? 0, /~ = (—/) ? 0 and ?'' is the dual fuzzy measure: ?'1 (A) =
?{?) - м(Ас). However the integral is not defined if the integrals of /+ and /~ are
both infinite.
The asymmetric integral is comonotone additive and hence shift invariant, i.e., for any
real constant c.
(a) j(c + f)d? = c?(?) + (a)Jfd?.

1364
P. Bemenuli et al.
Therefore, if inf / = с > — oo,
(a) I fdiL = ??(?) + I (/ - ?)??.
However, the homogeneity of the asymmetric integral holds in general only with respect
to multiplication by a constant с > 0 and fails if с < 0. Especially
(a) / b{c,C)d? =
c^{C) if с > 0,
????@ ifc<0.
The symmetric Choquet integral was originally developed as the Sipos integral A979)
and is defined for any (even non finite) fuzzy measure by
(s)
J fdn = j f+??- j ???
again avoiding terms oo — oo.
The symmetric integral is homogeneous with respect to multiplication by a constant с
positive or negative. However the symmetric integral is neither comonotone additive nor
horizontally additive.
For general fuzzy integral, related to a given pseudo-addition ? on [0, F], the extension
to the measurable functions with values in [— F. F] necessarily requires an appropriate
extension of the operations ? and ©. However, such extension with suitable properties
need not exist. Note that even in above discussed case of the Choquet integral, the
common addition defined on the extended real line is defective in the points (oo, -oo) and
(—oo, oo). When extending naturally the idempotent pseudo-addition ? = ? from [0, F]
to [- F, F], again the only idempotent extension is v. But now the neutral element will be
not more 0 but - F. This approach allows to extend, for example, the Sugeno integral to the
interval [- F, F]. However, keeping ?@) = 0, the Sugeno integral is always non-negative,
for non-positive functions as well. Recently Grabisch proposed an approach to define a
symmetric Sugeno integral on [— F, F]; nevertheless, the pseudo-addition used there is not
continuous and the monotonicity of the pseudo-multiplication is violated (Grabisch, 2000).
This integral is out of our framework.
Above mentioned problems exclude a general method of extending the general fuzzy
integral defined on [0, F] to [- F, F] keeping a reasonable number of properties and hence
we will not discuss this topic.
5. Examples
In this section, we introduce several examples of fitting pseudo-additions and pseudo-
multiplications and the corresponding fuzzy integrals. The first examples are about the
general form of pseudo-multiplication satisfying only the fundamental conditions (M1-M4).

Monotone set functions-based integrals
1365
To simplify the notations, as well as for the consideration of fuzzy measures not necessarily
continuous from below, we put in the following:
?/(?)=:?(G/(?)) and ?}(a) = : sup ?/(?).
a <x < F
In general for each fuzzy measure it is: ?*??) ^ ?/(a), but if ? is continuous from below,
then ?^· (?) = ?^(?).
EXAMPLE 5.1 {Continuation of Examples 3.1 and 3.4).
a®b = g-l(g(a)+g(b)). aOb = g~] (g(a)k(b)).
with g: [0, F] -»¦ [0, oo] continuous and increasing, g@) = 0, g(F) = oo, ?: [0, ?] ->
[0, ??'] left continuous non-decreasing, ?@) = 0.
The integral is a transformation of the standard Choquet integral:
j /0«?? = ^?(/(«?/)^(???)). E.1)
The function g acts as a change of scale on the interval [0, F] which is monotonically
transformed in the interval [0, oo]. The function ? acts as a change of scale on the values
of the fuzzy measure; the function ? ? ?: ? -> [0, ?'] is, as well, a fuzzy measure.
In particular, if ?(?(?)) = 1, then the integral satisfies the idempotence property:
/
???? = ? Vc e [0, F].
If F = ?, any element и е ]0, F[ may be assumed as the (unique) left unit. Then the
function ?„ and the ®-fitting pseudo-multiplication ©„ are uniquely determined:
?„ (&)=——, aO„b = g —— .
g(u) \ g(u) )
Putting h(a) =: g(a)/g(u), we obtain h(u) = 1 and
? ? b = h~l(h(a) +h(b)), ? ?„ b = /г (h(a)h(b)).
Moreover,
j /?„ ?? = ?-?(<[(???/)?·(???)\. E.2)
The integrals of the pseudo-characteristic functions Uc = b(u. C) reproduce the fuzzy
measure:
f
?????? = ?@.

1366
P. Benvenuti el al.
The integral E.2) is a generalization of the Pap g-integral A993) which is defined starting
from a ?-additive measure. Compare also Markova A996) and Mesiar A996).
EXAMPLE 5.2 (Continuation of Examples 3.2 and 3.5).
? ? b = ?"' ((g(a) +g(b)) A g(F)), aQb = g ((g(a)k(b)) A g(F)),
with g : [0, F] -> [0, g(F)] continuous and increasing, g@) = 0, g(F) < oo, ?: [0, ?] ->
{0} U [1, oo] left-continuous non-decreasing, ?@) = 0.
The integral is similar to the previous one:
j f?d? = g-i(g(F)A /(??/)?(???)?
Moreover, the integral satisfies the idempotence property if the range of ? is {0, 1), i.e.,
k(b) — 0 for b ^ bo, k(b) = 1 for b > bo, with fro < ?(?). In this case, the integral turns
out to be independent of the function g which characterizes the operation ? and we have:
f
f ??? = esssup(/) = supjjc: ?(?/(?)) > bo}-
???
No ?-fitting pseudo-multiplication with a left unit exists in this case (see Definition 3.1).
EXAMPLE 5.3 (Continuation of Examples 3.3 and 3.6).
a®b = aw b, aOb = h(a,b)
with h:[0,F] x [0,?] -> [0, F] left-continuous, non-decreasing in both variables,
/2@, b) = h(a,0) = 0. The distributivity (left and right) is assured by the monotonicity:
f
f?d?= у h(a^(Cf(a)). E.3)
яе[0.Л
If h(a, ?(?)) = a for all a e [0, F], then the integral is idempotent.
This example shows a generalization of the Sugeno and the Ralescu-Adams integrals
A980). Both use h(a,b) = a A b; the first on [0,1], the second on [0, oo]. Next, if
? = F — 1, then the integral E.3) generalize also the Weber's integral A986) in which
h(a, b) = ? (a, b) is a strict f-norm.
Reasonable candidates for v-fitting pseudo-multiplication are left-continuous f-norms
(with unit 1) or left-continuous uninorms (with unit и e ]0, 1 [).

Monotone setfimctions-based integrals 1367
Example 5.3.1. Let® = ? on [0, 1] ? [0, 1], assume on [0,1] ? [0, 1] that a Qb = a b
(or the same on [0, oo]). The associated integral
;
/??? = \J (fl/i(C/(fl)))
ae[0.1]
is idempotent whenever ?(?) = 1. It is also ©-homogeneous and reproducing, i.e., the
integral of the pseudo-characteristic function Ua = b( 1, A) results in the underlying fuzzy
measure. Observe that this integral is a special case of Weber's A986) generalization of
the Sugeno integral in [0,1], while on [0, oo] it generalizes the Shilkret integral A971).
Example 5.3.2. Let ? = ? on [0, F] ? [0, F], assume ? © 0 = 0 and ?? b = a on
[0, F]x]0, M]. The associated integral
r
f ??? = ess sup/ =: sup {?: ?(?/(?)) > 0}
is idempotent, ©-homogeneous but it doesn't reproduce the fuzzy measure.
Example 5.3.3. Let ? = ? on [0, F]2, for a e ]0, F[ we define ©„ : [0, F]2 -> [0, F]
by
\ a /\b if ? < a,
aOab=\
I b if a > a.
Then ©„ is a v-fitting pseudo-multiplication. All elements и е ]a, F] are left unity for
the pseudo-multiplication ©„; each corresponding basic function b(u,A) is a pseudo-
characteristic function. Moreover, ©„ is associative and distributive; so the resulting
integral is ©„ -homogeneous.
For a given measurable function /: ? -> [0, F] and a given fuzzy measure ? : ? ->
[0, F], the corresponding integral is given by
? /???? = ?*/(?)? I (f ??)?? = ?}(?) V j f ??. E.4)
Note that:
/V f fV
(/??)????, $ (f ??)??= ? (/??)????.
So, the first expression in E.4) correspond to the application of the ©-horizontal additivity.
Moreover, if $ f ?? > ?*(/) then it is also
S /??= V (aA/i(C/(a))) = i(/Aa)d/i
0<a<a

1368
P. Benvenuti el al.
and we obtain the second expression in E.4).
Observe also that ©„ = (л) ? (©*) (with notations of Corollary 3.6) and
I b it a > a.
We obtain again the second expression in E.4) by application of Corollary 4.9 because
/v /?*?? = ?*/(?).
Example 5.3.4. Let ? = ? on [0, F]2 and let ?:[0, F] -> [0, F] be an increasing
bijection. Put
aQb = (p{a) Kb, a,be[0,F].
The operation © is a v-fitting pseudo-multiplication and the integral is:
/ / ?? ??. = & ? ? f ??.
For the pseudo-multiplication ?? the only left unity is и = F which is as well the right
unity; so if ?(?) = F, the resulting integral is idempotent. The pseudo-multiplication is
associative and the integral is ©-homogeneous in the case ? = id only. In this case the
integral is just the Sugeno integral on [0, F].
Example 5.4. We assume for ? the general form C.1)
? ? ь = f Sk'\(gk(a)+gk(b)) A gk0k)) if ]a. b[ e ]ak. flt[2,
( a v b otherwise.
Let for some к e K, gk(fik) = oo. Consider a left-continuous non-decreasing function
?: [0, F] -> [0, oo], with ?@) = 0, and put d\ =: sup{fo: k(b) = 0), d2 =: inf{b: k(b) =
oo}. For a fixed h e K,v/c construct a non-decreasing system of left-continuous solutions
of the Cauchy equation putting (see Theorem 3.5):
for b^d] fh(a)=0,
0 ifa^ajt,
for d\ < b ? d2 Ma) = gbl(gl,Wh) A gk(a)Hb)) if a* < a < Д.,
??, ifa>^·,
for b>d2 /*(«>= (д, iffl>at.
We obtain the ?-fitting pseudo-multiplication ©;¦/,, see Figure 5.1.
aQkhb = Gh(gf.(a)k(b)) @ · oo = oo · 0 = 0), E.5)

Monotone setfiincrions-based Integrals
1369
where
g>) =
Fig. 5.1.
0 if a ^ oik:,
gk(a) if a* <a ^ &·, G/,(*) =
. oo if ? > Д,
0 if.r=0,
g-'U) ifO<*<g/,(fl,),
. /3,, if .v > #,(#,).
It is easy to verify that
/?(???)
= G,,(/ (lo^fC/f^'U)))^
E.6)
Really the expression E.6) gives a functional on ? which verifies (Ge. 1, 4, 5), therefore
it is identical to the integral.
Note that:
sup M(Cf(^'(*)))= ?* (<**) and inf M(C/-(gi7lW))=M/-(A).
Then the value of the integral is zero whenever ?^(?^) ^ d\. It is exactly ?/, if ?/($0 > d\
or ?^(?^) > <i2- Otherwise, the integration gives a value in the interval ]<*/,, ?/,].
Example 5.5. Let the pseudo-addition ®:[0, F]2 -> [0. F] possesses only one non-
trivial idempotent a e ]0, F[. Then there are two mappings
g\ :[0,a]-> [0, oo], g2 : [<*· Л-> [0, oo]
generating ? (see Proposition 3.2 and Figure 5.2). If any of these mappings is unbounded,
i.e., if either g\(a) = oo or gi(F) = oo, there exist a ?-fitting associative pseudo-
multiplication ? with left unit (see Proposition 3.7 and Example 3.10). Suppose that both

1370
P. Benvenuti el al.
Fig. 5.2. Pseudo-addition for Examples 5.5. 5.5.1. 5.5.2. 5.5.3.
g\ and g2 are unbounded, then there exist ?-fitting pseudo-multiplications © with left unit
element и е]0,a[ and, as well, with left unit element и е]а, F[. In both cases the unit
element is unique (see Corollary 3.9).
Example 5.5.1. Let и e]0,<*[. With no loss of generality we may suppose g\(u) = 1.
The general form of the ?-fitting pseudo-multiplications with left unit и is given by
aOb--
Г 0 if a = 0 or b = 0,
?
g, \g\(a)g\(b)) if0<a, b^a.
ft'diWfiW) ifO<a^a,a<b^F,
У 8?* Ш(а)к(Ь)) ifa<a^F,
where ?: [0, F] -> [0, oo] is a left continuous non-decreasing map such that ?@) = 0
and k(b) = oo for all b > a. See Figure 5.3 in which d\ = sup{b: k(b) = 0} > 0, and
di = inf{b: k(b) = oo) ^ a.
The pseudo-multiplication ©H can be obtained (see Corollary 3.6) as ©„ = (©?) ?
(?2) ? (©з), where (with notation of Example 5.4):
aO\b = G\ (g*\{a)g\(b)), aQ2b = G2{g^a)gUb)),
aQ3b = G2(g*2{a)k{b)).
The pseudo-multiplication ©|, ©2, ©3 are ?-fitting, nevertheless и is left unit for ©, but
not for ©?, never for ©2 or for ©3. We introduce three integrals:
h=f /02ЛМ = С2(/(?Г°/)Л(?2°М)).
h=j /???? = 02(?(82*?/)?(???)\

Monotone set functions-based integrals
1371
Fig. 5.3. Pseudo-multiplication for Example 5.5.1.
and we obtain (see Corollary 4.9):
I =¦ / / Ou dix = I\®h® h = h ? (/2 ? h).
Note that if ?*, @) < a and ess sup / < a, then h = /3=0 and
Moreover, if ?*??) > di, then / = /3 = F. Otherwise
/ = /2 ? /3 = 821По°°(82 о д)(С/(«Г' (*))) dx
+ j (?o?)(Cf(gTi(x)))d?\

1372
P. Benvenuli el al.
f A
Fig. 5.4. Pseudo-multiplication for Example 5.5.:
EXAMPLE 5.5.2. Let и e]a, F[. With no loss of generality we may suppose gi{u) = 1.
The general form of the ?-fitting pseudo-multiplications with left unit и is given by
aQb-.
0
ifa = 0or6 = 0.
g-[(g\(a)k(b)) ifO<as:a,
g\\g2{a)gx{b)) ifa<a^F,0^b^a,
g^\gl(a)g2(b)) if a < a, bs^ F,
where ? : [0, F] -> [0, oo] is a left continuous non-decreasing map such that ?@) = 0 for
all b ? a. See Figure 5.4 in which d\ = sup{b: k(b) = 0} > a.

Monotone set functions-based integrals
1373
The pseudo-multiplication © can be obtained by the pseudo-addition of three ?-fitting
pseudo-multiplications: ? = (??) ? (?2) ? (О.ч), where:
? ?? b = Gi (gf (?)?(?>)), a Q2 b = d (gj(fl)gf(fo)),
a03b = G2(g^a)g*(b)).
We introduce the integrals:
f Q2 ?? = ??{?{81 ? f)d(g*x ? ?)
?)?(?\??)
ч
ч
h = f /03<*m = G2(Y(,?2*°
and we obtain:
/=:/ /?» ^? = (/|?/2)?/3.
Note that if ?*?0) ^ d\ and ess sup/ ^ a then / = 0. Moreover, if ?*??) > a, then
I = h > a. Otherwise
I = I\®„h = gTl(j (? ? M)(C/(gf' U))) d.r
Example 5.5.3. The general form of the ?-fitting pseudo-multiplications with nghtunit
F is given by
f 0
aQb =
if a = 0 or b = 0,
*?'(*?(?)?|(&)) if0<a^a,
gr'(g2(a)A2(fc)) if<*<aiC F and 0 < b < <f\
. g^'(g2(a^3(W if ? <?^ F and 0 < fo iC F,
where ? ? : ]0, F] -> [0, 1], ?2: ]0, d*] -> [0, oo], ?-,: ]<**, F] -> [0, 1] are left continuous
non-decreasing maps such that: d\ = sup{b \ k\(b) = 0} > ? = sup{fo | ?2(/>) = 0}, and
X[(F) = k2(F) = 1. See Figure 5.5.
The pseudo-multiplication © can be obtained as ?„ = (??) ? (?2) ? (Оз), where:
aQib = Gi (gf (?)?,(?>)), ? ?2 fo = G, (??2*(?)?2(/>)),
а Оз b = G2(g2(a)k3(b))

1374
P. Benvenuti et al.
Fig. 5.5. Pseudo-multiplication for Example 5.5.3.
with notations of Example 5.4. We introduce the integrals:
h=j /???? = ??(?(§??/)?(????)\
h=\ /?2?? = ??(?(8*_?/)?(?2??)\
h= I /03^ = G2(i (*??/)?/(?3??)?
and we obtain
/=:/ /0»<?? = (/??/2)?/3.

Monotone set functions-based integrals
1375
Fig. 5.6. Pseudo-addition and pseudo-multiplication from Example 5.6.
Note that if ?^·@) ^ d\ and ?^-(?) ^ <fr then / = 0. Moreover, if ?^(?) > d*, then
/ = /3 > a. Otherwise / = /, ? h.
Example 5.6 (see Benvenuti and Vivona A996a, 1996b)). Let the pseudo-addition ? on
[0, F] be given by
a®b =
g-\g(a) + g(b)) ifa^a,b^a,
? V b
otherwise,
where a e ]0, F[ and g: [a, F] —>¦ [0, oo] is an increasing bijection. For a given unit
element и we may suppose with no loss of generality g(u) = 1. Any symmetric ©-fitting
pseudo-multiplication has the following form, where Г is an (arbitrary) left-continuous
triangular norm on [0, a]. See Figure 5.6.
aQb =
g \g(a)g(b)) ifa^a, b^a,
T{a,b)
а лЬ
if a < a, b < a,
otherwise.
Put ? = л. Then the resulting integral is given by
«4?|(«?/)?^)) ???}(?)>?,
$ fdn
where g* is defined similarly as in Example 5.4.
/
????
otherwise,
Example 5.7. Based on Example 5.6, we can introduce a one-parameter family of fuzzy
integrals including the Choquet and the Sugeno integrals as the limit members.

1376
P. Bemenuti et al.
Namely, for a given a e ]0, oof, put (for a, b e [0, oo])
a+b — a if a > a, b> a,
a®a b =
? ?? b =
av b
otherwise,
a + (a — a)(b — a) if a ^ a, b ^ a,
a Ab
The corresponding integral is
otherwise.
????? =
?+?/???? \??*((?)>?,
where /„ = (/ - a) V 0 and ?? = (? - a) V 0.
The limit members are: /?(/) = ^/^?, ?? =$/??.
Example 5.8. Let the pseudo-addition ? on [0, F] be given by
' g'\g(a) + g(b)) if ? ? a, b ? a,
a®b =
?? b
otherwise,
where a e ]0, F[ and g : [0, a] -> [0, oo] is an increasing bijection.
Any symmetric ?-fitting pseudo-multiplication © on [0, F] with the unit element
и = g~' A) can be written in the following form:
aQb =
0 ifa=0or/> = 0,
g~] (g(a)g(b)) if a iC a, b^a,
S{a,b) if a > a, b > a,
avb otherwise,
where S is an (arbitrary) left-continuous triangular conorm on [a, M].
Put S = v. Then the resulting integral is given by (using the notation from Example 5.4)
/
??? =
g '( f (g*°/)<*(?* ? ?)) ifesssup/sCa,
esssup/ ? ??-(?) otherwise.
6. Conclusions
Based on couples of a pseudo-addition ? and a pseudo-multiplication ©, we have
constructed a large class of general fuzzy integrals (on any interval of type [0, F] with

Monotone set functions-based integrals
1377
F e ]0, oo]), including the Choquet and the Sugeno integrals as distinguished members.
Several examples were given, including a one-parameter family of integrals with the
Choquet and the Sugeno integrals as limit members. Several applications, especially in
the area of aggregation operators and multi-criteria decision making can be expected.
Our approach allows to reduce the choice of an appropriate integral to the choice of an
appropriate pseudo-addition ? (for the processing of input values) and subsequently to
the choice of a suitable pseudo-multiplication ©. The pseudo-multiplication must satisfy
some fundamental conditions which are necessary in order to construct the integral. Other
supplementary conditions can be requested in order to obtain an integral with some
additional properties useful for particular applications.
Acknowledgement
The paper was written during the visit of the second author at the University "La Sapienza",
Rome. The support of University "La Sapienza", the G.N.F.M., the MURST of Italy, the
grants VEGA 1/8331/01 and 1/7146/20 are kindly announced.
References
Benvenuti, P. and Mesiar, R. B000a), Integrals with respect to a general fuzzy measure. Fuzzy Measures and
Integrals, M. Grabisch, T. Murofushi and M. Sugeno, eds, Physica-Verlag, Heidelberg, 205-232.
Benvenuti, P. and Mesiar, R. B000b), Pseudo-arithmetical operations as a basis for integration with respect to a
general fuzzy measure. Submitted.
Benvenuti, P. and Vivona, D. A996a), General theory of the fuzzy integral, Mathware and Soft Computing 3,
199-209.
Benvenuti, P. and Vivona, D. A996b), Integral with respect tofitzzx measure, Proc. IPMU '96, Vol. 1, 231-233.
Benvenuti, P. and Vivona, D. B000), Comonotone aggregation operators. Rend. Mat. 20, 323-336.
Benvenuti, P., Vivona, D. and Divari, M. B002), The Cauchy equation on I-semigroups, Aequationes Mathe-
maticae XX, 1-11.
Bouchon-Meunier, B. A998), Aggregation and Fusion of Imperfect Information, Physica-Verlag, Heidelberg.
Choquet, G. A953-1954), Theory of capacities, Ann. Inst. Fourier 5, 131-295.
De Baets, B. A998), Uninorms: the known classes. Fuzzy Logic and Intelligent Technologies for Nuclear Science
and Industry, Da Ruan et al., eds, Proc. Third Int. FLINS Workshop, World Scientific, Singapore, 21-28.
De Baets, B. A999), Idempotent uninorms, European J. Oper. Res. 118, 631-642.
de Campos, L.M., Lamata, M.T. and Moral, S. A991), A unified approach to define fuzzy integrals. Fuzzy Sets
and Systems 39, 75-90.
de Campos, L.M. and Bolanos, M.J. A993), Characterization and comparison of Sugeno and Choquet integrals.
Fuzzy Sets and Systems 52, 61-67.
De Giorgi, E. and Lena, G. A977), Line notion generate de convergence faible pour des fonctions croissantes
d ensemble, Ann. Sc. Nor. Pisa (IV) 4, 61-99.
Denneberg, D. A994), Non-Additive Measure and Integral, Kluwer Academic Publishers, Dordrecht.
Grabisch, M. A996), Fuzzy measures and integrals: a survey of applications and recent issues. Fuzzy Sets
Methods in Information Engineering: A Guided Tour of Applications, D. Dubois, H. Prade and R. Yager,
eds, Wiley, New York.
Grabisch, M. B000), Symmetric and asymmetric integrals: the ordinal cases. Proceedings IIZUKA, CD-room.
Grabisch, M., Murofushi, T. and Sugeno, M. A992), Fuzzy measures of fuzzy events defined by fuzzy integrals.
Fuzzy Sets and Systems 50, 293-313.
Grabisch, M., Murofushi, T. and Sugeno, M. B000), Fuzzy Measures and Integrals, Physica-Verlag. Heidelberg.

1378
P. Benvenuti et al.
Greco, G.H. A977), Integrate monotono. Rend. Sem. Mat. Univ. Padova 57, 149-166.
Halmos, P.R. A950), Measure Theory, Van Nostrand Reinhold Company, New York.
Imaoka, H. A997), On a subjective evaluation model by a generalized fuzzy integral. Int. J. Uncertainty,
Fuzziness, and Knowledge-Based Systems 5, 517-529.
Ishihachi, H., Tanaka, H. and Asai, K. A988), Fuzzy integrals based on pseudo-addition and multiplication,
J. Math. Anal. Appl. 130, 354-364.
Klement, E.P., Mesiar, R. and Pap, E. A996), On the relationship of associative compensatory operators to
triangular norms and conorms, Intemat. J. Uncertainty. Fuzziness Knowledge-Based Systems 4, 25-36.
Klement, E.P., Mesiar, R. and Pap, E. B000), Triangular Norms, Kluwer Academic Publishers, Dordrecht.
Klement, E.P, Mesiar, R. and Pap, E. B001), A geometric approach to aggregation, Proc. EUSFLAT '01,
Leicester.
Kolesarova, A. A993), A note on the ?-measure based integrals, Tatra Mountains Math. Publ. 3, 173-182.
Marinova I. A986), Integration with respect to a ?-measure. Math. Slovaca 36, 15-22.
Markova, A. A996), A note on g-derivative and g-integral, Tatra Mountains Math. Publ. 8, 71-76.
Matsushita, ? and Kambara, H. A996), Partition of attributes and fuzzy integral model for evaluation process,
Proc. IIZUKA '96, Vol. 1, 303-307.
Mesiar, R. A995), Choquet-like integrals, J. Math. Anal. Appl. 194, 477^488.
Mesiar, R. A996), Pseudo-linear integrals and derivatives based on a generator g, Tatra Mountains Math. Publ.
8, 67-70.
Mesiar, R. and Rybarik, J. A995), Pan-operations structure. Fuzzy Sets and Systems 74, 365-369.
Mostert, PS. and Shields, A.L. A957), On the structure of semigroups on a compact manifold with boundary,
Ann. of Math. 65, 117-143.
Murofushi, T. and Sugeno, M. A991), Fuzz\ t-conorm integrals with respect to fuzzy measures: generalization
of Sugeno integral and Choquet integral. Fuzzy Sets and Systems 42. 57-71.
Murofushi, T. and Sugeno, M. A993), Some quantities represented by the Choquet integral. Fuzzy Sets and
Systems 56, 229-235.
Nelsen, R.B. A999), An Introduction to Copulas, Lectures Notes in Statistica, Vol. 139, Springer, Berlin.
Onisawa, Т., Sugeno, M., Nishiwaki, Y, Kawai, H., and Harima ? A986), Fuzzy measure analysis of public
attitude towards the use of nuclear energy. Fuzzy Sets and Systems 20, 259-289.
Pap, E. A990), Integral generated by decomposable measure, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser.
Mat. 20A), 135-144.
Pap, E. A993), g-calculus, Univ. Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23 A), 145-156.
Pap, E. A995), Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht.
Pap, E. B002), Pseudo-additive measures and their applications. Handbook of Measure Theory, E. Pap, ed.,
Elsevier, Amsterdam, 1403-1468.
Ralescu, D. and Adams, G. A980), The fuzzy integral, J. Math. Anal. Appl. 75, 562-570.
Riecanova, Z. A982), About ?-additive and ?-maxitive measures. Math. Slovaca 32, 389-395.
Schmeidler, D. A986), Integral representation without additivity, Proc. Amer. Math. Soc. 97, 255-261.
Schweizer, B. and Sklar, A. A983), Probabilistic Metric Spaces, North-Holland, Amsterdam.
Shilkret, N. A971), Maxitive measures and integration. Indag. Math. 33, 109-116.
Sion, M. A973), A Theory of Semigroup Valued Measures, Lectures Notes in Math., Vol. 355, Springer, Berlin.
SipoS, J. A979), Integral with respect to apremeasure. Math. Slovaca 29, 141-145.
Suarez, F. and Gil, P. A986), Two families of fuzzy integrals. Fuzzy Sets and Systems 18, 67-81.
Sugeno, M. A974), Theory of fuzzy integrals and applications, Ph.D. doctoral dissertation, Tokyo Inst, of Tech.
Sugeno, M. and Kwon, S.H. A995), A new approach to time series modeling withfiizzy measures and the Choquet
integral, Proc. Yokohama '95, 799-804.
Sugeno, M. and Murofushi, T. A987), Pseudo-additive measures and integrals, J. Math. Anal. Appl. 122, 197-
222.
Tanaka, H. and Sugeno, M. A991), A study on subjective valuations of printed color images. Int. J. Approximate
Reasoning 5, 213-222.
Vitali, G. A925), Sulla definizione di integrate delle funzioni di una variabile, Ann. Mat. Рига ed Appl. IV, 2,
111-121. English translation: On the definition of integral of functions of one variable, Rivista di Matematica
per le scienze sociali 20 A997) 159-168.

Monotone set functions-based integrals
1379
Wang, Z. and Klir, G.J. A992), Fuzzy Measure Theory: Plenum Press, New York.
Weber, S. A983), A general concept of fuzz\ connectives, negations and implications based on t-norms and
t-conorms. Fuzzy Sets and Systems 11. 115-134.
Weber, S. A984), L-decomposable measures and integrals for Archimedean t-conorm, J. Math. Anal. Appl. 101,
114-138.
Weber, S. A986) Two integrals and some modified version - critical remarks. Fuzzy Sets and Systems 20,97-105.
Yager, R.R. and Rybalov, A. A996), Uninorm aggregation operators. Fuzzy Sets and Systems 80, 111-120.
Yang Quing Ji A985), The pan-integral on the fuzzy measure space. Fuzzy Math. 3, 107-114 (in Chinese).

CHAPTER 34
Set Functions over Finite Sets:
Transformations and Integrals
Michel Grabisch
LIP6, University of Paris VI, 4 Place Jussieii, 75252 Paris, France
E-mail: michel.grabisch@lip6.fr
Contents
1. Introduction 1383
2. Set functions over finite sets 1383
3. Transformations of set functions 1385
4. The Choquet and Sipos integrals 1390
5. ?-additive measures 1392
6. The ordinal case: the Sugeno integral 1395
References 1400
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1381

This page intentionally left blank

Set functions over finite sets: transformations and integrals
1383
1. Introduction
Measures in a finite setting are positive-valued set functions with some characteristic
properties (e.g., additivity for classical measures). Although the finiteness of the universe
considerably restricts the interest of the measure concept, it also allows other viewpoints,
coming from various fields of mathematics, as combinatorics, game theory, complexity,
etc., when viewing measures as particular set functions or pseudo-Boolean functions.
We intend to illustrate this viewpoint into this chapter. We will introduce various
transformations of set functions, and the notion of integral will appear naturally as an
extension of set functions. In a last part, we will consider set functions valued on linearly
ordered sets and corresponding transformations and integrals.
Throughout the paper, we assume a finite space N with ? elements, denoted simply
1,2, ...,n if there is no fear of ambiguity. In a similar way, s, t,... will denote the
cardinality of subsets S,T,... of N. We denote by л, ? the minimum and maximum
operators on R.
2. Set functions over finite sets
We consider real-valued set functions v.V(N) —>¦ R, and several particular cases. Set
functions vanishing on the empty set are called games, while capacities [3] or fuzzy
measures [32], which we will always denote by ?, refer to games which are monotonic
with respect to inclusion, i.e.,
ACS =>· ?(?)<:?(?).
Consequently, fuzzy measures assume only positive values. In applications, it is often
required in addition that ?(?) = 1.
For any set function v, the dual set function or conjugate set function of ? is defined by
v(S):=v(N)-v(Sc), VScN, A)
where Sc is the complement set of S.
A particular family of set functions of interest are the unanimity games. For any S с ?,
the unanimity game w.r.t. 5" is defined by
(?)·=|?, otherwise. B)
Another view of set functions which has its own interest is given by pseudo-Boolean
functions. We introduce briefly the topic here, more can be found in [12,17].
Any function /: {0, 1)" -> R is said to be a pseudo-Boolean function. By making the
usual bijection between {0, 1}" and V(N), pseudo-Boolean functions on {0, 1)" coincide

1384
?. Grabisch
with real-valued set functions on N. It has been shown by Hammer and Rudeanu [20] that
any pseudo-Boolean function can be written in a multilinear form:
/(?)=??(?)?\??, V^e{0, I}"· C)
TcN ieT
The coefficients a(T) will be explained below. The monomials П/бГ-*' correspond to
unanimity games uj. In terms of game theory, Equation C) gives the decomposition of a
game on the basis of unanimity games.
Note that C) can be put in an equivalent form, which is
/U)= ][>(Г)Д;с,, V*e{0,l}". D)
TcN ieT
A topic of interest concerns the extension of pseudo-Boolean functions to [0, 1]". An
obvious way to do this is to extend expressions C) and D) to the whole hypercube [0, 1]".
The first one is called the multilinear extension, while the second is the Lovasz extension.
The multilinear extension of /, given by
/(*):= ]? ? (Г)[];с„ Voce [0,1]", E)
TcN ieT
is the only multilinear function which extends /, hence its name (see Owen [27]). It
performs the classical linear interpolation of / in [0, 1 ]".
The Lovasz extension of / [21,31 ] is given by
/(*):= ? а(Г)Ддг/, Voce [0,1]". F)
TCN ieT
Defining the simplex ?? = [? e [0, 1]" | ???) ^ · · · ^ x* <.,,)} where ? is a permutation
on N, the Lovasz extension is the unique affine function which interpolates / between
the ? + 1 vertices of ??. By contrast, the multilinear extension interpolates between all
vertices.
As a preamble to the next section, we introduce informally some linear transforms over
set functions.
Mobius transform: in combinatorics, the Mobius transform is well-known (see, e.g.,
Rota [28]). For any set function v, its Mobius transform mv is defined by
muE:):= ^(-?^-'???). VS С N. G)
TcS
The inverse transform is the Zeta transform, expressed by
v(S) = ]Гш1,(Г), VSCN. (8)
TCS

Set functions over finite sets: transformations and integrals 1385
mv is called the Mobius representation of v. In game theory, this corresponds to
the dividends of a game. Comparing (8) and C), we see that ml'{T) is nothing else
than a(T). In other words, the Mobius transform is the coordinates of ? in the basis
of unanimity games. In Dempster-Shafer theory of evidence, mv is called the basic
probability mass assignment [29].
Interaction transform: it has been proposed by Grabisch [7], and is defined for any set
function ? by
IV(S):= ? (" ?'"'?'',' ? <-1>'~*?<*??>- *SCN- (9)
TCN\S {П S+l>- KcS
This definition extends in fact the Shapley value ?1' [30] and the interaction index I,j for
a pair of elements /, j in N, introduced by Murofushi and Soneda [24]. They are defined
by
???)-.= ? -(и~д71)!д!?*?w)-"(*>]. v/eN' A0)
ScN\i ""
hj:= ? -" 7 'J".?''' [v{SU[i,j}) - v(SU {/}) - v(SU{j}) + v(S)}.
ScN\[i.j\ {П '¦
(?)
We have /"({'}) = ??(?) and /L'({/',;'}) = /,y. /'' is often referred as the interaction
index. The interaction index has a natural interpretation in the framework of cooperative
game theory and multicriteria decision making. It has been axiomatized by Grabisch and
Roubens[19].
Co-Mobius transform: for any set function v, it is defined by
mv(S):= ? (-l)"~'v(T)= ]?(-1)'?(? \ ?), VScN. A2)
TDN\S TcS
m" corresponds to the commonality function of Shafer [29]. The analogy with the
Mobius transform can be noticed.
3. Transformations of set functions
We introduce formally the concept of transformation of a set function, as proposed by
Denneberg and Grabisch [6], borrowing the formalism of transformation and operators
used in combinatorics (see Berge [1]).
An operator is a two-place set function ? : V(N) ? V(N) -> R. The multiplication *
between operators and set functions is defined as follows, for every A, B,C С ?:
(? * ?)(?, ?) := ? ?(?, С)Ф(С, В),
CCN

1386 ?. Grabisch
(?*?)(?):= ]Г <P(A,C)v(C),
CcN
(?*?)(?):= ]? u(C)^(C, S).
CcN
The Kronecker's delta
1, ifA = fi,
?(?,?):=.?
1 0, otherwise,
is the unique neutral element from the left and from the right. If it exists, the inverse of ?
is denoted ?-1, satisfying ? *?~' = ?, ?-1 *? = ?.
It can be shown that the family
?:=\?-.?(?) ??(?)^?\?(?, ?)= 1 VA с ?, ?{?, ?) = 0 if ? <? ?\
of functions of two variables together with the operation * forms a group, and the inverse
?-1 &Q of ? eQ computes recursively through
?-\?,?)= 1,
?"'(?,?) = - ]? Ф~\А,С)Ф(С,В) ifA^B.
ACCgB
The Zeta operator Z(A, B), defined by
1, if А С S,
Z(A,B):=ln
1 0, otherwise,
and its inverse the Mobius operator correspond to our previous definitions, i.e., with former
notations,
m = v + Z~~], v = m-kZ. A3)
The next fundamental operator to introduce is the so-called inverse Bernoulli operator ?:
1
?(?,?):
, ifACfi,
\B\A\ + \'
0, otherwise.
The interaction index is recovered by
1 = Г*т. A4)
We turn now to a special class of operators, satisfying
?(?,?) = ?(&,?\?) for А С ?, A5)

Setfiinctions over finite sets: transformations and integrals
1387
i.e., they can be represented by an ordinary set function ?(?) := ?@, A), denoted with
the corresponding small Greek letter. In fact, the set of such operations forms an Abelian
group, as well as the corresponding set of set functions:
with operation * defined by
<p*i/r(A):= ]T<p(C)iA(A\C), AcN.
CCA
The neutral element S of g is
1, ifA=0,
8(A) :=,. . .
1 0, otherwise,
and the inverse of ? is denoted <p*~'. Since ? and ? have property A5), we can introduce
the corresponding Zeta function and Bernoulli function:
?(?)=? for all A e ?,
If moreover ? is a function only of the cardinal of sets, then we call it a cardinality function,
and the corresponding ? a cardinality operator. Note that ? and ? have also this property.
There is a general formula for the inverse of cardinality operators (which is also
cardinal). More specifically, if ?{?) = /(|A|), then ?*~\?) = /*~'(|A|), with /*_l
defined recursively by
/-40):= 1,
m-\
A6)
\ к Г
k=0
m-\ , ч
r-'(m) :=-? (™)/(/n-*)/*-'(*), meN.
?— ? V /
With this formula, we get ?*~] the Mobius function, and y* the Bernoulli function,
giving rise to the Bernoulli numbers.
Coming back to the interaction index, we have
/ = r*w = r*(u*Z~'),
which can be rearranged to obtain / = ? * 1.1 is called the interaction operator, and is
not in the group Q.
This formalism permits to compute easily, simply by combining and inversing operators,
all formulas between the different representations of set functions (Mobius transform,

1388
?. Grabisch
Table 1
Table of conversion between v, m and /
v(S) =
m(S) =
V
?(-\?~'?(?)
TcS
? (~1>v"' ^
m
TcS
m(S)
? ' ,'"'7'»
ks'-s+x
I
? ^??·)
гсл/
? B'-sHT)
TDS
1(S)
interaction index, etc.). These formulas are summarized in subsequent tables. We introduce
some notations. The Bernoulli numbers В к are defined by
*<:=-Edb(/)· *>0· and So = 1-
A7)
First numbers of the sequence are B\ = -1/2, Bi = 1/6, 63 = О, В4 = -1/30, 65 = 0,
etc., and 62^+1 =0 for all к > 0. The numbers ?[ are defined from the Bernoulli numbers
by
#:=E(*V-;' m = o,i,2....
A8)
7=0
First values of ?? are
k\l
0
1
2
3
4
0
1
1
4
{
2
1
6
1
3
1
6
3
0
1
6
1
6
0
4
1
30
1
30
2
75
1
30
1
30
This table has a property similar to the Pascal triangle, i.e., one coefficient ?^] is the sum
of the two above ftk, ?1^'. For other properties, see [8].
Another formalization is based on matrices, putting the coefficients ?(S) for all S С N
into a vector. Then, transformations leads to peculiar forms of matrices (see [16]).
We give some properties of the above introduced transforms (for more details, see [6,10,
12]).

Set functions over finite sets: transformations and integrals
1389
Table 2
Tables of conversion between m and other representations
m(S) =
V
? (-ly-'viT)
TDN\S
m
TDS
I
^2(-\)'-sB,.,l(T)
TdS
m(S) =
KS) =
m
? (-1)'ш(Г)
rc,v\.v
?(-l)'-V,G·)
> m (/ )
^ f-i+1
IDS
PROPERTY 1. For any set function ? and its conjugate v, we have
m"(S) = (-l)i+lm"(S), VScN, S^0,
w
D(S) = (-1)J+I ]Гт"СГ), V5"CN. 5^0.
A9)
B0)
PROPERTY 2. Аи;у set function can be decomposed on the set of unanimity games or their
conjugate:
B1)
B2)
B3)
v(S)= ]T m"B>r(S),
TCN
v(S)= ? (-l)'+]m4T)uT(S),
TCN.Tj=<fl
v(S)= ? m\T)UT(S).
TCN
PROPERTY 3. For any set function ? and its conjugate v, we have
?@) = ?(??-/?'@),
l'v(S) = -(-l)sIv(S), VScN, S^ii.
PROPERTY 4. Let ? be a fuzzy measure with ?(?) = 1. Then the interaction index l,j
ranges in [—1,+1]. /,y = 1 if and only if ? = u\jj], the unanimity game for the pair i, j.
Similarly, l\j = — 1 if and only if ? = й|,.у), the dual measure of the unanimity game for
the pair i, j.
PROPERTY 5 [2]. A set of 2" coefficients m(S), S С ?, corresponds to the Mobius
transform of a fuzzy measure if and only if
(i) w@) = O, EscnS)=1,
(ii) Y,ieTcSm(T) ^0, for all S<ZN, for alii e S.

1390
?. Grabisch
PROPERTY 6 [10]. A set of 2" coefficients I(S), S С ?, corresponds to the interaction
transform of a fuzzy measure if and only if
(») ?,-etfJ({'))=!,
(«0 Escwv fiiSsnT\J(SU {/}) > 0, V/ e N, VT с ? \ {/}.
4. The Choquet and Sipos integrals
Definition 1. Let / be a positive real-valued function on N, and ? a set function. Let us
denote /(/) by /,-, for every / in N, thus considering / as an element of (R+)". Then the
Choquet integral of / with respect to ? is given by
II
i=l
where ·(/) indicates a permutation on N so that /ц, ^ fi) ^ ·¦· ^ /(«>, and A(() :=
{(;),...,(n)}. Also /,o, :=0.
It has been shown by Chateauneuf and Jaffray [2], extending Dempster [4] (see also
Walley [34]), that the Choquet integral of positive integrands can be expressed using the
Mobius transform:
Cv{f)=Yjm\S)-f\f. B5)
ScN ieS
We recognize here the Lovasz extension of ? (see F)), which shows clearly the link
between the set function and the Choquet integral: the latter is an extension of the former,
which means that for any S С ?, if /5 is defined as fs = 1 whenever ; e S, and 0
otherwise, then Cv(fs) = v(S).
The next step is to define the Choquet integral for real-valued integrands. For any
/ e E", we introduce /+ := / ? 0, /~ := (-/) ? 0 the positive and negative parts of /,
i.e., such that / = /+ - /~. Two definitions exist [5,26]:
• the asymmetric integral, which is the usual definition of Choquet integral:
Cv(f)-=Cv{f+)~d,(f-), B6)
• the symmetric integral, which is also called the Sipos integral [33]:
Cv(f)-=Cv(f+)-CAD. B7)

Set functions over finite sets: transformations and integrals
1391
They satisfy for all / e (K+)'\ C,.(-/) = -C,-.(/) and C,(-/) = -C„(/), hence their
names. The explicit expression of the Sipos integral is, for any /el":
p-\
^.(/) = ?(/(?)-/(/ + ?))?'({A),...,@}) + /(?)?'({A),···.(?)})
/ = l
+ /(/,+ ?,?({(?+1),...,(«)})
+ ? (/(?)-/(?-?))«({? («)})
i=p+2
B8)
with /(!)<···< /(,,) < 0 < /(p+i) ^ ··· ^ /(»)· By contrast, expression B4) of the
Choquet integral does not change when / e R". This means that the Sipos integral behaves
differently on positive and negative numbers (true zero, ratio scale), while the Choquet
integral does not differentiate them (arbitrary position of the zero, difference scale) (more
on this topic and its relation to decision making in [15]).
We turn to the expression of these two integrals using the various transformations
introduced above.
PROPERTY 7. Let ? be a set function, m, in, I their Mobius, co-Mobius and interaction
transforms, and f a real-valued function on N. Then the Choquet integral of f w.r.t. ? is
expressed by
Cv(f)=Y,m(S)/\f,
ScN ieS
Cv(f)= ? (-1)?+?^E)\//?
ScN. S^0 ieS
^(/)=???*'-'/+<?>??-?
B9)
C0)
SCN^TDS
ieS
+ ? (-!M+| ?#-'/-<?> V/"
SCN. S^tA
C1)
¦TdS
ieS
with I+ indicating a restriction so that only terms with positive interaction are taken into
account, and similarly for I~.
PROPERTY 8. Let ? be a set function, m, m, I their Mobius, co-Mobius and interaction
representations, and f a real-valued function on N. Then the Sipos integral of f w.r.t. ? is
expressed by
ACN Li6A ieA
= ]T m{A)f\f+ ]T m(A)\J f,
C2)
ACN+
ieA
ACN~
ieA

1392
?. Grabisch
Uf)= ? (-1)|А|+|т''(А)\//'+ ? Ы)И|+|™"И)Д/;, C3)
АПЛ/+^0 ieA ?(??~?? ieA
W)= ? (-1)И1+1^'ЧА)Д/,+ ^ (-1),А|+|тг'И)\/^' C4>
АСЛ/+ ieA ACN- ieA
&(/> = ? ( ? %|/+Ми/?ЛД/,
АСЛ/+ ^ScN\A ' ieA
+ ? ? ? S|B|/+(AUS)W/,
ACN" ^BcN\A ' ieA
+ ? (-D|A|+I( ? ртпАивЛу/,
??\?+?<? ^BcN\A ' ieA
+ ? (-DIA|+I( ? PlBBlr(AUB)\/\f„ C5)
??\?~?<? ^BCN\A ' ieA
with N+ := {i eN \fj^ 0} and N~ := ? \ N+.
5. fc-additive measures
In this section, we focus on capacities (or fuzzy measures), although the concept of
?-additivity can be defined for any set function.
border additive measures or k-additive measures for short have been introduced by
Grabisch in an attempt to decrease the exponential complexity of fuzzy measures in
practical applications, since a fuzzy measure defined on a set of// elements requires 2" real
coefficients for its definition. A means which has been often used for this is to introduce
the property of decomposability: a fuzzy measure ? is decomposable if the measure of any
subset can be expressed as a function of the measures of each element in the set. Thus
we need only to define the distribution of ? over N, hence // coefficients instead of 2".
The most usual example in this category are additive measures. But it appears that this
is too drastic a simplification, which is too limitative, especially in multiattribute decision
making. One can think of a distribution defined not only for singletons, but also for pairs.
An adequate way to define this is to refer to pseudo-Boolean functions, and especially their
multilinear form C). An additive measure (defined by a distribution on singletons) has a
linear expression f(x) = ???'???- and tne coefficients a,'s (i.e., the Mobius transform)
are indeed the distribution itself. Then a fuzzy measure of which the multilinear extension
is limited to a degree 2 (or 3, etc.) can be expressed only by a distribution on singletons
and pairs (or also on triples, etc.). Equation C) will then provide the value of ? for every
subset.
Dehnition 2. A fuzzy measure ? is said to be k-additive if its Mobius transform satisfies
m(S) = 0 for any S such that s > k, and there exists at least one subset 5Of N of exactly
к elements such that m (S) ? 0.

Set functions over finite sets: transformations and integrals
1393
The following property of A;-additive measures is fundamental.
PROPERTY 9. Let ? be a k-additive measure on N. Then
(i) I (S) = m(S) = Ofor every S С N such that \S\ > k,
(ii) / E") = m(S) = m(S)for every ScN such that \S\=k.
Thus, A;-additive measures can be represented by a limited set of coefficients, either
m(S), \S\ < k, or I(S), \S\ sC k, or equivalently m(S), \S\ ij k, i.e., at most ?*=l ('')
coefficients.
The case of 2-additive measures is particularly appealing, since it is much more general
than additive measures, while remaining of low complexity. We detail properties of this
case in the sequel. A first fact is that any 2-additive measure ? can be expressed only by
?((/)),?({/, ;'}),forall /, ;' e ?, hence the notion of "distribution" for singletons and pairs
suggested above.
PROPERTY 10. Any 2-additive measure can be written as
?(?)= ? ?({/,;·})-(|?|-2)]??({/}), УД С ?, |?|>2. C6)
\i.j)eA ieA
Moreover, the interaction index /,y reduces to
??=?({?,?)-?({?)-?(?}). C7)
The Choquet and Sipos integrals have a particularly interesting expression for 2-additive
measure, in terms of the interaction transform.
PROPERTY 11. Let f eW and ? a 2-additive measure. Then
" ( 1
/„ >0 /,,<0 i=\ ^ ??\
C8)
сд(/)= ? (fiAfj)lu+ ? (/^/й
i.jeN+.Iij>0 i.jeN-.I,j>0
+ ? (???-)?/.7?+ ? (fi*fj)w
i.jeN+.Iij<0 i.jeN-.I,,<0
+ ?? ? m)+Ef{ ? m
ieN+ jeN~.I,j<0 ' ieN~ уеЛГ./,,-<()
+????-\?\?'? C9>
using previous notations.

1394
?. Grabisch
These expressions are useful for a deep understanding of these integrals, since all
coefficients are positive: thus the contribution of each term to the value of the integral
becomes clear. The following interpretation can be done:
• The Choquet integral is formed of conjunctive terms, disjunctive terms, and linear
terms. A conjunctive term appears whenever the interaction index is positive, while
a disjunctive one appears when the interaction index is negative. The linear part is
weighted by the Shapley value, minus all interactions (positive or negative). Putting
the Choquet integral into the form
сд (/) = ? aiJ (¦# л fj> + ? ^ (./·¦v /;) + ? и fi
ij i.j i
and remarking that due to ]?;. ?? = ?(?) = 1, we have
?^?+?^ + ?^1'
i-j iJ i
we conclude that any Choquet integral w.r.t. a 2-additive measure is a convex
combination of conjunctions, disjunctions and linear terms. The reciprocal also holds
since any coefficient in the combination can take the value 1.
• The Sipos integral, although more complicated, is very similar to the Choquet
integral, except that negative and positive numbers are processed differently. In fact,
disjunctions turn to conjunctions when the sign changes. Another remarkable fact is
that pairs (f\, /2) of values with different signs are not considered.
This interpretation is of high interest in decision making.
To conclude this section, we present a graphical interpretation of the Choquet integral
when ? = 2 [14]. In this case, formula C8) becomes:
Д i(/|V/2)/|2 + /|@| + ^/|2) + /j@2 + i/|2), if/|2<0,
which is in fact the general expression for Choquet integral when и = 2, since in this case,
at most 2-additive measures coincide with general fuzzy measures. Clearly, all possible
Choquet integrals are obtained when ф\, фг and 1\г vary on their domain. Using Property 6,
this domain is defined by
01 — 5/12 ^ 0, 02 - 5 /| 2 ^ 0,
01+5/12^0, 02+ 5-/12^0.
Recalling that ф\ + фг = 1, this domain can be represented with the (?\, /|?) coordinates
only, and is the shaded area on Figure 1. In other words, the Choquet integral is the convex
closure of the four vertices of Figure 1. We further study these vertices, and the two axes.
Let us consider the horizontal axis, where I a = 0. In the case ? = 2, this is equivalent
to say that the measure is additive, and thus the Choquet integral is a weighted sum,1
A weighted sum is any expression of the form g{x\ ,??) = w\.x\ + unxi, with w\. wj e M+ and w\ +wi = 1.

Set functions over finite sets: transformations and integrals
1395
!/2(t.;l+ii2i
Fig. 1. Interpretation in term of interaction index.
with weights ?({1}) and ?({2}), which coincide with ?\ and фг- Thus the horizontal axis
represent the set of all possible weighted sums.
Let us examine the vertical axis, where ф\—фг = \- Assuming /12 > 0 (the converse
case leads to the same result), formula D0) becomes:
C^(ai,a2) = з«A)A + /12) + з«B>A - /i2). D1)
We recognize here an ordered weighted average,2 with weights 5 A + /12) and ,A — hi)-
Moreover, since any OWA operator is such that ?\ = ?? = \ when и = 2, the vertical
axis is the locus of all possible OWA operators. The upper vertex (/12= 1) corresponds to
the minimum operator, and the lower vertex to the maximum operator, as it can be seen
from D1).
6. The ordinal case: the Sugeno integral
So far we have dealt with real-valued set functions. In this section, we address the case
of ordinal structures. We suppose that set functions are valued on a linearly ordered set E.
This corresponds in applications to evaluations on an ordinal scale, where only order makes
sense, and any arithmetical operation is forbidden. In this case, the Choquet and Sipos
2An ordered weighted average is any expression of the form g(x\, дгт) = u-ч jt( |) + wi-^B)· wi(n Ш|, шт б R
and ш 1 + u>2 = 1.

1396
?. Grabisch
integrals are of no use, and one has to turn to other integrals. The basic material here is the
Sugeno integral [32], defined as follows, for any function / valued on [0, 1]:
/1
<V/)--=V[/«>AM(A„,)], D2)
/=l
with same notations as for the Choquet integral B4). This definition uses only minimum
and maximum operators, so it can be used for any function and fuzzy measures valued on
a linearly ordered set. Using the residuated difference operator -*- defined by
fa, if a > b,
D3)
0, otherwise,
the above expression can be rewritten as [25]
/1
<V/):=V[(/(|->^/(/-i))AM(A(/))]. D4)
The similarity with the Choquet integral is striking, replacing sum by maximum, and
product by minimum. This shows that the Sugeno integral is a translation of the Choquet
integral in the ordinal world. Other types of integrals obtained by changing operators are
studied at length in [18,25].
In the sequel, we show how constructions similar to the cardinal case can be done in
this context, namely set functions transformations and the definition of symmetric and
asymmetric integrals.
As we need the notion of "negative" number on ordinal scales for the definition of
symmetric and asymmetric integrals, we begin by constructing a suitable scale with
suitable ordinal operators. Let ? be a linearly ordered scale with 2k + 1 degrees defined
by
e-k <e-k+\ < ·· <e-\ < e0 <e\ <¦¦¦ < ek-\ < ek
centered around e^. We speak of "negative" value for any degree with a negative index,
and denote E+ := {eo, e\,..., e^} and E~ := ? \ E+. By commodity, we denote eo by
(D and ek by 1. We introduce for every e, in ? its symmetrical element —e, defined
—ej := e-i (symmetric ordinal scale). We extend on ? the usual operations v, л defined
on E+, so as to have an algebraic structure like a ring. Denoting ©, © these extended
operators, called symmetric maximum and symmetric minimum, respectively, it would
be desirable to have properties like -(a © b) = (-a) © (—b), a © (-b) = a -"- b, and
-(a © b) = (-a) © b = a © (-b), where - is the residuated difference D3) properly
extended on E. Here, ©, © play the role of addition and product, respectively.

Set functions over finite sets: transformations and integrals
1397
Table 3
Definition of the symmetric maximum ?
a®)b
a <©
a = <D
a ><D
b <<D
a f\b
b
a^(-b)
b = <D
a
?
a
b><0
b^(-a)
b
a\ib
Table 4
Definition of the symmetric minimum ?
a@b
a <<D
a = <D
a ><D
b <<D
\a\A\b\
?
~(ал\Ь\)
b = <D
?
?
?
b><0
-(\а\лЬ)
?
а лЬ
The following definitions are suitable for our purpose [13]. The symmetric difference is
defined for any a, b on E+ by
¦b:=
a, if a > b,
<D, \ia = b,
—b, otherwise.
D5)
Using this definition, the symmetric maximum is defined in Table 3. Then, the symmetric
minimum is defined in Table 4. With these definitions, we can show the following result.
PROPERTY 12. The structure (?,©.©) has the following properties:
(i) © и commutative;
(ii) О и the unique neutral element ofQ, and the unique absorbant element of ©;
(iii) a © —a = (D,/or all a e E\
(iv) -(a©fe) = (-a)©(-/>);
(v) © is associative on E+ and on E~;
(vi) © E commutative;
(vii) 1 i5 the unique neutral element of<&, and the unique absorbant element o/©;
(viii) © и associative on E;
(ix) © 15 distributive w.r.t. © in E+ and E~.
The associativity of © and distributivity do not hold on ? in general.
The second step is to address set functions (in fact, we restrict to fuzzy measures here)
and their transformations. In this ordinal context, a fuzzy measure is defined naturally
as a set function ?:?>(?) -> E+ such that ?@) = <D, ?(?) = 1, and А с В implies
? (?) ^ ? (S). Following Berge [1], who presents the Mobius function as a powerful tool
for inversion formulas over posets, we define the ordinal Mobius transform of ?, denoted
by m v, as the solution of the equation:
?(?)=0)??»(?).
BcA
D6)

1398
?. Grabisch
In fact, unlike the cardinal case, there is no single solution to this equation. Previous studies
by the author [9,11] and related works of Mesiar [23] and Marichal [22] have shown that
the smallest positive solution is given by
mC(A):=(M(A)' ifM(A)>M(MO, Vi e A,
( 0, otherwise,
and the greatest one is simply m^ = ?. Any set function comprised between these two
solutions is a solution. We take as definition of the Mobius transform the lower bound,
which is a non-negative function, so that we can use from now on usual ? and л operators.
If there is no fear of ambiguity, the superscript? can be dropped.
Interestingly enough, the Mobius transform can be expressed as
"*v(A):= V ?(?)- V ?(?)
Вел вел
\A\B\ even |?\?| odd
which is an exact transcription of the cardinal case (see G)). However, if ? is not
monotonic, the above formula is no more solution of D6) in general.
The ordinal Mobius transform has many interesting properties. E.g., let us consider any
possibility measure ?, i.e., a fuzzy measure such that ?(? U ?) = ?{?) ? ?{?) for all
?, ? с ?¦ Then its Mobius transform m!J is
m'J({i}) = n({i}), VieN, and wG(A) = (D, VAcN, \A\>\.
Thus we have the same kind of result as with probability measures in the cardinal context.
Attempts have been done by Grabisch to define an interaction transform in this ordinal
context [11]. Changing in a suitable way operators in formula (9), and taking into account
some properties which are counterparts of the cardinal case (see below), the following
equation defines the ordinal interaction transform of a fuzzy measure ?:
/v(A):= V V M(BUC)- V M(fiUC)
CCA CCA
|A\C|even \A\C\ odd
BcN\A
For A = {i}, we get the ordinal Shapley value:
D7)
4>v0'):= V [M(AU/)-M(A)j. D8)
ACN\i
The ordinal Shapley value fulfills the following properties:
(Al) V,4=aX@ = M(A0 (sharing of ?(?0);
(A2) if i is such that ?(? U i) = ? (A) for every А С N \ i, then ?$(?) = <D (null
players);
(A3) if /, ;' are such that ?(? U i) = ?(? U ;') for every А с ? \ {i, j), then </#(/) =
?? (j) (symmetric players).

Set functions over finite sets: transformations and integrals
1399
These properties are counterparts of the properties of the (original) Shapley value [30].
However, contrary to what is wrongly claimed in [11], the "maxitivity" does not hold: in
general ?^?? ^?^,??^, where ?, ? are two fuzzy measures. Lastly, the ordinal interaction
transform can be expressed through the ordinal Mobius transform by
/v(A)= V m^(AUB).
BcN\A
D9)
Again, the similarity with the cardinal case has to be noted.
The last step is to address the definition of the Sugeno integral over E, the expression
over E+ being given by Equation D2), where / is any function from N to E+, and ? is a
fuzzy measure in the sense defined above. As in the cardinal case, it appears that the Sugeno
integral can be viewed as an extension of fuzzy measures (or pseudo-Boolean functions).
Specifically, any fuzzy measure can be expressed on the basis of unanimity games, by
?(?)= у (т$(В)Лив(А)),
BcN
E0)
which is the counterpart of C) and B1) in the ordinal case. The extension of this formula
on (E+)" gives rise to the Sugeno integral:
?„(/)= V "МА)л(Д/Л
ACN*- \'6A '
E1)
Remark the analogy with the Choquet integral (see B5)).
Based on formulas D4) and B8), the following expression defines the symmetric Sugeno
integral [13]:
5?(/):
©(/(/, ©?({0),...,(«)}))
./=1
© (л»©?({(?,···,(")}))
E2)
for any / e E", with /(n ^ · · · ^ /<,,> < <D ^ /(,,+ n ^ · · · ^ /(«>· Note that the above
expression is well-defined, i.e., there is no problem of associativity of ©, since @f=|
operates only on E~, and @"= +| only on E+. The symmetric Sugeno integral indeed
satisfies the property of symmetry:
5?(-/) = -5?(/)
E3)
for any fuzzy measure ? and any f e E".
The case of the asymmetric integral is not necessary to consider, if it can be thought, as
in the cardinal case, that the expression of the asymmetric integral does not depend on the

1400
?. Grabisch
position of the zero of the scale. Then, the zero can be put at the lower extremity, and we
are working on a scale with only positive values, where the usual definition of the Sugeno
integral works. Another possibility is to take as definition
4(?) = 5?(?+)^5?(?-),
where ?? stands for the asymmetric integral. But this needs a proper definition of Д, which
requires more structure on ? than what we have defined so far.
References
[1] C. Berge, Principles of Combinatorics. Academic Press. San Diego A971).
[2] A. Chateauneuf and J.Y. Jaffray, Some characterizations of lower probabilities and other monotone
capacities through the use ofMobius iirversion. Math. Social Sci. 17 A989). 263-283.
[3] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 A953). 131-295.
[4] A.P. Dempster. Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38
A967), 325-339.
[5] D. Denneberg, Non-Additive Measure and Integral, Kluwer Academic, Dordrecht A994).
[6] D. Denneberg and M. Grabisch, Interaction transform of set functions over a finite set. Inform. Sci. 121
A999), 149-170.
[7] M. Grabisch, k-order additive fiizzx measures, 6th Int. Conf. on Information Processing and Management
of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Spain (July 1996), 1345-1350.
[8] M. Grabisch, Alternative representations of discrete fuzzy measures for decision making. Intemat. J.
Uncertain., Fuzziness Knowledge Based Systems 5 A997), 587-607.
[9] M. Grabisch, k-additive and k-decomposable measures, Proc. of the Linz Seminar. Linz, Austria (February
1997), 88-94.
[10] M. Grabisch, k-order additive discrete fuzzx measures and their representation. Fuzzy Sets and Systems 92
A997), 167-189.
[11] M. Grabisch, On the representation of k-decomposable measures. 7th IFSA World Congress. Prague, Czech
Republic (June 1997).
[12] M. Grabisch, The interaction and Mbbius representations of fuzzy measures on finite spaces, k-additive
measures: a survey. Fuzzy Measures and Integrals - Theory and Applications, M. Grabisch, T. Murofushi
and M. Sugeno, eds. Physica-Verlag, Heidelberg B000), 70-93.
[13] M. Grabisch, Symmetric and asymmetric fuzzv integrals: the ordinal case, 6th Int. Conf. on Soft Computing
(Iizuka'2000). Iizuka, Japan (October 2000).
[14] M. Grabisch, A graphical interpretation of the Choquet integral, IEEE Trans. Fuzzy Systems, to appear.
[15] M. Grabisch and Ch. Labreuche, To be symmetric or asymmetric? A dilemma in decision making.
Preferences and Decisions under Incomplete Knowledge, J. Fodor, B. De Baets and P. Pemy, eds, Physica-
Verlag, Heidelberg B000), 179-194.
[16] M. Grabisch, J.-L. Marichal and M. Roubens, Equivalent representations of a set function with applications
to game theory and multicriteria decision making, Proc. of the Int. Conf. on Logic, Game theory and Social
choice (LGS'99), Oisterwijk. The Netherlands (May 1999), 184-198.
[17] M. Grabisch, J.-L. Marichal and M. Roubens, Equivalent representations of a set function with applications
to game theory and multicriteria decision making. Math. Oper. Res. 25 B) B000), 157-178.
[18] M. Grabisch, T. Murofushi andM. Sugeno, Fuzzy measures of fuzzy events defined by fuzzy integrals. Fuzzy
Sets and Systems 50 A992), 293-313.
[19] M. Grabisch and M. Roubens, An axiomatic approach to the concept of interaction among players in
cooperative games, Intemat. J. Game Theory 28 A999), 547-565.
[20] PL. Hammer and S. Rudeanu, Boolean Methods in Operations Research and Related Areas, Springer-
Verlag, Berlin A968).

Set functions over finite sets: transformations and integrals
1401
[21] L. Lovasz. Submodular function and convexity. Mathematical Programming. The State of The Art,
A. Bachem. M. Grotschel and B. Korte, eds, Springer. Berlin A983). 235-257.
[22] J.-L. Marichal, P. Mathonet and E. Tousset, Mesuresfloues definies sur une echelle ordinale. Working paper
A996).
[23] R. Mesiar, k-order pan-additive discrete fuzzy measures. 7th IFS A World Congress. Prague, Czech Republic
(June 1997), 488^90.
[24] T. Murofushi and S. Soneda, Techniques for reading fuzzy measures (III): interaction index, 9th Fuzzy
System Symposium, Sapporo, Japan (May 1993), 693-696 (In Japanese).
[25] T. Murofushi and M. Sugeno, Fuzzy t-conorm integrals with respect to fuzzy measures: generalization of
Sugeno integral and Choquet integral. Fuzzy Sets and Systems 42 A991), 57-71.
[26] E. Pap, Null-Additive Set Functions, Kluwer Academic Publishers, Dordrecht A995).
[27] G. Owen, Multilinear extensions of games. The Shapley Value. Essays in Honor of Lloyd S. Shapley, A.E.
Roth.ed.. Cambridge Univ. Press. Cambridge. UK A988). 139-151.
[28] G.C. Rota, On the foundations of combinatorial theory I. Theory of Mobius functions, Z. Wahrschein-
lichkeitsth. verw. Geb. 2 A964), 340-368.
[29] G. Shafer, A Mathematical Theory of Evidence, Princeton Univ. Press. Princeton, NJ A976).
[30] L.S. Shapley. A value for ?-person games. Contributions to the Theory of Games, Vol. II, H.W. Kuhn and
A.W. Tucker, eds, Annals of Mathematics Studies, Vol. 28, Princeton Univ. Press, Princeton, NJ A953),
307-317.
[31] I. Singer, Extensions of functions o/0-l variables and applications to combinatorial optimization. Numer.
Funct. Anal. Optim. 7 A) A984), 23-62.
[32] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. thesis, Tokyo Institute of Technology
A974).
[33] J. Sipos, Integral with respect to a pre-measure. Math. Slovaca 29 A979), 141-155.
[34] P. Walley, Coherent lower (and upper) probabilities. Technical Report 22, University of Warvick, Coventry
A981).

CHAPTER 35
Pseudo-Additive Measures and Their Applications
Endre Pap*
Institute of Mathematics, University of Novi Sad, Trg D. Obradovica 4, 21000 Novi Sad, Yugoslavia
E-mail: pape@eunet.yu, pap@im.ns.ac.yu
Contents
Introduction 1405
1. Pseudo-additive measures and integrals based on them 1406
1.1. Pseudo operations 1406
1.2. Pseudo-additive measures 1408
1.3. Pseudo-integral 1413
1.4. Pseudo-convolution 1415
1.5. The Riesz type theorem 1422
2. Applications 1424
2.1. Probabilistic metric spaces 1424
2.2. Fuzzy numbers 1426
2.3. Information theory 1428
2.4. System theory 1430
2.5. Optimization and morphism with the probability 1430
2.6. Pseudo-linear operators 1433
3. Idempotent integral as limit of g-integrals 1436
3.1. Pseudo-operations 1436
3.2. Measures and integrals 1438
4. Applications on nonlinear PDE 1440
4.1. Hamilton-Jacobi equation with non-smooth Hamiltonian 1440
4.2. Bellman differential equation for multicriteria optimization problems 1443
4.3. Stochastic optimization 1445
4.4. Option pricing 1448
5. Non-commutative and non-associative pseudo-operations 1449
5.1. Applications on nonlinear PDE 1452
5.2. Corresponding pseudo-measure 1454
6. Conditional distributive real semiring 1458
6.1. Conditional distributivity 1458
6.2. Hybrid probability-possibilistic measure and integral 1458
*The author wants to thank for the partial financial support of the Project in the Fields of Basic Research
"Mathematical models of nonlinearity, uncertainty and decision" A866) supported by Ministry of Science,
Technology and Development of Serbia.
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1403

1404 ?. Pap
6.3. Hybrid utility function 1463
References 1465

Pseudo-additive measures and their applications
1405
Introduction
It will be presented some results on special non-additive measures - pseudo-additive
(decomposable) measures and the corresponding integrals, which give a base for the so
called pseudo-analysis. There are applications in optimization problems, nonlinear partial
differential equations, nonlinear difference equations, optimal control, fuzzy systems,
For the range of a set function instead of the field of real numbers in Bacceli et al,
A992), Maslov et al. A992), Pap A995), Sugeno et al. A987) it is taken a semiring on the
real interval [a,b] с [—oo, +oo], denoting the corresponding operations as ? (pseudo-
addition) and ? (pseudo-multiplication). This structure is applied for solving nonlinear
equations (ODE, PDE, difference equations, etc.) using now the pseudo-linear principle
(Maslov et al,, 1992; Kolokoltsov and Maslov, 1997; Kolokoltsov, 2001; Pap, 1995),
which means that if щ and ui are solutions of the considered nonlinear equation, then
also ?? ? ? ? ? ?? ? «2 is a solution for any constants a\ and an from [a,b]. Based on
semiring structure it is developed in Pap A990, 1995, 2000) the so called pseudo-analysis
in an analogous way as classical analysis, introducing ?-measure, pseudo-integral, pseudo-
convolution, pseudo-Laplace transform, etc.
The advantages of the pseudo-analysis are that there are covered with one theory, and so
with unified methods, problems (usually nonlinear) from many different fields (system
theory, optimization, control theory, differential equations, difference equations, etc.).
Important fact is that this approach gives also solutions in the form which are not achieved
by other theories. In some cases it enables for mostly nonlinear equations to obtain exact
solutions in the similar form as for the linear equations.
Some obtained principles as for example the pseudo-linear superposition principle
allows us to transfer the methods of linear equations to the nonlinear equations.
Pseudo-analysis uses many mathematical tools from different fields as functional
equations, variational calculus, measure theory, functional analysis, optimization theory,
semiring theory, etc, and still is in the developing form. There are many investigations from
the theoretical point of view as well as in many new applications (related monographs:
Baccelli et al. A992), Grabisch et al. A995), Klement et al. B000a, 2000b), Kolokotsov
and Maslov A997), Maslov and Samborskij A992), Pap A995), Puhalskii B001), Wang
and Klir A992)),
The main problem in the application on nonlinear PDE is the identification of operations
? and ©. Goard and Broadbridge A997) have obtained a close connection with Lie
symmetry algebras (see Blumen and Kumei A989)). Namely, there are a number of
computer added algorithms for finding Lie symmetry algebra (Schwarz, 1988) and
therefore this relation can be used for finding © and ©.
There are presented also two further generalizations related the pseudo-analysis. First,
is related to further generalization of the real semiring structure to the case when
the operations ? and © are noncommutative and non-associative, and the (left) right
distributivity of © over ? plays a crucial rule. There is given a representation theorem
- Theorem 5.2 for such operations and in Theorems 5.6 and 5.3 we give a complete
characterization for generalized pseudo-addition and pseudo-multiplication. The approach
is based on the functional equation technique, given in Aczel A966). More details can be
found in Pap et al. B000).

1406
?. Pap
The second generalization is related to relaxation of the distributivity law, see Klement
et al. B000a, 2000b), which enables a generalization of the classical von Neumann-
Morgenstern utility theory A944) to hybrid probability-possibilistic utility theory (Dubois
et al., 2000).
1. Pseudo-additive measures and integrals based on them
1.1. Pseudo operations
Let ? be a ? -algebra of subsets of a given set X.
Now we restrict ourselves on special non-additive measures so called pseudo-additive
or decomposable measures (see Dubois et al. A982), Grabisch et al. A995), Klement and
Weber A991), Klement et al. B000a, 2000b), Murofushi et al. A991), Pap A990, 1993a,
1993b, 1995), Sugeno and Murofushi A987), Wang and Klir A992), Weber A984)). This
measures are based on pseudo-addition and serve as an important tool in pseudo-analysis.
We shall start with the operations on the real interval [0, 1].
A triangular norm ? (?-norm briefly) is a function ?: [0, 1 ]2 -> [0, 1 ] such that
(Tl) T(x,y) = T(y,x);
(T2) T(x,T(y,z)) = T(T(x,y),z);
(T3) T(x,y)^T(x,z)fory^z\
(T4) T(x, \) = x.
Example 1.1. The following are the most important i-norms
Тм(х,у) = т\п(х,у),
Tp(x,y) = xy,
Th(x,y) = тах@,* + у - 1),
?<? у) = { minU^ У) if тах(х,у)= 1,
10 otherwise.
The corresponding dual operation is given by
A triangular conorm S (?-conorm briefly) is a function S: [0, 1 ]2 -> [0, 1 ] such that
E1) S(x,y) = S(y,x);
E2) S(x,S(y,z)) = S(S(x,y),z);
E3) S(x,y) ? S(x,z) for у <;Z;
E4) S(x,0) = x.
We see that ?-norms and ?-conorms differs only by boundary conditions.
Example 1.2. The following are the most important i-conorms:
SM(x,y) = max(x,y),
Sp(x,y) = x + y-xy,

Pseudo-additive measures and their applications
1407
5ъ(-*,}0 = тгпA, * + }'),
5d(jc>>;)= Ыах(х,у) if min(.x,H = 0,
1 1 otherwise.
1 otherwise.
Many other important f-norms and f-conorms can be found in Klement et al. B000a,
2000b), Golan A992). The basic connectives in fuzzy logic and operations by fuzzy sets
are based on triangular norms and triangular conorms (Grabisch et al. A995), Klement et
al. B000a, 2000b)).
We extend now the considered interval for the previous operations. Let [a,b] be
a closed (in some cases semiclosed) subinterval of [—oo, -boo]. We consider here a
total order ^ on [a,b] (although it can be taken in the general case a partial order).
The operation ? (pseudo-addition) is a function ®:[a,b] ? [a,b] -> [a,b] which is
commutative, nondecreasing, associative and has a zero element, denoted by 0.
Let [a,b]+ = {?: ? e [a,b], ? ^ 0}. The operation ? (pseudo-multiplication) is a
function ?: [?, b] ? [?, b] ->¦ [?, b] which is commutative, positively nondecreasing, i.e.,
? ^ у implies ? ? ? ^ у ? ?, ? е [?, />]+, associative and for which there exist a unit
element 1 e [a, b], i.e., for each ? e [a, b], 1 ? ? = ?.
We suppose, further, 0 ? ? = 0 and that ? is a distributive pseudo-multiplication with
respect to ?, i.e.,
? ? (y ® z) = (? ? y) ® (? ? z).
The structure ([a,b],(B, ?) is called a sem/rmg (see Cuninghame-Green( 1979), Benvenuti
et al. B002), Maslov et al. A992), Pap A995)). It can be considered as a general algebraic
structure - semiring (?, ?, ?) on an arbitrary set ? endowed with the operations ? and ?
which satisfy the previously given conditions (Golan A992), Kuich A986)).
In this chapter we will take as basic the following special real semirings (using the
equality 0 ? ? = 0; we can consider also closed intervals in the following examples):
Case I. Pseudo-addition is idempotent and pseudo-multiplication is not idempotent,
e.g·,
(i) ? ®y = m\n{x,y], xQy — x +y,
on the interval ]—oo, -boo]. We have 0 = +oo and 1=0.
(ii) ? ? у = max{;t,)>}, xOy = x+y,
on the interval [—oo, +oo[. We have 0 = —oo and 1=0.
Case II. Semirings with pseudo-operations defined by monotone and continuous
generator g (Aczel, 1966;Mesiaretal., 1995; Pap, 1990, 1993a, 1993b, 1995). In this case
we will consider only strict pseudo-addition, i.e., such that the function ? is continuous
and strictly increasing in ]a, b[ ? ]?, b[.

1408
?. Pap
By Aczel's representation theorem (see Aczel A966)) for each strict pseudo-addition
? there exists a monotone function g (generator for ?), g : [a, b] -> [—00, 00] (or with
values in [0, 00]) such g@) = 0 and
ii®u=g-l(g(ii) + g(u)).
Using a generator g of strict pseudo-addition ?, we can define pseudo-multiplication ?
(Pap, 1990, 1993a):
This is the only way to define pseudo-multiplication ©, which is distributive with respect
to ? generated by the function g (Mesiar and Rybarik, 1995).
Case III. Both pseudo-addition and pseudo-multiplications are idempotent, e.g.,
(i) Let ? = max and © = min on the interval [-00, +00].
(ii) Let ? = min and © = max on the interval [—00, +00].
An idempotent semigroup (semiring, e.g.. Cases I and III) ? is called an idempotent
metric semigroup (semiring) if it is endowed with a metric d: ? ? ? -> Ш such that
the operation ? is (respectively, the operations ? and © are) uniformly continuous
on any order-bounded set in the topology induced by d and any order-bounded set is
bounded in the metric. Let X be a set, and let ? = (?, ©,d) be an idempotent metric
semigroup. The set B(X, P) of bounded mappings X -> P, i.e., mappings with order-
bounded range, is an idempotent metric semigroup with respect to the pointwise addition
(/ ? h)(x) = f(x) ? h(x), the corresponding partial order, and the uniform metric
p(/, h) = supx p(f(x), h(x)). If ? = (? ?, ©, d) is a semiring, then ?(?, ?) has the
structure of an P-semimodule, i.e., the multiplication by elements of ? is defined on
B(X, P) by (aQf)(x) = aQf(x). If X is a topological space, then by C(X, P) we denote
the subsemimodule of continuous functions in B(X, P). For X finite, X = {x\,..., *,,},
? & ?, the semimodules C(X, P) and ?(?, ?) coincide and can be identified with the
semimodule P" = {(a\,..., a„): ctj e A]. Any vector a e P" can be uniquely represented
as a linear combination a = ?"=| ?, © e-3, where {e/, ;' = 1,..., n\ is the standard
basis of P" (the ;'th coordinate of es is equal to 1, and the other coordinates are equal
to 0). As in the classical linear algebra, we can prove that the semimodule of continuous
homomorphisms F: ?" -> ? (in what follows such homomorphisms are called linear
junctionals on P") is isomorphic to P" itself. Similarly, any endomorphism ? : ?" -> ?"
(a linear operator on P") is determined by an P-valued ? ? ? matrix, see Cuninghame-
Green A979), Kolokoltsov B001), Maslov et al. A992).
1.2. Pseudo-additive measures
1.2.1. S-measures. Let X be a fixed non-empty set. We restrict first on [0, l]-valued set
functions.
Dehnition 1.3. Let S be a ?-conorm and let ? be a ?-algebra of subsets of X.
A mapping m:F-> [0, 1] is called an S-measure (decomposable measure) if it is

Pseudo-additive measures and their applications
1409
continuous from below, w@) = 0 and m is S-decomposable, i.e., for all А, В e ? with
А П В — 0 we have
m(AU B) = S(m(A),m(B)).
Remark 1.4. (i) If S is a left continuous f-conorm, then a set function m: X1 -» [0, 1]
satisfying w@) = 0 is an S'-measure if and only if for each sequence ( A„)„6n of pairwise
disjoint elements of ? we have
/ oo
m I
(J A„ = S m(A„).
\n=l /
(ii) If the set X is finite or countably infinite and S is a left continuous f-conorm,
then each S'-measure m :V{X) -» [0, 1] is uniquely determined by the values m({x}) with
xeX.
(iii) A set function m : ? -» [0, 1] is SM-decomposable if and only if for all А, В е X1
we have w(A U ?) = 5"M(w(A), m(S)).
Example 1.5. (i) Obviously, each measure m : ? -> [0, oo] with Range(w) с [0, 1] is
an S^-measure.
(ii) Let В be the ?-algebra of all Borel subsets of the real line. The function m:B^>
[0, 1] defined by m{A) = ????(?(?), 1), where ?:? -> [0, oo] is the Lebesgue measure, is
an S'L-measure, but not a measure.
(iii) Fixing a number ieN and defining the set function m : V(N) -> [0, 1 ] by
/card(A)
m(A) = min
where card(A) denotes the cardinality of the set A, it is easily seen that m is always an
5l-measure, and an SO-measure in the case к = 2, but not a measure. Generally, m is an
S-measure if and only if the restriction of S on the chain {0, \/k,...,k/k} is divisible
Archimedean discrete ?-conorm (see Klement et al. B000a, 2000b)), i.e., it coincide with
the restriction of S^ to that chain.
(iv) The function m:B^> [0, 1 ] defined by
m(A) = 1 — e
-?(?)
is an Sp-measure.
(v) For an arbitrary function /: X -> [0, 1], the set function w/ :V(X) -> [0, l]defined
by m/(A) = sup{/(;c): ? e A} is an ^M-measure (Maslov and Kolokoltsov, 1997; Pap,
1995; Shilkret, 1971; Sugeno, 1974).
(vi) The set function m : V([0, 1 ]) -> [0, 1 ] defined by
... [sup A if A is finite or countably infinite,
m(A) = { , ,
1 otherwise,

1410
?. Pap
is an ^M-measure.
(vii) Let ?: ? -> [0, 1] be a probability measure. Then for each a e ]-1, oof \ {0} the
set function ma : ? -> [0, 1] defined by
ma(A)=-U\+a)i'iA)- 1)
is an 5,swa-measure for the the Sugeno-Weber f-conorm Sswa(.x, y) = min(.x + у +
axy, 1), see Klement et al. B000a). Observe that we have mA = (s^w)-1 ? ?, where s^w
is the unique additive generator of Ssw<* which satisfies s^w( 1) = 1 · We remark that mi
is subadditive if and only if ? e] —1,0[, and superadditive if and only if ??]0, oof.
(viii) If m : ? -> [0, 1] is an S-measure and ?:[0, 1] -> [0, 1] an increasing bijection,
then ?-1 ?»?:?^ [0, 1] is an ^-measure, where So; is defined by
SOJ(x,y)=co-i(S(w(x),w(y))).
The S-measures presented in Example 1.5(i)-(iv), (vii) are special cases of S-measures
(decomposable measures) (Dubois and Prade, 1982; Weber, 1984), some of which can be
constructed as follows:
PROPOSITION 1.6. Let S be a continuous Archimedean t-conorm, s : [0, 1] -> [0, oo] an
additive generator of S, ?? ?-algebra of subsets of X and m ; ? -> [0, oo] a measure.
Then the function i'~"offl:I'^· [0, l]iian S-measure.
We introduce now a generalization of the concept of ?-finiteness of additive measures
(Klement et al., 2000a, 2000b).
Definition 1.7. Let 5" beai-conorm, ? aa-algebraof subsets of X and m : ? -> [0, 1]
an S'-measure.
(i) A set A e ? is called S-m-faithful if for each В e ? with ? с A we have
S(u, v) < 1 whenever и < m(B) and ? < m(A \ B).
(ii) A partition С = {Q: Q e X\ A: e ?} of X, where ? is a finite or countably infinite
index set, is called an S-m -partition if, for each к e K, the set Q is S-w-faithful.
(iii) The S'-measure w is said to be S-faithful if there exists an S-m-faithful partition С
of X.
Remark 1.8. LetS beai-conorm, ? aa-algebraof subsets of X and m : ? -> [0, 1] an
S-measure.
(i) If A e ? is S-w-faithful then each В e ? with ? с A is 5-w-faithful.
(ii) If we have S(x, y) = 1 if and only if max(;t, y) = 1 then X is S-w-faithful, and
consequently each partition С = {Q: Ck ? ?, к e K} of X, where К is a finite or
countably infinite index set, is an S-m -partition. Hence each SM-measure is Sm-
faithful and, if S is a strict ?-conorm, then each S-measure is S-faithful too.
(iii) Each A e ? with m{A) < 1 is S-w-faithful.
(iv) In case of a nilpotent ?-conorm S, the S-faithfulness of an S-measure m is weaker
than the requirement that X be w-achievable in the sense of Weber A984).

Pseudo-additive measures and their applications
1411
An interesting relationship between (?-additive) measures and S-measures with respect
to some continuous Archimedean f-conorm S is given as follows:
PROPOSITION 1.9. Let S be a continuous Archimedean t-conorm, s : [0, 1 ] -> [0, oo] an
additive generator of S, ? ? ? -algebra of subsets of X and m : ?1 —» [0, I] an S-measure.
Then s о m : ? —» [0, oo] is a measure if and only if X is S-m-faithful.
The assumption that X be S'-w-faithful cannot be dropped because of Example 1.5(H),
(iii). Observe that, in (ii), a set A e В is S^-m-faithful if and only if ?(?) ^ 1, and in (iii),
if and only if card(A) ^ k.
We only mention that, if S, s, ? and m are as in Proposition 1.9, X is not S-m-faithful
if and only if s о т is not a measure but a continuous from below set function vanishing
on empty set and (see case (NSP) in Weber A984)) there exist disjoint elements А, В & ?
such that
(s о m){A Ufi)<(so m)(A) |(jo m){B).
Turning to an example of a non-Archimedean f-conorm, recall that ^M-measures were
called (?-) maxitive measures in Shilkret A971), which appear in the context oi fractal
dimensions (see Falconer B002), this Handbook). Also the possibility measures, which
are special cases of upper probabilities and of plausibility measures and which have a
close connection to the theory of fuzzy sets, are of interest.
Since, for an arbitrary (even for an uncountable) index set /, we naturally have
E???)/6/?, =sup{a,-: i el}
for each family (a,)/6/ e [0, 1]', we may even consider the following special case of a
maxitive, i.e., ^M-measure.
DEHNITION 1.10. An ^M-measure m:V(X) -> [0, 1] is called a completely maxitive
measure if for each family (A,),-6/ of subsets of X we have
W(UA/) =suP{w(A/): ' e /)·
A completely maxitive measure ? :V(X) -> [0, 1] with ?(?) = 1 is called a possibility
measure.
Example l.l 1.
(i) For each function /: X -^ [0, 1], the 5м-measure m/ given in Example 1.5(v)is
a completely maxitive measure. Conversely, for each completely maxitive measure
m:V(X) -> [0, 1] there exists exactly one function /: X -> [0, 1], which is given
by f(x) = m({x}) (and which can be interpreted as the membership function of
a fuzzy subset of X in the sense of Zadeh (see Klement et al. B000a)), such that
m = m f.

1412
?. Pap
(ii) The ^M-measure m considered in Example 1.5(vi) is not completely maxitive.
(iii) One example of an SM-measure is related to the Hausdorff dimension defined
on V(W) (see, e.g., Falconer B002), this Handbook). First, for each subset
А с R" consider its diameter d(A) = sup{||;c - y\\ \ x, у е A) (where \\x\\ is the
usual Euclidean norm of ? e R"), and define for each ae]0,oo[ the function
Ha:V(R")->[0,oo]by
Ha(A) = \im inf
J^d(Akf: (Ak)keN e V(Rnf, А с |J Ak,
l/t=l k=\
d(Ak) ? ? for а\\ к eN\ J.
Then the Hausdorff dimension dim/y: ^(R") -> [0, oof is given by
dimw(A) = inf{ae]0,oo[: Я"(А)=0} (= sup{<* e]0, oof: Я"(А) = oo}),
and ^dimw :V(W) -> [0, 1] is an SM-measure (which is not completely maxitive).
1.2.2. ?-measure. We extend now the previous considerations for [a, /?]-valued set
function. Let X be a non-empty set. Let ? be a ? -algebra of subsets of X.
A set function w : ? -> [?, /?]+ (or semiclosed interval) is a ©-decomposable measure
if there hold m@) = 0 (if ? is not idempotent),
ffl(AUB)=m(A)$m(8)
for ?, ? e ? such that ? ? ? = 0.
In the case when ? is idempotent, it is possible that m is not defined on an empty set.
A ?-decomposable measure m is ?-?-decomposable if
Qa, =0m(A,)
/ /=l
hold for any sequence (A;),-6pj of pairwise disjoint sets from ?.
Further on, we shall suppose that m be a ?-?-decomposable measure and < is total
order on [a,b].
The construction of pseudo-integral is similar to the construction of the Lebesgue
integral.
Dehnition 1.12. A set function m: ? -> [a, b]+ (or semiclosed interval) is a
?-?-decomposable measure if there hold:
A) m@)=O;
B) w((J^| ?? = 0/ = i m(Ai) for any sequence {A,},6n of pairwise disjoint sets
from ?, where ?^, *,¦ = Нт,,^^ ?"=? ??- If ® is idempotent operation
condition A) and pairwise disjointness of sets can be left out.

Pseudo-additive measures and their applications
1413
1.3. Pseudo-integral
Let ([a, b], ?, ?) be a semiring (see the remark at the end of this section), ([a, b], ?)
and ([a,b], 0) are complete lattice ordered semigroups. Let interval [a,b] be endowed
with a metric d compatible with sup and inf (limsupjt,, = ? and liminf.*,, = ? imply
Vim„^ood(x,i,x) = 0) and let the metric d satisfies at least one of the following
conditions:
(a) d(x®y, x'®y')<:d(x,x')+d(y,y')\
(b) d(x®y, x'®y')^max{d(x,x'), d(y,y')}.
Both conditions imply:
d(x„, y„) -> 0 =>¦ d(x„ ? ?, у,, ?;)^0.
We suppose that metric d is also monotonic, i.e.,
x±z<y => d(x,y)^max.[d(y,z),d(x,z)}.
Example 1.13. Metric with property (b) on the semiring ([-oo, +oo[, max, +) is
ddx,y) = \e-x -e~y\.
Metric with property (b) on the semiring ([—oo, +oo], max, min) is
di(x,y) = | arctanjc — arctan;y|.
Both metric are monotonic.
The pseudo-characteristic function of a set A is:
1 for ? e A,
0 for ? f A,
where 0 is zero element for ? and 1 is unit element for O. For functions defined on X and
values in [a, b] we transfer pointwise the operations ? and 0.
A mapping s :X -» [a,b] is a simple function if it has the following representation
s = ?"=| ?, ? 1A., where ?, e [?, b], A, e ? and if ? is not idempotent then sets A,-
are disjoint. An elementary (measurable) function is mapping e.X -> [a, />] that has the
following representation
oo
е = фа, OIa,, A)
for a, e [a,b], Aj e ?,? e X and if ? is not idempotent sets A, are disjoint, when the
right-hand side of equality A) exist.

1414
?. Pap
Definition 1.14. Let ? be a positive real number and В с [a,b]. A subset {/f},6/ of set
? is a ?-net on ? if for each ? e В there exists If such that d(lf, ?) ^ ?. If we, also, have
If < x, then we shall call {If },6/ a /ower ?-иел If /f < /?_, holds, then {/(?} is monotone.
We choose a metric d which ensures that each bounded subset of metric space ([a, b], d),
is compact. Then this fact would imply the existence of a finite ?-nets in every bounded
subset of [a,b], e.g., in Example 1.13 for the semiring ([-oo, +oo[, max, +) for the metric
d\ we have that in the bounded set {x: d\ (x, — oo) ^ ?} there exists the following finite
?-net
/? _ ? 1?(? - ? ?) forO ^ ? < i
1 \ -oo for г = i
0,
where г'о = [?/?] + 1.
THEOREM 1.15. Let f : X —> [a, b] be measurable if pseudo-addition is idempotent, and
if not, let f be measurable such that for each positive real number ? exists a monotone
?-net in f(X). Then there exist a sequence (<pn)neN of elementary functions such that, for
each ? e X,
d(<pn(x), /(-*)) -> 0 uniformly as ? -> oo.
The proof of Theorem 1.15 that gives the construction of functions <p„ can be found in
Pap A990).
Definition 1.16. The pseudo-integral of a simple function s (elementary function e)
with respect to the ?-?-decomposable measure m is:
/ sQ<iffl = 0flj0m(Ai), (/ eQdm = QajOm(Ai)\.
The pseudo-integral of a bounded measurable function /: X -> [a, />], for which if ? is
not idempotent for each ? > 0 there exists a monotone ?-net in /(X), is defined by:
/ f Qdm = lim / <p„ ? dfi
Jx n^xJx
where (<^„)/6n is the sequence of elementary functions from theorem above.
The pseudo-integral over A, when A is an arbitrary subset of X, is given by:
/ fQdm= /
./A JX
fOdm= / AA ?/) ? dm,
where хд is pseudo-characteristic function of set A.

Pseudo-additive measures and their applications
1415
Let (G, +), G С ?.", where + is the coordinatewise addition.
Definition 1.17. The semiring B(G. [a.b]) consists in Cases I and III (at least pseudo-
addition is idempotent) of the bounded (with respect to the order in [a. b]) functions, and
in Case II (pseudo-addition has been represented by its additive generator g) of functions
/:G -> [a,b] with property g(|/|) e L\(G) (the space L\(G) consists of Lebesgue
integrable functions which satisfy the condition fG \f(x)\ dx < +00).
All previous considerations can be transferred to the case [a, b] с К", taking care that
the order in M." is a partial order.
1.4. Pseudo-convolution
Let G be subset of W and * a commutative binary operation on G such that (G, *) is a
cancellative semigroup with unit element e and G+ = {?: ? e G, ? ^ e) is a subsemigroup
of G. All considerations can be managed also for a general topological group G. We shall
consider functions whose domain will be G.
Definition 1.18. The pseudo-convolution of the first type of two functions /: G ->
[a, /?] and h:G-> [a, b] with respect to a ?-?-decomposable measure m and * e G+
is given in the following way
f®
Jg\
f*h(x)= / f(u)Odmh(v),
where Gx+ = {(?, ?): и * ? = ?, ? e G+, и е G+}, w/, = w in the case of
sup-decomposable measure m{A) = supv6A h{x), in the case of inf-decomposable measure m(A) =
infxeA h(x), and dm/, = h ? dm in the case of ?-decomposable measure m, where ? has
an additive generator g and g о m is the Lebesgue measure and / e S(G, [a, b]).
We consider also f/?e second type of pseudo-convolution when (G, *) is a group and the
pseudo-integral is taken over whole set G:
f*h(x)= / /(**(-0)©d»M0·
where (—f) is unique inverse element for t and jc e G.
Remark 1.19. When * is the usual addition on R and G = E, pseudo-convolutions of
the first and the second type, for jc e R+, are
(f*h)(x)= / f(x-t)Odmh(t),
J [0.x]
f®
= / fix
Jg
{f*h){x)= / f(x-t)Odmh(t),

1416
?. Pap
respectively (Pap and Stajner, 1999). For both types for Case II we shall use also the
notation
W-/
(f*h)(x)=l f{x-t)Oh(t)dm{t).
Remark 1.20. In Schweizerand Sklar( 1983) the convolution of two functions has been
defined as
f*h{x)= f f{x-t)dh{t) B)
J[0,x[
for all ? e ]0, oof, / * /?@) = 0 and / * h(oo) = 1. Convolution defined in such manner is
commutative, associative operation with identity.
Using this approach we can introduce a different type of pseudo-convolution of the first
type, e.g., for g-case when * = + and G = R:
f*h{x) = g-'U g(f(x-t))dg(h(t))\
Convolution B) can be transformed into the form
f*h{x)=i dn(f(u),h(v))
JSum\x}
where Sumfjc} = {(и, и): и + v < ?; и, ? е R+} and П(х,у) = х у. The main difference
between this definition and Definition 1.18 is existence of identity, which is not certain for
pseudo-convolution (see g-case).
Next definition considers cases when pseudo-addition ? is an idempotent operation.
Definition 1.21. Pseudo-delta function is given by
1 for ? = e,
??) =
K ' ? forjc^e,
where 0 is zero element for ?, 1 is unit element for ? and e is zero element for *.
We shall give some examples of pseudo integrals and related pseudo-convolutions for
Cases I—III. We restrict here to the case G С R·
Example 1.22. In this example we shall give the form of the pseudo-convolution of
the first and the second type for some characteristic cases (the relevant semirings are the
semirings that are mentioned in the section above) and for * = + and G = R.

Pseudo-additive measures and their applications
1417
Case I.
base i.
(i) For the semiring ([—oo, oof, max, +) pseudo-integral, with respect to
sup-decomposable measure m, m(A) = supv6A h(x), is given by
r®
/ fOdm = sup(f(x)+h(x)),
Jm ж
and the pseudo-convolution of the first type of the functions / and h will be
(f*h)(x)= sup (f(x-t) + h(t)),
??????
and the pseudo-convolution of the second type is
(f*h)(x) = sup(f(x-t) + h(t)).
Unit element for this pseudo-convolutions is the following pseudo-delta function
Лтах. + ,г,_{1(=0) if*=0,
w-|0(=-oo) ifx^O.
(ii) For the semiring (]—oo, oo], min, +) pseudo-integral is given by
/ fOdm=mf(f(x) + h(x)),
where the function h defines the inf-decomposable measure m. The domain of functions
will be [0, oo] (or some subset of [0, oo]) and the domain of the semiring can be any
subinterval of ] — oo, oo] which contains 0 and oo. The zero element for the ? is oo and
the unit element for the ? is 0. Pseudo-delta function is given by
min. + ,^_Jl(=0) if*=0.
Г'П' (*)-}0(=oo) ifjt^O.
Previous pseudo-delta function is the unit element for the operation of pseudo-convolution
of the first type
(f*h)(x) = Qmf<p(x-t)+h(t)).
and of the second type
(f*h)(x)=inf(f(x-t)+h(t)).
teR
We can consider in analogous way the more general semiring ([a, b], max, ?), such that
© is non-idempotent pseudo-multiplication.

1418
?. Pap
Case II.
Pseudo-convolution of the first type in the sense of the g-integral (see Pap A993a)), i.e.,
when the pseudo-operations are represented by a generator g as ? ? у = g~' (g(x) +g(y))
and ? О у = g~] (g(x)g(y)), is given in the following way
(f*h)(x)= / f(t)Qh(x-t)dm(t)
J[0..x]
= 8-*ПХ8(П»)-8(И(х-0)аГу
Pseudo-convolution of the second type in the same sense is
f®
(f*h)(x) = / f(t)Qh(x-t)dm(t)
Jm
= *"'(/ g(fU))-g(h(x-t))dt\
Case III.
For the semiring ([—oo, oo], max, min) pseudo-integral is given by
/ / О dm = sup(min(/(jc),/?(;c))),
JM хеш
where the function h defines the sup-decomposable measure m. The domain of functions
will be R (or some subset of R) and the domain of the semiring is [-00, 00] (or any
subinterval). The zero element for the ? is —00 and the unit element for the © is +00.
Pseudo-delta function is given by
«jmax.min.4_ J 1 (=+00) if*=0,
w~|0(=-oo) if jc ^0.
Pseudo-delta function is the unit element for the pseudo-convolution of the first type and
the pseudo-convolution of the second type:
(f*h)(x)= sup (min(/(;t - t),h(t))) the first type,
(/ * h)(x) = sup(min(/(;t - t), h(t))) the second type.
We can analogously consider the pseudo-integral for semiring ([—00, 00], min, max)
valued functions.

Pseudo-additive measures and their applications 1419
Example 1.23. Let ? *y = x + y +xy, G =R, /:K -> [0,1] and h:R -> [0, l].Let
? be arbitrary f-norm. The pseudo-convolution of the first type for ? = max, ? = ? and
? e G+ = [0, oo] (case (I)) is
f -kh(x) = sup[T(f(u),h(v)): и + v + uv = ? for u. ? e [0, oo]}.
Another way to write it down is
/**и,-.Я,|г(/(тТ7)Н1
or, in the sense of Definition 1.18
f*h(x)=[ f(±^L\odmh(t).
J[0.x] \1+'/
The basic properties of the generalized pseudo-convolution for an idempotent pseudo-
addition are given in the following theorem (see Pap and Stajner A999)).
THEOREM 1.24. Let J7 be a class of functions f such that f:G->[a, b], where (G, *)
is a commutative cancelative semigroup with unit element e. Let ? be continuous (up to
some distinguished points) pseudo-multiplication of the first or the second type on interval
[a,b].
Then the pseudo-convolution of the first type (second type for G a commutative group)
for the idempotent pseudo-addition (Cases I andlll) is commutative, associative operation
with the unit element <5®°.
Remark 1.25. When pseudo-addition ? is max, then, we can take left continuous
pseudo-multiplication instead of continuous pseudo-multiplication.
Restricting on Case I: P = (min, +), we have the convolution on B(G, P)
(f*h)(x)= M(f(y) + h(x-y)).
veG
This operation turns B(G, P) into an idempotent semiring, which will be denoted by
CS(G) and called the convolution semiring. Some subsemirings of CS(G) are of interest
in studying multicriteria optimization (Kolokoltsov, 2001). Let L denote the hyperplane
in R* determined by the equation
L=\(aJ)eRk: JV=o},
and let us define a function и e CS(L) by setting n(a) = maxy(-a^). Obviously, ? *? =n,
i.e., и is a multiplicatively idempotent element of CS(L). Let CS„(L) с CS(L) be the
subsemiring of functions h such that n*h=h*n=h. CS„(L) contains the function

1420
?. Pap
identically equal to 0 = oo and that the other elements of CSn(L) are just the functions
that take the value 0 nowhere and satisfy the inequality h(a) - h{b) ^ n(a - b) for all
a,b e L. In particular, for each h e CS„(L) we have
\h{a) — h{b)\ ^ max|a^ — ЬЦ = \\a — b\\,
j
which implies that h is differentiable almost everywhere. Closely related to this
convolution semiring is the semiring in the following example.
Example 1.26. Pareto order (a = (a\,..., a„) ^ b = (b\,..., b„) if and only if ?, ^ bi
for all i = 1,..., n) defines in R" the structure of an idempotent semigroup. For any subset
А с Шк, by Min(A) we denote the set of minimal elements of the closure of A in Rk. Let
P(Rk) denote the class of subsets А с Шк whose elements are pairwise incomparable,
P(Rk) = {Ac Rk: Min(A) = A}.
Obviously, P(Rk) is a semiring with respect to the operations ? ? ? ?? = Min(A| U A2)
and A1 © A2 = Min(Ai + A2); the neutral element 0 with respect to addition in this
semiring is the empty set, and the neutral element with respect to multiplication is the
set whose sole element is the zero vector in Rk. The semiring P(R.k) is isomorphic to the
semiring of normal sets, i.e., closed subsets N С IR*· such that b e N implies a e N for
any a ^ b; the sum and the product of normal sets are defined as their usual union and
sum, respectively. Indeed, if N is normal, then Min(N) e P(Rk); conversely, with each
A e P(Rk) we can associate the normalization Norm(A) = {a e Rk | Bb e A: a ^ b).
The last two semirings CS„ (L) and P(Rk) are closely connected, as shows the following
proposition that is a specialization of a more general result given in Samborskii and
TarashchanA990).
THEOREM 1.27. The semirings CS„(L) and P(Rk) are isomorphic.
For functions with values in semirings of type II we have the following
THEOREM 1.28. Pseudo-convolution of the first or the second type for g-case (Case II)
/5 commutative and associative operation while G is whole set of reals and * is the usual
addition on E.
We consider now the question when the generalized pseudo-convolution satisfies the
cancellation law, see Pap and Stajner A998). First we give a example which shows that
generally this law is not true.
Example 1.29. Take functions / and g with values in the semiring (]-oo, +00], inf, +)
given by h(x) = ex\ f(x) = e~x; k(x) = e~lx. Then
(f*h)(x)= inf (f(y) + h(z))=0
\+Z=X

Pseudo-additive measures and their applications
1421
and
(k * h)(x) = inf (k(y) + h(z)) = 0,
although f фк.
The following theorems from Zagordny A994) give us some positive results concerning
the (inf, +) pseudo-convolution from Example 1.29.
THEOREM 1.30. If f,k,h:W -> IU {+00} are proper lower semicontinuous convex
functions such that h is strictly convex and
.. 4x)
hm = 00
|.r|->-oo |JC I
then
f -kh =k*h implies f=k,
i.e., * does not have a zero divisor for this particular class of functions.
THEOREM 1.31. If f,k,h:R" -> IU {+00} are proper lower semicontinuous convex
functions. Moreover, suppose h is uniformly convex. Then
f *h =k*h implies f = k.
Now, let consider pseudo-operations represented by its generator g. It is well-known
that Titchmarsh's theorem give us a wide class of functions for which generalized pseudo-
convolution with generator g(x) = ? and * = + does not have a zero-divisor. With the
same method of proof we can prove more general theorem. Let ?|oc be a class of functions
with real values that are Lebesgue integrable over every compact subset of E+.
THEOREM 1.32. Let g be a generator for pseudo-operations ? and ?. If compositions
g о / and g oh are in C\oc, then generalized pseudo-convolution
f*h{x) = g-'(j g(f{t))-g(h(x-t))dt\
does not have a zero-divisor.
Definition 1.33. A function /: [0, +oo[^· [a, b] is an idempotent with respect to the
generalized pseudo-convolution * if / * / = /.
We shall discuss here the idempotents of the pseudo-convolution of first type and
* = +, using Markova-Stupftanova A999). First we remark that if there exists a neutral

1422
?. Pap
element <5?·? with respect to the pseudo-convolution, then it is it idempotent. Also,
the zero-function f{x) = 0, ? e [0, +oo[, is an idempotent of the pseudo-convolution.
We remark that if the pseudo-addition ? has only trivial idempotents, then the pseudo-
convolution has no *-idempotents up to zero-function, which we shall call trivial
idempotent. Therefore we reduce on the case when ? = sup.
THEOREM 1.34. Let ([0, 1], sup, T) be a continuous semiring, where ? is a continuous
Archimedean t-norm and let g be an additive generator of T. Then a function f is
an idempotent with respect to the corresponding pseudo-convolution * if and only if
/@) e {0, 1} and the composite g о / is a subadditive function on [0, +oo[. Moreover,
if /@) =0 and f is a non-trivial ^-idempotent, then limv^o+osup/(.x) = 1.
THEOREM 1.35. (Necessary condition) Let * be a pseudo-convolution based on the
semiring ([a, b], sup, inf). Then if f is an idempotent with respect to the pseudo-
convolution *, then for every с е [a, b] we have
/(c)+/(c)C/«\
where fu"> = [x e [0, +oo[: f(x) ^ c}.
(Sufficient condition) If for every с е [a, b] we have
f(c)+f(c) = /"¦',
then f is an idempotent with respect to the pseudo-convolution *.
1.5. The Riesz type theorem
The Riesz theorem in functional analysis establishes a one-to-one correspondence between
continuous linear functionals on the space of continuous real functions on a locally
compact space X vanishing at infinity and regular finite Borel measures on X, see Diestel
B002), this Handbook. Analogous correspondence exists in idempotent analysis.
We restrict here our consideration to the case of the semiring ? = (min, +) using
Kolokoltsov B001). Proofs, generalizations and references could be found in Kolokoltsov
and Maslov A997).
Idempotent measures are absolutely continuous, i.e., any such measure can be
represented as the idempotent integral of a density function with respect to some standard
idempotent measure. Let us formulate this fact more precisely. Let Co(X, P) denote the
space of continuous functions /: X -> ? on a locally compact normal space X
vanishing at infinity, i.e., such that for any ? > 0 there exists a compact set К с X such that
d@,f(x)) < ? for all ? e X\K. The topology on Co(X, P) is defined by the uniform
metric d(f, g) — supx d(f(x), g(x)). The space Co(X, P) is an idempotent semi module. If X
is a compact set, then the semimodule Co(X, P) coincides with the semimodule C(X, P)
of all continuous functions from X to A. The homomorphisms Co(X, ?) —* ? will be
called pseudo linear functionals on Co(X, P). The set of pseudo linear functionals will be
denoted by Cq (X, P) and called the dual semimodule of Co(X, P).

Pseudo-additive measures and their applications
1423
THEOREM 1.36. For any m e C^(X. P) there exists a unique lower semicontinuous and
bounded below function f : X -> ? such that
m(h) = Mf(x)Qh(x) for every heCo(X,P). C)
Conversely, any function f: X -*¦ ? bounded below defines an element m e C^(X, P)
by C). At last, the functionals m^ and mj2 coincide if and only if the functions f\ and /2
have the same lower semicontinuous closures; that is, CI f\ = CI /1, where
(С1/)(д0 = supf^U): ??/,?? C(X, P)}.
We can define an idempotent measure ?/ on the subsets of X by ? f(B) = inf{/(.v): ? e
?}. This measure is ?-?-decomposable. Equation C) specifies a continuation of m/ to
the set of ?-valued functions bounded below. As in the classical analysis, we say that such
functions are integrable with respect to the measure ? f and denote the values taken by m j-
on these functions by the idempotent integral
r®
mf(h)= / h{x)dvf(x) = mif(x)Qh(x).
J ? v
This integral is just the pseudo-integral, which was defined as the limits of the
corresponding idempotent Riemannian (or Lebesgue) sums. Theorem 1.36 is equivalent
the statement that all idempotent measures are absolutely continuous with respect to the
standard idempotent measure m\.
Let Co(K") be the space of continuous functions /: R" -> ? (? is of type (I)(i), (III))
with the property that for each ? > 0 there exists a compact subset К с Ш" such that
d@, infv6K„^ /(*)) < ?, with the metric D(f, g) = supv d(f(x), g(x))- Let C0S(R") be
the subspace of Co(K") of functions / with compact support supp0 = [x: f(x) ? 0}.
One can develop the concept of weak convergence and the corresponding theory of
generalized functions. In particular, for X = R", simple delta-shaped sequences can be
constructed of smooth convex functions; e.g., ?™,?'+(?) is the weak limit of the sequence
(fn)neN defined by f,(x) =n(x - yJ. Therefore each linear functional (or operator) on
Co(K") is uniquely determined by its values on smooth convex functions.
The dual semimodul (Co(K"))* is the semimodul of continuous pseudo-linear P-
valued functionals on Co(K") (with respect to pointwise operations). Analogously the
dual semimodul (Cq((E"))* is the semimodul of continuous pseudo-linear P-valued
functionals on Cqs(E") (with respect to pointwise operations).
We shall need the following representation theorem, which is a consequence of
Theorem 1.36, see Kolokoltsov and Maslov A997).
THEOREM 1.37. Let f be a function defined on R" and with values in the semiring ? of
type (I)(i) or (III) and afunctional mj-: CqS (?") -> ? is given by
r®
m/(h)= fOdmh=M(f(x)Oh(x)).

1424
?. Pap
Then
A) The mapping f н» m/ is a pseudo-isomorphism of the semimodule of lower
semicontinuous functions onto the semimodule (C'0S (R"))*·
B) The space Cq(R") is isometrically isomorphic with the space of bounded functions,
i.e., for every m ft,m f2 e Cq (R") we have
suprf(/i (x), f2(x)) = sup{d(mfl (h), mh{h)): h e C0(R"), D(h, OK 1}.
.V
C) The functionals m ft andmj\ are equal if and only if Cl/i = CI /i, where
C\f(x) = $ир{ф(х): ? e C(R"), iA < /}·
We remark that Theorem 1.37 is not valid for the semimodule C(R, P) of bounded
continuous functions defined on R.
2. Applications
2.1. Probabilistic metric spaces
We shall show that the basic notion of the theory of probabilistic metric spaces, the triangle
function, is based on the pseudo-convolution of the first type.
A distance distribution function is a function F, F:[0,+oo] -> [0, -boo], such that
F@) = 0 and F(+oo) = 1 and it is left continuous on ]0, +oo[. The family of all distance
distribution functions is denoted by ?+. Now we shall introduce an operation ? on ?+.
DEHNITION 2.1. A triangle function ? is a binary operation on ?+ that is commutative,
associative and non-decreasing in each place, and has ?? as identity, where
. . Г 0 if jc = 0,
[1 if 0 < ? < +oo.
The following definition introduce the basic space (see Schweizer and Sklar A983)).
DEHNITION 2.2. A probabilistic metric space is a triple (?,?,?) where ? is a
nonempty set, T:M2 -> ?+ is given by (p,q) м> Fpq, ? is a triangle function, such
that the following conditions are satisfied for all p, q. r e M:
(i) PpP =??;
(?) Fpq ^?0??? p^q;
(iii) Fpq = FqP\
(iv) Fpr ^T(Fpq,F4r).
Let F, ? e ?+ and u, ?, ? e [0, oo]. Taking for triangle function ? = ??, where
tt(F, H){x) = sup{T(F(M), H(v)): и + ? =?],

Pseudo-additive measures and their applications
1425
(pseudo-convolution of the first type with respect to (max, T) and * = +) for a left
continuous ?-norm ? we obtain a special important probabilistic metric space, the Menger
space. The fact that function zj is triangle function yields from Theorem 1.24 (also, see
Schweizer and Sklar A983)). It is important to obtain rich source for different triangle
functions which would enable the construction of new probabilistic metric spaces. One of
the useful construction goes in the following way.
Definition 2.3. С is the class of all binary operators L on [0, +oo[ which satisfy the
following conditions
(i) L maps [0, +oo[2 onto [0, +oo[;
(ii) L is non-decreasing in both coordinate;
(iii) L is continuous on [0, +oo[2 (except possibly at the points @. +oo) and (+oo, 0)).
We introduce now, using the family С the following binary operation on ?+.
Definition 2.4. For a ?-norm ? and L e ?, a function zT L defined on (?+J and with
values in [0, +oo[ is given by
tt.l(F, G)(x) = sup{r(F(M), G(u)): L(u, v) = x].
In a special case L(x, y) = ? + у, we obtain ??? = *T·
The following theorem guarantees under mild conditions the triangularity of zj.l (see
Schweizer and Sklar A983)).
THEOREM 2.5. If ? is left continuous t-norm and Lfrom С is commutative, associative,
has 0 as identity and satisfy the condition
if и ? < mi and v\ < ит, then L(u\, v\) < L(ui, vj), D)
then zt.l is a triangle function.
Condition D) is weaker than strictly increasingness of L in each place. So min and max
satisfies it, although they are not strictly increasing in each place.
Example 2.6. zT max is a triangle function for any left continuous ?-norm T.
The left continuity of Г in the preceding theorem is only sufficient but not necessary
condition.
It can be proved in a similar way that for ? = min and ? = S. where S is continuous
?-conorm that functions
zs(F, H)(x) = inf{S(F(ii). H(v)): и + ? = ?}.
zs.l(F, H){x) = inf{S(F(ii), H(v)): L(u. v)=x]
are triangular functions, where the binary operation L has all properties from previous
theorem and F, ? e ?+.

1426
?. Pap
Example 2.7. rs.mjn is a triangle function for any continuous ?-conorm S.
Example 2.8. Let F, ? e ?+ and ? e [0, oo]. If we consider convolution from
Remark 1.20, i.e.,
F*tf@) = 0 and F*tf(oo)=l,
F*H(x)= J F(x-t)dH(t) for;te]0,oo[,
which has identity ??, we have obtained an triangle function.
Remark 2.9. The g-case of pseudo-convolution of functions F and ? is not an triangle
functions due to the lack of identity for that particular operation of generalized pseudo-
convolution.
For a general method for solving nonlinear equations, i.e., fixed point theory, in
probabilistic metric spaces see Hadzic and Pap B001).
2.2. Fuzzy numbers
Generally, a fuzzy number is a fuzzy subset (represented by its membership function
? a :R -> [0, 1]) of the real line R, see Dubois et al. B000). That is the most basic
definition but in practice some less general definitions are being used. If A is a fuzzy
set, its membership function ?? (?) will be identified with A{x).
Definition 2.10. Let A be a fuzzy subset of the real line. A is a fuzzy number if it is
normalized and convex, i.e., if A(x) = 1 for some ? e R and for all x,y,z eR such that
? < у < ?, it is A(y) ^ min(A(;c), A(z)).
If A is a fuzzy number, then set Aa = {x e R: A(x) ^ a) for a e ]0, 1] is the ?-cut of
fuzzy number A.
We will give a few examples of the fuzzy numbers that are usually used in applications.
Example 2.11.
(i) An LR-fuzzy number A = (a,a,fi)L.R is a function from reals into the interval [0, 1]
satisfying
A(x) =
(a-x\
LI ? ??? — ?^?^?,
. 0 otherwise,

Pseudo-additive measures and their applications
1427
where a e R is the center, a > 0 is the left spread, ? > 0 is the right spread of A, L
and R are non-increasing and continuous functions from [0,1] to [0, 1] satisfying
L@) = R@)= 1 andL(l) = tf(l)=0.
(ii) An LR-fuzzy interval A = (/, r, a, /S)z..«. for / ^ r, is a function from reals into the
interval [0, 1] satisfying
if ? ^ / — a,
if ?-? ? ? ? I,
ifl^x^r,
ifr^x^r+?,
ifr + ???,
where L and R are non-increasing and continuous functions from [0, 1] to [0, 1]
satisfying L(Q) = R@) = 1 and L(\) = R(\) = 0, / and r are called the left and
right peak, a > 0 is the left spread, ? > 0 is the right spread of A. Specially, for
L(x) = R(x) = 1 — ? trapezoidal fuzzy interval is obtained.
The arithmetical operations with fuzzy numbers are based on Zadeh's extension
principle (see Юетет et al. B000a)): Let ? be an arbitrary but fixed ?-norm and * a
binary operation on R. Then the operation * is extended to fuzzy numbers A and В by
A*TB(z)= sup T(A(x),B(y))
.V*Y=:-
for ? e R- Some usual operations with fuzzy numbers are following: Addition is obtained
for * = +: ?®? B(z) = supA.+v=. T(A(x), B(y)). Subtraction for * = -: ? ?? ? (?) =
supv_v=- T(A(x), B{y)). Multiplication for * = ·: A QT B(z) = supv.v=. T(A(x), B(y)).
Quotient for * = /: А Сдт B(z) = supv/v=_ T(A(x), B(y)). Multiplication and addition are
generalized pseudo-convolutions of the second type based on semiring ([0, 1 ], max, Г) but,
due to the fact that subtraction of reals and quotient of reals are not commutative operations,
subtraction and quotient of fuzzy numbers are not generalized pseudo-convolutions.
As a result of a Theorem 1.24 addition and multiplication of fuzzy numbers for ? being
left continuous ?-norm are commutative and associative operations.
In practical computation it is natural to require preserving the shape of fuzzy numbers
during the arithmetic operations. Preserved shape allows us to control the resulting spread
while working with LR-fuzzy numbers or LR-fuzzy intervals (see Dubois and Prade
A988), Markova-StupfianovaA997), Mesiar A997)). In the sense of generalized pseudo-
convolution, preserving the shape means that generalized pseudo-convolution is closed
operation on certain family of functions. The following corollaries that result from some
Propositions from Dubois and Prade A988) will give us the resulting spreads for addition
based on the strongest Гм and the weakest Tq triangular norm.
A(x) =
@
L
1
R
lo
1-х
? — r

1428
?. Pap
COROLLARY 2.12. Consider ? LR-fuzzy intervals A, = (/,, r,, a,, Pi)l.r, г = 1,...,и,
then the sum ф"=| A, based on Гм (pseudo-convolution of the second type on semiring
([0, 1], 5"m, 7m)). where 5"м = max, is given by
II ? II II II II \
??·= ?''.?"·?«'·??- · E)
/=1 \?=? ?=? ?=? ?=? I LR
COROLLARY 2.13. Consider n LR-fuzzy intervals
A, =(//, r/, ?,-, Pi)l.r, /" = 1, — и,
?/геи ?/ге яиш ф"=. A, based ои Го (pseudo-convolution of the second type on semiring
([0, 1], max, To)) is given by
II ? II II \
? At'= ??/(' ?r''' m'ax ?/'max a ) ^^
1 = 1 \i'=l i=l '~ '~ / LR
Formulas E) and F) give the limits for addition of ? fuzzy numbers based on an
arbitrary ?-norm ? ((®T)"=iAj). The limit ?-norm based addition preserves the shape
of incoming fuzzy numbers. However, for some special shapes, there exist some special
?-norms preserving that shape (see Klement et al. B000a)).
Next interesting issue in fuzzy number theory is whether fuzzy numbers are comparable.
Set of fuzzy numbers can be ordered partially. The idea is to extend operations min and
max in the sense of Zadeh's extension principle:
???(?,?)(?)= sup min(A(jt),B(}')),
min(.v.v)=:
MAX(A,B)(z)= sup min(A(;c), B(y)).
max(.t.\)=c
Since min and max are continuous operations MIN(A, B) and ???(?,?) are fuzzy
numbers and are pseudo-convolutions of the second type for ? = max and ? = min.
Properties of new operations MIN and MAX are commutativity, associativity, existence of
idempotent element, absorption, distributivity. The first two properties for MIN and MAX
follow from Theorem 1.24 and the rest of those properties is easy to prove.
2.3. Information theory
The information theory based on the notion of a probability has been introduced in 1948 by
Wiener and Shannon (see for more details Sander B002) in this Handbook). Let (?, ?, p)
be a probability space. The mapping J : ? -> [0, oo] is called an information measure if
for an event A e ? we have J (A) = — logp(A).
Axiomatization of information measure has been introduced in 1967 by Kampe de Feriet
and Forte (see Klement et al. B000a)).

Pseudo-additive measures and their applications
1429
Definition 2.14. Let (?, 27) be a probability space. Then the following mapping
J : 27 -> [0, +oo] is called an information measure if it satisfy the following conditions:
A1) 7(.?) = 0, 7@) = +oo;
A2) ? с A implies J(B) ^ 7(A);
A3) if A and ? are independent in the sense of Banach-Marczewski, then J (? ? ?) =
7(A) + 7(?);
A4) J (A U ?) = FG(A), У (?)) whenever ? ? ? = 0 for some two-place function F;
A5) ./«?? A<) = linW+DC ./«?? A/)·
Remark 2.15. Two events A and ? are independent is Banach-Marczewski style if they
are contained in independent sub-?-algebras ? and ?? of 27, respectively. Two sub-?-
algebras ? and N oi ? are independent if and only if for all 0 ^ A e ? and 0 ? В е Л/"
is АП ? ^0.
The function F that appears in the previous definition is called a composition law.
Usually, for F are required:
A) continuity,
B) symmetry (F(x, y) = F(y,x)),
C) associativity (F(F(jc, }>), z) = F(jc, F()·, z))),
D) monotonicity (F is non-decreasing in both components),
E) neutral element (F(jc, oo) = jc).
Schweizer and Sklar A969) proposed that information J should be treated as
a random variable on [0, oo] instead as a non-negative real number. The idea is
similar to that of Menger's generalization of metric spaces. So, we have distribution
function K(A):[0,oo] ->¦ [0, 1] that is a left-continuous non-decreasing mapping with
K(A)@) = 0. Now, let 7(A) be a random information on [0, oo] for each AeA, K(A, x)
the probability that the information given by the realization of A is less than x. Axioms
(Il)-(I5)can be rewritten in the style of Schweizer and Sklar A969). The additivity axiom
has been translated into the following form
A4') for ?, ? e ? that ? ? ? ? 0 it is
K(Al) ?) =?(?(?), ?(?)),
where ? is a continuous, associative, commutative, non-decreasing two-place
function on the system of distributions on [0, oo], with neutral element 0 and
annihilator 1.
Let S be a continuous ?-conorm. Then functions
?5(?, G,x) = S(H(x),G(x)), ts(H.G,x)= sup S(H(u),G(v)),
where и, ?, ? e [0, +oo] and H, G are distributions on [0. +oo], fulfill the required axiom.
The second one is an pseudo-convolution of the first type with respect to (max, S).

1430
?. Pap
2.4. System theory
Pseudo-convolutions are as important tools in the system theory as the classical convolution
was (see Baccelli et al. A992), Maslov and Samborskij A992)).
Let us consider arbitrary set of events. There is a numbering mechanism which assigns
a number to each event in order of these events taking place. It starts from an initial finite
value ко e ? (where ? is set of all integers) which can, also, be negative value. There is,
also, a special type of event that corresponds to ticks of an absolute clock. These ticks are
numbered in increasing order, starting from some value ?? e Z.
Mapping from ? into ? such that graph is the set of all pairs (event number, clock value)
is dater associated with a certain type of event.
Definition 2.16. A signal и is a mapping from (R,+, ·) into ([—oo, oo], max, +).
When a signal is a nondecreasing function it is called a dater.
Dater d may not be strictly increasing since events of the same type may occur
simultaneously. The convention is that
'
d(k) =
oo when к < ко.
+oo event never took place,
any finite value otherwise.
Domain of d may be extended by setting d(—oo) = —oo and d(oo) = oo. The semiring for
the range of d is ([ —oo, oo], max, +). Now, pseudo-addition is the conventional pointwise
maximum: (d\ ? di)(k) = max(d\(k),d2(k)). Pseudo-multiplication is: (d\ ? ат)(к) =
suP/ez(^i @ +di{k - /)) and that is the pseudo-convolution of the second type (integral is
taken over whole group Z) for ? = max, ? = + and * = +.
One of elementary systems is shift ?4. This system maps inputs to outputs according to
the equation у (к) = и (к - g) for all к, where и is a signal.
Definition 2.17. A linear system S is called shift-invariant if it commutes with all shift
operators, that is if for all и and for all c, S(rc(u)) = rc(S(u)).
Interesting part is that in the shift-invariant case the input-output relation can be
expressed as follows: y(k) = supseU(h(k - s) + u(s)). That is, the pseudo-convolution of
the second type (for ? = max, ? = + and * = +) and it plays the role of the convolution
in conventional system theory.
2.5. Optimization anamorphism with the probability
First, we state the problem of finding the maximum of the utility function
f\(X\)+f2(X2)+--- + fN(XN),
on the domain D = {(x\,X2, ¦ ¦ -,xn)'· x\ +X2 ? b*yv =¦*· *i ^ 0, ? = 1,..., ?].

Pseudo-additive measures and their applications
1431
We can rewrite this problem in such manner that it represent generalized pseudo-
convolution of the first type applied on N functions. Semiring that is used for this particular
problem as a range of this pseudo-convolution is ([0, oof, max, +)¦
This type of problem often occurs in the mathematical economics and operation research
(see Baccelli et al. A992), Bellman and Dreyfus A962)) and it can be solved by applying
the pseudo-Laplace transform, the pseudo-exchange formula and inverse of pseudo-
Laplace transform (see Pap and Ralevic A998)).
DEFINITION 2.18. The pseudo-character of the group (G, +), GcM" is a continuous
(with respect to the usual topology of reals) map ? : G -> [a, b], of the group (G, +) into
the semiring (fa, b], ?, ?), with property
Их+У)=Их)ОИУ), x.yeG.
The map ? = 0 is trivial pseudo-character.
The forms of pseudo-character in the special cases can be found in Klement et al.
B000a), Pap and Ralevic A998). Interesting case for us is ([a,b], ?, ?) = ([0, oof, max, +)
and then pseudo-character has the form ?(?, с) = с ¦ ?, for each с е R, where we have taken
the dependence of the function ? also with respect to the parameter c.
DEFINITION 2.19. The pseudo-Laplace transform ?e(/) of a function / e B(G, [a,b])
is defined by
(?e/)(?)(z)=i Hx.-z)Odmf(x).
JGn|0,3C|"
where ? is the pseudo-character.
When at least pseudo-addition is idempotent operation we can consider the second type
of pseudo-Laplace transform:
(?®f)(№)= f H{x,-z)Odmf(x\
Jg
i.e., pseudo-integral has been taken over the whole G.
The forms of the pseudo-Laplace transform are known for three special cases and can
be found in Pap and Ralevic A998). We shall restrict to the special important case is
([0, oof, max, +) and then pseudo-Laplace transform has the following form
(?max/)(z) = sup(-«+ /(*)),
.v^O
and in и-dimensional case
(?max/)(z)= sup (-г,*, z„x„+fW).
л,^0./=| /?

1432
?. Pap
The important result that has been proved in Pap and Ralevic A998) (see Kolokoltsov and
Maslov A997)) is the pseudo-exchange formula, that transforms operation of generalized
pseudo-convolution of the first or second type to pseudo-multiplication:
??(/?*/2) = ??(/?)???(/2),
where functions f\, /2 belong to B(G. [a, b]).
In order to solve the problem from the beginning of this section we need the next theorem
(see Pap and Ralevic A998)).
THEOREM 2.20. //?*(/) = F for semiring ([0, oof, max, +), then there exists (?e)~',
inverse of pseudo-Laplace transform, and it has the following form:
((C®y\F))(x)=mfo(xz + F(z)).
Now, let f(x) = maxD (/1 (x \) + /2 (??) ? l· /n (*n))- This is the pseudo-convolution
of the first type of functions /1, · · ·, /yv with respect to (max, +)· Applying the pseudo-
Laplace transform (for the case ? = max and ? = +), the pseudo-exchange formula and
inverse of pseudo-Laplace transform we obtain
N
Cmm(f) = Cm^(fdxu + ---+fN(XN))=^Cm'd^f^
and finally the solution
/(*)= i(?™*)-' ^?m-(/0J(^) = mini^ + ^?max(/,)(z)j.
There is an interesting correspondence principle between probability theory and
stochastic processes on the one hand, and optimization theory and decision processes on the other
hand (Baccelli et al., 1992; Kolokoltsov and Maslov, 1997). Then the Markov causality
principle corresponds to the Bellman optimality principle. For the development of the
theory of optimization processes in the spirit of stochastic processes, where the notions of
optimization martingales and optimization measurability play the main role, see Akian et al.
A998), Kolokoltsov B001), Kolokoltsov and Maslov A997), Puhalskii B001). Formally,
in the case of (space and time) homogeneous processes, the connection between Markov
and Bellman processes is given by the Cramer transform (—?m,n) о logo?+ (Akian et al.,
1998). In general, Bellman processes present a kind of semiclassical approximation to the
Markov processes (Kolokoltsov and Maslov, 1997; Kolokoltsov, 2001; Puhalskii, 2001).
We are not going into details in this direction, but we shall explain here only the
connection between probability and idempotent measures, which is given by the large deviation
principle. To this end, let us recall first the following general definition. Let ? be a
topological space and ? be the algebra of its Borel sets. One says that a family of probabilities

Pseudo-additive measures and their applications
1433
(??), ? > 0, on (?, ?) obeys the large deviation principle if there exists a rate function
I: ? -> [0, oo] such that
A) /is lower semi-continuous and i?(i = {a>e ?: ? (?) ? a} is a compact set for every
? < oo,
B) - limsup^oelogPftC) ^ infw6c /(?) for every closed set С С ?,
C) -liminfj^oelogPHi/) ^ infw6(y /(?) for every open setU с ?.
Obviously m(A) = ????6? ? (?) is then a ?-min-additive measure with respect to the
operation ? = min function on ?. Therefore, it is naturally to generalize the previous
definition in the following way (Akian, 1999; Puhalskii, 2001). For any Borel set A let
Pout(A) = limsupelogPf(A). P,n(A) = \immfs\ogP?(A).
One says that (??)?>? obeys the weak large deviation principle, if there exists a positive
idempotent measure m on (?, ?) such that
A) there exists a sequence (i2„)„6N of compact subsets of ? such that ??(?^) -> 0 =
+oo as ? -> oo, where С stands for the complimentary set of C,
B) m(C) ^ -Pout(C) for every closed С С ?,
C) m(t/) ^ -P'n(U) for every open U (??.
Applying Theorem 1.36 and its generalizations one can prove (see details in Akian
A999), Puhalskii B001)) that the large deviation principle and its weak version are actually
equivalent for some (rather general) "good" spaces ?. One can obtain also an important
correspondence between the tightness conditions for probability and idempotent measures.
2.6. Pseudo-linear operators
Every pseudo-linear operator on the semimodule of functions with values in a semiring
with idempotent addition is an integral operator, see Maslov and Samborskii A992),
Kolokoltsov B001), Pap and Teofanov A998). Let ? = (min, +) and X a locally compact
normal space. An P-linear continuous operator on (or homomorphisms of) semimodules
Co(X) is continuous mapping ? : Co(Y) -> Q(X) such that
B(a ? h ? с ? g) = a Q B(h) ? с ? B(g)
for every а, с е A and h, g e Со(У).
THEOREM 2.21. Let B:Co(Y) -> Co(X) be a continuous ?-linear operator. Then
there exists a unique function b:XxY—>P (called the integral kernel of B) lower
semicontinuous with respect to the second argument and such that
(Bh)(x) = mfb(x,y)Oh(y). G)
у
This result is actually a direct consequence of Theorem 1.36. Properties of the kernel
b(x, y) are known, which are necessary and sufficient for G) to define a continuous linear
operator. One can also specify the properties of b under which the corresponding operator

1434
?. Pap
acts on the spaces of continuous functions with a compact support, or on the space of
functions having a limit at infinity. Moreover, one can describe the properties of the kernel
b that ensures that the corresponding operator is compact (or completely continuous), in
the usual sense, i.e., it carries each set bounded in the metric to a precompact set. One easily
sees that if X is compact, then for any continuous b, G) defines a completely continuous
linear operator.
The group of invertible linear operators is very slender. For example, any invertible nxn
matrix with elements in ? is the composition of a diagonal matrix and a permutation of
the standard basis of the free semimodule P". More generally, the following result holds,
see Kolokoltsov B001).
Theorem 2.22. Let
B:C0(Y)^Co(X) and D:C0(X) -»· C0(Y)
be mutually inverse ?-linear operators. Then there exists a homeomorphism ? : X —» У
and continuous functions ?: X -» ? and ?:? -*¦ ? nowhere assuming the value 0 such
that ?(?) ? ? (? (?)) = 1 and the operators В and D are given by the formulas
(Bh)(x) = <p(x)Oh(a(x)), (Dg)(y) = V(>0 ©·?(<*"' (>0)
for every heCo(Y), geCo(X)-
The spectrum of general compact linear operators in idempotent analysis is also very
slender.
THEOREM 2.23. Let X be a compact set, and let В be a P-linear operator on C(X, P)
with a continuous integral kernel b, which is nowhere equal to 0 = oo. Then В has a
unique eigenvalue ? ф 0 and a (not necessarily unique) eigenfunction h e C(X, P), which
is nowhere equal to 0, such that Bh = ?©/? = ?+h.
The application of this result to the optimization theory is based on the following two
corollaries.
COROLLARY 2.24. Let an operator В satisfy the conditions of Theorem 2.23, so that its
eigenvalue ? is unique and the eigenvector h is nowhere equal to 0. Let f ^ h + с with
some constant с Then
Bmf(x)
lim =?.
m—*oc m
COROLLARY 2.25. Under the conditions of Corollary 2.24, if ? > 0, then the Neumann
series
/0?(/)??2(/)?··· (8)
is finite, that is, is equal to the finite sum ? ? B(f) ? · · · ? Bk (f)for some к.

Pseudo-additive measures and their applications
1435
The ?-series (8) represent the solution of the equation g = Bg® f which describes for
the stationary optimization problem corresponding to the Bellman operator B.
We consider now operators homogeneous in the sense of the semiring ? = R U {+00},
i.e., operators В on function spaces such that
В (a +h)=a+B(h)
for any number a and any function h. The theory of such operators is closely related
to game theory and there is an analog of the eigenvalue theorem for such operators
(Maslov and Samborskii, 1992). This analog is applied to the construction of the turnpikes
in stochastic games. For simplicity, we only consider the case of a finite state space
X = {1,..., ?} in detail, using Kolokoltsov B001).
We define an antagonistic game on X. Let pij (?, ?) denote the probability of transition
from state i to state j if the two players choose strategies a and ?, respectively (a and ?
belong to some fixed metric spaces), and let bis (?, ?) denote the income of the first player
from this transition. The game is called a game with value if
? ?
minmaxVV^a, fi)(h> +bjj(a. ?)) = max пипУ^ ;>;,((*, /S)(/?y + bij(a, ?))
с/ ? ? ?
for all у e R". The operator В :R" -> R" such that B,(y) is equal to (9) is called the
Bellman operator of the game. Applying the dynamic programming method Bellman and
Dreyfus A962), we can prove that the value of the A:-step game defined by the initial
position i and the terminal income h e R" of the first player exists and is equal to Bkt (h).
It is obvious that the operator В has the following two properties:
B(ae + h) = ae + B(h), for every a e R, h e R", e = A,..., 1) e R", A0)
\\B(h)- B(g)\\ < ||/i-g||, for every h,geR". A1)
where \\h\\ = max|/?'|. These two properties are characteristic of the game Bellman
operator. Namely, as it was proven by Kolokoltsov in Maslov and Samborskii A992), each
map satisfying A0) and A1) can be represented in form (9). Another characterization
of homogeneous nonexpansive maps was obtained in Crandall and Tartar A980): if
S:R" -^ R" satisfies A0), then В is nonexpansive, i.e., it satisfies A1), if and only if
it is order-preserving.
In the investigation of homogeneous operators, it is useful to define the quotient space ?
of the space R" by the one-dimensional subspace generated by the vector e = A,..., 1).
Let ?: R" -> ? be the natural projection. The quotient norm on ? is defined by
II Л(/?)|| = inf \\h +ae\\ = - (max/2^ - minhj).
" " aeM 2\ j j '
It is clear that ? has a unique isometric (but not linear) section S: ? -> R". The image
5"(?) consists of all h e R" such that max; h' = - min^ h'. By virtue of properties A0)
and A1) of ?, the continuous quotient map ?.? -> ? is well defined.

1436
?. Pap
Further, some additional technical assumptions on the transition probabilities:
3<5 > 0: V/,;, a 3/3: ?^(?,?)^?, A2)
3i > 0: Vi,./ 3m:V<*,/3: ????(?,?)>?, pjm(a,fi)>S. A3)
Let all |/>iy(a, ?)\ be bounded by some constant С The proof of the following simple
fact can be found in Kolokoltsov B001), Maslov and Samborskii A992) and Kolokoltsov
and Maslov A997).
PROPOSITION 2.26. (i) If A2) holds and ? < \/n, then В maps each ball of radius
R ^ C<5 centered at the origin into itself.
(ii) If A3) holds, then
\\B(H)-B(G)\\ <c A -щн -G\\, for every H,Ge0.
As a direct consequence of this proposition, the fixed point theorems, one obtains the
following result.
THEOREM 2.27. (i) //A2) holds, then there exists a unique ? e R and a vector h e K"
such that B(h) = ? + h and for all g e Ш." we have \\B'"g — mk\\ ^ \\h\\ + ||й - g||, and
lim,n^oo „ =?.
(ii) If A3) holds, then h is unique {up to equivalence), and
lim Son(B'"(g)) = Son(h), for every g e E".
3. Idempotent integral as limit of ^-integrals
The properties of decomposable measures and the corresponding integrals are closely
related to the properties of the pseudo-additions on which they are based. Generally, there
is quite different behavior between two classes of decomposable measures: one which
is based on generated pseudo-addition (g-case, see Pap A993a, 1995)) and other based
on idempotent pseudo-operation (sup and inf, see Baccelli et al. A992), Kolokoltsov and
Maslov A997), Maslov and Samborskii A992), Pap A995)).
3.1. Pseudo-operations
Starting from a semiring of (II) type ([a, b], ?, ©), where
x®y = g~]{g(x)+g(y)) and xQy = g'](g(x)g(y))
with a generator g: [a, b] -> [0, +oo]. For ? e ]0, oof the function gA (the function g on
the power ?). Then gk is a generator of the semiring (fa. b], (Bi, ©a), where
х®ьУ = №)~1(8к(х)+8к(У)) A4>

Pseudo-additive Measures and their applications
1437
and
x ?? У = (/)"' (g4x)gHy)) =xOy< A5)
since (gk(x))~i = g~\x]/A). Hence gA is a generator of the semiring ([a,b], ®?, ?).
Example 3.1. Let g: [-oo, +oo] -> [0, +oo] be given by g(x) = ex. Then
* ? у = ln(ev + ey) and xQy=x+y
are operations defined on [-oo, +oo]. We have 0 = -oo and 1 = 0. We have for gA(x) =
ln(eAv +ek>)
? ?? у = and ? О у = ? + у.
?.
We have by Mesiar and Pap A999).
THEOREM 3.2. Let g: [a,b] -> [0, -foo] be a strictly decreasing generator ofthe semiring
([a, b], ?, O) of the (II) type and gA the function g on the power ? e ]0, oof. Then gk is
a generator of the semiring {[a, b], ??, ?) and for every ? > 0 and every (x, у) е [a, b]
there exists ?? such that
\x ?? У - inf(;c, y)\ < ?
for all ? ^ ??. For g increasing the same result holds for sup.
The same result holds for ? -> — oo changing only the monotonicity of g.
COROLLARY 3.3. By the same supposition as in Proposition 3.2 every semiring
([a, b], inf, O), where © is generated by g, and a sequence {?,,} with ?„ -> oo as ? -> oo
is the limit of the sequence of semirings (fa, b], ®x„, ??„) ?* ?/? -*¦ oo. w/геге ??„ and
Qx„ are given by A4) and A5), respectively.
We can partially obtain also a semiring of the type (III) as a limit of a family of semirings
of type (II) (Mesiar and Pap, 1999).
Example 3.4. We can partially obtain the sup-inf case (on interval [0, +oo]) if we put
h„(x)= (exP(-(n/*)") forjr^n,
[ x/{ne) for ? > n.
Then we take the sequence of semirings {([0, +oo]. ?„, ?,?)}- where
x®ny = h~^(hn{x) + hn{y)) and ? Q„ у = h~\h„{x)hn{y)).

1438
?. Pap
Taking ? -> -boo we obtain ? ®„ у -» sup(;c, v) and for all ?, у finite ? ©„ у -> inf(jc, у).
If X or у is oo and the other is non-zero then ? ? у = oo. The continuous extension of the
limit O,, on open square is just inf. Unfortunately, there cannot exist generators leading to
inf on the whole square as always we have for X non-zero oo ? ? = oo.
3.2. Measures and integrals
First, remark that if m is a ?-decomposable measure, where ? has a generator g, then
g о m is an additive measure. Denote by ? the usual Lebesgue measure on R. The following
problem will be investigated:
For ®„ generated by g„ (we have ®„ -> sup) let m„ be ?-?„-decomposable measures.
Let m„ -> w. Is then w sup-decomposable measure?
We have m„ = g~' ? ?„, where ?„ is a ?-additive measure. Note that then ?„ = g„om„.
Suppose that ?„ is absolutely continuous with respect to ?. Denote by hn the Radon-
Nikodym derivative
_ rf(g„ ow„)
?/?
Then
™,,(A) = g~ll / /?,,^?
where the usual Lebesgue integral is taken into account.
First we shall give an example.
Example 3.5. Let g: [a, b] -> [0, +oo] be an increasing continuous generator of the
semiring ([a, b], ?, ?) of type B). Let m be a ©-decomposable measure on a measurable
space (?, ?). Let a family of pseudo-additive measures {m-A} be given such thatgAow^ =
g о m for all ? e ]0, +oo[. Then
mk=g-]((g°m)]l'A)
is a ®i-decomposable measure, ? e ]0, +oo[. Taking ? -> +oo we obtain m\^> M, where
? (A)-.
where g '(!) = !.
0 ifw(A) = 0,
1 ifw(A)>0,
Example 3.5 shows that the resulting sup-decomposable measure can be reduced to a two-
valued set function. However, more general sup-decomposable measures can be obtained,
too (Mesiar and Pap, 1999).

Pseudo-additive Measures and their applications
1439
THEOREM 3.6. Let m be a sup-decomposable measure on ([0, +oo], ??([0, +oo]), where
m(A) = esssup((p(x): ? e A),
?
where ?: [0, +oo] -> [0, +oo] is a continuous densit): Then for any generator g there exist
a family {mi} of ^^-decomposable measures on ([0, +oo), B). where ®A is generated
by gk, ? e ]0, +oo[, such that
lim mi = m.
?—>+oo
Remark 3.7. Note that on the intervals sup = ess sup. For the case of arbitrary Borel
set, e.g., one point set, we need in the transforming from interval to arbitrary Borel set an
essential supremum.
Each sup-decomposable measure which is limit of g-measures (?-measures, where ? is
generated by g) fulfills the countable chain condition (CCC). Theorem 3.6 shows that some
special members of the class of sup-decomposable measures fulfilling CCC are limits of
g-measures.
Problem: is this true for every sup-decomposable measure fulfilling CCC?
Then density ? need not be continuous. Theorem 3.6 gives only sufficient conditions
and for measurable ? it was not yet shown, at least by our approach. We can formulate
three problems:
(a) Is Theorem 3.6 true for every measurable density ??
(b) Is any CCC sup-decomposable measure limit of g-measures?
(c) Is any CCC sup-decomposable measure representable as ess supM ? for some
measurable density ??
It is obvious that (a) and (c) would imply (b).
For any continuous function / : [0. +oo] -> [0, +oo] the integral
/
sup
f Qdm
can be obtained as a limit of convenient g-integrals (Mesiar and Pap, 1999).
THEOREM 3.8. Let ([0, +oo], sup, ©) be a semiring with О generated by generator g.
Let m be same as in Theorem 3.6. Then there exists a family {mA} of ^^-decomposable
measures, where (Bi is generated by gA, ??]0.+??[, such that for every continuous
function f : [0, +oo] -> [0, +oo]
fSUp
/sup pis;.
fQdm= lim / f Qdmy
= ^JsT(fg^ofedx).

1440
?. Pap
4. Applications on nonlinear PDE
We consider here the nonlinear PDE, so called Hamilton-Jacobi-Bellman equation
du(x,t) /Эй \
H[—,x,t)=0, A6)
dt \dx'
(see Maslov and Samborskii A992), Kolokoltsov and Maslov A997), Pap A993b, 1997b,
1997c, 1999), Pap and Ralevic A998)). Hamilton-Jacobi equations are specially important
in the control theory. Unfortunately, usually the interesting models are represented by
Hamilton-Jacobi equations in which the non-linear Hamiltonian ? is not smooth, for
example the absolute value, min or max operations. Hence we can not apply on such cases
the classical mathematical analysis. There is so called "viscosity solution" method (see
Lions A982)) which gives upper and lower solutions but not a solution in the classical
sense, i.e., that its substitution into the equation reduces the equation to the identity. Using
the pseudo-analysis with generalized pseudo-convolution it is possible to obtain solutions
which can be interpreted in the mentioned classical way.
4.1. Hamilton-Jacobi equation with non-smooth Hamiltonian
4.1.1. Two examples. We start with two examples to illustrate how can be applied the
pseudo-linear superposition principle on some non-linear partial differential equations.
Example 4.1. An important nonlinear partial differential equation is the Burgers
equation for a function и = u(x,t). Burgers A948), Hopf A950) and Cole A951) (see
also Ames A965)) investigated as a model of turbulence the following equation
dv dv с d2v ,,_.
dt dx 2 ox1
where с is a parameter. Putting ? = ди/дх in A7) and integrating with respect to X we
obtain the equation
Эй 1 /Эи\ с д~и .,„.
Э7 + 2Ы -2Э^=°< A8)
for ? е Ш and t > 0, with the initial condition u(x,0) = uq(x), where с is the given
positive constant, and which models the burning of a gas in a rocket, see Ames A965).
We shall apply on this equation by the g-calculus with the generator g(u) = e~"/c. Then,
the corresponding pseudo-addition is
u®v = -c\n(e-H/c+e-v'c)
and the distributive pseudo-multiplication и ? ? = и + v.

Pseudo-additive Measures and their applications 1441
Then for solutions u\ and иг the function (?? ? и\) ? (?2 ? иг) is also a solution of
Burgers equation.
The solution of the given initial problem is
c /-? tx _ s\i
u(x,t) = -\nBnct)Q Qu0(s)ds,
???) ? /
where the integral is given by
f f(x)dx = -c\n(f e-flx)'cdx\
The operator L: и -> ко is self-adjoint with to respect to scalar product
f®
(u, ?)? = I u(x)Ov(x)dx.
Jm
Example 4.2. Taking с -> 0 in the Burgers equation we obtain Hamilton-Jacobi
equation
Эй 1/Эич
э7 + 2Ы =°·
Then for solutions u\ and «2 the function (?? ? и ?) ? (?2 ? иг), where
и ? ? = min(w, ?) and и © ? = и + ?,
is also a solution of the preceding Hamilton-Jacobi equation.
4.1.2. General case. We extend now the pseudo-superposition principle to a more
general case, see Kolokoltsov and Maslov A997), Pap A997b, 1999).
THEOREM 4.3. If u\ and иг are solutions of the Hamilton-Jacobi equation A6), where
? e C(E"+2) anddu/дх is the gradient of u, then (? ? ???)?(?2??2) is also a solution
of the Hamilton-Jacobi equation A6), with respect to the operations ? = min and © = +.
We consider now the following Cauchy problem for Hamilton-Jacobi (-Bellman)
equation
Эй /Эи\ ,._,
—+//(—j=0, и(дг,0) = ио(дО, A9>
where xeE", and the function ? :R" -> R is convex (hence by boundedness of ? it
is also continuous). For control theory the important examples of the Hamiltonian ? are
non-smooth functions, e.g., max and | · |. The approach with pseudo-analysis avoids the

1442
?. Pap
use of the so called "viscosity solution" method, which does not give the exact solution
of A9) (see Lions A982)). We apply now the methods of pseudo-analysis.
For that purpose we define the family of operators {R,}, >o, for a function щ(х) bounded
from below in the following way
u(t,x) = (R,u0)(x)= mf(uo(z)-tC®(H)(^^\), B0)
where С is considered on the whole R". The operator Rt is pseudo-linear with respect to
? = min and ? = +. We denote by L (or L(q), q e R") the transformation L{H){q) =
supp6R„ (pq — H(p)), which is called also Legendre transform of H. Note that
L(H)(q)=-C®(H){q),
where C®(H)(q) = mfpew(-pq + H(p)). We note that L is also convex for ? convex,
but it can be discontinuous-
First we suppose that mo is smooth and strongly convex. We shall use the notations {x, y)
and || ? || for the scalar product and Euclidean norm in R", respectively. For a function
F: R" -> [-oo, +00] its subgradient at a point и e R" is a point w e R" such that F{u)
is finite and
{w,v-u)+F(u)^F(v)
for all ? e R". Then we have by Kolokoltsov and Maslov A997).
LEMMA 4.4. Let щ{х) be smooth and strongly convex and there exists S > 0 such that
for all X the eigenvalues of the matrix u'q(x) of all second derivatives are not less than ?.
Then
A°) For every ? e R", t > 0, there exists a unique ?(t,x) e R" such that (x - ?(?, x))/t
is a subgradient of the function ? at the point u'Q{%(t,x)) and
(RlUO)(x) = uo(Ht,x))+tL(H)(X~^t,X)\ B1)
B°) The function ?(?, ?) for t > 0 satisfies the Lipschitz condition on compact sets, and
???\^0?(?,?)=?.
C°) The Cauchy problem A9) has a unique C1 solution given by B1), and
ди
д~х
uch;
ди
~dt
м@
(',*)
=«;>(?('.
у problem
+ я(
,*) =
' ди\
???)~
"oW.
?)).
= ?,
B2)

Pseudo-additive measures and their applications
1443
is the adjoint problem of the Cauchy problem A9). The classical resolving operator R* of
the Cauchy problem B2) on the smooth convex functions by Lemma 4.4 is given by
(R;u0)(x) = M(uo(t)+tL(H)
We note that /?* is the adjoint of the resolving operator Rt with respect to bipseudo-linear
functional
f®
/ fOhdm.
Then we can introduce as in the theory of linear equation (see Pap and Takaci B002)) the
notion of generalized weak solution (using Theorem 1.37).
Definition 4.5. Let щ be a bounded from below function m0:IR" ->MU {+oo} and
w„0 the corresponding functional from Cq(R"). The generalized weak pseudo solution
of Cauchy problem A9) is a continuous function from below (Rt uq)(x) which is defined
uniquely by m/?,„„(<?) — muo(/?*<p) for all smooth convex functions ?.
We can construct the solution for the case when mo is a smooth strictly convex function
by Lemma 4.4. Then it follows by Theorem 1.37 and Definition 4.5.
THEOREM 4.6. For an arbitrary function щ(х) bounded from below the weak pseudo-
solution of the Cauchy problem A9) is given by
(R,u0)(x) = (R, CI ио)(дО = inf/ci ii0(z) + tC®{H)
where
CI/(Jt) = sup{!/.(*): r/,eC(R"), ? ? /}·
4.2. Bellman differential equation for multicriteria optimization problems
We present results from Kolokoltsov and Maslov A997), Kolokoltsov B001) obtained for
the controlled process in W specified by a controlled differential equation ? = f(x, v)
(where ? belongs to a metric control space V) and by a continuous function ? e B(R." ?
V,Rk), which determines a vector-valued integral criterion
*(*(¦))= ? ?(?(?),?(?))??
Jo
?))¦
(?))

1444
?. Pap
on the trajectories. The problem of finding the Pareto set ?, (?) for a process of duration t
issuing from X with terminal set determined by some function ?? e S(R", Шк), i.e., can
be described by
a>t(x) = Min\J(<P(x(-))Qa>o(x(t))), B3)
*(¦)
where x(-) ranges over all admissible trajectories issuing from X. Proposition 1.27 enables
to encode the functions ?, e B(R", P(Rk)) by the functions (t, x, а) н» и,
m:R+ xR" ? L-> R.
The optimality principle implies the following equation, which is valid modulo 0(? ) for
small ?:
u(t, x,a) = Min(/?T(p(v ,V)*u(t — ?,? + ??(?)))(?).
Using the representation C) of hTiptXA,) and the fact that ? is the multiplicative unit in
CSn(L) we obtain
u(t, x,a) = min(r^(;c, v) + u(t — ?, ? + Ax(v), a — ???{?, ?))),
к
where ? = к~ 2_J'PJ аП(^ 4>L = ? — 4>e-
j=\
Substituting Ajc = xf(x, v) into this equation, expanding S in a series modulo 0(?2), and
collecting similar terms, we obtain the equation
Эй / Эй ди \ „ ^
— + max[<pL (x, v) — - f(x, v) — - Щх, ?) = 0. B4)
dt ? \ da ox )
Despite of the presence of a vector criterion has resulted in a larger dimension, this equation
coincides in form with the usual Bellman differential equation. Therefore, the generalized
solutions can be defined on the basis of the idempotent superposition principle, as in
Section 4.1. Then we obtain the main result of this subsection, Kolokoltsov and Maslov
A992).
THEOREM 4.7. The Pareto set ?,(?) B3) is determined by a generalized solution и, е
?, CS„(L)) of B4) with the initial condition
u0(x) = hao(x)eB(R",CS„(L)).
The mapping Res '¦ «0 ^ ut is a linear operator on S(R", CSn{L)).

Pseudo-additive measures and their applications
1445
Remark that B(R", S„(L)) is equipped with the CS„(L)-valued bilinear inner product
{h,f)=mfx(h*f)(x).
We apply the Pontryagin's maximum principle to the problem in question based on the
following observation. Let R be the usual resolving operator for generalized solutions
of the Cauchy problem for B4), so that R acts on the space B(R" ? L, ]-oo, +00]) of
]-oo, +oo]-valued functions bounded below on R" ? L. There is an embedding
m:B(R",CS„(L)) -> B(R" ? L, ]-oo, +00]),
which is an idempotent group homomorphism, i.e., preserves the operation ? = min. We
have that the diagram
Res
B(R",CS„(L)) »- B(R",CSn(L))
R
B(R" ? L,]-oo, +00]) —*- B(R" ? L,]-oo, +00])
commutes. Namely, for smooth initial data this follows from the fact that a smooth
solution of B4) always defines optimal synthesis. This implies commutativity for general
initial conditions, since the operators Res and R are uniquely defined by their action on
smooth functions and by the property that they are homomorphisms of the corresponding
idempotent semigroups, i.e., preserve the operation ? = min. This implies the following
result.
PROPOSITION 4.8. u(t,x,a) is the minimum of the functional
? ?(?(?), ?(?)) dz + fc^Ud »(?(?)) B5)
Jo
defined on the trajectories of the system
i = f(x,v), a = -<pL(x.v) B6)
in R" ? L issuing from (x, a) with free right endpoint and fixed time t.
4.3. Stochastic optimization
The method considered in Section 4.1 for constructing generalized solutions to the
Hamilton-Jacobi-Bellman equation, can be used in more general situations, for stochastic
or infinite dimensional generalizations. Here we formulate some results obtained in
Kolokoltsov A998b, 2001), where one can find the proofs as well as the applications of

1446
?. Pap
these results to the construction of WKB-type asymptotics of stochastic pseudodifferential
equations. We consider the equation
du + H it, x, -^ J dt + (c(t, x) + g(t, x) — J о dW = 0, B7)
where ? e R", t ^ 0, W = (W',..., W") is the standard и-dimensional Brownian motion,
see Brooks B002), this Handbook (o, as usual, denotes the Stratonovich stochastic
differential), u(t,x,[W]) is an unknown function, and the Hamiltonian H(t,x,p) is
convex with respect to p. This equation is called the stochastic Hamilton-Jacobi-
Bellman equation. First, we explain how this equation appears in the theory of stochastic
optimization. Second, we develop the stochastic version of the method of characteristics
to construct classical solutions of this equation. Third, on the basis of the methods of
idempotent analysis (and on analogy with the deterministic case), we construct a theory of
generalized solutions of the Cauchy problem for this equation. We start from the controlled
stochastic dynamics which is described by the equation
dx =f(t,x,v)dt + g(t,x)odW,
where the control parameter ? belongs to some metric space V and the functions /
and g are continuous in t and ? and Lipschitz continuous in X. Let the income along
the trajectory ?(?), ? e [t,T], defined by the starting point ? = x@) and the control
[?] = ?(?), ? e [t, T], be given by the integral
lf(x,[v],[W])= f b(z,x(T),u(T))dT+ ? c(T,x(T))odW.
We will find an equation for the cost (or Bellman) function
u(t, ?, x, [W]) = sup(ll(x, [v], [W]) + u0(x(T))),
?
where the supremum is taken over all piecewise smooth (or equivalently, piecewise
constant) controls [v] and mo is some given function (terminal income). Our approach
is based on the following fact: if we approximate the noise W in some stochastic
Stratonovich equation by smooth functions, then the solutions of the corresponding
classical (deterministic) equations will tend to the solution of the given stochastic equation.
For smooth functions W, we have the dynamics
i = f(T,x,v)+g(T,x)W(T)
and the integral income
j [b(z, ?{?), ?(?)) + c(t,x(t))W(t)] dx.

Pseudo-additive measures and their applications
1447
The Bellman equation for the corresponding deterministic (nonhomogeneous)
optimization problem is given by
ди / ди\ ( ди\ о
— +sup{b(t,x,v) + f(t.x,v)—\ + lc(t,x)+g(t,x)—)W(t)=0.
Rewriting this equation in the stochastic Stratonovich form, we obtain B7) with
H(t,x, p) = sup(b(t, ?, v) + pf(t,x, v)).
V
We restrict on the following two particular cases.
(i) Let с = 0 and g = g(t) be independent of X. Differentiating B7), we obtain
ди (дН ЭН д2и\ д2и
дх \ дх др дх1 / дх1
and using the relationship
w о dW = w dW + - dwdW
2
between the Ito and the Stratonovich differentials, we obtain the equation for и in the Ito
form
/ du\ 1 ? д~и ди
du + H(t,x,ir)dx + -g--^dt+g—dW=0.
\ дх/ 2 дх1 дх
For the mean optimal cost function и, this implies the standard second-order Bellman
equation of the stochastic control theory:
ди ( ЭиЛ 1 -, д'и
— + ? ?, jc, — + - g2 —^ = 0.
dt \ dxj 2s dx2
(ii) For g = 0 B7) reduces to
du + H(t,x,-^-}dt+c(t,x)dW=0, B8)
since in this case the Ito and the Stratonovich differential forms coincide.
Particularly, we formulate a result for the case of B8) with ? and с that do not explicitly
depend on t.
THEOREM 4.9. For an arbitrary initial function uq(x) bounded below, there exists a
unique generalized solution of the Cauchy problem for B8)/or all t ^ 0.

1448
?. Pap
Approximating nonsmooth Hamiltonians by smooth functions and defining the
generalized solutions as the limits of the solutions corresponding to the smooth Hamiltonians,
Kolokoltsov A998b, 2001) found (on analogy with the deterministic case) that formula for
generalized solutions remains valid for nonsmooth Hamiltonians.
4.4. Option pricing
We shall use a part of Kolokoltsov A998b, 2001) for describing the subject. The well-
known Black-Sholes and Cox et al. A979) formulas are basic results in the modern theory
of option pricing in financial mathematics. They were obtained by means of stochastic
analysis. Various generalizations of these formulas were proposed using more sophisticated
stochastic models for common stocks pricing evolution. The systematic deterministic
approach to the option pricing leads to a different type of generalizations of Black-Sholes
and Cox-Ross-Rubinstein formulas characterized by more rough assumptions on common
stocks evolution. This approach reduces the analysis of the option pricing to the study of
certain homogeneous nonexpansive maps, which are "strongly" infinite dimensional: they
act on the spaces of functions defined on sets, which are not (even locally) compact.
Kolokoltsov A998a) has shown what type of generalizations of the standard Cox-Ross-
Rubinstein and Black-Sholes formulas can be obtained using the deterministic (actually
game-theoretic) approach to option pricing. Characterizing the class of homogeneous
nonexpansive maps which appear in these formulas, considering first a simplest model
of financial market with only two securities in discrete time, then its generalization to the
case of several common stocks, and then the continuous limit.
A simplest model of financial market deals with only two securities: the risk-free bonds
(or bank account) and common stocks. The prices of the units of these securities, В = (Bk)
and S = (irrespectively, change in discrete moments of time к =0, 1,... by the recurrent
equations Bk+\ = pBk, where ? ^ 1 is a fixed number, and Sk+i = &+i &, where & is an
(a priori unknown) sequence taking value in a fixed compact set McR. Denote by и and
d respectively the exact upper and lower bounds of ? (и and d stand for up and down)
with 0 < d < ? < и. There were investigated two cases:
(i) ? consists of only two elements, its upper and lower bounds и and d,
(ii) ? consists of the whole closed interval [d,u].
Case (i) corresponds to the Cox-Ross-Rubinstein model and case (ii) stands for the
situation when only minimal information on the future evolution of common stocks pricing
is available.
An investor is supposed to control the growth of his capital in the following way. Let
Ck-1 be his capital at the moment к - 1. Then the investor chooses his portfolio defining
the number у* of common stock units held in the moment к — 1, and we have
Ck-1 = YkSk-1 + (Ck-1 - YkSk-1).
where the sum in brackets corresponds to the part of the capital laid on the bank account
(all operations are friction-free). The control parameter y* can take all real values, i.e.,
short selling and borrowing are allowed. In the moment к the value ijk becomes known and

Pseudo-additive measures and their applications
1449
thus the capital becomes equal to
Ck = YkHkSk-\ + (Ck-1 - YkSk-1)p.
The strategy of the investor is any sequence of numbers ?\,..., ?? such that each ?-}
can be chosen using the whole previous information: the sequences Co, · · ·, Cj-\ and
SO,..., Sj-\. The investor, selling an option by the price ? = Co should organize the
evolution of this capital (using the described procedure) in a way that would allow him to
pay to the buyer in the prescribed moment ? some premium f(Sn) depending on the price
S„. In the case of the standard European call option, which gives to the buyer the right to
buy a unit of the common stocks in the prescribed moment of time ? by the fixed price K,
the function / has the form
f(S„) = nax(Sn-K,0).
Therefore the income of the investor will be C„ - f(Sn). The strategy ?\,..., y„ is called
a hedge, if for any sequence ?\,..., ?? the investor is able to meet his obligations, i.e.,
C„ - f(S„) ^ 0. The minimal value of the initial capital Co for which the hedge exists is
called the hedging price P/, of an option. The hedging price P/, will be called correct (or
fair), if moreover, C„ - f(Sn) = 0 for any hedge and any sequence ?y. The correctness
of the price is equivalent to the impossibility of arbitrage, i.e., of a risk-free premium for
the investor. It was in fact proven in Cox et al. A979) (using some additional probabilistic
assumptions on the sequence ?]) that for case (i) the hedging price P/, exists and is correct.
It is known that when the set ? consists of more than two points, the hedging price will not
be correct anymore. Kolokoltsov A998a) proved using exclusively deterministic arguments
(idempotent analysis) that both for cases (i) and (ii) the hedge exists and is the same for
both cases whenever the function / is nondecreasing and convex (possibly not strictly).
Some generalizations of the standard Black-Sholes formula (characterized by more rough
assumptions on the underlying common stocks evolution) were obtained which reduce the
analysis of derivative securities pricing to the study of homogeneous nonexpansive maps,
which act in infinite dimensional spaces.
5. Non-commutative and non-associative pseudo-operations
We present in this section results obtained in Pap and Vivona B000) related the relaxation
of the properties of pseudo-addition and pseudo-multiplication and the application of
obtained results on nonlinear PDE.
DEnNlTlON 5.1. We call real operations ? and ? generalized pseudo-addition and
generalized pseudo-multiplication (from the right), respectively, if they satisfy the
following conditions:
(i) ? and 0 are functions from C2(E2),
(ii) the equation t ? t = ? for given ? is uniquely solvable,

1450
?. Pap
(iii) ? is right distributive over ?:
(Dr) (*?H?? = (.???)?()>?;:)·
Changing in the previous definition in (iii) that ? is left distributive over ?:
(D,) zQ(x®y) = (zQx)®(zQy).
we obtain generalized pseudo-addition and generalized pseudo-multiplication (from the
left), respectively. Since all considerations are analogous we shall take in the future
considerations only on the right case.
We shall give a representation theorem for generalized pseudo-addition and generalized
pseudo-multiplication. For that purpose first we shall prove several lemmas. First of all for
a function of two real variables и =и(х, у) we shall use the following notations
Эй Эй Эй
(и)\= — =и_х, (иJ= — =иу, (к)|2 = т—г- =".rv
дх ду дхду
The following representation theorem is obtained in Pap and Vivona B000).
THEOREM 5.2. Two real operations ? and ? from C2(E2) with (х<Эу)\ ^Oforallx, y,
are generalized pseudo-addition and pseudo-multiplication {from the right), respectively,
if and only if they are representable by
uQz = h-](f(z)h(u)+a(z)), B9)
and
Х®у = И~]((И(х)ШИ(у)), C0)
where f and a are arbitrary functions from C2(E), the function h is given by
h(u) = ("{e-W^'^ds,
Jy
the operation ?' is given by ? ?' у = ?~' (? ? y), where ? is defined by ? (?) = ? ? ? and
ЕВ satisfies the functional equation
(u EB v)f(z)+a(z) = (uf(z)+a(z)) EB (vf(z)+a(z)).
THEOREM 5.3. If® and ? are generalized pseudo-addition and pseudo-multiplication
(from the right), respectively, and additionally that
(a) x © 0 = a (constant) for every x;

Pseudo-additive measures and their applications
1451
(b) there exists yo such that (x Qyo)\ ? 0; then we have
x®y = h-\h{x) + h{y))x 0у = Г' (f(y)h(x)), C1)
where (for ?, ? arbitrary non-zero constants)
h(x) = (Y®PJ[{^^dx, C2)
J (?®?J
and f is an arbitrary function fmm C"(R) and /@) = 0.
Remark 5.4.
(i) Theorem 5.3 holds also for ?, 0 e C1, then also / e C1.
(ii) As a consequence of Theorem 5.3 we obtain that ? is also commutative and
associative. Specially, for / = h we obtain the g-calculus, see Pap A993a, 1995).
Definition 5.5. For given generalized pseudo-addition ? and generalized pseudo-
multiplication О (from the right) we shall call any function h from Theorem 5.2 in B9)
and C0) and Theorem 5.3 in C1) and C2) a generator of operations ? and ©.
Now we can give a complete characterization of generalized pseudo-addition and
pseudo-multiplication
THEOREM 5.6. Let ? and © be generalized pseudo-addition and pseudo-multiplication,
respectively, with (? ? у) \ ? Ofor all x, y.
Then they can be represented in one of the following forms
(i) X®y= h~\h(y) + i/(h(X) - h(y))),
(ii) ? ? у = h-l(h(y) + Y(h(x) - h(y))),
xQy = h-\a{y)-h{x)),
with ? odd function;
(iii) ? ? у = й-1 (h(y) +a(h(x) - h(y))),
xOy = h-i(h(x)f(y)+a(y)),
where ?(?) = t ???,? is a constant and ЕБ, a, f, h are functions from Theorem 5.2.

1452
?. Pap
5.1. Applications on nonlinear PDE
First order equations. We shall consider the following nonlinear PDE of the first order,
see Goard etal. A997),
c\(x,y)u_x +c2(x,y)uy = F(u), C3)
where c\, ci and g are given functions. Taking
dt
о-/
' F(t)
we introduce using (i) from Theorem 5.6 generalized pseudo-addition ? and pseudo-
multiplication © (from the left) for arbitrary ?
и ? ? = h~\h(v) + \j/e(h(u) - h(v))),
и О ? = /г (h(v) +?(?)),
respectively, for an arbitrary function a, and ?^(?) = G~](s + G(x)) for arbitrary
function G from C1. Then by Goard et al. A997) we have that if и ? and иг are solutions
of Equation C3) then also for every ?
mi ? u2 = /г (h(u2) + \l/e(h(ui)- h(ui)))
is a solution of Equation C3).
For arbitrary but fixed real number a and a solution и of Equation C3) we have that also
for every function a
а<Эи =h~*(h(u)+a(a))
is a solution of Equation C3). Summarizing, we have that if u\ and иг are solutions of
Equation C3) and a\ and a2 arbitrary but fixed real numbers, then also for every ? and
every a
(?? ???)?(?2?)
is a solution of Equation C3).
Taking specially c\(x, y) = c2(x, y) = 1 and F(u) = u2 we obtain the equation
"л + «у = " ·
Then we have h{u) = -\/u. In this case we have for the operations ? and О
1
1/?-?A/?- 1/и)
1
и 0 ? =
1/? — а(и)

Pseudo-additive measures and their applications 1453
where ?(?) = - \n(e' + 1), by Goard et al. A997), the same statement as for the general
case.
Multi-dimensional aggregation of solutions. We can consider the extension to
«-dimensional case of Equation C3) for F ? 0
" 9
Ycl(x],...,x„)-^ = F(u) C4)
?[ dx'
considering the m-dimensional pseudo-superposition principle. We have that if и ?...., u,„
are solutions of Equation C4) then also the generalized geometric mean of solutions
G(u ? ,...,«,„) = /-' (/(«?)"' ···/(«,„)"'")
and the generalized arithmetic mean of solutions
A(« , u„) = (/-' (fl,/(ii,)"+·· ¦+a„J(umY)L,\
where f{x) = exp(/i dt/ F(t)), a; ^0 (i = 1,.. .,m) and ? ? ? \-am = 1, and ? ^0,
are solutions of Equation C4).
Second order equations. We shall need the following special class of operations ? and ?.
THEOREM 5.7. Let ? be an arbitrary but fixed positive real number. If к is a strictly
monotone positive function and belongs to C~, then operations defined by
x®y=k~] (?k(x) + k(y)), ? ? у = к~' (к(хУк( у))
are generalized pseudo-addition and pseudo-multiplication (from the left), respectively,
which belong to the class (i) in Theorem 5.6 (introducing an operation ?' by ? ©' у =
У ? ?).
Specially for ? = 1 we obtain a semiring which is based on a generator (g-calculus).
Example 5.8. We consider again the Burgers equation A8). We introduce generalized
pseudo-addition and generalized pseudo-multiplication (from the left) from Theorem 5.7
given by
u®v = к~*^к(и)+к^)). uQv = k~\k(u)ek(v)),
respectively, for special k(u) = e~"^2A). Then by Goard et al. A997) we have that if u\
and и 2 are solutions of Equation A8) then also for every ? > 0
и ? ? «2 = -2? 1?(?*>-"'/|2?) + е'"-/ак))
is a solution of Equation A8).

1454
?. Pap
For arbitrary but fixed real number a and a solution и of Equation A8) we have that also
for every ?, ?© и =?? + u is a solution of Equation A8). Summarizing we have that if и ?
and И2 are solutions of Equation A8) and a\ and сь arbitrary but fixed real numbers, then
also for every ? > 0, (? ? ? u\) ? (?2 ? u2) is a solution of Equation A8).
Note that if и is a solution and a is an arbitrary real number then и © ? is a solution of
Equation A8) if and only if ? = 1, which gives us the g-calculus.
Example 5.9. Consider the following PDE of the second order
? ? ?
????](?\,..., x„)ux,Xj + Y^bk(x\,..., x„)uXk +c(x\,..., x„)u
?'=? j=\ i=l
. ? ?
= - ???'^??' ¦ ¦¦'Xn)uXiuyj, C5)
" '=1 7=1
where ?,-,-, fa, с are given functions and ? is a constant different from 1. Taking
k(x) = ? for ? > 0 and ? ? 1 we consider generalized pseudo-addition and pseudo-
multiplication (from the left) from Theorem 5.7
и ? ? = (??'"? + ?'-?)?/(|-?), и © ? = и"?,
respectively.
If u ? and И2 are positive solutions of Equation C5) then also for every ? > 0
и ? ? И2 = k'] (sk(u ?) + k(u2))
is a positive solution of Equation C5).
For arbitrary but fixed positive real number a and a solution и of Equation C5) we
have that a © и = aeu is a positive solution of Equation C5). Taking ? ? (? ? ? иг) with
a = \????\]/? and ? = \?\ \/\?2\ we obtain Levin's A970) result.
We note that if и is a solution and a is an arbitrary positive real number then и ? ? is a
solution of Equation C5) if and only if ? = 1, which gives us the g-calculus.
5.2. Corresponding pseudo-measure
In this section we shall consider special class of the generalized pseudo-operations ?
and О initiated by Theorem 5.7.
Dehnition 5.10. Let ? be an arbitrary but fixed positive real number and к a positive
strictly monotone continuous function defined on R or [0, oof, then operations defined by
x e у = Г' (ek(x) + k(y)), ? ? у = k~' (k(xfk(y))
are called к -pseudo-addition and A:-pseudo-multiplication (from the left), respectively, on
? or [0, oof.

Pseudo-additive measures and their applications
1455
Since generalized pseudo-operations are not necessarily commutative nor associative
operations we need to define pseudo-sum of ? elements and pseudo-sum of pseudo-
products.
Definition 5.11. Let ? and ? be operations from Theorem 5.7 and a, e R (or a, e
[0,oo)) i e {1,2,...и). Then
0 щ = (· · · ((?, ? a2) ? a3) ? · · ·) ? a„ = k~' I ^?"-1 ?(?,) J.
i=l \?=? /
Now, pseudo-sum of pseudo-products is
?(?, 0ft) = r' ij2e"^k(a,Y ¦ *(ft)V
i = l \i=l /
Let С be a family of semiclosed subintervals of R, then С is semiring of sets, i.e.,
A) 0eC,
B) ?, ? eC=> АП5 eC,
C) for all ?, ? e С there exists С ?,..., C„ from С such that Cj П C, = 0 for i ^ ;' and
A\« = U*=|C*.
Let ]?,/>] с R be an arbitrary but fixed semiclosed interval, let ? be an и-partition of ]a, b],
a = xo < x\ ^ · ·· ^ xn =/?,andletC]o.ft] =C(~){]a,b]}. We denote by C? h] the subfamily
of the family Су,.ь\ which consists of all subintervals ]*,-, x,+ \ ] obtained by the partition ?
and the interval ]a, b].
If a = xq < x\ 4. ¦ ¦ ¦ < jt„ is the «-partition ? of the interval [a,b], the (n + l)-partition
p' will be obtained if we keep all the points from previous partition and add one more point
and renumerate the points of the new partition in the increasing order. After s-repetition of
the preceding procedure we obtain (n + s)-partition pis).
First, we shall define by Pap and Stajner-Papuga B001) ?-measure on C'^ fc) with respect
to a given partition p.
Definition 5.12. A mapping ?? :C!'(l h, -> [0, oo[ is called ?-measure for given
«-partition ? if
Mp(]^.*,-+.])=*-|(igS?). C6)
where i e {0, ?,.,.?- 1), ?;,(]?,?]) = k~\b - a), and if for some у e]a,b] such that
¦*; < У < *i+b we have for ?,,-: C? ,;| -+ [0, oof
Мр'(]*.-.У])=*-'(^?) and дИ]^.+|])=Г'(^т)' C7>
where p' is the (n + l)-partition obtained from ? partition plus the point y.

1456
?. Pap
Remark 5.13. Equation C7) transform ©-measure ?? to ©-measure ??· on (n + 1)-
partition p' that differ from starting «-partition at most one point y, in such manner that
following equality is valid
?-?{]??,??+\]) = ?,,'(]??, у]) ® ??'(]}·,??+\]). C8)
We remark that with respect to ?? unchanged are the values m p'{]xj,Xj+\\) for j > i;
the values mp'(\xj,y]) and тр'(\у,х,+ \]) are given by C7), and for j < i we have
mp'(]Xj,Xj+]]) = k-](k(mp(]Xj,Xj+i])/s).
Now, for any interval ]c, d] с ]a, b] we can obtain the value ??"{?, d]) by ?? and, at
most, (n + 2)-partition.
THEOREM 5.14. Let ?? be ?-measure from the previous definition, then following holds
(a) ?,,@) = *-'(()),
(b) MlXo]*/.*'¦+!]) = ®i=o Vp(]xi,Xi+\])-
Now, the ?-integral can be constructed. The construction of ?-integral of continuous
function is analogue to the construction of Riemann's integral through the Riemann's
integral sums. In order to construct ?-integral we shall need the sequence of partitions
of some interval [a, b].The ?-measure ?? defined for и-partition will give us ?-measure
??· defined for (« + l)-partition ? by procedure given in C7). Repeating this procedure
s-times we obtain ?-measure ? <,> defined for (« -M)-partition pis).
DEnNlTlON 5.15. Let / : [a,b] -> [0, oo[ be a continuous function. The ?-integral of
function / is defined by
/(?.?) /«+.V-I \
???.?= lim ?(/(%?H?A?''·?'+?1))
Sx.b\ ??,(,)(].?,..?? + ,|)^*-1@)\ ;=0 /
(if the limit exists).
The ?-integral over the interval [a,b] does not depend on the partition of interval [a, b].
THEOREM 5.16. The ?-integral ofa continuous function f :[a, b] -+ [0, oo[ with respect
to ???)? any partition ? over the interval [a, b] is given by:
/¦(?.?) / rh \
/ /???=?? (k(f(t)))f dt). C9)
J[u.b\ \Ja /
Since the integral on the right side of C9) does not depend on the partition of interval
[a,b], we shall denoted the right side of C9) by f^f f.
For ? = 1 the g-integral is obtained.

Pseudo-additive measures and their applications
1457
Next theorem will give us some basic properties of ?-integral. First, we shall introduce
a transformation [·]* in the following manner:
[/]*(*) =*_?(*?/?(/(*»).
where / is a continuous function that maps interval [a, b] into [0, oof. If this
transformation is applied on constant a from [0, oof we obtain:
[?]*=*-'(*?/?(?)).
THEOREM 5.17. Let f andh be continuous functions that maps interval [a, b] into [0, oof
and let a be a constant from [0, oof, then:
(b)«oCf)/ = Cf)(^o/).
We can introduce a generalization of g-derivative from Pap A993a).
Dehnition 5.18. Let / e С' [a, b]. Generalized derivative of / is
if it exists.
The generalized derivative have the following properties
@ C_f Г-е) + k(f(a)) = k(f(x)) and (jj^f /)<'¦'> = *(/(*)).
(ii) (/? a)"'f) = /('-f) ?
causing the representation of ?-integral obtained in Theorem 5.16, we can introduced a
generalization of ^-convolution.
Dehnition 5.19. Let /, h : [0, oof-* [0, oof be continuous functions. The pseudo-
convolution / * h : [0, oof—> [0, oof of functions / and h is given by
f*h(x) = r](j\k(f(x-t))f(k(h(t))ydt\
THEOREM 5.20. Let f, g:[0, oof-* [0, oof be continuous functions. Then the pseudo-
convolution of functions f and h has the following presentations:
?-(?.?)
f*h(x)= ([/]*(*-0Ой@)¦
J 10.x]
It follows that pseudo-convolution is a commutative operation.

1458
?. Pap
6. Conditional distributive real semiring
6.1. Conditional distributivity
A uninorm U (respectively ?-norm T) is a commutative, associative, monotone binary
operation on the unit interval [0, 1] and for some 0 < e < 1 we have U(x,e) = ?
(respectively T(x, 1) = x) for all ? e [0, 1]. Throughout this section, let U be a left-
continuous uninorm or ?-norm and S a continuous ?-conorm such that U is conditionally
distributive over S, i.e., they satisfy the property (CD):
U(x,S(y,z)) = S(U(x,y),U(x,z))
for all x,y,z e [0, 1] such that S(y,z) < 1. In this way ([0, 1], 5", U) is a conditionally
distributive semiring.
For ?-norm U = ? there was obtained in Klement et al. B000a) the following
characterization
THEOREM 6.1. A continuous t-norm ? is conditionally distributive over a continuous t-
conorm S if and only if there exist a e [0, 1], a strict t-norm T* and a nilpotent t-conorm
S* such that the additive generator s* of S* satisfying s*(l) = 1 is also a multiplicative
generator of T* such that
T(x,y) =
?·7?(?) forx,ye[0,a],
a + (l-a)r*(f^,g) forx,ye[a,l],
min otherwise.
(t-norm ordinal sum) where T\ is an arbitrary continuous t-norm and
[ max otherwise,
(t-conorm ordinal sum).
If S is a strict ?-conorm generated by an additive generator s, then the only distributive
left-continuous uninorms with respect to S are associative compensatory operators (see
Klement et al. A996)) generated by multiplicative generator с ¦ s, ce ]0, oof (with the
convention 0 · oo = 0).
6.2. Hybridprobability-possibilistic measure and integral
Based on the structure ([0,1], S, U) with conditional distributivity it is developed in
Klement et al. A996, 2000a, 2000b) the so called E", i/)-integral.
We introduce now an integral based on S'-measure, see Section 1.2.1, and a conditionally
distributive pair E", U). Denote by ? the set of all .^-measurable functions from X to

Pseudo-additive measures and their applications
1459
[0, 1]. As usual, a measurable step function ?: X ->· [0, 1] is a measurable function which
assumes only finitely many values, and the set of all step functions will be denoted S. If
Range(<p) = {a\,a2, ¦ ¦ -,an} with ?, ? ?, whenever i ? j, and if A, = ?~] ({a,}), then
there is a canonical representation of ? given by
/1
?= S i/(a,,lA,), D0)
where
iA(X) = {e0
e for ? e A,
otherwise.
We have U(a, 1д) = ???, where
1 for ? e A,
10 otherwise.
Observe that representation D0) is independent of the ?-conorm S under consideration.
As classical measure theoretic result we have
THEOREM 6.2. For each measurable function f:X -*¦ [0, 1] there exists a sequence
(<Pn)n6N of step functions such that ?„ / f uniformly.
Dehnition 6.3. Let m : ? -+ [0, 1] be an 5"-faithful 5"-measure.
(i) Given an 5"-w-faithful partition С = {Ск- к е К) the E", i/)-integral of a
measurable step function ?: ? -> [0. 1] (which is represented as in D0)) is defined
by
flS.U) ? ? \
/ q>dm= S I S i/(flj,m(A,nCi)) )¦
(ii) The (S, i/)-integral of a measurable function / : X -> [0, 1] is defined by
•(S.f) /-(S-f)
/ / iiw = lim / ?„ dr,
Jx h^^Jx
where (<p„)fieN is a sequence of step functions such that ?„ / f (which exists by
Theorem 6.2).
(iii) The (S, i/)-integral of a measurable function / : X -> [0. 1] over a set A e ? is
defined by
•(S.f) /-(S.i;)
/ /dw= / U(lA,f)dm.
J A JX

1460 ?. Pap
Integral of a step function given in Definition 6.3(i) is independent of the choice of
the 5-m-partition and the E, i/)-integral of a measurable function in Definition 6.3(iii) is
independent of the choice of the sequence (<p„)„eN °f steP functions such that ?„ / f.
The basic properties of E, i/)-integral are contained in the following theorem.
THEOREM 6.4. Let m : ? -> [0, 1] be an S-faithful S-measure and f,geM-
(i) Iff ^g then
MS,U) MS.U)
/ fdm ? gdm.
Jx Jx
(ii) If А, В e ? with ? ? ? = 0 then
/ fdm =S\ \ fdm, / fdm).
Jaub \Ja Jb J
(hi) IfU(S(f, g), a) = S(U(f, a), U{g, a)) for all a e [0, 1] then
/ S(f,g)dm = sll fdm, I gdmj.
Remark 6.5. It should be mentioned that for a ?-norm U = ? the condition in
Theorem 6.4(iii) is fulfilled for all f,geM only if S equals 5м- Because of conditional
distributivity of Г over 5 (iii) in Theorem 6.4 also holds if S(f(x),g(x)) < 1 for all ? e X.
For the E, i/)-integral some classical convergence theorems such as the monotone
convergence theorem and Fatou's lemma, can be proven, see Klement et al. B000b).
THEOREM 6.6 (Beppo Levi). For every non-decreasing sequence of measurable functions
(fn)neN, ie-, fn ^ /(,+ i, with Ит^эс f„ = fe M, we have
~(S.U) MS.U)
I f dm = lim / /„ dr,
Jx "^xJx
THEOREM 6.7 (Fatou lemma). Let (/„) be a sequence of measurable functions. Then
•(S.G) MS.U)
I liminf/,, dm < liminf / /„ dm.
Jx "^°° "^x Jx
The E, i/)-integral is an extension of 5-measure, see Klement et al. B000b).
Theorem 6.8.
(i) For each f e Л4 the function m / : ? —> [0, 1 ] defined by
(S.U)
mj(A) = I f dm,
J a
is an S-faithful S-measure.

Pseudo-additive measures and their applications
1461
(ii) For all f,geMwe have
f(S.U) riS.U) flS.U)
/ fdmg= gdmf= t/(/,
J X J X J X
g)dm.
Theorem 6.8(i) shows that the (S, i/)-integral is a proper extension of the underlying
S'-measure since, in the case where / is the constant function with value 1 on X, we have
rrif = m. We remark that for the S'-measure m : V(X) -> [0, 1] specified by m{A) = 1 for
all non-empty subsets A of X and for SM-measure given by
M/(A) = sup{/(.x): xeA]
we always have ?/ = m f.
For each S-faithful S'-measure m : ? —>¦ [0, 1] we consider the subfamily Mm of ?
defined by
M,„ = {f e M: X is S-m/-faithful).
Obviously, for each ^M-measure m: ? ->· [0, 1] we have Mm = M. We have that the
(S, i/)-integral is {/-homogeneouson the family Mm-
THEOREM 6.9. Let m : ? -+ [0, 1] be an S-faithful S-measure. Thenfor every a e [0, 1]
HS.U) / r(S.U)
/ U(a,f)dm = Llla, / fdm\.
For special uninorms or ?-norms U and ?-conorms S, we obtain some integrals which
are well-known in the literature.
Example 6.10. (i) Let m: ? -> [0, 1] be an ^-measure which is also a (?-additive)
measure. Then it is a S^-faithful S^-measure and the (Sl, Tp)-integral of a function f &M
coincides with the classical Lebesgue integral of /.
(ii) For each left-continuous uninorm or ?-norm i/, for each S^-measure m : ? —> [0, 1 ]
and for each function / e ? we have
/.
(SM.G)
fdm = sup{U(a,m({xeX: f(x)^a}))\ae[0, 1]}.
IX
If m is a completely maxitive measure then we have
•(SM.f)
/,
fdm = sup{U(f(x),m({x})): xe X).
?
Remark 6.11. (i) The procedure for the construction of the E", i/)-integral used in
Definition 6.3 works also if we replace uninorm or ?-norm U by an operation *: [0, !]"—>·

1462
?. Pap
[О, 1] which is left-continuous, monotone, has an annihilator 0 and a left neutral element
e e @, 1 ], and which is conditionally distributive over S. Moreover, Theorem 6.4(i), (ii) as
well as Theorems 6.6 and 6.7 still hold.
(ii) The restriction to S'-measures and measurable functions whose ranges are subsets
of [0, 1] causes no loss of generality. Let [a, b] с [-oo, oo] such that ([a, b], ©) is an
/-semigroup with neutral element a (see Klement et al. B000a)). Consequently, there is
some strictly increasing bijection ?: [a, b] —>¦ [0, 1] such that the continuous r-conorm S
defined by
S(x,y) = w(w-\x)®w-\y))
is isomorphic to ?. Let * be a binary operation on [0, 1] as in (i), which is conditionally
distributive over S, and define the operation ? on [0, 1] by
xQy = ?(?~\?)* ?-1 (}>))·
Observe that ? is a left-continuous monotone operation on [a, b] which has annihilator
a and possesses some left neutral element, and which is conditionally distributive
over ?. Then for an ?-measurable function / : X -> [a, b] and an ©-faithful ©-measure
m : ? —> [a,b] can be introduced in a complete analogy to our construction the integral
jf-Q) fdm (compare Section 1.3, Maslov and Samborskii A992), Pap A990)), and it
can be represented by the (S, *)-integral via
(S.*)
/ fQdm=w'][j (wof)d(coom) \.
Example 6.12. (i) For [a,b] = [0, oo] we obtain the integral introduced in Sugeno
and Murofushi A987), and for ([a, b], ?. ?) = ([0, oo], +, ¦) we again come back to the
classical Lebesgue integral.
(ii) If the operation ? in the semiring ([a, b], ?, ?) is not idempotent, then the
operations ? and ? are generated by some uniquely determined strictly increasing
bijection g : [a, b] —> [0, oo] via
^©^^"'(gW + gOO).
xOy = g-*(g(x)g(y)).
The corresponding (?, ©)-integral was studied in Section 1.4, Pap A990) (called g-
integral in Pap A993a, 1993b)), and it has the special form
J fOdm = g-](J(g0f)d(gom)
where the integral on the right hand side is a Lebesgue integral. Under some additional
conditions, also a generalized derivative (g-derivative)can be defined (Pap, 1993a, 1993b)
which turns out to be useful when solving some differential equations.

Pseudo-additive measures and their applications
1463
(iii) If 5 is a continuous Archimedean ?-conorm with an additive generator s : [0, 1] —>
[0, oo] and if we replace the ?-norm ? by the binary operation * given by
a * b = s~ (a ¦ s(b))
and if we assume that X is w-achievable if s(l) < 1 (in the sense of Weber A984)), we
can repeat the whole construction of Definition 6.3 for the E, *)-integral, and we obtain
the integral introduced in Weber A984).
6.3. Hybrid utility function
In order to generalize decision theory to non probabilistic uncertainty, one approach is to
generalize mixture sets. In the paper by Dubois et al. B000) there are characterized the
families of operations involved in generalized mixtures, due to a previous result on the
characterization of the pairs of continuous ?-norm and ?-conorm such that the former is
conditionally distributive over the latter, see Theorem 6.1.
A basic notion in probability theory is independence. The main issue in probabilistic
independence is the existence of special events A\,...,A„ such that P(A\ ? ¦¦¦ ?
??) = ?'/=? ?(??- Such events are called independent events. In order to preserve
the computational advantages of independence, any operation * for which it could be
established that P(A\ П- ¦ ¦ П A„) = *'.'=| P(Aj), would do. However the Boolean structure
of sets of events and the additivity of the probability measure, impose considerable
constraint on the choice of operation *.
In the paper by Dubois et al. B000) is studied the possible operations * when changing ?
for a pseudo-additive (decomposable) measure based on a ?-conorm 5. A first remark is
that it is natural to require that * be a continuous triangular norm. If A = X is a sure event,
then A and X are independent, and it follows that m(A П X) = m{A) * m{X) = m{A) *
1 = m(A). Commutativity and associativity of * reflect the corresponding properties for
conjunctions. It is also very natural that * be non-decreasing in each place and continuous.
We try to find which triangular norms can be used for extending the notion of independence
for pseudo-additive measures in the sense of a prescribed triangular conorm. Since the term
independence has a precise meaning in probability theory, we shall speak of separability
in the framework of 5-measures. Two events A and В are said to be ^-separable if
т(АП B) = m{A)*m{B) for a triangular norm *. It turns out by Dubois et al. B000) that
the only reasonable pseudo-additive measures admitting of an independence-like concept,
are based on conditionally distributive pairs E, T) of ?-conorms and ?-conorms in the form
(Em, 5l), (T\, Tp))a, see Theorem 6.1, namely.
(a) probability measures (and * = product);
(b) possibility measures (and * is any i-norm);
(c) suitably normalized hybrid set-functions m such that there is a e ]0, 1[ which gives
for A and В disjoint
(m{A) +m(B) - a if m(A) > a, m{B) > a,
max(m(A), m{B)j otherwise,

1464
?. Pap
and for separability:
m(Af)B) =
а + ША)-а)шВ)-а) if m(A) > a, m(B) > a,
1-я
rm(A) mi B) >
а-Г|(«1,^) ifin(A)<a,in(B)<fl,
min(m(A), m(B)) otherwise,
Any probability distribution on a finite set X can be represented as a sequence of binary
lotteries. A binary lottery is 4-uple (A,a,x,y) where А с X and a e [0, 1] such that
? (A) = a, and it represents the random event that yields X if A occurs and у otherwise.
Let ? be a probability on X such that p, = P(xj), x, e X. Assume X = {x\,xi,xi} then ?
can be described by the following. The binary tree is obtained as follows: First partition X
into {? |} and {x2, хз} with probabilities ? \ and pi + рз, respectively, then partition ? \ {? ?}
into {*2} and {*з} with probabilities pi/ipi + Pi) and pj/ipi + pi), respectively. The
two trees are equivalent, provided that the probability of jc, is calculated by performing the
product of weights on the path from the root of the tree until the leaf x.
More generally, suppose m is a S'-measure on X = {x\,xi,xy\ and rrij = т{{х(}).
Suppose we want to decompose the ternary tree into the binary tree so that they are
equivalent. Then the reduction of lottery property enforces the following equations
S(y\,\>2)=\, ?(?,??) =W2, ?(?,??) = тт,,
where ? is the triangular norm that expresses separability for S'-measures. The first
condition expresses normalization (with no truncating effect for ?-conorm S allowed). If
these equations have unique solutions, then by iterating this construction, any distribution
of a S'-measure can be decomposed into a sequence of binary lotteries. This property is
basic in probability theory since it explains why probability trees can be used as a primitive
notion for developing the notion of probability after Shafer A996). Turning the S'-measure
into a sequence of binary trees leads to the necessity of solving the following system of
equations
<*l = ?(?, ?>|), ?? = ?(?, ??), S(v\,vi)=\ D1)
for given ? ? and <*2- Assuming that T\ = min we have solved D1) completely in Dubois et
al. B000) and exhibited the analytical forms of (?, v\, vi).
We define the set ?$.? of ordered pairs (?, ?) in the following way
*s.a = {(<*,?): (?,0)?]?, 1[\ ?+?=\+?]
U {(?, ?): min(a, ?) ? a, max(a, ?) = 1}.
A hybrid mixture set is a quadruple (<?, M, T, S) where Q is a set, (S, T) is a pair
of continuous ?-conorm and ?-norm, respectively, which satisfy the condition (CD) and
? :Q2 ? ?$.? —> Q is a function (hybrid mixture operation) given by
M(x,y-a,fi) = S(T(a,x),T(fi,y)).

Pseudo-additive measures and their applications
1465
It is enough to restrict to the case ({Sm, Sl), {T\,Tp))a. Then it is easy to verify that ?
satisfies the axioms (M1)-(M5) on 0s.a, where
(Ml) M(x, y; 1,0) = л;
(M2) M(x, y; ?, ?) = M(y. ?; ?, a);
(M3) M(M(x, y; ?, ?), у; ?, ?) = ?(?, у; ? (?, ?), ?(?(?, ?), ?(?, 1)));
(?4) ?(?,?;?,?) = ?;
(?5) ?(?(?, у; ?, ?), ?(?. у; ?. ?); ?, ?) = ? (?, y; S(T(a, ?), ? (?, ?)). 5"(?(/3, ?),
?(?, ?))) holds for all ?. у e Q and all (?, ?), (?. ?), (?, ?) e 4>s.a ¦
This kind of mixtures exhausts the possible solutions to (M1)-(M5).
Let us show the main appeal of (M1)-(M5). Let (S. T) be a pair of continuous i-conorm
and ?-norm, respectively, of the form
((SM,SL),(T\,TP))a.
Let u\,u2 be two utilities taking values in the unit interval [0, 1] and let ??, ?? be two
degrees of plausibility from ?$.? ¦ Then we define the optimistic hybrid utility function by
means of the hybrid mixture as
U(u |, м2; ? ?, ??) = S(T(u ?, ? ? ), T(u2, ?2)).
In the paper by Dubois et al. B000) it is examined in details this utility function. Although
the above description of optimistic hybrid utility is rather complex, it can be easily
explained, including the name optimistic. Putting together the results of this paper, the
utility of a n-ary lottery can be computed by decomposing the S'-measure into a sequence
of binary trees and applying the above computation scheme for hybrid utility recursively
from the bottom to the top of the binary tree expansion.
More details and proofs of theorems stated in this paper can be found in Dubois et al.
B000). It remains a pending problem to find a corresponding axiomatization for hybrid
utility as was done for classical utility theory in von Neumann and Morgenstern A944)
and possibility utility theory in Dubois et al. A998).
Cox's well-known theorem, see Cox A946), which justifies the use of probability
for treating uncertainty, was discussed in many papers. Some relaxed axioms have
been proposed, enabling non-additive functions to be admissible solutions. It is clear
that the family of pseudo-additive measures exhibited here is worth studying in Cox
relaxed framework, since a natural form of conditioning can be defined on the basis
of the triangular norm in the pair (S, T) satisfying (CD), leading to an almost regular
independence notion exhibited here.
References
Aczel, J. A966), Lectures on Functional Equations and their Applications, Academic Press, New York.
Akian, M. A999), Densities of idempotent measures and large deviations. Trans. Amer. Math. Soc. 351 A1),
4515^4543.
Akian, M. Quadrat, J.-P. and Viot, M. A994), Bellman processes. 11th International Conference on Analysis
and Optimization of Systems: Discrete Event Systems. Lecture Notes in Control and Inform. Sci., Vol. 199,
Springer, Berlin.

1466
?. Pap
Akian. M., Quadrat. J.-P. and Viot. M. A998). Duality between probability and optimisation, Proc. Intern.
Workshop "Idempotency", Bristol. October 1994. J. Gunawardena. ed.. Cambridge University Press.
Cambridge, 331-353.
Ames, W.F. A965), Ad hoc Exact Techniques for Nonlinear Partial Differential Equations, Nonlinear Partial
Differential Equations in Engineering, Academic Press, New York.
Azencott, R., Guivarc'h, Y. and Gundy. R.F. (eds) A978). Ecole d'e'te' de Saint Flour 8, Lecture Notes in Math.,
Springer, Berlin.
Baccelli. F, Cohen, G.. Olsder, G.J. and Quadrat. J.-P. A992). Synchronization and Linearity: An Algebra for
Discrete Event Systems, Wiley, New York.
Bellman, R.E. and Dreyfus, S.E. A962). Applied Dynamic Programming. Princeton University Press. Princeton.
NJ.
Benvenuti, P., Mesiar. R. and Vivona. D. B002). Monotone set functions-based integrals. Handbook of Measure
Theory. E. Pap, ed., Elsevier. Amsterdam. 1329-1379.
Blumen. G.W. and Kumei. S. A989). Symmetries and Differential Equations, Springer. New York.
Brooks, J.K. B002). Stochastic processes and stochastic integration in Banach spaces. Handbook of Measure
Theory, E. Pap, ed.. Elsevier. Amsterdam, 449-502.
Burgers, J.M. A948), ? mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1. 171.
Choquet. G. A953-1954), Theory of capacities. Ann. Inst. Fourier (Grenoble) 5. 131-292.
Cole, J.D. A951), On a quasi-linear parabolic equation occurring in aerodynamics, Quat. Appl. Math. 9, 225-
236.
Cox, R. A946), Probability, frequency, and reasonable expectation. Amer. J. Phys. 14 A). 1-13.
Cox. J.C.. Ross, S.A. and Rubinstein. M. A979). Option pricing: A simplified approach, J. Financ. Econom. 7,
229-263.
Crandall, M.G. and Tartar. L. A980), Some relations between nonexpansive and order-preserving mappings, Proc.
Amer. Math. Soc. 78. 385-390.
Cuninghame-Green. R.A. A979). Minimax Algebra. Lecture Notes in Economics and Math. Systems. Vol. 166.
Springer, Berlin.
Denneberg. D. A994), Non-Additive Measure and Integral, Kluwer. Dordrecht.
Diestel. J. B002). The Riesz Theorem, Handbook of Measure Theory. E. Pap. ed.. Elsevier. Amsterdam. 401^t47.
Dubois. D., Godo, L., Prade, H. and Zapico. A. A998). Making decision in a qualitative setting: from decision
under uncertainty to case-based decision. Proceedings of the Sixth International Conference Principles of
Knowledge Representation and Reasoning (KR '98), A.G. Cohn. L. Schubert and S.C. Shapiro, eds, Morgan
Kaufman, San Francisco, 594-605.
Dubois, D.. Pap, E. and Prade H. B000), Hybrid probabilistic-possibilistic mixtures and utility functions.
Preferences and Decisions under Incomplete Knowledge, J. Fodor. B. de Baets and P. Pemy. eds, Springer,
Berlin, 51-73.
Dubois, D. and Prade. H. A982). A class of fuzzy measures based on triangular norms. Intemat. J. Gen. System
8,43-61.
Dubois, D. and Prade, H. A988). Fuzzy numbers: an overview. Analysis of Fuzzy Information, Vol. 2. J. Bezdek.
ed.. CRC Press. Boca Raton. 3-39.
Dubois, D. and Prade, H. B002). Qualitative possibility functions and integrals. Handbook of Measure Theory.
E. Pap, ed., Elsevier, Amsterdam, 1469-1522.
Dubois, D., Kerre, E., Mesiar, R. and Prade, H. B000). Fuzzy interval analysis. Fundamentals of Fuzzy Sets,
D. Dubois and H. Prade. eds, Kluwer. Dordrecht, 483-582.
Falconer, K.J. B002), Fractal measures. Handbook of Measure Theory. E. Pap, eds, Elsevier, Amsterdam, 1037-
1054.
Freidlin, M.I. and Wentzell, A.D. A984), Random Perturbations of Dynamical Systems. Springer, New York.
Gaubert, S. and Gunawardena, J. A998). The duality theorem for min-max functions. C. R. Acad. Sci. Paris Ser. I
326. 43^48.
Goard, J.M. and Broadbridge, P. A997). Nonlinear superposition principles obtained Lie symmetry methods,
J. Math. Anal. Appl. 214, 633-657.
Golan, J.S. A992), The Theory of Semirings with Applications in Mathematics and Theoretical Computer
Sciences, Pitman Monogr. Surveys Pure Appl. Math., Vol. 54. Longman. New York.

Pseudo-additive measures and their applications
1467
Golan, J.S. A999), Power Algebras over Semirings, With Applications in Mathematics and Computer Science,
Kluwer, Dordrecht.
Gondran. M. A996). Analyse M1NPLUS. C. R. Acad. Sci. Pans Ser. I 323, 371-375.
Grabisch, M.. Nguyen. H.T. and Walker. E.A. A995). Fundamentals of Uncertainty Calculi with Applications to
Fuzzy Inference, Kluwer, Dordrecht.
Gunawardena, J. (ed.) A998). Proc. Intern. Workshop "Idempotency", Bristol, October 1994. Cambridge Univ.
Press, Cambridge.
Hadzic, O. and Pap, E. B001), Fixed Point Theory in Probabilistic Metric Spaces. Kluwer, Dordrecht.
Hopf, E. A950). The partial differential equation u, + imv = ???? v.Comm. Pure Appl. Math. 9. 201-230.
Inselberg, A. A970), Noncommutative superpositions for nonlinear operators. J. Math. Anal. Appl. 29, 294-298.
Jones, S.E. and Ames, W.F. A967), Nonlinear superposition. J. Math. Anal. Appl. 17, 484-487.
Klement, E.P, Mesiar, R. and Pap, E. A996), On the relationship of associative compensatory operators to
triangular norms and conorms. Intemat. J. Uncertainty. Fuzziness Knowledge-Based Systems 4. 25-36.
Klement, E.P, Mesiar, R. and Pap, E. B000a), Triangular Norms. Kluwer, Dordrecht.
Klement, E.P, Mesiar, R. and Pap. E. B000b). Integration with respect to decomposable measures, based on a
conditionally distributive semiring on the unit inten-al. Intemat. J. Uncertain. Fuzziness Knowledge-Based
Systems 8. 701-717.
Klement, E.P. and Weber, S. A991), Generalized measures. Fuzzy Sets and Systems 40, 375-394.
Kolokoltsov, V.N. A998a), Nonexpansive maps and option pricing theory. Kibemetika 34 F), 713-724.
Kolokoltsov, V.N. A998b), Stochastic Hamilton-Jacobi-Bellman equation and stochastic WKB method, Proc.
Intern. Workshop "Idempotency", Bristol, October 1994, J. Gunawardena, ed., Cambridge Univ. Press,
Cambridge, 285-302.
Kolokoltsov. V.N. A998c), Semiclassical Approximation for Diffusion and Stochastic Processes, Monograph.
Kolokoltsov, V.N. B001), Idempotent structures in optimization, J. Math. Sci. 104 A). 847-880.
Kolokoltsov, V.N. and Maslov, VR A992), Bellman's differential equation and Pontryagin's maximum principle
for multicriteria optimization problem, Dokl. Alcad. Nauk SSSR 324 A). 29-34.
Kolokoltsov, V.N. and Maslov. VR A997). Idempotent Analysis and Its Applications, Kluwer. Dordrecht.
Kuich, W A986), Semirings, Automata. Languages, Springer, Berlin.
Levin, S.A. A970), Principles of nonlinear superposition. J. Math. Anal. Appl. 30. 197-205.
Lions, PL. A982), Generalized Solutions of Hamilton-Jacobi Equations, Pitman, London.
Markova-Stupnanova, A. A997), T-sum of L-R fiizzv numbers, Fuzzy Sets and Systems 85. 379-384.
Markova-Stupnanova, A. A999), ? note to the idempotent functions with respect to pseudo-convolution. Fuzzy
Sets and Systems 102. 417^421.
Maslov, VR A987), ? new superposition principle for optimization problems, Uspekhi Mat. Nauk 42 C), 39^48.
Maslov, V.P. and Samborskii, S.N. (eds) A992). Idempotent Analysis. Adv. Sov. Math., Vol. 13, Amer. Math.
Soc., Providence, RI.
Mesiar, R. A997), Triangular-norm-based addition offiizzy intewals. Fuzzy Sets and Systems 91. 231-238.
Mesiar, R. and Pap, E. A999), Idempotent integral as limit of g-integrals. Fuzzy Sets and Systems 102, 385-392.
Mesiar, R. and Rybarik, J. A995), PAN-operations. Fuzzy Sets and Systems 74, 365-369.
Murofushi, T. and Sugeno, M. A991). Furry t-conorm integral with respect to fuzzy measures: Generalization of
Sugeno integral and Choquet integral. Fuzzy Sets and Systems 42, 57-71.
von Neumann, J. and Morgenstem. O. A944), Theory of Games and Economic Behavior, Princeton Univ. Press,
Princeton, NJ.
Pap, E. A990), Integral generated by decomposable measure, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak.
Ser. Mat. 20A), 135-144.
Pap, E. A993a), g-calculus, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 23 A). 145-150.
Pap, E. A993b), Solution of nonlinear differential and difference equations, Proc. of EUFIT'9.3, Vol. 1,498-504.
Pap, E. A994), Nonlinear difference equations and neural nets, BUSEFAL60, 7-14.
Pap, E. A995), Null-Additive Set Functions. Kluwer. Dordrecht.
Pap, E. A997a), Pseudo-analysis as a base for soft computing. Soft Computing 1,2, Springer, Berlin.
Pap, E. A997b), Decomposable measures and nonlinear equations. Fuzzy Sets and Systems 92, 205-222.
Pap, E. A997c), Solving nonlinear equations bx non-additive measures. Nonlinear Analysis 30, 31^40.
Pap, E. A999), Applications of Decomposable Measures. Handbook Mathematics of Fuzzy Sets-Logic, Topology
and Measure Theory, U. Hohle and S.R. Rodabaugh, eds, Kluwer, Dordrecht, 675-700.

1468
?. Pap
Pap, E. B000), Pseudo-convolution and its applications. Fuzzy Measures and Integrals, Theory and Applications,
M. Grabisch, T. Murofushi and M. Sugeno. eds, Physica-Verlag, 171-204.
Pap, E. and Ralevic, N. A998), Pseudo-Laplace transform. Nonlinear Analysis 33, 533-550.
Pap, E. and Stajner, I. A998), Generalized pseudo-convolution, Proc. Seventh Intemat. Conf. IPMU '98 Paris, La
Sorbonne, 1216-1222.
Pap, E. and Stajner, I. A999), Generalized pseudo-convolution in the theory of probabilistic metric spaces.
information, fuzzy numbers, optimization, system theory. Fuzzy Sets and Systems 102, 393^415.
Pap, E. and Stajner-Papuga, I. B001), Pseudo-integral based on non-associative and non-commutative pseudo-
addition and pseudo-multiplication, Intemat. J. Uncertainty, Fuzziness, Knowledge-Based Systems, 159-167.
Pap, E. and Takaci, A. B002), Generalized derivatives. Handbook of Measure Theory, E. Pap, ed., Elsevier,
Amsterdam, 1237-1260.
Pap, E., Takaci, Dj. and Takaci, A. A997), Partial Differential Equations through Examples and Exercises,
Kluwer, Dordrecht.
Pap, E. and Teofanov, N. A998), Pseudo-delta functions and sequences in the optimization theory, Yugoslav J.
Oper. Res. 8, 111-128.
Pap, E. and Vivona, D. B000). Non-commutative and non-associative pseudo-analysis and its applications on
nonlinear partial differential equations, J. Math. Anal. Appl. 246, 390-408.
Paris, J.B. A994), The Uncertain Reasoner's Companion, A Mathematical Perspective, Cambridge University
Press, Cambridge.
Puhalskii, A. B001), Large Deviations and ldempotent Probability. CRC Press.
Samborskii, S.N. and Tarashchan, A.L. A990), Semirings occurring in multicriteria optimization problems and
in analysis of computing media, Sov. Math. Dokl. 40, 441^45.
Sander, W. B002), Measures of information. Handbook of Measure Theory, E. Pap, ed., Elsevier. Amsterdam,
1523-1565.
Schweizer, B. and Sklar, A. A969), Measures aleatoires de I'information. C. R. Acad. Sci. 269 A. 721-723.
Schweizer, B. and Sklar. A. A983), Probabilistic Metric Spaces. North-Holland, Amsterdam.
Schwarz, F. A988), Symmetries of differential equations: from Sophus Lie to computer algebra, SIAM Rev. 30,
450-481.
Shafer, G. A996). The Art of Causal Conjecture. The MIT Press, Cambridge, MA.
Shilkret, N. A971). Maxitive measure and integration. Indag. Math. 33, 109-116.
Sugeno, M. A974), Theory of fuzzy integrals and its applications, Ph.D. thesis, Tokyo Institute of Technology.
Sugeno, M. and Murofushi. T. A987), Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122. 197—
222.
Wang, Z. and Klir, G.J. A992), Fuzzy Measure Theory, Plenum Press, New York.
Weber, S. A984), L-decomposable measures and integrals for Archimedean t-conorm, J. Math. Anal. Appl. 101.
114-138.
Zagordny. D. A994), The cancelation law for inf-convolution of convex functions. Studia Math. 110 C), 271-282.

CHAPTER 36
Qualitative Possibility Functions and Integrals
Didier Dubois and Henri Prade
IRIT-UPS. 118 Rome de Narbonne. 3/062 Toulouse. France
E-mail addresses: dubois@irit.fr. prade@irit.fr
Contents
1. Introduction 1471
2. Set-relations and ordinal belief structures 1472
2.1. Basic postulates of confidence relations 1472
2.2. Basic examples of confidence relations 1473
2.3. Representing confidence relations by set-functions 1475
3. Qualitative possibility theory 1477
3.1. Possibility distributions 1478
3.2. Information content of a possibility distribution 1479
3.3. Interpretation of possibility distributions 1481
3.4. Refined possibility relations 1483
4. Conditional possibility and plausible inference I486
4.1. Qualitative conditioning 1487
4.2. Plausible inference with a possibility distribution 1488
4.3. Universal possibilistic entailment 1491
4.4. Probabilistic interpretations of plausible inference 1494
5. Independence in qualitative possibility theory 1496
5.1. The paradoxes of probabilistic independence 1496
5.2. The minimum rule in possibility theory 1498
5.3. Possibilistic independence between variables 1499
5.4. Possibilistic event independence based on conditioning 1502
5.5. Qualitative independence and belief revision 1505
6. Qualitative integrals 1506
6.1. Possibilistic integrals 1507
6.2. Axiomatics for possibilistic integrals 1510
6.3. Sugeno integrals 1513
6.4. Comparing functions without commensurateness 1515
7. Conclusion 1518
References 1518
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1469

1470
D. Dubois and H. Prade
Abstract
Measure theory relies on numerical set-functions. Here we consider relations induced by
set-functions on a set of events. Such relations considered as primitive objects are more
natural than set-functions for the purpose of modelling subjective uncertainty. Starting from a
set of events and a preordering relation among events, some properties are described, that
such relations should obey in order to be natural candidates for representing uncertainty.
Well-known families of confidence relations on finite sets are surveyed, such as comparative
probabilities, first proposed by De Finetti. An important particular case is studied in more
details: comparative possibility measures. Their importance stems from the fact that they can
be simply induced from a complete preordering between elements of the referential. Such
orderings play an important role in Artificial intelligence, in theory revision and nonmonotonic
reasoning. Several issues are discussed like information contents and conditioning. It is shown
that possibility orderings are representable by means of idempotent measures or maxitive
set-functions, called possibility measures, taking values on finite chains. The compared
expressive powers between such set-functions and possibility orderings are discussed. Notions
of conditioning and independence and integrals on finite ordinal scales are presented.

Qualitative possibility functions and integrals
1471
1. Introduction
Measure theory relies on numerical set-functions. However, fields such as decision theory
and artificial intelligence consider numerical set-functions as too sophisticated notions for
the representation of uncertainty viewed from a subjective point of view. The purpose of
uncertainty modelling includes the representation of belief. Quantifying subjective belief
without resorting to objective frequency observations is a very difficult task in practice, and
a more natural approach is to model belief by means of relations on a set of events. Such
relations, considered as primitive objects, are more natural than numerical set-functions
since it sounds more reasonable for human agents to assert that an event is more likely than
another, than to precisely describe to what extent an event is likely. This chapter, whose
topic is related to but not central to measure theory, has but a limited ambition. It proposes
an introduction to the comparative counterpart to set-functions, their representations in
terms of numerical set-functions, and their connections, and pointers to a rather recent
literature which finds its origin in the works of some 20th century scholars such as De
Finetti A937), Koopman A940) and Ramsey A963). In particular, this paper has no
pretence of being a piece of genuine pure mathematics, only a gateway to interesting
mathematical issues.
Starting from a set of events and a preordering relation among events, some properties
are described in Section 2, that such relations should obey in order to be natural candidates
for representing subjective beliefs. Well-known families of confidence relations on finite
sets are surveyed, such as comparative probabilities, first proposed by De Finetti A937)
and possibility orderings, which are representable by means of idempotent or maxitive
set-functions, called possibility measures, taking values on finite chains.
This important particular case is studied in more detail in Section 3 devoted to qualitative
possibility theory. Its importance stems from the fact that possibility orderings on events
can be fully characterized by a complete preordering between elements of the referential.
Such orderings play an important role in Artificial intelligence, in theory revision
and nonmonotonic reasoning. Several issues are discussed further in this section, like
information content of possibility distributions and refinements of possibilistic orderings
using lexicographic notions.
Section 4 introduces the ordinal version of conditioning. Its links to non-monotonic
inference are explored in depth, by means of representation theorems. Some connections
to probabilistic reasoning are also outlined. Section 5 introduces qualitative counterparts to
probabilistic independence on finite ordinal scales and shows that in the qualitative setting
the language of independence is equivalent to possibility theory. Connections to belief
revision theory are pointed out.
Section 6 describes qualitative counterparts to integrals, focusing on possibility integrals,
and generalizing them to any qualitative monotonic set-functions, thus leading to the
so-called Sugeno integrals. Properties of such integrals are described and representation
results are surveyed, in the scope of decision theory. Lastly, the problem of constructing
a partial ordering relation on a set of functions with the same domain and range, from
the knowledge of complete preorderings on these domain and range is discussed. This
construction can be viewed as an attempt to defining a purely ordinal counterpart to an
integral.

1472
D. Dubois and H. Prade
2. Set-relations and ordinal belief structures
Let S be a finite set of elements called "states". States encode descriptions of possible
situations, states of affairs, etc. Subsets of states are called events. The finiteness
assumption is made for the sake of simplicity, and is natural when the description of
situations is achieved by means of a formal language such as propositional logic. Then,
S is the set of interpretations of the language. It comes down to understanding propositions
as ultimately true or false. Here we consider a syntax-free environment, from a logical
point of view. In particular, a proposition is understood as the set of states where it is true.
A relation, denoted ^L, on the Boolean algebra 2s of events A, B,C,... can be called
a set-relation, in analogy with set-functions. We shall focus on a particular type of set-
relations called confidence relations, adapted to the operational purpose of representing
beliefs. Then, a relation ^ ? is attached to an agent, and A ^ ? ? means "the agent has at
least as much confidence in event A as in event B". This relation describes an epistemic
state that refers either to a particular situation (for instance, the agent's belief that the next
person he will meet today is a close friend rather than an unknown person), or to a
collection of situations. In the latter case, the confidence relation pertains to concepts of
normality, typicality, as for instance in the statement "it is more typical that birds fly than not".
2.1. Basic postulates of confidence relations
With this interpretation in mind there are some naturally expected properties for set-
relations modelling relative likelihood:
reflexivity: A^l A;
completeness: A ^? ? or ? ^? ?;
transitivity: A^L В and B^LC ^ A^LC.
Under these properties (the first of which is redundant, mathematically speaking), the
relation ^ ? is a complete preordering; also called a weak order. While reflexivity is natural
because A ^ ? ? is understood as a weak form of dominance, the two other properties are
more debatable. They are natural in the scope of representing confidence orderings by
means of numerical set-functions, for which these two properties are bound to hold when
comparing the levels of confidence of events. Transitivity is also a way of enforcing a form
of rationality in pairwise comparisons of events in terms of relative likelihood by an agent.
Violating transitivity would be strange. Let the strict part of ^z. be denoted by >l, where
A >i В means that ?^? ? holds but not В ^l A. When both hold, this is denoted by
? ~?. ?. This relation is an equivalence relation that partitions 25 into families of equally
likely events. The quotient set 2s/~^ forms a chain.
At this point, some comments are in order. First, regarding completeness, one may in
the principle admit that two events be incomparable in terms of likelihood, and use partial
preorderings instead of weak orders (Halpern, 1997). Second, one may start from a strict
confidence relation: > ? irreflexive and transitive so as to express strict dominance in terms
of likelihood. Then weak dominance ?^? ? defined as "not (B >l A)" is not transitive,
generally.

Qualitative possibility functions and integrals
1473
Next come properties that express interactions between the relation and the Boolean
structure of its support. Stating from the intuition that if a proposition implies another,
the agent should be at least as confident in the former as in the latter, and understanding
implication as set inclusion, the following postulate makes sense:
Coherence with deduction: if А с ? then B^A.
From this postulate and transitivity, the limit conditions S1 ^z. A ^z. 0 follow. S is
understood as the ever sure event (the tautology), and 0 the ever false event (the
contradiction). Then the following condition will also be requested
non-triviality: S >l 0-
Lastly there are properties which involve symmetric set theoretic operations like union
and intersection. The weakest such property is as follows:
Weak stability: If (A U ?) ? С = 0 then A > L В implies AUC^t ? U C.
It expresses a minimal stability of the relation forbidding a strict dominance reversal
when augmenting the size of the two sets A and В in the same way. Interestingly,
using simple contraposition and some renaming, it is equivalent, to a conjunctive stability
property: A >L В implies ? ? С ^L В П С, if (А П В) U С = 5".
Overall we shall use the following notion:
Dehnition 1. A confidence relation is a set-relation which is a weak order, which is
non-trivial, coherent with deduction, and weakly stable.
2.2. Basic examples of confidence relations
Historically, the oldest set-relation is the comparative probability relation, first suggested
by De Finetti A937), and exploited by Savage A972) in decision theory:
Dehnition 2. A set-relation ^p is called a comparative probability iff
(i) ^p is a non-trivial weak order;
(ii) VA, A>p0 (consistency);
(iii) if А П (? U С) = 0 then: В ^Р С <S> AU В ^Р AU С (pre-additivity).
A comparative probability is a confidence relation. The preadditivity condition is
stronger than weak stability since it supports a cancellation property, clearly compatible
with the additivity property of probability measures. It is a strong stability condition since
augmenting or diminishing the size of the two sets A and В in the same way strictly
preserves the ordering relation. The coherence with deduction is a corollary of Definition 2.

1474
D. Dubois and H. Prade
So far we did not consider set complementation and the behavior of set-relations
with respect to this connective. Comparative probabilities are stable with respect to
complementation, in other words, they are self-dual relations:
A^pB <s> ?>?? (self duality).
A more recent confidence relation was introduced by Lewis A973) in the setting of
a modal logic of counterfactuals, and independently rediscovered by one of the authors
(Dubois, 1986): the comparative possibility relation.
Dehnition 3. A set-relation ^/7 is a possibility relation iff
(i) ^/7 is a non-trivial weak order;
(ii) VA, A >n 0;
(iii) VA, В ^п С =>· A U В ^п A U С (disjunctive stability).
В^п С reads В is at least as possible (that is, at least as plausible for the agent) as C, for
reasons that will become clear below. Again, the disjunctive stability is stronger than weak
stability, but it cannot be compared with the preadditivity property. This kind of set-relation
has been more recently used by Grove A988) in the setting of belief revision theory.
Possibility relations are not self dual. The dual set-relation defined A ^-n В о· В ^п A is
called a necessity relation (Dubois, 1986). It can be independently defined as follows:
Dehnition 4. ^N is a necessity relation iff
(i) ^-n is a non-trivial weak order;
(ii) VA, S^N A;
(iii) B^NC^-AnB^NAnC (conjunctive stability).
В >yv С reads В is at least as necessary (that is, certain for the agent) as С The
conjunctive stability is also stronger than weak stability, but it cannot be compared with
the preadditivity property, nor to disjunctive stability. The dual of a necessity relation
is a possibility relation and conversely. The duality between possibility and necessity
relations is a counterpart of the duality between possibility and necessity modalities in
modal logic Hughes and Cresswell A968). Possibility and necessity relations can actually
be axiomatized in the framework of modal logic, as first shown by Farinas del Cerro and
Herzig A991), based on conditional of Lewis A973) (the so-called VN conditional logic).
Other attempts include Boutilier A994), Farinas del Cerro et al. A994), Klir and Harmanec
A994), Hajek and Harmancova A994), Hajek et al. A994).
It has been pointed out (Dubois and Prade, 1991) that a necessity relation is very close
to an epistemic entrenchment relation after Gardenfors A988), except that in the epistemic
entrenchment framework, the stronger condition S >n A, for ? ? S, is requested instead
of (ii) Definition 4, i.e., non-tautological statements are always strictly less certain than
tautologies. This is natural in the framework of Gardenfors' belief revision theory, where
any non-tautological sentence may disappear upon a revision process.

Qualitative possibility functions and integrals
1475
The set-relations that are at the same time any two of possibility, necessity or probability
orderings reduce to deterministic (truth-like) set-relations, defined as follows:
3!s°eSVA,B, A> В iff s° e A, s° e B, and A ~ В otherwise.
The state s° is such that S ~ {s0} > 0, and (s) ~ 0 otherwise for s ^ i°. So, for any
event A, A~{i°}ifi°eA, and A ~ 0 otherwise. The name "deterministic" stems from
the idea that the agent knows that the real state of the world is 5° so that any event can be
told true or false by this agent with certainty. This is the case of no uncertainty.
2.3. Representing confidence relations by set-functions
As opposed to set-relations that offer a tool for a relative representation of belief, set-
functions are the natural tool for absolute representations of belief. Let L be a totally
ordered scale with bottom 0 and top 1. L can be a finite chain, the set of integers, the
unit interval, for instance. A set-function is a mapping ? from the set of events 2s to L. It
assigns to each event its level of confidence. Some natural conditions must be fulfilled by
a set-function that models the epistemic state of an agent:
Definition 5. A confidence function is a set-function ? such that
(i) <p@) = 0, <p(S) = 1 (limit conditions);
(ii) if А с ? then ?(?) ^ ?(?) (monotonicity).
The limit conditions are self explanatory and the monotonicity expresses the coherence
of confidence functions with deduction see Grabisch B002) for a survey (Chapter 34 in this
Handbook). A set-function ? induces a set-relation >? such that В ^l A iff ?(?) ^ ?(?).
Conversely a class С of set-functions ? is said to be a representation of a class C' of set-
relations ^ i provided that for any set-relation ^ r in C', there is a set-function ? in С such
that
?(?)>?(?) iff B^rA.
This is the natural bridge between absolute and relative representations of uncertainty.
A lot of works exist on the representation of comparative probability. See Fine A973), and
Fishburn A986a) for surveys. Of course for the representation of comparative probability
by numerical probabilities, one must choose L = [0, 1]. In the infinite case it is possible to
represent comparative probabilities by means of probability measures, provided that some
continuity axiom is added to the basic properties of comparative probabilities (like
Savage A972) did). However, in the finite setting, there is a famous counterexample of Kraft
et al. A959) showing that while any probability measure induces a comparative
probability, the converse is false, namely there are comparative probabilities which are not repre-
sentable by any probability measure. This negative result can be addressed by finding a
stronger substitute to preadditivity that guarantees probabilistic representations. However,
proposed axioms turned out to be either too strong or unnatural, from a modeling point of
view.

1476
D. Dubois and H. Prade
On the contrary, possibility and necessity relations on finite sets can be represented
by specific classes of confidence functions, and the choice of the scale L is arbitrary
(Dubois and Prade, 1998). It can be a finite chain or the unit interval, or even the set
of integers. The only set-functions capable of representing any possibility relation are
possibility functions (also called possibility measures, Zadeh A978)) that are maxitive
set-functions ? satisfying Л@) = 0,17(S) = 1 and
П(А U ?) = тах(Л(А), П(В)) (maxitivity).
Any maxitive set-function induces a possibility relation and conversely any set-function
representing a possibility relation is maxitive (Dubois, 1986). Similarly, the only set-
functions capable of representing any necessity relation are necessity functions (also called
necessity measures, Dubois and Prade A980)) that are "minitive" set-functions N verifying
N@) = O, N(S) = I and
N(A П В) = min(N(A), N(B)) (minitivity).
When L is embedded into the unit interval, these set-functions are unique up to a
monotonically increasing transformation. The duality property between possibility and
necessity can be expressed, in the case of set-functions, using an order-reversing bijection
? of L on itself, namely ?(?) = v(N( A)), where A is the complement of set A. Namely
? is a possibility function if and only if its dual ?'' such that ?'(?) = ?(?( A)) is a
necessity function. Representing epistemic states by means of necessity or possibility
functions, rather than using comparative notions, is a matter of convenience. These set-
functions were first suggested as useful models of uncertainty in terms of possibility by the
economist Shackle A961). He called the quantity N( A) the degree of potential surprise
of A.
The decomposability properties of probability, possibility and necessity functions
make them attractive because of their structural simplicity. Probability functions ? are
completely determined by their distribution on S. Similarly, possibility and necessity
functions are completely determined by so-called possibility distributions on S, which
are mappings ? from S to L such that n(s) = 1 for some state s. In order to keep this
structural simplicity for more general set-functions, the following decomposability axiom
can be adopted: there exists an operation * on the unit interval such that
ifAnfi = 0 then<p(AU ?) = ?(?) *?(?) (disjunctive decomposability).
Such non-additive representations have been introduced by Dubois and Prade A982) and
Weber A984), and studied at length by Pap A995) see Chapters 23 (Butnariu and Klement,
2002) and 35 (Pap, 2002) of the Handbook for surveys. Similar ideas have been proposed in
the literature of decision theory under the name "distorted probabilities" (Yaari, 1987). An
interesting question is then, to find a relational counterpart of disjunctively decomposable
measures. The following axiom, a weak form of preadditivity we call "almost disjunctive
stability", was proposed by Dubois A986):
if An(fiUC) = 0andfi^C thenAUS^AUC

Qualitative possibility functions and integrals
1477
It is weaker than both disjunctive stability and preadditivity. Results in Dubois A986),
refined by Chateauneuf A996), indicate that indeed the only numerical counterparts of
almost disjunctive stable confidence relations are disjunctively decomposable measures.
Of course set-functions dual to disjunctively decomposable measures can be studied too
and satisfy a conjunctive counterpart to the above axiom.
Other well-known set-functions can be described at the relational level: probability
envelopes and belief functions. Upper and lower probabilities (Walley, 1991) can be
used as degrees of plausibility and certainty, respectively. This is the path followed by
Shafer A976) and Smets and Kennes A994) for whom degrees of subjective belief can be
expressed by means of two set-functions induced by a third one called basic probability
assignment, which is formally a random set, i.e., a probability distribution m on 2 such
that w@) = 0. A belief function is a numerical set-function Bel defined by
Bc\(A) = E{m(B), B^A].
The dual set-function P1(A)= 1-Bel( A) = ?{??(?). ВП А фЩ is called plausibility
function see Grabisch B002) (Chapter 34 in this Handbook). These set-functions subsume
probability functions (obtained when A is a singleton whenever m(A) > 0). Numerical
possibility (respectively necessity) functions are special cases of plausibility (respectively
belief) functions (obtained if, whenever m(A) > 0 and m(B) > 0, А с В or В с A). Wong
et al. A990) have proposed families of set-relations that can always be described by belief
functions. Such relations obey the following axiom, for disjoint events А, В, С:
A U S >Bei В =>· A U В U С >Bei BUC.
Comparative plausibility relations satisfy the dual axiom:
A U В U С >pi В U С =>· A U В >pi В.
Fishburn A986b) also adopts a purely ordinal view of more general classes of upper
and lower probabilities, based on interval-orderings. However his axioms for Shafer
belief functions are too technical to be meaningful. Research on the determination of belief
function-based orderings is continuing. A natural line of research is to carry an ordering
relation satisfying Definition 2 from a set of events to another one, through a multiple-valued
mapping; it comes down to using Dempster's A967) definition of belief functions based
on multiple-valued mappings where the basic probabilistic assignment is changed into an
ordering relation (Wong et al., 1993).
3. Qualitative possibility theory
The simplest confidence relations are comparative possibilities and necessities, because
they can be fully characterized by complete preorderings of elementary states. Hence the
computational complexity of representations of such confidence relations is linear in the
number of states while it is exponential in general for confidence relations. In particular,
comparative probabilities cannot be defined in this way, contrary to numerical probabilities.
The simplicity of possibility theory makes it an attractive tool for representing knowledge.
In this section, we present two kinds of qualitative representation of possibilistic

1478
D. Dubois and H. Prade
knowledge: comparative possibility distributions and finitely-scaled ones. We show that
they are not exactly equivalent, and that the latter are more expressive than the former.
The comparison of possibility distributions in terms of their informational content leads
to different results in each case. Lastly the interpretation of possibility distributions is
discussed, and the basic blocks of possibilistic reasoning are laid bare. It is claimed that
possibility theory is a natural setting for plausible reasoning, that is, reasoning under
incomplete information, as if the state of the world were systematically normal.
3.1. Possibility distributions
A possibility relation ^/7 and its dual necessity relation >yv are fully characterized by a
so-called comparative possibility distribution (cpd) ^? which is a complete preordering
on the set of states S. Namely, define ^? as
s>Rs' iff H^nis')-
Then the possibility and necessity relations for non empty sets are recovered as follows:
A^n В iff 3i e A, Vs' e B, s ^? s',
A^NB iff 3seB, Vi'eA, ?,??'·
However the cpd does not tell anything about the possibility relation linking any event
A to the empty set 0. Namely, in order to recover the full possibility relation, one has to
add 0 ^/7 A or A >/7 0 for each subset A, according to whether event A is impossible or
not.
A pair (S, ^тг) where S is a finite set and ^ is a complete preordering will be called
a possibility structure. The set of states of affairs is then partitioned via ^? into a totally
ordered set of ? + 1 clusters of equally possible states S = Eq U E\ U · · · U E„ by means of
the equivalence relation ~? induced by the symmetric part of °^?. This set of equivalence
classes is linearly ordered by the strict part of the possibility relation
?0 >n E\ >n ¦¦¦ >n E„.
The set ?0 contains the most possible states that will be considered as the normal ones.
The set E„ contains the least possible states and, by convention, starting from a given cpd,
these states are not fully impossible (E„ >n 0), unless the contrary is explicitly said. The
family ?o>/7 E\ >n ¦¦¦>n E„ is called a well-ordered partition (wop) by Spohn( 1988).
Another kind of qualitative possibility distribution is the finitely-scaled one defined as a
mapping ? from S to a finite chain L. Let Я be a set-function representing a possibility
relation ^/7, and N its dual possibility function. Then the finite chain L possesses a number
of elements equal to the number of equivalence classes of the symmetric part of ^/7. The
possibility distribution (pod) ? is then defined as n(s) = T7({s}), Vs e S. Then we can
consistently recover
?(?) = max n(s), N(A) = minu(n(s)),
s?A s?A
where ? is the order-reversing map on L. The cardinality of L is ? + 2, generally, one per
element of the wop, plus 0 (except if En ~n 0, and then it is ? + 1). In order to ensure that

Qualitative possibility functions and integrals
1479
n(S) = 1, it is requested that n(s) = 1 for some state, that is, some state is fully possible,
normal. The possibility distribution is then said to be normal.
There are several noticeable types of possibility distributions:
- deterministic possibility distributions such that n(s) = 1 for some state s, and n(s') =
0 for all 5 ^5'. For all events А, П(А)= 1 > ?( A) = 0or ?( A) = 1 > ? (A) = 0;
- linear possibility distributions: they are such that the corresponding cpd is a total
ranking of S; the associated WOP contains only singletons. They are deterministic by
default, in the sense that for all events ?, ?(?) > ?( A) or ?( A) > ?(?);
- non-dogmatic possibility distributions: they are such that n(s) > 0 for all states; no
state is totally excluded;
- subnormal possibility distributions: they are such that n(s) < 1 for all states; no state
is totally possible, every state is somewhat abnormal. It expresses a state of partial
inconsistency. The logical contradiction corresponds to a subnormal pod n(s) = 0,
Vie 5".
Are cpd and finitely-scaled pod equivalent representations? Not exactly. The relative
expressive power of the set of complete preorderings on S and the set of mappings ? from
5" to L is different from several point of views.
First, an ordering ^л of states defined on S cannot express that some states are
impossible, as pointed out above. With an absolute scale this is easy: one just has to
let n(s) = 0 for impossible states. In particular, it is easy to represent deterministic
information with a pod. On the contrary, it is not possible to express this deterministic
situation with a cpd alone. One needs the whole possibility relation on the power set
Moreover, it is possible to capture the idea of internal conflict with a pod. A subnormal
? is interpreted as a partially inconsistent epistemic state, and the level ConsO) =
max565 7r(i) is the degree of consistency of ?. It is easy to see that for any event A,
min(N(A), N( A)) = u(ConsO)) > 0. A partially inconsistent epistemic state is one for
which an event and its contrary are somewhat certain. On the contrary, cpds cannot be
compared with regard to the notion of internal conflict. Two possibility distributions ? and
? may correspond to the same cpd(^T= ^?), while we may have ConsO) < Сош(д).
In the scope of information fusion, it is better to have scaled representations under the
form of pods with the same range, that can be aggregated because they are commensurate.
In contrast, ordering relations are not commensurate: how to combine a confidence relation
^tt and a preference relation ^ ? on SI As first established in social choice theory (Moulin
A988), for instance), there are very few reasonable aggregation schemes, if any. The
situation is easier if there is a common scale.
Lastly, notions of quantity of information contained in a possibility structure cannot be
equivalently defined with absolute and relative representations, as seen in the next section.
3.2. Information content of a possibility distribution
Now that a model of an epistemic state has been described, the next question is to qualify
epistemic states in terms of information content. Namely, how informative is an epistemic
state? This notion is crucial in the face of an incompletely specified epistemic state. Then

1480
D. Dubois and H. Prade
one might be interested in finding the least informative possibility ordering compatible
with the explicit knowledge.
A natural idea is to define an ordering relation between epistemic states, whose
meaning is "more informative than". Let ^T and ^;< be two cpds corresponding to wops
?oU?| U · · · U E„ and Fq U F\ U · · · U Fm. A first natural idea is to use the notion
of refinement of a relation. Namely: a cpd ^? is said to refine ^T iff Vs, s' if s ^? s'
then s ^? s'. In terms of wops, one gets more refined by splitting equivalence classes
Ei. Namely, for all / = 0,..., n, E, = F}¦ U ¦ ¦ ¦ U Fi for some j and k, j <^k. The wop
Fq U F\ U · · · U Fm refines F() U E\ U · ¦ · U E„. The least refined cpd is unique and is
obtained when Eq = S: states cannot be distinguished according to plausibility. This cpd
brings no information at all. The most refined cpds are such that V/', ?, = {s} for some
s: they are all linear possibility orderings (no ties): they are informative as the agent can
always point out the most plausible one among any two states.
Comparing two normal L -valued pods leads to a different natural concept namely the
specificity ordering: ? is said to be less specific than ? in the wide sense iff ?(s) ^ ?(?),
Vs. It means that any state is at least as possible according to ? as according to ?. It can be
checked that any event is at least as sure according to ? as to тт. The least specific pod is
such that every state is fully possible: n(s) = 1, Vs. It corresponds to a fully ignorant agent.
The most specific pods are such that ??(?) = 1, for some unique s, and n(s') = 0, Vs' ? s,
they are the deterministic pods: the agent precisely knows what is the state of affairs.
Interestingly, linear possibility distributions are no longer the most specific ones. They are
less specific than deterministic pods. However, the least specific pod does correspond to
the least refined cpd.
It is interesting to try and relate the two notions of informational content comparisons,
the scale-based and the ordinal one. A notion of comparative specificity can be defined as
follows: >? is ordinally less specific than >(J, in the wide sense, if and only if, in terms of
their associate wops:
V/= l,...,min(w,«), F0U F| U---U F; C?0UE| U---U ?;.
Here the idea is that the specificity of an epistemic state is decreased by moving states
up towards more plausible elements of a wop. Maximally specific wops are also the most
refined ones, and the least specific wop is also the least refined. However ordinal specificity
and refinement are different orderings of cpds.
Note that if ~^? is less specific than ^? in the above sense, then there exist two pods ?
and ?, ordinally equivalent to >„¦ and ^;< respectively, such that ? is less specific than ?
in the wide sense. However these pods cannot be arbitrarily chosen: one may easily find
two pods ? and ?, ordinally equivalent to >T and ^;1 such that ? is not less specific than
? even if ^n is ordinally less specific than >;;. Even more, it is easy to see that if ? is
less specific than ?, then ^? may fail to be less specific than ^/(. The best example is
when ? is a deterministic pod while ? is a linear pod with n(s) = ?^) = I. Then ? is
strictly less specific that ?. However the wop induced by ? is (j)UX\|s), while the one
induced by ? is a maximally refined partition, which is ordinally strictly more specific than
(i)US\(s). This is partly due to the lack of expressiveness of wops, regarding impossible
states. However the same phenomenon occurs if ? is such that Vs' ? s, n(s') = a > 0 and
?(?') > a.

Qualitative possibility functions and integrals
1481
These remarks show that L -valued possibility distributions are more expressive and
more flexible than cpds, in terms of information contents. Besides the notion of ordinal
specificity is only apparently ordinal. In actual fact, it presupposes the use of a scale
L — {?? = 1 > ? ? > ¦ ¦ ¦ > ?„ > · ¦·} where, by convention, states in ?, have plausibility ?,.
In the case of the two-element wop (s)US\ [s\, states other than s are always supposed
to have plausibility ? ?.
So it seems at first glance that the notion of refinement is a more natural tool for
comparing the information content of cpds than ordinal specificity. Unfortunately, as
shown in Benferhat et al. A999), the problem of finding the least informative cpd obeying
constraints of the form А >п В cannot be solved using the least refined cpd obeying those
constraints. Such a cpd generally does not exist. However a unique least specific cpd always
exists, if the constraints are not conflicting.
The principle of minimal specificity argues that every proposition that is not known to
be impossible is supposed to be possibly true, thus leading to maximize the possibility
of states, each time this possibility is unknown. This principle of minimal commitment is
cautious and conservative. This is why wops of the form Eq U E\ are interpreted in such
a way that states in Eq are fully plausible and states in ? \ are as plausible as can be even
if less plausible than those in Eq. It contrasts with other kinds of informational principles
such as the closed world assumption (CWA) according to which any proposition not known
to be true is supposed to be false. The principle of minimal specificity and the CWA are
actually at odds with respect to each other. The former is more similar to the principle of
maximal entropy in probability theory, although this probabilistic principle rather suggests
maximal disorder or conflict between pieces of information, and is not concerned by the
preservation of information incompleteness. It leads to introducing default independence
assumptions. On the contrary the principle of minimal specificity prohibits the introduction
of arbitrary additional default information.
3.3. Interpretation of possibility distributions
A possibility distribution is supposed to represent the state of knowledge of an agent
regarding the state of affairs, either the current one (as of now) or the usual one (generally).
The complete preordering of states tells what is likely from what is less likely. Like a
probability distribution, it may represent generic information as to what is usually the
case and what is less usually the case, except that we do not presuppose the existence
of explicit statistics, nor must we refer to a well defined population of cases. Under
this view, s ^ s' means that state s is at least as normal or typical as s'. Alternatively
the pod may only refer to the current situation and what is currently the most plausible
one, according to the agent's subjective beliefs. Let ?(?) = [s. n(s) = 1} be the core of
the pod. In the following we assume normal pods. It coincides with the set ?? of most
plausible, or most normal states in the corresponding wop (according to whether we adopt
the point of view of specific or generic knowledge). This model involves a presupposition,
which is considered to be characteristic of possibilistic reasoning, namely that the agent
believes that, by default, and in the absence of any other information, the truth lies in
S(n), until getting new evidence that it is not the case. That is, the agent expects the

1482
D. Dubois and H. Prade
current state of affairs to lie in S(n). With this postulate in mind, the possibilistic ordering
possesses a clear meaning: В ^п С means that the expected situation where В occurs is
at least as plausible as the expected situation where С occurs. This is really comparing
propositions on the basis of their most normal models. This view drastically differs from
the probabilistic representation of belief, which cumulates likelihood of occurrence of all
situations realizing an event.
Now the dual necessity ordering precisely describes what are the propositions the agent
accepts as true by default in the face of the available knowledge modelled by a pod. They
are the propositions that are true in all normal, or maximally plausible situations, i.e., A
such thatSOr) с A, which is equivalent to A >yv 0- Events A such that A >yv 0 correspond
to accepted beliefs of the agent.
It is easy to prove that for any event A, if A >yv 0 then 0 ^/v A. If the agent accepts
one proposition, (s)he cannot at the same time accept its negation. As a consequence there
are three epistemic situations only regarding event A: either A >n 0, or A >yv 0, or both
0 ^yv A and 0 ^/v A. At one extreme end, there are the sure events A such that A ># S.
In the case of a non-dogmatic pod, the only sure event is S, the tautology. At the other
extreme end, there are the impossible events A such that A^n S (equivalently 0 ^/7 A).
Again, in the case of a non-dogmatic pod, the only impossible event is 0, the contradiction.
There is a third category of events, namely the ones the agent does not know anything
about, called the unknown (or uncertain) events, such that 0 ># A and 0 ^n A. These are
the propositions the agent is agnostic about. If the pod is deterministic (hence dogmatic)
there are only sure events and impossible events, and no unknown events. If the pod is
linear and non-dogmatic, then again there are no unknown events, but events are more or
less accepted or rejected.
Noticeably, accepted events are closed under deduction. This is due to the following
obvious properties
A >n 0 and Ac В imply ? >? 0,
A > ? 0 and В > yv 0 imply А П ? > # 0.
The set of accepted propositions induced by a pod forms a belief set in the sense of
Gardenfors A988). More generally, fixing any accepted belief С >n 0. the set {A ^ C),
called a cut, also forms a belief set for any C. This approach is in agreement with the idea
of "reasoning as if the current situation were normal", since, by considering the belief set
induced by a pod, the agent is jumping to conclusions obtained from the assumption that
the situation is as normal as can be.
In summary, the possibilistic representation of belief leads to sharing the set of
propositions, or events into 3 subsets:
accepted beliefs A: A >^ 0;
rejected beliefs TZ: A such that S >n A (or A >^ 0);
agnostic beliefs V: A such that neither A >n 0 nor A >yv 0 hold.
This partition is similar to the one in classical logic where a belief set is the set of
consequences of a set of formulas, the rejected beliefs are the negation of accepted beliefs
and the ignored beliefs are sentences which cannot be inferred, nor their negations. So

Qualitative possibility functions and integrals
1483
the possibilistic setting is like the one of classical logic: ? is deductively closed, and
incompleteness is captured. Unlike classical logic, ? is ranked in terms of certainty, and
1Z is ranked in terms of impossibility. Any cut of ? is deductively closed as well. This
feature confirms the interpretation of the necessity relations in terms of acceptance: the
agent reasons with accepted beliefs by means of classical logic, as if they were true.
But the complete preordering in A captures the idea that some accepted beliefs are more
entrenched than others, following Gardenfors, and that some beliefs may be more easily
given up than others upon the incorporation of new pieces of information.
3.4. Refined possibility relations
Two events A and В cannot be discriminated using the dual pair (?, П) as soon as ? (A) =
?(?) and N(A) = N(B), which occurs quite often, namely, as soon as тахл6д n(s) =
тах.5бВ n(s) < 1 or тахл^д n(s) = тах^д n(s) < 1. The equality ?(?) = ? (?) may
be due to a high value of П(В ? A). However, in the spirit of probability theory, this weight
should not affect the comparison between A and B, since it is common to both. The same
reasoning_applies to П( В ? А) with respect to the equality N(A) = N(B). Only В ? A
and ? ? A should matter in telling A from B. Let us introduce the relation >riL on 2 ,
which is different from, although closely related to the original necessity and possibility
orderings modelled by N and its dual ?:
A>nLB iff П(ВПА)>ЯEПД),
A^nLB iff not (B>nLA).
The relation >nt is a refinement of both possibility and necessity orderings in the
following sense:
N(A) > N(B) => A >nL В. П(А) > П(В) => A >nL B.
Of course, it may happen that both N(A) = N(B) and П(А) = П(В) hold and that
A >nL B. Note that ^/7 and ^riL coincide when restricting to mutually exclusive
events A and В since then П(А П ?) = 0. More generally ^/7 and ^nt coincide
whenever ? (А ? B) < min(/7(A П В), П( А П ?)) which is to say when the property
?(? ? ?) = тт(Л(А), П(В)) does not hold, a possibilistic counterpart to independence
(see Section 5). Notice that the relation A ~til В iff neither A >r\L B_ nor ? >_?? A
hold on 25 is not transitive since Я(А П ?) = ?(?~ ? ?) and ? (В ПС) = П(ВГ)С)
do not imply ?(? ? С) = П( А П С). This is because тах(л-?, ?}) = тах(тг4, щ) and
тахОз.тгб) = тахОз, ??) do not imply тах(Л4.Л7) = тах^т.л^) denoting ?, =
? (Si), A = {52.53,55}, В = {s2,S4,S(,}, С = {53,54.57} (even if ? takes on two values,
0 and 1 only).
Interestingly, the refined possibility and necessity relation ^/7/, satisfies not only
coherence with deduction, but also the preadditivity condition (iii) of comparative
probabilities, and the relation ^riL turns out to be self-dual see Dubois, Fargier and Prade
A998b). Moreover if ? is a proper subset of A then A >nt В always holds for non-
dogmatic possibility distributions.

1484
D. Dubois and H. Prade
So, the relation ^til possesses all the properties of a comparative probability, except
for the transitivity of indifference. In the following, the possibilistic relation ^riL will
be called possibilistic likelihood and A ^riL В reads "A is at least as likely as ?",
so as to emphasize the probabilistic-like self-duality. However, because of the lack of
transitivity of the indifference relation, we cannot generally represent such relations by
means of probability measures,' nor can we assume properties that are usually derived from
comparative probabilities. For instance, the following property of comparative probabilities
is NOT satisfied in general by possibilistic likelihood:
If АПВ = 0апс1СПО = 0,
then A ~p С and B~P D
<S> AU B^pCUD.
Counter-example. 5" = [a,b,c,d,e, f.g.h); A = [a,b,c]; В = {e, f, g}; С =
[b, g, h}; D = {c, d, e]. Suppose that ?(?) = n(f) = л(с) = n(g) = 1; ??) = ?(?) =
л(е) = л(И) = 0.
П(АГ\С) = П({а,с}) = \ and П(С ? ? ) = n{{g, h}) = 1: A ~nL С,
n(BD~D) = n({f,g}) = \ and Я(ОП ?) = n({c,d}) = 1: ? ~nL D,
П((АиВ)Г)СГI>) = n({a,f})= 1 > Я((С U D) П АПб)
= n({d,h}) = 0.
Thus: AU В >nL CUD.
Moreover, the following very strong, non probabilistic property holds for the
possibilistic likelihood:
For A, S, С disjoint, AUC >?? ? and BUC >nL A imply С >nL AU B.
Hence, although similar to a comparative probability relation, possibilistic likelihood
noticeably differs from it.
In terms of well-ordered partitions, the possibilistic likelihood can be described as
follows:
A > n L В iff there is an Ek such that ? ? S ? Ek ? 0 and ВПАП?(=0, while
А П В П ?? = 0 and ? ? ? ? ?,¦ = 0 for all /' < ?. In particular if ? is uniform (total
ignorance), ? >/7?. ? if and only if ? is a proper subset of A. The possibilistic likelihood
coincides with the inclusion relation. At the opposite, if ? is linear, then for any ? ? ?,
either A >nt В от В >nt ?.
More generally the relation A >ni В can be generated from the restriction of the
regular possibilistic relation to disjoint events (this is a partial ordering) completed using
the preadditivity property for non disjoint events.
'This representation is not always possible with standard comparative probability on finite sets (Fishbum.
1986a).

Qualitative possibility functions and integrals
1485
The problem of axiomatically characterizing these uncertainty relations remains open.
However, they belong to a more general family of what can be called partial additive
set-relations ^i on events, that are characterized by the same axioms as comparative
probabilities but for the completeness of ^/.. A large class of such relations can be
defined by means of incompletely specified numerical weights ? = (p\,..., p„) assigned
to elements of S, and a set of constraints on the p'ts defining a domain ? of feasible-
tuples p. Then define (Lang, 1996):
S>z.A iff Y^p(s)^Y^p(s). Vpe/i.
seB seA
By suitably restricting ?, the possibilistic likelihood relation can be recovered
(Benferhat et al., 1998).
Similarly, one can further refine the possibility relations between events A and В
by considering a lexicographic ranking of reordered elements in A and B. First let us
define the leximax ordering of vectors. Let V and W be two vectors having m L-valued
components respectively denoted u, and ш,-. Denote V > leximax W whenever
Зк, гю[ц = V[i] for/ < к and ищ > w\k\
where W[\] > W[i] > ¦·· > w\m] and t>[i[ ^ f|2| ^ ¦ ¦ · ^ i>[w| result from a decreasing
reordering of components. Now if a is the vector (?? a,„) such that, denoting S =
{?|,?2. ¦¦¦,Sm}:
aj = ??(?,) if Si e A,
a; = 0 otherwise,
then define an ordering among events by A >/7Lex ? ?· ? >leximax ^· This relation will
be called "possibilistic lexicographic likelihood" ("leximax" likelihood for short). The
complement ^riLex of this relation is a complete preordering of events which refines the
possibilistic likelihood
^ruin terms of well-ordered partitions, the leximax likelihood can be described as follows:
A>/7LexS iff 3Ek: \АПЕк\>\В Г) Ek\ and \??\??\ = \? ? Et\
for all i < k.
The leximax likelihood relation can be generated from the well-ordered partition
{Eq, ..., Ep] of S as follows
@) {*}>/7Lex0;
(i) ifi,i'e ?,, then {i}~/7Lex {i'l:
(ii) if s e Ei, and А с Ei+\ U --·? ?,, then {s} >/7Lex A;
(iii) apply additivity: В >nLe\ С О A U ? >/7Lex A U C.
For a uniform possibility distribution the leximax likelihood relation coincides with
the comparative probability relation that is induced by a uniform probability. This is

1486
D. Dubois and H. Prade
not surprising in view of the fact that the leximax likelihood relation is a comparative
probability in the usual sense. Even more, any leximax likelihood relation can be
represented by a special kind of probability measure. Indeed, consider the well-ordered
partition {?0 E„] of S. Let /„ = |?„| and /, = \E,\(fi+\ + 1), for / < ? decreasing
to 0. Let К be the sum of the ffs. Define a probability distribution ? such that p(s) =
fi/(K -\Ei\) if seEi. Then
S^/7LexC О P(B)^P(C).
The possibilistic likelihood and the leximax likelihood relations coincide for linear
possibility distributions. Under the linearity assumption possibilistic likelihood relations
can be represented by a very special kind of probability measures introduced by Snow
A994, 1999) and also studied in Benferhatet al. A999). It is called an atomic-bound system
by the first author.
Definition 6. The atomic bound system on S is the set of probability distributions ?
such that p(s) > 0, Vs, each inducing a linear ordering > on S, whereby s > s' if and only
if p(s) > p(s') and such that for each interpretation s, p(s) > ]ГД... ?>?. p(s').
A probability distribution in an atomic-bound system is such that the probability of an
interpretation is much bigger than the probabilities of each less probable interpretation.
Such a distribution is called a big-stepped probability (bsp) by Benferhat et al. A999)
and is such that VA с 5", 3s e A, P((s}) > P(A \ {s}) so that ? (A) > P(B) if and
only if тахFд P(s) ^ maxv6« P(s): the magnitude of P(s) can never be attained by
summing the probabilities of states that are individually less probable than s. Moreover
P(s\) > P(s2) > ¦¦¦ > P(s,,-\) > P(s„) for states s\,...,s„ of nonzero probability.
Clearly a lot of probability measures are ruled out by these conditions.
Example. 5" = {a, b, c, d, e] and ? = (?/0.6, b/0.3, c/0.06/, d/0.03, e/0.01).
The above defined probability is an example of a bsp when \E,¦ | = 1 for all /. Given a
linear pod ? which induces an linear ordering of states > rs that coincides with the ordering
induced by a bsp p, it is obvious that ?(?) ^ ?(?) ? P(A) ^ P(B).
In fact the possibilistic likelihood relation family marginally intersects the set of
comparative probabilities (and leximax likelihoods), while leximax likelihoods form a
subclass of comparative probabilities.
4. Conditional possibility and plausible inference
In this section, a qualitative possibilistic counterpart of conditional probability is presented.
While this notion is very simple, it is shown that it leads to a form of plausible reasoning
that has been axiomatized in the literature of artificial intelligence by Lehmann and
colleagues (Kraus et al., 1990; Lehmann and Magidor, 1992), and in philosophy of science
by Adams A975) and Lewis A973), under very different frameworks and motivations.

Qualitative possibility functions and integrals
1487
This view of plausible reasoning is also captured by the so-called preferential inference
proposed by Shoham A988). It accounts for the idea that tentative conclusions derived in
the absence of complete information can be questioned, and even refuted, by the arrival
of new information. This non-monotonic behavior is also the one found in Gardenfors
A988) theory of belief revision. At the heart of these various systems and application
areas, possibility orderings are at work.
4.1. Qualitative conditioning
Let ^/7 be a possibility ordering, and С an event, supposed not to be impossible, which
characterizes a context of interest. A possibilistic ordering ^cn restricted to the subsets of С
can be induced as follows: A is at least as possible as В in the context С > /7 0 (denoted by
A ^ сп В) if and only if А П С ^ /7 ? П С. The relation ^ cn is called a conditional possibility
relation. This kind of conditioning can be applied to any relation between events.
For finitely-scaled normal possibility measures valued on L, the following quantitative
conditional possibility function has been defined by Dubois and Prade A988):
jl ifA^0. Л(С) = Л(АПС)>0,
\П(АГ\С) if П(А Г) С) <П(С).
It is also, by virtue of the minimal specificity principle, the greatest solution to the
Bayesian-like equation
?(? ? С) = тт(Л(А|С), П(С)).
The possibility distribution associated with Л(- \ С) is given by
л(и\С)
1 if л(и) = П(С). и еС,
л-(и) if л(и) < П(С), и е С,
10 ifu<?C.
Note that if П(С) = 0 then ?{- \ С) is still a solution to the Bayesian-like equation and is
equal to the characteristic function of C, which is simply substituted to ?. The only
difference between conditional possibility and conditional probability is that the renormalization
via division cannot be carried out. It is changed into a simple move of the most plausible
elements in A to 1. Ramer A989) noticed that only one element with possibility П(С)
needs to receive possibility 1 after conditioning by A. in order to minimize the change in
the possibility values.
The purely ordinal view of conditional possibility theory agrees with the finitely scaled
form of conditioning in the sense that if a possibility measure ? represents a possibility
ordering ^n then the conditional possibility measure П(- \ С) represents the conditional
possibility ordering ^cn.
Qualitative conditioning makes sense in a finite setting only. Indeed this form of
conditioning applied to infinite numerical settings creates discontinuities, and the infinite maxi-
tivity axiom is then not preserved by conditioning (see De Baets et al. A999)). Especially, if

1488
D. Dubois and H. Pnide
?(?) = supiFfi тг(и) holds, П(В\А) = supueBn(u\A) may fail to hold for non-compact
events В (De Cooman, 1997). The dual conditional necessity orderings can be
unsurprisingly defined as follows:
A^CN В if and only if ? >^ A.
The relation A^CN В reads "A is at least as necessary as В in the context С >n $"· It is
equivalent to С U A ># С U ?, not to А П С ^yv ? ? С. The latter is indeed equivalent to
A^yvS or A^n С and is not a proper translation of conditioning as focusing on reference
class С For L-valued necessity functions on L, the conditioning of necessity functions by
duality property corresponds to the following definition: N(A\C) = ?(?( A\C)).
It is easy to interpret the condition N(A\C) > 0 (or A >^ A) by the sentence "A is an
accepted belief in the context C". Indeed a basic property of qualitative conditioning is,
IfC>/70, thenN(A|C) >0 iff AnC>/jSnC,
that is, A and С are conjointly more plausible than С and the negation of A. It easily
follows that the set A(C) = {A: N(A\C) > 0} of conditionally accepted beliefs is
deductively closed. But it may be the case that N(A) > 0 and N(A\C) > 0. It means that
while A is an accepted belief, this belief is rejected, and even refuted in the context C. This
is clearly a non-monotonic behavior, that comes close to a form of plausible commonsense
inference described in the literature of non-monotonic reasoning, as seen later.
One may be tempted to consider that qualitative conditioning on a finite scale is
more expressive than ordinal conditioning. This is only partly true because qualitative
conditioning cannot express a graceful attenuation of belief, due to the following
property (Dubois et al., 1997):
THEOREM 1. IfN(A) > 0 then either N(A\B) ^ N(A) orN(A\B) = 0.
When ? becomes known, A is either at least as sure as it was, or is canceled (N(A\B) =
N(?|?) = 0) or refuted by В (N(A~\B) > 0). Hence, the absolute value of N(B\A)
is not meaningful on a finite scale L. When N(A\B) > N(A), one may be tempted to
consider that the belief in A is reinforced by the knowledge of B. However if a further
piece of plausible knowledge С comes to be known, that somewhat contradicts A, then,
N(A\B П С) = 0. This problem does not occur with quantitative conditional possibility
defined by
?(?|?) = ?(? ? ?)/?(?), ?(?\?) = 1 - ?(?\?)
for which N (A) > N(A\B) > 0 is a feasible situation.
4.2. Plausible inference with a possibility distribution
Shoham A988) proposed to understand the default inference of a proposition A from a
proposition С in terms of the truth of A in all the best models of C, where the term "best"

Qualitative possibilin functions and integrals
1489
means the maximal models of С in terms of some prescribed relation on the set of states.
In the following, this relation is considered to be a non-dogmatic possibility preordering
of states ^?, and the best models of a proposition С ? 0 are understood as the maximal
models according to this (complete) preordering. These are the most plausible (normal)
states when A is true. The restriction to non-dogmatic orderings enables any non empty
subset of states С to be considered as a not fully impossible context for conditioning. Let
us define a possibilistic consequence relation (=T between propositions, induced by the
knowledge of the cpd ^n as follows:
С\=ЛА iff СГ\А>пСГ)А.
The notation С \=? A, which reads "A is a plausible consequence of С (in epistemic
state ?)", is a logical (semantic) substitute to the condition N(A\C) > 0, which stresses
the close connection between the idea of conditioning in uncertainty theories and the notion
of deduction in non-classical logics. It is easy to see that it exactly matches the intuition of
Shoham A988) since it holds that С (=т A iff A is true in the most plausible worlds (in the
sense of ^?) where С is true. Using an absolute representation of the possibility ordering
on states: С \=? A iff Vs e C, if n(s) = П(С). then s e A. This is pictured on Figure 1.
This inference is less demanding than the usual deduction of A from С that requires that
all models of С be models of A. It has much intuitive appeal as a reasonable representation
of how people make inferences in real life situations, when they have a clear idea of what
is normal and what is not. In order to figure out what can be said about the state of the
world when the (incomplete) piece of information С is available, an agent will assume the
situation is as normal as can be (in the context delimited by C) and will jump to conclusions
that can be made under this normality assumption.
In order for the inference С \=? A to qualify as a consequence relation of some kind,
it is important to characterize it by its properties. For instance it is well known that the
standard consequence relation С (= A in logic (corresponding to an inclusion relation of С
into A) satisfies the following (semantic) characteristic properties:
reflexivity: С \=C;
monotonicity: С \= A implies В П С (= A;
transitivity: if С (= В and В (= A then С \= A.
The relation \=? violates all these properties. In contrast, noteworthy properties of (=,T
are as follows:
С (=7г С if С ф 0 (restricted reflexivity);
? preferred worlds in С
Fig. I. C plausibly implies A.

1490
D. Dubois and H. Prade
0 (=7г A never holds (nihil ex absurdo);
if С ? 0, then С (=7г 0 never holds (consistency preservation);
if А с ? and С \=? A then С (=? В (right weakening rule RW);
if С (=7г A and С (=? ? then ? П С (=л· A (cautious monotony CM);
if С (=7г ? and ? П С (=7г A then С (=.т A (cut, or weak transitivity);
if С (=7г A and С (=? ? does not hold, then В DC \=? A (rational monotony RM);
if ? П С (=7г A then С (=?г ? U A (half of deduction theorem);
if ? \=? ?; ? (=? С then ?\=? ВПС (Right AND);
if ? (=? С; ? (=? С then A U ? (=,т С (Left OR).
Let us comment on these properties so as to assess their cognitive appeal. Reflexivity is
restricted to non contradictory propositions. Indeed, the notion of "most plausible model"
is not mathematically defined on an empty set. Intuitively С = 0 corresponds to an agent
receiving inconsistent information that prevents him from deriving any sensible conclusion.
The impossibility to plausibly infer in a conflicting epistemic state is stressed by the "nihil
ex absurdo" property. Consistency preservation states that in a coherent context the agent
should not infer contradictions. Right Weakening is easily checked from the definition of
(=тг. It means that the agent is allowed to make standard inferences from plausible beliefs.
The subsequent properties embody the notion of plausible inference in the presence of
incomplete information. Namely, they describe the properties of default deduction under
the assumption that the state of the world is as normal as can be. The crucial rules are
Cautious Monotony and the Cut. Cautious Monotony claims that if in the context C, the
normal course of things is that A and В hold, then assuming that В and С hold should not
lead us to situations that are exceptional for the context C: A should still normally hold.
This property gives some conditions when the monotonicity С \=? A implies В П С \=? ?
holds namely, the condition С \=? В. The Cut is the converse rule: If В is believed in
context С and A usually holds in the context defined by С and В then, one should take it
for granted that A normally holds in context C. The set of these two properties, Cautious
Monotony and Cut, is also called cumulativity by Makinson A994). Rational monotony,
also introduced by Makinson, is a strengthening of weak monotonicity. The condition
under which С \=? A implies В П С Кт A is weakened: it is enough that in the context C,
the negation of В is not believed. Rational Monotony is not a condition of the same nature
as the other ones, because it uses the negation of a plausible entailment (which differs from
the plausible entailment of a negation - remember there are three epistemic attitudes with
respect to propositions).
The remaining properties are not specific to plausible inference, and hold for standard
inference as well. Only half of the deduction theorem holds. Indeed the other half leads to a
reduction of the context that is not in agreement with the nonmonotonic nature of plausible
inference. The Left OR property enables disjunctive information to be handled without
resorting to separately reasoning on cases. The Right Weakening rule, when combined
with the Right AND, just ensures that the set A(C) = {A: С \=? A] is deductively closed.
The set of plausible inferences ?(?) = {(С, А), С \=? A), considered as default rules,
form what Kraus et al. A990) call a rational extension.
In fact it has been proved (under various guises, e.g., Lehmann and Magidor A992),
Gardenfors and Makinson A994)) that any consequence relation |~ satisfying the above
properties and inducing a complete plausibility preordering on S is representable by the

Qualitative possibility functions and integrals
1491
possibilistic consequence relation (=T. There is a representation theorem for possibilistic
entailment proper that can be stated as follows (Benferhat et al. A997)). It must first be
noticed that given a qualitative possibility ordering ?, the following equivalence hold:
A>nB iff ???\=??}.
Let |~ be a consequence relation on 25 ? 25; define an induced partial relation on subsets
as ?>?? iff AUBhfi-
THEOREM 2. A consequence relation |~ satisfies restricted reflexivity, consistency
preservation, right weakening, rational monotony, Right AND and Left OR if and only if A > ^ В
is the strict part of a possibilistic ordering of events.
This theorem precisely describes the nature of possibilistic inference, and the intuitive
character of these properties suggest it is an appealing model of plausible reasoning.
4.3. Universal possibilistic entailment
In the previous section, a cpd on states induces a set of pairs of events (C, A) corresponding
to inferences of the form "in context C, A is typically true". It has been pointed out that
the set of beliefs of the agent, obtained as plausible consequences in a given context, is
a deductively closed set of propositions. In this section we address the converse problem.
Namely suppose a confidence relation ^ is given, that represents the epistemic state of an
agent. Define an accepted belief A as one such that A >i A, and an accepted belief A in
the context С as one such that АПС >l АПС Suppose the set of accepted beliefs in any
non contradictory context, induced by the confidence relation, forms a deductively closed
set. What does it enforce on the nature of the confidence relation? The main assumptions
we shall use concerning a confidence relation are transitivity of its strict part > ? and its
monotony with respect to inclusion. The latter is the orderly property of Halpern A997).
Axiom MI. If А с A' and В' с В then A >L B^-A'>L B'.
Since a belief set is deductively closed, the compatibility between the confidence relation
and the deductive closure property requires that any consequence of an accepted belief is
an accepted belief and that the conjunction of two accepted beliefs is an accepted belief
(Dubois and Prade, 1995b), hence the two properties:
• Consequence stability (CS): if А с ? and ?>? A then В >l B.
• AND rule: if A > L A and В > L ? then Anfi>z.AUfi. _
Since the set of accepted beliefs when С holds, Al(C) = {А: А П С >l А П C}
is supposed to be closed under deduction the above axioms must be generalized into
conditional versions of CS and the AND rule:
• Conditional consequence stability (CCS):
if Acfi and СПА>^СПА then СП В >LCn~B,

1492
D. Dubois and H. Prade
• Conditional AND rule (CAND):
if С П A>L С П A~ and СП В >L С Г\~В then С ? ? ? ? >tCll(AUS).
CCS and CAND give back the CS and AND rules when С = 5". In Dubois et al. A998a),
confidence relations satisfying transitivity and inclusion monotonicity CAND (CCS is
implied by MI) are characterized. These kinds of confidence relations are called acceptance
relations.
It is easy to prove that as soon as transitivity and inclusion monotonicity for >l are
assumed, the other properties come down to the following axiom introduced in Dubois and
Prade A995b):
Acceptance Axiom (Ac). VA, В, С three disjoint events,
AUB >L С and AUC>L В =>· A>i5UC.
The CCS property is recovered from Axiom MI and the CAND property is equivalent
to Ac in presence of Axiom MI. Thus, we can define acceptance preorder relations ^i
on a set of events as those relations whose strict part satisfies transitivity, MI and Ac.
The relation > l induced by an acceptance preorder will be called a partial acceptance
order. Let hi. be the consequence relation induced by this acceptance preorder (C|~z.A
iff С П ? >? С П A). Then we get the following result related to Friedman and Halpern
A996) results (see Dubois, Fargier and Prade A998a)):
THEOREM 3. If the strict part of any confidence preorder satisfies transitivity (T),
inclusion monotonicity (MI) and the acceptance axiom (Ac), then it induces a consequence
relation that also satisfies restricted reflexivity, Right AND, MI, RW, CM, CUT, Left
OR, and CP. Conversely, if a consequence relation |~ satisfies CAND, OR, RW, CM,
CUT, restricted reflexivity, and CP, then the confidence relation it defines via (A > l В iff
A U ? \^? ?) is a partial acceptance order on disjoint events.
Note that this is almost like the properties of the consequence relations induced by
possibility orderings except that rational monotony is replaced by the weaker cautious
monotony. These properties form the so-called system ? proposed by Kraus et al. A990)
who call the consequence relations, satisfying CAND, OR, RW, CM, CUT, and reflexivity,
"preferential". Here reflexivity is replaced by CP, only non contradictory contexts are
considered and reflexivity is restricted to such non-contradictory propositions.
Another result bridges the gap between acceptance orderings and possibility theory,
namely, Dubois and Prade A995a) have shown that an inference relation |~ is preferential
if and only if there exists a set of nondogmatic possibility relations ? such that if ? ? 0:
Ah-? iff V>/7e ?. ??)? >? ???.
Hence the only way of representing epistemic states of an agent under the form of
a deductively closed set induced by a confidence relation is, under mild assumptions,

Qualitative possibility functions and integrals
1493
by means of a family of possibility orderings. From the point of view of the induced
consequence relation, going from a single possibility ordering to a family thereof naturally
leads to the loss of the rational monotony property, due to the presence of a negated
condition in RM. Only cautious monotony remains. Denote A (=vT В the consequence
relation induced by any family VJ of possibility orderings. This consequence relation
is called universal possibilistic entailment, and the set A(VJ) = {(?. ?): ? (=?? ?} is
called a preferential extension. It can be viewed as a set of default rules. It is the intersection
of rational extensions ?(?) induced by each ? in VJ.
The universal possibilistic entailment is preferential. The maximal set VJ( |~) of
possibility orderings that yield a preferential extension of the form VJ{T) is usually
different from and contains ?. Benferhat et al. A999) investigates the structure of VJ( h)
from a lattice-theoretic point of view. First, they define the maximum of two comparative
possibility distributions, and show that this maximum operation is internal to VJ{ \~~), and
that it computes the least upper bound of the two comparative possibility distributions in
the lattice-theoretic sense, for the specificity ordering.
Dehnition 7. Let ^ and ^?· be two comparative possibility distributions on a
set 5", WOPi^,) = (?o ?„) and WOP(^,T) = (E'() ?',,) being their well-ordered
partitions. Then Max(^T. ^T) denotes the unique comparative possibility distribution
whose associated well-ordered partition is obtained from the set {?,',' ^т\щп.п'^
defined as follows
Eq = ?oU Eq,
E'l = (Ek UE'k)-( U E'A iork=\ min(/7. n),
\i=\.k-\ I
by eliminating the empty ?[' and renumbering the non-empty ones in sequence.
The following properties of (VJ( h). Max) have been established:
• Max is associative.
• Max(^,^)e?J(l·)·
• Maxj^Tr, ~^?·\ is less specific than ^ and ^-,
• If ^тг* is less specific than both >T and ^v (in the wide sense) then it is less specific
than Maxj^Tr, ^?·}.
It is generally not true that ^,T and ^T- refine Max{^,T. ^v) (in a non-trivial way). To
summarize, the set VJ( h) is an upper semi-lattice under operation Max and VJ( h)
possesses a least specific comparative possibility distribution ?* (see Lehmann and
Magidor A992), Benferhat et al. A998)). The set ?(?*) is called the rational extension
induced by VJ( И (hence by VJ).
However VJ( |~) has several most specific comparative possibility distributions. A most
specific distribution corresponds to a linear ordering on S, or equivalently. to a maximally
refined (or informative) comparative possibility distribution. Benferhat et al. A999) have
proved that the preferential extension A(VJ) can be generated from the set VJl( h) of
linear cpd in VJ( h). neglecting other possibility orderings, namely (А, В) е A{VJ) iff
VneVJL(H,A\=n B.

1494
D. Dubois and H. Prade
4.4. Probabilistic interpretations of plausible inference
It is tempting to interpret plausible inference in a probabilistic way. Namely, in the spirit
of acceptance orderings, the plausible inference of В from A might be interpreted as "the
majority of A are B" that is, P(B | A) > 1/2. Unfortunately, the set of plausible inferences
in this sense, say [В: Р(В | A) > 1/2} is not closed under deduction. Some properties of
preferential inference are violated under this interpretation, like Cautious monotony and
Left Or. This is shown by the following counter-examples:
Counter-example for CM.
1% of students are male", i.e., P(m \ s) > 0.5,
1% of all students will pass the exam", i.e., P(g \ s) > 0.5,
but: % of male student pass the exam", i.e., P(g \ s П m) < 0.5.
Counter-example for OR.
P(C | A) = 38/66 > 0.5, and P(C \ B) = 38/66 > 0.5, but P(C \ A U B) = 44/100 <
0.5.
In order to maintain the semantic link between probabilistic deduction and plausible
inference it is necessary either to increase the threshold of plausible inference from 1/2 to
a number close to 1, or to restrict the class of probabilities involved. There are two existing
approaches: infinitesimal probabilities, and big-stepped probabilities.
Adams A975) and more recently Pearl A988) have proposed to interpret the default
statement "Generally, if A then B" as P(B | A) ^ 1 - ? where ? is a positive real
number arbitrarily close to 0. Let Ve — [?: ?(?, | A,) ^ 1 - ?, / = 1 ?] be a
set of probability distributions which are induced by a set of statements of the form:
"Generally, if A, then ?,". Define the infinitesimal closure of these statements as the
set ?(??) = {(?, ?): VP e VF, P(B | A) ^ 1 - ?(?)} where ???^??(?) = 0. Let
? — {тг, ?, \=? ?,-, / = 1 ?] be the set of cpds induced by the same set of statements
expressed as possibilistic inferences. A result in Lehmann and Magidor A992) comes down
to proving that the preferential closure ?(?) coincides with A(Ve).
However the interpretation of plausible inference in terms of arbitrarily high
probabilities is not so satisfactory at the intuitive level. An alternative probabilistic semantics is in

Qualitative possibility functions and integrals
1495
terms of big-stepped probabilities defined in Section 3.4. Under a bsp, each proposition A
possesses a "usual" model (it occurs more probably that the disjunction of other models
of A). Namely let ? be a linear pod and ? be a bsp inducing the same total ordering of
states as the pod. The ordering of events induced by the associated possibility and
probability measures are the same because if A and В are two subsets of S, sa and sb being
the unique states with maximal probability, P(A) > P(B) is equivalent to p(sa) > P(sb)-
In particular, it implies that А \=л В iff P(B \ A) > 1/2. This remark leads to a standard
probabilistic semantics for universal inference
THEOREM 4 (Benferhat et al. A999), Snow A999)). Let V be a set of big-stepped
probabilities, and VJ be the corresponding set of linear pods, then define: A (=pbS В
as WP e V, P(B | A) > 1/2. It holds that A (=Pbs В ifandonly if ? (=?,? ?.
This result is at the same time good news and bad news for possibility theory. It is
good news because qualitative possibility possesses a deep link to probabilistic reasoning.
However the above results also point out the limitations of ordinal representations of
uncertainty and of the kind of plausible reasoning, based on acceptance and deductive
closure advocated in this section.
These limitations are closely related to the so-called "Lottery paradox" (Kyburg, 1988).
Assume that the statement "A is accepted in context C" is modelled by the property
? (A | C) > a where ? is a fixed threshold close to 1. This model of acceptance fails
as illustrated by the following counterexample; consider a lottery game where ? (say
1 000000) lottery tickets are sold. Let А,, В denote the respective events "player / loses"
and "all tickets are sold". There is one winner ticket so that the following probabilities can
be estimated: Prob(A, | ?) = 1 — l/п. For ? high enough, the statement A, is accepted.
Now suppose universal inference is applied, then, using the right AND rule, plausible
inference from all В \=yn A,, V/, leads to conclude that ? (=?,? ??=?.« Ai· mat is, all
players lose. However, ?,=? n A — 0. since there is one winning ticket, so that in any
case Prob(P|, = |.„ A; \ B) — 0. Ordinal possibilistic reasoning fails in this situation. This
counterexample has been used to refute the well-foundedness of plausible inference based
on acceptance, as opposed to probabilistic reasoning.
However the paradox will not be observed if a big-stepped probability is used, since then,
{A, P (A | C) > 0.5 (remains consistent and deductively closed. The reason for the paradox
is the presence of a supposedly purely random phenomenon in the lottery game, namely
a uniform winning probability on the set of players that bought a ticket. In this situation
there is nothing typical. The only high probability events are those with many possible
realizations, and they are globally inconsistent. On the contrary, big-stepped probabilities
account for random phenomena where precise events are observed much more frequently
than other ones. Plausible reasoning based on jumping to likely conclusions makes sense
only for such situations, not for the lottery game.
Bsp's seem to be rare, since the constraints bearing on them are very strong. However
in the scope of knowledge representation, the set of states is built in such a way that states
represent meaningful clusters of actual situations that can be described using the chosen
language. The assumption behind the existence of a bsp is thus that natural languages are
built in such a way that they can be used to describe typical course of things. Based on this

1496
D. Dubois and H. Prude
assumption, one may even think of deriving a bsp from statistical data in the following way:
given a sample space ? and a probability measure P, find a "linguistically meaningful"
partition Son ? (in other words, a language), informative enough, such that the coarsening
of ? on S is a big-stepped probability. This point of view suggests a way of bridging the
gap between statistics and symbolic induction.
5. Independence in qualitative possibility theory
The notion of independence has been given a precise definition in the probabilistic
framework. Other notions of independence exist, such as logical independence between
events. A and В are said to be logically independent if none of the pairs (?, ?), (?, ?),
(A, B), and (A, B) is inconsistent. This notion is very weak. Logical independence is
implied by stochastic independence. Probabilistic independence has been questioned as
to its appropriateness for representing epistemic notions of relevance. When a probability
distribution has a frequentist flavor, the meaning of stochastic independence is clear. A and
В are independent events if A occurs as often when В occurs as when not, and conversely.
But this explanation becomes debatable when the probability distribution is subjective,
that is, represents the cognitive state of an agent. Possibilistic independence is a difficult
and composite topic whose full elucidation is a big challenge for future research. First,
one must distinguish between independence between binary variables and independence
between events. Moreover the situation of independence in quantitative possibility theory
is significantly different from the one in qualitative possibility theory.
5.1. The paradoxes of probabilistic independence
The standard definition of probabilistic independence is via the Multiplication Law: events
A and С are probabilistically independent iff Prob( А П C) = Prob( A) · Prob(C). It satisfies
the properties:
(symmetry) If С is independent of A then A is independent of C.
(negation stability) If С is independent of A then С is independent of A.
(truth) A and S are independent.
Symmetry justifies saying "A and С are independent" instead of "A is independent of
C". However interpreting the Multiplication Law in terms of independence is far from
being straightforward. It seems more natural to define independence in terms of invariance
with respect to conditioning: С is probabilistically independent of A iff Prob(C|A) =
Prob(C). It turns out that, for positive distributions, this conditioning-based notion of
independence is equivalent to the multiplication law. It follows from the axioms of
probability theory that independence defined in this way is a symmetric relation, and
that it is not sensitive to negation. However, the symmetry property cannot be guessed
by just looking at the equality Prob(C|A) = Prob(C), and negation stability cannot be
guessed at all. Both are obtained as a by-product of the additivity axioms of probability
theory. They are not intrinsic to the intuitive concept of independence. Moreover, the above

Qualitative possibility functions and integrals
1497
properties (symmetry, negation, truth) are not enough to completely characterize the notion
of probabilistic independence (Fine, 1973).
In qualitative possibility theory, the two definitions of independence do not coincide.
However, when conditioning is defined in the numerical setting via the product rule, then
the product-based symmetric independence notion П(А П ?) = ? (A) ¦ ?(?) is trivially
equivalent to П(В\ А) = П(В) for positive values and is thus much closer to probabilistic
independence.
Note that there are no simple properties governing the interplay of probabilistic
independence with conjunction and disjunction. For instance it is not true that if A is
independent of С and В is independent of С then А П В is independent of C, nor A U В
is independent from C. Neither does it hold that if A is independent of С and A is
independent of B, then A is independent of С П ?, nor A is independent from CUB.
A formal objection to the multiplication Law has been given by Keynes A921), cited
in Gardenfors A978). According to Keynes, the following property should be accepted: If
С depends on A, and В does not contradict A then С depends on А П B. The intuition
behind this postulate is that since A is an argument that affects C, the conjunction of A and
В should also be considered as relevant to C, since А П В contains a relevant part, even if
P(C) = P(C I An S). Keynes notes that his postulate is not validated by the Multiplication
Law. He proposes a stronger definition of probabilistic independence that does it: С is
independent of A iff there is no В such that A implies В and Prob(C | ?) ? Prob(C).
Carnap A950) has shown that this definition leads to a trivial notion of independence: It
entails that С is independent of A only if A = S. A special case of Keynes postulate is the
so-called conjunction criterion for dependence (called conjunction criterion for relevance
in Gardenfors A978)):
(CCD) If С depends on A, and С depends on В and А П ? ? 0 then С depends on
???.
Gardenfors A978) has also suggested a conjunction criterion for independence dual to
(CCD):
(CCI) If С is independent of A, and С is independent of В then С is independent of
???.
Moreover Gardenfors A978) considers that the notion of relevance should satisfy the
following minimal requirements:
(Rl) If A is relevant for С then С is not independent of A.
(R2) If A is relevant for С then A is relevant for C.
(R3) S is not relevant for С
(R4) If С is contingent (= neither S nor 0) then there is some A that is relevant for C.
These postulates equate relevance with dependence (i.e., the negation of independence),
insist on negation insensitivity (so that 0 is not relevant for C). As seen later, negation
insensitivity is not valid for possibilistic independence, and the postulate that there is no
middle way between relevance and independence is then questionable. Gardenfors shows
that under (Rl )-(R4), CCI + CCD is equivalent to Keynes postulate. Since the conjunction
of CCI and CCD leads to trivializations, one has to choose one of them. Gardenfors
criticizes the Multiplication Law of probability theory because it does not guarantee any
of these principles. He argues that CCD should be given up because it sanctions some
events as relevant to С while intuitively they are not. This is due to postulate (R2).

1498
D. Dubois and H. Prude
Namely assume A is relevant for C, then generally A U D is also relevant for С even
if D is not. Due to (R2), AUOis also relevant for С so that by CCD, we conclude
that D is also relevant for C, even if it is not! This remark leads Gardenfors A978) to
accept CCI and he proposes a probabilistic concept of independence that satisfies it: С
is independent of A iff Prob(A) = 0, or Prob(C|A ? ?) = Prob(C) for all В such that
Prob(A П ?) > 0 and Prob(C|fi) = Prob(C). Note that Gardenfors' independence relation
is non-symmetric.
An alternative attitude is, rather than rejecting CCD, to accept both CCD and CCI and
drop (R2). This is allowed for in the possibilistic framework. We show that in an ordinal
setting where uncertainty is described by ordering the states of the world according to their
plausibility, we capture properties similar to CCI and CCD in terms of disjunction, with
much simpler definitions of independence than the above probabilistic one.
5.2. The minimum rule in possibility theory
Zadeh A978) introduced a symmetric definition of independence called "non-interactivity"
between possibilistic variables that is not based on conditional possibilities. This notion has
also been studied by Nahmias A978) for events, under the name "unrelatedness".
Dehnition 8. A and В are unrelated propositions in Nahmias's sense (denoted by ALB)
iff ? (? ? ?) = ?????(?(?), ? (?)).
This definition is clearly the possibilistic counterpart of the multiplication rule of
probability theory. It is interesting to characterize the constraints induced by unrelatedness
on the ordering of events ???, ???, ???, ??? respectively (that correspond to
interpretations if A and В are encoded as literals in logic). The following property is easily
obtained: A and В are unrelated if and only if ?(? ? ?) ^ тт(Л(А ? ?), ?{ ? ? ?)).
Clearly, A and ? are "related" means that А П ? is an implausible situation (since in
any case it holds that П(А П ?) < тт(Л(А), ?(?))), i.e., A and В are (more or less)
mutually exclusive: it implies that A and В are "related" if and only if N( B\ A) > 0 and
N(A\B) >0.
So the terms "related" and "unrelated" proposed by Nahmias are not perfectly
appropriate. The joint statement N( B\A) > 0 and N( A\B) > 0 indicates an epistemic
form of mutual exclusion between A and В since when A is true it is normally the
case that В is true and conversely. On the contrary when A and В are "unrelated", the
two propositions are allowed to be true together. So, the minimum rule is a weakening
of the logical notion of consistency between A and ?, and we shall call it "epistemic
consistency", although the case when ? (A) = 0 or П(В) = 0 is encompassed. This notion
is not very demanding. Moreover this notion is local in the sense that it is sensitive to
negation: if A and В are epistemically consistent, it does not say anything about the other
conjunctions of events A and В, В and A, B and A.
Other properties of "epistemic consistency" are as follows (Dubois et al., 1994):
A) AJ_C iff CJ_A (symmetry).
B) If ALB and ALC then A LB U C.

Qualitative possibility functions and integrals
1499
C) If AJ_fi U С then A±B or AJ_C.
D) 0J_A.
E) S±A.
F) AJ_A.
G) AJ_Aiff Л(А) = 0 or Я( A) = 0.
(8) A UCLA.
Facts B) and C) are disjunction-oriented. However, none of the two conjunction criteria
(CCD) and (CCI) are valid with epistemic consistency. The proof of these facts is easy
using the max-decomposability of possibility measures and the mutual distributivity of
min and max. Note also that fact D) sounds strange in view of the "epistemic consistency"
interpretation of J_. In fact A±B means that "either A and В are non-mutually exclusive
or at least one of them is impossible". This remark also sheds light on fact G). Facts
F) and (8) are certainly strange properties for an "independence" relation, and motivate
the search for a stronger notion. De Cooman A997, Part III) adopts the possibilistic
independence between events, so as to solve the paradox of Nahmias definition whereby
A is unrelated with itself. Lastly, there seems to be no way of characterizing possibility
theory by means of J_, the reason being that we cannot express ? (A) = 1. Therefore,
we conjecture that (just as for probabilistic independence) J_ cannot be axiomatized
alone.
5.3. Possibilistic independence between variables
Enforcing epistemic consistency between all values of binary variables corresponds to a
generalization of logical independence. It contrasts with the situation in probability where
independence between events is insensitive to negation. The variables are said to be non-
interactive (Zadeh, 1975).
PROPOSITION. Two binary variables .x and у are non-interactive (П(.х. у) — тт(Л(.*).
? (у)) far all ? e {A, A }, у е {?. ? } iff there is a choice of values ? e {A, A] and
у e {?, ? }, such that П(х. у) ^ П(.х. у) ^ ? (.?. у) = ?(?. у).
When ? takes on values in {0. 1} only, the non interaction between .x and у means
that the subset of {?. ?) ? {?. ?) with possibility 1 is itself a Cartesian product of
subsets of {A, A } and {?, ? }. Logical independence is when it is {A. A } ? {?, ? } itself.
Actually, non-interactivity is a decomposability property of the joint possibility distribution
on {A, A } ? {?, ? } into its projections combined by the minimum rule.
More generally, let ? and у be two variables (on U and V respectively) linked together
through a fuzzy restriction on the Cartesian product U ? V, encoded by a possibility
distribution, ?,.?, called a joint possibility distribution. The possibility distribution ??
representing the induced restriction on the possible values of ?, can be calculated as the
projection of ??-? on U defined by Zadeh A975):
??(??) = П([и) ? V) — sup7Tv.v(". v).

1500
D. Dubois and H. Prade
Generally, ???- ^ ?????(??, ??). When equality holds, ?,·.? is then said to be min-
separable. Note that projection is also in accordance with the principle of minimum
specificity, since ?? (и) is calculated from the highest possibility value of pairs (x, y) where
? = u. As long as a possibility distribution represents a state of incomplete information,
non-interactivity expresses a lack of knowledge about the links between ? and y.
Conversely, now consider two variables ? and у with possibility distributions ?? and ??
respectively. The principle of minimum specificity leads us to define the joint possibility
distribution ?* ,. induced by these two pieces of information as the "non-informative
product"
?*? = minO.y, лу)
as a consequence of the inequalities ??.?(?, ?) ^ л>(и), Vu, and лх_у(и, ?) ? ??(?), Vm,
which hold for the unknown joint possibility ??.?, since ?? does not presuppose anything
about у while ???- does. Note that if there is an unknown link between ? and y, then
?* ? provides upper bounds on degrees of possibility, i.e., the conclusions deduced from
the uninformative product are always correct but might be less informative that those
deduced from ??_,¦· Indeed some values (и, v) may be considered as possible according
to ?* r, while when ??.? < ????(??.??), they are not so. This upper bounding effect
is in agreement with the entailment principle which states that if "x lies in A" then "x
lies in S", provided that А с ?, or Vm, A(u) ^ B(u) more generally. Note that while
in probability theory, independence plays the same role as non-interactivity in possibility
theory (probabilistic variables can be assumed to be stochastically independent by virtue
of the principle of maximum entropy), stochastic independence does not lead to bounding
properties as non-interactivity does. This is because stochastic independence assumes an
actual absence of correlation while non-interactivity expresses a lack of knowledge. The
projection and the non-informative product extend to several variables and are part of the
calculus of fuzzy relations (Zadeh, 1975).
The non-interactivity of variables implies that the possibility and necessity measures
become decomposable with respect to Cartesian products (? ? ?) and the associated co-
products (A + ? = ? ? ? ), respectively:
? (? ? ?) = гат(Д, (А), ПУ(В)),
? (? + В) = max(yVv(A), Ny(B)),
where A and В are subsets of the universes U and V (the ranges of ? and )¦) and Д, and Nx
are possibility and necessity measures based on the normalized possibility distribution ??.
Possibilistic counterparts of probabilistic graphical representations have been proposed.
Bayesian networks encode joint probability distributions, by decomposing them on the
basis of an arbitrary ordering of the variables, as a product of conditional probability
distributions. This decomposition is done in an efficient way, exploiting conditional
independence assumptions so as to identify simple acyclic directed graphs representations
(DAGs). The local propagation schemes described by Pearl A988) become very efficient
if the directed graph has no cycles. These schemes are made possible due to the properties

Qualitative possibility functions and integrals
1501
of the conditional independence relations. A conditional independence relation is a ternary
relation relating disjoint subsets of variables ?. ?. ?. denoted (X±Y \ Z) which spells
out "X is independent of ? given Z" and is usually understood in the probabilistic sense.
The basic properties of these relations are known as "semi-graphoid axioms" (see Pearl
A988)):
Symmetry: (X±Y | Z) => (Y±X \ Z).
Decomposition: (X U W±Y | Z) =>· (X±Y \ Z) and (W±Y | Z).
Weak union: (X U W±Y | Z) => (X±Y \Z\JW).
Contraction: (W±Y | Z) and (Х±У \Z\JW)=>(XU W±Y | Z).
While symmetry is easy to understand, decomposition implies that if X is independent
of ? given ? then any variable in X is independent of ? given Z. Weak union implies
that learning something about a variable in set X that is independent of ? should not affect
our knowledge about Y. The contraction axiom implies that, if a variable x' is independent
of ? and if ? is independent of Y, when we learn about x' then it should be that the two
variables ? and x' are independent of Y. Probabilistic independence satisfies an additional
axiom:
Intersection: (X±Y \ZUW) and (X±W \ ? U Y) =>· (Х±У UW|Z).
This axiom together with the ones above equips the independence relation with a
so-called graphoid structure. The graphoid structure enables a directed acyclic graph
Q = (N,A) whose node set N contains variables and A contains directed arcs describing
causal links between variables to be read in terms of independence properties. Namely
it provides topological separation properties between disjoint sets of nodes of the graph
that precisely reflect conditional independence relations among the corresponding sets of
variables (See Verma and Pearl A990)).
In the possibilistic setting, conditional possibility distributions can be used as a basis
for a decomposition of joint possibility distributions, whose structure can be reflected by a
DAG Q = (N, A)- Namely using ordinal conditioning,
?(?\ xn) = min ?(jc, | Par(x,))
/=l и
where Par(jc() = [x; (x,x,) e A). The corresponding conditional independence notion is
of the form
n(x,y \z) = mm(n(x \?),?(? \ ;)).
This kind of conditional independence based on minimum leads to a semi-graphoid
structure. Changing minimum into product (Dempster conditioning) recovers the full
graphoid structure. For details the reader is referred to De Campos et al. A995), De Campos
and Huete A999). In the min-based framework, hypergraph representations sound more
adapted. Recent activity on possibilistic networks rather deals with the problem of learning
from data (Gebhardt and Kruse, 1996, 2000; Sanguesa et al., 1998a, 1998b). Benferhat
et al. B002) study how to encode directed possibilistic graphs in possibilistic logic.

1502
D. Dubois and H. Prade
Other decomposable representations for ordinal possibility distributions on Cartesian
products of universes have been much less studied and prove to be much more restrictive
than in the finitely scaled or numerical setting due to the non-commensurateness of
orderings (see Ben Amor et al. B000)).
5.4. Possibilistic event independence based on conditioning
Other notions of qualitative independence between events have been introduced based on
ordinal conditioning (Dubois et al., 1994, 1997). They are not equivalent, but stronger than
the minimum rule in possibility theory.
A first proposal is to define В as independent of A when the conditional measure of
В given A is equal to the unconditional measure of B. Here we have two uncertainty
functions ? and N. Hence we can define independence either as ?(? \ A) = П(В) or as
N(B | A) = N(B). But since N(B \ A) = N(B) is equivalent to ?( ? | A) = ?( ?) one
of them is enough. Let us consider the equality N(B \ A) = N(B) in order to express the
positive statement that A is irrelevant to (the level of acceptance of) B. It turns out that this
notion of irrelevance is rather heterogeneous because it expresses the disjunction of two
distinct forms of independence according to whether ? is a priori believed or not:
(i) N(B\ A) = N(B) = 0.
(ii) N(B | A) = N(B) > 0.
The first condition holds if and only if
1 = тах(Я( ? ? ?), ? (? ? ??)) and ? (? ? б) ^П(АГ)В).
The second condition holds if and only if Л (? ? ?) > ? (? ? ~?) > ? ( ? ? ?).
The two situations (i) and (ii) correspond to (almost) reversed orderings of conjunctions.
Moreover in (i), В is either ignored or a priori rejected both a priori and in the context A,
again a heterogeneous situation. Worse, the invariance of the degree of certainty via
conditioning is not stronger than the minimum rule: N(C\A) = N(C) does not imply
AJ_C. For instance if П(А П С) > Я(АПС) and Л( АПС) > Я(АПС) then/V(C|A) =
N(C) = 0 while A±C obviously does not hold. Instead, N(C\A) = N(C) implies AJ_C
so that П(С\А) = ? (С) is stronger than AJ_C.
Now, in possibility theory, the knowledge about В is expressed by the pair (N(B),
N(B)) and includes the three situations where С is accepted, rejected or unknown. It
leads to recognize three situations of independence in the absolute form:
(Al) N(B_\ A) = N(BJ >0(henceyV(fi | A) = N(~B) = 0).
(Al') N( В | A) = N( В ) > 0 (hence N(B |_A) = N(B) = 0).
(II) N(B | A) = N(B) = N(B\A) = N(B) = 0. _
Condition (Al) (respectively (АГ)) means that believing В (respectively: B) is not
affected by A, while the third condition means that A does not inform about B. In
the first and the second situation we shall speak of absolute epistemic independence В
(respectively B) vis a vis A, where the term "absolute" refers to the stability of the level of
acceptance, and the term "epistemic" indicates that it is assumed for this notion that В is

Qualitative possibility functions and integrals
1503
an accepted belief. Absolute epistemic independence N(B \ A) = N(B) > 0 implies A±B
since ? (А П ?) > П(А П В) holds, and it is stronger.
The property (II), which is never met in the probabilistic case, means that in the presence
of A, B, which was originally ignored, is still ignored. In this case, we shall speak of
uninformativeness of A about В (or C, equivalently). Condition (II) holds iff Л (А П ?) =
Л(АП ?)= 1 or Л( АП ?) = Л(АП ?) > Л(АП В) = Л(АП ?). It is obvious that
when A does not inform about В then A±B holds (since Л(АПв) = Л(АПв)). So
uninformativeness is stronger than A±B. The contrary of uninformativeness is that the
agent starts accepting В when learning A. We shall speak of epistemic causality.
These notions are not symmetrical, and the first two are sensitive to negation. A is then
said to be irrelevant to С if any one of the three above conditions (??), (??) and (II) holds.
Irrelevance is thus taken as a generic notion of invariance of the level of belief with regard
to evidence.
The absolute epistemic independence may be seen as too strong. Recall that in the
ordinal setting the preservation of absolute values does not mean much. Especially
the situation where N(B) > N(B | A) > 0 is provably impossible as recalled earlier.
There cannot be any form of dependence expressing a belief attenuation effect. So the
reinforcement N(B | A) > N(B) > 0 does not make much sense. Moreover, absolute
epistemic independence enforces the inequality ?(??)?)^?(??)?) which implies
that in the context where В would be false, it is forbidden to conclude that A can be
accepted. Indeed, the absolute epistemic independence property reads: В is absolutely
independent of A if and only if N(B | A) > 0 and not (N( A \ В) > 0). This may be
viewed as counterintuitive for an agent declaring that his acceptance of В is "independent"
from the knowledge of A and not wishing to commit himself when В is false.
A milder notion of independence is that if В is accepted unconditionally, then if A is
true, В remains accepted; then we do away with any commitment in the case when В would
turn out to be false. Hence the following definition of qualitative epistemic independence:
Dehnition 9. The acceptance of В is qualitatively independent of A (denoted ? ?> ?)
iff N(B | A) >0andyV(B) > 0.
It expresses that В is an accepted belief independently of A and only requires ? (? ?
?) > ? (? ? ?) and ? (?) > ? (A). It is non-symmetric and sensitive to negation. It
can be verified that this notion implies, but is stronger than, the condition ?(? f] В) =
тт(П(А),П(В)).
An antonym notion of qualitative relevance, denoted A ~> В can be defined as not
(А ф> В) and S ф> В. А «> В is not equivalent to not (A j=> B) so that Gardenfors
(R2) does not hold. It is such that A *«> В is equivalent to N(A) > 0 and N(A | ?) = 0.
There are two situations, according to whether В refutes A (N( A \ B) > 0) or simply
cancels it (?(?\ ?) = 0).
These notions have been studied in detail and axiomatized by Dubois et al. A997). They
examine the validity of Keynes-Gardenfors criteria of Section 1 in the ordinal setting of
possibility theory. Namely the following requirements:
(CCD) If A «> С and В «> С then А П В «> С.
(CCI) If А ф> С and В ф> С then ? ? ? ?> С

1504
D. Dubois and H. Prade
They also consider symmetric counterparts of (CCD) and (CCI):
(CCD-r) If A %> В and A^> С then A «> В П С.
(CCI-r) If ? ?> В and ? ф> С then A ^> В П С.
and the corresponding properties changing conjunction into disjunction ((DCD), (DO),
etc.).
(DCI) If А ф> С and В ф> С then A U В ?> С
(DCI-r) \i Аф> В and А ^> С then А ф> В U С
(DCD) ИА^> С and ? *«> С then A U ? ^> С.
(DCD-r) If А ^> В and А «> С then А «> ? U С.
First the minimum rule J_ satisfies the four above criteria concerning disjunctions.
Moreover (CCD-r), (CCI-r), (DCI), (DCD) hold for absolute and qualitative epistemic
independence. The qualitative independence has the following stronger properties:
А ф> В П С iff ? ?> В and А ф> С.
АГ\Вф>С iff А ^> С and ? ^> С.
They are (DCI) and (CCI-r) with equivalence. This is natural if qualitative independence
is considered in terms of belief revision: if I continue to accept В ПС upon learning A, I
should continue to accept С and В as well. Overall, eight axioms are used to completely
characterize qualitative independence, namely:
(QI1) S ф> S (tautologies do not undermine tautologies).
(QI2) If А ф> С then ? ?> ? U С (right weakening).
(QI3) If S ?> A then ??> A (if С is believed, then it cannot undermine itself).
(QI4) If ? ?> ? then 5" ?> ? (if A is known not to undermine В then ? is a priori
believed).
(QI5) ??> A never holds.
(QI6) If А ф> С and В ф> С then A U В ?> С (left OR rule, i.e., (DCI)).
(QI7) If A U ? ф> С then A ^> С or A U В ф> A (similar to rational monotony).
(QI8) If А ф> В and ? ?> С, then A ?fe- В П С (stability under conjunction for
acceptance, i.e., (CCI-r)).
Note that S ?> A means N(A) > 0. Especially, (QI4) says that when believing В does
not depend on А, В must be an a priori belief. (QI2), (QI3) and (QI7) are typical of
qualitative independence. (QI1), (QI2), (QI4), (QI5), (QI6) and (QI8) are valid for absolute
epistemic independence. Now the following characterization theorem holds, showing the
axiomatic equivalence of ?> with possibility theory (Dubois et al., 1997):
THEOREM 5. Let ?> be a relation on events, and N a mapping from the set of events
to [0, 1] such that А ф> В iff N(B | A) > 0 and N(B) > 0. Then ф> is a qualitative
independence relation iff N is a necessity measure.
The proof uses a remarkable property that relates qualitative independence and
possibility orderings, namely for any possibility ordering >n, the following equivalence
holds:
VA,C,AUC^>C iff A>nC

Qualitative possibility functions and integrals
1505
For such pairs of events of the form (A U C. C), qualitative independence A U С ?> С
is equivalent to absolute epistemic independence.
Hence it is possible to axiomatize qualitative relevance as well. One uses the fact that
A U В ~> В iff П( В) > ? (?) ^ ?{ A). It can also be done by translating the relevance
relation back to the qualitative independence one, and the result is closely related to the
one of Farinas del Cerro and Herzig A996) in their setting of theory revision.
5.5. Qualitative independence and belief revision
Several notions of independence and relevance defined above, among which qualitative
dependence, can be fully expressed in the framework of revision of propositional theories
also called belief sets (Gardenfors, 1988). A central problem for the theory of belief
revision is what is meant by a minimal change of a state of belief. As pointed out in
Gardenfors A990), "the criteria of minimality that have been used in the models for belief
change have been based on almost exclusively logical considerations. However, there are
a number of non-logical factors that should be important when characterizing a process
of belief revision." Gardenfors focuses on the notion of independence (he considers it as
synonymous to the term 'irrelevance') and proposes the following criterion: If a belief state
К is revised by a sentence A, then all sentences in К that are independent of the validity
of A should be retained in the revised state of belief.
This seems to be a very natural requirement for belief revision operations, as well
as a useful tool when it comes to implementing belief change operations. As noted by
Gardenfors, "a criterion of this kind cannot be given a technical formulation in a model
based on belief sets built up from sentences in a simple propositional language because the
notion of relevance is not available in such a language". However the above criterion does
make sense in the ordinal setting of possibility theory.
We suppose given a belief set ?, that is, a set of propositions closed under deduction, and
an AGM revision operation * (Gardenfors, 1988). К * A represents the result of revising
К by a proposition A. According to Gardenfors and Makinson's characterization theorem,
if the revision operation satisfies the so-called AGM postulates (Alchourron et al., 1985)
then К and * can be represented equivalently by an epistemic entrenchment ordering,
which in turn is nothing else than a qualitative necessity ordering (Dubois and Prade, 1991).
Conversely any qualitative necessity ordering leads to an AGM revision operation. Namely,
given a necessity function ?, the set К = {С, N (С) > 0} is a belief set. This is because N
is an acceptance function (Dubois and Prade, 1995b) that is, {C. N(C) > 0} is closed under
conjunction and logical entailment. Moreover, it can be proved that the revision operation
* is described in terms of possibility theory as follows: С belongs to К * A is equivalent
to N(C\ A) > 0 (Dubois and Prade, 1992) (also equivalent to N( A U C) > N(A)).
If we translate the various definitions of possibilistic independence and relevance of this
section in terms of revision we get:
Qualitative independence: ? ?> С iff С e К and С е К * A.
Relevance: A cancels С iff С е К and С <?K*A and С ? К * A. A refutes С iff С е К
and С е К * A.

1506
D. Dubois and H. Prade
Epistemic causality: A justifies С iff С ф К and С е К * ?.
Uninformativeness: A does not inform about С iff С ? К, С ? К, С ? К * A and
С <?K*A.
Qualitative independence exactly corresponds to Gardenfors' above requirement for
revision-based independence.
The operation opposite to revision is contraction. Contracting a belief set К by a
proposition A means deleting A from К as well as propositions that possibly enable A
to be derived from К - {A}, so as to obtain a belief set К - A that does not contain A.
The Harper identity (Gardenfors, 1988) defines contraction in terms of revision as follows:
К — A= К Г) (К * A), i.e., first revise К to accept A and keep only those formulas in
К * A that are in К. Conversely К *A = Cn((K - A)U{A}), where Cn is the consequence
operation. This is Levi's identity whereby revising by A means contracting A first and then
adding A. Companion definitions of qualitative independence and relevance relations ?>?
and ss>(. can be associated to a contraction operation "-" via the following definition:
Аф>,С iff CeKmdCeK-A iff С е К - A,
A^>CC iff CeK and C?K-A iff N(C) > 0 and N(A) > N(A U C).
It is easy to check that Levi and Harper's identities can be written in terms of
independence relations as follows: А ф>с С iff ? ?> С and A «*><- С iff A ^> C. Farinas
del Cerro and Herzig A996) have proved the equivalence between axioms characterizing
possibilistic relevance ~>( and an AGM contraction operator. Similarly, postulating the
equivalence between ??> С and С е К * A, it can be proved that axioms (QI1)-(QI8)
are equivalent to the AGM postulates for revision.
6. Qualitative integrals
Mathematical notions similar to integrals can be defined for finitely-scaled set-functions.
Of course, such "integrals" will not be additive, and the role of the addition and the
multiplication will be played by the maximum and the minimum indifferently. Moreover,
qualitative integrals can be defined only for functions ranging on the same scale as the set-
function. Since we shall stick to the finite setting, we shall get qualitative counterparts of
weighted averages for possibility and necessity functions. Considering more general set-
functions, so-called Sugeno integrals are obtained. They are the qualitative counterparts of
Choquet integrals.
Qualitative integrals can be used to evaluate the likelihood of fuzzy events. Events,
which we would like to evaluate the uncertainty of, are often ill-defined: e.g., we would
like to estimate whether somebody is "rich" without being able to describe in precise
quantitative terms what it means. Possibility and fuzzy set theory jointly offer a simple
tool for modelling both the partial lack of available information and the ill-definition of
relevant events. Fuzzy events are represented by a function from the set of states S to a
totally ordered scale L (L-fuzzy sets, Goguen A967)). Set-functions representing partial
belief can be generalized by extending the usual concept of "event" to fuzzy events via

Qualitative possibility functions and integrals
1507
qualitative integrals and the algebra of events to sets of functions, so as to accommodate
fuzzy events. Algebras of fuzzy events are no longer Boolean.
Decision theory is also a framework where these notions of integral make sense, and
can be axiomatized. However in the purely ordinal case, defining the counterpart of
integrals is tricky due to commensurateness problems between orderings. In Savage's
A972) framework, the consequence of a decision depends on the state of the world
in which it takes place. For instance, in a restaurant one must choose between dishes
without being totally sure that the food is good. If 5 is a set of states and X a set of
possible consequences, a decision is a mapping from S to X. The decision-maker has some
knowledge of the actual state and some preference on the consequences of his decision. The
usual approach to decision-making under uncertainty is based on representing uncertainty
on states by a unique probability distribution and the decision-maker by a utility function
on consequences. Then decisions are selected on the basis of expected utility. When
uncertainty is just due to a lack of information, and only the result of the present decision
matters, the idea of expectation is less attractive. For this reason a purely possibilistic
approach to decision under uncertainty that addresses this kind of situation has been
developed (Dubois, Prade and Sabbadin, 1998, 2001). It exploits possibilistic and Sugeno
integrals as preference functionals serving as decision rules.
6.1. Possibilistic integrals
Consider a pod ? defined on a finite set S and ranging on a totally ordered scale L. Let /
be a function from S to L. Let 1 /(i)^a denote the characteristic function of {s; f(s) ^ a].
Note that the obvious identity holds:
f(s) = maxmin(a, 1 flf)>a(s)).
C/€L
So / can always be considered as a multistep function. Then possibility and necessity
integrals can be defined in the usual style from the possibility and necessity functions ?
and N induced by ?, as:
n(/) = maxmin(ci, Я(/ ^?)), N(f) = maxmin(a, N(f ^a)).
C/€L C/€L
These expressions reduce to what Zadeh A978) and Dubois and Prade A980)
respectively call possibility and necessity of fuzzy events, noticing that function / is the membership
function of a fuzzy set:
Я(/) = maxmin(/(i), n(s)), N(f) = minmax(/(i), v(n(s))),
where ? is the order-reversing function of L, noticing that Я(/ ^ a) = v(N(f < a))
and that maxa€L min(a, N(f ^ a)) = min„gi. max(a, N(f > a)). The possibility and
necessity integrals can be viewed as prioritized versions of the maximum and the minimum,

1508
D. Dubois and H. Prade
the possibility values тг(и) acting as weights (Dubois and Prade, 1986), while the usual
integral reduces to a weighted average.
Possibility and necessity integrals have been used as a semantic matching process
between the representation of the available information (?) and the description of some
request in the form of a fuzzy event F. Possibility theory enriches the symbolic pattern
matching paradigm of Artificial Intelligence, allowing for graded semantic matching
(Cayrol et al., 1982; Dubois et al., 1988). Given a collection of objects for which we have
some (possibly incomplete) knowledge about the value of an attribute x, represented by
means of a possibility distribution for each object, the set of objects such that "x is F" is
an ill-known set described by a pair of fuzzy sets, namely the fuzzy set of objects which
more or less possibly satisfy the requirement "x is F" and the fuzzy set of objects which
more or less necessarily (i.e., certainly) satisfy it. Membership grades to such fuzzy sets
are calculated using possibility and necessity integrals.
For two-valued functions fA, that can be used as characteristic functions of a subset A
of S, the expression of the qualitative integrals is rather simple: assume fA(s) = a if s e A
and fA(s) = ? if s ? A , with a > ?. Observe that
n(fA) = max(/3, тт(Я(А), a)) = min(a, тах(Я(А). ?)),
N(fA) = max(/3, min(yV(A), a)) = min(a, max(/V(A), ?)).
Note that the possibilistic integrals of the binary function are the medians of {а, Я (A),
?], and {a, N(A), ?} respectively, thus contrasting with usual integrals, which are based on
averaging. This view of possibilistic integrals as medians remains valid in the general case,
as can be seen from the definitions based on level-cuts of /. Assume S = {s\, jt, ..., s,,,},
and denote F, = {s„{i),.. .,s„(,n)}, where ? is a permutation such that /(??(?)) ^
/(¦VB)) ^ ¦ ¦¦ ^ f(sn(m))- ri(f) is actually the median of 2m — 1 terms f(s\),..., f(s,„),
n(F2),-..,n(Fi„), and N(F) is obtained similarly.
As another illustration, the value f(s) of function / may represent the utility of
some decision made in state s. The possibilistic integral N(fA) is a kind of pessimistic
preference functional that is easy to understand: if the agent is sure enough that A occurs
(N(A) > a) then the utility of the decision is a. If the agent has too little knowledge
(max(jV(A), N( A)) < ?), then cautiousness prevails and the utility is /3,_the worst case.
Of course the same happens if the agent is at least somewhat certain that A occurs. If the
agent's certainty that A occurs is positive but not extreme, the preference functional N(fA)
reflects the certainty level and is equal to N(A). The possibilistic integral Я(/А) is a kind
of optimistic preference functional: the value of Я(/А) is high (= a) as soon as the agent
believes that obtaining ? is possible enough (Я(А) > a).
In particular, suppose ? is the characteristic function of a subset ? of states. In other
words, the agent only knows that the state of the world is in E, then the possibilistic
integrals reduce to:
Я(/) = тах/E), N{f) = mmf{s)
s e ? jet"
which coincide with well-known maximax and maximin criteria in decision theory. More
generally, the inequality Я(/) ^ N(f) holds, often with strict inequality, so that N(f) can

Qualitative possibility functions and integrals
1509
be called a pessimistic possibilistic integral and Л(/) an optimistic possibilistic integral.
These functionals are special cases of Sugeno integrals studied below. The optimistic
criterion has been first proposed by Yager A979) and the pessimistic criterion, under the
form of a maximin disutility, by Whalen A984), and also used in Inuiguichi et al. A989).
The main properties of possibilistic integrals are:
- monotonicity: If / ^ g (statewise) then Л(/) ^ 17(g) and N(f) ^ N(g);
- maxitivity of Л(/): Let / ? g be the function from S to L defined, for any s e S, by
/ ? g(S) = max(/(i), g(s)). Then Л(/ ? g) = max( Л(/), Л(#));
- minitivity of N(f): Let / л g be the function from S to L defined, for any s e S, by
/ л g(s) = min(/(i), g(s)). Then N(f Ag) = min(/V(/), N(g));
- homogeneity properties: Consider truncated functions /?? and / л ? for any ? e L:
weak minitivity of Л(/): Л(/ ? ?) = тт(Л(/), ?),
weak maxitivity of N(f): N(f ??) = max(jV(/), ?).
More general forms of possibilistic integrals have been studied. The generalized optimistic
possibilistic integral can be defined by:
Л*(/) = sup f(s)*n(s)
where * is a conjunctive-like operation on L called a semi-norm (De Cooman, 1997). This
is a fuzzy conjunction that is not necessarily commutative or associative, but is increasing
in the wide sense, and such that 1 * ? = ? * 1 = ?. Then, Л*(/) = П(А) when / is
the characteristic function of a set A, and, Л*(/) = Л(/) when * = min. Similarly, the
generalized pessimistic possibilistic integral can be defined by:
?,(/) = ??(?E),/E))
where / is an implication operation on L, that is, decreasing in the first place and increasing
in the second place (in the wide sense), and such that /A,0) = 0, /@,0) = /A, 1) =
/ @, 1) = 1. Viewing function / as the membership function of a fuzzy set F and choosing
/ of the form ? (?, ?) = ? (a * ?(?)), the identity ? (A) = v(N(A)) is extended to fuzzy
events, i.e., n*(F) = v(N*(~F)) for the usual fuzzy set complementation Fc'(¦) = v(F(-)),
where у denotes an order-reversing function on L. If operation * is increasing in the wide
sense in both places, the characteristic properties of possibility and necessity measures
with respect to disjunction and conjunction of fuzzy sets still hold (respectively defined by
the pointwise maximum and the minimum of membership functions).
??? v 8) = тах(Л*(/), Я»(^)). /V»(/ л g) = min(yV*(/), NAg))-
Moreover, if ? * ? ^ min(a, ?), then Л(/) > W(/), V/ also holds (if ? is normalized).
Inuiguichi and Kume A994) provide a systematic study of possibility and necessity
measures of fuzzy events with various implications. They point out three kinds of fuzzy
necessity integrals:

1510
D. Dubois and H. Prade
- strong ones such that N(f) = 1 if and only if n(s) > 0 implies f(s) = 1 (the support
of ? is contained in the core of /); this is when ?(?,?) = max(l -?, ?), for instance;
- medium ones such that N(f) = 1 if an only if ? ^ /; this is when ? (?, ?) = 1 as soon
as a < ?, especially residuated implications, see Bouchon-Meunieret al. A999);
- and weak ones such that N(F) = 1 if and only if n{s) > 0 implies f(s) > 0.
Shilkret A971) integral is defined for L = [0, 1] by
/prod fdn = sup а П(/> a).
It is an optimistic possibilistic integral where the minimum is replaced by a product
and values lie in the unit interval. In that case, it is easy to see that / , f dil = Л*(/)
with * = product. This notion is also studied by Agbeko A991) under the name "optimal
average". More generally, r-norm- and conorm-based possibility integrals have been
studied by De Cooman et al. A992), Yan A993,1994), De Cooman and Kerre A996). They
correspond to the following integral-like schemes, called normed and conormed possibility
integrals, respectively:
f*fdn = SUp„;>(,<* * П(/ > <*)-
f±fdN = Ma>0a±N(f>a).
The first of these expressions is equal to Л*(/) if * is a triangular norm. The second
one is of the form N/ (/) for / = v(a) J_ ? if J_ is a triangular conorm-like operation. On
such a basis, De Cooman A997, Part I) outlines a theory of possibilistic integration, where
possibility levels are taken in a complete lattice, and the operation * is a semi-norm. He
proves possibilistic counterparts of Radon-Nikodym and Fubini theorems. In a series of
papers, Mesiar A992, 1994, 1997) studies the representation of possibility and Shilkret
integrals via Markov kernels. See also Benvenuti, Mesiar and Vivona B002) (Chapter 33
in this Handbook).
6.2. Axiomatics for possibilistic integrals
It makes sense, if information is qualitative, to represent the incomplete knowledge of
an agent on the state by a possibility distribution ? on S with values in a plausibility
scale L and the agent's preference on the set of consequences X of decisions by means
of another possibility distribution ? with values on a preference scale U. The utility of a
decision d whose consequence in state s is ? = d(s) e X for all states s, can be evaluated
by combining the plausibilities n(s) and the utilities ?(?) using the two quantitative
criteria based on possibilistic integrals for evaluating the worth of decisions, provided that a
commensurability assumption between plausibility and preference is made (Dubois, Prade
and Sabbadin, 2001).
Maximizing the pessimistic integral means to find a decision d, all the highly plausible
consequences of which are also highly preferred. The definition of "highly plausible" is
decision-dependent and reflects the compromise between high plausibility and low utility

Qualitative possibility functions and integrals
1511
expressed by an order-reversing map between L and U that make utility values and
possibility values commensurate. The pessimistic integral is small as soon as there exists a
possible consequence of the decision which is both highly plausible and bad with respect to
preferences. This is clearly a risk-averse and thus a pessimistic attitude. However, contrary
to the maximin Wald criterion, the pessimistic possibilistic criterion focuses on the idea
of usuality and relies on the worst plausible consequences induced by the decision. Some
unlikely states are neglected by a variable thresholding and the threshold is determined by
the commensurateness mapping between utilities and possibilities. In contrast, the other
criterion corresponds to an optimistic attitude since it is high as soon as there exists a
possible consequence which is both highly plausible and highly prized.
The pessimistic criterion has been axiomatically justified by Dubois and Prade A995c)
in the style of Von Neumann and Morgenstern utility theory; see Dubois et al. A999) for
the simplified set of axioms given below. The idea is that if the uncertainty on the state is
represented by a finitely scaled pod ?, each decision induces, on the set of consequences
X. a possibility distribution such that ?(/(?) = ?(?~](л)). So ranking decisions comes
down to ranking possibility distributions on X. Assume the decision-maker supplies an
ordering between possibility distributions on X. thus expressing his attitude in front of
risk, that is, in front of various possibilities of happy and unhappy consequences in X. Let
nj(x) e L be the plausibility of getting ? under decision d. The question is to know what
kind of axioms on the ordering between possibility distributions on X make it representable
by the ranking of decisions according to the above pessimistic or optimistic criteria. Let
(?/?, ?/?') denote the "qualitative lottery" yielding ? with plausibility ? e L and ?' with
plausibility ? e L. Of course, ???3?(?, ?) = 1. The following are the pessimistic axioms:
A) The set of possibility distributions is equipped with a structure (!>=. ~. >-) of
complete preordering, where >.~. > are the weak preference, indifference and
strict preference.
B) Independence: ?\ ~ ?? =>· (?/??. ?/?) ~ (?/??. ?/?).
C) Continuity: if ? > ?' then 3? e L. ?' ~ A/?. ?/?).
D) Reduction of lotteries: (?/(?/?. ?/?'), ?/?') ~ (?/?. ?/?') where ? = min(a, ?)
and ? = ??3?(?, ????(?, ?)).
E) Risk aversion: ? ^ ?' =>· ?' )? ?.
The risk aversion axiom states that the less informative is ?. the more risky the situation
is perceived by the agent: for a pessimistic decision-maker, the worst epistemic state is total
ignorance. Continuity says that the utility of ? goes down if the uncertainty about ? raises.
It can be proved that if the knowledge is represented by a subset A of possible states, then
3x e A . ? ~ A. This property, violated by expected utility, suggests that contrary to it, the
pessimistic utility is not based on the idea of average and repeated decisions, but makes
sense for one-shot decisions. The reduction of lotteries suggests that if lottery (?/?, ? /?')
is embedded in the lottery (?/?, ?/?') at place ?, ? obtains in two sequential steps with
possibility min(a, ?) and ?' obtains either directly with possibility ? or in two steps with
possibility ????(?, ?).
The reduction of lotteries property is due to the fact that the whole setting relies on
a qualitative counterpart of the structure of probabilistic mixtures that underlies the Von
Neumann and Morgenstern approach (Herstein and Milnor, 1953). This is due to the
closure of the set of possibility measures under weighted maxima: if ? and ?' are

1512
D. Dubois and H. Prade
possibility measures, then тах(/(Л), /'(Я')) is a possibility measure again (Dubois and
Prade, 1990) where / and /' are monotonic transformations of the possibility scale, with
/@) = /'@) = 0, and max(/(l), /'A)) = 1. This is the only aggregation that preserves
possibility measures.
Let ? с L2 be the set of ordered pairs (?. ?) such that max(a, ?) = 1. A possibilistic
mixture set is a pair (G, M) where G is a set of possibilistic lotteries, ? is a function
from G ? G ? ? to G that computes for each pair of possibilistic lotteries ? and ?' and
each pair of weights (?, ?) in ? another element of G denoted ?(?, ?'; ?, ?) such that
(Dubois etal., 1996):
Ml: ?(?,?'\ 1,0) = ?;
M2: ?(?,?';?,?) = ?(?',?;?.?);
?3: ?(?(?, ?'; ?, ?), ?': ?, ?) = ? (?.?'; min(a, ?), max(min(/3, ?), ?)).
Ml says that lottery (?/?,?/?1) is ?; ?2 says that lotteries (?/?,?/?') and
(?/?', ?/?) are the same; M3 is axiom 4 (which can be dropped if M3 is adopted as a
computation rule). M1-M3 imply the following:
M4: ?(?,?;?,?) = ?\
M5: ?(?(?, ?'\ ?, ?), ?(?, ?'; ?, ?); ?, ?)
= ?(?, ?'; max(min(a, ?), min(y. ?)), max(min(/S, ?), min(<5, ?))).
The last expression explains the form of the optimistic criterion. The form of the other
criterion is obtained by working with degrees of necessity (ordered pairs (?, ?) such that
min(a,/3) = 0).
An alternative axiomatic justification of possibilistic integrals has been developed in the
style of Savage (Dubois, Prade and Sabbadin, 2001). The problem is one of representing
a complete preordering ^ over the set of functions from a finite set S to a finite set X by
means of a functional whose form is dictated by the properties of this preordering. The
functions d in D = Xs represent the set of potential decisions of an agent, X being the set
of consequences of decision, d(s) being the consequence of choosing decision d in state
s. Let А с S be an event, / and g two functions in Xs, and denote by f Ag the function
such that
fA8(s)-[g(s) if si A.
A set of axiom for the pessimistic possibilistic integral is as follows:
SI: (D, » is a complete preorder.
Let > denote the strict part of this complete preorder.
S2: There exist two functions d and d' where d > d' (non triviality).
S3: If fx and fy denote constant functions such fx(s) = ? and fy(s) = y, Vs e S,
then for any function d e Xs and any event A: fx > fy implies fx Ad > fy Ad.
S4: If d > d' and d > d" then d > d' ? d", where d' is a constant function.
S5: lfd' >d and d" > d then d' a d" > d.
THEOREM 6 (Dubois et al. A998a)). If the preference relation over D satisfies the five
above axioms then there exists a possibility distribution ? on S, a preference function ?
on X and an order-reversing function between plausibility and preference scales such that
d > d' if and only if ?(???) > ?(???') using the pessimistic possibilistic utility.

Qualitative possibility- functions and integrals
1513
The optimistic criterion can be justified likewise, moving the constant act condition from
axiom S4 to S5. In fact axioms SI, S2, S3 are enough to show that the uncertainty on states
is captured by a monotonic (Sugeno) set-function. We have also proved that SI, S2, S3,
S4, and S5 restricted to constant acts d', are enough to show that the preference relation
> is representable by a Sugeno integral (Dubois. Prade and Sabbadin, 1998). S5 ensures
the min-decomposition of the uncertainty measure (which is then a necessity function).
The pessimistic nature of S5 can be guessed from its equivalent form: d' Ad > d implies
d ^ d Ad'. If decision d is improved by adopting d' when A occurs it means that A is sure
enough to occur and so, there is no way of improving decision d by changing it in case A
does not occur (the agent neglects this possibility of non-occurrence).
6.3. Sugeno integrals
In this section, the possibilistic integrals are extended to more general confidence functions,
also called fuzzy measures (Sugeno, 1974). Fuzzy measures are monotone increasing (with
respect to set inclusion) set-functions. Namely if A is a subset of В it is assumed that
?(?) ^ ?(?), where ?(?) is a degree of confidence of A. In his thesis, Sugeno A974) was
inspired by fuzzy logic operations in formalizing the concept of integral of a function F
valued on a totally ordered scale L, with respect to a fuzzy measure ?. Replacing addition
and multiplication of numbers by max and min, respectively, he arrived at the following
definition where / ^ a = [u e U: f(u) ^ a) is the level-cut of the function from S to L
(Sugeno, 1974, 1977):
f f ??= sup min(a, ?(/ ^ a))·
For a careful comparison of Sugeno integral and Lebesgue integral, see Ralescu and
Adams A980). Note that if ? varies on a numerical, continuous scale, this definition can
also be written as
jfdq>= inf max(a. ?(/ ^ a))·
Consider the case of a finite set S = (s\ , s,,,}. The Sugeno integral reduces to the
following expression:
ffd<p = max mm(f(s„a)).<p({s„{i).....s„{llu}))
i = l »i
= min max(f(s„ii)).<p({s„{i-\) s„im)}))
l = \ ???
using an increasing reordering of vector (f(s\). f(sm)) according to a permutation
?. f f ?? is actually the median of 2m — 1 terms \f(s\) f(s„,).<p({sa{2) s„{m)}),
¦ ¦ -,??????)})} (Dubois and Prade, 1980). Another form of Sugeno integral is an extension
of Boolean polynomials:
f f d<p= max mini min/(.?). ?(?)) = minmax(max/(i). ?(?)\.
A V.se.A / ? V (?д /

1514
D. Dubois and H. Prade
Viewing / as a utility function, these forms suggest that Sugeno integral achieves an
optimistic (respectively pessimistic) compromise between the level of confidence in each
event and the degree of utility of the worst (respectively best) state where the event occurs
(respectively does not occur).
Possibility and necessity integrals are particular cases of Sugeno integrals, where ?
is a possibility measure and a necessity measure respectively. Namely / fail = 17(f)
and / fdN = N(f); see for instance Dubois and Prade A980), Inuiguichi et al. A989),
Grabisch et al. A992). Other special cases of Sugeno integral are the ordered weighted min
and max (see, e.g., Dubois et al. A988)). If ?(?) only depends upon the cardinality of A,
so that <?({??(,-),..., ??(„?)}) = ?? (and фт+\ = 0) one gets of the form:
OWmm(u) = min, = | „, max@, + |, f(s„U))),
OWmax(u) = max, = | ,„ min(</>,-, f(s„{i))).
whose values coincide with the median of 2m - 1 terms f(s\) f(snl),Ф2, ¦ ¦ -,фтН
? (A) = 1 if | A | ^ к and 0 otherwise, it corresponds to the requirement that a decision must
be good enough at least in any к states. This will be modelled by m — к + 1 weights equal
to 1, i.e., ф, = 1 if 1 ^ i ^ m - к + 1, ф,¦ = 0 for i > m - к + 1. More generally the set
of weights can be derived from a fuzzy quantifier of the form "most": a fuzzy subset Q of
integers s.t. ??@) = 0, ??(/') ^ ??(/ + 1), ??(??)= 1; then define 0, = 1 - ??(/ - 1),
for /' = 1,..., m + 1.
For a detailed study of Sugeno integral and its characterization as an aggregation
operation, see Marichal B000), Dubois, Prade and Sabbadin A998). For the mathematics
of continuous Sugeno integral, see Wang and Klir A992), Pap A995).
Sugeno integral satisfies the following properties, where ??? denotes the function /
such that f(s) = ? if s e A and ? otherwise:
- Increasing monotonicity (in the wide sense): if / ^ h then f f ?? >//? ??.
- Idempotency: if / is a constant function and f(x) = ?, then j f ?? — ?.
- Non compensation. For any ACS and ? e S, /\????, /????? e [?(?), ?}.
- Weak minitivity and maxitivity
// /\??? = mini j f ??. ?), // ? ??? = maxi f f ??, а).
- Comonotonic minitivity and maxitivity: if / and h are comonotonic that is, Vs, s1 e S,
f(s) > f(s') implies h(s) ^ h(s'), then
//????? = ?\??(//??,/????) and // ? \??? = maxi j f ??, $\???\.
The last property emphasizes the similarity between Sugeno integrals and Choquet
integrals which are additive with respect to comonotonic functions. Then, we have the
following three characterizations of Sugeno integral, see Marichal A998, 2000):
THEOREM 7. Let M(f) be ? functional from Lx to L. The following assertions are
equivalent

Qualitative possibility functions and integrals
1515
(i) ? is increasing, idempotent, and поп compensatory,
(ii) ? is increasing and weakly minitive and maxitive,
(iii) ? is increasing, idempotent, and comonotonic minitive and maxitive,
(iv) there exists a confidence measure ? such that M(f) =//??.
Similar axiomatization of Sugeno integrals have been obtained in the Savage setting by
Dubois, Prade and Sabbadin A998), restricting S4 and S5 above to when d' is a constant
act. De Campos et al. A991) and Ralescu and Sugeno A996) prove representation theorems
for Sugeno integrals on the basis of comonotonic minitivity and maxitivity.
Numerical variants of Sugeno integrals exist such as Shilkret integral introduced above,
changing minimum and maximum into more general operations. Later on, generalized
variants of Sugeno integral with t -norms and conorms in the finite setting have been
considered by Suarez Garcia and Gil Alvarez A986), Grabisch at al. A992), De Cooman
and Kerre A996).
6.4. Comparing functions without commensurateness
Let S and X be two finite sets (interpreted as a state space and a consequence space in
decision theory). Assume there is a cpd >T on S, and another complete preordering ^ ? on
X. The former describes the agent's knowledge and the latter describes the preferences of
the agent on consequences. Let the set of functions Xs be equipped with a strict preference
relation >, that is, a transitive and irreflexive relation. Let ^ denote the associated non strict
relation such that / ^ h if and only if h > f does not hold. The relation > represents the
agent's preferences among decisions. The question addressed in this section is to what
extent such a relation on Xs can be constructed from >T on S and ^/> on X. This problem
is easily solved if ^л on S and ^/> on X are modeled by functions ? and / from S to
L and X to L respectively, where L is a totally ordered set, using possibility integrals for
instance. Here we do not assume the existence of L.
To this aim, a set of axioms, that slightly differs from the ones of Savage, is proposed by
Dubois etal. A997, 2000):
• S'|: (Xs, >) is a transitive, irreflexive, partially ordered set.
• St: (Xs, >) satisfies the sure-thing principle: f Ah ^ gAh if and only if /Ah' >
gAh'.
• WSTP: if fAg > g and gAf ^ g then f)?g.
• S3: VA с 5", A not null, (fx > fy)A if and only if л ^Р у.
• S4: Vx,y,x',y' e X s.t. ? >p y,x' >P y',xAy )^xBy <?> х'Ау' !>= x'By'.
• S'y 3x, y, ? three constant acts such that fx > fx > fz-
Axioms of the form S, are originally from Savage. Axioms of the form S- are variants of
Savage axioms. Note that the weak preference relation !>= is not transitive. The sure-thing
principle enables two notions to be simply defined, namely conditional preference and null
events. Function / is said to weakly preferred to g, conditioned on A if and only if V/?,
f Ah ^ gAh. This is denoted by (/ !>= g)A. An event A is said to be null if and only if V/,
v?. (/ > g)A holds. WSPT is called the weak sure thing principle (Grant et al., 1997). It
is a consequence of the sure thing principle if the weak preference relation ^ is transitive.

1516
D. Dubois and H. Prade
Again, we identify the set of constant functions [fx, ? e X} and X. The preference on
X can be induced from (Xs, !>=) as follows. Given (Xs, », the preference relation ^p on
X is of the form Vjc, у e ?, ? ^p у if and only if fx !>= fy.
A likelihood relation on 25, that is, a binary relation >l.w among events can be defined
by A >lx\ В iff xAy > xBy for ? >p y. Property S4 enables events A and В to be
consistently compared by fixing ? > ? у arbitrarily when selecting binary events ? Ay and
xBy. All relations >(..„¦ are then the same and denoted >l-
Since we are working in a purely ordinal setting, the relative position of / and g with
respect to relation > should more generally not depend upon the particular values f(s)
and g(s), but only on the fact that f(s) >p g(s) or not. Changing / into /' and g into
g' such that the relative positions of images of each state is not altered should not affect
the preference between functions. Two pairs of functions (/, g) and (/', g') will be called
statewise order-equivalent whenever Vs e S, f(s) ^/> g(s) if and only if f'(s) ^p g'(s).
This is denoted (/, g) = (/', g'). It leads us to postulate the following requirement (Fargier
and Perny, 1999):
Ordinal Invariance axiom. @1). V/, /', g, g', if (/, g) = (/', g') then (/^o
The following results can be proved (Fargier and Perny, 1999; Dubois et al., 2000).
THEOREM 8. Under assumptions that 01 holds, and that relation ^ on acts is reflexive, it
follows that S2, and S4 hold too.
Function / is better than g in state s if f(s) > P g(s). Denote [f >p g]= {s, f(s) > ?
g(s)} the set of states where / is better than g.
THEOREM 9. Under the assumption of ordinal invariance, if the weak preference on
acts is reflexive and the induced weak preference on consequences is complete, the weak
preference relation on X is complete and transitive, and the only possible definition of the
ordering on functions is via the so-called likely dominance rule:
f>g&[f>pg]>L [g >p f] where >p is the projection of>onX (due to S3) and
>L is the projection of > on events (due to S4).
The latter result enables the partial ordering on functions (acts) to be reconstructed from
a likelihood relation on events and a preference relation on consequences.
It can be proved that if a possibilistic likelihood relation is chosen for >t, then the
induced partial ordering on functions does satisfy S\, St, S3, WSTP, S4, S^ and 01.
Actually, starting with a regular possibilistic ordering ^/7 on events, lifting it to acts
using the likely dominance rule and reprojecting the relation on acts back to events,
yields the possibilistic likelihood relation refining ^/7 (see Section 3.4). Moreover, starting
from a strict preference ordering satisfying S'h S3, WSTP, S'5 and 01, the induced

Qualitative possibility functions ami integrals
1517
likelihood relation is preadditive and satisfies two already encountered properties that are
characteristic of the acceptance partial orderings (see Section 4.3):
A >l A and AC; В imply В >l В for А, В, С disjoint.
AUC>ifiandfiUC>tA imply С >LAU В.
These properties imply that >i is generated by a family of cpds on S (Dubois et al.,
1997, 2000) and lead to non-monotonic inference rules in the sense of Kraus, Lehman and
Magido (Friedman and Halpern, 1996).
However the class of relations on Xs thus induced is rather poor. This is because the
considered problem is almost the same as the one addressed by Arrow A951) when he
studied the theory of voting in the ordinal setting and proved a famous impossibility
theorem. Assume S is a set of voters and XlS a set of candidates. The set (Xs. ^s) is
the preference profile of voter s, defined by / ^s g iff f(s) ~^p g(s). Then the likely
dominance rule is a generalized Condorcet rule. For instance suppose we start from a single
cpd on S. Then the following relations are obtained:
Suppose all states are equally possible. Then the Pareto ordering is recovered
f>g iff f(s)^P g(s)ys апаЪ. f(s)>P g(s).
This is known as the unanimity rule in voting theory. If there is a linear order on states
¦Л >? ^2 >? ¦ ¦ ¦ >? sn then the ordering of functions is defined by:
f>g iff 3fc such that/(л,-) ~/>g(i,-).V' <kmaf(sk) >Pg(sk).
In other words, the ordering of the functions is the one induced by the most plausible
discriminant state under a lexicographic scheme. It corresponds to the presence of a
sequence of more or less important dictators in the corresponding voting procedure. In
the general case of a well-ordered partition of S: {??· E\ E„}, the functions in Xs
must be ordered in the following way: / > g iff
3k such that f(s) ~P g(s), V.v e ?,, V/ < k.
and/E) ^p g(s),Vs e Ek and 3j e Ek, f(s) >p g(s)
where ~/> is the equivalence relation induced from the complete preordering ^p. It
corresponds to an oligarchy Eq of most plausible states where the unanimity rule applies;
otherwise, if / and g are statewise preferentially equivalent on E\, the ordering decision
is made by ?2; if there is no unanimity when comparing / and g on E\, then / and g
are not comparable. These relations, viewed as decision rules are either too adventurous,
or too undecisive (Mas-Colell and Sonnenschein, 1972). It is one more piece of evidence
showing the difference between the qualitative settings with and without and underlying
totally ordered scale. More general results along this line are proved in Dubois, Fargier and
Perny B002).

1518
D. Dubois and H. Prade
7. Conclusion
This chapter has proposed an overview of set-functions valued on a totally ordered scale,
with emphasis on the simplest of such set-functions, found in possibility theory. The
corresponding relations on a power set describing the relative confidence of events have
been studied. It has been shown that the qualitative counterpart of many measure-theoretic
and probabilistic notions such as conditioning, independence and integrals still make
sense in a finite ordinal setting. These notions bear strong connections with nonmonotonic
reasoning and decision theory.
References
Adams. E.W. A975). The Logic of Conditionals. Reidel. Dordrecht.
Agbeko, K. A991). Optimal average. Ann. Univ. Sci. Budapest. Sect. Сотр. 16. 5-10. (Full version: Report
73/1990. Mathematical Institute. Hungarian Academy of Sciences. Budapest.)
Alchourron, C.E.P.. Gardenfors. P. and Makinson, D. A985). On the logic of theory- change: Partial meet functions
for contraction and revision. J. Symbolic Logic 50. 510-530.
Ben Amor, N.. Benferhat. S.. Dubois. D.. Geffner. H. and Prade. H. B000). Independence in qualitative
uncertainty frameworks, Proc. of the 7th Intemat. Conf. on Principles of Knowledge Representation and
Reasoning (KR2000). Brekenridge, CO A1 avril-15 avril 2000). 235-246.
Arrow, K. A951). Social Choice and Individual Values. Wiley. New York.
Benferhat, S., Dubois. D. and Prade. H. A997). Nonmonotonic reasoning, conditional objects and possibility
theory; Artificial Intelligence 92. 259-276.
Benferhat, S., Dubois. D. and Prade. H. A999). Possibilistic and standard probabilistic semantics of conditional
knowledge, J. Logic Comput. 9, 873-895.
Benferhat, S. et al. A998). Л general approach for inconsistency handling and merging information in prioritized
knowledge bases. Proc. of the Intemat. Conference on Principles of Knowledge Representation and Reasoning
(KR'98). Trento. Italy. June 1998.
Benferhat, S.. Dubois, D.. Garcia. L. and Prade. H. B002). On the transformation between possibilistic logic
bases and possibilistic causal networks. Intemat. J. Approx. Reason. 29. 135-173.
Benvenuti, P.. Mesiar. R. and Vivona. D. B002). Monotone set functions-based integrals. Handbook of Measure
Theory, E. Pap. ed.. Elsevier, Amsterdam. 1329-1379.
Bouchon-Meunier, В., Dubois. D.. Godo. L. and Prade. H. A999). Fuzzy sets and possibility theory in
approximate and plausible reasoning. Fuzzy Sets in Approximate Reasoning and Information Systems.
J.C. Bezdek. D. Dubois and H. Prade. eds. The Handbooks of Fuzzy Sets Series. Kluwer. Boston. 15-190.
Boutilier. С A994). Modal logics for qualitative possibility theory: Intemat. J. Approx. Reason. 10. 173-201.
Butnariu. D. and Klement. E.P. B002). Triangular norm-based measures. Handbook of Measure Theory. E. Pap.
ed., Elsevier. Amsterdam. 947-1010.
Carnap. R. A950). Logical Foundations of Probability. Chicago.
Cayrol. M.. Farreny. H. and Prade. H. A982). Fuzzy pattern matching. Kybemetes 11. 103-116.
Chateauneuf, A. A996). Decomposable capacities, distorted probabilities and concave capacities. Math. Soc.
Sci. 31. 19-37.
De Baets. В.. Tsiporkova. E. and Mesiar. R. A999). Conditioning in possibility with strict order norms. Fuzzy
Sets and Systems 106, 221-229.
De Campos. L.M.. Gebhardt. J. and Kruse. R. A995). Axiomatic treatment of possibilistic independence.
Symbolic and Quantitative Approaches to Reasoning and Uncertainty, С Froidevaux and J. Kohlas. eds.
Lecture Notes in Artif. Intel.. Vol. 946. Springer. Berlin. 77-88.
De Campos, L.M.. Lamata. M.T. and Moral. S. A991). Л unified approach to define fuzzy integral. Fuzzy Sets
and Systems 39. 75-90.
De Campos. L.M. and Huete, J.F. A999). Independence concepts in possibility theory. Part I: Fuzzy Sets and
Systems 103. 127-152: Part II: Fuzzy Sets and Systems 103, 487-505.

Qualitative possibility functions and integrals
1519
De Cooman. G. A997), Possibility theory - Part I: Measure- and integral-theoretics groundwork: Part 11:
Conditional possibility: Part III: Possihilistic independence. Intemat. J. Gen. Systems 25 D). 291-371.
De Cooman, G. and Kerre. E. A996), Possibility- and necessity integrals. Fuzzy Sets and Systems 77. 207-229.
De Cooman. G.. Kerre. E. and Vanmassenhove. F.R. A992). Possibility theory and integral-theoretic approach.
Fuzzy Sets and Systems 46. 287-300.
De Finetti. B. A937). La prevision, ses his logiques et ses sources subjectives, Ann. Inst. H. Poincare 7, 1-68.
Dempster. A.P A967). Upper and lower pwbabilities induced by a multivalued mapping, Ann. Math. Statist. 38.
325-339.
Dubois. D. A986). Belief structures, possibility theory and decomposable confidence measures on finite sets.
Comput. Artificial Intelligence (Bratislava) 5 E). 403^416.
Dubois. D., Fargier. H. and Prade. H. A996). Refinements of the ma.x-inin approach to decision-making in fuzzy
environment. Fuzzy Sets and Systems 81. 103-122.
Dubois. D.. Fargier. H. and Prade. H. A997). Decision-making under ordinal preferences and comparative
uncertainty, Proc. of the 13th Conf. on Uncertainty in Artificial Intelligence. D. Geiger and P.P. Shenoy. eds.
Providence. RI, Aug. 1-3. 1997. Morgan Kaufmann. San Francisco. CA. 157-164.
Dubois, D.. Fargier. H. and Prade. H. A998a). Comparative uncertainty, belief functions and accepted beliefs.
Proc. of the 14th Conf. on Uncertainty in Artificial Intelligence. Madison. WI. G. Cooper and S. Moral, eds,
Morgan Kaufmann, San Francisco. CA. 113-120.
Dubois. D.. Fargier, H. and Prade. H. A998b), Possihilistic likelihood relations. Proc. of 7th Intemat. Conference
on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU'98. Paris).
Editions Medicales et Scientifiques. Paris. 1196-1202.
Dubois. D.. Fargier. H.. Pemy. P. and Prade. H. B000). Qualitative Decision Theory: From Savage's Axioms to
Non-Monotonic Reasoning. IRIT Research report n° 02-04R. Toulouse.
Dubois. D.. Fargier. H. and Pemy. P. B002). On the limitations of ordinal approaches to decision-making. Proc. of
the 8th Intemat. Conf. on Principles of Knowledge Representing and Reasoning (KR2002). Toulouse. France.
April. 2002, Morgan Kaufmann, San Francisco. CA. 133-144.
Dubois. D.. Farinas del Cerro. L., Herzig. A. and Prade. H. A994). An ordinal view of independence with
application to plausible reasoning. Proc. of the 10th Conf. on Uncertainty in Artificial Intelligence, Seattle.
WA, July 29-31, R. Lopez de Mantaras and D. Poole, eds, 195-203.
Dubois, D.. Farinas del Cerro, L.. Herzig. A. and Prade, H. A997). Qualitative relevance and independence:
A roadmap. Proc. of the 15h Inter. Joint Conf. on Artificial Intelligence (IJCAI'97. Nagoya. Japan, August.
23-29), 62-67. Extended version: A roadmap of qualitative independence. Fuzzy Sets, Logics and Reasoning
about Knowledge. D. Dubois, H. Prade and E.P Klement. eds. Applied Logic Series. Kluwer, Dordrecht, 1999.
325-350.
Dubois, D., Fodor, J.C.. Prade. H. and Roubens, M. A996), Aggregation of decomposable measures with
application to utility theory. Theory Decis. 41. 59-95.
Dubois. D., Godo. L., Prade. H. and Zapico, A. A999). On the possihilistic decision model: from decision under
uncertainty to case-based decision. Intemat. J. Uncertain., Fuzziness and Knowledge-Based Systems 7 F),
631-670.
Dubois. D. and Prade. H. A980). Fuzzy Sets and Systems, Theory and Applications. Academic Press. New York.
Dubois, D. and Prade. H. A982). A class of fuzzy measures based on triangular norms, Intemat. J. Gen. Systems
8,43-61.
Dubois D. and Prade H. A986), Weighted minimum and maximum operations. Inform. Sci. 39. 205-210.
Dubois, D. and Prade, H. A988), Possibility Theory. Plenum Press. New York.
Dubois, D. and Prade, H. A990). Aggregation of possibility measures. Multiperson Decision Making Using Fuzzy
Sets and Possibility Theory. J. Kacprzyk and M. Fedrizzi. eds. Kluwer. Dordrecht. 55-63.
Dubois, D. and Prade. H. A991). Epistemic entrenchment and possihilistic logic. Artificial Intelligence 50. 223—
239.
Dubois. D. and Prade. H. A992). Belief change and possibility theory. Belief Revision, P. Gardenfors, ed.,
Cambridge University Press. UK. 142-182.
Dubois, D. and Prade, H. A995a). Conditional objects, possibility theory and default rules. Conditionals: From
Philosophy to Computer Science. G. Crocco. L. Farinas del Cerro and A. Herzig. eds. Oxford University Press.
311-346.

1520
D. Dubois and H. Prade
Dubois. D. and Prade. H. A995b). Numerical representations of acceptance, Proc. of the 11th Conf. on
Uncertainty in Artificial Intelligence. P. Besnard and S. Hanks, eds. 149-156.
Dubois. D. and Prade. H. A995c). Possibility- theory as a basis for qualitative decision theory. Proc. of the Inter.
Joint Conf. on Artificial Intelligence (IJCAI'95. Montreal. August). 1924-1930.
Dubois, D. and Prade, H. A998), Possibility theory: qualitative and quantitative aspects. Quantified
Representation of Uncertainty and Imprecision, Handbook on Defeasible Reasoning and Uncertainty Management
Systems - Vol. 1, P. Smets, ed.. Kluwer, Dordrecht, 169-226.
Dubois, D., Prade, H. and Sabbadin. R. A998). Qualitative decision theory with Sugeno integrals. Proc. of the
14th Conf. on Uncertainty in Artificial Intelligence. Madison. WI. G. Cooper and S. Moral, eds, Morgan
Kaufmann. San Francisco. CA. 121-129. Also in: Fuzzy Measures and Integrals. M. Grabisch, T. Murofushi
and M. Sugeno. eds, Physica-Verlag. 2000. 314-322.
Dubois. D., Prade, H. and Sabbadin. R. B001). Decision theoretic foundations of possibility theory. Eur. J. Oper.
Res. 128, 459^478.
Dubois. D., Prade, H. and Testemale, С A988), Weighted fuzzy pattern matching. Fuzzy Sets and Systems 28.
313-331.
Fargier. H. and Pemy. P. A999). Qualitative models for decision under uncertainty without the commeiisurability
assumption, Proc. of the 15th Conf. on Uncertainty in Artificial Intelligence, Providence, RI, K. Laskey and
H. Prade, eds. Morgan Kaufmann. San Francisco, CA, 157-164.
Farinas del Cerro, L. and Herzig. A. A991). A modal analysis of possibility theory, Proc. of the Inter. Workshop
on Fundamentals of Artificial Intelligence Research (FAIR'91). Smolenice Castle. Czechoslovakia. Sept. 8-12.
1991. Ph. Jorrand and J. Kelemen. eds. Lecture Notes in Comput. Sci.. Vol. 535. Springer. Berlin. 11-18.
Farinas del Cerro, L. and Herzig. A. A996). Belief change and dependence. Proc. of the 6th Conf. on Theoretical
Aspects of Rationality and Knowledge (TARK'96), De Zeeuwse Stromen. The Netherlands, March 17-20,
1996, Y. Shoham, ed., Morgan Kaufmann, San Francisco. CA. 147-161.
Farinas del Cerro, L., Herzig, A. and Lang. J. A994). From ordering based non-monotonic reasoning to
conditional logics, Artificial Intelligence 66. 375-393.
Fine, T.L. A973), Theories of Probability, Academic Press, New York.
Fishbum. P. A986a), The axioms of subjective probabilities. Statist. Sci. 1. 335-358.
Fishbum, P. A986b). Interml models for comparative probability on finite sets. J. Math. Psych. 30. 221-242.
Friedman. N. and Halpem. J. A996). Plausibility measures and default reasoning. Proc. of the 13th National
Conf. on Artificial Intelligence (АААГ96. Portland). 1297-1304. To appear in J. Assoc. Comput. Mach.
Gardenfors. P. A978), On the logic of relevance. Synthese. 351-367.
Gardenfors. P. A988). Knowledge in Flux. MIT Press. Cambridge.
Gardenfors. P. A990), Belief revision and irrelevance. PSA 2. 349-356.
Gardenfors, P. and Makinson. D. A994). Non-monotonic inference based on expectations, Artificial Intelligence
65. 197-245.
Gebhardt, J. B000). Learning from data: possibilistic graphical models. Abductive Reasoning and Uncertainty
Management Systems. Handbook on Defeasible Reasoning and Uncertainty Management Systems - Vol. 4.
D. Gabbay and R. Kruse, eds. Kluwer. Dordrecht. 315-389.
Gebhardt, J. and Kruse. R. A996). Automated construction of possibilistic networks from data. Appl. Math.
Comput. Sci. 6. 529-564.
Goguen. J.A. A967). L-fuzzy sets. J. Math. Anal. Appl. 8. 145-174.
Grabisch. M. B002). Set functions over finite sets: transformations and integrals. Handbook of Measure Theory.
E. Pap, ed.. Elsevier. Amsterdam. 1381-1401.
Grabisch. M.. Murofushi, T. and Sugeno. M. A992). Fuzzy measure officzy events defined by fuzzy integrals.
Fuzzy Sets and Systems 50. 293-313.
Grant, S.. Kajii. A. and Polak. B. A997). Weakening the sure-thing principle: decomposable choice under
uncertainty, Workshop on Decision Theory, Chantilly. France. June 12-14. A revised version. "Decomposable
Choice under Uncertainty", is to appear in J. Econom. Theory.
Grove. A. A988). Two modellings far theory change. J. Philos. Logic 17. 157-170.
Halpem. J. A997), Defining relative likelihood in partially-ordered preferential structures. J. Artificial
Intelligence Res. 7. 1-24.
Hajek, p. and Harmancova. D. A994). A qualitative fi<zzv possibilistic logic. Intemat. J. Approx. Reasoning 12.
1-19.

Qualitative possibility· functions and integrals
1521
Hajek, P.. Harmancova, D.. Esteva. F., Garcia, P. and Godo. L. A994), On modal logics for qualitative possibility
in a fuzzy setting. Proc. of the 11th Conf. on Uncertainty in Artificial Intelligence. R. Lopez de Mantaras and
D. Poole, eds. Morgan Kaufmann. San Francisco. CA. 278-285.
Herstein. I.N. and Milnor. J. A953). An axiomatic approach to measurable utility. Econometrica 21. 291-297.
Hisdal. E. A978). Conditional possibilities independence and noninteraction. Fuzzy Sets and Systems 1. 283—
297.
Hughes. G.E. and Cresswell. M.J. A968). An Introduction to Modal Logic. Methuen. London.
Inuiguichi. M.. Ichihashi, H. and Tanaka. H. A989). Possibilistic linear programming with mensurable
multiattribute value functions, ORSA J. Comput. 1. 146-158.
Inuiguichi, M. and Kume. Y. A994). Necessity measures defined by level set inclusions: Nine kinds of necessity
measures and their properties. Intemat. J. Gen. Systems 22. 245-275.
Keynes. J.M. A921). A Treatise on Probability MacMillan. London.
Klir. G.J. and Harmanec. D. A994), On modal logic interpretations of possibility theory. Intemat. J. Uncertain..
Fuzziness. and Knowledge-Based Syst. 2. 237-245.
Koopman. B.O. A940), The bases of probability, Bull. Amer. Math. Soc. 46. 763-774.
Kraft. C.H.. Pratt. J.W. and Seidenberg. A. A959). Intuitive probability on finite sets. Ann. Math. Statist. 30.
408^419.
Kraus, S.. Lehmann. D. and Magidor. M. A990). Nonmonotonic reasoning, preferential models and cumulative
logics. Artificial Intelligence 44, 167-207.
Kyburg. H. A988), Probabilistic inference and nonmonotonic inference. Uncertainty in Artificial Intelligence.
Vol. 4. R. Schachter et al.. eds. North-Holland. Amsterdam. 319-326.
Lang, J. A996), Conditional desires and utilities. Proc. European Conference on Artificial Intelligence (ЕСАГ96,
Budapest). Wiley. New York. 318-322.
Lehmann. D. and Magidor. M. A992). What does a conditional knowledge base entaiP.. Artificial Intelligence 55
A), 1-60.
Lewis. D. A973), Counter/actuals. Basil Blackwell. London.
Makinson. D. A994). General patterns in nonmonotonic reasoning. Nonmonotonic Reasoning and Uncertain
Reasoning. Handbook of Logic in Artificial Intelligence and Logic Programming -Vol. 3, Oxford University
Press. UK. 35-110.
Marichal, J.-L. A998), Aggregation operators for multicriteria decision aid. Ph.D. thesis. Institute of
Mathematics. University of Liege. Liege. Belgium.
Marichal. J.-L. B000), On Sugeno integral as an aggregation function. Fuzzy Sets and Systems 114. 347-365.
Mas-Colell, A. and Sonnenschein. H. A972). General possibility theorems for group decisions. Rev. Econom.
Stud. 39, 185-192.
Mesiar, R. A992). Characterisation of possibility measures offiizzy events using Markov kernels. Fuzzy Sets and
Systems 46. 301-303.
Mesiar, R. A994). On the integral representation offiizzy possibility measures. Intemat. J. Gen. Systems 21,
109-121.
Mesiar, R. A997). Possibility measures, integration, and fuzzy possibility measures. Fuzzy Sets and Systems 92.
191-196.
Moulin. H. A988). Axioms of Cooperative Decision Making. Cambridge University Press. Cambridge. MA.
Nahmias. S. A978), Fuzzy variables. Fuzzy Sets and Systems 1 B). 97-110.
Pap. E. A995). Null-Additive Set-Functions, Kluwer. Dordrecht.
Pap, E. B002). Pseudo-additive measures and their applications. Handbook of Measure Theory. E. Pap, ed.,
Elsevier. Amsterdam. 1403-1468.
Pearl. J. A988), Probabilistic Reasoning Intelligent Systems: Networks of Plausible Inference, Morgan
Kaufmann. San Mateo. CA.
Ralescu. D. and Adams. G. A980). The fuzzy integral. J. Math. Anal. Appl. 75. 562-570.
Ralescu. D. and Sugeno, M. A996). Fuzzv integral representation. Fuzzy Sets and Systems 84. 127-133.
Ramer. A. A989). Conditional possibility measures. Cybernet. Systems 20. 233-247.
Ramsey, F.P. A963), Truth and probability. Studies in Subjective Probability. H. Kyburg and H.E. Smolker, eds.
Wiley. New York. 61-92.
Roubens, M. and Vincke. P. A985). Preference Modelling. Lecture Notes in Econom. Math. Systems. Vol. 250.
Springer. Berlin.

1522
D. Dubois and H. Prade
Savage. L.J. A972). The Foundations of Statistics, Dover. New York.
Sanguesa. R..Cabos, J. and Cortes. U. A998). Possibilistic conditional independence: A similarity-based measure
and its application to causal network learning. Intemat. J. Approx. Reason. 18, 145-167.
Sanguesa. R.. Cortes. U. and Gisolfi. A. A998), A parallel algorithm for building possibilistic causal networks.
Intemat. J. Approx. Reason. 18. 251-270.
Schmeidler. D. A986), Integral representation without additivity. Proc. Amer. Math. Soc. 97 B). 255-261.
Shackle. G.L.S. A961). Decision, Order and Time in Human Affairs. 2nd edn, Cambridge University Press. UK.
Shafer. G. A976). A Mathematical Theory of Evidence. Princeton University Press. Princeton. NJ.
Shilkret, N. A971). Maxitive measure and integration. Indag. Math. 33, 109-116.
Shoham, ? A988), Reasoning about Change - Time and Causation from the Standpoint of Artificial Intelligence,
The MIT Press. Cambridge. MA.
Smets, P. and Kennes, R. A994). The transferable belief model. Artificial Intelligence 66. 191-234.
Snow. P. A994). The emergence of ordered belief from initial ignorance, Proc. 11th National Conf. on Artificial
Intelligence (AAAI-94. Seattle. WA), 281-286.
Snow. P. A999). Diverse confidence levels in a probabilistic semantics for conditional logics. Artificial
Intelligence 113, 269-279.
Spohn. W. A988). Ordinal conditional functions: A dynamic theory of epistemic states. Causation in Decision.
Belief Change and Statistics. W. Harper and B. Skyrms. eds. 105-134.
Suarez Garcia. F. and Gil Alvarez. P. A986). Two families offuzzx integrals. Fuzzy Sets and Systems 18. 67-81.
Sugeno. M. A974), Theory of fuzzy integrals and its applications. Ph.D. thesis. Tokyo Institute of Technology.
Japan.
Sugeno, M. A977). Fuzzy measures and fuzzy integrals -A sunvy. Fuzzy Automata and Decision Processes.
M.M. Gupta. G.N. Saridis and B.R. Gaines, eds. North-Holland. Amsterdam. 89-102.
Verma. T. and Pearl. J. A990). Causal networks: semantics and expessiveness. Uncertainty in Artificial
Intelligence, Vol. 4. R.D. Schachter et al.. eds. North-Holland. Amsterdam. 69-78.
Walley. P. A991), Statistical Reasoning with Imprecise Probabilities. Chapman and Hall.
Wang. Z. and Klir. G. A992), Fuzzy Measure Theory. Plenum Press. New-York.
Weber, S. A984). Decomposable measures and integrals for Archimedean t-conorms. J. Math. Anal. Appl. 101.
114-138.
Whalen. T. A984). Decision making under uncertainty with various assumptions about available information.
IEEE Trans. Systems Man Cybernet. 14. 888-900.
Wong. S.M.K.. Yao. Y.Y. Bollman. P. and Burger. H.C. A990). Axiomatization of qualitative belief structure.
IEEE Trans. Systems Man Cybernet. 21. 726-734.
Wong, S.M.K.. Yao, Y.Y. and Lingras, P. A993). Comparative beliefs and their measurement. Intemat. J. General
Systems 22. 69-89.
Yaari. M.E. A987). The dual theory of choice under risk. Econometrica 55. 95-115.
Yager. R.R. A979), Possibilistic decision making. IEEE Trans. Systems Man Cybernet. 9. 388-392.
Yan, B. A993), Semiconormed possibility integrals for application-oriented modelling. Fuzzy Sets and Systems
57, 239-248.
Yan, B, A994). Seminormed possibility integrals for application-oriented modelling. Fuzzy Sets and Systems 61,
189-198.
Zadeh. L.A. A975), Calculus of fuzzy restrictions. Fuzzy Sets and Their Applications to Cognitive and Decision
Processes, L.A. Zadeh, K.S. Fu. M. Shimura and K. Tanuka. eds. Academic Press. New York. 1-39.
Zadeh, L.A. A978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1. 3-28.

CHAPTER 37
Measures of Information
Wolfgang Sander
Institute for Analysis, Technical University of Braunschweig. Pockelsstr. 14, 38106 Braunschweig, Germany
E-mail: w.sander@tu-bs.de
Contents
Introduction 1525
1. Probabilistic and non-probabilistic information measures 1526
2. Branching inset information measures 1531
3. Recursivity and generalized additivity 1534
4. Recursive measures of multiplicative type 1536
5. Regular weighted (/,m)-additive measures of degree (?. ?) 1540
6. Subadditive information measures for random vectors 1543
7. Information measures in a theory of evidence 1546
8. Measures of fuzziness 1550
9. Weighted entropies 1555
10. Summary '561
References 1562
HANDBOOK OF MEASURE THEORY
Edited by Endre Pap
© 2002 Elsevier Science B.V. All rights reserved
1523

This page intentionally left blank

Measures of information
1525
Introduction
There is no doubt that the concept of information plays nowadays an important role since it
has so many aspects and applications in different areas, see for example the books of Aczel
and Daroczy A975), Behara A990), Goodman and Nguyen A985), Martin and England
A981), Grabisch et al. A995), Guiasu A977), Taneja( 1989), Klirand Yuan A995).
Another commonly used word for information is uncertainty. Notice that we can
interpret decrease of information as increase of information. This was the reason of saying
that a measure of the amount of uncertainty is the same as a measure of the amount of
information.
And still a third expression occurs in the connection with information and uncertainty:
the name "entropy", which was originally introduced in physics by R. Clausius in 1864,
but which was proposed by John von Neumann "because no one knows what entropy really
is, so in a debate people using this word will always have an advantage".
The most common approach of modeling uncertainty is based upon some form of
probability theory. In this framework a measure of the amount of uncertainty (entropy)
should use all kinds of information that is provided, which here means: dependence upon
the outcomes of an experiment and the corresponding probabilities.
But in the early 1980s new types of uncertainty-based information were investigated,
due to different deficiencies of information like vagueness, ambiguity, non-specificity,
divergence, etc., mainly in fuzzy measure theory and fuzzy set theory.
So there are many different mathematical frameworks dealing with uncertainty or
information. We here restrict to
- the non-probabilistic concept,
- the probabilistic concept,
- the concept of a fuzzy measure,
- the concept of a fuzzy set.
Moreover, in this paper we want to focus on attempts of combining the above different
concepts so that we arrive at so-called "total measures of uncertainty" or "total information
measures" because we feel that there is now more need for generalized measures of
information and uncertainty within systems due to randomness, evidence and fuzziness
(see also Aczel and Daroczy A978), DeLuca and Termini A972), Forte A977), Klir and
Mariano A987), Ramer A987), Pal and Bezdek A994)).
The idea is to define an information measure by a set of intuitively, natural properties.
These properties can rather often be represented by a system of functional equations and
functional inequalities the solutions of which finally lead to characterization theorems
for total uncertainties. This axiomatic procedure is probably a most effective and a most
useful one for practical applications (see also Aczel and Daroczy A975) and Ebanks et al.
A998)).
We don't claim completeness in any respect, in contrary, we restrict ourselves to discrete
information measures, and also here, only a choice of the (in our opinion) relevant facts
and the bibliography is presented.

1526
W. Sander
1. Probabilistic and non-probabilistic information measures
In Sections 2-6 we want to present results in a "mixed" theory of information - inset
theory for short (the name was chosen because the idea was born at a meeting at the Ecole
Normale Superieure de L'Enseignement Technique - ENSET - near Paris, but inset can
also be understood as "in set"), where the measures of information may depend on both
the probabilities and the events. Thus the probabilistic and non-probabilistic concepts are
here combined.
The first pioneering result in this direction is due to Aczel and Daroczy A978) and was
the basis for many researchers for a long time. It turns out that nearly all problems in this
field are solved, and we will present the complete solutions.
To do this we start with some preliminaries like notations and important definitions.
But we also repeat here briefly results on well-known probabilistic and non-probabilistic
information measures.
The events are considered as elements of a ring В of subsets (containing, with any two
sets, their union and difference, and thus also their intersection and the empty set 0) of some
given set (universe) ?. Note that В is an algebra if for each ? e В there exist ?\? e B.
Fixing к to be an arbitrary, but fixed positive integer and putting
J = ]0, \[k, R0 = [0,oo[. R+ = ]0.oo[, A.1)
we denote the set of all finite (и-агу) disjoint classes of nonempty events (which form
a partition of ?, in case of an algebra B) and the set of A:-tuples of discrete, complete
probability distributions - without zero probabilities - of length ? (и = 2, 3,...) by
T>„ = {?=(?,,...,?„): ?,ей\@), ?,П ?,= 0, ? ? j, ij=\,...,n)
A.2)
(огЯ„= ? = (?,, ...,E„)eV„: \J ?,¦. = ? 1 A.3)
and
? " 1
??=\? = {?\,...,?„): ??, = 1 p\,...,p„eJ , A.4)
I i=l J
respectively.
If in A.2), A.3) and in A.4) empty sets and zero probabilities are allowed then we denote
these sets by
V„, Я„, Г„, A.5)
respectively.

Measures of information
1527
Here 1 represents the /<:-vector A. 1 1), and all operations on vectors are to be done
componentwise. We remark that ? e ?„ is а (к х /<:)-matrix where the
P,=(P\,,---,Pk,)TeJ, 1</?? A-6)
(? denotes the transpose) are column vectors and the
Pj = {Pj\,...,pjn), l^J^k. A.7)
are row vectors. We agree on writing ? e ?„ as
P = (P\,P2,...,Pn) = (P\,Pi Pk). A-8)
although /> = (/>,,..., Pk)T.
Dehnition 1.1. A sequence (/„) of functions
/„:2)„xf„^l or I„ :?„ ? ??-> R. ? = 2,3 A.9)
is called a (^-dimensional) inset information measure (or IIM, for short) on the open
domain. If in A.9) T>„ is replaced by V„ then (/„) is called an IIM on the closed domain.
If к = 1 then it is convenient to consider (/„) as a measure of uncertainty, for instance,
in the outcome of an experiment (or message or market situation, etc.) where the possible
events E\,..., E„ have probabilities p\ p„, respectively.
Let us first give some examples of important information measures in the probabilistic
case where we have no dependence upon the events:
Hartley A928) essentially introduced the following entropy
HZ(p\,...,Pn) = \og\{i:Pi>0. \^i^n}\ A.10)
which depends only upon the number of nonzeros in ? e ?,,.
The other principal entropy is probably the most famous entropy (it is the entropy for
the theory of communications), due to Shannon A948):
/7
H„(P) = -Yipi\ogpi. A.11)
Further important entropies are the entropies of degree ?, ? eM,
я-(Р)=(B^-1Ш=1Р?-1). «? U (U2)
\Н„(Р), а=\

1528
W. Sander
(see Daroczy A970)), and the entropies of degree (?, ?), ?. ? e R (see Behara A990) and
Sharma and Taneja A975))
/,e./J=|B'--2'-/')-|E?=l(pr-pf). ?? ?, A13)
Note that
lim H%(P) = H„(P),
a^ I
Н^Р) = Щ and Н^-Р) = НР
so that #,, is a two-parametric family generalizing the Shannon entropy and also the
entropy of degree a.
Examples of so-called deviations (k = 2) and information improvements (k = 3) are the
inaccuracy of Kerridge A961)
/1
Kn(P\, Pi) = ~??? log p2i A.14)
1 = 1
and the information improvement of Theil A967)
//
Ln(P\,Pi,P3) = yiP4l°S—· 0·15)
?? p*
respectively.
Here K„ is given by observing a true probability distribution P\ as Pi, whereas the
interpretation of L„ is that a true probability distribution P\ will be predicted by Pi, and
then this prediction will be revised by ??,. Note also that
K„(P,P) = H„(P).
Continuing this last procedure it seems natural to consider information measures depending
upon к probability distributions since situations like repeated revision of data collections
occur rather often in weather forecast, economics and experiments.
If, for example, zero probabilities are allowed in A.15) then we have to require that
p\i =0 whenever pit = 0 or рц = О, and also the awkward conventions
01og0 = 0, 01og-=0. A.16)
To avoid these restrictions in "closed domains", information measures on the open domain
are investigated (and there are well-known techniques to extend results on an open domain
to one on an "half-closed" or "closed" domain, if necessary; see Aczel and Ng A983)).

Measures of information
1529
Let us remark that the proofs of characterization theorems of information measures
become much more difficult on an open domain than on the closed domain.
Now we briefly comment non-probabilistic information measures. For a survey see, for
example, the books of Guiasu A977) and Grabisch et al. A995). The two main approaches
are due to Ingardenand Urbanik( 1962) and to Kampede Feriet and Forte A963, 1967).
Without going into details the first two authors have shown that information and
probability are not separated things: starting with information content I{E\) then in a
canonical way p(E\) occurs, and reversely. The connection is given by
/(?,) =-logP(?,). ?(?|) = «?"?". A.17)
In Kampe de Feriet and Forte A963, 1967) the principle of localization is used defining
the information measure / on a measure space (?.?) supplied by the events (instead of
defining the information of the whole experiment). Here I :B—> [0. oo]. and if E\, ?S are
disjoint elements of В then / satisfies
I(E\ U E2)= F(I(E\ ),/(F2)) O·18)
where F: [0, oo]2 -> [0, oo] is a so-called composition law, that is F is a ?-norm on the
interval [0, oo] (F is associative, commutative, monotonic increasing in each place, and
oo is neutral element of F). We get immediately (see, for example. Schweizer and Sklar
A983)):
F is a composition law iff for any continuous and strictly increasing function ?: [0.1 ] —»
[0, oo] the function
Т(х,у) = ф-*[Р(ф(х),ф(у)]. л, ? e [0.1]. A.19)
is a ?-norm (on [0, 1]). Thus we can generate composition laws from f-norms using
Е(х,у) = ф[Т(ф-](х),ф-](у))]. л-.уе[0,оо]. A.20)
Let us consider two examples (where с > 0 is a fixed constant).
A) If T(x,y) = max(.v +y - 1,0), ф(х) = -flogd -л). 0 '(-?) = 1 - e~cx then
A.20) implies that
F(jc,y) = max^--log(e-<v+e-<v).oY *.ve[0.oo]. A.21)
is the Wiener-Shannon law with parameter с
B) If T(x, y) = xy, ф(х) = - 1/log.v, 0 (.v) = e"l/v then we arrive at the hyperbolic
law
F(x,}>)=—^, *.ve[0.oo]. A-22)
x+y
To connect the local point of view (information of an event) with the global point of view
(information of the whole experiment) we may assume that (?.?. ?) is a probability

1530
W. Sander
space and we consider a partition (? \ ,...,?„) of ?. Then the amount of information that
is provided when one among these events occurs is one of the numbers
/(?,), /(F2), ..., /(?„) A-23)
while
P(?i), P(E2), .... P(E„) A-24)
are the corresponding probabilities. Putting p, = P(F,) the expected information
uE-p)=!'{pfEi) pfk) pS.))=P'/(?') (L25)
is the amount of information one expects to receive from the experiment to find out which
of the events E\,..., E„ will occur.
Finally, let us introduce some notations which will be needed in the following sections.
Let U denote the setoff unit vectors и,- = @, 0, l,0,...,0)eR', 1 ^ /' ^ k.
If X is a set then V(X) = [А: А с X) is the power set of X.
If Pe r„,then
D„ = \(p\,...,p„): pte J, ^,еЛ. A-26)
We define for PeThQe Г,„, F e ?>,, F e Vm (F e П,, F e ?>„,)
P* Q = (p\q\, ¦ ¦ ·. P\qm, ¦ ¦ ¦, p,q\ ¦ ¦ ¦ ¦, p,qm) e П,„ A.27)
and
? ? F = (F, ? F,,..., ?\ ? F,„,..., ?, ? F,,..., ?, ? F,„) e V,m A.28)
(F ? F e Л,,„, respectively). Moreover F ? F ^ 0 means that F, ? Ej? 0, 1 ^ /' ^ /,
1 ? j ^ w.
We also agree on writing
pa = (Pi,...,Pk)iai ai):=pT---Ptk (L29)
for all p, a e R*. Note that p"eR whereas p" e R* for и е N. For instance,
p2 = (P\,.--,Pk)-(P\,--.,Pk)=(pl--;Pl)- (L3°)
A map g: В \ {0} -> R is additive if
S(F, UF2) = g(F|)+g(F2), (F|,F2)eP2· d-31)

Measures of information 1531
Moreover mappings aJ,m:J -> R are called additive, logarithmic and multiplicative if
a(x+y) = a(x) + a(y). (i.v)eO:. A.32)
l(x-y) = !(x)+l(y), x.yeJ. A.33)
m(x ¦ y) = m(x) ¦ m(y), x.yeJ, A-34)
respectively. The general form of these additive, logarithmic and multiplicative functions
are given by
к
а(дс) = а(дс|,...,/*) = J]fl/(jr/), x e J. A-35)
i=l
*
Kx) = l(x\,...,lk) = Yili(xi). xeJ. A.36)
i=l
к
m(x) = m(x],...,lk) = Y[mi(xi). xeJ. A-37)
i=l
where each a,, /;, w, is additive, logarithmic and multiplicative, respectively on {{p,q):
p,q, ? +q e ]0, l[},on ]0, oo[2, or on ]0. 1[2.
So the corresponding regular (say measurable) functions are given by
a(x) = cQx, l(x) = cQlogx. m{x) = xl. ceJ. A.38)
Here с ? jc is the usual inner product in Rk and log.? is defined componentwise (where
log=log2):
cQx = (c\,...,ck)(x\ xk) = c\x\ ? hckxk· A-39)
logjr = (logor logjrjt). A-40)
Finally we denote by
к к
pOCOq = YiYiPiCijqj A.41)
1 = 1 1 = 1
the multiplication of vectors p, q with a real-valued (к х к)-тгХпх С = (с,/).
2. Branching inset information measures
Definition 2.1. An inset information measure (/„)

1532
W. Sander
A) has the branching property if there is a function G : Vi_ ? Di -> R such that
Pi ·· Pn I \P\+P2 Pi ·· Pn I \P\
for all (?, P) e P„ ? ?„ (и = 3,4,...),
B) is /-symmetric for some / e N, / > 3 if
? ? ··· E/\ _ л (??{\) ··· ??(/)
~?(?) -?1" , (?, Р)еР/хГ/, B.2)
Pi ··· ?// V^iD '" ???/)/
for all permutations ? on {1, 2 /}.
In the following we agree on the notation
In(E,P) = I„(E;P) = I„(E] '" E" V (?,/>) e P„ ? ?„. B.3)
\?? ·· P«/
The branching property is one of the most fundamental axioms for any information
measure having the following interpretation. In an experiment A with possible outcomes
E\,..., E„ and corresponding probabilities p\...., p„ the left-hand side of B.1) is the
uncertainty of A. The right-hand side contains as first summand the uncertainty of the
experimentB with outcomes ? ? UEi, ?3 E„ and corresponding probabilities ? ? +P2,
??,,..., p„. So we are uncertain whether E\ or Ei in the outcome will occur. To overcome
the difference of the uncertainties of A and В a third experiment С will be performed to
decide whether E\ or Ei will occur. This is taken into account by the second summand ?
on the right-hand side of B.1): ? depends only upon E\, E->_ and p\, P2 which is weak
assumption.
The second property which will be needed for our first main characterization theorem
is the symmetry of an IIM: Each //, / ^ 2, is invariant under a simultaneous change in the
order of the events and the corresponding probabilities. From a mathematical point of view
only a weaker symmetry condition is needed.
It is interesting that it is sufficient to use only these two properties, the branching
property and the 3- and 4-symmetry for a first important characterization theorem of an
IIM (/„).
Before we present this result we look more closely at the functional equation B.1) with
? =4 and see that B.1) doesn't require the value of ? on all of ?? ? A>, if Sis an algebra
with atoms. It requires values of ?(?\, ?2; p\, pi) only for those
(??, ?2) to which ?3, ?4 exist with (?|. ?2, ?3, ?4) e P;
or there exist E\ \. ?12 satisfying
?i=?iiU?|2, (?ц,?|2)еГЬ and (?,,, ?,2, ?3) e P.v

Measures of information
1533
Denoting this domain of ? by V ? Di it can be seen that if В either is not an algebra or is
a non-atomic algebra then V may be replaced by pairs
(??, ??) to which there exists an ?3 such that (?|, ?2. ?3) ? ?>з·
Moreover, if В is not an algebra then D = ZV
Due to technical difficulties of this type of domains the occurring functions in the
following Theorem 2.2 are not defined on V\ ? J, but on appropriate subdomains which
will not be given explicitly. They will be indicated by some "primes" on D|. and the
interested reader is referred to the original paper of Ebanks A990), where also some
refinements of the following result are presented.
THEOREM 2.2. A) Let В be a ring of sets. An IIM (/„) on V„ ? ?„ is 3- and Asymmetric
and has the branching property, if and only if there exist maps b:T>\ ? J —> ? and
с: V'[ -> ? such that
/„(?, P) = ][>(?,. Pi) +c((j ?,) B.4)
for all (?, ?) e V„ ? ?„, ? ??,? 2.
B) If in A) В is an algebra and I„ is defined on ?? ? ?„ then I„ has the form B.4)
with с = 0, that is
?
I„(E,P) = Yjb(El.Pl) B.5)
i = l
has the so-called sum-form property with generating function b.
Remember that nearly all known information measures have this sum-form property.
Thus it is surprising that this result holds without any regularity condition.
The proof is rather long, and is based on many earlier work. We mention here the
pioneering papers of Jessen et al. A968) and of Ng A974). For a survey in the purely
probabilistic case we refer to the book of Ebanks et al. A998).
Let us give a very brief outline of the proof to indicate the occurring difficulties and the
dominant role of functional equations in this area.
Assuming that (/„) satisfies B.1). the 4-symmetry of (/„) leads to the so-called
fundamental equation of IIM with the branching property
?,U?2 ?? /E\ E2
Р + Я r ) \p q
?,U?, ?? (?? ?? B6)
P + r q ) \p r )
for all (p,q,r) e Dj and all {?\.??,?^) for which there exists an E\ with (E\,Ei.
?3, ?4) e T>4.

1534
W. Sander
Moreover, the 3-symmetry leads to
'?|U?i ЕЛ Л(Е\ Е2
b ¦ . . - ¦
Р+Я r ) \p q
= /?,U?, E2\ (Et ЕЛ 2
So the key is to find the solution ? of B.6) (see Ebanks A989)) because then, substituting
the form of ? into B.7) the form of h can be derived, and finally the form of (/„) can be
derived inductively via B.1). (We remark that keeping (?|, ??, ? у,) е Рд temporarily fixed
in B.6) we arrive, defining i/r,: D2 -> К in a canonical way, B.6) leads to the functional
equation
?\ (? + q,r) + ?2(?, q) = ЫР, О + ЫР + г, q) B.8)
for all {p, q, r) e D^, which contains simultaneously 4 unknown functions. Having
determined these 4 functions and freeing (?|. Ei, ?-0 again, the form of ? can be found.)
3. Recursivity and generalized additivity
In Section 2 it turned out that all 3- and 4-symmetric IIM (/„) on T>„ ? ?„ with the
branching property have the form
I„(E,P) = Ylb(Ei,pi)+cl\jEiY О 2. C.1)
?=? \/=l /
for some functions b:V\ ? J -> R and c:V'[ -> R (and с = 0 if (/„) is defined on
T>„ x П„).
The problem is to determine b explicitly since up to this point b is an arbitrary map.
Now there are two main streams to get the explicit form of b:
A) One possibility is to require a special form of ? in B.1) This will lead to a natural
condition, the so-called
Recursivity of multiplicative type. C.2)
B) Another possibility is to impose additional conditions, and we consider here
Generalized additivity conditions. C.3)
Let us explain the ideas behind C.2) and C.3).
Remembering the interpretation of the branching property (after Definition 2.1). The
difference
lJE> ·· E")-lJEiUE2 E* ·· E") C.4)
\P\ ·· Pn I \P\+P2 Pi ·· Pn )

Measures of information
1535
in B.1) will now be recaptured by an experiment С having two outcomes E\. ?2 with
conditional probabilities
q\ = ¦ . 92 = ; C-5)
P\+P2 P\+P2
(so that <7i + qi — 1). On the other hand the experiment С must only be performed in case
of E\ U Ei occurs, the probability of which is p\ + /r>. Thus we assume that ? is the
uncertainty of the experiment С multiplied with a function of p\ + pi'-
?' E2) = M(p]+p2).I2(E> E2). C.6)
P\ Pi) У F' \q\ qi)
Moreover there is a convincing argument to assume that ? in C.6) is not an arbitrary, but
a multiplicative function (for details see the paper of Aczel and Ng A983)). This leads to
Dehnition 3.1. A) An IIM (/„) is called recursive of multiplicative type ? if (/„)
satisfies B.1) with the function ? given in C.6) with a multiplicative map M.
B) If in addition M(x) = xa, ? e J then /„ is called recursive of degree a, a e Rk.
For example, the Shannon entropy H„ is recursive of degree 1,
- the entropy of degree ? {? 1) is recursive of degree a,
- the inaccuracy of Kerridge is recursive of degree A.0), and
- the information improvement of Theil is recursive of degree A.0,0).
In the next section we will present all 3-symmetric IIM (/„) which are recursive of
multiplicative type ? (without and with regularity conditions).
Let us remember the idea of additivity.
We now assume that we have two independent experiments A and В which are given by
(E, P) eT>i ? ? and (F. Q) e T>„, ? ?,„, respectively. Here independence means that we
can perform the combined experiment
(EnF,P*Q)eV/m ? r,m.
Now, additivity says that the information obtained from experiment С is the sum of the
amount of information received from A and B:
//„,(?¦ П F, ? * Q) = I,(E, P) + I,„(F, Q). C.7)
? e ?/, Q e Гт, ? П F ? 0. More generally we assume that the information obtained
from experiment С is a weighted sum of the amounts of information obtained from
experiments A and В for some weight functions w\ and u'::
//,„(?¦ П F, ? * Q) = w\(P) ¦ I,AF. Q) + w2{Q) ¦ //(?. P), C.8)
? e ?/, Q e ?,?, ? П F фЪ. Since the operations П and * are associative it can be shown
(only assuming that the range {/„ (P„ ? ?„): ? = 2, 3 } has infinite cardinality) that w\

1536
W. Sander
and wi have the sum-form property with multiplicative generating functions:
/ m
w](P) = YjM\(Pi), w2(Q) = J2M^Pj)· P*rhQerm. C.9)
i=l j=\
Here M\,Mi:J -> Ш are multiplicative functions.
Note that C.8) and C.9) go over into C.7) if M\ and Mi are projections, that is
M(pi) = M(pu, · · ·, pi,) = Psi for some s,
M(qj) = M(q\j,..., qmj) = q,j for some t.
Dehnition 3.2. A) For a fixed pair (l,m), I ^ 2, m ^ 2, we say that an IIM (/„) is
weighted (/, w)-additive of type (?\, ??) if it satisfies
/ m
//„(?¦ П F, ? * 0 = ]? ??? (?) · /„(F. ?) + J] ??2(?>) · ??(?. ?) C.10)
?=? 7=1
for all ? e ??, Qe ?,„, F e D,, F e D,„, F П F ^ 0, where ? ?, ??2 : J -> ? are
multiplicative.
B) If in C.10)
M\(p) = pa, M2(q) = qfi for some a, /3 e R* (p, q e У), C.11)
then (/„) is called weighted (/, w)-additive of degree (a, /3).
In this survey we will present all regular (here measurable) solutions of C.10) in the
case C.11) if (/„) satisfies the sum-form property B.5).
We remark that we require C.10) only for a fixed pair (/, w) and not for all / > 2, m > 2.
Moreover, it turns out that the case / = 2, m = 2 is the most difficult and surprising one (in
both cases) since unexpected solutions occur.
4. Recursive measures of multiplicative type
THEOREM 4.1. Let В be a ring of sets ?/?. An IIM (/„) is 3-symmetric and recursive of
multiplicative type ? (? ? 0) if and only if it is given by
/,,(F, ?) = ? f(E,)M(Pi) - /I \J E,¦ ) +?(?, ?) D.1)
i=l \?=| /

Measures of information
1537
for (?, P) e V„ ? ?,„ ? = 2, 3,..., where
E'=|L(?„/>). ifM=\,
?(?,?)= ?"=]1(?,)?(??), if ? is additive, D.2)
0, otherwise.
Here l:J->Ris logarithmic, L:V\ ? J -> 1 is additive in the first and multiplicative
in the second component {on appropriate domains), and f.g:V\ -> R are arbitrary
functions.
If В is an algebra with atoms then for ? = 2 the domain ofI2 is given by V ? ?2 (instead
ofT>2 ? ??) with
V=V2\\(E\,E2): E\ is an atom and E\ U E2 = ?]. D.3)
Again, let us indicate the proof.
Substituting the special form C.6) of ? into B.7) - and using that p+q+r=\-we
arrive at the so-called fundamental equation of recursive inset information of multiplicative
type:
ф(Е]иЕ2,Еур) + М(\-р)ф[Е].Е2: Я
\-p
= ф{Е\ UE?,E2;p) + M(\ -р)ф(е\.Еу.-^-Л D.4)
for (?|, E2; p) eT> ? J, where
ф(ЕиЕг,р):=Ь[,Е] ЕЛ. (Ei.E2\p)eT>xJ. D.5)
V1"^ ? )
A key step in solving D.4) is to solve the functional equation
K{E\ U E2, E3) + K(E\. E2)= K(Et U ?3. E2) + K{EU ?3) D.6)
for all (?|, ?2, Ey) e V3, if К : V -> R.
It is interesting that only by the additional form C.6) of ? the function b in B.5)
(now defined on T>\ ? J instead of V\ ? J) can be given explicitly (note that if (/„) is
3-symmetric then (/„) is 4-symmetric so that (/„) indeed has the form C.5)).
Let us now give some examples of Theorem 4.1.
A) If ? ? 1 and ? is not additive then the case / = constant = с gives the probabilistic
information measures
/„(/>) = с(]ГМ(р,)-1). D.7)

1538 W.Sander
If we assume recursivity of type ? (?) = pa (with some a priori given a e Шк) then we
arrive without any regularity conditions at the regular solutions
1п(Р) = с1?щрЛ-1\=с(?ра11р%---р%-\ D-8)
Some well-known special cases of D.8) are:
- (for k = 1) the entropy of degree ?, ? el:
1п(Р) = сГ?рГ-\ D·9)
- (for к = 2) the inaccuracy of degree a, a e E:
1»(П = сГ?р1р*2Г-Ч- D1°)
- (for к = 3) the divergence of degree a, a e E:
/»(/,) = с(Ер|^2Г1р'Г-Ч· DЛ1>
B) If ? is additive (and nonzero) then ? is a projection. In this case it is not trivial to get
the regular solutions from the representation in Theorem 4.1. The reason is the following.
Let us assume к = 1 in D.2). Then M(p) = ? and the Shannon entropy H„(P) is
very regular having the sum form ??=\ b(pi) with the regular function b(x) = —x log*.
Consider now the very "irregular" function
c(x) = —x log* + d(x)
where d: ? -> ? is any nontrivial derivation (that is, d satisfies d(x + y) = d(x) + d(y)
and d(xy) = xd(y) + yd(x) so that we arrive at ?"=] d(pi) = d(\) = 0. But a regular
(say measurable) derivation is continuous (being additive) and thus linear, which means
d(x) = ? d(\) = 0 for all ? el). Introduce the function
c(x) d{x)
b{x) — xl{x) where l(x)= = —logjrH ·
? ?
Then / is logarithmic and ??=\ KPi)M(pi) can be very regular for some irregular
function /.
If (/„) is measurable, that is, only the function ? defined in D.5) is measurable in ? for
fixed (?|, ?2). then some additional machinery is needed to prove that
/„(?, ?) = ? f(Ei)M(Pi) - /I (J ?< ) + ?(? °^gPi)M(pi) D.12)

Measures of information
1539
for some constant vector A e Rk (see, for instance, Jarai and Sander A997) for a more
general result and the book of Ebanks et al. A998) for historical remarks and further
references).
Thus we get in the probabilistic case (/ = constant) that D.12) reduces to the last sum
in D.12) from which we get for example - with appropriate constants - Hn(P), Kn{P\, Pi)
or Ln(P\, P2, P?) (cf. A.11), A.14) and A.15)).
C) The real surprise of Theorem 4.1 comes in the case ? = 1:
We get an unexpected term
/7
]Tl(?,,/>) D.13)
? = ?
which disappears if either zero probabilities are allowed or there is dependence on events
(due to the fact that L is logarithmic in the probabilities and additive in the events). If again
(/„) is measurable then we get
? / " \ "
/„(?, P) = ? /(?,)- /I (J ?, +]Ta(?,)Ologp, D.14)
i = l \/ = l / i'=l
for some additive set function a:T>\ -» Rk (see Ebanks A984)). Note that if a =
(??,...,?*) for additive set functions aj\V\ -> R, \ ? j ? k, then (with our
notations A.6) and A.7)) we have
/7 ? k П к
]Tl(?,, p,) = ]??(?,) ©log?, = ]? ]??,(?,I?§?„ =: ]?/,/,(?. />,·).
?' = ? ?'=? j = \ 7=1 y' = l
D.15)
Finally we remark that many authors have contributed to these results, and it took a long
time until the above Theorem 4.1 was proved by Ebanks et al. A988). A complete list of
(nearly) all contributors can be found, for example, in Sander A987) and in the book of
Ebanks etal. A998).
It is interesting that we can get the recursive inset entropies also by the well-known
"how to keep the expert honest" method (see the book of Aczel and Daroczy A975), Aczel
A980) and Kannappan and Sahoo A989)):
An expert predicts q\,..., qn as probabilities of the events E\...., E„ which in reality
are p\,..., p„ (here it is assumed that E, e В where В is not an algebra). The expert agrees
to be paid only after the occurrence of the event and that his payoff will be /(?,) if the z'th
event happens. So his expected gain is ?''_| Pif(Pi)· 1° order to keep the expert honest
the payoff function / will be chosen so that the expert's earnings will be maximum if he
predicts the "true" probabilities, that is we have
77 ?
]? Pif(Ej, q,¦) < ]T pi/(?,, pi)
/=l i=l

1540
W. Sander
or more generally
77 77
/ = l ? = ?
for a fixed я ^ 3, ? > 1 and for all ? eT>„, ? e ?„.
Without any assumption on / it was shown by Aczel A980) and Kannappan and Sahoo
A989) that the last two inequalities lead to the inset entropies
I(E Р)=\сН"(П + Я=\я№)· a = l>
'Л ' \cHX(P) + r;=iP7f(E0 + b. a>\,
where b, с are constants.
5. Regular weighted (/, m)-additive measures of degree (?, ?)
Dehnition 5.1. An IIM (/„) on ?„ ? ?„ has the measurable sum-form property if there
exists a map b: T>\ ? J e ? satisfying B.5) for ? = 2. 3.... such that ? -> b{\, p) is
measurable on J for each E\ &T>\.
We remark that rather often the sum-form property will be considered as an "artificial"
property but we have pointed out that this property is a consequence of the very natural
(and weak) conditions:
Branching property and weak symmetry.
Let us now fix /, m e N, / ^ 2, m ^ 2 and ?, ? e Шк.
In this section we present all weighted (/, w)-additive IIM (/„) of degree (?, ?) which
have the measurable sum-form property.
Substituting B.5) and C.11) into C.10) we arrive at the fundamental equation of
weighted (/, w)-additive inset information of degree (?, ?):
I m
???(?, П Fj^iq^-q^biEi. ?,)) - ?? (b(Fj¦, qj))) = 0 E.1)
/ = l y = l
for all ? en,,F e ?,„, ? ? F ? 0 and for all ? e ?, Q e ?,„.
The problem is to solve this functional equation for fixed / ^ 2, m ^ 2:
If b is known then we get /„ from B.5).
The main result here is a result of Gross A993) and Gross and Sander A991) which is
an extension of the first results in this area by Ebanks A986, 1994) who investigated the
case (/, m) = B, 3) and ?, ? e U.
THEOREM 5.2. Let В be a ring of sets, let I. m e N, / ^ 2, m ^ 2 and let ?, ? e Rk be
arbitrary, but fixed.

Measures of information
1541
An IIM (/„) on ?„ ? ?„ has the measurable sum-form property and is weighted (I, m)-
additive of degree (?, ?) if and only if
A) there is a purely probabilistic information measure (J„) with J„ : ?„ —* ?,
? — 2,3,..., which has also the measurable sum-form property (with a purely
probabilistic generating function) and which is weighted (I. m)-additive of degree
(?, ?), and
B) if in the case ?, ? e U there is an additive set function a:V\ -+Rk such that
IAE,P)=UP) + \T!Ua{E')OXOgP'· ?-?"?· <5·2>
10. otherwise,
if В is a non-atomic algebra and
I„ (?, P) = J„(P) if В is not an algebra. E.3)
Ebanks A994) remarked that if В is an algebra with atoms the same methods as in
the proof of Theorem 5.2 can be applied but this case was not investigated because of its
complexity and the lack of applications.
The case E.3), where В is not an algebra, is uninteresting from the inset point of view:
it reduces to the purely probabilistic case, that is /„(?. P) = J„(P) is given by
?
J„(P) = Y^f(Pi) for some function/: J -» ?. E.4)
? = ?
where ? e ??, and where / satisfies
/ 111
? YXfiPW) - ??/(Л") - P?f(qj)) = 0· E-5)
i = l 7=1
If ? is a non-atomic algebra we have again the purely probabilistic case with the exception
of ?, ? e U where surprisingly the same term occurs like in Section 4 (see the last sum
in D.14)).
Since the solutions of E.5) are known in nearly all cases (for further details we refer to
the survey article of Ebanks et al. A997)) we will present the most interesting solutions.
Again, in the case / = m = 2 the behavior of the solutions is somewhat surprising
since new unexpected solutions arise, namely polynomials up to degree 3 if ?. ? in U
and polynomials up to degree 5 if a = ? (in the case к = 1).
If ?, ? e U we get from E.2)
/„(?, P) = J2Dd ? Pi ~9dO PI +ЫО Pi) + 0m -l-m+ n)b
i = l
II /I
+ ^2piOCO\ogPi+^2a(Ei)Q\ogpi. E.6)
i = l 1 = 1

1542
W. Scolder
Here a : V\ -> R* is an additive set function, b eR, d = (d\,... ,dk) eRk and С = (c,y)
is a (& ? A:)-matrix. Moreover, a = 0 if/w ^ 6.
This shows that the first sum in E.6) occurs only in the exceptional case / = m = 2. Note
that /„ can be rewritten as (putting d\ = -jdj, 1 < /' ^ k)
к к к
I„(?, P) = 2J2d',H*{P,) - 3 ]????^??) + - ? ?»(?,)
? = ? ?' = | ? = ?
к к к
+ ?????(?.) + ? ? cuK„(Pi,Pj)
?
+ {lm-l-m + \)b + Y^a(E,H logp/. E.7)
Let us assume for the moment that the set function a = 0. Then all measures J„
satisfying E.4), E.5) and the usual normalization condition /2G, 4) = 1 for all / ^ 3, w ^ 3
are given by
¦/„(/>) =
E"=iP;OCOlogp,, 1OC0U-I, ?,???,
TH=\^(bO\ogPi), bO\ = -2a~], a = peRk\U,
.B*-°-2*-?)-?(?!!^??-?!=???)' ???,???? or ?^?.
Here fo is a A ? A:)-vector and С = (c,y) is a (? ? A:)-matrix.
In this last formula for Jn the first sum represents a natural к -dimensional analogue of
the Shannon entropy, whereas the last two sums are a natural ^-dimensional analogue of
the entropies of degree (?, ?) (cf. A.13)).
Finally we remark that it is an interesting open problem to determine the general solution
of E.5) if ?, ? e U (in the case a <? U or ? <? U the general solution of E.5) is known!!).
As an indication of the occurring difficulties we consider in the one-dimensional case
the map /: @, 1) -> R defined by
D2(x)
/00 = —^, *e@.1). E.8)
?
where D is a nonzero real derivation. Then a calculation shows that / satisfies E.5) with
a = ? = 1 for all fixed pairs (/, m) of integers, / ^ 2, m ^ 2. Thus / is not measurable
(even nonnegative), but it is not of the form E.6) (with a = 0). On the other hand it
was pointed out in the book of Ebanks, Sahoo and Sander that all remaining problems
in the characterization of functions / satisfying E.5) (and more generalized functional
equations) if the following one functional equation
/(W)+ /(/>(!-?))+/((l-/>)?) + /((l-/>)U-?))=0, P,q e J, E.9)

Measures of information
1543
can be solved. Here
f(p) = d(p)+aOp + b E.10)
where a e Шк, b e R and d: J -> R is a non-measurable solution of E.9).
Summarizing the results in Sections 4 and 5 we can say that from the two results in
Theorems 4.1 and 5.2 we can derive nearly all relevant and well-known characterizations
for (inset) information measures.
6. Subadditive information measures for random vectors
Let us briefly remember the concept of subadditivity of information measures using an
appropriate notation.
Let ?, ? be two real-valued discrete random variables (d.r.v., for short) on a probability
space (?, ?, P) with values in two finite sets of real numbers
i* ·=
?(?) = {?],...,?„] and ??:=?(?) = {}?,...,?„,}, F.1)
respectively. Let (??,?>(??), ??) and (??,?(??), Qy) be the probability spaces
induced by X and Y, so that
pi = P(X=xi) = Px(xi), 1^ /<;„,
q]=Q{Y = yj)=QY(yj), \<:j^m. F.2)
Definition 6.1. The amount of uncertainty induced by the random vector (X, Y) with
range
Аху.Ву = \(х\,у\),...,(х„,ут)\, \AX ? BY\=nm F.3)
and the joint probability distribution
Pxy = {Рц = P(X =x,,Y = }'j): l^i^n, K/^m) F.4)
is defined by
, ?? ^R ? ч_/ {(х\,У\) ·· (хп,У„,)\ /6сч
1„??(?? ? ??, ???) = 1„т I „ )· УЬ--1)
\ Р\\ ¦¦¦ Рпт )
We will denote Inm also as the information measure of (X, Y).
Definition 6.2. If Inm is the information measure of (x, y) Then /„,„ is (n, w)-subaddi-
tive if
Inm(Ax ? ??, ???) < I„(AX, Px) + Iln(BY, Qy)
F.6)

1544 W.Sander
or
, ((х\,У\)
V Pw
\P\
where
m ?
?? = ??4' 1? = ??'?' l^i^n,l^j^m. F.8)
7=1 ?=?
Now the property of subadditivity is self-explanatory.
Aczel et al. A974) gave a simultaneous characterization of the Shannon entropy (H„)
and the Hartley entropy (#,*) (see A.10)). More generally, it could be proven that every
probabilistic one-dimensional information measure J„ : ?„ -> Ш which is
A) /-symmetric for all / e N,
B) (/, 2)-subadditive for all / e N, / > 2, and
C) (/, w)-additive for all /, m e N, / ^ 2, w > 2
has the form
/„(/>) = A ·#„(/>)+0(л), РеГ„, и = 2,3,..., F.9)
where ? ^ 0 is a constant and ? is an arbitrary additive number theoretical function:
ф(пт)=ф(п)+ф(т), п^2,т^2. F.10)
This result is interesting because it was difficult to derive an equality from the inequality
(in the definition of subadditivity). In the paper of Forte and Gupta A985) an inset version
of the above result is shown, but the main steps of the proof can already be found in another
paper of Forte A977). For simplicity (and to avoid long and heavy notations) we present
their result in the case of a 2-random vector instead of а к -random vector.
The following properties are assumed:
(Al) Subadditivity. /„„, is subadditive for all ? ^ 2, m ^ 2.
(A2) Additivity. If X and ? are independent then the uncertainty about (?? ? ??, ???)
should be maximal (cf. C.7)):
I„m(Ax ? ??, Px * QY) = I„(AX, ??) + ?,,???, Qy). F.11)
Note that now ??? = ?? * Qy.
(A3) Weak expansibility. If pnj = 0 for all 1 ^ ;' < m or pim = 0 for all 1 ^ /' < ? then
/„„, is weak expansible if
, {(х\,У\) ··· (*„,?», )\ , {(х\,У\) ·¦· (Хп-\,Ут)\
\ Pi ? ¦·· Pnm ) V Pi ? ¦·· P(n-\)m )
(Xi,}'i)
PU
Pn)
iyl
(х„,Ут)
Pnm
¦ ¦ ¦ Ут
¦ ¦ ¦ Чт
F.7)

Measures of information
1545
or
T {(Х\,У\) ··· (Xn,y„,)\ _ , f(x\,y\) ··· (хп,Ут-\)
V ?? ? ··· Pnm ) V Pi ? ·¦· Аи/л-п
(A4) Partial symmetry. Let (X, У) and (X', У") be two independent random vectors
with the same ranges ?? = ??· and ?? = ??· where \??\ = ?, \??\ = т.
Then ?????' = ??? ¦ ???' and it seems naturally to assume that for all fixed
(r,s) and (t,u)
, ? (¦*/-, }\, xt, y„) · · · (??. y„, .v,-, у ?) ·
V",M- prsP'tH ··· a„p;s
? ,/'" С*/" > У-? > *f > У» ) ··" (jfi. ,V||.^/·. V.v) ••? F 12")
/?-//?-
PtuPrs ¦¦¦ PrsP,,,
Here the dots in F.12) indicate the unchanged part of the domain of /„:,„:. Thus
only the probabilities PrsP,,, and plup'rs are exchanged.
(A5) Boundedness from below. Define p, s = 1 - ? and p!u = ? for some fixed 1 ? r,
t ^ ? and 1 ^s, и ^m and p,y = 0 otherwise. Then the functions
?^??,?(?? ? BY, PXY) on [0,1] F·13)
are assumed to be bounded from below.
Now we are ready to present the main result of Forte and Gupta A985) (in a slightly
different but equivalent form).
THEOREM 6.3. The measures of uncertainty (?,,??) satisfy the properties (Al)-(A5)/or
all n,m e N if and only if
him (A x x ??, ???) = -а ]Г ]Г pulog ??
? = ? 7=1
+ ]? F/(jti. · · ·, Xi)Pi + ? F>(-V| · · · ·' yJ)qJ
i=l 7=1
+ G„(.vi, ....*„) + Я,„(У|,..., y,„) F.14)
for all ??? such that pij > 0, 1 ^ /' ^ //, 1 ^ j ^ m, where a is a nonnegative real
constant, Ft: №.'' -> E, / = 1, 2,..., Pi, qj satisfy F.8), G„ : ?? -> К аиЛ Я,„ : By -> R.
The idea of the proof is the same like in the paper of Aczel et al. A974) but is now
much more technical. Let us mention that the investigations lead to a generalized version
of Equation D.4) where in the purely probabilistic case the function ? will be replaced by
4 different functions (see, for example, Jarai and Sander A997)).

1546
W. Sander
Let us finally present the following the following special case of Theorem 6.3. If we
require a slightly extended version of expansibility and if the boundedness condition will
be replaced by a continuity condition at 0 then F.14) reduces to
him {? ? ? ??,???)
? m ? m
= -aYJYJPij\ogplj+YjF(xl)pl+YjF(yj)qj F.15)
i = l 7=1 /=l 7 = 1
for some function F: ? -> ? where the first sum is of course the Shannon entropy and the
last two sums represent the expected information (cf. A.25)).
7. Information measures in a theory of evidence
A more general concept than the concept of measure - the concept of a so-called fuzzy
measure - takes into account that in many applications no "additivity" is present:
Fuzzy measures are monotonic increasing functions m : V(X) —> [0, 1 ].
Here X is an arbitrary set. These fuzzy measures occurred already in other fields of
mathematics, take for example capacities or characteristic functions in cooperative game
theory.
Thus a fuzzy measure assigns to each subset of a universal set a value signifying
the degree of evidence or belief that a particular element belongs to the subset. The
monotonicity says that, if we know with some degree of certainty that the element in
question belongs to a set, then the degree of certainty that it belongs to a larger set
(containing the former set) cannot be smaller.
In the following we drop the monotonicity and consider a set X (which is assumed to be
finite here) and a basic assignment
m:V(X)^[0, 1] G.1)
such that
m@) = O, ]T m(A)=\ G.2)
???,,??)
where
f,„(X) = {AcX:m(A)>0} G.3)
is the set of all focal elements. Any set function m satisfying G.1) is called a belief
structure, which presents an approach to representing uncertainty and imprecision within
the available evidence.
Here m(A) is interpreted as the degree of evidence or belief that a specific element ?
of X belongs to the set A. Together with two quantities, called belief of A (collecting all

Measures of information
1547
focal elements of subsets of A)
Bel(A)=Y^m(B), AcX, G.4)
BcA
and plausibility of A (collecting all focal elements of sets which are not disjoint with A)
Pl(A)=l-Bel(A)= ]T m(B). А С X, G.5)
a basic assignment m forms the core of the Dempster-Shafer theory (Dempster A967) and
Shafer A976)); A denotes the complement of Л in X).
Note that m need not to be additive or monotonic. The basic assignment defines a
probability measure on V(X) but (in general) not on X since the focal sets neither need to
be disjoint nor form a partition of X.
Whenever all focal elements are singletons then we can define a probability measure
P:V(X)-> [0, l]via
P(A) = ]T/w ({*}), ACX G.6)
.veA
(since a probability measure is uniquely determined by the values on the finite set X). Thus
Bel(A) (and ?I (A)) are probability measures iff m({x}) = Bel({x}) (m({x}) = Pl({x}))
and m{A) = 0 for all subsets ? of X with \A\ > 1.
In the Dempster-Shafer theory also measures of uncertainties are investigated:
- Uncertainty about which element A e T,„(X) contains a specific element ? e X and
uncertainty about the location of ? within A.
- Measures which contain both types of uncertainties will be called measures of total
uncertainties.
In several papers total information measures were proposed (see the book of Klir and
Yuan A995) and Ramer A993)), for example,
T(X,m) = - ? m(A)logi ]T m(B)———\
D(in)
- ]P m(A)log(m(A) G.7)
AeF,„{X)
? (in)
or
T(X,m) = - ? w(A)log|A|- ]T w(A)logw(A). G.8)
AeF,„{X) ???,,,??)
Nun) Hon)

1548
W. Sander
In G.7) and G.8) the last term is the Shannon entropy applied to focal elements and is
a measure of non-specifity (since H{m) is not specific about ? e A for all sets A with
\A\>\).
In G.7), D(m) is a measure of conflict among given focal elements satisfying
?\???\ ,„„,
т(В)*——^аР1(А) G.9)
BeF„,(X)
and thus
]T m(A)Bel(A) ^ D(m) <: ]T m(A)Pl(A). G.10)
D(m) expresses the sum of individual conflicts of focal elements m(B) (each of which
is multiplied with the degree of subsethood \A П B\/\B\ of the set A in the set ?) with
respect to a particular focal element m{A).
Moreover, p(A) defined in G.9) is a probability measure so that T(X, m) in G.7) has
the form E.2) of a (one-dimensional) additive inset entropy
T(X,m) = - ]T m (A) log p(A)- ]T m(A)\ogm(A). G.11)
AeF„,(X) A?F,„(X)
In G.8) N(m) is a weighted mean of a Hartley-type information and was characterized
by Ramer A987) and Klir and Mariano A987). But the author is aware of only one
characterization result concerning total uncertainties in the theory of evidence. We present
the main result of Sander A997).
The main idea is to reformulate the property of subadditivity appropriately and then to
combine it with ideas presented in the last section.
If X is a finite set of cardinality ? we introduce
X„ = {x\,...,x„}, Xk = {x\,...,Xk} and
X„-k={xk+i,...,x„], G.12)
where к < ?, ? > 2.
Dehnition 7.1. In a belief structure on X„ the amount of total uncertainty is measured
by a real-valued function T„ depending upon the ordered tuple
??„ =(A|,A2,...,A2") G·13)
of the 2" subsets A,- of X„, 1 < /' ^ 2", and a basic assignment
???? = (m |, mi, ¦ ¦ ¦, mp) G.14)
where w, = w(A,).

Measures of information
1549
We now identify T(X„) := P(X„) \ {0} with
~P(X„) = (F(Xk) ? {0}) U ({0} ? P(X„_*)) U (P(Xk) ? T(X„-k)). G.15)
Notethat|7>(X„)|=2* - 1 +2"-k - 1 +Bk — 1)B" A — 1) = 2" — 1.
Definition 7.2. A sequence (Г„) is subadditive - or (T„) satisfies (A'l)-if for all ? e N,
и ^ 2 and for all 1 < к ^ ? - 1 the condition
~P(Xk) ? T>(X„-k) С ?,,?„ (??) implies
ii(^„mx») < 7*(А^,/я^) + Тп-к(АХп_к,тХп^) G.16)
where »??„ is a given basic assignment and mXk and т^_, are the usual marginal basic
assignments, that is, the projections of mXn on Xk and X„-k, respectively (cf. F.8))
In the same manner like (Al) is replaced by (A'l), the axioms (A2)-(A5) can be
modified by (A'2)-(A'5) (where in the symmetry axiom an additional assumption is
included) and we arrive at the following result of Sander A997).
THEOREM 7.3. In a belief structure on X„ the amount of total uncertainty satisfies the
properties of (A'l) subadditivity, (A'2) additivity, (A'3) expansibility, (A'4) symmetry and
(A'5) one-side boundedness if and only if there exist real constants a, b e E, a ^ 0, and a
?-additive function g:V(X„) -> К such that
T„(AXn,mXn)= -a ]P m(A)\ogm(A)
A?F„,(X„)
+ ? m(A)g(A) + b-log\Fm(X„)\. G.17)
???,„{?„)
Here the P-additivity (P stands for powerset) of g means that
g(AxB) = g(A) + g(B), AeP(Xk), BeP(X„-t). G-18)
for all 1 ? к -? ?- 1, ? ^ 2 (identifying A with ? ? {0} and ? with {0} ? ?, cf. G.15)).
Now we present some remarks and examples concerning Theorem 7.3.
(a) In G.17) the last term is nothing else but the "Hartley entropy" (compare
log\fm(X„)\ with H* (see A.10)). It represents the non-continuous part of the
total uncertainty. In different applications we can have different interpretations for
this term. For example, when determining the center of a certain pain the term
b ¦ log|:F,„(X„)| can be interpreted as the "intensity of the pain" which is an
additional effect besides the localization of the center of the pain.
(b) We don't need a continuity assumption for deriving G.17), the above rather weak
boundedness assumption is sufficient to prove the result.

1550
W. Sander
If for example, T„ has values in the nonnegative reals then the boundedness
assumption is satisfied. Thus in this case the result is true assuming only
(A'1MA'4).
(c) If in D) g{A) = log \A\ and if Tn is continuous at 0 (this means
Mm ? ( A] Al A* '" Al-"\ _ ? (A\ Al- A* '" Al"\
,,l™+ "\l-u и 0 ... 0 J' "{ 1 0 0 .·· 0 )
(? e ?)) then b = 0 and G.17) goes over into G.8).
Moreover, if in addition a = 0 then we get a characterization theorem for
J2A€yrm{X)m(A)log\A\ with a specified continuity assumption, which was not
known before (see the paper by Ramer A987)).
(d) If we assume that m{A) > 0 only on singletons then our sequence (?„) reduces to
a sequence (T„) dependent only upon X„ and a positive probability distribution Pn
on X„. Thus we arrive at the inset case, and in particular we obtain the result of
Aczel et al. A974) (with g = 0).
For further examples and results we refer to the papers of Dubois and Prade A987),
Ramer and Lander A987), Ramer A993), and the book of Klir and Yuan A995), where,
for example, possibility and necessity measures are discussed (when all focal elements in
the sums in G.10) are nested).
8. Measures of fuzziness
Let X be a finite set. A fuzzy set Л in X is a generalized subset of X which is characterized
by a membership function fA which associates to each ? e X a number of the "valuation
set" [0, 1], that is, /a(x) represents the grade of membership of jc in A. When ? is a set
whose membership function has the valuation set {0, 1} (with /a(jc) = 1 or /л(х) = 0
according to whether ? does or does not belong to A) then fA reduces to the well-known
characteristic function of a set Л с X. Since a fuzzy set is completely characterized by its
membership function /д we define:
Any map / from X into [0, 1 ] is called a fuzzy set. (8.1)
The fuzzy sets were first introduced by Zadeh A968) and since then fuzzy set theory
became an attractive new mathematical theory, see, for example, the books by Dubois
and Prade A980), Klir and Yuan A995) and Grabisch et al. A995).
In the following we first discuss "natural axioms" for so-called measures of fuzziness,
and then we report on attempts to combine probabilistic and fuzzy uncertainties.
If/,ge[0, \]x mdhe[0, if then 1 -/, fvg, f/\g e [0, 1]X and fxg e [0, \]XxY
are given by
A-/)(*)= 1-/(*),
(/v *)(·*) = тах{/(д0,*(д0}.

Measures of information
1551
(fAg)(x) = mm{f(x).g(x)}.
(fxg)(x,y)=f(x)g(y).
moreover
?(?)=??(?) (8-2)
xeX
is the power of /, which is a natural extension of the notion of cardinality for crisp sets.
Now we assign a nonnegative number to each fuzzy set / e [0, \]x which characterizes
the degree of fuzziness of /.
DEHNITION 8.1. If X = {л-1, ...,x„] is a finite set then every function ?: [0. l]x ->
Eo := [0, oo [ satisfying
M(f) = M,,(f(x\l...,Axn)). /e[0. l]x. (8.3)
for some mapping M„ : [0, 1]" -> R(> is called a measure of fuzziness or a fuzzy entropy.
This definition says that a fuzzy entropy can be regarded as an entropy in the sense that
it measures the uncertainty about the presence or absence of a certain property over the
set X. In case of a finite set X a measure of fuzziness M(f) is determined by the function
values of the fuzzy set /.
The first list of desiderata for measures of fuzziness was presented by DeLuca and
Termini A972):
(PI) Sharpness: ??(/) = 0 iff /(X) с @, 1). / e [0, 1]X.
(P2) Maximality: M{f) is maximum iff /(X) = \, / e [0. \]x.
(P3) Resolution: ?(/) ^ ?(/*) where /* is a "sharpened" version of /.
(P4) Symmetry: M(f) = M(\ - /), / e [0, \]x.
(P5) Valuation: M(f ? g) + M(f лg) = M(f) + M(g), f,ge [0, l]x.
Here /* is a "sharpened" version of / means that \f* - \\~^ \f - {\- The properties
(Pl)-PD) seem quite natural, (P5) states that the sum of the degrees of fuzziness doesn't
change if / and g are exchanged with the two operations of the lattice ([0, 1]?, ?, л),
/ ? g and / л g.
As examples, Deluca and Termini A972) introduced
?
ж/) = -?/(*.-) log/и-) (8·4>
i = l
and showed that
H\f):=H(f) + H(\-f) (8.5)
satisfies (P1)-(P5). There is an interesting connection between #'(/) and the Shannon-
type functional H(m) defined in G.7) the idea of which goes back to Forte's personal
communication in the paper by Ebanks A983):

1552
W. Sander
Let us interpret f(x) as the probability that ? e X possesses a certain property P,
and think of the process of deciding whether x\ ,x„ do or do not possess ? as an
experiment. If we assume that f(x) and f(y) are independent for ? ? у, х, у е X, then /
induces a basic assignment on ^(Х) by
m{A)=Y\f{x) Y\ {I-f(x)), ACX, (8.6)
x€A .\eX\A
and a calculation shows that H(m) = H'(f), that is (with the convention 0 log 0 = 0)
— 2_. rn(A)\ogm(A)
AeFm(X)
= - ]T f(x)logf(x) - ]TA - /U))log(l - f{x))
.xeX .\eX
= E5(/w) (8·7)
xeX
where
S(t) = H2(t, 1 -i)=-ilogi-(l -i)log(l -0, re [0,1], (8.8)
is the so-called Shannon function.
Concerning characterizations for fuzzy entropies, it turns out that (P5) plays a key role
because of the following result (Sander A989)):
If X = {jc, ,..., x,,} and ?: [0, l]x -> E() is a fuzzy entropy then ? satisfies (P5) iff
there exist generating mappings <pj: [0, 1] -> Eo A ^ /r ^ n) such that
?
M(/) =]>>(/(*,)). (8-9)
? = ?
If M„ in (8.3) is a symmetric function in its variables then all functions ??, ? i ^ и,
coincide (see Ebanks A983)) and we get with ?, =?
?' = ?
In a recent paper, Ebanks A998) replaced (P5) by
(P6) Localization: There exists a continuous map ?: [0, 1 ]2 -> R such that
??(/) = ??(/') + p[f(x,). f\xi)] (8·1 D
whenever / and /' differ at only one point x, e X.

Measures of information
1553
This states that the difference in the degrees of fuzziness of / and of /' depends only on
their values at the single point where they differ. The condition in (8.11) is a branching-type
property for "fuzzy uncertainties" (like in (P5) also "fuzziness" is "exchanged").
It is interesting that (P5') is again equivalent to the existence of a generating (continuous)
function ?:[0, 1] -> ? such that M(f) has the sum-form property (8.10) (see Ebanks
A998)).
Starting from the representation (8.9) or (8.10) we have numerous proposals for
measures of fuzziness since only the functions 0, and ? must be specified and characterized.
For a survey see the book of Dubois and Prade A980) and the survey article of Pal and
Bezdek A994), which contains several lists of interesting measures of fuzziness.
We here present a recent result of B. Ebanks which is based on the following observation.
The functional H(f), given in (8.4), satisfies
H(fxg) = P(g)H(f) + P(f)H(g) (8.12)
(and also (PI) and (P5), but not (P2)-(P4)), which is an weighted additivity condition
where the weights correspond to the "size" of / and g.
Thus he assumes more generally that the amount of fuzziness of / ? g is a function of
M(f), M(g) and of the "sizes" P(f), P(g):
(P7) Direct Product Composition Law: There exists a function G:M+xR+^R+ such
that
M(f xg) = G(P(f), P(g), M{f), M(g)). (8.13)
Moreover, a natural symmetry condition is supposed:
(P8) M(fxg) = M(gxf). (8.14)
The property (P7) - together with (P8) - seems to be a more natural assumption than (8.12)
and the two generalizations of (8.6), considered in Ebanks A983) and Sander A989).
The following result is the announced (slightly modified) result of Ebanks A998) which
generalizes a result of Sander A989).
THEOREM 8.2. (I) A measure of fuzziness M{f) satisfies (PI) and (P6)-(P8) ifand only
if ? has either the form
?
M{f)=-aYjf(x,)\ogf(x,) (8.15)
i = l
for some a > 0, or else there is some a e @, 1) U A. oo) such that
?
M{f) = cYj(f(xir-f(xl)) (8.16)
1 = 1
with some constant с (с < 0 if a > 1, с > 0 //0 < a < 1).

1554
W. Sander
B) The only measure offuzziness satisfying (P1)-(P4) and (P6)-(P8) is given by
?
M(f) = bYjf{xl)(\-f{xl)) (8.17)
i=\
for some constant b > 0.
For some further results we refer to two papers by Ebanks A983, 1998).
Let us now consider an experiment with outcomes E\,..., E„ (where ?, are elements
of a ring В of sets of a universal set ?) and corresponding probabilities p\,..., p„. If
in addition a fuzzy set /: X = {E\ ,...,?„} -> [0, 1 ] is given then we have two kinds of
uncertainty:
A) The uncertainty of random nature concerns the particular event E, which will occur.
The average uncertainty will be measured by Shannon's entropy H„.
B) The uncertainty of "fuzzy nature" is related to the interpretation of ?, as 1 or 0. This
amount of uncertainty can be measured by
S(f(Xi))
(see (8.8); note that S:[0, 1] -> [0, 1] satisfies S@) = S(\) = 0, S is concave on
[0, 1 ] having a unique maximum at \ with S( \) = 1).
Thus the statistical average information in taking a decision A or 0) on the possible
outcomes of the probabilistic experiment is
II
?(?,?) = ???5(???)), PeT„. (8.18)
?'=?
Therefore a total entropy Hlol can be defined as
Hm(f,P)=H„(P) + L(f, P)
? ?
= -^p,logp,+^AEo/)(?,). (8.19)
1=1 ?=?
This seems to be an interesting total entropy for several reasons:
A) If there is now fuzziness, that is /(X) С {0, 1} then
Hm(f,P) = H„(P). (8.20)
B) If there is no uncertainty, that is pj — 1 and p, = 0 if ?'? j then
Hlol(f,P) = (Sof)(Ej). (8.21)

Measures of information
1555
C) IfP = (I ... I)then
1
Hl0l(f, ?) = Щ{Р) + - Y(So /)(?,). (8.22)
П
1 = 1
D) If we don't know the probability distribution ? and if ? = 1 and E\ = ? then we
get
H101(f,P)=(Sof)(Q). (8.23)
E) There is a connection with inset entropies which indicates a characterization of #I01.
The difference between the two uncertainties in (8.19) and (8.23) is the amount
Pi ··· Pn
/(?,) ··· /(?„)
= -I> log ? +? A (^ о /)(?,) - (So /)( [J ?,- ) <8·24)
of information gained from the knowledge of the probability distribution, and it has exactly
the form of a recursive inset entropy.
Because of Theorem 4.1 it seems naturally to introduce a one-parametric family of total
entropies (Hl) where H^{ = ??? if a = 1 and
+ ? № о /)(?,)- (So/) [J ?,) (8.25)
i=l \?=?
if ? ? 1. It is an open problem to chatacterize H"n, a e R.
For further total entropies combining probabilistic and fuzzy uncertainties and
applications of these entropies we refer to Pal and Bezdek A994).
9. Weighted entropies
Let us consider the following example which comes from an unpublished manuscript of
B. Forte.
A real estate agent receives (commission and bonus) $ te, if he sells a house at the date ?,.
A measure of the information content of the statement

1556
W. Sander
"?, = the house was sold on the date T, and the probability for that to occur is pf
should take into account the gain w, and the probability p,. A good candidate is
I(E()=-cwilogpi, KD«. (9.1)
Thus the measure of expected information (see A-25)) leads to the so-called weighted
entropy
?
I„{w\,...,w„;p\,..., p„) = - ]P WiPi log p,. (9.2)
/ = l
Here w; e [0, oo[ denotes the subjective importance of different events of a ring В of
sets, that is w, = w{E;) for some function w:B -> [0,oo[. This weighted entropy was
introduced and characterized by Belis and Guiasu A968).
The idea was that the occurrence of an event ?, is connected with two types of
uncertainty.
- the quantitative type related to the corresponding probability p, of an event ?,, and
- the qualitative type relative to the utility w,- = »(?,) which the experimenter has in
mind.
Special cases of (9.1) are for example (if ? e ?„):
?
In(w,...,w;p\,...,p,,)= -w]P PiTog pi, (9.3)
i = l
/,,(^,...,^) = -X>logp,, (9.4)
\P\ PnJ f~^
/-i-r?L----rA-) = E"?· (95)
V log ?, logp,,/ f^
In the following we denote и-tuples (w\,..., w„) of RJj or R'| by V and define:
A weighted entropy (/„) is any sequence of functions where
/„:Rgxr„-»-Ro. (9.6)
We now present some axioms which are natural extensions of axioms characterizing the
entropies of degree a, so that in particular we get the result of Belis and Guiasu A968):
(Bl) Continuity. /"(»], w2; p, 1 — p) is continuous on [0, 1] for all им, хиг е Ко·
(B2) Symmetry. We have for all permutations on {1, ..., n]
/"(»!,..., w„; p\, ...,pn)= I" (w„ \,..., ???„; ??\, ...,???).

Measures of information 1557
(B3) Hw'=(pa]w[ +p>2)/(pi +p2)then
" \P\ рг ръ ?,,
_ ,a ( w' W2 Ws wA,„a,a( W] W2 \ /q 7ч
_/"U+W P2 P3 ?») ??-\7$?- PW+P*)
for all WelJ.Pe T„ with p\ + p2 > 0.
(B4) Uniformity. If ? = (^, ..., 1) then
/,?(? " "i"^L,)W'+"' + W" (9-8)
for some function L" : N \ {0} -> R+ and for all WeKJ, that is /" is proportional
to the mean value of the weights.
(B5) Normality anddecisivity. LaB) = 1 and I%(w\, w2; 1,0) = 0.
In (B2) the definition of w is a natural one since, e.g., for a = 1 this means that
W(E,UE2)=P|W(?|) + P2W(?2) (9.9)
P\ + Pi
is satisfied by utilities in decision theory, or in game theory.
The following result for a = 1 is due to Belis and Guiasu A968) whereas the case
??1+,?^1, can be found in Kriimpelmann A991) (I was not aware of another
reference):
THEOREM 9.1. LetaeR+.A weighted entropy (/,?) with /,?: R? ? T„ satisfies (Bl)-
(B5) iff
П
Iaa(W,P) = -2x-aYdwip^\0gPi. (9.10)
i = l
If a = 1 then we obtain (9.2) (without using (B5)).
In a similar way the weighted entropies of degree (?, ?) сап be characterized. We refer
to the paper by Dial and Taneja A981).
In the characterization of weighted entropies using weighted additivity conditions we
get similar to the considerations in Section 3 the functional equation
Ih„(V *W,P*Q,) = /,(V, P)I„,(W, Q)+g,„(W, Q)I,(V, P), (9.11)
? e ?/, Q e T,„, V e M!+, ff e R'J, where V * W = (v\ w\,..., v,wm) e R1"' and
g,„ : R™ ? T,„ -»¦ R0, /,: R'+ ? ?, -»· R0.

1558
W. Sander
If /„, // and gm have the sum-form property with generating functions I, f,g:R+ ?
[0, 1] -> R, respectively, then (9.11) leads to the functional equation
/ m
^^{HviWj^iqj)- g(wj,qj)I(Vi,pi)- f(vl,pi)I(wj,qj)) = 0. (9.12)
i=l 7 = 1
In Sander A985) it was shown that all measurable solutions /, /, g of Equation (9.12)
(assumed for / = m = 1 or for some fixed pair (l,m), I > 3, m > 3) if ? e [0, 1]',
? e [0, 1]'", V e R'+, W e R'^ are given by
I!
^pfuflogpfuf. (9.13)
? = ?
II
fc^^ufsinOog^uf). (9.15)
i = l
where k, a, b, c, d are constants (? ? 0). The same result was obtained in Dial and Taneja
A981) if / = gin (9.12).
There is another interesting result (proved in the theory of gambling) because it leads
again to recursive inset entropies (see Meginnis A976)).
If W is a utility function defined on a ring В of sets (that is, W : В -> К) satisfying a
certain compound gamble rule and if /„ has the form
II
/,I(W,P) = ^/(W(?,),A) (916)
i = l
for some regular function / then /„ is given by
,(W p)_\ ?- = , PiW(E,) +a LIU, ?, log Pi, « = 1.
where ? is a constant.
Here the first term for a — 1 is the expected gain whereas J.R. Meginnis interprets the
second sum as the "joy in gambling".
Weighted entropies have many applications, for instance, Gil et al. A989) introduced
and characterized the following information measures
? - ?!'-1 Pi log(w,7?(W)), a = 1,
IH(W,P)=\ , -'" , (9.18)
(where E(W) = ?? = ? wiPi) an<J presented applications to statistics.

Measures of information
1559
Different weighted entropies were proposed by Zadeh A968)
II
M{f,P) = -Yjf{El)Pl\ogPl (9.19)
1 = 1
where /: {E\,..., E„} -> [0, 1 ] or by expressions of the form
?
/,,(W,/) = ]>>,#(/(?,)) (9.20)
i=l
for ffeMJ and some function g : [0. 1] -> E.
These last two measures are not yet characterized but they were applied for big classes of
functions (see, e.g., Pal and Dutta Majunder A986)). To be convinced that these measures
are "good" measures some efforts should be made to characterize them.
We close this section by considering information measures which are defined on
? " 1
?„ = \(??,...,?„): ?#<1. Pi>0\. (9.21)
I 1 = 1 J
that is, probability distributions are allowed to be incomplete.
The question arises (we formulate here only the one-dimensional case) of characterizing
real-valued information measures of the the form
/,,WP) = W ? u-j *(/,(/>,¦))) (9·22)
where ?: ? -> ? is a strictly monotonic function, and where (/„) is polynomially additive:
/„„(V * W, ? * Q) = G{Ii(V, P), I,„(W, Q)) (9.23)
(V e E'+, W e R™, PeA,,Qe ?,„\ (9.23) is valid for all /, m e N). Here G:RxR^R
is a polynomial in two variables.
The property (9.21) of an information measure (/„)is called ^-average, and is well-
known (see, for example, the book by Aczel and Daroczy A975)). The condition on G
implies (assuming a fullness condition of the range of (/„)) that (9.23) has only the form
I,m(V *W,P*Q)= I,(V, P) + I„,(W, Q) + ?/,( V, P)Im(W, Q) (9.24)
for some ? e ? (see Behara and Nath A973)).
In this generality the problem is unsolved, in particular if no regularity assumption
is made. To indicate that this problem may lead to interesting information measures we
consider a special case, where the weights are given by
PS
ил = _Г ., <5eE, (9.25)
Lj = \Pj
so that /„ : ?„ -> ?.

1560
W. Sander
The following result is based on a result of Veres A982) (see KrumpelmannA991)).
Theorem 9.2. Let S e R. An information measure (/„) with I„ : A, -> R, ? e N, й
ш ?-average with weights w, = ??/?/ = ? P^ ш^ /'? polynomially additive satisfying the
normalization condition I\ (\) = 1 /y
/„(/>) =
(??=,?-'«(-??8?))/??=,???.
^??((??=,?-2"?(-|08?'-,)/??=??-).
:(((??=,^2"'.(-?8?))/??=|^)
54lOg(l+A)/y
у eM, ? = -1,
у = 0, ? = 0,
У^О, ? = 0,
у = 0, ?^?,
?>-1,
1), у^О, ?^?,
?> -1,
(9.26)
w/геге у e R, ? e [— 1, ??[ a/id h is an arbitrary additive function satisfying h( 1) = 1.
We denote /„ in the following by U„'k' saying that it is an entropy of order 1 - у of
type(l-log(l+A),a).
Concerning continuous solutions of Theorem 9.2 we get that 1/%'*' is continuous if h
is continuous. Thus (because of h(\)= 1) we get h{x) = x. With the bijections
]-l,oo[-
we get the following result
у -> 1 — ?,
(9.27)
THEOREM 9.3. Let S e R. A nonconstant information measure (/„) with I„ ¦ ?,, -> R,
? e N, where I\ is continuous is an ?-average with weights u>, = pf /?/ = ? Pj an^ 's
polynomially additive satisfying the normalization condition I\ (^) = 1 iff
I„(P) =
?(-?7=????<*?)/?7=???.
^????-?/^-'?/??-???).
B(|-^(-?"=?^?°8?.·)/?:=?/'?_?))
21 —/? ?
? =1,0=1,
??? 1,0=1,
? г'-/1-!
1 (((??=???+?-,)/??=???),,_/',/(,",,,-?). «* 1.0*1.
(9.28)
w/геге ?, /3 e R. We denote I„ by ?" , the entropy of order a of type {?, <5).
We remark that H„' is 3-parametric generalization of H„ since
lira„^i Д, ' = H„ (? constant),
lim/j^ ? ?"?-& = H"u& (a constant).

Measures of information
1561
Thus this result contains many results as special cases: Renyi entropies and the results
characterized by Sharma and Mittal A975), Van der Pyl A976) and Arimoto A971) (put
5= 1 and 1/a = 2-/3).
10. Summary
In this paper we were mainly interested to give a survey on "total entropies", which are
useful and may have applications. A useful measure here means that it comes from a real
problem, or it has natural and intuitive properties which lead to a characterization (see also
AczelA984)).
Of course, there are other "justifications" like the maximum entropy principle. For
details we refer to the books by Guiasu A977) and Kapur A989).
Thus we have gathered some interesting results which perhaps can be a basis for further
research, and mainly in Sections 7-9 there is room for additional work which seems worth
to be done.
Only sometimes we have given applications, but there are a lot of hints and references
for those who are interested. The emphasis was on the mathematical background without
which there are no applications.
We close with two remarks.
A) We have not reported on so-called continuous analogous for entropies. Let us only
mention that for instance the "continuous" analogue of the Shannon entropy #„,
Hp = - j p(x)\ogp(x)dx A0.1)
Ja
for some continuous probability density function ? is not the limit of the Shannon
entropy for discrete distributions
?
-5>(дг,-Iоер(дг,-). (Ю.2)
i = l
but the approximating sums of the integral A0.1) are inset entropies
¦]Tp(&)logp(S/)(.x/ -*,--i)
1 = 1
?
^F(xi)-F(xi-i)l F(Xi)-F(Xi-i)
= -> log (x,--Xj-\)
^ Xi-Xi-i X,~Xi-\
1 = 1
? ?
= - ? ? log ?, + ? ^ l0SKEi), (Ю.З)
i=l i=l
where ?,¦ e ]jc,- - jc,_i ], F is the probability distribution function (Ff = p, F(a) = 0,
F(b)= 1) with probabilities p, = F(Xj) - F(x,-_i) A ^ /' ^n), sets ?,¦ = ]*,¦_?,*,-]
with union [a, b] and Lebesgue measure /(?,¦) = Xj — x\-\.

1562
W. Sander
See Aczel A978) for details and further applications, and the books by Guiasu
A977) and Kullbacket al. A980) for "entropy as variation of information".
For characterization theorems in the "continuous" case we refer to Forte and
Sastri A975), Forte and Hughes A989) and Ebanks A998).
B) Probably this field of research will never come to an end. Always new and interesting
things are coming up. Let us mention here an idea of combining probabilistic
concepts and concepts from cooperative game theory (which in some sense is
"an application" of fuzzy measures). If m is a fuzzy measure on a finite set
N = {1,..., n) (or: the characteristic function of a cooperative game on N) then
the Shapley value of m is given by
ф(т)= (</>|(m), ...,<Mm)), Ф-Ю, l]P(yV)-> R",
4>i-.[0, lflN)^R, A0.4)
where
*i \ ТГ (\?\-\?\-1)\\?\\, ,
4>i(m)= у -— (m(TUi)-m(T)). A0.5)
t—i \N\\
TcN\[i) '' '
But a calculation shows that the Shapley value is a probability measure on N and
we can ask for an interpretation and characterization of
?
Н(Ф(т)) = -]Г </>,-(/и) log <fc(/w). A0·6)
i = l
For more details and similar problems we refer to Marichal B002) and Kojadinovic et al.
B002).
References
Aczel, J. A978). A mixed theory of information. VI: An example at last: A proper discrete analogue of the
continuous Shannon measure of information (and its characterization). Univ. Beograd. Publ. Elektrotehn.
Fak. Ser. Mat. Fiz. 602-633. 65-72.
Aczel, J. A980). A mixed theory of information. V How to keep the (inset) expert honest. J. Math. Anal. Appl.
75. 447^453.
Aczel. J. A984). Some recent results on information measures, a new generalization and some 'real life'
interpretations of 'old' and new measures. Functional Equations: History. Applications and Theory. J. Aczel. ed.,
Reidel. Dordrecht.
Aczel, J. and Daroczy, Z. A975). On Measures of Information and Their Characterizations. Academic Press,
New York.
Aczel, J. and Daroczy, Z. A978). A mixed theory of information. I. Symmetric, recursive and measurable entropies
of randomized systems of events, RAIRO Informat. Theor. 12. 149-155.
Aczel, J., Forte. B. and Ng, C.T. A974). Why the Shannon and Hartley entropies are 'natural'. Adv. Appl. Probab.
6. 131-146.
Aczel, J. and Ng. C.T. A983). Determination of all semisymmetric recursive information measures of
multiplicative type on ? positive discrete probability distributions. Linear Algebra Appl. 52/53. 1-30.
Arimoto, S. A971). Information theoretical considerations on estimation problems. Inform. Control 19. 181-194.
Behara, M. A990), Additive and Nonadditive Measures of Entropy. Wiley. New York.

Measures of information
1563
Behara. M. and Nath. P. A973), Additive and non-additive entropies of finite measurable partitions. Probability
and Information Theory. II. Lecture Notes in Math.. Vol. 296. Springer. Berlin. 102-138.
Beli§, M. and Guia§u. S. A968). A quantitative-qualitative measure of information in cybernetic systems. IEEE
Trans. Inf. Theory IT-14. 593-594.
Daroczy. Z. A970), Generalized information functions. Inform. Control 16. 36-51.
DeLuca. A. and Termini, S. A972), A definition of nonprobabilistic entropy in the setting of fiizzy sets theory.
Inform. Control 20, 301-312.
Dempster, A.P. A967), Upper and lower probabilities induced bx multivalued mappings, Ann. Math. Statist. 38.
325-339.
Dial. G. and Taneja, I.J. A981), On weighted entropy of type (?. ?) and its generalizations. Appl. Math. 26.
418^425.
Dubois. D. and Prade, H. A980), Fuzzy Sets and Systems, Academic Press. New York.
Dubois. D. and Prade, H. A987), Properties of measures of information in evidence and possibility theories.
Fuzzy Sets and Systems 24 B). 161-182.
Ebanks, B. A983), On measures offuzziness and their representations, J. Math. Anal. Appl. 94, 24-.37.
Ebanks, B. A984). Measures of inset information on open domain. 111. Weakly regular, symmetric, ?-recursive
entropies, C. R. Math. Rep. Acad. Sci. Canada 6. 159-164.
Ebanks. B. A986). Measures of inset information on open domains, 11. Additive inset entropies with measurable
sum property. Probab. Theory Related Fields 73, 517-528.
Ebanks, B. A989). Generalized characteristic equation of branching information measures. Aequationes Math.
37, 162-178.
Ebanks, B. A990), Branching inset entropies on open domains, Aequationes Math. 39. 100-11.3.
Ebanks. B. A994). Additive k-dimensional inset entropies with measurable sum property. J. Math. Anal. Appl.
187, 952-960.
Ebanks. B. A998), Localizable composable measures offuzziness, Publicationes Math. Debrecen 52, 343-355.
Ebanks, В., Kannappan. PI. and Ng. C.T. A987). Generalized fundamental equation of information of
multiplicative type. Aequationes Math. 32. 19-31.
Ebanks, В.. Kannappan, PL and Ng. C.T. A988), Recursive inset entropies of multiplicative type on open domains,
Aequationes Math. 36. 268-293.
Ebanks, В., Kannappan. PL, Sahoo. P. and Sander. W. A997). Characterizations of sum form information
measures on open domains. Aequationes Math. 54. 1-30.
Ebanks, В.. Sahoo, P. and Sander. W. A998). Characterizations of Information Measures, World Scientific.
Singapore.
Forte. B. A977). Subadditive entropies for a random variable. Boll. Un. Mat. Ital. В E) 14. 118-133.
Forte, B. and Gupta, R. A985). Additive and subadditive entropies for discrete n-dimensional random vectors.
Aequationes Math. 28. 269-287.
Forte, B. and Hughes, W. A989). A representation theorem for entropies with the branching property.
Aequationes Math. 37, 219-232.
Forte, B. and Sastri. C.C.C.A. A975). Is something missing in the Boltzmann entropy?. J. Math. Phys. 16, 1453-
1456.
Gil. M.A.. Perez. R. and Gil. P. A989). A family of measures of uncertainty involving utilities: definition.
properties, applications and statistical inferences. Metrika 36. 129-147.
Goodman. I.R. and Nguyen, H.T. A985), Uncertainty Models for Knowledge-Based Systems. Elsevier.
Amsterdam.
Grabisch. M.. Nguyen, H.T. and Walker. E.A. A995). Fundamentals of Uncertainty Calculi with Applications to
Fuzzy Inference. Kluwer. Dordrecht.
Gross. H. A993). Inset-lnformationsmasse. Ph.D. thesis, TU Braunschweig. Germany.
Gross. H. and Sander, W. A991). Determination of all additive inset information measures with measurable sum
property, Tech Rep., Technical University Braunschweig 5, 1-28.
Guia§u, S. A977), Information Theory with Applications. McGraw-Hill, New York.
Hartley, R.V.L. A928), Transmission of information. Bell System Tech. J. 7, 535-563.
Ingarden. R.S. and Urbanik. K. A962). Information without probability, Colloq. Math. 9. 1.31-150.
Jarai. A. and Sander, W. A997), A regularity theorem in information theory, Publ. Math. Debrecen 47, 1-13.

1564
W. Sander
Jessen, В., Karpf, J. and Thorup, A. A968). Some functional equations in groups and rings. Math. Scand. 22,
257-265.
Kampede Feriet, J. A963), Theorie de Г Information. Principe du Maximum de VEntropie et ses Applications a la
Statistique et a la Mecanique. Publications du Laboratoire de Calcul de la Faculte des Sciences de l'Universite
de Lille, Lille.
Катаре de Feriet, J. and Forte, B. A967), Information et probabilite 1, 11. Ill, C. R. Acad. Sci. Paris Ser. A 265,
110-114, 142-146,350-353.
Kannappan, PI. and Sahoo, P. A989), On a generalization of the Shannon functional inequality, J. Math. Anal.
Appl. 140, 341-350.
Kapur, J.N. A989), Maximum-Entropy Models in Science and Engineering, Wiley Eastern. New Delhi.
Kerridge, D.F. A961), Inaccuracy and inference, J. R. Stat. Soc. Ser. В 23, 184-194.
Klir, G.J. and Mariano, M. A987), On the uniqueness of possibilistic measure of uncertainty and information.
Fuzzy Sets and Systems 24, 197-219.
Klir, G.J. and Yuan. B. A995), Fuzzy Sets and Fuzzy Logic. Theory and Applications, Prentice Hall, Upper Saddle
River, NJ.
Kojadinovic, I.. Marichal. J.-L. and Roubens. M. B002). An axiomatic approach to the definition of the entropy
of a discrete Choquet capacity, Manuscript.
Kriimpelmann, M. A991), lnformationsmasse und Distanzmasse. Master thesis, Technical University,
Braunschweig, Germany.
Kullback, J., Keegel, J.С and Kullback, J.H. A980). Topics in Statistical Information Theory. Lecture Notes in
Statistics, Vol. 42. Springer. Berlin.
Marichal, J.-L. B002), Entropy of discrete Choquet capacities, European J. Oper. Res. 137, 612-624.
Martin, N.F.G. and England, J.W. A981), Mathematical Theory of Entropy, Addison-Wesley, Reading, MA.
Meginnis. J.R. A976), A new class of symmetric utility rules for gambles, subjective marginal probability
functions, and a generalized Bayes rule. Bus. Econ. Stat. Sec. Proc. Amer, Stat. Assoc.. 471^476.
Ng, C.T. A974), Representations for measures of information with the branching property. Inform. Control 25,
45-56.
Ng, C.T. A979), Measures of information with the branching property over a graph and their representations.
Inform. Control 41, 214-231.
Pal, N.R. and Bezdek, J.C. A994). Measuring fuzzy uncertainty. IEEE Trans. Fuzzy Systems 2, 107-118.
Pal, S.K. and Dutta Majumder, D. A986). Fuzzy Mathematical Approach to Pattern Recognition. Wiley. New
York.
Pap, E. B002), Pseudo-additive measures and their applications. Handbook of Measure Theory. E. Pap, ed..
Elsevier, Amsterdam, 1403-1468.
Ramer, A. A987), Uniqueness of Information Measure in the Theory of Evidence. Fuzzy Sets and Systems 24.
183-196.
Ramer, A. A993). Total uncertainty of possibility assignments. Proc. 2nd IEEE Int. Conf. on Fuzzy Systems. San
Francisco, 1220-1224.
Ramer, A. and Lander, L. A987). Classification of possibilistic uncertainty and information functions. Fuzzy Sets
and Systems 24 B), 221-230.
Sander, W. A985), On a sum form functional equation, Aequationes Math. 28, 321-326.
Sander, W. A987), The fundamental equation of information and its generalizations, Aequationes Math. 33,
150-182.
Sander, W. A989), On measures offuzziness. Fuzzy Sets and Systems 29. 49-55.
Sander, W. A997), A characterization theorem in the Dempster-Shafer theory. Abstracts of Linz-Seminar:
Enriched Lattice Structures for Many-Valued and Fuzzy logics, Linz, 47-50.
Schweizer, B. and Sklar, A. A983). Probabilistic Metric Spaces. North-Holland, Amsterdam.
Shafer, G. A976), Л Mathematical Theory of Evidence, Princeton University Press, Princeton.
Shannon. C.E. A948), A mathematical theory of communication. Bell System Tech. J. 27. 379^423, 623-656.
Sharma, B.D. and Mittal, D.P. A975). New non-additive measures for discrete probability distributions, J. Math.
Sci. 10, 28-40.
Sharma, B.D. and Taneja, I.J. A975), Entropy of type (?, ?) and other generalized measures in information
theory, Metrika 22, 205-215.

Measures of information
1565
Taneja, I.J. A989), On generalized information measures and their applications. Adv. Electron. Electron Phys.
76,327^413.
Theil, H. A967), Economics and Information Theory, North-Holland. Amsterdam/Rand McNally. Chicago.
Van der Pyl, T. A976). Axiomatique de /'information d'ordre a et de type ?. С. R. Acad. Sci. Paris Ser. A 282,
1031-1033.
Veres, S. A982), On afunctional equation in connection with information theory, Aequationes Math. 24, 179-
186.
Zadeh, L.A. A968), Probability measures of fuzzy events, J. Math. Anal. Appl. 23, 421—427.

Author Index
Roman numbers refer to pages on which the
refer to reference pages.
Aaboe, A. 25
Aarnes, J.F. 842, 866
Abraham, P. 174
Abramov, L.M. 1227, 1229, 1231
Aczel, J. 972, 1007, 1405, 1407, 1408, 1465. 1525,
1526, 1528, 1535, 1539, 1540, 1544, 1545,
1550, 1559, 1561, 1562,7562
Adams, E.W. 1486, 1494, 1518
Adams, G. 1332, 1366, 1378, 1513, 1521
Adams, J.F. 90, 120
Adams, R.A. 1243, 1244, 1259
Adamski, W. 506, 526, 783, 783
Adian, S.I. 96, 120
Adler, R.J. 1044, 1051
Adler, R.L. 1225,1231
Agbeko, K. 1510, 1518
Ajupov, S.A. 741
Akcouglu, M.A. 153,774
Akian, M. 1432, 1433, 1465,1466
Albeverio, S. 1304, 1319, 1327
Alchourron, C.E.P 1505, 1518
Aleksjuk, V.N. 174
Alexiewicz, A. 610
Alfsen, E.M. 227, 231, 245, 438, 444, 512, 526
Aliprantis, CD. 714, 716, 741, 796, 797, 805, 822
Almgren, F.J., Jr. 1028, 1032, 1034, 1035
Alo, R. 660, 666
Ambrosio, L. 1034, 1035
Amemiya, I. 814, 822
Ames, W.F. 1440, 1466, 1467
an der Heiden, U. 507, 512-514, 518, 526
Anantharaman, R. 439, 444
Andersen, E.S. 1319,1327
Andersen, K. 25
Anderson, R.M. 1304, 1305, 1310, 1313, 1314,
1316, 1317, 1319, 1323, 1327
Ando,T. 127.174
Andrews, K.T. 543, 559, 560, 580
Anger, B. 512, 520, 526, 526
ir (or his/her work) is mentioned. Italic numbers
Aniszczyk, B. 677, 700, 773, 783
Antosik, P. 128, 135, 146, 149, 150, 157, 162, 170,
174, 175
Anzai. H. 739, 741, 1221,1231
Appell, J. 807, 823
Arbeiter, M. 1050, 1051
Arias de Reyna, J. 440, 444, 741, 741
Arimoto, S. 1561.1562
Arkeryd. L. 1327
Armstrong, Т.Е. 741, 742
Arnold, K. 1102, 1127
Amoux, P. 1227, 1228. 1232
Arrow, K. 1517, 1518
Artstein, Z. 628, 639, 643, 646, 647, 650, 651, 653,
654. 656, 662-666, 666
Arveson, W. 776, 783
Asai, K. 1331, 1378
Ascherl, A. 724, 742
Ash, R.B. 517, 519, 520, 526
Asplund, E. 510, 526
Assani, I. 1214, 1232
Attouch, H. 655, 664. 665, 667
Aubin, J.R 619, 666. 667
Aumann, G. 508, 511 -513, 524, 525, 526
Aumann, R.J. 619, 636-638, 662, 667, 949, 1007
Aversa. V. 238, 245, 699, 700
Azencott, R. 1466
Babiker, A.G. 1152-1154, 1170-1172, 1175, 1177,
1178
Baccelli, F. 1405, 1430-1432, 1436, 1466
Bachelier. L. 342
Bagchi, S. 660, 667
Balder, E. 638, 657, 662, 663, 667
Ban, A.I. 905, 906
Banach, S. 85, 90,92, 117, 120, 224,245, 436,444,
783, 1093, 1096, 1112, 1121, 1127, 1263, 1290
Bandt. С 436, 444, 1112, 1121, /127
Baraki, G. 1121. 1127
1567

1568
Author Index
Barbati, A. 629, 633, 634, 637, 667
Barbieri, G. 914, 935, 940, 944, 949, 1006, 1007,
1007
Barcenas, D. 634, 667
Baron, M.E. 25
Bartle, R.G. 81, 398, 439, 444, 564, 580, 595, 610,
1083, 1090
Basile, A. 261, 262, 292, 741, 742, 944
Bator, E.M. 558, 580, 581
Batt, J. 514, 520,526
Bauer, H. 68, 81, 956, 961, 963, 1007
Bayod, J.M. 1304, 1328
Beck. A. 576, 581
Becker, H. 118, 120, 1110, 1115-1117, 1121,
1123,7/27
Bedford, T. 1044,705/
Beer, G. 629, 633, 634, 637, 655, 657, 667
Behara, M. 1525, 1528, 1559, 1562, 1563
Beli§, M. 1556, 1557, 1563
Bellman, R.E. 1431, 1435, 1466
Bellow, A. 1133, 1170, 1171, 1178
Belluce, L.P. 956, 958, 1008
Beltrametti, E.G. 866, 866
Ben Amor, N. 1502, 1518
Benes, V.E. 638. 667, 776, 783
Benferhat, S. 1481, 1485. 1486, 1491, 1493, 1495,
1501, 1502, 1510, 1518
Bennett, M.K. 171, 176, 834, 867
Benvenuti, P. 288, 292, 1332, 1333, 1343, 1347-
1349, 1351, 1375, 1377, 1407, 1466. 1518
Berberian, S.K. 175, 520, 527, 1112, 1127
Berg, С 520, 527
Berge, С. 1385, 1397, 1400
Bergelson, V. 1202, 1205, 1211, 1232
Bergh, J. 813, 823
Bemau. J. 918, 944
Bemik, V.I. 1044, 1051
Berti, P. 263, 292
Besicovitch, A.S. 228, 233, 242, 245, 680, 700,
1017, 1020, 1022, 1024, 1035, 1044, 1051
Bessaga, С 438, 440, 444, 553, 581
Bezdek, J.C. 1525, 1553, 1555, 1564
Bharucha-Reid, A.T. 1177, 1183
Bhaskara Rao, K.P.S. 81, 175, 261, 262, 292, 730,
732, 742, 769, 779, 780, 783, 786, 940, 944,
1178
Bhaskara Rao, M. 81,175, 730, 732, 742,940, 944,
1178
Bianconi, R. 1306, 1327
Bichteler, K. 508, 509, 511-513, 522-525, 527,
1140, 1154, 1178, 1179
Bierlein, D. 724, 742
Bigard, A. 879, 906
Billingsley, P. 313, 337, 342, 647, 667, 767, 769,
783
Bilyeu, R.G. 581
Bing, R.H. 1290
Birkhoff, G.D. 19, 25, 175, 269, 292, 534, 581,
797, 823, 830, 836, 866, 871, 906, 956. 1008,
1193, 1232
Bishop, E. 437, 444
Bismut, J.M. 639, 667
Bjomsson, O.J. 771, 783
Blackwell, D. 772, 779, 783, 783, 784, 992, 1008
Bliedtner, J. 229, 245, 245,1179, 1319, 1320,1328
Blum. JR. 1205, 1232
Blumberg. A. 679, 700
Blume, F. 1223, 1232
Blumen, G.W. 1405, 1466
Blumenthal, R.M. 342
Bobillo Guerrero, P. 513, 526, 527
Boccuto, A. 287, 292
Bochner, S. 260, 266, 293, 399
Boehme, Т.К. 1254, 1259, 1260
Bogdan, V.M. 513, 527, 742
Bogdanowicz, W.M. 506, 513, 527
Bogoliouboff, N.N. 1117.1127, 1199, 1234
Bolanos, M.J. 1332, 1377
Bollman, P. 1477, 1522
Bombieri, E. 1035, 1035
Bongiomo, B. 82, 234, 245, 371, 383, 399, 581,
592-599, 610
Bonsall, ?.???,444
Boole, G. 834, 866
Borkowski, L. 1008. 1009
Bouchon-Meunier, B. 1333, 1377, 1510, 1518
Bourbaki, N. 25, 82, 256, 293, 399, 527, 749, 762,
940, 944, 1057-1059, 1062. 1072, 1086, 1090
Bourgain, J. 438, 441. 444, 557, 568, 581, 1198,
1210, 1213, 1232
Bourgin, R.D. 438, 444
Boutilier, С 1474, 1518
Bowen, R. 1044, 1051, 1225, 1232
Boyer, C.E. 26
Boyvalenkov, P. 228, 245
Brehmer, S. 512, 527
Breiman, L. 297, 308, 313, 342, 1222, 1232
Broadbridge, P. 1405, 1452, 1453, 1466
Brook, С 742
Brooks, J.K. 127, 128, 139, 149, 153, 175, 283,
284,293, 297, 313, 330, 342, 342,399,441,444,
451,452, 454,455,458,469,477^479, 482,485,
492, 495, 500, 501, 581, 1446, 1466
Brothers, J.E. 1013, 1035
Brown, G. 1050, 1051
Brown, R. 342

Author Index
1569
Bruckner, A.M. 194, 201, 206, 225, 231, 232, 235,
242-244, 245, 247, 596, 610, 677, 687-689, 700
Bruckner, J.B. 194, 235, 243, 244, 245, 247
Brudno, A.A. 1226, 1232
Buczolich, Z. 245, 590, 591, 600, 604, 606, 610
Bukhvalov, A.V. 813, 823
Bullen, PS. 592, 598, 610
Bunce, L.J. 842, 866
Bungart, L. 510, 526
Burger, H.C. 1477, 1522
Burgers, J.M. 1440, 1466
Burke, M.R. 87, 120, 757, 758, 762, 1133, 1137,
1144, 1153, 1156, 1157, 1159, 1179
Burkill, J.C. 598, 599, 610
Burkinshaw, O. 714, 716, 741, 796, 797, 805, 822
Bumes, J.A. 628, 666
Burzyk, J. 1259, 1259
Busemann, H. 205, 231, 241, 245
Butnariu, D. 876, 879, 906, 913, 918, 925, 927,
930,940, 944,949,951,952,954-957,964, 965,
973, 983, 988, 991. 997, 1008, 1476, 1518
Byrne, C.L. 638, 667
Cabos, J. 1501,7522
Cafiero, F. 127, 130, 153, 175
Calderon, A.P. 1099,1127
Campbell, J.T. 1213, 1232
Candeloro, D. 260, 263, 275, 276, 278, 279, 283,
284, 289, 290, 293. 477, 478, 501, 524, 527
Cao, S.S. 608, 610
Caratheodory, С 82, 240, 246, 693, 700, 747, 762,
889, 907
Carlson, Т. 724, 742,1179
Carnap, R. 1497, 1518
Carothers, D.C. 1179
Carrillo, M.D. 259, 293
Carrington, D.C. 598, 610
Cartan, H. 1113,7/27
Cassels, J.W.S. 654, 667
Cassinelli, G. 866, 866
Castaing, С 284, 293, 619, 622, 630-633, 638,
639, 643, 657, 662. 664, 666, 667
Cawley, R. 1050. 1052
Cayrol, M. 1508, 1518
Ceccherini-Silberstein, T. 85, 96, 120
Cembranos, P. 440, 444
Cerf. R. 666, 667
Chacon, R.V. 153, 175, 297, 313, 330, 342, 342,
1195, 1206, 1232
Chakrabarti, S. 677, 701, 1181
Chakraborty, N.D. 581, 583
Chang, C.C. 175, 830, 853, 860, 866, 878, 879,
905, 907, 913, 914, 944, 956, 1008, 1267, 1290
Chateauneuf, A. 1389, 1390, 1400, 1477, 1518
Chatterji, S.D. 268, 273, 274,293, 581, 1175, //79
Chelidze, V.G. 594, 610
Chemoff, H. 992, 1008
Chew, T.S. 594, 598, 600, 602, 610. 611, 613
Chictescu, I. 524, 527, 581
Chistensen, J.R.P. 520, 527
Chlebik, M. 1045,7052
Choban, M.M. 753, 762
Choe, G.H. 1206, 1232
Choirat, С 643, 664-666, 667
Choksi, J.R. 748, 749, 752, 753, 755, 758, 760,
762, 1175,7/79
Chong, K.M. 813, 823
Choquet, G. 437, 438, 444, 527, 652, 655, 668,
1331, 1377, 1383, 1400, 1466
Choukain-Dini, A. 660, 668
Chovanec, F. 171, 177, 834, 867, 889, 907
Christensen, E. 842, 866
Christensen, J.PR. 1114, 1127, 1147, 1168, 1179
Chuaqui, R.B. 117. 120, 1096, 1100, 1102, 1127
Cichon, J. 1106,7/27
Cicogna, G. 513, 527
Ciesielski, K. 117,120,682, 700, 1124, 1127,1127
Cignoli, R. 878, 879, 905, 907, 944
Cilin, V.I. 741
Clauzure, P. 662, 667
Clemens, J.D. 1110, 1127
Clifford, A.H. 951, 1008
Climescu, A.C. 950,1008
Cohen, G. 1405, 1430-1432, 1436, 1466
Cohen, P.J. 1290
Cohn, D.L. 82, 516,527, 769,778, 784, 1169,1179
Cole, J.D. 1050, 1051, 1052, 1440, 1466
Collet, P. 1050, 1052
Collins, H.S. 552, 581
Colombeau, J.F. 1254,1259
Colubi, A. 657, 668
Comfort, W.W. 242,246, 754, 762, 1279, 1290
Constantinescu, С 128, 144. 175, 509, 510, 527,
742
Contreras, M.D. 441, 444
Conze, J.-P. 1211,/2i2
Cook, A.T./75
Cooke, R. 842, 866
Comfeld. I.P 1189, 1191, 1206, 1209, 1210, 1221,
1232
Corson, H.H. 534, 581
Cortes, U. 1501, 1522
Coste, A. 643, 646, 647, 660, 668
Collar, M. 1214, 1232
Couvreux, J. 655, 665, 668
Cox, J.C. 1448, 1449, 1466
Cox, R. 1465, 1466

1570
Author Index
Crandall, M.G. 1435, 1466
Cressie, N. 653, 662. 668
Cresswell, M.J. 1474, 1521
Csomyei, M. 700, 700
Cuninghame-Green, R.A. 1407, 1408, 1466
Cutland, N.J. 1328
Cutler, CD. 1043, 1048. 1052
Dal Maso, G. 656, 664, 665, 668
Dalgas, K.P. 1152,7/79
Dall'Aglio, G. 776, 784
d'Andrea, A.B. 170. 175
Dang-Ngoc-Nghiem 1102, 1127
Daniell. P.J. 507, 510,527
Darmawijaya. S. 598, 611
Daroczy, Z. 1525, 1526, 1528, 1539, 1559, 1562,
1563
Darst, R.B. / 75, 772, 784
Dashiell, F.K. 175
Dauben, J.W. 119,720
Daures, J.P. 660, 668
Davies, R.O. 591,611, 1047, 1052
Davis, W.J. 272, 293
De, A. K. 691.70/
de Amo, E. 522, 524. 527
De Baets, B. 954, 1008. 1352. 1377, 1487, 1518
de Branges, L. 438, 444
de Campos, L.M. 1332. 1377. 1501, 1515. 1518
De Cooman, G. 1488, 1499, 1509, 1510, 1515,
1519
DeFinetti, B. 1471, 1473,75/9
De Giorgi, E. 656, 664, 668, 1025, 1028, 1034,
1035. 1035. 1331, 1377
de Glas. M. 956, 1008
de Guzman. M. 226, 228. 231, 240-242, 245. 246,
677,691,70/
de Korvin, A. 583, 660, 666
de la Harpe, P. 85, 96,120
de la Vallee Poussin, Ch. 247
de Leeuw, K. 437, 444
DeLia, A. 513. 527
de Lucia, P. 170, 173. 174,175,176. 742, 836.867,
941,944
de Maria, L.J. 176
dePossel, R. 212, 230, 231,247
Debreu, G. 619, 622, 626, 643. 647, 668
Dedekind, R. 85. 120
Defant, A. 437, 444
Dekiert, M. 773, 784
Dekker, T.J. 103. 104. 120
Dekking, F.M. 1223, 1232
delJunco, A. 1229, 1231, 1233
Dellacherie, С 301, 342, 386, 393, 399, 451, 453,
477, 491, 494, 495, 501, 630, 668
DeLuca, A. 1525, 1551, 1563
Dempster, A.P. 1390, 1400, 1477, 1519, 1547,
1563
Denjoy, A. 589, 599. 611, 683, 686-688, 700
Denneberg, D. 252, 293, 512, 518. 527, 1331,
1332, 1334, 1336, 1339, 1363, 1377, 1385,
1388, 1390, 1400, 1466
Deuber, W.A. 96, 120
Devlin. K.J. 1290
Di Nola, A. 889, 898, 907, 944, 956, 958, 1008
Di Piazza, L. 234, 245, 581, 592-599, 601, 603,
610,611
Dial, G. 1557, 1558, 1563
Diaz, S. 440,441,444, 574, 575. 581
Diaz Carrillo, M. 513, 514, 522, 524-526, 527
Dichtl, M. 832, 866
Dickman, M.A. 1290
Diestel, J.J. 82, 128, 150, 151, 153, 154, 176. 266,
272,293, 362, 379,399,437,439^*41, 444,445,
520, 527, 5.33, 536, 539, 540, 542, 572, 581,
627, 636,646,668,709,712, 741, 742, 937, 944,
1081,/090, 1422, 1466
Dieudonne, J. 128, 163, 176, 1081, 1083, 1090,
1179
Dilworth, S.J. 570, 572, 582
Dimitrov, D.B. 540, 582
Dinaburg, E.I. 1225, 1232
Dinculeanu, N. 56, 82, 146, 175, 176, 347. 356.
360-364, 368, 371, 372, 379, 383,399, 432.440.
445, 451, 452, 454, 455, 458, 469,478, 479, 482,
485.492,495, 500,501,524, 527, 560, 581, 582.
646, 668, 1058, 1060, 1080, 1085, 1090. 1177,
1178, 1179
Dim G. 805, 823
Ditkin, V.A. 1258, 1260
Ditzen, A. 1120,//27
Divari, M. 1347, 1349, 1377
Djvarsheishvili. A.G. 594, 610
Dobbertin, R. 1050, 1052
Dobrakov, I. 176, 399, 442, 445, 534, 582, 742.
1085, 1090
Dobric, V. 576. 582
Dodson, M.M. 1044. 1051, 1052
Dominguez-Menchero, J.S. 657, 668
Dong. W.L. 660, 668
Donoghue, W.F. 1179
Doob, J.L. 779. 784, 1321, 1322, 1328
Dorofeev, S.V. 843, 866
Doss, H. 6.39, 668
Doss, S. 638, 668
D'Ottaviano, I.M.L. 878, 879, 905. 907, 944
Dougherty, R. 108, 120, 1114, 1127
Douglas, J. 1027, 1035

Author Index
1571
Dowling, P.N. 440. 445
Drake, F.R. 1267. 1284, 1285, 1290
Drewnowski, L. 127, 137, 153, 176, 534, 541-543,
572, 573. 575.582, 643. 644, 668, 705, 706, 723,
738,740,741, 742,944
Dreyfus, S.E. 1431. 1435, 1466
Drisch. T. 844, 866
Dubins. L. 110,720
Dubois, D. 954, 1008, 1406, 1410, 1426, 1427,
1463-1465, 1466, 1474, 1476, 1477, 1481,
1483, 1486, 1488, 1491-1493, 1495, 1498,
1501-1505, 1507, 1508, 1510-1517, 1518-
1520, 1550. 1553, 1563
Dudley, R.M. 82
Dumitrescu, D. 905, 907, 954, 1008
Duncan, R. 1115,1179
Dunford, N. 82. 127, 129, 146, 149, 152, 153, 176,
266, 269, 279, 293,399.439,444. 513, 514, 520,
528, 539, 561, 564, 580.582, 669, 740, 742, 944,
1059, 1083,7090, 1144,7/79
Dupacova, J. 665, 669
Dutta Majumder. D. 1559, 1564
Dvoretzky, A. 571, 582, 949, 993, 1006, 1008
Dvurecenskij. A. 171, 174, 176, 830. 831, 835,
836, 840, 842-845, 848, 850, 851, 857, 858, 860,
862, 865, 866, 866, 867, 879, 898, 907, 919, 944
Dye, HA. 1102,7/27
Dynkin, E.B. 297, 329, 332. 342, 639, 669
Easton, W.B. 1290
Ebanks, B. 1525, 1533, 1534, 1539-1541, 1551-
1554, 1562, 1563
Edgar, G. 533, 536-538, 540, 543, 559, 560, 582
Edgar, G.A. 206, 246. 437, 445, 781, 784, 1039,
1042, 1049, 1050, 1052, 1133, 1170, 1177,
1178, 1179
Edwards, C.H., Jr. 10, 16. 18,26
Edwards, R.E. 1083, 1085, 1090
Egghe. L. 582
Eggleston, H.G. Ill, 120
Egorov. D.F. 72, 82
Ehrenpreiss, L. 1252,1260
Eifrig, B. 1179
Eigen, S.J. 755, 762
Eilers, M. 844. 867
Einstein, A. 342
El Amri. K. 637. 669
Elek. G. 95, 120
Elliott, G.A. 878, 907
Ellis. H.W. 1140, 1179
Eisner, J. 522, 528
Elstrodt, J. 26, 68, 82
Elton, J. 438. 445
Emch, G.G. 878. 907
Emmanuele, G. 440, 445, 575, 582
Ene, V. 234, 246
Engelking, R. 82. 760. 762
England, J.W. 1525, 1564
Erben, W. 534, 582, 1149, 1179
Erdos, P. 749, 762. 1104, 1127
Erochin, V.D. 772, 784
Ershov, M.P. 781, 784
Es-Sahib, A. 639. 669
Esteva, F. 1474, 1521
Etemadi, N. 656. 669
Evans, L.C. 1013, 1025, 1035
Evans, L.E. 225, 228, 240, 246
Evstigneev, I.V. 639, 643, 669
Ezzaki. F. 657, 659, 667, 669
Faden. A. 779, 784
Faires, B. 176,581
Falconer. K.J. 246, 1013, 1022, 1035, 1039, 1040,
1042-1044, 1046, 1048-1051, 1052, 1411.
1412, 1466
Fan, A.H. 1048, 1052
Farag, H.M. 1025, 1035
Farah, I. 1179
Farell, R.H. 1118, 1127
Fargier, H. 1483, 1488, 1492, 1502-1504, 1512,
1515-1517, 1519,1520
Farinas del Cerro, L. 1474, 1498, 1502, 1505, 1506,
1519, 1520
Farkova, J. 176, 742
Farmaki, V. 582
Farreny, H. 1508, 1518
Fatou, P. 1143,7/79
Federer, H. 182, 205, 220, 225, 226, 234, 246,
1013, 1015, 1017-1019, 1024-1026, 1028,
1031-1033, 1035, 1039,/052
Fefferman, C.H. 524, 528
Fejzic. H. 604, 611
Feldman, J. 1122, 1128
Feledziak, K. 801, 805, 813, 815, 823
Feller, W. 82, 205, 231, 241, 245, 297, 342, 343
Fenstad, J.E. 1304, 1319, 1327
Ferenczi, S. 1198, 1221. 1232
Fernandez, A. 440, 444, 574, 575, 581
Filipczak, M. 690, 700
Filipczak, T. 690, 691, 700
Fillmore, PA. 1179
Filter, W. 510, 527
Fine, T.L. 1475. 1497, 1520
Fink, J.P 582
Fishbum, P. 1475, 1477, 1484, 1520
Fisk, D.L. 501
Flachsmeyer, J. 520. 528

1572
Author Index
Fleischer, I. 594, 611, 944
Fleissner, R.J. 596. 610
Fleissner. W.G. 1281, 1282, 1290
Fleming, W.H. 1028, 1031-1033, 1035
Florencio, M. 440, 444, 573-575, 581, 582
Floret, K. 437. 444, 507, 521, 523, 528
Fodor, G. 1290
Fodor, J.C. 1519
Foias, С 379. 399, \\??,1179
Folmer, ?. 501
Fomin, S.V. 1117, 1128, 1189, 1191, 1206, 1209,
1210. 1221, 1232
Foran, J. 596. 610. 690. 700
Foreman, M. 108, 119,120, 121
Forte, B. 1525, 1529. 1544. 1545, 1550, 1562.
1562-1564
Foulis, D.J. 171,176, 178, 833, 834, 867
Fourie. J.H. 437, 444
Frank, M.J. 949, 951, 952, 1008
Frankiewicz. R. 677, 700, 1179
Frankowska. H. 619, 667
Frechet, M. 638, 669, 705, 742
Freedman, D. 297, 301, 343
Freidlin, M.I. 1466
Freiling, С 599, 604. 611
Fremlin, D.H. 29, 42, 44, 48-50, 59, 63, 65,
66, 68, 73-75, 81, 82, 257, 293, 516, 528,
533, 535, 548, 550, 553, 557, 559, 568, 581-
583, 608, 611, 747-751. 753, 755-757, 760,
761. 762, 763, 796, 797, 801. 805, 823, 889,
899, 906, 907, 919, 944, 1102-1104, 1108,
1124, 1128, 1133-1135, 1138-1140, 1150-
1157, 1159-1161, 1163. 1165, 1170. 1177,
1179, 1288, 1290,7290
Freniche, F.J. 128, 144,176, 440, 445, 575, 583
Freudenthal, H. 797, 822, 823
Fric, R. 907
Friedman, N.A. 1093, 1099, 1128, 1189, 1221,
1232, 1492, 1517,7520
Friedrichs, K. 1239,7260
Frisch, U. 1049,1052
Fu, S. 598,6/7
Fuchs, L. 836, 867, 879, 898, 907
Fuglede, B. 1180
Fiiredi, Z. 228, 246
Furstenberg, H. 755, 763, 1187, 1199, 1201, 1202,
1208. 1209. 121 \, 1232, 1233
Futas, E. 512, 528
Galileo Galilei 85,121
Gamelin, T.W. 438, 445
Gao, S. 1110, 1121, 1127, 1128
Gapaillard, J. 1140, 1180
Garcia, F. 176
Garcia, L. 1501, 1510,1518
Garcia, P. 1474, 1521
Gardenfors, P. 1474, 1482, 1487, 1490. 1497, 1498,
1505, 1506, 1518,1520
Gardner. R.J. 101, 110, 115. 116, 121. 747. 748,
763, 111, 784, 1180
Garg. K. 232. 246
Gariepy, R.F. 225, 228. 240,246, 1013, 1025,1035
Garling, D.J.H. 439. 445
Garsia, A.M. 1195, 1233
Gaubert, S. 1466
Gebhardt. J. 954, 1009, 1501, 1518, 1520
Geffner, H. 1502, 1518
Geitz, R. 542, 543, 550, 556, 583
Gelfand, I.M. 539, 583, 1247. 1260
George, С 82
Georgescu, G. 830, 860, 861, 865, 867, 889, 907,
944
Georgiou, P. 1154, 1180
Gerla, B. 904, 907
Getoor, R.K. 342
Ghoussoub, N. 583
Gibson, F.G. 692, 700
Gil, M.A. 657, 668, 1558, 1563
Gil, P. 1332, 1378, 1558, 1563
Gil Alvarez, P. 1515, 1522
Gil de Lamadrid, J. 439, 445
Gillman, L. 1162, 1180
Gine, E. 647, 669
Girardi, M. 570, 572, 582
Gisolfi, A. 1501,7522
Gitik, K. 1288,7297
Giuntini, R. 834, 867
Giusti, E. 1013, 1025, 1034, 1035, 1035
Glasner, E. 1208, 1231, 1233
Gleason, A.M. 830, 841, 867
Glicksberg, I. 514, 520, 528
Gnedenko, B.V. 767, 779, 784
Goard, J.M. 1405, 1452, 1453, 1466
Godel, K. 1265,7297
Godet-Thobie, С 643, 647, 666, 669
Godin, V. 754, 763
Godo, L. 1465, 1466, 1474, 1510, 1511, 1518,
7579. 7527
Goffman, С 683, 689, 692, 693, 707
Goguen, J.A. 1506,7520
Golan, J.S. 1401,1466,1467
Goldman, A. //SO
Gondran, M. 1467
Goodearl, K.R. 867. 878, 879, 890, 904, 907
Goodman, I.R. 1525, 1563
Goodman, T.N.T. 1225. 1233
Goodwyn. L.W. 1225, 1233

Author Index
1573
Gordon, H. 242, 246
Gordon, R.A. 236, 246, 583, 591. 592, 594. 595,
597, 598, 608, 611
Gordon, Y. 439, 445
Gottwald, S. 949, 950,1008
Gould, G.G. 506, 513, 518, 528
Gowurin, M. 399
Grabisch, M. 954, 1008, 1331, 1333, 1343, 1364,
1377, 1383, 1385, 1388, 1390, 1391, 1394,
1396-1399, 1400, 1405-1407, 1467, 1475,
1477, 1514, 1515,1520, 1525, 1529, 1550,1563
Graf, S. 1044,7052, 1133, 1136, 1137, 1139, 1144,
1147, 1149, 1151, 1173, 1177, 1178, 1178,1180
Graham, C.C. 443, 445
Grant, S. 1515,7520
Grattan-Guinness, I. 26
Gravereaux, B. 451, 501
Graves, W.H. 583
Graziano, M.G. 830, 857. 860, 867
Greco, G.H. 252, 293. 512, 518, 528, 1331. 1378
Greechie, R. 832, 867
Greenleaf, F.P. 1118.1128
Grekas, S. 522, 528, 750, 751, 753-756, 758-761,
762,763, 1133, 1153, 1154, 1180
Greschonig, G. 1120, 1128
Gretsky, N.E. 776, 784
Greuling, H. 834, 867
Gribanov, Yu.I. 798, 823
Grigorchuk, R.I. 85, 96, 120
Grimeisen, G. 534, 582
Gross, H. 1540,1563
Grothendieck, A. 135, 176, 437, 439, 445, 550,
583, 1081-1085, 1087, 1088, 1090
Grove, A. 1474,7520
Gruenhage, G. 1150, 1180
Gryllakis, С 748, 750, 751, 755, 758, 759, 761,
763, 1133, 1153,1180
Guariglia, ?. ПО, 176
Gudder, S. 866, 867, 907
Guiasu, S. 1525, 1529, 1556, 1557, 1561, 1562,
1563
Guivarc'h, Y. 1466
Gulisashvili, A. 583
Gunawardena, J. 1466,1467
Gundy, R.F. 1466
Gunson, J. 842, 867
Gunzler, H. 506, 509, 513, 514, 516, 518, 525, 526,
527, 528
Gupta, R. 1544, 1545, 1563
Gurevich, B.L. 510, 529
Guselnikov, N.S. 176
Haar, A. 1110, 1128
Habil, E.D. 176
Hackenbroch. W. 774, 784
Hadjiev, D. 741
Hadzic, O. 1426, 1467
Hagler, J. 436, 445
Hagood, J.N. 279, 293
Hagood, J.W. 524, 528
Hahn, F.J. 1199,1233
Hahn, H. 176
Hahn, M.G. 647, 669
Hajek, P. 949, 950, 1008, 1474, 1520, 1521
Hajian, A. 1093, 1099, 1100, 1128
Hall. M., Jr. 97, 121
Halmos, P.R. 50, 59, 82. 177, 251, 255, 291, 293,
399, 520, 528, 748, 752, 763, 767, 769, 784,
889, 907, 956, 992, 995-997, 1004, 1005, 1008,
1060. 1061, 1071. 1090, 1095, 1100. 1101,
1113, 1128. 1140, 1180, 1189, 1208, 1214,
1233, 1337, 1339, 1378
Halpem, J. 1472. 1491, 1492, 1517. 1520
Halsey, T.C. 1049, 1052
Hammer, PL. 1384,7400
Hanf, W.P 7297
Hansel, G. 775. 784,1180
Hansell, W. 7290
Hansen, J.C. 654, 666
Hanson, D.L. 1205,/2i2
Hardt, R. 1013, 1028, 1035
Harima, ? 1331, 1378
Harmancova, D. 1474, 7520, 7527
Harmanec, D. 1474, 7527
Harrison, J.M. 331, 335, 343
Hart, S. 647, 650, 651, 656, 666, 669
Hartley, R.V.L. 1527, 756i
Hashimoto, H. 699, 707
Hashimoto, K. 583
Hasselblatt, B. 1044, 1053, 1189, 1199, 1233
Haupt, O. 683, 707
Hausdorff, F. 26, 121, 1039, 7052
Hawkins, T. 22, 25, 26
Haydon, R. 777, 546, 548, 558. 583, 776, 784
Hayes. C.A. 181, 187, 201, 203, 206, 225, 226,
231, 238, 241, 242, 244, 245. 246
Heath, T. 8, 26
Hebert, D.J. //SO
Heilio, M. 575. 583
Heinich, M.H. 568, 583, 639, 669
Heisenberg, W. 829, 867
Hejcman, H. 177
Hejcmann, J. 711, 742, 940, 944
Heller, G. 1170-1172, 1175, 1178, 1180
Helmer, D. 583
Hensgen, W. 440, 445
Hensley, D. 1044, 1052

1574
Author Index
Henstock, R. 188, 201, 236, 246, 589, 591, 592,
598, 600, 608-610, 611, 612
Herer, W. 638, 669, 705, 723, 742
Herstein, I.N. 1511,752/
Herzig, A. 1474, 1498, 1502, 1505, 1506, 1519,
1520
Hess, С 285. 293, 623. 628. 629. 633. 634. 637.
639, 643, 647, 650, 654-657. 660. 662-665.
667-670
Hewitt, E. 98, 121. 752, 763. 1059. 1060, 1090.
1106, 1113, 1126, 1128
Hiai. F. 619, 629, 635, 637-641. 643. 645. 646.
651, 654, 656. 657. 660. 663, 670
Hildenbrand, W. 619, 628, 647, 662, 670
Himmelberg. C.J. 629. 632, 670
Hirsch, M. 110,720
Hisdal, E. 1521
Hobson, E.W. 679. 701
H0egh-Krohn, R. 1304, 1319. 1327
Hoffmann. D. 509. 512. 522, 525, 528
Hoffmann, K. 1143, 1149, 1180
Hoffmann-J0rgensen, J. 575, 576, 583. 742, 775,
776, 778, 784, 1114, 1130, 1175, 1180
Hofmann, K.H. 754, 762
Hohle, U. 949, 954, 956, 1008
Holmes, R.B. 550, 583
Hopf, E. 1093, 1096,1128, 1195,1233, 1440,1467
Hormander, L. 1244, 1247, 1260
Horst, E. 844, 867
Host, B. 1209, 1231, 1233
Howard, E.J. 246
Howroyd, J.D. 1046, 1047, 1052, 1053
Hrbacek, K. 1293
Hu, X. 1050,1053
Hudetz, T. 907
Huete, J.F. 1501, 1518
Huff, R.E. 272, 293, 541, 543, 583
Hughes, G.E. 1474,752/
Hughes, W. 1562,1563
Hukuhara, M. 628, 670
Hulanicki, A. 1108. 1124, 1128
Hunt, BR. 1051, 1053
Hurd, A.E. 1297. 1313, 1328
Hutchinson, J.E. 1043, 1044,1053
Ichihashi, H. 1509. 1514.1521
Imai, Y. 830. 856, 867
Imaoka, H. 1332. 1378
Ingarden, R.S. 1529, 1563
Inoue, H. 653, 672
Inselberg, A. 1467
Inuiguichi. M. 1509. 1514.1521
Ioffe, A.D. 631.670
lonescu Tulcea, A. 82, 211, 293, 363, 399, 538,
545, 583, 611, 701, 755, 756, 763, 1133,
1139, 1141, 1143, 1146-1148, 1151, 1153-
1158, 1172, 1174, 1175, 1177, 1178,1180,1181
lonescu Tulcea, С 82, 111, 293, 363, 399, 538,
545, 583, 611, 701, 755, 756, 763, 1133,
1139. 1141. 1143. 1146-1148, 1151. 1153-
1158, 1172, 1174. 1175. 1177. 1178, 1181
Iorgulescu, A. 830, 860. 861. 865. 867
Iseki. K. 177, 246. 830. 856. 867
Ishihachi, H. 1331.1378
Ito. K. 297. 313. 331, 343
Ito. Y. 1093. 1099, 1128
Jacobs, K. 506, 528
Jaffray. J.Y 1389. 1390. 1400
Jahnke, H.N. 26
Jajte, R. 848. 867
Jaker. A.Sk. 581, 583
Jakubik. J. 879. 898. 907, 914. 944
James. R.C. 550. 583
James, R.D. 599. 612
Janicka, L. 565, 572, 573, 583
Jarai, A. 1545, 1563
Jarchow, H. 439, 444
Jarnik, J. 246, 593, 598, 601-605, 612, 613
Jarvenpaa, M. 1047, 1053
Jasmski, J. 770, 784
Jech.TJ. 1266-1268, 1277, 1284, 1285, 1291
Jedrzejewski, J.M. 686, 701
Jeffery, R. 693, 701
Jensen, M.H. 1049,7052
Jensen, R.B. 1291
Jessen, B. 241, 246, 693, 701, 773, 779, 784, 786,
1319, 1327. 1533, 1564
Jewett, R.I. 1200, 1233
Jewett, R.S. 127, 139,775
Jirina, M. 1175, 77S7
Johnson, R.A. 291. 292.293, 748, 763, 1133, 1156.
1181
Johnson, W.B. 436. 445
Jones. F.B. 7297
Jones. N.D. 1225,/2ii
Jones. S.E. 1467
Josefson. B. 436. 445
Jovanovic, A. 1275, 7297
Joyce. H. 1047, 1053
Junnila. H.J.K. 7290
Jureckova. M. 890. 899. 907. 943, 944
Jurkat, W.B. 603. 607, 672
Just, W. 87. 97. 727. 757. 762, 1133, 7779
Kacian. A. 907
Kadanoff. LP. 1049, 7052

Author Index
1575
Kadec', M.I. 439, 445
Kadets, V.M. 570, 583
Kahane, J.-P. 597, 612, 1044, 1047, 1050, 1053
Kajii, A. 1515,7520
Kakutani, S. 436, 445, 749, 752, 763, 1099, 1100,
1126,1128, 1163, 1181, 1194, 1235
Kalikov, S.A. 1209, 1233
Kalitvin, A.S. 813, S24
Kallianpur, G. 772, 774, 784
Kalmbach, G. 177, 866, 867
Kaloshin, V.Y. 1051, 1053
Kalton, N.J. /77, 440, 444, 572, 573, 5S3
Kambara, H. 1332, 1378
Kampe de Feriet, J. 1529, 1564
Kanamori, A. 1268,7292
Kannan, V. 1107, 1128
Kannappan, PI. 1539-1541, 1563, 1564
Kantorovich, L.V. 776, 784, 797, 822, 823
Kaplansky, I. 932, 944
Kapur, J.N. 1561,7564
Karatzas, I. 343
Kargapolov, M.I. 1109, 1128
Karlin, S. 343
Karolis, P. 760, 763
Karpf, J. 1533,7564
Karush, J. 110,720
Katok, A. 1044, 1053, 1189, 1199, 1233
Katz, V.J. 26
Katznelson, Y. 1201, 1202, 1211, 1233
Kaufman, R. 1046, 1053
Kawada, Y. 1093, 1096, 1098, 1100, 1102, 1128
Kawai, H. 1331,7373
Keane, M. 842, 866, 1222, 7233
Kechris, A.S. 118, 720, 1094, 1110, 1113, 1115-
1118, 1121, 1123,1127,1128
Keegel, J.C. 1562,7564
Keimel, K. 879, 906
Keisler, H.J. 1267, 7290, 7297, 1302, 1325, 1328
Kellerer, H.G. 775-777, 784
Kelley, J.L. 82, 252, 293, 522, 524, 528
Kendall, M.G. 619, 666, 670
Kendall, W.S. 666, 672
Kennes, R. 1477, 7522
Kenyon, H. 181,231,245,246
Kerre, E. 7466, 1510, 1515, 7579
Kerridge, D.F. 1528, 7564
Ketonen, J. 7297
Keynes, J.M. 1497,7527
Khan, M.A. 638, 670
Khararazishvili, A.B. 1104-1106, 1124, 1126,
1127,7727,7725
Khintchine, A.I. 237, 246, 598, 672, 679, 707,
1200, 7233
Khurana, S.S. 438, 445, 742
Kifer, Y.I. 1120,7725
Kindler, J. 516, 528
King, J.L. 1050, 7053, 1208, 7233
Kingman, J.F.C. 1195, 1196, 7233
Kirchheim, B. 1015, 1022, 1024, 1034, 7035
Kirk, R.B. 748, 763
Klawonn, F. 954, 7009
Klei, HA. 638, 643, 647. 670
Klein, E. 619, 670
Klement, E.P. 876, 879, 906, 908. 913, 918, 925,
927, 930, 940, 944, 949-952, 954-958, 961,
965, 970, 973, 991, 1008. 1009. 1332, 1344,
1352, 1378, 1405-1407, 1409-1411, 1427,
1428, 1431, 1458, 1460, 1462, 7467, 1476, 1518
Klimkin, V.M. 128, 777
Klir, G.J. 1331, 1332, 1343, 1351, 7379, 1405,
1406, 1468, 1474, 1514, 7527, 7522, 1525,
1547, 1548, 1550, 7564
Klis, С 777
Kluvanek, I. 399, 513, 528. 583. 1085, 7090
Knight, F.B. 330, 343
Knorr, W.R. 5, 26
Knowles, G. 583
Knowles, J.D. 748, 763. 1152, 7/7S
Kochen, S. 842, 867
Kodaira, K. 752, 763, 1126, 7/2S
Kohlberg, E. 647, 650, 669
Kohnheim, A.G. 1225,7237
Kojadinovic, I. 1562, 7564
Kolesarova, A. 1331, 1378
Kolmogorov, A.N. 26, 767, 779, 784. 829, 867,
905, 908, 1188, 1210, 1214, 1219, 7233, 7234
Kolokoltsov, V.N. 1405, 1408, 1409, 1419, 1422,
1423, 1432-1436, 1440-1445, 1448, 1449, 7467
Kolzow, D. 207, 244, 246. 257, 259, 293. 1133,
1140, 1146, 1172, 1181
Komlos, J. 583
Konig, D. 97, 98, 727
Konig, H. 439, 446, 511, 516-518, 520, 522, 528
Koopman, B.O. 1471,7527
Корка, F. 171, 777, 834, 867. 908
Komfeld, I. 1205,7232
Kotz, S. 776, 784
Koumoullis, G. 748, 750, 763. 772, 784
Kowalski, O. 1021,7035
Kraft, C.H. 1475,7527
Krasnoselskil, M.A. 797, 813, 819, 823
Kraus, S. 1486, 1490, 1492, 7527
Krawczyk, A. 1108, 1125, 7/2S
Krein, S.G. 813, 823
Krengel, U. 1204, 1205, 7234
Krieger, W. 1102, 7729, 1200, 1220, 7234
Kriimpelmann, M. 1557, 1560, 7564

1576
Author Index
Krupa, G. 654, 657, 660, 671
Kruse, R. 639, 671, 954, 1009, 1501, 1518, 1520
Kryloff, N.M. 1117, 1127, 1199, 1234
Kubota, J. 594, 612
Kubota, Y. 613
Kudo, H. 619, 67/
Kuich, W. 1407, 1467
Kuipers, L. 114,72/
Kullback, J.H. 1562, 1564
Kume, ? 1509,1521
Kumei, S. 1405, 1466
Kunen, K. 1266, 1291
Kunita, H. 477, 501
Kunze, M.A. 516,529
Kunze, R.A. 1112, 1121,1129
Kupka, J. 150, 169, 177, 561, 583, 756, 763, 1133,
1145, 1156, 1161, 1177,//S/
Kuratowski, K. 631, 671, 768, 769, 784, 1291
Kurepa, D. 1291
Kuriyama, K. 908
Kurzweil, J. 246, 589, 592, 593, 598, 601-605,
608, 612, 613
Kushnirenko, A.G. 1223,1234
Kussmaul, A.U. 399, 451, 478, 484-486, 499, 500,
501
Kutateladze, S.S. 797, 823
Kuz'minov, V. 763
Kwon, S.H. 1331, 1378
Kyburg, H. 1495, 1521
Labreuche, Ch. 1391, 1400
Labuda, I. 741, 742
Lacava, F. 944
Lacey,M. 1212, 1234
Laczkovich, M. 85, 89, 91, 96, 98, 101, 103,
105-107, 110-117, 120, 121, 1095, 1099, 1103,
1129, 1263, 1291
Ladyzhenskaya, O. 1044, 1053
Lahiri, B.K. 677, 701,1181
Lamata.M.T. 1332, 1377, 1515, 1518
Lander, L. 1550, 1564
Landers, D. 153,177, 742
Lang, J. 1474, 1485,1520, 1521
Larsen, L. 682, 700
Larson, L. 682, 700
Lavie, M. 647, 660, 671
Leader, S. 203, 246, 594, 613
Lebesgue, H. 26, 82, 589, 613, 679, 701, 1142,
1143, 1181
Lee, P.Y 591-594, 598, 600, 601, 611, 613
Lee, T.Y. 600, 613
Leese, S.J. 634, 671
Lehmann, D. 1486, 1490, 1492-1494, 1521
Lehn, L. 724, 742
Leibman, A. 1202,1232
Leibniz, G.W. 16,26
Leinenkugel. С 511, 528
Leinert, M. 511,516, 528
Lembcke, J. 723. 724, 742, 775, 784
Lepellere, M.A. 914, 944
Lesigne, E. 1211, 1232
Lena, G. 1331, 1377
Lettieri, A. 889, 907, 944
Levi, B. 1260
Levin. S.A. 1454, 1467
Levin, V.I. 776, 785, 1154, 1178, 1181
Levy, A. 1291
Levy, P. 298, 343
Lewis, D. 1474, 1486, 1521
Lewis, P.W. 441. 444, 446, 581, 1085,1090
Li, S. 660, 671
Liao, K.C. 594. 598, 613
Liapounoff, A.A. 992, 1006, 1009
Lifshits, Je.A. 797, 823
Lima, A. 439, 446
Lindenbaum, A. 86, 87, 121
Lindenstrauss, J. 813, 823, 992, 1009
Lindstr0m,T. 1304, 1319. 1327
Ling, CM. 951,/009
Lingras, P. 1477. 1522
Lions, PL. 1440, 1442,1467
Lipecki, Z. 572, 573, 575, 582, 705, 724, 726, 743,
1178, 1181
Liu.G.Q. 590, 591, 613
Livshits, A.N. 1228, 1235
Lloyd, S.P 1152, 1181
Lodkin, A.A. 842, 868
Loeb, PA. 228, 229, 245, 245, 246, 1179, 1297,
1303, 1305, 1306, 1308-1317, 1319, 1320,1328
Loeve, M. 297. 343
Lofstrom.J. 813, 823
Lomonosov, V. 438,440, 444, 446
Loomis, L.H. 513, 525, 526, 528
Loomis, L.L. 1121, 1129
Lopes, A.O. 1050,1053
Lopez-Diaz, M. 657. 668
Lorent, A. 1023, 1035
Lorentz, G.G. 813, 823
Los, J. 1108, 1123,1129
Losert, V.L. 758.759, 763,1133, 1152, 1153, 1156,
1177, 1181, 1182
Lovasz, L. 97,121, 1384, 1401
Lu, J. 600, 613
Lu, S.P 591, 592, 598. 613
Lubotzky, A. 85. 121
Lugovaja. G.D. 845, 846, 868
Lukasiewicz, J. 879, 908, 950, 1009

Author Index
1577
Lukes, J. 677, 686, 701
Lusin, N.N. 592, 613, 679, 687, 700, 701
Luther, N.J. 290, 293
Luu, D.Q. 583, 660, 671
Luxemburg, W.A.J. 169, 170, 177, 796-798, 805,
813, 814, 817, 822, 823, 824, 1288, 1291, 1313,
1328
Lyashenko, N.N. 654, 661, 671
Macdonald, A.L. 813, 824
Macheras, N.D. 277,294, 534, 547, 583,586, 755-
757, 763, 764, 1145, 1150-1152, 1159-1162,
1164-1168, 1171,1182, 1183
Mackey, G.W. 830, 868, 1113, 1129
Maeda, S. 842, 844, 868
Magidor, M. 1291, 1292, 1486, 1490, 1492-1494,
1521
Maharam, D. 177, 256, 277, 293, Ы1, 701, 705,
723, 743,749,752,755, 764, 774,781, 785,905,
908, 1139, 1145, 1163, 1173, 1114,1182
Maher, R.J. 1150, 1151, 1154, 1157, 1181, 1182
Maitland Wright, J.D. 1182
Maitra, A. 783, 784, 785
Makinson, D. 1490, 1505, 1518, 1520, 1521
Malgrange, B. 1252, 1260
Malicky, P. 905, 908
Mallory, D. 772, 785
Maly, J. 677, 686, 701
Mandelbrot, B.B. 1039, 1044, 1049, 1050, 1053
Mandl, P. 343
Mangani, P. 914, 945
Manning, A. 1044,1053
Marcinkiewicz, J. 241, 246, 599, 613, 693, 701,
813,524
Marczewski, E. 770, 774, 785, 1108, 1123, 1129,
Wll, 1182
Marczewski (Szpilrajn), E. 785
Marechal, O. 1178, 1182
Mariano, M. 1525, 1548, 1564
Marichal, J.-L. 1383, 1388, 1398, 1400, 1401,
1514,752/, 1562,7564
Marinova, I. 1331, 1378
Markechova, D. 905, 908
Markova-Stupnanova, A. 1366, 1378, 1421, 1427,
1467
Marra, V. 905, 908
Marstrand, J.M. 1020-1022, 1024, 1035, 1036,
1046, 1053
Martellotii, A. 260, 263, 275, 276, 278-284, 286,
293
Martin, N.F.G. 677, 701, 1525, 1564
Mas-Colell, A. 1517,752/
Maslov, V.P. 1405, 1407-1409, 1422, 1423, 1430,
1432, 1433, 1435, 1436, 1440-1444, 1462,1467
Massopust, PR. 1044. 1053
Matheron, G. 652, 666, 671
Mathonet, P. 1398, 1401
Matousek, J. 106, 121
Matsuda, M. 567, 584
Matsushita, Y. 1332, 1378
Manila, P. 225, 228, 229, 240, 246, 1013, 1017,
1020-1022, 1024, 1036, 1039, 1040, 1042,
1044-1047, 1053
Mattioli, J. 666, 672
Matvejchuk, S.M. 842, 868
Mauldin, R.D. 109, 121, 1044, 1050, 1052, 1053,
1104, 1127, 1153, 1182
Maurey, B. 439. 446
Mawhin, J. 603, 613, 614
Maynard, H.B. 270, 279, 294, 524, 528
Mazur, S. 436, 444
Mazurkiewicz, S. 87, 121
Mc Shane, E.J. 628, 671
McAloon, K. 1292
McAndrew, M.H. 1225, 1231
McCluskey, H. 1044, 1053
McCutcheon, R. 1202, 1232
McGehee, O.C. 443, 445
Mckean, H.P., Jr. 297, 313, 331, 343
McLeod, R.M. 236, 246, 591,614
McMillan, B. 1222, 1234
McMinn, T.J. 1182
McMullen, C.T. 1044, 1053
McShane, E.J. 188, 238, 246, 247, 512, 528, 591,
596,614, 1057, 1073, 1076, 1090
Meaya, K. 661, 671
Mecke, J. 666, 672
Medvedev, F.A. 26
Meginnis, J.R. 1558, 1564
Menas, Т.К. 1292
Mendoza, J. 583, 608, 611
Menger, K. 949, 950,1009
Mercourakis, S. 754, 755, 763, 764
Mertens, J.P. 662, 670
Merzbach, U.C. 26
Merzljakov, Ju.I. 1109, 1128
Mesiar, R. 288, 292, 898, 899, 905, 906, 908, 952,
954,957,973,975,983,1008,1009, 1332, 1343,
1344, 1351, 1352, 1362, 1366, 1377, 1378,
1398, 1401, 1405-1411, 1427, 1428, 1431,
1437-1439, 1458, 1460, 1462, 1466, 1467,
1487, 1510, 1518, 1521
Metivier, M. 399, 451, 478, 482, 490, 498, 501
Meyer, P.-A. 301, 342, 386, 393, 399, 437, 444,
451, 453, 477, 491, 494, 495, 501, 630, 668
Meyer-Nieberg, P. 797, 824
Mibu, Y. Wll, 1129

1578
Author Index
Michalak, A. 572, 573, 584
Miction, G. 1050, 1051
Mijajlovic, Z. 1291
Mikusinski, J. 128, 156, 177, 1254, 1258, 1259,
1260
Mikusinski, P. 513, 527
Miller, A.W. 770, 785
Milman, M.M. 813, 824
Milnor, J. 1511, 1521
Misiurewicz, M. 1095, 1099, 1129
Mitchell, W. 1291
Mitial, D.P. 1561, 1564
Mityagin, B. 1205, 1232
Moedomo, S. 274,294, 584
Mokobodzki, G. 1133, 1139, 1152,1183
Molchanov, IS. 666, 671
Molto, A. 177
Monakov-Rogozkin, A. 1174, 1175, 1183
Monge, G. 776, 785
Montagna, F. 898, 899, 908
Montgomery, D. 752, 764
Montgomery-Smith, S.J. 177
Moore, C.C. 1122, 1128
Moore, E.F. 1020, 1022, 1036
Moore, G.H. 119,/2/
Moral, S. 1332, 1377, 1515, 1518
Morales, P. 170, 175,176
Moran, P.A.P. 619, 666, 670
Moran, W. 842, 866, 1171, 1183
Morayne, M. 245, 247
Morgan, F. 1013, 1028, 1036
Morgenstern, O. 950, 1009, 1406, 1465, 1467
Morris, A.S. 755, 764
Morse, A.P. 181,201,225,226,228,231,241,242,
245,246,247, 1020, 1036
Morters, P. 1045, 1053
Mosco, U. 656, 664, 671
Mostert, PS. 1343, 1344, 1378
Mostowski, A. 1291, 1292
Moulin, H. 1479, 1521
Moussa, P. 1050, 1052
Muldowney, P. 610, 614
Mullins, CW. 822, 824
Mundici, D. 830, 853, 854, 856, 868, 878, 879,
889, 898-900, 904-906, 907, 908, 914, 919,
944, 945
Muni, G. 513,529
Mufioz Rivas, P. 513, 514, 525, 526, 527
Munroe, M.E. 82, 247, 570, 584
Murofushi, T. 290, 294, 1331-1333, 1343, 1362,
1377, 1378, 1385, 1396, 1400, 1401, 1405,
1406, 1462, 1467, 1468, 1514, 1515, 1520
Musial, K. 277, 280-282, 293, 294, 533, 534, 537,
541, 547, 549, 550, 553, 554,556, 561-563, 565,
566, 569, 576-580, 581-586, 636, 671, 755-
757, 764, 771,774, 779, 785, 1151, 1152, 1160,
1161, 1164-1168, 1172, 1182,1183
Mycielski, J. 86,91,92, 107-109,/2/, 122, 1111,
1121, 1129, \2Ю,1292
Nachbin, L. 708, 743, 1113, 1129
Nadkarni, M.G. 774, 785, 1115, 1117, 1122, 1129
Nahmias, S. 1498, 1521
Naimark,M.A. 1059, 1090
Nakanishi, S. 608, 609, 614
Nakano, H. 797, 824
Nath, P. 1559, 1563
Natkaniec, T. 692, 700
Navara, M. 832,868, 879, 898, 906, 908, 950, 957,
958,973,975,983, 1009
Neal, D. 477, 501
Negrepontis, S. 1279, 1290
Nelsen, R.B. 951, 1009, 1332, 1378
Neubrunn, T. 879, 890, 899, 905, 909
Neugebauer, C.J. 683, 689, 693, 701
Neveu, J. 517, 529, 625, 639, 660, 671, 767, 769,
785
Newton, D. 1223,1234
Ng, C.T. 1528, 1533, 1535, 1539, 1544, 1545,
1550, 1562-1564
Ng, W.L. 601, 613
Nguyen, H.T. 652, 671, 814, 824, 1405-1407,
1467, 1525, 1529, 1550,1563
Nguyen, NT. 652, 671
Niederreiter, H. 114, 121, 122
Nikodym, O.M. 127, 135,177, 184, 247, 251, 294,
743
Nikol'skii, S.M. 1244, 1260
Nirenberg, L. 1253, 1260
Nishiura, T. 683, 689, 693, 701
Nishiwakj, Y. 1331, 1378
Nissenzweig, A. 436, 446
Nonnenmacher, D.J.F. 603, 604, 607, 612, 614
Norberg, T. 652, 671
Novikov, A. 247
Nowak, M. 801, 805, 813-815, 823, 824
Nowik, A. 1106,1129
Nygaard, O. 439, 446
Nyikos, P.J. 1281,/292
Oberle, R.A. 742
Ochoa.J.P 441,446
Odell, E. 585
Ogura, Y. 660, 671
Oharu, S. 583
Oja, E. 439, 446
Okada, S. 513, 529

Author Index
1579
Olsder, G.J. 1405, 1430-1432, 1436, 1466
Olsen, L. 1049-1051, 1053
Ol'shanskii, A.Yu. 96, 122
O'Malley, R.J. 685, 690, 693, 701
O'Neil, R. 813,824
O'Neil.T.C. 1050, 1051, 1052. 1053
Onisawa.T. 1331, 1378
Orey, S. 500, 501
Orlicz, W. 146, 177
Omstein, D.S. 1102,1129. 1195, 1206, 1221, 1222,
1227, 1228, 1232,1234
Osswald, H. 1323, 1324, 1328
Ostaszewski, K.M. 608, 614, 682, 684, 700. 701
Ostroy, J.M. 776, 784
Owen, G. 1384, 1401
Oxtoby, J.C. 117, 122, 678, 683, 684, 687, 701,
749, 755, 762, 1113, 1114, 1126, 1128, 1129,
1136,1183
Pachl, J.K. 771-775, 781-783, 785, 1175-1177,
1183
Pacquement, A. 598, 610
Pal, N.R. 1525, 1553, 1555,1564
Pal, S.K. 1559, 1564
Pallares, A.J. 585
Pallu de la Barriere, R. 643, 647, 668, 671
Panchapagesan, T.V. 1057-1059, 1070, 1081,
1083-1086, 1088-1090, 1090
Panti, G. 878, 908
Pap, E. 128, 135, 145, 146, 150, 156, 157,
160, 162-164, 169, 173, 174, 175-178, .350,
399, 512, 529, 836, 867, 908, 941, 944, 952,
1009, ПАЛ, 1252, 1254, 1259, 1260, 1331,
1332, 1336, 1340, 1343, 1344, 1351, 1.352,
1363, 1366, 1378, 1390, 1401, 1405-1411,
1414, 1416, 1418-1420, 1426-1428, 1431-
1433, 1436-1441, 1443, 1449-1451, 1455,
1457, 1458, 1460, 1462-1465, 1466-1468.
1476, 1514,1521,1564
Papageorgiou, N.S. 639, 643, 647, 666, 671, 672
Papangelou, F. 506, 529
Paplauskas, A.B. 26
Papoulis, A. 241,247
Paris, J.B. 1468
Parisi, G. 1049, 1052
Parry, W. 1192, 1197, 1200, 1204, 1205, 1210,
1219,1221,1234
Parthasarathy, K.R. 1111, 1113, 1120,7/29
Parthasarathy, T. 632, 670
Paszkiewicz, A. 842, 868
Paterson, A.L.T. 85, 95-97, 122, 1118, 1129
Patzschke, M. 1050,705/
Pauc, C.Y. 181, 187, 201, 203, 206, 225, 226, 231,
238, 241, 242, 244, 245, 246, 683, 701
Paul, P.J. 440, 444, 573-575, 581, 582
Pawlikowski, J. 119, 122
Pearl, J. 1494, 1500, 1501, 1521, 1522
Peck, C.V. 755, 764
Pelc, A. 117, 120, 1106-1108, 1124, 1127, 1127,
1129
Pelczynski, A. 438^*40, 444, 446, 55.3, 581, 1085,
1090
Pellaumail, J. 399, 451, 478, 498, 501, 1175, 1183
Penconek, M. 118, 122, 1105, 1129
Peres, Y. 1044, 1054
Perez, R. 1558, 1563
Perkins, E.A. 1050, 1054, 1328
Pemy, P. 1515-1517, 1519, 1520
Perron, O. 589, 614
Pesin, I.N. 26
Pesin, YB. 1044, 1049, 1050, 1054
Petersen, K.E. 1187-1189, 1193, 1194, 1196-
1198, 1204-1207, 1210, 1212-1214, 1217,
1220-1222, 1232,1234
Petrovicova, J. 905, 908
Pettis, B.J. 146, 178, 269, 293, 533, 535, 540, 559,
570, 573, 585
Petunin, Ju.I. 813, 823
Peyriere, J. 1044, 1050, 1051, 1053, 1054
Pfanzagl, J. 779, 785
Pfeffer, W.F. 2.34, 236, 245, 247, 505-508, 510,
515, 516, 519, 521,529, 590, 603, 604, 606, 608,
610, 613, 614, 748, 763
Phakadze, S.S. 1124, 1129
Phelps, R.R. 272, 293, 438, 446
Phillips, R.S. 135, 149, 178, 534, 570, 585
Piasecki, K. 879, 908
Pier, J.-P. 26
Pietsch, A. 28.3, 294, 439, 446
Pincus, D. 126.3, 1288, 1292
Pirogov, S.A. 1120, 1128
Piron, С 842, 868
Pisier, G. 439, 446
Plante, J.E. 1118, 1129
Plappert, P. 100, 122
Plebanek, G. 556, 558-560, 585, 775, 776, 785
Plummer, M.D. 97, 121
Poincare, H. 1200, 1234
Pol, R. 534, 585
Polak, B. 1515, 1520
Pollard, D. 518, 529
Poncet, J. 1121, 1129
Popa, D. 585
Poprugenko, G. 992, 993, 1009
Portenier, С 512, 520, 526, 526
Portinko, N.I. 331, 343
Possani, С 1306,1327

1580
Author Index
Postnikov, A.G. 1199.1234
Povolotskii, A.I. 813, 824
Prade, H. 954, 1008, 1406, 1410, 1426, 1427,
1463-1465, 1466, 1474, 1476, 1481, 1483,
1486, 1488, 1491-1493, 1495, 1498, 1501-
1505, 1507, 1508, 1510-1517, 1518-1520,
1550, 1553, 1563
Prandini.J. 1306, 1327
Prasad, V.S. 755, 762, 764
Pratelli, M. 451,502
Pratt, J.W. 1475, 1521
Preiss, D. 106, 122, 234, 238. 245. 247. 596, 597,
599, 610. 614, 1020-1022, 1024, 1025, 1035.
1036,1045-1047, 1053,1054
Preston, C.J. 178
Price, J.F. 754, 764
Prikry,K.L. 724, 741, 742, 1133, 1156,1181, 1277.
1291, 1292
Procaccia, I. 1049,1052
Protter, P. 502
Prudnikov, A.P. 1258, 1260
Przymusinski, T. 1292
Ptak, P. 178, 866, 868
Pucci, S. 289, 290, 293
Puhalskii, A. 1405, 1432, 1433, 1468
Pujie, W. 594, 614
Pulmannova, S. 178, 851, 858, 866, 867, 868, 889,
907, 908
Puri.M.L. 647, 654, 661,672
Pustylnik, E.I. 813, 819, 823
Qu, Y.S. 517, 529
Quadrat, J.-P. 1405, 1430-1432, 1436, 1465, 1466
Rachev, ST. 776, 785
Rachunek, J. 860, 868
Rado, R. 97, 122
Rado, ? 1027,1036
Radon, J. 133, 178, 436, 446
Radon, M. 251,294
Rahe, A.M. 1231, 1233
Raikov, D.A. 1113, 1129
Rainwater, J. 438, 446
Raj'u, S.R. 1107, 1128
Ralescu, D.A. 647, 654, 661, 672, 1332, 1366,
1378, 1513, 1515,752/
Ralevic, N. 1431, 1432, 1440, 1468
Ramachandran, D. 767, 770, 774, 776-783, 784-
786, 1095, 1099, 1129
Ramakrishnan, S. 783, 785
Ramer, A. 1487, 1521, 1525, 1547, 1548, 1550,
1564
Ramsey. F.P. 1471. 1521
Randall, C.H. 178. 833, 867
Randolph, J.F. 1020, 1036
Randrianantoanina, N. 441, 446
Rao, B.V. 769, 783
Rao, K.M. 502
Rao, M.M. 178, 399. 517. 523, 529. 812, 813, 824
Rashid, S. 1305, 1313, 1316, 1327
Raynaud de Fitte, P. 639, 647, 657, 672
Regazzini, E. 263, 292
Reidi, R. 1044, 1050. 1053
Reinhardt, W.N. 1268, 1292
Remus, D. 754, 762
Remy, M. 773, 783, 786
Ren, Z.D. 812, 813, S24
Renyi, A. 82, 1205. 1234
Ressel.R 520,527. 748. 764
Retherford. S.R. 441, 447
Rice, N.M. 813, 823
Richter, H. 619, 637. 672
Rickart, C.E. 127, 129, 135, 178, 709, 743
Riddle, L.H. 544, 546. 547, 554, 557, 560, 566,
569, 585
Riecan, B. 512, 529, 856, 868, 879, 889, 890, 898,
899, 905, 907-909. 914, 945
Riecanova, Z. 1331, 1378
Rieffel, M.A. 270. 294. 542, 585
Riesz, F 436, 446. 692, 701. 797, 822, 825
Rigo, P. 263, 292
Rinkewitz, W. 1154, 1175, 1183
Rinne, D. 599, 604, 611
Robbins. H.E. 619. 672
Roberts, Ch.E. 583
Roberts, J.W. 177. 1121, 1129
Roberts, R. 660, 666
Robinson, A. 1297, 1328
Robinson, R.M. 103, 122
Rockafellar. R.T. 631, 656, 664, 672. 1004, 1009
Rodine, R.H. 781.7S6
Rodriguez-Piazza, L. 440, 444
Rodriguez-Salinas. ?. ??83
Rogers, C.A. 206, 247. 1013, 1036, 1039, 1047,
1048, /052, 1054
Rogers, L.C.G. 343
Rogge. L. 153, 177
Rokhlin (Rohlin). V.A. 780. 786. 1208, 1220, 1221,
1227, 1229, 1231, 1234
Ro.se. A. 945, 949. 1009
Rosenblatt, J.M. 97, 102, /22, 1102, 1129, 1211,
1234
Rosenfeld, M. 201, 245
Rosenkrantz, W. 331, 343
Rosenthal. H.P. 128, 150, 178, 436, 446, 545, 556,
557. 566, 567, 585
Ross, D.A. 652, 672. 1183

Author Index
1581
Ross, K.A. 98, 121, 752, 763, 1106. 1113, 1126.
1128
Ross, S.A. 1448, 1449, 1466
Rosser, J.B. 945, 949, /009
Rota, G.C. 1384, 1401
Roubens, M. 1383, 1385, 1388, 1400, 1519, 1521,
1562, 1564
Rowbottom, F. 1292
Royden, H.L. «2,505,507, 510,516,517,520,529
Rubinstein, M. 1448, 1449, 1466
Rudeanu, S. 1384,7400
Rudin, V.N. 513, 529
Rudin, W. 50, 82, 347, 399, 802, 804, 825, 1059,
1090, 1198, 1199, 1234
Rudolph, D.J, 1189, 1209, 1212, 1218, 1219, 1222,
1229-1231, 1233, 1234
Ruelle, D. 1044, 1054
Ruess, W. 552, 581,583
Ruess,W.M.441,445
Ruschendorf, L. 776-778, 785, 786
Ruspini, E. 905, 909, 954, 1009
Rusu, D. 677, 701
Rutickii, Ya.B. 813, 823
Ryan, R. 437,446, 1183
Rybakov, VI. 278, 279, 294, 533, 540, 560, 566,
568, 573, 585
Rybalov, A. 1352, 1379
Rybarik, J. 905, 908, 909, 954, 1009, 1351, 1378,
1407, 1408, 1467
Rybarikova, E. 907
Ryll-Nardzewski, С 565, 566, 585, 631, 671, 770,
771, 774, 779, 784-786, 1104, 1129
Rynne, B.P. 1044, 1052
Ryzhikov, V.V. 1209, 1234
Saab, E. 440, 441, 444, 446, 544, 546, 547, 554,
557, 560, 566, 569, 583, 585
Saab, P. 440, 441,444, 446, 585
Sabbadin, R. 1507, 1510, 1512-1515, 1520
Sacks, G.E. 1292
Saeki. S. 157. 175. 178
Sahoo, P. 1525, 1533, 1539-1541. 1552, 1553.
1563. 1564
Saint Raymond. X. 1045,1054
Saint-Pierre, J. 639, 672, 1175, 1183
Saint-Raymond. J. 633. 672
Sajid, S. 639. 672
Sakai, K. 1100. 1129
Saks. S.26. 127. 135. 178. 184. 206. 232. 234. 240,
243. 247, 255. 294. 436. 446. 598. 614. 692. 701
Salama. I. 1198, 1234
Saleski. A. 1223, 1234
Salinetti. G. 647. 652. 655. 672. lib. 784
Salvati. S. 176
Samborskii, S.N. 1405, 1407, 1408, 1420, 1430,
1433, 1435, 1436. 1440. 1462, 1467. 1468
Sambucini, A.R. 280-282, 284, 286, 293
Sander, W. 1428, 1468. 1525, 1533, 1539-1541,
1545, 1548, 1549, 1552, 1553, 1558, 1563. 1564
Sanguesa, R. 1501,7522
Santalo, L. 666, 672
Sapounakis, A. 1152, 1183
Sarkhel, D.N. 691,70/
Sarymsakov, T.A. 741
Sastri, C.C.C.A. 1562. 1563
Savage. L.J. 1473. 1475, 1507. 1522
Sazenkov. A.N. 178
Sazonov, V 772, 786
Scepin, E.V 760, 764
Schachermayer. W. 128. 178, 441,445, 585
Schaefer, H.H. 640, 672, 797, 805, 825
Schafke, F.W. 509. 511-513, 522, 524, 525, 528,
529
Scheinberg. S. 684, 685, 701
Schief, A. /183
Schmechel, N. 954, 1009
Schmeidler, D. 512, 529, 643, 647, 662, 668, 672,
1331, 1378. 1522
Schmidt, K. 1120, 1128, 1129
Schmidt. K.D. 774, 786, 945, 954, 959, 983, 1009
Schmidt, W. 114, 122
Schmitt, M. 666, 672
Schreiber. B.M. 1177. 1183
Schurger, K. 654, 672
Schurle. A.W. 592,614
Schuss, Z. 591,6//
Schwabik. S. 603, 604, 612
Schwartz, J.T. 82, 127, 129, 146, 149, 152, 153,
176,399, 439,444, 513. 514,520, 528, 561, 564,
580, 582, 669, 740, 742, 944, 1059, 1083, 1090,
1144,1179
Schwartz, L. 266, 279, 293, 439, 446, 520, 529,
1139. /183. 1260
Schwarz. F. 1405. 1468
Schweizer. B. 776, 786. 949. 951. 1009. 1332.
1352. 1378. 1416. 1424, 1425. 1429. 1468.
1529. 1564
Scott. D.S. 1268. 1291. 1292
Segal. I.E. 257. 294. 516. 529. 1112, 1121. 1129,
1140. 1183
Seidenberg. A. 1475. 1521
Seifert, C.J. 439, 445
Semadeni, Z. 797. 805. 825
Semenov, E.M. 813. 823
Sentilles, D. 543, 585. 586
Sen. R. 643, 664-666. 667
Serra, J. 672

1582
Author Index
Serres, M. 26
Sersouri, A. 585
Shackle, G.L.S. 1476, 1522
Shafer, G. 1385, 1401, 1464, 1468. 1477, 1522,
1547, 1564
Shannon, C.E. 1222, 1234. 1527, 1564
Shapiro, L.B. 760, 761, 764
Shapley, L.S. 949, 1007, 1385, 1399, 1401
Sharma, B.D. 1528, 1561, 1564
Shelah, S. 583, 757, 764, 1133, 1139, 1144, 1157,
1173, 1179. 1183. 1288, 1291, 1292
Shepp, L.A. 331, 335, 343
Sherman, G.A. 87, 89, 91, 122
Sherstnev, A.N. 843, 845, 866. 868
Shieh, N.-R. 1050, 1054
Shields, A.L. 1343, 1344, 1378
Shields, P.C. 1222, 1235
Shilkret, N. 1367, 1378, 1409, 1411, 1468, 1510,
1522
Shilov, G.E. 510, 529, 1247, 1260
Shiryayev, A.N. 82, 1191, 1235
Shoenfield, J.R. 1292
Shoham, Y. 1487-1489, 1522
Shorn, R.M. 770, 772, 775, 779, 780, 786
Shraiman, B.J. 1049, 1052
Shreve, S. 343
Shulman, V. 776, 784
Shwartz, L. 748, 764
Sierpinski, W. 85-87, 90, 100, 103, 119, 121, 122,
586, 679, 701, 770, 786, 1292
Sikorski, R. 586, 797, 825, 831, 868, 890, 909
Silver, J.H. 1122, 1129, 1274, 1284, 1292
Simha, R.R. 752, 762
Simon, B. 846, 868
Simon, J. 1035, 1036
Simon, K. 1044, 1054
Simon, L. 1013, 1028, 1035, 1036
Simonnet, R. 517,529
Simonovits, M. 96, 120
Simpelaere, D. 1050, 1054
Sinai, Y.G. 1189, 1191, 1206, 1209, 1210, 1219,
1221, 1232, 1234,1235
Singer, I. 379, 399, 440, 446, 1384, 1401
Sion, M. 723, 743, 1085, 1090, 1146, 1184, 1333,
1378
Sipos, J. 512, 519, 529, 1331, 1332, 1364, 1378,
1390, 1401
Siu-Ah Ng 1328
Sklar, A. 949, 951, 1009, 1332, 1352, 1378, 1416,
1424, 1425, 1429, 1468, 1529, 1564
Skvortsov, V.A. 245, 592-595, 597-599, 608, 610,
614
Smets, P. 1477, 1522
Smith, B. 441,446
Smorodinsky, M. 1222, 1233, 1235
Snobar, M.G. 439, 445
Snow, DO. 1140, 1179
Snow, P. 1486, 1495, 1522
Sobolev, A.V. 797, 823
Sobolev, S.L. 1239, 1260
Sobolevskii, P.E. 813, 819, 823
Sokal, A.D. 779, 786
Solecki, S. 245, 247, 1106, 1114, 1129, 1130
Solodov, A.P. 608, 614
Solomon, B. 1032, 1034, 1036
Solomyak, B. 1044, 1054
Solovay, R.M. 49, 82, 1263, 1268, 1286, 1288,
1292
Soneda, S. 1385, 1401
Sonnenschein, H. 1517, 1521
Sonntag, Y. 655, 657, 672
Sos, V.T. 96, 120
Sparre-Anderson, E. 773, 786
Specker, E.R. 842, 867
Spohn.W. 1478, 1522
Srinivasan, T.P 522, 528
Stajner-Papuga, I. 1416, 1419, 1420, 1455, 1468
Stefansson, G.F. 541, 556, 558, 586
Stegall, C.P 272, 294, 441,447
Stegall, Ch. 536, 586
Steinhaus, H. 1114, 1130
Steinlage, R.C. 436, 447, 1121, 1130
Stepan, J. 776, 783
Stepanoff, W.W. 687, 701
Stone, C.J. 330, 343
Stone, M.H. 508, 529, 1136, 1139, 1173, 1183
Stout, E.L. 438,447
Stoyan, D. 666, 672
Strassen, V. 775,776, 786
Straus, E.G. 86, 122
Strauss, W. 277, 294, 534, 586, 755-757, 763,
764,1140,1145,1150-1154, 1159-1162, 1164-
1168, 1170-1172, 1175, 1178, 1182-1184
Strieker, С 502
Stromberg, K. 1059, 1060, 1090
Stroock, D.W. 297, 343, 507, 508, 516, 517, 529,
774, 786
Stroyan, K.D. 1304, 1328
Struik, D.J. 16,26
Suarez, F. 1332, 1378
Suarez Garcia, F. 1515, 1522
Sucheston, L. 153, 174
Sudakov, V.N. 776, 786
Sugeno, M. 288, 290, 294, 949, 1009, 1331-1333,
1343, 1362, 1377, 1378, 1383, 1396, /400,
1401, 1405, 1406, 1409, 1462, 1467, 1468,
1513-1515, 1520-1522

Author Index
1583
Sullivan, D. 1041, 1054
Sullivan, J.M. 228, 247
Sun, T.C. 1177, 1183
Sun, Y.N. 638, 670, 1322-1324, 1326, 1327, 1328
Swanson, L. 1231, 1233
Swart, J. 437, 444. 520, 527
Swartz, С 128, 135, 148-150, 157, 162, 170, 175.
177, 178,441.447
Swierczkowski, S. 1287, 1292
Szemeredi, E. 1201, 1235
Szpilrajn, E. 1094, 1107, 1123, 1124, 1126, 1130
Takaci, A. 1244, 1252, 1259, 1260, 1443, 1468
Takaci, Dj. 1244, 1252, 1259, 1260. 1468
Talagrand, M. 438, 441, 444. 447. 533, 535, 537,
543, 546, 548-550, 553, 554, 557, 559, 561, 568,
569,575, 576,578,580,581.583.586. 751, 757,
758,764, 1133, 1153, 1156, 1158, 1159, 1169,
1170, 1177, 1178, 1179. 1184
Tall, F.D. 682, 684, 686, 691, 701, 1281, 1283,
1292, 1293
Tanaka, H. 1331, 1378, 1509, 1514, 1521
Tanaka, S. 177
Taneja, I.J. 1525, 1528, 1557, 1558, 1563-1565
Tarashchan, A.L. 1420, 1468
Tarski, A. 85, 87, 90, 97, 98, 100, 110, 117, 120,
122, 879, 908, 909, 1093, 1096, 1099, 1103,
1127, 1130, 1274, 1291, 1292
Tartar, L. 1435, 1466
Taylor, A.E. 508, 510, 520, 529
Taylor, H. 343
Taylor, M. 776, 786
Taylor, R.L. 653, 672
Taylor, S.J. 233, 247, Ы1, 694, 695, 701, 1041,
1044, 1048, 1050, 1053, 1054
Telgarsky, R. 1104, 1129
Temam, R. 1044, 1054
Teofanov, N. 1433, 1468
Terepeta, M. 694, 696, 698, 699, 701
Termini, S. 1525, 1551, 1563
Terpe, F. 520, 528
Testemale, С 1508, 1514, 1520
Theil,H. 1528, 1565
Thiam, D.S. 513, 529, 643, 647, 672, 673
Thiele, H. 954, 1009
Thomas, E. 570, 572, 573, 576, 586, 1057, 1085-
1089, 1090
Thompson, A. 619,670
Thomson, B.S. 194, 231, 233-235, 237, 239, 243,
244,245,247, 598, 599,610,611,614,615, 687,
701, 1260
Thorup, A. 1533, 1564
Thouvenot, J.-P. 1231, 1235
Tiser, J. 1022, 1025, 1036, 1045, 1054
Tits, J. 96, 122
Tkacenko, M.G. 760, 764
Tomczak-Jaegermann, N. 439, 447
Tong, A.E. 439, 447
Tonge, A. 439, 444
Topsoe, F. 517, 518, 529, 773, 786, 1114, 1130
Tortrat, A. 586, 1171, 1184
Tousset.E. 1398, 1401
Touzani, A. 284, 293
Traynor, ? 153, 176, 178, 705, 738, 743, 944, 945,
1136, 1184
Tricot, C, Jr. 122, 233, 247, 1041, 1045, 1054
Triebel, H. 1244, 1260
Troallic, J.P. 775, 784
Trotter, H.F. 309, 343
Troyer, R.J. 677, 701
Truss, J.K. 101, 122
Tsiporkova, E. 1487, 1518
Tsukada, M. 67i
Tweddle, I. 148, 178
Tzafriri, L. 813, 823
Uemura, T. 653, 673
Uhl, J.J., Jr. 82, 128, 150, 151, 153, 154, /76,266,
272, 274,293, 294, 362, 379,399, 439,445, 533,
536, 539,544, 546, 547, 554,557,560, 566,572,
581,582,584-586, 627, 636, 646,668,709, 712,
741, 742,937, 944, 1081, 1090
Ulam, S.M. 261, 294, 1212, 1235, 1268, 1284,
1285, 1293
Ulger, A. 441,447
Umegaki, H. 619, 629, 635, 637-641, 643, 660,
670
Urbanik, K. 1529, 1563
Urbanski, M. 1044, 1050, 1053
Urbina, W. 634, 667
Uspenskii, V.V. 760, 764
Ustunel, S. 477, 502
Valadier, M. 284, 293, 619, 622, 630-633, 639,
641, 643, 664, 666, 667, 673, 781, 786, 1175,
1184
Valdivia, M. 520, 529
Valko, S.98, 121
Van Cutsem, B. 619, 639, 643, 673
van Dantzig, J. 1110, 1130
VanderPyl.T. 1561, /565
van der Waerden, B.L. 26, 1110, 1130, 1201, 1235
van Dulst, D. 582
van Neerven, J.M.A.M. 585
Van Vleek, FS. 632, 670
Vanmassenhove, F.R. 1510, 1519
Varadajan, VS. 520, 529

1584
Author Index
Varadarajan, VS. 559, 586, 747, 764, 866, 868,
890,909, 1118, 1130
Varadhan, S.R.S. 297, 343
Varga, R.S. 1191, 1235
Vasak, L. 559, 586
Vath, M. 789, 795, 798, 805, 813, 814, 821, 822,
823, 825
Veech, W.A. 755, 764
Vera, G. 585
Veres, S. 1560, 1565
Verma.T. 1501, 1522
Vershik, A.M. 1228, 1235
Vervaat, W. 652, 672
Vesterstr0m, J. 1177, 1184
Vetterlein, T. 866, 867
Vickers, J.A.G. 1044, 1052
Vincke, P. 1521
Vind, K. 643, 647, 673
Viot, M. 1432, 1465, 1466
Vitale, R.A. 653, 661, 666, 673
Vitali, G. 89, 122, 133, 178, 1103, 1130, 1331,
1378
Vivona, D. 288, 292, 1332, 1333, 1343, 1347-
1349, 1351, 1375, 1377, 1407, 1449, 1450,
1466, 1468, 1518
Vladimirov, D.A. 706, 743, 1102, 1130
Vladimirov, VS. 1247, 1250, 1260
Volcic, A. 255, 291, 292, 294, 510, 523, 524, 527,
529,530, 1184
Volkmer, H. 743, 932, 945
Volpert, A.J. 1025, 1036
von Neumann, J. 91-93, 104, 105, 107, 122, 277,
294, 830, 866, 868, 871, 906, 950, 1009, 1097,
1102, 1110, 1130, 1136, 1137, 1139, 1141,
1153, 1173, 1183, 1189, 1193, 1212, 1214,
1233-1235, 1406, 1465, 1467
von Weizsacker, H. 538, 586, 1136, 1137, 1139,
1144, 1147, 1157, 1172, 1174, 1178, 1180, 1184
Vopenka, P. 1293
Vrabelova, M. 909
Vulikh, B.Z. 918, 945
Vybomy, R. 592, 593, 610, 613
Wagner, D.H. 631,638, 673
Wagner-Bojakowska, E. 694, 696-699, 701, 702
Wagon, S. 85-87, 89-91,93,95-97, 101-104, 107,
108, 111, 115, 117, 119, 122, 1095, 1096, 1100,
1118, 1123, 1130
Wajch, E. 748, 751,760, 763
Wald, A. 949, 993, 1006, 1008
Waldschaks, G. 774, 786
Walker, E.A. 1405-1407, 1467, 1525, 1529, 1550,
1563
Walley, P. 1390, 1401, 1477, 1522
Walsh, J.B. 331, 3.35, 343
Walters, P. 1204, 1235
Wang, P.J. 594, 615
Wang, S.P. 1120, 1130
Wang, Z.P. 178, 660, 668, 673, 1331, 1332, 1343,
1351, 1379, 1405, 1406, 1468, 1514, 1522
Ward, A.J. 247
Warner, S. 707, 739, 743, 929, 945
Waterman, D. 683, 692, 701
Weakley, C.B. 743
Weber, H. 128, 144, 156, 157, 160, /75,705,711,
718, 72,3, 724, 7.38, 741, 742, 743, 913-915,932,
935, 938, 944, 945, 949, 1006, 1007, 1007, 1010
Weber, K. 509, 510, 527
Weber, S. 956, 1008, 1331, 1332, 1345, 1366,
1367, 1379, 1406, 1410, 1411, 1463, 1467,
1468, 1476, 1522
Weglorz, B. 1106, /127
Wehrung, F. 100, 108, 116, 119, 121-123
Weil, A. 26, 1113, 1114, 1120, 1130
Weil, W. 662, 673
Weir, A.J. 505, 510, 511, 514, 516, 520, 523, 530
Weiss, B. 1211, 1227, 1228, 1232, 1233
Weiss, H. 1050, 1054
Weiss, T. 770, 784
Wentzell, A.D. 1466
Wenzel, J. 439,446
Werner, E. 585
Wets, R.J.-B. 647, 652, 655, 656, 663-665, 666,
669, 672
Weyl.H. 1198, 1199, 1235
Whalen, T. 1509, 1522
Wheeden, R. 700, 702
Wheeler, R.F. 586
White, B. 1034, 1036
White, H.S. 1225, 1226, 1235
Whiteside, D.T. 26
Widom, H. 82
Wiener, N. 298, 343, 1194, 1196, 1213, 1235
Wierdl, M. 1211, 1212, 1234
Wijsman, R.A. 632, 673
Wilczynski, W. 682, 694, 697, 699, 700-702
Wilhelm, M. 512, 530
Williams, D. 343
Williams, S.C. 1044, 1052, 1053
Williamson, J.H. 82
Willis, G.A. 95, 123
Wills, J.M. 114, 122
Wils, J. 1177, 1184
Wintner, A. 1213, 1235
Wisjman, R.A. 632, 673
Wittaya Naak-in 592, 613
Wloka, J. 1244, 1260

Author Index
1585
Wnuk, W. 805, 825
Wojdowski, W. 691, 702
Wqjtaszczyk, P. 439,447
Wolfenstein, S. 879, 906
Wolff, M.P.H. 1297, 1328
Wolfowitz, J. 949, 993, 1006, 1008
Wong.S.M.K. 1477, 1522
Wright, J.D.M. 842, 866, 899, 909
Wu,T.S. 1120, 1130
Wyler, O. 956, 1010
Xue, X.H. 660, 673
Yaari, M.E. 1476, 1522
Yager, R.R. 1352, 1379, 1509, 1522
Yan, B. 1510, 1522
Yand, J.S. 1120, 1130
YangQuingJi 1332, 1379
Yannelis, N. 662, 673
Yao, YY. 1477, 1522
Yeadon, F.J. 842, 868
Yor.M. 451,502
Yosida, K. 1194, 1235
Yuan, B. 1525, 1547, 1550, 1564
Zaanen, A.C. 26. 169. 170. 177. 523, 530. 796-
798, 805, 813, 814, 819, 822, 823-825
Zabrelko, P.P. 805, 807, 813, 814, 819, 822, 823-
825
Zadeh, L.A. 945, 949, 953, 956, 1010, 1476, 1498-
1500, 1507, 1522, 1550, 1559, 1565
Zagordny, D. 1421, 1468
Zahorski, Z. 689, 702
Zajicek, L. 677, 679, 686, 701. 702
Zakrzewski, P. 117, 118, 123. 1100-1105, 1107,
1108,1125-1127, 1128-1130
Zalinescu, С 655, 657, 672
Zame, W.R. 776, 784
Zapico, A. 1465. 1466, 1511, 1519
Zbierski, P. 1179
Zermelo, E. 119, 123
Ziat, H. 657, 673
Ziemer, W.P. 225, 228, 240, 247, 677, 701, 1013,
1025,1036
Zinn, J. 647,669
Zippin, L. 752, 764
Zygmund, A. 241, 246, 599, 613, 693, 700, 701,
702, 1143, 1184

Subject Index
?-orthogonality, 171
®-step representation, 1354
®-sum, 171,837
? functor, 874
<5-ring, 30
?-bracket, 509
?-independent partitions, 1221
? -almost everywhere (?-a.e.), 41
? -chained, 731
?-conegligible set, 38
?-integrable function, 56
?-integrable set, 39
?-locally, 514
?-measurable set, 39
?-measurable step functions, 41, 351
?-negligible sets, 38
?-null set, 262
?-topology, 708
?-uniformity, 934
?-continuous, 936
?-additive, 748
?-additive measure, 1134
?-additive product, 749
.^-measurable function, 35, 348
?-Borel measure, 1060
?-Borelset, 1058, 1081, 1082
? -algebra, 30
? -bounded, 1060, 1061
-set, 1058
? -complete MV-algebra, 873
?-homomorphism, 877
?-ideal, 798, 1096
(a-)order continuity, 716
?-order continuous, 922, 929
- operator, 819
- submeasure, 716
? -representation integral, 518
?-Riesz homomorphism, 799
?-ring, 30
?-subadditive function, 713, 921
?-submeasure, 713, 921
?-density topologies, 693-699
? ?-saturation, 1303
a.e. topology, 690
Aarnes theorem, 844
absolute value of a linear functional, 1058
absolutely
- p-summable sequence, 426
- continuous, 251, 285, 288, 997
- continuous ?[,-measure, 991
-continuous functional, 523
absorbing, 306
absorbing subset, 285
abstract
-Li'(l) spaces, 510
- Daniell-Loomis spaces, 526
- Fubini theorem, 521
- Lebesgue integral, 516
- Riemann integral
- improper, 513
- proper, 513
- Riemann^-integrable functions, 514
- Riemann-type approach, 520
AC, 1265
acceptance preorder relations, 1492
action, 87
action of a group, 1094
- m-free, 1095
- by measurable transformations, 1093
- free, 1094
- transitive, 1094
addition of observables, 886
additive
- measure, 1333
- set-valued function, 643
adic transformation, 1228
admissible density, 1138
admissible extension of a Daniell system, 510
admissible linear liftings, 1144
admissibly generated lifting, 1139
AF C*-algebra, 871
1587

1588
Subject Index
affine function, 413
Aleksandrov theorem, 79
algebra of sets, 30, 1526, 1533, 1537, 1541
-non-atomic, 1533
almost perfect, 808
almost strong
-density, 1149
-lifting, 1149
- lifting property, 1150
almost-invariant set, 1095
o?-almost perfect, 808
o?-perfect, 808
ambiguous set, 86
amenable group, 92
ample product system, 521
analytic space, 769
Andersen and Jessen theorem, 1319
aperiodic Markov shift, 1206
aperiodic transformation, 1227
approximate
- finitely additive Radon-Nikodym property, 275
- functional Radon-Nikodym, 524
- limit, 686
- Radon-Nikodym-Bochner theorem, 277
- tangent m -plane, 1019
- tangent vectors, 1018
approximately continuous, 686-690
- functions, 677
approximation property, 430
Archimedean
- Riesz space, 792
- t-norm, 951
area minimizing current, 1034
arithmetic progression, 1201
arithmetical mean of observables, 886
AS LP, 1150
associate space, 815
associativity, 1351
asymmetric integral, 1390
atom, 43, 628, 646, 729, 797, 920
atomic
- 7"L-measure, 998
- lattice, 729, 920
atomistic lattice, 729
atomless lattice, 729, 920
attractor, 1043, 1044
Aumann integral, 636
Aumann-Pettis integrable, 636
Aumann-Pettis integral, 636
auxiliary function, 1049
average density, 1045
axiom
- K, 535
- L,535
-Martin's, 1287
?-exceptional sets, 509
Baire
- a -algebra, 275
-lifting, 1139
- measure, 522, 748, 1060, 1061, 1071, 1134
-set, 1058, 1081-1084
- strong liftings, 757
Banach
- algebra, 407
-density, 1201
- function space, 813
- lattice, 406
- measure, 90
Banach-Schroder-Bemstein theorem, 90
Banach-Tarski paradox, 90
band, 798
Bartle-Dunford-Schwartz
- integral, 277, 280
-representation theorem, 1083
bases associated with a lifting, 206
basic
-assignment, 1546, 1547
- function, 1334
-probability mass assignment, 1385
- values of Choquet integral, 1336
- values of general integral, 1352
- values of Sugeno integral, 1340
basically disconnected space, 875
basis
-decomposable, 187
- pointwise character, 187
BCK-algebra, 856
belief, 1546, 1547
- function, 1477
- revision, 1505
Bellman operator, 1435
Bernoulli
- numbers, 1388
-shift, 1190, 1198, 1206, 1220
Besicovitch-Morse property, 228
Bessaga-Pelczynski theorem, 155
big-stepped probability, 1486
bimorphism, 898
bipartite graph, 97
bipolar, 282
Birkhoff ergodic theorem, 654
Biting Lemma, 153
Blackwell space, 779
block, 832
Blumenthal 0-1 law, 305

Subject Index
1589
Bochner
- integrable, 275
- integrable functions, 268, 273, 509, 524
-integral, 266, 351,354
- Lebesgue space, 806
- type derivative, 284
Boolean algebra, 873
- of projections, 573
Boolean partition of unity, 903
Boolean valued satisfaction, 1266
Borel
- ? -algebra, 31
-?-field, 268
- function, 35
-lifting, 1139
-measure, 519, 522, 1014, 1060, 1110, 1134
- Radon measure induced by a positive linear
functional, 1059, 1061, 1062
- regular, 1040
-set, 31, 1058
- space, 1095
bound, lower, 791
bound, upper, 791
boundary of current, 1029
bounded
-complex Radon measure, 1071
- linear functional in /CG")*, 1071
- linear lifting, 1149
- Radon operator, 1087
-set, 711
- variation, 269, 285
- order
- from above, 791
- from below, 791
--operator, 818
--set, 791
Bourbaki integral, 506
Bourgain property, 544
box dimension, 110
Brooks-Jewett theorem, 139
Brownian
- local time, 309
- motion, 1044, 1050, 1297, 1317
- natural stretched, 332
- normalized, 298
- - stretched, 332
-path, 1041
Burgers equation, 1440
Busemann-Feller bases, 205
C*-algebra, 407
e.g. ?-algebra, 767
c.l.d., 1139
- product measure, 72
Cafiero uniform exhaustivity theorem, 130
cancellation law, 99, 101
canonical extension, 368
- of additive measures, 370
Cantor
-diagonal argument, 1264
- function (Devil's Staircase), 50
-set, 50, 1039, 1043
- theorem, 1284
capacities, 1383
capacity
- approach, 517
- functional, 652
Caratheodory, 10.39
-construction, 1014
-extension, 257, 517
-extension theorem, 1303
cardinal
-compact, 1267
-hige, 1279
-Mahlo, 1285
- medium compact, 1266
- real valued compact, 1276
- real valued strongly compact, 1276
- rel valued large, 1276
-RV large, 1275
- strongly compact, 1266, 1275
- supercompact, 1267, 1275
- weakly compact, 1266
cardinal
- invariants, 1275
- norm,1267
cardinality
-function, 1.387
-jump, 1279
- operator, 1.387
cascade, 1199, 1224, 1226
Castaing representation, 631
categorical equivalence, 874
category measure, 11.35
Cauchy sequence, 624
Central Limit Theorem, 886
centre, 916
Chacon transformation, 1206
Chacon-Omstein theorem, 1195, 1197
chained, 728
- Гм-clan, 1006
- uniform space, 940
change of variables, 68
character of a measure space, 1126
characteristic function, 876
charge, 768
- extension property, 778

1590
Subject Index
Choquet
- integral, 252, 289, 1336, 1390, 1393-1395
- theory, 1314
classical representation, 1335, 1354
closed
-convex hull, 620, 621
- order, 796
- subset, 263
closure
- of a Daniell system, 509
- theorem, 1033
CMEA, 1283
co-Mobius transform, 1385, 1391
commonality function, 1385
commutativity, 1351
comonotone
- ®-additivity of general integral, 1353
- additivity of Choquet integral, 1337
-function, 1334
- maxitivity of Sugeno integral, 1341
compact, 1301
- charge, 770
- class, 769
- measure, 770
-Polish G-space, 1117
- range property, 554
compactness, 278
-theorem, 1033, 1266
comparative
- possibility distribution, 1478, 1493
-probability, 1473
compatible element, 832
complement of a fuzzy set, 954
complete
- algebra, 255
-ergodicity, 1206
- Dedekind, 793
--?-, 793
- super, 793
- measure, 515
- MV-algebra, 873
completeness, 261
completion
- of FN-topologies, 727
-of MV-algebra, 933
- regular, 748
- regular measure, 1134
complex Radon measure induced by a positive
linear functional, 1063
complex-valued function
- integrable, 59
- measurable, 59
composite observable, 885
compressible equivalence relation, 1122
conditional
- entropy, 1216
- entropy of a partition of unity, 902
- expectation, 67, 273, 523, 893
- information, 1216
- possibility, 1486
- possibility function, 1487
cone, 790, 836
- positive, 790
confidence
- function, 1475
-relation, 1472, 1473
congruence property, 1041
conjecture of J.R. Choksi, 747
conjugate set function, 1383
conservative ergodic, 1197
consistent
-density, 1159
-lifting, 1158
content, 33
continued fractions, 1044
continuity
- of Choquet integral, 1338
- of general integral, 1353
- of Sugeno integral, 1341
continuous
- measures, 263
- operator, order, 819
--?-,819
- projective system, 1163
continuum displacement, 1286
controlled convergence theorem, 593
convergence
-a.e., 888
- a.e., in an MV-algebra, 888
- in distribution, 652, 888
- in distribution, in an MV-algebra, 888
- in measure, 71, 888
- in measure, in an MV-algebra, 888
- in the mean, 57, 352
- in the mean of order p, 354
- order, 796
- theorems, 390
convex measures, 995
convexification property, 653
convolution
- of functions, 77
- of measures, 81
- semiring, 1419
coordinate
- random variable, 885
- random vector, 885
corresponding ?-group with unit of an
MV-algebra, 875

Subject Index
1591
Corson space, 534
countable
- chain condition, 1101
- convexity, 58, 352
- equivalence relation, 1122
countably
- 7"-additive function on a ?-tribe, 959
- equidecomposable sets, 117, 1096
- paradoxical set, 117, 1096, 1103, 1104, 1115,
1117,1122
- supported, 634, 637
counting measure, 36, 258, 291
Cousin's lemma, 235, 589
covariance function, 661
cover, 1040
covering relation, 185
crisp set, 953
critical point, 1266
CRP, 554
crude
- inner envelope, 517
- outer envelope, 517
cumulative hierarchy, 1265
current, 1029
cutting-and-stacking construction, 1206, 1227
cylinder, 885
d-bounded set, 172
D-poset
- complete, 172
- quasi-?-complete, 172
Daniell
- integral, 506
- lattices, 506
- lower integral, 507
- mean, 508
- system, 506
- upper integral, 507
Daniell-Loomis integrals, 524
Daniell-Stone theorem, 514
Darboux
- lower sum, 60
- property, 993
- sum, 20
- upper sum, 60
De La Vallee Poussin theorem, 243
decisivity, 1553
decomposability, 1392
decomposable, 638
decomposition, 292
-(D), 1139
-(ND), 1139
- of a measure space, 1138
-Jordan, 818
Dedekind
- complete, 793
--?-, 793
- - super, 793
- vector lattice, 513
definable, 1265
defining sequence, 276, 286
degree of membership, 953
?j-condition, 812
Dempster-Shafer theory, 1547
densities, 1017
density
- function, 881
- point, 677-686
- property for ?, 212
- respecting coordinates, 1159
- topology, 677, 682-697, 1147
deniable, 270, 272
denting point, 270
dependent choice, 1264
derivation, 1538, 1542
derivation basis, 187
derivative, 1302
deterministic possibility distributions, 1479
deviation, 1528
Diagonal Theorem, 162
- Antosik's, 161
- Mikusinski-Antosik-Pap's, 161
diameter, 1040
- based packing measure, 1042
difference poset (D-poset), 171, 834
differentiable form, 1028
differentially equivalent, 202
differentiation
- basis, 597-599
- theory of Daniell integrals, 523
diffusion, 305
- canonical form, 321
- regular, 306
dimension decomposition, 1048
Diophantine approximation, 1044
direct
-density, 1145
-lifting, 1145
- sum property, 795
directed
- downwards, 791
-set, 791
- upwards, 791
discrepancy, 112
discrete
- measure, 36, 768
- spectrum, 1214, 1221

1592
Subject Index
disintegrable, 782
disintegration, 782
disjoint, 799
- complement, 799
- measure-preserving systems, 1230
dispersion
- of an observable, 882
- point, 678-686
distance function, 620
distribution, 1248
- of a measurable multifunction, 652
- of a multifunction, 648
distributivity under subtraction, 891
divergence of degree a, 1538
dividends, 1385
divisible
- MV-algebra, 878
- probability MV-algebra, 880
division, 609
Dixmier's condition, 95
Doleans
- function, 480
- measure, 397
domain, 629
dominance relation, 978
dominated
-convergence theorem, 812
- ergodic theorem, 1196
Doob-Meyer decomposition, 397
Dorofeev-Sherstnev theorem, 844
Downward Transfer, 1300
downwards, directed, 791
drastic product, 951
dual
-basis, 190
- nuclear space, 283, 284
-order, 818
-set function, 1383
- space, 270
duality, 282
- space, 777
Dunford integral, 538
Dunford-Schwartz integral, 514
dynamical system, 1044, 1187
Dynkin system, 32
? -ergodic measure, 1123
? -invariant set, 1122
effect algebra, 834
Effros
-?-field, 629, 633, 648, 649
- measurable, 629, 631
Egorov theorem, 71, 790
elementary
- embeddings, 1266
- integral system, 508
Elliott's partial addition, 872
embedding, 626
entire topology, 804
entropy, 1188, 1214, 1224
-convergence rates, 1223
- function, 902
- fuzzy, 1551
-Hartley, 1527, 1548, 1549
- of a partition of unity, 902
-of degree a, 1527, 1538
-of degree (?. ?), 1528, 1542
-of order a of type (?, у), 1560
-Shannon, 1527, 1535, 1551, 1561
-total, 1554
- weighted, 1554
epigraphical
- convergence, 663
- multifunction, 664
equally weighted independent random variables,
1322
equiconsistency, 1268, 1274
equicontinuous norm, 811
equidecomposable sets, 86
equidistribution theorem, 1198
equimeasurable
- function, 807
- set of functions, 798
equivalent projections, 871
ergodic, 1196
- Hubert transform, 1213
-hypothesis, 1187, 1188
- Szemeredi theorem, 1201
- theorem, 1193
- theory, 1187
ergodicity, 1189, 1196
essential
- boundary, 1026
- integration, 512, 513
- measures, 256
- Radon integrals, 513
- supremum, 43
- upper functional, 513
- variation, 595
essentially i'-integrable functions, 513
evanescent set, 386
exact dimension, 1048
excess, 621
exhaustion, 524
- theorem, 798
exhaustive, 709, 922, 935
expectation of an observable, 882

Subject Index
1593
expected information, 1530, 1556
exponentially bounded group, 88
exposed points, 263
extension
- of ? -additive measure, 369
- of an invariant measure, 1107
- of positive measures, 368
- property, 778
external, 1299
F(S), 848
factor map, 1230
factorization, 783
faithful
- measure, 900
- state, 880
Fatou property, 805
Fatou's lemma, 55, 58, 373, 507
Feller's theorem, 333
filtration, 386
finite
- measure property, 35, 795
- subset property, 795
- variation, 268
finitely
- 7"-additive function, 958
- additive measure, 253, 260-263, 285, 514, 837,
900
firefly in a box, 834
first category, 875
Fischer, Riesz-property, 811
fixed point, 1100, 1118
flat norm, 1030
Fleissner theorem, 1282
FN-topology, 706
focal element, 1546
Fourier transform, 77
- exchange formula, 78
- inverse, 78
fractal, 1039
frame function, 842
Frank t-norms, 952
Fubini theorem, 74
fuge, 1287
full
- integral, 512
- tribe, 876
function
-АССд, 598
-AC*. ACG*. 592
- ACS,ACGS, 592
- absolutely continuous, 64
-additive, 1531, 1537
- almost bounded, 1272
- Borel measurable, 48
-continuum displacement, 1286
- convex. 67
- essentially bounded, 69
- incompresible, 1272
- Lebesgue integrable, 59
- Lebesgue measurable, 48
-logarithmic, 1531, 1537
- lower density, 1149
-multiplicative, 1531, 1536, 1537
-nowhere bounded, 1272
- of an observable, 885
- upper density, 1149
function-lifting, 1140
functional, 313
-determined by a complex measure, 1075, 1077
fundamental equation
- 11M with the branching property, 1533, 1534
-of recursive 11M, 1537
-of weighted additive 11M, 1540, 1541
fundamental solution, 1251
- of the Cauchy-Riemann operator, 1252
- of the heat equation, 1254
- of the Laplace operator, 1254
- of the one-dimensional wave equation, 1253
- of the two-dimensional wave equation, 1253
fuzzy
- integration, 252
- measure, 288, 290, 1333, 1383, 1546
- measure space, 288, 1333
-set, 517, 953, 1550
- subset, 953
G-? -bounded set, 1100
G-absolutely negligible set, 1124
G-atom, 1097
G-bounded set, 1096
G-equidecomposable sets, 88
G -invariant a -algebra, 1093
C/-invariant (linear) lifting, 1157
G-negligible set, 10%
G-partition, 1108
G-space, 1093, 1094
game, 1383
- with value, 1435
gauge, 589
- essential, 595, 606
- function /i, 1041

1594
Subject Index
Gauss
- measure, 1191
- Green theorem, 1027
Gaussian
- multifunctions, 660
- system, 1191
GCH, 1265
Gel' fand
- derivatives, 278
- integrable, 277
- integral, 277, 538
Gel'fand-type derivative, 277
general
- duality theorem, 777
- fuzzy integral, 1352
generalized
- Bartle-Dunford-Schwartz representation
theorem, 1083
- derivative
- distributional, 1249
- in the Sobolev sense, 1241
- dynamical system, 902
- product, 898
generated
- (r-)tribe, 956
- ?-measure, 977
- tribe, 876
generator, 903
generic
- extensions, 1266
- ultrafilter, 1266
geometric measure theory, 1044
Gleason theorem, 841
graph, 629
graph-measurable, 630
graphoid structure, 1501
Greechie logic, 832
Greenleaf's problem, 107
Gribanov, Luxemburg theorem, 795, 817
Grothendieck space, 431
group
- of Borel automorphisms, 1122
- of homeomorphisms, 1110, 1121
- of isometries, 1110, 1121
- of permutations, 1094
group-valued measure, 882
Holder
- exponent, 1047
-inequality, 71,356, 815
H6rmander formula, 622
Haar measure, 1093, 1106, 1110, 1112, 1113
- existence of, 1111
Haar null set, 1114
Hahn decomposition, 40, 252, 254, 262, 523
halo properties, 225
harmonic measure, 1314
Hashimoto topology, 699
Hausdorff
-dimension, 1041, 1042, 1046, 1412
- distance, 621, 655
- measure, 255, 1015, 1039, 1042, 1046, 1050
- outer measure, 1040
- type, 1050
Heaviside's function, 1240
helical transform, 1213
Henstock's lemma, 591, 608
Hewitt-Yosida theorem, 153
Hubert space, 269, 871, 1192
HOD, 1265
HODR, 1265
homogeneity
- of Choquet integral, 1337
- of Sugeno integral, 1340
homogeneous G-space, 1120
homomorphism, Riesz, 799
- normal, 799
-?-, 799
Hopf-Kawada theorem, 1097, 1099, 1100
horizontal
- ®-additivity of general integral, 1353
- additive decomposition, 1338
- additivity of Choquet integral, 1338
- integral, 517
- maxitivity of Sugeno integral, 1341
Hudetz entropy, 904
hull-kemel topology, 873
hyper-stonian space, 751, 1135
hyperfmite set, 1302
/ continuous
-at oc, 518
-at 0,518
/-integrable function, 510
/-measurable function, 510
/-summable function, 507, 514
ideal, 798, 800
- null, 820
-?-, 798
- of an MV-algebra, 873
- space, 806
idempotent
- integral, 1423
- semimodule, 1408
- on pseudo-linear functionals, 1422
image measure, 47
imaginary part of ?, 1058

Subject Index
1595
improper integrable functions, 526
inaccuracy, 1528, 1535, 1538
indecomposable integral current, 1032
indefinite integral, 251, 359
independence, 883
- space, 779
independent
- complement, 780
- multifunctions, 648
- observables, 883
- sequence of observables, 884
indicator, 876
- function, 623
induced transformations, 1229
inductive limit locally convex topology, 1057
inequality, Holder, 815
infimum, 791
infinitary
- distributivity law, 873
- languages, 1266
infinite product measure, 885
infinitesimal, 1300
- numbers, 1298
information, 1187
- function, 1215
- improvement, 1528, 1538
- measure, 1429
--additive, 1535, 1544, 1548, 1549
- - branching, 1532, 1534, 1540
- expansible, 1544, 1549
--inset, 1527
- non-probabilistic, 1529
- on closed domain, 1528
- on open domain, 1528
- - polynomially additive, 1559
--probabilistic, 1528
--recursive of degree a, 1535, 1541, 1555, 1558
--recursive of multiplicative type, 1535, 1536
--subadditive, 1543, 1549
- - sum form, 1533, 1536, 1540, 1541
--symmetric, 1532-1534, 1536, 1545, 1549,
1551,1556
--weighted additive, 1535, 1536
injective tensor norm, 411
inner
- product, 1531
- product space, 848
- regular, 519
instantaneous, 305
integrable, 287
- function, 251
- multifunction, 635, 657
- observable, 882
integrably bounded multifunction, 635, 654, 655
integral, 286, 375, 819, 1302, 1310
- C, 596
- M|,604
- ?-regular Henstock-Kurzweil, 601
- BV, 606
- F, 606
- GP, 603
- PU, 605
- i>, 607
- p, 602
- g,606
- approximate, 597
- approximate symmetric, 599
- bilinear form, 412
-Bochner, 351, 354
- current, 1031
- dyadic, 597
- flat chain, 1032
- Henstock-Kurzweil, 590, 600, 608, 609
- indefinite, 66, 359
- induced by measure, 515
- McShane, 596
- measure, 251, 252
- metric, 525
- Nonnenmacher, 604
- normal, 819
- norms, 524
- of step function, 349
- Pfeffer, 604
- regular Henstock-Kurzweil, 602
- repeated, 73
-representation, 514
- stochastic, 390, 461
- symmetric, 598
- with respect to a finitely additive measure, 93
integralgeometric measure, 1016
integration
- lattice, 524
- with respect to a measure with finite variation,
357
interaction
-index, 1385, 1386, 1388, 1389, 1393, 1394
-operator, 1387
-transform, 1385, 1390, 1391, 1393
internal, 1299
-description, 1299
intersection of fuzzy sets, 953
interval
- effect algebra, 835
-exchange, 1192, 1206, 1221, 1228
-MV-algebra, 891
invariant
- mean, 95

15%
Subject Index
- measure, 900
- measure extension theorem, 92
- set, 1094
inverse
- Bernoulli operator, 1386
-density, 1145
- lifting, 1145
irrational rotation, 1198
isomorphism
-invariant, 1189, 1218
- metric, 1188
- spectral, 1189
isoperimetric
- inequality, 1027
- theorem, 1032
iterated
- function system (IFS), 1043
- (local) integral norms, 522
- upper and lower integrals, 521
Jacobian, 69
Jensen's inequality, 67
joining, 1230
joint observable, 883
Jordan
-decomposition, 41, 265, 818, 1058
- upper norm,525
- m-integrable functions space, 525
Julia sets, 1044
?-additive measures, 1392
К -automorphism, 1209, 1220
К -tight, 724
(K) property, 534
К weak precompactness, 545
Keisler's Fubini theorem, 1323, 1325
kernel, 953
Kolmogorov
-complexity, 1225
- construction, 884
- probability space, 885
Kolmogorov-Sinai entropy, 903
Kolmogorov-Sinai theorem, 903, 1219
Komlos theorem, 657
Kothe space, 813
Krieger generator theorem, 1200, 1220
«-group, 853, 874
LP -spaces, 523
Laplace transform, 79
large deviation principle, 1433
- weak, 1433
large set, 1103
lattice, 791
- norm,806
--pseudo-, 801
- - quasi-, 806
lattice-order of an MV-algebra, 873
law of large numbers, 575
Lebesgue
-completion, 1059, 1063, 1066, 1073-1075, 1077
- completions, 1076
- density, 1136
- density theorem, 63, 678, 692
- dominated convergence theorem, 23, 58, 507
-integral, 23,59, 1313
- measurable set, 48
- measure, 23, 251, 258, 268, 291, 769, 1093,
1094, 1102, 1105-1107, 1123, 1124, 1314
- measure on ?'", 48
- outer measure, 48
- PIP, 559
-set, 182
- spectrum, 1206
- theorem, 355, 376
Lebesgue-Bochner space, 806
Lebesgue-Radon completion, 1089
Lebesgue-Stieltjes integral, 379
left unit, 1349
Legendre transform, 1050
lifting, 756, 1135, 1304
- compact map, 1170
- respecting coordinates, 757
- theory, 524
- topology, 1147
limited, 1299, 1300
linear
- differentiation, 259
- extensions of set functions, 508
-lifting, 1142
- lifting respecting coordinates, 1159
- possibility distributions, 1479
Lipschitz
- mapping, 1040
- neighbourhood retract, 1032
local
- convergence in measure, 514
- dimension, 1047
- integral metric, 524, 525
- null set, 795
localizability, 523
localizable, 794
- measure. 1134
-uppergauge, 524
localization. 1529, 1552
localized Lebesgue-Radon completion, 1089

Subject Index
locally
- commutative action, 103
-determined measure, 1134
- finite graph, 97
- integrable function, 522
- solid topology, 800
- - entire, 804
Loeb
- measure, 1304
- measure space, 1304
Loomis system, 514
Loomis-Sikorski theorem, 877
Lorentz space, 807, 810
Lorentz-Luxemburg theorem, 816
lottery paradox, 1495
Lovasz extension, 1384, 1390
lower
-bound, 791
- density, 682, 692, 1136
- variation, 984
lower-upper aproximant, 518
Lukasiewicz
- t-conorm, 951
- t-norm, 951
Lusin theorem, 43, 81
Luxemburg
- Lorentz theorem, 816
-norm, 807
Luxemburg-Gribanov theorem, 795, 817
Lyapunov convexity theorem, 637
.M(M). 873
ш-almost everywhere convergence, 896
m-almost everywhere property, 372
m-measurable function, 372
m-negligible set, 372
m-set, 1023
OT-complete homomorphism. 1266
Mobius
- operator, 1386
-representation, 1385
-transform, 1384-1387, 1389-1392
Maharam measure, 257
Marcinkiewicz space, 807
Marczewski's problem, 108
marginal measure, 773
marginal problem, 775
Markov
-kernel, 961
- property
- - strong, 300
-shift, 1190, 1198, 1206, 1220
- time, 300, 305
Martin boundary theory, 1314
martingale, 273, 396
-convergence theorem, 894, 1319
- square integrable, 397
mass of current, 1029
maximal
- ergodic theorem, 1194
- function, 1194
- ideal, 873
-inequality, 1195, 1320
maximality, 1551
maxitive measure, 1334, 1411
-completely, 1411
Mazur property, 534
meager set, 875
mean ergodic theorem, 1192
mean functional, 665
mean variation, 480
measurable, 251
-cylinders, 75
- function, 789
- Implicit Function Theorem, 633
- multifunction, 629
- network, 1150
-partition, 1190, 1214
- rectangle, 1323
- space, 34
-sets, 36,515, 1095
-space, 34, 1095
measure, 36, 251, 916
- G-ergodic, 1095
- absolutely continuous, 65
- additivity, 1267
-algebra, 255, 1269
- associated to a function, 382
- atomless, 43
- Cantor, 80
- continuous, 839
- countable-cocountable, 44
-diffuse, 1048
-diffused, 1095
- Doleans, 397
- dominated, 534, 839
- finitely additive, 900
- image, 47
- indefinite-integral, 66
- induced by integrals, 514
- inner regular for the compact sets, 80
-invariant, 1093, 1095
- invariant on a Boolean algebra, 1102
- localizable (Maharam), 43
- locally determined, 43
- locally finite, 80
- nonatomic, 1110

1598
Subject Index
- normal, 1274
- normalized, 789
-of fuzziness, 1550, 1551, 1553
- on BCK-algebra, 858
- on projections, 901
-onX, 1095
- optional, 394
- outer, 38
- outer regular, 81
- predictable, 394
- product, 883
- purely atomic, 43
-quasi-invariant, 1095
- Radon, 80
-regular, 79, 1267
- self-affine, 1050
-self-similar, 1049, 1050
- semi-finite, 43, 1107, 1114, 1127
- semiregular, 1107
- singular, 65, 839
- space, 36, 515
--complete, 38
- completion, 39
- Stieltjes, 46
- stochastic, 394
- strictly localizable (decomposable), 43
- topological, 80
- totally finite, 36
- truly continuous, 65
- uniform real-valued, 1275
- uniformly ? -additive, 355
- universal on a set, 1102
- usual, 76
-vector-valued, 1050
measure-preserving
-flow, 1187, 1196
-system, 1187
- transformation, 897, 903, 1093, 1187
measure-theoretic
-boundary, 1026
- exterior normal, 1026
membership function, 953
metric approximation property, 430
metric outer measure, 1040
Mikusinski's
- differential operator, 1256
-integral operator, 1256
- operator field, 1255
mild mixing, 1208
minimal homeomorphism, 1199
minimal self-joinings, 1231
minimal surface, 1034
minimal unbounded function, 1272
Minkowski
- addition, 620, 623, 625
- functional, 282
mixed norm space, 817
mixing, 1189
mixture problem, 781
model, 1265
modular
- function, 1112
-space, 813
moments of an observable, 882
monocompact
- class, 772
- measure, 773
monoid, 99
monotone
-class, 30, 31
-convergence theorem, 58, 373, 507
-differentiation, 259
- function on a ?-tribe, 959
- sequence, 876
monotonically irreducible ?-measure, 977
monotonicity
- of Choquet integral, 1336
- of general integral, 1352
- of Sugeno integral, 1340
monotonous lifting, 1140, 1149
monotony, 1274
morphism of unital i-groups, 874
Mosco convergence, 656
multifractal, 1049
-measure, 1049, 1051
-spectrum, 1049-1051
multifunction, 285, 622, 629
multilinear extension, 1384
multimeasure, 286, 643
multiple recurrence, 1201
multiplicative MV-algebra, 898
multiset, 905
Murray-von Neumann order, 871
mutual independence, 1327
MV-algebra, 853,872,913
- ? -complete, 873
- boolean, 873
- complete, 873
- divisible, 878
- faithful state, 880
- ideal, 873
-interval, 891
- multiplicative, 898
-observable, 881
- product, 891
- semisimple, 873
- spectral topology, 873

Subject Index
1599
- state, 879
- weakly ? -distributive, 875
- with product, 891
*N,c, 1299
narrow convergence, 647
natural
- scale, 306
- time, 341
necessity
- measures, 1476
- relation, 1474
net bases, 206
Nikodym
-convergence theorem, 132
- Radon theorem, 821
NMSC, 1281-1283
non-additive integration, 512
non-continuity integration, 513
- process, 525
non-dogmatic possibility distributions, 1479
non-lattice integration, 511
non-singular transformation, 1095
nonatomic, 646
- 7"L-measure, 998
- measure, 768
- part, 628
- vector r^-measure, 998
nondogmatic possibility relations, 1492
nonmeasurable set, 1094, 1102, 1103. 1105-11
1123, 1124
nonperfect measure, 771
nonstandard
- extension, 1299
-natural numbers, 1299
-real numbers, 1299
norm,706,1267
- equicontinuous, 811
- lattice, 806
- Luxemburg, 807
- pseudo-, 801
--lattice, 801
- quasi-, 806
- - lattice, 806
normal
-current, 1030
-ideal, 1272
-integral, 819
- Riesz homomorphism, 799
- singular operator, 819
normalization, 1557, 1560
normalized measure, 789
norming
- space, 365, 381
- subspace, 348
nuclear
- map, 283
- spaces, 282, 283
null
-function, 510, 1307
- ideal, 820
- set, local, 795
numerical index of contraction, 203
o-topology, 800
observable, 881
- addition, 886
- arithmetical mean, 886
- composite, 885
- dispersion, 882
- expectation, 882
- independence, 883
- independent in probability MV-algebra, 883
- independent sequence, 884
- integrable, 882
-joint, 883
- mean, 882
- moment, 882
- product, 886
- square integrable, 882
- variance, 882
- zero, 896
OD, 1265
OML, 831
OMP, 831
open set condition, 1043
operator
- absolutely p-summing, 419
- order continuous, 819
--?-,819
- positive, 818
- regular, 818
-singular, 819
- unconditionally converging, 423
optimistic hybrid utility function, 1465
optional
- ?-algebra, 386
- measure, 394
- process, 386
- projection, 393
- stochastic measure, 393
orbit, 1094
-equivalence relation, 1122
- of a state, 1188
order
- bounded
- from above, 791

1600
Subject Index
- from below, 791
- - operator, 818
- set, 791
- closed, 7%
- continuous, 922, 929, 935
-continuous operator, 819
--?-,819
- convergence, 796
-dual, 818
- separable, 792
- topology, 7%
order-complete, 287
order-continuous systems, 506
order-determining system, 835
ordered
-semigroup, 836
- weighted average, 1395
ordinal, 1264
-interaction transform, 1398
- invariance, 1516
- Mobius transform, 1397
-Shapley value, 1398
ordinary density point, 691
Orlicz space, 807
Orlicz-Pettis theorem, 146
orthoalgebra, 833
orthogonal lifting, 1140
orthogonally closed subspace, 848
orthomodular
- lattice, 831
- poset, 831
outer measure, 38, 255
packing, 1042
-dimension, 1042, 1046
-measure, 1041, 1042
Painleve-Kuratowski
- convergence, 654, 655
- lower limit, 655
- upper limit, 655
pairwise independence, 1327
paradox, 85
- of infinity, 85
paradoxical set, 87
part, regular, 811
partially ordered vector space, 790
partition, 589
- a-fine, 589
- McShane, 5%
- partial, 589
-of unity, 901
perfect
- almost, 808
--a-, 808
-a-, 808
- matching, 97
- measure, 771
- probability space, 771
- semi-, 808
perimeter, 1025, 1026
Pettis
- derivative, 274, 278
- integrable, 274
- integral, 274, 538
- integral property, 558
Pietsch domination theorem, 420
PIP, 558
Plateau problem, 1027
plausibility, 1547
- function, 1477
plausible inference, 1486, 1488
PMEA, 1281, 1282
points of density, 678
Poisson process, 1316
Polish
- G-space, 1115
-group, 1110
-space, 663, 1110
polynomial recurrence, 1202
positive
- cone, 790
-contraction, 1195
- linear functional, 1058, 1059, 1062, 1064, 1066,
1071, 1074, 1084
- operator, 818
possibilistic
- event independence, 1502
-independence, 1499
-integrals, 1507, 1510
-likelihood, 1484, 1485
- likelihood relation, 1516
possibility
-distribution, 1478, 1481, 1488
-measure, 1411, 1476
-relation, 1474
pre-density, 1140
predictable
-a-algebra, 386
- measure, 394
- projection, 393
preferential extension, 1493
preideal space, 806
pretopological measure space, 1134
primitive product measure, 72
principle
-Vopenka, 1266, 1268, 1284

Subject Index
1601
probabilistic
-independence, 14%
- metric space, 1424
probability, 33
- ?-measure, 959
- distribution, 883
- measure, 517
- MV-algebra, 880
- MV-algebra with product, 892
- space, 1187
problem
-of coefficients, 599
- of primitives, 589, 5%
process, 387
- p-summable, 388
- adapted, 386
-cadlag, 452
-evanescent, 387
- indistinguishable, 387
- locally p-summable, 391
- modification, 387
- with finite variation, 392
product
-density, 1158
-joint refinement, 901
- lifting, 758, 1158
- linear lifting, 1158
-measure, 75,521,748, 883
- of observables, 886
- system, 521
-transformation, 1221, 1226
projection
- equivalent, 871
- property, 778
- theorem, 632
projective
- limit of a self-consistent family of densities
(liftings), 1164
-tensor norm, 411
prolongable, 1088, 1089
- Radon operator, 1087-1089
- Radon vector measure, 1089
proper ideal in MV-algebra, 873
proper Riemann ?-integrable functions, 514
properly measurable function, 549
property
- Baire, 1287
- direct sum, 795
- Fatou, 805
- finite measure, 795
- finite subset, 795
- Riesz-Fischer, 811
- V, 423
primings, 186
pseudo-
- addition, 1343
- Boolean functions, 1383-1384
- convolution
- of the first type, 1415
- of the second type, 1415
-difference, 1345
-multiplication, 1346
- MV-algebra, 860
-norm, 706, 801
--Riesz. 801
pure 7"; -measure, 976
purely
- atomic part. 628
- finitely additive measure, 736
- ???-?-order continuous, 926, 938
- unrectifiable set, 1017
qualitative conditioning, 1487
quantum logic, 831
quasi-
- compact measure, 771
- complete space, 284
- dyadic, 750
- norm, 806
- Radon measure space, 1135
- singular, 291
quasimartingale. 480
^-regular. 1062
-complex measure, 1060
- measure, 1060
^-regularity, 1085
r topology, 690
r-dimensional volume
- of half open interval, 47
radical, 854
radius of a subset, 653
Radon
- integral, 520
-measure, 508, 519, 520, 1014, 1314
-operator, 1086-1089
- probability space, 1289
-regular, 1060-1062, 1064, 1066, 1068-1071,
1073-1076
- regularity, 1073
- vector measure, 1087-1089
Radon-Nikodym
-derivative, 66, 251, 257, 258, 268
- property, 252, 269, 272, 362
-theorem, 361,522, 647, 821
random
- ergodic theorem, 1212

1602
Subject Index
- sets, 623
- vector, 883
- walk
- natural stretched, 336
- stretched, 336
range, 263, 282
rank one transformation, 1198
rate function, 1433
rational extension, 1490, 1493
real
-form, 801
- part of a linear functional, 1058
real-valued-measurable cardinal, 1103
rearrangement invariant, 807
rectifiable, 1044, 1045
- current, 1031
- measure, 1020
-set, 1017
recurrence, 1188, 1200
reduced boundary, 1026
refined possibility relations, 1483
refinements of partitions, 1215
reflection principle, 1317
reflexive, 272
region
- starshaped, 1244
regular
-conditional probability, 778
- distribution, 641
- linear functional, 520
- measure, 519
- operator, 818
-part, 811
- space, 811
- state (charge), 852
relative cancellation property, 857
relatively compact, 274
relatively dense set, 1200
Rellich's lemma, 1244
representation by Radon integrals, 520
representing measure, 1088, 1316
-of «, 1084
residuated difference, 1396
residuation in Elliott's addition, 872
resolution, 1551
return-times theorem, 1213
rich multimeasure, 645
Riemann
-integral, 19,60
-sums, 589
- functionally small, 592
- locally small, 592
Riemann-integrable functions space, 525
Riemann-Liouville operators, 1256
Riemann-Stieltjes integral, 508
Riemann-type integral, 287
Riesz
- homomorphism, 799
--a-, 799
- normal, 799
- operations, 791
- pseudo-norm, 801
-space, 791
- Archimedean, 792
- of measurable functions, 792
- space (vector lattice), 69, 282, 287, 406
-representation theorem, 519, 1306, 1313
- theorem, 523, 790
Riesz-Dedekind complete, 69
Riesz-Fischer property, 811
Riesz-Herglotz theorem, 1314
right
- distributivity, 1351
-unit, 1348
ring of sets, 30, 1526, 1533, 1540
Rokhlin tower, 1229
Rosenthal's lemma, 150
Rudin-Keisler order, 1279
Rybakov control, 279, 282, 284
?-bounded measure, 279
?-dimensional Hausdorff (outer) measure, 1040
?-dimensional packing (outer) measure, 1042
5-integrability, 1304
5-integrable, 1305, 1310
-lifting, 1311
5-measure. 1408
saturated measure, 515
saturation, 1271
scalar
- integrability, 538
- measurability. 535
scalarly
- concentrated measure, 556
- measurable, 634
- uniformly absolutely continuous, 281
scaling property, 1041
SDP-property, 1083, 1085
segment property
- of a region, 1244
selection, 631,645, 648
selector, 1103
self-affine functions, 1044
self-affine set, 1044
self-conformal set, 1044
self-consistent family
-of densities, 1163

Subject Index
1603
-of liftings, 1163
-of linear liftings, 1164
self-dual bases, 207
self-joining, 1231
self-similar set, 1043, 1044
selfsupporting measure, 545
semi-finite measure, 256, 1134
semi-perfect, 808
semiconvex vector 7"L-measure, 993
semigenerated G"-)tribe, 957
semimartingale, 398
semimodule
-dual, 1422
seminorm, 706
semiring, 30, 1407
semisimple MV-algebra, 873
semivariation, 706
-of a function, 380
- of a vector measure, 364
separable, 270
-a-algebra, 768
- dual, 273, 278
- dual spaces, 274
- order, 792
separably valued function, 267
separately additive density, 1160
separating subalgebra, 874
sequence
- asymptotically- ACG*, 595
- control convergent, 594
- entropy, 1223
- equi-convergent, 594, 603
- of set functions
- uniformly absolutely continuous, 133
- uniformly ACG*. 594
- uniformly ACG$, 594
set
- a, -definable, 1273
-constructible, 1265
set function
-?-additive, 33
-?-finite, 34
- ?-triangular, 157
- continuous from above, 33
- absolutely continuous, 133
-additive, 33, 1530, 1541
- continuous from below, 33
-exhaustive, 129
-exhaustive (s-bounded), 34
- extended real valued, 32
- (finite) real-valued, 32
-finitely additive, 33
- measure, 33
- monotone, 32
- nonnegative, 32
- normalized, 32
- positive, 32
-purely finitely additive, 152
- subadditive, 32
- submodular (concave), 32
- superadditive, 32
- supermodular (convex), 32
- totally monotone, 33
- triangular, 157
set-valued
- central limit theorem, 662
- conditional expectation, 639
- expectation, 623
- Fatou Lemma, 663
- integral, 623
- martingale, 658
- measures. 643
- strong law of large numbers, 652
sets of finite perimeter, 1025
Shannon function, 1552
Shannon-McMillan-Breiman theorem, 1222
Shapley value, 1385, 1394
Shapley-Folkmann inequality, 653
sharpness, 1551
Sierpinski sets, 770
signed measures, 265
simple
- function, 1334
- multifunction, 623-625
singular, 291,723
- 7"i,-mea.sure, 992
- element, 836
- operator, 819
- normal, 819
Sipos integral, 1390-1395
skew product, 1227
slice convergence, 656
small set, 1103
smooth equivalence relation, 1123
Sobolev
- lemma, 1244
-space, 1242, 1243
solid, 798, 800
- locally topology, 800
- - entire, 804
- set. 706
space
- associate, 815
- Banach function, 813
- ideal, 806
-Kothe, 813
- Lebesgue-Bochner, 806

1604
Subject Index
-Lorentz, 807, 810
- Marcinkiewicz, 807
- mixed norm, 817
- modular, 813
- normal Moore, 1281
- Orlicz, 807
- partially ordered vector, 790
- preideal, 806
- product measure, 1269
- regular, 811
-Riesz, 791
- Archimedean, 792
- of measurable functions, 792
- symmetric, 807
spectral
-invariant, 1189
- topology, 873
speed measure, 307
splicing, 774
splitting subspace, 848
square integrable
- martingale, 397
- observable, 882
stabilizer, 1109
stable family of functions, 548
standard, 1299
- Borel G -space, 1114
- Borel group, 1113
- Borel space, 768, 1110
- measure space, 780
- part, 1300
- part map, 1312
- part mapping, 1316
state, 832, 854, 863
- in an MV-algebra, 879
-in Hilbert space, 871
- on a tribe, 876
-space, 1188
state-morphism, 855
stationary random process, 1189
step
- function, 34, 348
-representation, 1335
Stieltjes integral, 383
stochastic
-integral, 390, 392, 461
- locally p-summable, 392
- interval, 386
-matrix, 1191
- measure, 394
- process, 387
Stone
-extension, 275
- representation theorem, 1080
- space, 275
Stone-Weierstrass theorem, 417
Stonian, 505
- Daniell system, 515
- lattice cone, 518
stopping time, 386
-predictable, 386
Stratonovich stochastic differential, 1446
strict t-norm, 951
strictly
- localizable, 259
- localizable measure space, 1138
- positive state, 880
strong
- approximate continuity, 693
- Daniell lattices, 506
- density, 692, 1147
- dual space, 282
- duality space, 777
- extension of measure, 1126
- law of large numbers, 889
-lifting, 1147
- lifting property, 756
- Lusin condition, 593
- mixing, 1203
- mixing of higher order, 1209
- unit, 835, 874
- Vitali assumptions, 198
strongly
- Blackwell space, 779
- comparable measures, 257
- equivalent measures, 1102
- integrable, 286
- lifting compact map, 1170
- lifting compact topological space, 1170
- measurable, 267, 626, 654
- measurable multifunction, 623, 625, 629
subadditive, 706
-ergodic theorem, 1195
- function, 921
- law, 902
- measures, 512
- sequence, 1195
submeasure, 706
subnormal possibility distributions, 1479
Sugeno integral, 282, 288, 1340, 1396-1400, 1513
summability
- of processes with integrable semivariation, 395
-of the stochastic integral, 391
super Dedekind complete, 793
superadditive set function, 357
superatomic Boolean algebra, 730
superstructure, 1297

Subject Index
1605
support, 953
- function, 620
- of a continuous function, 1239
-of a function, 792
-of a set, 792, 793
- of a topology, 805
supramenable group, 89
supremum, 791
Suslin topological space, 633
symmetric
- difference, 1397
-integral, 1390
-maximum, 1396-1397
-minimum, 1396-1397
- space, 807
- Sugeno integral, 1399
Szemeredi's theorem, 1201
?-clan, 956
?-disjoint, 954
?-measure, 959
?-partition, 954
Г-tribe, 956
?-valuation, 958
t-norm, 950
tangent measure, 1021, 1045
Tarski's circle-squaring problem, 110
tensor quadratic variation, 471
theorem
- Central Limit, 886
- dominated convergence, 812
- Egorov, 790
- exhaustion, 798
- Hahn-Banach, 1288
- Kolmogorov-Sinai, 903
- Loomis-Sikorski, 877
- Lorentz-Luxemburg, 816
- Luxemburg-Gribanov, 795, 817
- martingale convergence, 894
- Nikodym convergence, 132
- Radon-Nikodym, 821
- Riesz, 790
- strong law of large numbers, 889
- Vitali's convergence I, 812
- Vitali's convergence II, 813
- weak law of large numbers, 889
thick subspace, 768
Titchmarsh's theorem, 1255
Tonelli theorem, 74
topological
- entropy, 1224
-ergodicity, 1210
- measure space, 1134
- probability space, 748
- strong mixing, 1210
- weak mixing, 1210
topologically amenable group, 1117, 1118
topology
- compatible with density, 1147
- locally solid, 800
- - entire, 804
- o-, 800
- order, 7%
total
-uncertainty, 1547, 1548
- variation, 706, 985
totally
- measurable, 286
- ordered measure space, 1152
trace, 1245
- of a function, 1245
- ?-algebra, 767
- operator, 841
- probability, 767
tracial state, 901
transfer principle, 1298
translation invariant lifting, 1155
transportation problem, 777
transversal, 1103
triangular
-conorm,950
-conorm corresponding to a t-conorm, 950
- functional, 160
- norm,950
tnbe. 876
Trotter's theorem, 309
truncation properties, 518
Turing machine, 1225
type
- I convergence, 1259
- ? convergence, 1259
- Maharam, 1288
- semigroup, 98
W-bounded set, 172
n-integrable, 1086
и-measurable, 1088
ucc topology, 1314
ultrarilter
-?-weakly normal, 1279
- measure, 709
- nonregular, 1277
- nonregular countably incomplete, 1277
- uniform, 1267
- topology, 709
-weakly normal, 1267
unanimity game, 1383, 1384, 1389

1606
Subject Index
uncertainty, 1547, 1554
unconditional convergence, 643, 645
unconditionally convergent
-map, 1085
-operator, 1085
underlying lattice of an MV-algebra, 873
uniform absolute continuity
- of the indefinite integral, 355
uniform
-convergence, 376
- integrability, 1310
-measure, 1021, 1045
- MV-algebra, 927
uniformity, 1557
uniformly ? -additive set of positive measures, 367
uniformly exhaustive, 941
- measures, 732
unimodular group, 1112
union of fuzzy sets, 953
unique ergodicity, 1199
unit, weak, 807
unitary operator, 1189
universal
- independence space, 779
- possibilistic entailment, 1491
- strong lifting property, 1150
- Turing machine, 1226
universally
- measurable, 630
- measurable function, 557
- measurable set, 1157
unlimited, 1298-1300
unrelatedness, 1498
upper
-bound, 791
- density, 1136
- envelopes of upward-filtering families, 520
- functional, 512
-gauge, 525
- integral, 513
- norm,524
- S-norm,511
--S-norm, 524
- variation, 984
upwards, directed, 791
usual conditions for filtration, 386
USLP, 1150
?-integrable functions, 513
valuation, 900, 1551
van der Waerden's theorem, 1201
variance of an observable, 882
variation, 39, 193, 267
-function, 45, 46, 381
- of a function, 44, 380
- of a multimeasure, 645
- of vector measure, 357
variational
- characterization of the integral, 208
- measure, 593, 597, 599
- principle, 1224
vector
- T\ -measure, 993
- lattice, 505
-measure, 1083
- locally bounded, 366, 370
- quadratic variation, 471
-state, 841
Vitali, 1286
-convergence theorem I, 812
-convergence theorem II, 813
- covering theorem, 183, 679
-set, 1103
- theorem, 63, 376
Vitali-Hahn-Saks theorem, 133
von Neumann's paradox, 104
W*RNP,561
weak
-convergence, 1305, 1313
- extension of measure, 1126
- halo properties, 231
- law of large numbers, 889
- mixing, 1200, 1203
- mixing of higher order, 1208
- normality trace, 1279
- precompactness, 545
- Radon-Nikodym set, 554
- RNP, 554
- unit, 807
-uppergauge, 525
- Vitali property, 230
weak*
- Radon-Nikodym property, 561
- scalar integrability, 538
- scalar measurability, 535
-topology, 1305, 1313
weak**
- Radon-Nikodym set, 554
- RNP, 554
weakly
-? -distributive i-group, 875
-a-distributive MV-algebra, 875
-a-distributive probability MV-algebra, 880
- p-summable, 426
- Bernoulli partition, 1221

Subject Index
1607
-compact, 1089
- compact Radon operator, 1087, 1088
- compact Radon vector measure, 1089
-comparable measures, 258
- complete spaces, 274
- measurable function, 267
- wandering set, 118, 1099
weight, 842
well-ordered partition, 1478
Wiener measure, 1319
Wiener-Shannon law, 1529
Wiener-Wintner theorem, 1213
Wijsman
- convergence, 656
- topology, 632
WMEA, 1283
Young function, 807
Young-Fenchel transform, 663, 665
zero observable, 896
Zeta
- operator, 1386
-transform, 1384