OHAPTEE I.
The integral In an abstract space.
$ 1. Introduction. Apart from functions having as argument
a variable number, or system of n numbers (point in w-dimensional
space), we shall discuss in this book functions for which the inde-
independent variable is a set of points. Functions of this kind have
occurred already in classical Analysis, in several important particular
cases. But they only began to be studied in their full generality
during the growth of the Theory of Sets, and in close relation to
the parts of Analysis directly based on that theory.
If we are, for instance, given a function f[x) integrable on every
interval, then by associating with each interval I the value of the
integral of j{x) over I", we obtain a function F(I) that is a function
of an interval. Similarly, by taking multiple integrals of functions
/(a5j, x2, ..., x,,) of n variables, we are led to consider functions of
more general sets lying in spaces of several dimensions, the argu-
argument I of our function F[I) being now replaced by any set for which
the integral of our given function f(a\, a?2, ..., xn) is defined.
We dwell on these examples in order to emphasize the natural
connection between the notion of integral (in any sense) and that
of function of a set. Needless to say, there are many other examples
of functions of a set. Thus in elementary geometry, we have for
instance, the length of a segment or the area of a polygon. The
class of values of the argument of these two functions (the length
and the area) is in the first case, the class of segments and in the
second, that of polygons. The problem of extending these classes
gave birth to the general theories of measure, in which the notions
of length, area, and volume, defined in elementary geometry for
a restricted number of figures, are now extended to sets of points
S Salts. Ttieon of the Integral. 1

'2 CHAPTKK I, The integral in an abstract N|>a.cts. **"
of much greater diversity. It is, nevertheless, remarkable thai, thasu
researches arose fur less from problems of Geometry than from
their connection with problems of Analysis, above, all with liho
tendency to generalize, and to render more precise,, the notion of
definite integral. This connection has occasionally found expression
even in the terminology. Thus du Bois-.Reymond called integrable
the sets that to-day are, said to be of measure zero in the Jordan sense.
The theories of measure have, in the course of their development,
been modified in accordance with the changing requirements of the
Theory of Functions. In our account, the most important part will
be played by the theory of H. Lebesgue.
Lebesgue'w theory of measure has made it possible to dis-
distinguish in Euclidean spaces a vast class of sets, called measurable,
in which measure, has the property of complete additwity — by this
we mean that the measure of the rum of a sequence, oven infinite,
of measurable sets, no two of which have points in common, is equal
to the sum of the measures of these Rets. The importance of this
class of sets is due to the fact that it includes, in particular, (with
their classical measures), all the sets of points occurring in problems
of classical Analysis, and further, that the fundamental operations
applied to measurable sets lead always to measurable sets.
It is nevertheless to be observed that the ground was prepared
for Lebesgue's theory of measure by earlier theories associated
with the names of Cantor, Stolz, Harnack, du Bois-Boymond,
Peano, Jordan, Borel, and others. These earlier theories have,
however, to-day little more than historical value. They, too, wnre
suitable instruments for studying aud generalizing the notion of
integral understood in the classical sense of Riemann, but their
results in this direction have been largely artificial and accidental.
It is only Lebesgue's theory of measure that makes a decisive step
in the development of the notion of integral. This is the more re-
remarkable in that the definition, of Lebesgue apparently requires
only a very small modification of a formal kind in the dofinitiou
of integral due to Biemann.
To fix the ideas, let us consider a bounded function f(x, y)
of two variables, or what comes to the same thing, a bounded func-
function of a variable point defined on a square Kn. In order to deter-
determine its Biemann integral, or more precisely, its lower Riotnann-
Darboux integral over Kn, we proceed as follows. We. divide the
Introduction.
square Ko into an arbitrary finite number of non-overlapping
rectangles Rlt R*, .... R,,, and we form the sum
A.1)
i—i
where v, denotes the lower bound of the function / on Rt, and m{Bi)
denotes the area of Z?,. The upper bound of all sums of this form is,
by definition, the lower Iiiemcum-Darbotuc integral of the function /
over KQ. We define similarly the upper integral of / over Eo. If
these two extreme integrals are equal, their common value is called
the definite Riemann integral of the function / over Ko, and the
function / is said to be integrable in the, Riemann sense over Ka.
The extension of measure to all sets measurable in the Lebesgue
sense, has rendered necessary a modification of the process of Eie-
mann-Darboux, it being natural to consider sums of the form A.1)
for which (iw),_i. ¦>...-,« is a subdivision of the square KQ into a finite
number of arbitrary measurable sets, not necessarily either rectangles
or elementary geometrical figures. Accordingly, ni{Ri) is to be un-
understood to mean the measure of Ri. The »,¦ retain their former
meaning, i. e. represent the lower bounds of / on the corresponding
sets R/. We might call the upper bound of the sums A.1) interpreted
in this way the. lower Lebesgue integral of the function / over Ko.
But actually, this process is of practical importance only for a class
of functions, called measurable, aud for these the number obtained
as the upper bound of the sums A.1) is called simply the definite
Lebengue integral of / over Ko. What is important,' is that the func-
functions which are measurable in the sense of Lebesgue, and whose
definition is closely related to that of the measurable sets, form
a very general class. This class includes, in particular, all the func-
functions integrable in the Biemann sense.
Apart from this, the method of Lebesgue is not only more
general, but even, from a certain point of view, simpler than that
of Kiieniann-Darboux. For, it dispenses with the simultaneous
introduction of two extreme integrals, the lower aud the upper.
Thanks to this, Lebesgue's method lends itself to an immediate
extension to unbounded functions, at any rate to certain classes
of the latter, for instance, to all measurable functions of constant
sign (of. below § 10). Finally, the Lebesgue integral renders it per-
permissible to integrate term by term sequences aud series of functions
in certain general cases where passages to the limit under the iu-

4 CHAPTER I. The integral in an abstract space.
tegral sign were not allowed by the earlier methods of integration.
The reason for this is to be found in the complete additivity of L«-
besgue measure. The fundamental theorems of Lebesgue (of. bolovr
§ 12) stating the precise circumstances under which term by term
integration is permissible, are justly regarded by Ch. J. de la Valise
Poussin [I, p. 44] as one of the finest results of the theory.
Lebesgue's theory of measure has, in its turn, led naturally
to further important generalizations. Instead of starting with area,
or volume, of figures, we may imagine a mass distributed in the
Euclidean space under consideration, and associate with each set
as its measure, (its "weight" according to Ch. J. de la Vallee
Poussin [I, Ohap.VI; 1]), the amount of mass distributed on the sot.
This, again, leads to a generalization of the integral, parallel to
Lebesgue's, known as the LobeBgue-Btie.lt jes integral. .In order
to present a unified account of the latter, wo shall consider in this
chapter an additive class of measurable Rets given a priori in au
arbitrary abstract space. We Khali suppose further, that, in this
class, a completely additive measure is determined for the mea-
measurable sets. These hypotheses determine completely a corresponding
method of integration in the Lebesgue sense. All the essential prop-
properties of the ordinary Lebesgue integral, except at most those im-
implying the process of derivation, remain valid for this abstract integral.
Prom this point of view, in a more or less general form, the Lebesguo
integral has been studied by a number of authors, among whom
we may mention J. Badon [1], P. J. Daniell [2], O. Nikodym [21
and B. Jessen [1]. For further generalizations (of a somewhat
different kind) see a.lso A. Kolmogoroff [1], S. Boc liner [1],
G-. Pichtenholz and L. Kantorovitch [1], and M. Gown-
tin [1].
§ 2. Terminology and notation. Given two sets A and B,
we write AC.B when the set A is a subset of the set B, i. e. when
every element of J. is an element of B. When we have both A C B
and BCA, i. e. when the sets A and B consist of the same ele-
elements, we write A = B. Again, aeA means that a is an element
of the set A (belongs to A). By the empty set, we mean, the set without
any element; we denote it by 0. A set A is enumerable if there exists
an infinite sequence of distinct elements «„ a3, ..., a,,,... consisting
of all the elements of the set J..
[§ 2] Terminology and notation. 5
Given a class S3I of sets, we call sum of the sets belonging to
this class, the .set of all the objects each of which is an element of
at least one set belonging to the class S3I. We call product, or common
part, of the sets belonging to the class % the set of all the objects
that belong at the same time to all the sets of this class. We call
difference of two sets A and B, and we denote by A — B, the set
of" all the objects that belong to A without belonging to B.
Given a sequence of sets [A,,] — a finite sequence Av A2, —, A,,,
or an infinite sequence Av A2,..., A,,,... — we denote the sum by
SAi, by y, At, or by A1 + Ai-\-...+Aa, in the finite case, and by
i ' i--1
CO
~Ah by EAh or by A1 + A2+...-\-An+... in the infinite case.
Similarly, merely replacing the sign 2 by 77, we have the expres-
expression for the product of a sequence of sets. If the sequence \An) is
infinite, we call upper limit of this sequence, the set of all the ele-
elements a such that a e A,, holds for an infinity of values of the index n.
The set of all the elements a belonging to all the sets A,, from some n
(in general depending on a) onwards, we call lower limit of the se-
sequence [A,,]. The upper and lower limits of the sequence [A^n^ia,...,
we denote by lim. sup J.,, and liminf A,, respectively. We have
B.1)
CO O CO
liminf An = ? J7 A»C.U EAn = MmsupAn.
l n—k
If lim sup J.,, = b'minf A,,, the sequence {A,,} is said to be convergent;
its upper and lower limits are then called simply limit and denoted
by limJ.,,.
n
If, for a sequence [An\ of sets, we have An(Z.An+\, for each n,
the sequence {Alt) is said to be ascending, or non-decreasing; if, for
each n, we have J.,,+] C A,,, the sequence [A,,] is said to be descending
or non-inereasing. Ascending and descending sequences are called
monotone. We see directly that every monotone sequence is con-
convergent, and that we have lim A,,— SAn for every ascending se-
n n
quence {A,,), and liml,,= I/i» for every descending sequence [An\.
Finally, given a class S of sets, we shall often call the sets
belonging to (?, for short, sets ((?). The class of the sets which
are the sums of sequences of sets ((?) will be denoted by ?,-. The
class of the sets which are the products of such sequences will
be denoted by (?„ (see F. Hausdorff [II, p. 83]).

CHAPTER I. Tim integral in mi nimtrnd; h\mm\.
§ 3. Abstract space A'. In the rest of this chapter, a ml A'
will be fixed and called space. The, elements of .V will bo callad points.
If A is any set contained iu A' the set -V — A will bo called cr/w-
plement of A with respect to X; the expression "with respect, to A'"
will, however, generally be omitted, since sets outside the .space? A'
will not be considered. The complement of a set A will bo demoted
by GA. We evidently have, for every pair of sets A and H,
C.1) A—B = A-OB,
and for every sequence [X,,\ of sets
n x,, = c-sox,,, x xn = o // ox,,
n a it "
C.2)
lira sup X,, = 0 lira inf OX,,, lira inf X,, — 0 lim ku |> <1 A",,.
71 II « II
In the space A'we shall consider functions ol' a set, and functions
of a point. The values of these functions will always be real numbers,
finite or infinite. A function will be called finite, when it. (whuiyigh
only finite values.
To avoid misunderstanding, let us agree that when infinite func-
functions are subjected to the elementary operations of addition, subtrac-
subtraction etc., we make the following conventions: a-\~( +co)=( -j-oo)-|-a- ^oo
for a^Toc; (+oo)+(—coj = (_co) + (+oo) = (±oo)_ (±oc)=();
«• (±oo)=(+oa) • a—±oQ and a ¦ (±°°)=( + oo) • a=:foo, according
asa>0 or a<0; 0 ¦ (±°o)=(+oo) -0=0; a/( + oo)=<); ft/0=|-1.;o.
We call characteristic function cF(ai) of a set E, the function
(of a point) equal to 1 at all points of the set, and to 0 everywhere
else. The following theorem is obvious:
C.3) If -E=vj&\, and E,-^=0 whenever i^k, then cA,(.c) = V(i («).
If \E,,) is a monotone sequence, of sets, the sequence of their
characteristic functions is also monotone, non-decreasing or non-in-
non-increasing according as the sequence \En) is ascending or descending.
If [U,,] is any sequence of sets, A and, B denoting its upper
and lower limits respectively. we have
e.j (a)==lim sup cA.
and cB{x) = lim inf
jo that, in order that a sequence of sets (_&'„) cmrmje to « mi K,
it is necessary and sufficient that the sequence of their chatiuier-
istic functions {a/in(x)) converge to the function 'cv,(a;)|.
Additive classes of sets.
A function assuming only a finite number of different values
on a set E is called a simple function on E. If cv *>..., /.'„ are all
the distinct values of a simple function f(:v) on a set E, the
function f{x) may on E be written in the form
n
a>)=^ 0kCF (as) where E=
E^-\-...-rEk ami ErE,= Q for t±j.
The function / given by this formnla over the set E will be denoted
toy l^i, -Bj; t'2, Ef,...; vn, E,,\.
Tile notion of eluiKK'terwtic ftui<;tioii is ilue to (Mi. J. .le la, Vallee
Poussiu [1] ami [I, p. 7].
§ 4. Additive classes of sets. A class $? of sets hi the space A*
will be called additive if (i) the empty set belongs to 9?, (ii) when
a set X belongs to J? so does its complement OX, and (iii) the
sum of a sequence {Xil of sets selected from the class 9?, belongs
to the class $.
The classes of sets, additive according to tins definition, are sometimes
termed completely additive. We get tlie definition of a class of sets additive in,
the weak sense if we replace tin1 condition (iii) of the preceding definition
Ijy thp following: (iii-l)is) 1fte sum of Urn seta helonging to 9E also belongs to it.
The sets of an additive class 9? -mOl be called sets measurable (§E),
or, in accordance with the definition given in § 2 (p. 5), simply
sets BE). We see at once that, on account of the conditions (i) and (ii),
the space A', as complement of the empty set, belongs to every
additive class of sets. Making use of the relations B.1), C.1), and C.2),
we obtain immediately the following:
D.1) Theorem. If $? is an additive class of sets, the sum, the
product, and the two limits, upper and lower, of every sequence of sets
measurable (96), and the difference of two sets measurable (9?), are also
measurable (9E).
In later chapters we shall consider certain additive classes of
sets that present themselves naturally to us, in connection with
the theory of measure, in metrical or in Euclidean spaces. In the.
abstract space A', about which we have made practically no hypo-
hypothesis, we can only mention a few trivial examples of additive
classes of sets, such a.s the class of all sets in A", or the class of all
finite or enumerable sets and their complements. Let us still mention
one further general theorem:

01IAPXKK I. Thcs integral in an abstract space.
D.2) Theorem. Given any class SCR of sets in X, there exists
always a smallest additive class of sets containing 3)J, i. e. an
additive dams 9l0 Z) 99? contained in every other additive elas,t that
contains SOL
For let 91() be the product of all the additive classes that con-
contain 301. Such classes evidently exist, one such class being the class"
of all sets in A". We see at once that the class 5ft0 thus defined has
the required properties.
§ 5. Additive functions of a set. In the rest of thin chapter
we suppose that a definite additive dass St' of sets is fixed in the space X.
In accordance with tins hypothesis, we may often omit; the symbols
in our statements, without causing any ambiguity.
A function of a wot, tf>(X), will be called additive function of
a set (??) on a set E, if (i) E is a set (??), (ii) the function </»(X)
is defined and finite for each set XC.E measurable (9?), and if
(iii) 0BX,,) = 3§(Zn) for every sequence {X,,} of sets (?) con-
tained in E and such that Xt • X*=0 whenever i fclc. For simplicity,
we shall speak of an "additive function" instead of an "additive
function of a set (S?)" whenever there is no mistaking the meaning.
An additive function of a set (9E) will be called monotone on E
if its values for the subsets (9?) of E are of constant sign. A non-
negative function #(9:) additive and monotone, will also be termed
non-decreasing, on account of the fact that, for each pair of seta
A and B measurable (??), the inequality AQB implies $(B) =
= A>(A)-\-<f>(B—A) ^ ®{A). For the same reason, non-positivo
monotone functions will be termed non-increasing.
E.1) Theorem. If <Y>(x) is an additive function on a set W, then
,) =lhn <1
for every monotone sequence \X»} of sets (9E) contained in E. If §(X)
is a non-negative monotone function, then
E.3) f(liminfX,,Xlirninf*(J,,) and #(lim sup A^)>lim trap </>(X,)
for every sequ&nee [X,,] of sets (9E) in M.
Additive funotions of a set.
Proof. Let (I,,j,,, i,2,... be a sequence of sets (?) contained inE.
If [X,,) is an ascending monotone sequence, then
Jim X, = j? Xn = Xx + v1 (z,,+1 — Xn),
n 7i—l n^ 1
and consequently, ®{X) being an additive function oa U,
<P(nmXJ = ff (Xx) + v 0{x,1+1 — X,,) =
71 n^= 1
= lim [ <P (Xi) +'v'(P (x*+i — X;,)] = lira S» (Xa).
If {X,,) is a descending sequence, the sequence {E — X,,} is
ascending, and, by the result already proved,
U> (E)—$ (lim X,,)= d \\\m(E—X,,)] = lim 0 {E—X,1)=0 (E)~iim0{X,,),
ii n n n
from which E.2) foEows at once.
Finally, if [X,}f is any sequence, but $(X) is a non-negative
monotone function, we put
{5.4)
for n =
The sets Y,, are measurable (??) on account of D.1), and form an
ascending sequence. We therefore have, by the part of our theorem
proved already
E.5) <P(limYJI) = limfl»(rn).
How, it foEows from E.4) that YnCXi, and so, §(?„)< $[X»),
for each to. On the other hand, lim inf Xn — lim Yn, and therefore
n n
the first of the relations E.3) is an immediate consequence of E.5).
We establish similarly (or, if preferred, by changing Xn to E—X,,)
the second of these relations, and this completes the proof of the
theorem.
Every function of a set #(X), additive on a set E, can easily
be extended to the whole space A". In fact, if we write, for instance,
3>1(X)= 0{X-E) for every set X measurable (s2), we see at once
that CPj(X) is a function additive on the whole space X,\ that co-
coincides with <I>(X) for measurable subsets of E and vanishes for
measurable sets containing no points of E. We shall call the func-
function ^i(X), thus defined, "the extension of 0(X) from the set E to
the space A".

10
(JHAPTEE, 1. The, integral in im abstract spuou.
§ 6. The variations of an additive function. The, upper
and lower bounds of the values that a function of a set *(.Y), ad-
additive on a set JU assumes for the, measurable subsets of t.his sef, M,
will be called upper variation and lower variation of the function <l>
over E, and denoted by W((f';E) and W(<C; 7i') respectively. Hineo
every additive function vanishes for the, empty set;, we. evidently
have W(*;II)<I)<W(M), The number W{<!>; J0)-\~ \W('l>; E)\
will be called absolute variation of the function <7> on JU and denoted
by W(®;E).
{6.1) Theorem. If </>(X) in an additive function on n mi li, its
variations over E are always finite.
Proof. Suppose that W($;iJ) = +°°- We shall show firstly
that there then, exists a sequence {E,,\,,. i,a,... of sets (9?) mich that
F.S
For let us choose Ex — E and suppose the, sets E,, for n— \,'J,,...,h
defined so as to satisfy the conditions F.2). By the second of tho.se
conditions with n = It, there exists a measurable set A. C Eh such that
F.3) \(I{A)\^\(b(Eil)\ + l.
If W((T<; J.) = oo, we have only to choose E^\ = A in order to
satisfy the conditions F.2) for «=& + i. J.f, on the other hand,
W{0;A) is finite, we must have W( #>;-$/,. — A) = + oo, and, by (li..'i),
|#(-E/<—A-)|>|^(^)| — |^(jBA)|>ft, so that the conditions (l>.2)
will be satisfied for n = k+l, if we choose E,lir^=E,, — A. The
sequence {En\ is thus obtained by induction.
Now, on account of Theorem 5.1 and of the third of the con-
conditions F.3), we .should have the equality #(luniJ,,)=lim#Gi!,,)==oo.
" It
and since every additive function of a set is, by definition, finite,
this is evidently impossible. Q. E. D.
It follows from the theorem just, proved that every function
#(X) additive on a set E is not only finite for i,he subsets (ft)
of E, but also bounded; in. fact, the values it resumes an; bomidwi
in modulus by the finite number W(«P; K).
f§ 0]
The variations of an adilitivu function.
11
Theorem 6.1 can be further completed as follows:
F.4) Theorem. For every function F(X) additive on a set E<
the variations W(§\X), W($;X) and W{<T>;X) are also additive
functions of a set (96) on E, and toe have, for ever;/ measurable set
F.5)
<25(X) = W(*; X) + W((P; X).
Proof. To fix the ideas, consider the function _Q1(J) = W(ff';Z).
Since this function is finite by Theorem 6.1, we have to show
that for every sequence \Xtl) of measurable sets contained in E,
and such that X,- ¦ Xi, = 0 whenever i =]= It,
F.6)
O C Y ) — 3 O I T )
For this purpose, let us observe that for every measurable set XC-X,,
we have (f>(X) = 3&(X-X,,) ^. 2 QX(X,,), and hence
ii n
n n
On the other hand, denoting generally by Y'„ any measurable set
variable in X,,, we have QX{2X,,)^$>B Yn) = 2$(YU), and
therefore also Qx{2X,,)^>3Q^X,,). Combining this with F.T) we
get the equality F.6).
Finally, to establish F.5), we observe that for every measurable
subset Y of X we have 0( Y) = @(X)—<1>{X— Y)<0(X)—W(<f>jX),
andsoW(CP;XX^(X)—W($; X). Similarly W(&;X)^$(X)—W($;X).
These two inequalities give together the equality F.5), and the
proof of Theorem 6.1 is complete.
It follows from this theorem that every function of a set
additive on a set E is, on E, the difference of two non-negative
additive functions. The formula F.5) expresses, in fact, <1>{X) as
the sum of two variations of #(X), of which the one is non-negative
and the other non-positive; this particular decomposition of an
additive function of a set will be termed the Jordan decomposition.

12 CHAPTEB I. The integral in an abstract space.
We can now complete Theorem 5.1 as follows:
F.8) Theorem. If $(X) is adMtive on a set E, we have
$(Iim.5r,,)=lini$(Z,,) for every convergent sequence, {X,,} of nets (96)
contained in E.
In fact, making use of the Jordan decomposition, we may
restrict ourselves to non-negative functions ®{X), and for these
Theorem 6.8 follows at once from the second part of Theorem B.I.
$ 7. Measurable functions. Given an arbitrary condition,
or property, (F) of a point x, let us denote generally by R[( P")]
.V
the set of all the points so of the space considered thai; fulfill this
condition, or have tins property. Thus, for instance, if f(x) denotes
a function of a point defined on a set E and a 1h a real number,
the symbol
G.1) E[a!eJS?;/(ai)>a]
.V
denotes the set of the points x of E at which f(oc) > a.
A function of a point, f{x), defined on a set E, -will be termed
measurable (Sc), or simply function (96), if the set E, and the set G.1)
for each finite a, are measurable (96). It is easy to see that
G.2) In order that a function f(w) be measurable on a measurable
set E, it suffices that the set G.1) should be so for all values of a he-
longing to an arbitrary everywhere dense set B of real numbers [the
same holds with the set G.1) replaced by the set E[ase.E;
.V
In fact, for every real a, the set B contains a decreasing
sequence of numbers {r,,\ converging to a. We therefore have
(a5)>r;,] and, each term of the sum
2
'1=1 A"
on the right being measurable by hypothesis, the same holds for
the sum itself (cf. Theorem 4.1).
Every function f{x) measurable on a set E, can be continued
in various ways, so as to become a measurable function on tho
whole space Ar. For definiten.ess, we shall understand by tho <?«?-
tension of the function f(x) from the set IS to the apano A",' |;h« fu ac-
action fo{%) equal to f(%) on E and to 0 everywhere dm. For brevity,
we shall often deal only with functions meammiblo on tlut wholo
[§71
Measurable, functions.
13
space A', but it is easy to see that all the, theorems and the
reasonings of this, and of the succeeding, § could be taken relative
to an arbitrary set (96).
The equations
E [/(a) < a] = OE [/(as) > a], E [/(as) ^ a] = 17 E \f(cc) > a—-1
x x x n=l .v L ni
E[/(a>)<a] = OE [/(as) ^ a], B[/(»)=a] = B[/(as)>a] ¦ B[/(*)<«],
X -V -V X X
B[/(as)<+oo]=fB[/(ai)<»], B[/(as)> —oo]= vB[/(as)> —»],
x n=K -v /iT=i -v
E[/(as)=+oo]=CE[7(as)<+oo], E[/(sj) = — oo] = CE [/(<b) > — oo]
X iV .T -V
show that for every measurable function f[x) and for every number a,
the left hand sides are measurable sets. Conversely, in the definition
of measurable function, we may replace the set G.1) by any one
of the sets E[/(*)>a], E[/(aj)s^a] or E[/(*)<«]; this follows at
once from the identity
B[/(as)>a] = f E
A" 71=1 X
a] =
To any function f{x) on a set E, we attach two [functions f{x)
and f{x) on E, called, respectively, the non-negative fart and the
non-fositive fart of f(x) and defined as follows:
f(x) — f(x) or 0 according as f(x) > 0 or f(x) < 0,
f{x) = /(*) or 0 according as f[x) < 0 or f(x) ^ 0.
We see at once, that in order that a function be measurable on a, set E,
it is necessary ami, sufficient that its two parts, the non-negative and
the non-positive, be measurable.
Eeturning now to the notions of characteristic function, and
simple function introduced in § 3, we have the theorem:
G.3) Theorem. In order that a set E be measurable (96), it is ne-
necessary and sufficient that its characteristic function be measurable.
More generally, in order that, on a set E, a simple furiotion f(x) be
measurable (96), it is necessary and sufficient that, for each value
of f(x), the points at which this value is assumed on E, should con-
constitute a measurable subset of IS.
Another theorem, of great utility in applications, is the fol-
following:

14
OHAl'TliK J- Tlio integnil in an ;tl>sta.iet, h|>;kmi.
G.4) Theorem. Every function f(x) that is measurable (9Q and
non-negative on a, set JE, is the limit of a non-deereasing sequence
of simple functions, finite, mca^umble and wm-negalive on 14.
In fact, if we write for each positive integer n and for -.r.tti,
/«(«) =
¦ , if f(,®)^*h
the functions f,,{ae) thus defined are evidently simple and uori-nogativo,
and, on account of Theorem 7.3, measurable on B. Further, as is
easily seen, the sequence {/„(*)' is non-decreasing. Finally limf,,(x)==f{x)
for every xeB; for, if /(*) <+°°, we have, aa hoou as n exceeds
the value of fix), the inequalities i)-C.f{x) — f,,{x) ^.i/'2", while,
if
we have
f,,(x) — n for w—1,2, ..., and ho
§ 8. Elementary operations on measurable functions.
We shall now show that elementary operations effected on meamir-
able functions always lead to measurable functions.
(8.1) Theorem. Given two measurable functions f{x) and g(x), the sets
E [/(<»)> 0@)], E [/(*)> g(se)] and E [f{w)=(/{x)],
X X X
are measurable.
The proof follows at onee from the identities
n
-|-oa -\-ao
B [f(x) >g(x)]= T v b \f(x)> ^
X II——OO 777—1 .V I "<•
'['«>=]¦'['
m
• E
•v x x x x
(8.2) Theorem. If the function f{x) is measurable, \f{x)\" in also
a measurable function.
For a>0, the proof is a consequence of the identity
E [|/(o>) |" > a] =E [f(x) >ai«] + B [/(»)< _«i«l,
which is valid for every «>0, while for a < 0 its loft hand side
coincides with the whole space and therefore constitutes a mea-
measurable set. For e<0, the proof is similar.
EleunmUry operations ou uieasiu-ahle functions.
15
(8.3) Theorem. Every linear combination of measurable functions
with constant coefficients represents a 'measurable function.
The. identities
E[a-/(ffi)+?>a] = E[/(»)>^=^| for a>(),
x x L a J
JS [a • f{x)
= B
/or u < 0,
valid for every function /(a?) and for all numbers ft, a 4=0, and /J,
show, in the first place, that«. /(*) + A is a measurable function,
if fix) is measurable. It follows further, from Theorem 8.1 and
from the identities:
for «
that if /(a?) and g(x) are measurable functions, so is a -fix) -\- j3 ¦ g(x).
(8.4) Theorem. The product of two measurable functions f(x)
an,d gix) is a measurable function.
Measurability of the product f-g is deiived by applying Theorems
8.2 and 8.3 to the identity fg = iiif+ g)i — (f~g)i], the com-
completion of the proof, by taking into account possible infinities
of / and g, being trivial.
(8.5) Theorem. Given a sequence of measurable functions (/n(a5)j,
the functions
upper bound/„(*), lower bound /,, (x), lim sup /„(*) and litninf/„(*)
n » 17 71
are also measurable.
The measurability of hix) = upper bound f,,{x) follows from
it
the identity ~E[h(x) ~>a] = -S'E[/n(a5)>a]. For the lower bonnd, the
x n x
con*espondmg proof is derived by change of sign.
Hence, the functions 7(,,(a;)=upper bound \j,,+\{x), fn+i(x),...]
are measurable, and the same is therefore true of the function
lim sup f,,{x) — lira h»(x) = lower bound lhn[or). By changing the sign
77 II II
of fn(x), we prove the same for lim inf.

16
GHAPTEft 1. Tlie integral iu an abrtrnci, H
§ 9. Measure. A function of a set f<(X) will ho called h #
¦wre (S?), if it is defined and non-negative for every set (9?), iind if
for every sequence (X,} of sets (9c) no two of which have -point*
in common. The number ,u.(X) is then termed, .for every set .A' mesi-
surable (9c), the measure (ft) of X. If every point of a wot 70, except
at most the points belonging to a subset of E of measure (,«) zoto,
possesses a certain property F, we nha.ll say that the condition V
is satisfied almost everywhere, (p) in B, or, that aim-oat every (//,) point
of E has the property V. We shall suppose, in the sequel of
this Chapter, that, just as the class 9c was chosen once for all,
a measure /( corresponding to this class is also kept fixed. Accordingly,
we shall often omit the symbol (//) in the expressions "measure
(//,)", "almost, everywhere (ft)", etc. Olcariy /((.?) s^/«( V) for any
pair of sets X and 7 measurable (9c) such thai; XC Y, and
for every sequence of measurable sets {2',,J.
A measure may also assume infinite valuta, and in theroforo. mil in gen-
general an additive function according to the definitiou of § 5.
The results established in this chapter concerning perfectly arbitrary meas-
measures will be interpreted in the. sequel for more special theories of measure, (for
instance, tkoBe of Lebesgue and Carathdoclory). For the present, we shall
"be content mentioning a few examples.
Lot us take for 9c, the class of all sets in a Bpaee X We obtain a l.rivial
example of measure (S?) by writing //(X)=i-:O identically, (or else u (X) -|-cx->)
for every set XCX. Another example consists in choosing an elonumt a iti A"
and writing ,»(X) = 1 or ,»(X)=0, according as »sl or not. In tlm cane of mi
enumerable space X, consisting of elements av as, ...,a,h..., the gonwul I'orni
of a measure ,»(X) defined for all subsets X of A" in /i(J.)~2'k,,fi,{J{) whore
{hn} is a sequence of non-negative real numbers and /«(X) is equal do 1 ov 0 ac-
according as aneX or not. It follows that, every measure defined for nil snlwote
of an enumerable space, and vanishing for the sets that consist of a Mingle point,
vanishes identicaEy. The similar problem for spaces of higher potencies is nmcli
more difficult (see S. Ularn [1]). For a space of the potency of the continuum
see also S. Banach and C. Kuratowski [1], '?. Szpilrujn |.i], W. Hiorpinski
[I, p. 60], W. Sierpinski and E. Szpilrajn [1],
We shall now prove the following theorem analogous to The-
Theorem 5.1:
(9.1) Theorem. If {X,,} is a monotone ascmtdini/ Net/wnec of
measurable sets, then lim/<(X,,) == ^(liniX,,). The m,<m<: hold* lor
monotone descending sequences provided, however, thai [/(A",) | do.
[§ 9] Measure. 17
More generally, for every sequence [Xa] of mensurable sets,
(9.2) it (lim inf X,,) < lim inf u (X,,)
and, if further /j.(SX,,)~\- oc,
(9.3) /i (limsup X,,) ^liin ,snp it (X,,),
II 1!
so that, in particular, if the sequence [X,,] converges and its sum has
finite 'measure, lim f((X,,) = \i(lim X,,).
n n
Proof. For an ascending sequence (X,,),,^!,-.>,... the equation
lim//(X,,) = /((limX7,) follows at once from the relation
II 11
lim X,, = 2 Xn = Xx + V (X,,+i — Xn),
and if the sequence {X,,) is descending and /'(X^^oo, then the
measure /<(X) is an additive function on the set Xt and consequently
the required result follows from Theorem 5.1.
In exactly the same way, if for an arbitrary sequence [Xn]
of measurable sets, ^Xn is of finite measure, the measure n{X)
is an additive function on this set, and the two inequalities (9.2)
and (9.3) follow from Theorem 5.1. To establish the first of these
inequalities without assuming that the sum of the sets X,, has finite
measure, we write as in the proof of Theorem 5.1
Since the sequence is ascending, and X,,CX,, for every n, we have
/((lim inf X,,) = pi (lim Yn) = lim n (Y?, X lim inf u (X,,).
We conclude this § with an important theorem due to D. Ego-
roff, concerning sequences of measurable functions (cf. D. Ego-
roff [1], and also W. Sierpinski [3], F.Eiesz [2; 3], H. Hahn
[I, pp. 556—8]). We shall first prove the following lemma:
(9.4) Lemma. If E is a measurable set of Unite measure (ft) and
if {fn{%)) is a sequence of finite measurable functions on E, eon-
verging on this set to a- finite Measurable function f(x), there exists, for
each pair of positive numbers ?, rj, a positive intpfter^ and a, mea-
measurable subset H of E such that ii(E)<rj and
for every ti ~> N and every xeHJ-—H.
S. hiikx. Thi'ory nf Hit' Integra].

IS
CHAPTliU 1. Tim integral in an iibstmot spaco.
Proof. Let us denote generally by ?'„, tho Hubsoti of .&' con-
consisting of the points x for which (9.5) holds whenever n>m. Thiw
defined, the sets i?iH are measurable and form a monotone adi
sequence, since for each integer m, wo have
Further, since !/„(«)! converges to f(x) on the whole of ./<?, we have
E=T,Em, and so, by Theorem 9.1, fi(E) = lim^ (.$„,), i. e.
in '»
lim^(J?—-Bm) = 0, and therefore, from a sufficiently largo m0
III
onwards, fi(E—7?,,,) < t\. We have now only to choose JV ¦—«»<,
and E=E—.Em,,, and the lemma is proved.
(9.6) Eyoroff's Theorem. If E is a measurable set [of finite
measure (/*) and if {fn(x)) is a sequence of measurable functions finite
almost everywhere on E, that converges almost everywhere on this set to a
finite measurable function f(x), then there exists, for each e~>Q,
a subset Q of E such thai /<•(-??—Q)< s and suo'h that the converg-
convergence of {fn(x)) to f(x) is uniform on Q.
Proof. By removing from E, if necessary, a set of measure (,«)
zero, we may suppose that on E, the functions fn(os) are everywhere
finite, and converge everywhere to f(x). By the preceding lomma,
we can associate with each integer m > 0 a set J2",,', C •$ such that
pi{Hn,) < f/2'" and an index Nm such that
(9.7) \fn(w) — f{x)\< 1/2'" for n>Nm and for xe'E—Em.
Let us write Q = E— ? Hm. We find
i
. (E—Q)
]
and since the sequence /„(#) converges uniformly to f(x) on lih«
set Q on account of (9.7), the theorem is proved.
H 10]
Integral.
19
The theorem of Egoroff can be given another form (cf. if. Lusin [I, p. 20]),
and, at the same time, the. hypothesis concerning finite measure of B can he
slightly relaxed.
(9.8) If E is the sum of a sequence of measurable, sets of finite measure (,«) and
if {fn(x)} is a sequence of measurable functions finite almost everywhere on this set,
converging almost everywhere on E to a finite function, then the set S can be expressed
as the surn of a sequence of measurable sets H, Eu E%,... such that t<(H) = 0 and
that the sequence (/i(j>)} vonoerges uniformly on eaah of the sets En.
For the proof, it suffices to take the case in which the set E is itself of fin-
finite measure. With this hypothesis, we can, on account of Theorem 9.6, define
n
by induction a sequence {2j?*}a=i,->.... of measurable sets such that n (E—^Et) -?- I/ft,
A=l
and that tho sequence [fn[x)} converges uniformly on the set E( for each fc. Choosing
oo
E = E—-^Em, we have it(H)=Q, and the theorem is proved.
As we may observe, the hypothesis that the set E is the sum of a sequence
of sets of finite measure, is essential for the validity of Theorem 9.8. For this pur-
purpose, let us take as a space Xo tho interval [0, 1], and as au additive class i?0
of sets, that of all subsets of Xo. Further, let us define a measure /<„ by writing
ao(X)= co whenever the set Xe9:0 i8 infinite and »0(X)=?i, if X is a finite
set and n denotes the number of its elements. The sets of measure (»0) zero then
coincide with the empty set. Finally, let {gn(x)\ be an arbitrary sequence of fun-
functions, continuous on the, interval [0, 1], converging everywhere on this interval,
but not uniformly on any subinterval of [0, 1].
To Justify our remark concerning Theorem 9.8, it sufficeB to show that
the interval -.\T0=[0,1] is not reprcsentable as the sum of a sequence {En} of
sets such that the sequence of functions {gi(x)\ converges uniformly on each of
them. But if such a decomposition were to exist, we might suppose firstly —
since the functions g»[x) are continuous — all the sets E, closed. Then, however,
by the theorem of Baire (cf. Chap. II, § 9) one of them at least would contain a
subinterval of [0, 1]. This gives a contradiction, since by hypothesis, the 86-
quence {gn(x)\ does not converge uniformly on any interval whatsoever.
§ 10. Integral. If we are given in the space X an additive class
of sets 9? and a measure ,a defined for the sets of this class, we
can attach to them a process of integration for functions of a
point. In fact:
(i) If f(x) is a fnnction (X) non-negative on a set E, we shall
understand by the definite integral (9E, .«) of f(x) over E the up-
upper bound of the sums n
where {Ek\k--i,-i „ is an arbitrary finite sequence of sets (¦?) such
that E — E1JrEa + ...-j-E,, and ErEk=b for i-\=Jc, and where
i1*, for i=l,2,...,n, denotes the lower bound of f(x) on Ek.
2*

20
CHAPTER 1. Tin- integral in mi nbHtriMtt
(ii) If f[x) is an arbitrary function measurable (9t*) on a sod Ii,
we shall say that /(,¦») possesses a definite integral (9c, /r) <wer M, if
one at least of the non-negative fimotiom f(a>) an.d —/(<») (ef. §3)
possesses a finite integral over IS according to definition (i). And,
if this condition is satisfied., we shall understand by tlio definite
integral [% p) of the function f{x) over E tlio difference between
the integral of f\x) and that of —Hsu) over E. The definite integral
(9t*, ^) of f(x) over E will be written (9E) / /(«) d(i{x). If this integral
is finite, the function /(»¦) is said to be inlegrablc. (SE, /.(-). lAor every
function f{x) possessing a definite integral over a, Bet I<], we evi-
evidently have
&)ffdp =
is
ff
f)df< - (?) ffdfi\- m
° i
We see at once that the two definitions (i) and (ii) aro c o m-
patible, i. e. that they give the same value of the integral to any
non-negative measurable function. Moreover:
A0.1) If g = {»1} Zx; «j., X2; ...;»»„ Xm\ is a simple non-negative func-
function on the set JS=X1 + X2+ ... + Xm, the sets Xt being mmsur-
able (9?), ihm
For, if {JB/V-,1,2 „ is an arbitrary subdivision of E into a, finite
number of sets (S?) without points in common, and if wj denotes
the lower bound of g(x) on Eh we have wj^Zv, whenever 'E} • X,=\- 0.
Hence | f 1] |
J1
and therefore f
i
The opposite inequality is ob-
obvious, since the sets X1,Xi,...,Xm themselves constitute a subdivision
of E into a finite sequence of sets (9t) on which fch« valites of g
are »1,«2,-...,ym respectively.
[§
Fundamental properties of tie integral.
21
§ 11. Fundamental properties of the integral. We shall
begin with a few lemmas concerning integration of simple functions.
As in the preceding §§, the symbols % /i etc. will often be omitted.
A1.1) Lemma. 1° For every pair of functions g(x) and Ji(os), simple,
non-negative, and measurable (9?) on a set IE, we liave
A1.2)
(xfid/i(cc)= fg(x)du(x)+ fh(x)du(x).
2° If the function f(x) is simple, non-negative, and measurable C6)
on the set A-\-B where A and B are sets (S6) without common points, then
A1.3) /' f(x) d/i (x) = ff(x) d/i (x) + f f{x) d,u(x).
.4-1« A B
Proof. As regards 1°, let
g = [gu Gt; g%, (?2;...; gn, 0,,} and h = [hv
where E = G^
%;...; hm, Hm),
We then have, by A0.1),
/' [g(x) +}i(w)] dp (x) = v y
¦1 J1
121 II
= f g(co) da (x) + f h(x) <Zp (x).
ft -
As regards 2°, if E=A+B and f = [fvQi,h,Qil...;fmQn},
where E = Q1+Qs+...+Q!l, we have
(f{x) &!i(X) = ?fr!'(Qi) = Vfrp(A-Qt) + ffr^B-
A1.4) Lemma. If [g,,{x)\ is a non-decreasing sequence of functions
that are simple, non-negative, and measurable (S?) o« a sei JS, and if,
for a function h{ic), simple, non-negative, and measurable, on E,
we have lim g,, (x) ~^ li(x) on E, then
A1.5)
lim / gn [x) dfi (x) 5* / H®) ^ («)
" k k

22
CHAPTER I. The integral in im itbRtaict spams.
Proof. Let h = {%, Bt; vz, Ba;...; vm, T<],,,}, where
0 < % < »2 < ... < «m ««/, K —Ml I" ^V I' ••¦ I ¦ ^-
We may suppose vx > 0, for, otherwise, wo .should have
I hd/t = j'hdn, and, since / g,,d[i.s~ j u,,d/i, we could roplaee
the set E by the set E—EL on which ft (a?) does uot vanish any-
anywhere. Further we shall assume first that v,,, <. + co-
Let us choose an arbitrary positive number e < vt> and lot
us denote, for each positive integer n, by Q,, the set of the points
x of E for which gn(x)>h{x) — c The seta Qn evidently form an
ascending sequence converging to B, and, by Theorem 9.1, we have
)->//(.?). This being so we have two cases to distinguish:
(i) /((E) =|= °°- We then can find an integer nn such that for
n0 we have [<(E — Q,,)<.*> «in<i therefore, by Lemma ,11.1,
hdfi—vm-ii(E—Q,,)—
1—Ki
and, passing to the limit, making first n -> oe, and then, t > 0,
we obtain the inequality A1.5).
(ii) ft(J!)=oo. Then, since / gnd\i~^{vx — e)ji.(Q,,), we obtain
k
I™1 g,,dfi=oo, so that the inequality A1.5) is evidently satisfied.
Suppose now v,,,= + oo. Then by A0.1) and by what has
already been proved, 1imfg,,d(i>v-p(Bm)+?vrH(Bi) for any
» ii N-i
finite number v, and consequently for v = + oo = «„, also; whence,
in virtue of A0.1) the inequality A1.5) follows at once.
A1.6) Lemma. If the functions of a non-decreasing sequence
\gn(x)} are simple, non-negative, and measurable (S?) on a set E, and if
g(x) = lim g,,(x), then lim/' gn(x) d,u (a?) = f g(x) dfi {x).
Proof. Let E1,Ez,...,Enl be an arbitrary subdivision of B
into a finite number of measurable sets, and let v1} vs, ..., vm be
the lower bounds of g{x) on these sets respectively. Let. us write
«={«!, Bx) v2,Ez;...; vm, Em). We evidently have lim gn (*) ~ i
on E, and hence, by Lemma 11.4 and by Theorem 1,0.1
[§"]
Fundamental properties of the integral.
23
lim f gn du ^ /' v dfl = vv. fl (jj.).
" k 'e '='
It follows that lim \ g,,dft ^ j gd;i, and since the opposite
" k k
inequality is obvious, the proof is complete.
We are now in a position to generalize Lemma 11.1 as follows:
A1.7) Theorem. The relation A1.2) holds for every pair of functions,
g(x) and li(x), non-negative and measurable (SE) on the set E, and the
relation A1.3) holds for every function f(x) non-negative and measur-
measurable (S6) on the set A-\-B, where A and B are sets E?) without
points in common.
Proof. By Theorem 7.4 there exist two non-deereasing se-
sequences [gn{%)\ and {hn(x)\ of simple non-negative functions mea-
measurable (9E) on E, such that g(a?) = limgn(%) and h{x) = liralin(x).
n n
Now, by Lemma 11.1 A°), we have / (g,,+ h,,)d[i= / gndfi+ j hndp
is be
and hence, making n^-oo, we obtain, on account of Lemma 11.6,
the relation A1.2). Similarly, if we approximate to f{x) on A-\~B
by a non-decreasing sequence of simple non-negative functions and
make use of Lemma 11.1 B°), we obtain the relation A1.3).
A1.8) Theorem. 1° For any function measurable (9?), the integral over
a set of measure zero is equal to zero. 2° If the functions g(x) and
h(x) measurable on a set E are almost everywhere equal on E, and
if one of the two is integrable on E, so is the other, and their integrals
over E have the same value. 3° If a function f(x) measurable ($?)
on a set E has an integral over E different from + °°> *^e se^ °f the
•points x of B at which f(x) = + oo has measure zero. In particular,
if the integral of f(x) over E is finite, the function f{x) is finite
almost everywhere on E.
Proof. We obtain at once part 1° of this theorem by making-
successive use of the definitions (i) and (ii) of § 10.
As regards 2°, it is evidently sufficient to consider the case
of non-negative functions g(x) and h(x). If we denote by Bt the
set of the points x of E at which g(w) =j= h{x), we have by hypothesis
1) = 0, and, on account of A°) and of Theorem 11.7, we obtain
/ gdfi = I gdp = I hdfi = j Ii d/i, as required.
f.' /!',

24
CHAPTER f. This iiitenriil in an nlwl/i'iioli
Finally, as regards 3°, let us suupoiw that 1'or a function /(*¦)
measurable (SE) on-Ewe have /(,«) = +00 on a not /{¦'„( I<) of ponitivo
measure. We then have / /dp > // dii'.^n-u (A?,,) lor every •»,, and
Ho I f d.u = + 00. Consequently, the integral ol: /(,«) ovor A1, it it
re
exists, is positively infinite, and this completes the proof.
We now generalize Lemma 11.1 A") and also complete Tho-
orem 11.7, as follows:
A1.9) Theorem of distrtbutivity ofthe inter/i-al. Every linear
combination with constant ftoeffioisnU, a-g(tr,) \-b-h(x) of two fnno-
tions g(x) and h(x), inlegrable] (9c, ft) over a set M, in alxo integral)!,?,
over E, and we have
A1.10)
I [ay-\-bh)dii, — a j gdfi-\~b j hd/i.
Proof. By Theorem 11.8 C°), the set of the pouiliH ill. which
either of the functions g(x) and h(x) is infinite, has moaHure hoco,
and if we rejjlace on this set the values of both functions by 0, wo
shall not affect the values of the integrals appearing in the relation
A1.10). We may therefore suppose that the given functions g and h
are finite on E. Further, the relations
agdji = ajgd.u,
blidii —b I
being obvious, we need only prove the formula A.1.10) for tho
case a=b—l. Finally, the set E can bo decomposed into four
sets on each of which the two functions <j(x) and h(x) ar« of cons-
constant sign. So that, on account of Theorem 11.7, we may assumo
that the functions g(x) and h[x) are of constant sign on tho whole
set H. S"ow, by the same theorem, the relation
(U-ll) /' (g + h)du = f g d(i + /' h dn
k k k
holds whenever the functions g and h are both iion-imgiitivc or
both non-positive on E, and it only remains, therefore, to hIiow
that this relation is valid when g and h have, on /¦?, oppoHik) hi^iih,
the one, g{x) say, being non-negative, the otli«r, //,(,•«), rion-pOHiiiivo.
Fundamental properties of tie integral.
25
This being so, let Hx and E2 be the sets consisting of the points
x of E for which we have g(io) + Ji.(x)'^>Q and g{x)+7i(a>)<.0}
respectively. The functions g, g+ h, and —h are non-negative on E1
and we therefore have, by Theorem 11.7,
fgdf<= f(g+Ji)dfi+ f(—h)du= f{g+h)dii— fhdu.
E, ?, it, E, E,
Similarly
— I hdu— I (— h) du = I (—g—h) dfi -f- / gdu = — j {g+h)dfi + j gdji.
k, li, E, R, E. E.
Therefore, for i=l, 2, we have /(ff+A) du = j gdp -{-jftdn, and
by Theorem 11.7 we obtain the relation A1.11).
A1.12) Theorem on absolute hiter/rabiHf//. 1° In order thai
a function f(so) measurable C:) on a set E should be integrable (SE, «)
on E, it is necessary and sufficient that its absolute value should be
so. 2° If, for a function g(so) vieasurable (S?) on a set E, there exists a
function k(x), integrable C6, ,») and such that \g(x)\^.h(w) on E,
tlien the function g(x) also is integrable on E; in 'particular, every
function measurable (SE) and bounded on a set E of finite measure (p)
is integrable (9E, ») on E.
Proof. As regards 1°, we have by Theorem 11.7
f\f\dp=f}dn+J(—j)dp,
and integrability of |/j is therefore equivalent to that of / and that
of —/ holding together, i. e. to integrability of /.
As regards 2°, we have the inequalities gr(iBX| flr(aj)|^A(as) and
— g{®) < 1/7(^I ^ ft('y) ori ^; aim> since Hx) iR: fey hypothesis, inte-
integrable on E, it follows that the same is true of the non-negative func-
functions g and —g, and therefore of the function g{x).

26
CHATTEE I. The integral in :ui .-itwl-riusfc n\v.u:u.
As an immediate consequence of Tlieowm 1.1.3 a we Uavo the
following theorem, known as the
A1.13) First Mean Value Theorem, Given, on a nut M, a
function f{x) bounded and measurable, (9c) on E and a fundion //(a?)
integrahle (% p) on E, the function f(x)-g{x) in integrable on B
md there exists a number y lying between the bound.* of /(.?;) on
E, such that
A1.14)
f\f(x) \g{a>)\
f \g{a>)\dfi{a>).
Proof. If* we denote by m and M respectively the lower mid
the upper bound of f(x) on E, and make uho of Theorem Jl.lU,
we verify successively, that tho functions (|M'|+|to|) •!//(*)!; [/((«) //(*¦)[,
f{x)\g{x)\ and f(x) g{x) are intograble on E. Further, wo have
m\g(x)\ ^ f(®)-\g{x)\ ^ ilf \g (x)\ over .B, and, tiiorol'oro also,
-\g\d;t^Mj\g\dfi, and ro choosing yH.//-!.<7l*'l
m
(or, if the denominator vanishes, an arbitrary y between m and /If),
we obtain the formula A1.14) with m^.
§ 12. Integration of sequences of functions. In this §,
we shall establish some theorems on term by term integration of
sequences and series of functions.
A2.1) Theorem. If the functions of a sequence {g,,[x)) are finite and
integrable C", <i) on a set H of finite measure, and tho neguenee
converges uniformly on 13 to a function g(x), tlien the function g(x)
also is integrable over JE, and we have
A2.2)
lim / g,,{x) dfi(x) = f
Proof. By Theorem 8.5, the function g(x) is measurable (S?)
on JE. The functions g(x)—gn{x) are therefore all measurable also,
and, further, since the sequence {g,,{x)} converges uniformly to g(x)
the functions g(x)—g,,(x) are all bounded, at any rate from Home
value of the index n onwards. These functions are tlms, by The-
Theorem 11.12 B°), integrable on E, and it follows, by Thoorotii V\.%
that the function g (x) = [g (x) — #„(.*)] -|- ^(a,) ia mte^rable too.
Pinaly, denoting by e,, the upper bound of |</(flJ) —//,,((/i)| on E,
we have
[512] Integration of sequences of functions.
fg(x)d(i{x)— I gn{x)diA,(x)\^ j \g(x)~g,l(x)
k 'e k
and this establishes the relation A2.2) since, by hypothesis, ?,,->0
and // (E) =)= °°-
Keither the theorem thus established, nor its proof, contains,
at bottom, anything new, as compared with the similar result for
the classical processes of integration of Cauchy, or of Eiemann.
We now pass on to the proof of theorems more closely related to
Lebesgue integration. Among these theorems, a fundamental part
is played by the following one, which is due to Lebesgue:
OS
A2.3) Theorem. Let f(x)==y fn{x) be a series of non-negative functions
measurable (S?) on a set JE. Then
A2.4) [ f(x) di*{x) = y ffn(x)d{i(x).
E "=1 E
Proof. Erom Theorem 11.7, we derive in the first place, that
fdu^z [y fnda] = y, / fndu for every m, and so
i e "~J ' "=li
A2.5)
To establish the opposite inequality, let us attach, in accordance
with Theorem 7.4, to each function f,,(x) a non-decreasing sequence
{gn\x)}i!=i,%... of simple functions measurable and non-negative on 18,
in such a manner that lim g^(x)=fn{x) for n—1, 2,... Let us write
h
h
sk(x) = y. gf>(x). The functions sk{x) are clearly simple, measurable,
and non-negative, on E, and they form a non-decreasing sequence.
m .
Further, for each m, and for fc>»i, we have yg\ }{x) < sk{x) < f(x).
Making Tc—>oo we derive y fi(x) ^lim sk{x) ^if{x) for every m,
/=i *
and so, f(x)=MmSk(x). Therefore, by Lemmas 11.6 and 11.1 A°),
fi
f f da = lim /' s* d(i = lim v fgf dp ^ J / /,¦ du,
E * E * ;=1 k i=l k
and this, combined with A2.5), gives the equality A2.4).

CHAPTER I. The integral in mi itlmtraet Hpawi.
Theorem 32.3 may also lie stated in liho following form:
A2.6) Lebesgne's Theorem, on intef/ration of monotone se-
sequences of functions. If \f,,(x)} in a mm-dacreamuj sequence of
non-negative functions measurable (9?) on a set H, and j{x)---lim fn{w),
then I fix) du [as) = lim j f,,(x) d/i- («).
'e " k
j
" k
Proof. If we write g,,{as) = f,,\ i — /«(*), wo obtain
and the functions g,,{x) will bo non-nogativo and mosiHurablo on HI
so that by Theorem 12.3
/ fdii = / Ad" + v / fl«&n = lim / [fi I-^'f/Jrf/' - linn / //(.rf//.
A1 A' " ' A" '< /V " I * /.;
Q,. 11 f).
A2.7) Theorem, of adMUvity for the inte<jra,L If {[<]„) i» a
sequence of sets measurable (9?) no two of which hwmi cotn/mon points,
and D = 5] B,,, then
/'/dii = v /'
A2.8)
for every funotion f(x) possessing a definite integral {finite or infi
over E.
Proof. It is clearly sufficient to prov« AU.H) in. th« mm of
a function /(*) non-negative on ii1. anppoHin^ thin to \m (,Iio cjwo,
let us write /„(»)=/(») for otffl,,, and /„(;»)«=0 Cor meM—fl,,. Wo
then have f{x)=^f,,{x) on B, and, the funetioiiH /„ b«intf in<«wimU)U>
and non-negative, we may apply LofooHgno'B Thoorom 12.H. Tliis
gives, by Theorem 11.7,
If a function f(x) has a definite iniHigra,! (%,u) ovor a h«1, K,
then /(«) also has a definite integral ovor any sulmoli of /(/ uiou-
surable (9?). We may tlieroforo aHaoc.iiUi(» wilili ill l;ho funoliion of
a set (9-) defined as follows:
A2.9) F(X)=ff(x)dfl,(x) where XQ.XI and X <X.
12]
Integration of sequences of functions.
The latter will be called the indefinite integral B6, p) of f(x) on E.
It follows from Theorem 12.7 that, whenever the function f(x)
is integrable (96, /0 on JE, its indefinite integral is an additive func-
function of set (9?) on JE.
We end this § with two simple but important theorems. The
first is known as Fatou's lemma, and appears for the first time
(in a slightly less general form) in the classical memoir of P.Eatou
[1, p. 375] on trigonometric series. The second is due to Lebesgue
[5, in particular p. 375], and is called the theorem on term by term
integration of sequences of functions; cf. also Ch. J. de la Vallee
Poussin [1, p. 445—453], E. L. Jeffery [1] and T. H. Hilde-
brandt [2].
A2.10) Theorem (Fatou's Lemma). If {/„(#)) is any sequence
of non-negative functions measurable (S6) on a set IS, we have
j lim inf /„ {as) du {as) ^ lini inf j f,, (%) du («)¦
k " " e
Proof. Let us write ^-(ic) = lower bound [/i(a;),/i+i (a;),//+2(a;),...]
where i=l, 2,.... Thus defined {#,(*)} is a non-decreasing sequence
of non-negative functions measurable on JE7, and converges on the
set I? to lirninf f,{x). We therefore have, by Lebesgue's Theorem 12.6,
/'liminf /,(«) du{x) — lim / gt{x) du
' ' ' k
f,(x) dfi{x).
A2.11) Z,ebes(/ue's Theorem on term by term inteyration.
Let \fn(x)} be a sequence of functions measurable (9E) on a set E,
fulfilling, for a function s{%) integrable (9E) on E, the inequality
\f,,(x)\ < s(x) for n=l, 2, ... Then
lim inf [ fndn^ / Mm inf fnd(i,
" k k "
lim sup / /,,5ft< / nmsup/nfZ//.
A2.12)
If, further, the sequence {/„} converges on E to a function f, the sequence
is integrable term by term., i. e. we have
A2.13)
lim / /„ dp = / / a!^.
" f. b

CHAPTKK I. 'I'lit' integral in ;in iiliHtriMsl,
Proof. Let a(x) = lim iuf j,,{x) and lei; k(x) --• lim .sup /„(,-«).
We may clearly suppose s(a:) < + °o throughout M. Wo then derive
from Eatou's Lemma 12.10, lim inf /(« + /„)#>/ {ft [-<j)d{i and
" A' «
HmM f{8—f,,)dii^ f[s—h)dfi, which gives at oneo the rela-
" /; /V
tions A2.12).
Further, if lim/„(«)=/(a;), wo derive from (.1U.12) the relation
lim inf //„#>/ /$/*> lim sup / /„«!// which givos (ihe equality
» ii ii " k
A2.13).
§ 13. Absolutely contiimoiis additive function** of a set.
The fact that the indefinite integral of a function in(n^rn.I)lo (ft", //,)
on a set B is, on E, an additiro funoliion of a Heii (9?), humm the
problem of characterizing directly the additive FunediojiH ox,|>t'OH-
sible as indefinite integrals.
If we restrict ourselves to the ljoboHgutt intiogvul (if 1'iimstioiin of ;i r<wl var-
variable, we may regard indefinite integrals <tt) tunotionw of tin. intiu'val, or, what
tomes to the same thing, as functions of ii veal variable. In tluili <mino, n, imciin-
sary and sufficient condition fur a function to ho exprcHHihlo w fcho imlufiDite
integral of a real fxmction was given, in 1004, Hliill by 1jo1>ohj?uo [T, ]>. 121), foot-
footnote]. A little later (in 1905), G. Vitali [1| oxplioitly (liHtiii^iiiwhod tlui hIiihk
of functions possessing the Lel>e8guo jiroporty by iiitrcxliir-inpf th« namn <A "uh-
solutely continuous funotions".
Tlia condition of Lebosgue ami Vitali wan labor uxtamdod to i'tiiicliioim of
a set by J. Radon [1] (of. also P. J. Danioll [2]). But Radon coimiriitrod only
additive fnnctions of sets measurable in the Rorol hoiiho iti I'luoliddiin Hpn.coH,
and only measures determined by additive lunotiann of intorvaln (c,f. holiw (Ihap-
ter III). The final form of the condition of LoboHguo-Vitji.li, iw friv(^i) in
Theorem 14.11 below, is due to O. Nikodym [2].
An additive function of a set (??) on a sot J'J, will ho Haul to
be absolutely continuous (% ji) on B, if tho funet.ion v
for every subset (9?) of E whoso mmmro (ft) in zero. An, jwldidi
fonetion <25(Z) of a set (S6) on a set B will bo tertnod nnyular (%
on B, iE there exists a subset EaC J® raeuHurablo \C&), of meaNiifo
eo h tht <Z»(Z) ih idil
, aC euHurablo \C&), of me
zero, such that <Z»(Z) vanishes identically on, M-~ /{,'„
for every subset JT of
following statements a,re at oaoe oliviouw:
moiwuru.ble
i. (J.
The
[§
Absolutely continuous additive functions of a set.
31
A3.1) Theorem. 1° In order that an additive function of a set (S?) on a
set B should be absolutely continuous (S?, ,m) [or should be singular]
it is necessary and sufficient that its two variations, the upper and
the lower, should both be so. 2° Every linear combination, with con-
constant coefficients, of two additive functions absolutely continuous
[or singular] on a set E is itself absolutely continuous [or singular]
on B. 3° If a sequence {0>n{X)) of additive functions, absolutely
continuous [singular] on a set E, converges to an additive function
0(X) for each measurable subset X of E, then tlw function $(X) is
also absolutely continuous [singular]. 4° If a function, of a set (9E)
is additive and absolutely continuous [singular] on a set E, the
function is so on every measurable subset of E. 5° If E—V,En,
where {E,,} is a sequence of sets (S6), and if an additive function
on E is absolutely continuous [singular] on each of the sets E,,, the
functioji is absolutely continuous [singular] on the whole set E.
6° An additive function of a set cannot be both absolutely conti-
continuous and singular on a set E, without vanishing identically on E.
For sets of finite measure, it is sometimes convenient to apply
the following test for absolute continuity:
A3.2) Theorem. In order that a function ®{X) additive on a set E of
finite measure, be absolutely continuous (S?? ii) on E, it is necessary
and sufficient that to each ?>0 there correspond an »/>0, such that
:u(X)<i] imply \$(X)\<? for every set IC-E measurable (9t).
Proof. It is evident that the condition is sufficient. To prove
it also necessary, let us suppose the function $(X) absolutely con-
continuous in E. We may assume, replacing if necessary, ®(X) by
its absolute variation, that 0(X) is a non-negative monotone func-
function on E. This being so, let us suppose, if possible, that there
exists a sequence {E,,},,-..-ax- of measurable subsets of E, such that
!i(E,,)<l/2" and that ${En)>ri0, where % is a fixed positive number.
Let us write Eo = lim sup En. For every n, we then have
/'(^oX^W^1/2"! and therefore «(_B0)=0. On the other
hand, by Theorem 5.1, we have ®{E0) > lim sup $(-&»Ms%• Tais
n
is a contradiction, since ®(X) is absolutely continuous, and the
proof is complete.

32
(JHAPTKK I. 'l'ln' iut;<if?rnl hi itn iiJ
A3.3) Theorem. In order that <t junction ^(X), addition on, it, m
E, be singular (%p) on XJ, it in tmmmry mid xu/fidmd Ikal fm
each ?>0 there exist a, set, ZC® measurable (9t) and fidjiWinij Uu
two conditions u [X) < ft, W( <I>; M — X) <,, k
Proof. The condition is clearly rioeesHary. To proves il, mil'-
ficient, let us suppose that for each n there, is a sot X,,(^]<] meas-
measurable (?) such that MX,) < .1/2" and W(tf>; ./</ — A';,)<M/2", and
let us write J70 = lim sup X,,. We then have
A ii
/i{Xk)<';j/2"
for each «, and so /i(JBo) = O. On the otluvr hand, by Theorem r>,.l.
•we have W(<f>; JE — J?o) *=: lim itif W(<P; 0— A',,) — I). The. function
'i'(Z) is therefore, singular on Ii.
§ 14. The Lebesgxie decomposition ot an additive
function. Before proving the. rcwnlt ininouucwl in the ])r(«!(Hlinjji' §,
we shall establish some auxiliary theorems. We Ix^in with 1:1 io follow-
following theorem due to I-I. Ilahu \\, p. -1011 ((if. also W. Mierp in Hk i [;i 11):
A4.1) Theorem. If §(X) is an additive funation of a sat (ft) on a
set M, there exists always a net PBJH niewnnrablc, (S?), molt that
W(#;JP)= 0 = W(#;J? —P), or, what mmen to the mine thing,
such that <25(Z)^5=O for every meamraUk net X.Q.P and <!>(X)<^i)
for every measurable set X C JS — P.
Proof. For each positive integer n, we cIiooho ;i, ned !<]„
that $(KOzW{$;B)—1/2". By Theorem (i.-i we then have,
A4.2) W(#;?„)>—1/2" and W{<P; Ii— /(/„)< 1/2".
Writing P=lmiDiH,,, we see that IS—I'=lim sup (Ii— !<]„) Q v(li—Ii,,)
" » » in
for every m, and therefore, by A:1.2),
which gives F(«Z>;1! —P)=0. On the other hand, the lower var-
variation W(<I>;X) is a non-positive monotone function of a measurable
set IC-E, and, by Theorem 5.1. and the first inequality A4.2),
we must have the relation |W( <I>; P)\ ^ iim Uif | W( tf'; ft1,,I ¦¦ I), which
n
gives W((J;P)= o and eompletos the proof.
[§ 14] Tlia T^ebesgue decomposition of an additive function.
33
A4.3) Lemma. If $[X) is a non-negative additive function of a,
set (S6) on a set IS, there exists, for each a > 0, a decomposition
of E into a sequence of measurable sets without common points,
E,E1,E2, ...,En,... such that ji{E) = 0 and that
A4.4) a ¦ (n — 1) ¦ fi(X) ^${X) ^m- u(X)
for every set X C En measurable (SE).
Proof. By Theorem 14.1, there exists, for each positive in-
integer n, a measurable set An such that $(X) — an- u(X)^Q for
every measurable set XQA,, and $(X)— «»'|i(I)<0 for every
DO
measurable set X C2$ — An. Write Bn = J? Ak. Any measurable
CO
subset X of Bn may be represented in the form X = J? Xk, where
Xk are measurable sets, XkCZAk for ~k=n, n-f-1, ..., and Xi-Xj=0
for i^ j; and so ^(X)=^<25(ZA)^2a&-MX>)>a«-."(X)- We obtain
thus a descending sequence of measurable sets {Bn} such that
®(X)^an-[i(X) if XCBn, Xe%
(U'5) l(IK«i.(i(I) if XCE—Bn, Xe%
the second relation being obvious, since H—Bn is a subset of
E—An.
Let us now write E1=JE—B1, Ea=Bn-l—Bn for 71=2,3,...,
and I=]im?,,. Thus defined the sets E, Ev E2, ...,E,,,... are mea-
surable and without common points, and H=E-{-^lI!n. Taking
into account the relations A4.5), we see at once that the inequality
A4.4) holds whenever X is a measurable subset of En. Finally, H(ZBn
for each positive integer n, and therefore, by the first of the rela*
tions A4.5), we get §(H)^an- f{B), which requires u[H) — 0 and
completes the proof.
A4.6) Theorem. If E is a set (SE) of finite measure (,«)> or>
more generally, a set expressible as the sum of a sequence of sets (SE)
of finite measure, every additive function of a measurable set $(X)
on E is expressible as the sum of an absolutely continuous additive
function ^'(X) and a singular additive function &(X) on JE. Such a de-
decomposition of $>(X) on E is unique, and the function W{X) is, on E,
S. Silks. TliiMiry til1 tin- Iiitccml. 3

34
CHAPTER I. Tim integral in an aliMtrai'.li
the indefinite integral of a function inlctjnMc {% //) on M. If <J>(X)
is a non-negative monotone function on E, no an the mmwpondvng
functions W(X) and (H)(X).
Proof. Since every additive function of a set is the difference)
of two non-negative functions of the same kind (of. § 0), wo may
restrict ourselves to the case of a non-negative ®(X). Further, wo
shall assume to begin with that the set E has finite measure,. By the
preceding lemma, there exists, for every positive integer m, a de-
decomposition of-B into a sequence of measurable setw II.\'"\ E\'"\ 'fi-z"\ ...,
without common points and subject to the conditions:
tU.7) E = Hlm)
E
\m)
« 0,
A4.8) 2-m.(»—l).^Z)<fl»(A')<2-"'.»./i(X)f if• X
We therefore have, for all positive integers n, n, and k,
2-7"-«-M4"')-4m+1))^^D'")-4;"+1))>2 '" ' • (k—1).fiitf',"'-Iti"]\
and
O, and that
-4"' M))=-<> whenever
from which it follows that Bm—ft+1)
(l—2n+2)l*(Mr)-Etn+n)^0. Hence
either lc>2ri+l, or 7c<2w—2.
We may therefore write
This being so, let E=^H{'")+ yj Q^"\ We write /("')(;«),,==2 »<-{n— i)
for aseE^—H, n=l,2,..., and /('"'(a;) = 0 for well. We thus obtain
a sequence {/(ffl)(a>)} of non-negative functions measurable (9?) on the
set JE?. By A4.9) we have clearly \f'"-^[x)—fi'»\x)\^2 '" on M, so
that the sequence {fm\x)) converges uniformly on '.E to a non-neg-
non-negative measurable function f[x).
The set E being of measure zero, we have, by A4.7) and A,4.8),
for every measurable set ZC-E and for every positive integer m,
®(X)>®(X.H)+?2-<» ¦ (n—1) ¦ n(X.'W^)~: 0{X-U) \- f /"»>*<,
and A>
[§
Tlie Lebesgue decomposition of an additive function.
35
Hence, making m->oo, we derive $(X)= / f(X)du(x)Jr§(X-H).
x
This decomposition, so far established subject to the hypo-
hypothesis that the set JB is of finite measure, extends at once to sets
expressible as the sum of an enumerable infinity of sets of finite
oo
measure. In fact, if E=]?An, where the An are sets (9?) without
common points and of finite measure, then, by what we have al-
already proved, there exists on Au a non-negative function f,,(x) in-
tegrable on A,,, and a measurable set HnQAn having measure
zero, such that &{X-An)= ffndii + $(X-Hn), for w=l, 2,... If we
now write E=yjHn, and f(x)=fn(x) for xeAn, we obtain a measur-
able set E(ZE of measure zero, and a function f(X), non-negative
and integrable on E, such that, "by Theorem 12.7, for every meas-
measurable set XQJS
C4.10)
Now, the indefinite integral vanishes for every set of measure zero,
and therefore is an absolutely continuous function; on the other
hand, we have <7>BM?)=0 for every measurable set X(^E—E.
Thus, since the set E has measure zero, formula A4.10) provides
a decomposition of <D(X) into an absolutely continuous function
and a singular function. Finally, to establish the unicity of such
a decomposition, suppose that ®(X) = W1(X)+@1(X)=W2(X)+&2(X)
onJS, the functions W^X) and W^X) being absolutely continuous, and
the functions ®X{X) and @Z(X) being singular. Then W1(X)—Wi(X)=
= ©2(X)—®X[X) identically on E, whence by Theorem 13.1 B°
and 6°), we have S'1(Z)=f2(Z) and ©1(X)=fe»2(Z), and this com-
completes the proof of our theorem.
The expression of an additive function as the sum of an ab-
absolutely continuous function and of a singular function will be termed
the Lebesgue decomposition. The singular function that appears in
it is often called the function of the singularities of the given function.
From Theorem 14.6, we derive at once
3*

36
CHAPTER 1. The integral in an uJffltmct wpaco.
A4.11) Theorem of lladon-JNikodym. if h) is a net of finite
measure, or, more generally the sum of a miqiicnm of setn of finite mea-
measure (//), then, in order that an addiliw, funotion of a net (9?) on 'K be
absolutely continuous on E, it is- neecsstary and xufjic-ient that this
function of a set be the indefinite integral of some inteyrablr function
of a point on E.
Tha hypothesis that the Rot li in the wmi of mi ;U most enumerable infin-
infinity of sats of finite measure, plays au essential part in the ansortion of the theorem
of Radon-Niltodym, just as in Theorem O.K. To see thin, Jot ilh talu* again the
interval [0, 1] as our space Xv and lot the class 9ct of all mibmttH of @, 1] that
are measurable in the LaboBgue sense (of. below ('hap. Ill) be oni" fixed additive
elasa of seta in tlie space Xt. A measure //, will be defined by tiikinp /it(X) -oo
for infinite sets and /t1(X)=n for finite nets 'with u elements, Thiw being ho, l,ke
sets (Xi) of measure (//l) zero coincide with tho empty not, and liluircforo, ev(*ry
additive fun otion. of a set (9E,) on XL is alwolntely eonliiinioiiH (9^, /ij). In ]iartienlar,
denoting by A(X) tha Labesgue measure, for every set Ji'i-Sf,, wv Him l,liat /I(A')
is absolutely continuous (S?v'(<i) on Xr We Rhjill hIiow l.lial -I(A') is not an imld-
finite integral (%, /ir) on Xr Suppose indeed, if poHHi'ble, tbal,
for every set X e&v the func.tion g[a>) being iutograble (i€,,/ij) on Xv Hine.e
A(X) is non-nagative, wa may suppose that g(x) is no too. Ij<»t 14—li\g(x) -0]
X
and JBn=E[j(a!)>l/n] for »=1, 2, ... Wo have /I(A", H)-~ liidn.^i), so that
a-, -/¦:
A(B)=A(X1)—l and this requires the sot 14 to be noii-oiiuinortilild. Mince N--21<}n,
n
the same must be true of B,,,, for some positive integer ¦»¦„. Thus
which is evidently a contradietion.
§ 15. Change of measure. Any non-negativo additiiyo fimc-
tdon v{X) of a set (9E) may clearly bo regarded as a meamiro cor-
corresponding to the giyen additive class 9t. Whon such ti, funefcion
v{X) is defined only on a set E, vra can always continuo it (oil. § R, p.tt)
on to the whole space. The terms measure (»-), in l,«grti,'l (9E,v)
etc. are then completely determined for till wots (Sf), hut., in 'thin
case, it is most natural to consider only tJic Hii!>H«tH of ID for wliicli
the function v{X) was originally given.
[§ 15] Change of measure. 37
A5.1) Theorem-. Whenever, on a set ~E measurable (9?), we have
A5.2) v(X) = (9?) fg{co)dn(x) + 9{X)
k
where ${X) is a non-negative fnnetion, additive and singular (%p)
on E, and where g[x) is a non-negative function integrable (9E,/t)
over J3, then also
A5.3) (9E) ff(a>)dv(a>) = (9E) f f(x)g(co) du(x) + (SE) ff(w)d&(-x)'
X X X
for every set XQEmeasurable^) and for every function f(x) that possesses
a definite integral (SE,v) overX. If, further, the function f(x) is integrable
(??,?') over E, the formula A5.3) expresses the Lebesgue decomposition
of the indefinite integral / fdv on E, corresponding to the measure fz,
x
the function 6(X) = I fd& being the function of singularities (SE,ft)
k
of the indefinite integral j f dv.
x
Pro of. We may clearly assume that f{x) is defined and non-
negative on the whole of the set E. We see at once that, for each
set TC-E measurable (S6), foy{x)dv{x) =v(Y)= fg{x)dfi(<B) + d{Y) =
Y 7
= / cY(%) g(x) dpi(%) + j cy(x)d&(w), and hence also that for every
y y
finite function li(x) simple and measurable (SE) on a set X(ZE,
A5.4) fh{x)dv(x) = fji(x)g(x)du{x) + / h(x)dd-(oo).
xx x
Let now [lin(x)\ be a non-decreasing sequence of finite simple
functions measurable (9?) and non-negative on X, converging to the
given function f(x). Substituting Jin(x) for h{x) in A5.4) and making
h-^-oo, we obtain A5.3), on account of Lebesgue's Theorem 12.6.
If, further, f(x) is integrable (% v) over E, the identity just es-
established shows at once that the product f(x)g[x) is integrable (9?, fi)
over E and hence, that the indefinite integral ffgdy. is absolutely
continuous (SE, p) 'on E. On the other hand, the function 6{X)
vanishes on every set on which the function &(X) vanishes, and
therefore, is singular (% fi) on E together with &(X). This completes
the proof.

3g CHAPTTCK 1. The integral in sin aliKi.raot hjukh'. ta
The wide scope of Theorem Ifi.l is due to the fact, that, if
fi(X) and v(X) are any two measures iiNHOoiated with the Hume class
S? of measurable sets and we have at the same time ji{M) < -|-oo,
and v{B) < + oo, for a set E a % then the measure ?' can be repre-
represented on B in the form A5.2), when* 0@) is a function intogmble
(9-, n) over the set J3 and 5(X) is a non-negative function, swlditive
and singular (?,.») on the same set (cf. Th. 1-d.O). Heuee, with the
above hypotheses and notation, in order that j f <lv ¦.= j f <j <Ljtl
k x
should hold identically on JH, it is necessary and sufficient that
the indefinite integral ffdv be absolutely eontinuous (9c, ^m.) 011.JH.
k
This condition is clearly satisfied whemvver the meaHivro »'(A') is
itself absolutely eontmuoiis (S?,//).

CHAPTEB II.
Caratheodory measure.
§ 1. Preliminary remarks. In the preceding chapter, we
supposed given a priori a certain class of sets, together with a meas-
measure defined for the sets of this class. A different procedure is usually
adopted in theories dealing with special measures. We then begin
by determining, as an outer measure, a non-negative function
of a set, defined for all sets of the space considered, and it is only
a posteriori that we determine a class of measurable sets for which
the given outer measure is additive.
An abstract form of these theories, possessing both beauty
and generality, is due to C. Caratheodory [I]. The account that
we give of it in this chapter, is based on that of H. Hahn [I, Chap. VI],
in which the results of Caratheodory are formulated for arb-
arbitrary metrical spaces. This account will be preceded by two §§
describing the notions that are fundamental in general metrical
spaces.
§ 2. Metrical space. A space M is metrical if to each pair
a and b of its points there corresponds a non-negative number Q(a, b),
called distance of the points a and b, that satisfies the following
conditions: (i) g{a, &) = 0 is equivalent to a=b, (ii) Q{a, b) = Q(b, a),
(iii) Q(a, b)-\-Q[b, e)"^Q(a, c). In this chapter, we shall suppose
that a metrical space M is fixed, and that all sets of points that
arise, are located in M.
The notation that we shall use, is as follows. A point a is limit
of a sequence [a,,} of points in M, and we write a = lima,,, if
n
limp (a, «,,)=0. Every sequence possessing a limit point is said to

CHAPTE.R II. Canitlxiodory
be convergent. Given a set M, the upper bound of the numbera (>{a, b)
subject to aeM and beM in called diameter of M and in denoted
by 3(M). The set M is bounded if 6(M) in i'inide. For a cIumh 9Jt
of sets, the upper bound of the numbers <3(M) subject to M <-<$R
is denoted by 4(9Jl) and called eharaetemlde number of TO.
By the distance g(a, A) of a point a and a net A, we mean the lower
bound of the numbers (>{a, x) subject to no a A, and by Hut ditlaneo
q(A,B) of two sets A and B, the lower bound of the numbei'K (i(m, y)
for a? e A and j/ e?.
We call neighbourhood of a point a with radius ¦>•>(), of open
sphere &(a; r) of centre a ami radius r, the noI; of all pointa :v mioh
that q{a,x)<,r. The set of all points x mieh lihat Q(n,ii>)^lr in
called dosed sphere of centre a and radius r, and. in denoted by ii(a; r),
A point a is termed point of aeaumulation of a sat, A, if every
neighbourhood of a contains infinitely many points of: A. The net
A' of all points of accumulation, of A. is termed derived not, of A.
The set A-\-A\ that we denote by A, is termed closure of J., If yj.~ Af
the set J. is said to be closed. The points of a Hot, other than its
points of accumulation, are termed isolated. A net is isolated, if all
its points are isolated. We call, perfect, any (flowed set not containing
isolated points.
A point a of a set A is said to be an. internal point of A, if. there
exists a neighbourhood of « contained, in. A. Tho Hot of till the internal
points of a set A is called interior of A and denoted by A". The Hot
-4-—-4.° is termed boundary of A. If A = 4", the net ./I in said
to be open. Two sets A and Ji are called non-overlapping, if
A?o5J°
The class of all open sets will be denoted by © and that of
all closed sets by $• In accordance with the convention adopted
in § 2 of Chap. I, p. 5, open and closed sets will also be termed
sets (©) and sets (<$) respectively. We see at once that the com-
complement of any set (©) is a set ($) and viee-verna.
The sum of a finite number or of an infinity of open net-H, as
weU as the common part of a finite number of hhcIi aetH, in always
an open set. Any common part of a finite number, or of nn infinity,
of closed sets, and also any sum. of a finite number of huc.1i moIin,
.§2]
Metrical space.
41
are closed sets. Nevertheless, the sets E<r) and (®(j) (cf. Chap. I, § 2,
p. 5) do not in general coincide with the sets {%) and (©), although
every set E) is clearly a set {%„) and, at the same time, a set (®,s);
for, if F is a closed set and 0,, denotes the set of the points x such
that q{x,F)<ljn, we have F—TIG,,, where Gn are open. The eor-
n
responding result for the sets (©) is obtained by passing to the
complementary sets. Moreover, it follows that any set expressible
as the common part of a set (^) and a set (©) is both a set (%„) and
a set (©a).
We shall denote by SB, the smallest additive class that includes
all closed sets (of. Chap. I, Th. 4.2). This class, clearly, includes also
all sets (©a) and {%„). The sets (93) are also termed measurable (93)
(in accordance with Chap. I, § 4, p. 7). They are known as Borel sets.
We shall also give a few "relative" definitions having re-
reference to a set M. The common part of II with any closed set
is closed in M; we see at once that, for a set PCJ^ to be closed
in M, it is necessary and sufficient that P—M-Ps i. e. that the
set P contains all its points of accumulation belonging to M. Sim-
Similarly, any set expressible as the common part of M and an open
set is termed open in M.
Any set of the form Jf-S(a; r), where aeM and r>0, is called
'portion of M. If every portion of M contains points of a set A,
i. e. if A^)M, the set A is said to be everywhere dense in if. If a set B
is not everywhere dense in any portion of M, i. e. if no portion
of M is contained in B, the set B is said to be non-dense in M. In.
other words, a set B is non-dense in M, if, and only if, each portion
of M contains a portion in which there are no points of B. It fol-
follows at once that the sum of a finite number of sets non-dense in
the set itf is itself non-dense in M. The sets expressible as sums
of a finite or enumerably infinite number of sets non-dense in M
are termed (according to E. Baire [1]) sets of the first category in M,
and the sets not so expressible are termed sets of the second category
in M. In all these terms, the expression "in if" is omitted when
M coincides with the whole space; thus, [by "non-dense sets",
we mean sets whose closures contain no sphere and by "sets of
the first category", enumerable sums of such sets.
A set M is called separable, if it contains an enumerable
subset everywhere dense in M.

42
CHAPTER If. V/M'tMiiodory
§ 3. Continuous and semi-continuous functions. II f(a!)
is a function of a point, defined, on a set A. containing the point; a,
we shall denote by KA{f;a;r) and m.4(/;a;r), respectively, the
upper and lower bounds of the values assumed by f(w) on l;hc pot>
tion J.'S(a; r) of the set A. When r tends to 0, these two bounds
converge monotonely towards two limits (finite or infinite) whic:li
we shall call respectively maximum, and minimum of the function
f(co) on the set A at the point a, and denote by MU(/; a) and mA(f; a).
Their difference oA[f; a) = M.A(f; a)— mA{f; a) will bo called
lation of /(a?) on A at a. We clearly have
C.1) mA(f;a) *Zf{a) <M^(/; a) for every point (if A.
If f(a)=mA{f; a), the function /(*') in said to be, Hmow nemi-
continuous on the set A at the point a; .similarly, if f(a)~ M.A(f; a),
the function f[x) is upper sem/i-continuous on A at a. \f both con-
conditions hold together, and if f(x) is finite at tilio point a, i. c. if
nu(/; «)=Ma(/; a)^=oo, the function /(a?) in termed eoKtirmoua <m A
at the point a. Functions having the appropriate property at all
points of the set A, will be termed simply lower semi-continuous,
or %pper semi-continuous, or continuous, on A. In all these terms
and symbols, we usually omit all reference to A., when the latter
is an open set (in particular, the whole apace), or when A is kept
fixed, in which case the omission causes uo ambiguity.
From these definitions we conclude at onco that, if f{x) is
upper semi-continuous, the function —f{x) is lower semi-continuous,
and vice-versa; and further, that, if two functions are upper (or
lower) semi-continuous, so is their sum (supposing, of course, that
the functions to be added do not assume at any point; infinite values
of opposite signs).
C.2) Theorem. For every function /(,«) defined on a, net A, the
set of the points of A at which f(x) is not continuous on A, is the com-
common part of the set A with a set (%a).
Proof. Let us denote by Fn the set of the points x of A at
which either /(aj)=±oo, or oA{f;x)^l/n. The set F^2F,, con-
consists of all the points of A at which the function f(x) j/not con-
continuous. How it k easy to see that each of the sets F,, in closed, in. A,
i. e. that Fn=A-F,,. Therefore F is the common part of; A and the
set SF,,, which is a set ($„).
[S3]
Continuous and semi-continuous functions
C.3) Theorem. For a function of a point f[x) to be upper [lower']
semi-continuous on a set A, it is necessary and sufficient that, for
each number a, the set
C.4) E [x <¦ A; f(x) > a] [E [x e A; f{x) ^ a]]
-V .V
be closed in A, i. e, expressible as the common part of A with a set ($)•
Proof. We need only consider the case of upper semi-contin-
semi-continuous functions, as the other case follows by change of sign.
Let f(x) be a function upper semi-continuous on A, a an ar-
arbitrary number, and xoeA a point of accumulation of the set C.4).
For each r>0, the sphere S(a;0; r) then contains points of that set,
and this requires M4(/; a;0; r)>a and so JhLA[f; cco)~^>a. Since by
hypothesis M.A(f; a?0) = /(#„), we derive f(xa) ^ a, so that x0 belongs
to the set C.4). This set is thus closed in A.
Suppose, conversely, that the set C.4) is closed in A for each a.
Since the relation M_4(/; x) — f(x) is evident for any x at which
/(a?) = + oo, let xa be a point at which /(a?0) < + oo, and a any
number greater than f{xQ). The set C.4) is closed in A and does not
contain x0, and so, for a sufficiently small value ra of r, contains
no point of the sphere 8>(xo;r). Thus MA{f; oso)^.MA(f;xQ;rll)^.a for
every number a~>f(x0), and hence M.4(/; ,*0)^/(o;0), which, by C.1),
requires MA(f; a?0) = f(x0).
An immediate consequence of Theorem 3.3 (cf. Ohap. I, § 7,
particularly p. 13) is the following
C.5) Theorem. Every function semi-continuous on a set C3) is
measurable C3) on this set. More generally, if SE is any additive
class of sets including all closed sets [and so all sets measurable (93)),
every function semi-continuous on a set (S?) is measurable (9E) on
this set.
§ 4. Carath6odory measure. A function of a set F(X),
defined and non-negative for all sets of the space M, will be called
outer measure in the sense of Carathe'odory, if it fulfills the fol-
following conditions:
(Ci) r(Z)^. F{Y) whenever Jl',
(C) r[SXi)^2 P(Xi) for each sequence {XI} of sets,
i i
(G3) T{X+ Y) = F{X) + F{ Y) whenever q{X, Y) > 0.

44 AHAPTHR. If. ('iirntlieodory inciinurc.
It should he noted that of tliewc Uircti coiidHioii*, the liml, one, only, haw
a metrical character. Now in Hub §, hh well iih in Uie §§ f> mid E, wo nlui.ll uho
only properties (CJ and (C2) of Ui« Camtlieodoi'y meantim. Hoimi all l,he re-
results of these §§ remain valid in u perfectly iirbitniry ulmtnict Hpaco,
In order to simplify the wording, we nhall Huppose, iji the
rest of this chapter, except in § 8 which in concerned with oer'Uin
special measures, that an outer (liiratheodory measure P{X) is
uniquely determined in the space considered.
A set B will be termed measurable, with reaped to Ike given
outer measure F(X), if the relation. F(P+Q)---~ l\P)-\-T(Q) holds
for every pair of sets P and Q contained, respectively, in the not K
and in its complement 0/5; or, what amounts to the same, if
F(X)—F(X-JE)-\-F(X-(yM) holds for every Met A'. By condition ((!a)
this last relation may be replaced by the inequality 7'(.\')^
The class of all the sets that are tneamirable witli roHpect to /',
will be denoted by 2/\ We see at once that thin class includes all
the sets X for which F(X)~Q (iu particular, it inclndeH the empty
set). Moreover it is clear that complements of hcI.n Br) are also
sets Br).
The main object of this § is to establish the additivity of
the class 2t< (in the sense of Chap. I, § 4) and to prove that the
function T(X) is a measure (g'r) in the sense of dhap. I, § 9.
This result will constitute Theorems 4.1 and 4.5.
D.1) Theorem. If 8 is the sum, of a veqiumee {X,,)n i,2,... of mis (ilr)
no two of which have common points, the set A' is' aywin a net (iiij and
T(8) = 2 F[X,,); more generally, for each set Q
D.2) F(Q) = 3 F(Q -X,,) -|- F(Q . (W).
n
li
Proof. Let Si,= 2X,,. We begin by proving iinhictively that
all the sets & are measurable with respect to P, and that, for each k
and for every set Q,
D.3)
Suppose indeed, that Sp is a set Br) and that; i;he ine(|iui,lit,v (<!..'*)
holds for every set Q, when h = p. Hince X,,n is, by hyp.ithesiH,
a set {2r) and 8p-Xp+i = 0, we then .have
[§ 4]
Caratheodory measure.
45
= r(Q-Xp+l)
. GXP+VSP) + T(Q ¦
¦ C8P) =
• Sp
p+1
. X,,)
and this is D.3) for h — p + 1. In view of condition (O2), p. 43,
it follows further that r{Q)^T{Q-8p+i) + r{Q-C8p+^), which
proves that Sp+i is a set Br)-
Combining the inequality D.3), thus established, with the in-
inequality F(Q-C8n)~^r(Q-C8), we obtain, by making fc—>oo, the
inequality F{Q)^ jf-F^.Z,,) +r(Q-G8), and from this D.2)
follows on account of condition (C2).
Finally, the same condition enables us to derive from D.2)
that r(kQ)^r(Q.8)+I\Q-G8), and this shows that S is a set Br)
and completes the proof.
D.4) Lemma. The difference of two sets (?/<) is itself a set (fir).
Proof. Let Xe2r and Ye2r, and let P and Q be any two
sets such that PC-?—Y and (?CC(X— T). Write Q1 = §-Y
and (?2=(?-0Y. Making successive use of the three pairs of in-
inclusions QtCr, 0»CCY; PCI, g,C0(-T —r)-0YCCZ; and
we find JT(P)+JT((?) = 7I(P)+r(Q1)+r((J,)=
), which shows that X—Y is a set Br).
D.5) Theorem. 2r is an additive class of sets in the space M.
Proof. We have already remarked (p. 44) that the empty
set and that complements of sets Br) are sets Br). To verify the
third condition (ill) for additivity (cf. Chap. I, § 4, p. 7), let us
observe firstly that, on account of Lemma 4.4 and of the identity
X-Y = X — CY, the common part of any two sets Br) is itself
a set Br)- This result extends by induction to common parts of
any finite number of sets Br) and, with the help of the identity
H Xi— C// CXh we pass to the similar result for finite sums of
sets. Finally, if X is the sum of an infinite sequence {Xn},,=i, %...
of sets B/0, we have X=S,+^ (8n+i — 8n) where S^Jlj. Sow,
(T---1 *=1
clearly, of the sets 8± and 8n+i — 8,t, no two have common
points, and, moreover, by the results already proved, they all
belong to the class 2r. Consequently, to ascertain that X is a set (Sir),
we have only to apply Theorem 4.1. The class 2r is thus additive.

46 (.'JIAPTKR II. (ln,nil.hri)(loi'y menmnv..
Theorem 4.5 connects the considerations of this chapter with
those of the preceding one. Thus, in accordance with the, conven-
conventions adopted in Ohap. I, pp. 7 and 1 (i, the sets {Sir) may bo termed seLs
measurable Bv), and F(X) may, for Xt-Sir, bo regarded as a
measure associated with the class Sir. This class, together with,
the measure F, determines further the notions of functions
measurable B;'), of integral {2i;F), of additive fnuol.ion
of a set Br) absolutely continuous Br,I'), and the. other
notions defined generally in Ohap. I. Since, the outer measure F
determines already the class Sir, we shall omit in the sequel the
symbol representing this class, whenever the notation makes expli-
explicit reference to the outer measure; thus we shall say "function
integrable (/')" instead of "function integrablc {2r, Z1)" and the
integral (F) of a function /(a?) over a set M will be denoted simply
by ff{cc)dF{%), instead of by (Sr) j'f{ai)dF{tr).
k k
In accordance with Ohap. I, § 0, the value taken by P{X) for
a set X measurable B/') will be termed measure {F) of X; when
X is quite arbitrary, this value will be called its outer measure (I1).
If Eo is a subset of a set tJ such that F{13— ./?„)=(), then
for any function /(a?) on E the measurability Br) of / on E is
equivalent to its measurability B/') on .130. This remark and
Theorem 11.8, Chap. I, justify the following convention:
If a function /(a?) is defined only almost everywhere (/')
on a set 23, then, 230 denoting the set of the points of [<] at which
/(a?) is defined, by measurability {Sir), iwlegrability (F) and integral (I1)
of f on the set 23 we shall mean those on the seb Eo.
Let us note two further theorems.
D.6) Theorem. Given an arbitrary set 23, (i) F{.E-2 X,,)=2r{2<)-X,,)
for every sequence (X,,} of sets measurable Br) no two of which have,
common points, (ii) F{E-limX,,) = lim F B3-X,,) for every ascetiding
sequence {X,,} of sets measurable {Sir), and this relation remains valid
for descending sequences provided, however, that 2'{J<]-X1)~\~oq,
(iii) more generally, for every sequence \XH) of nets measurable {Sir)
r(J.liminfX7,)<liminfr(i7-Z,,), and, if further F(l<}-2 A,,) -..[•- oo,
then also F{E- lim. sup X,,) > lim sup F{E- A',,).
The. operation (A).
47
Part (i) of this theorem is contained in Theorem 4.1, and
parts (ii) and (iii) follow easily from (i) (cf. Chap. T, the proofs of
Theorems 5.1 and 9.1).
A part of Theorem 4.6 will be slightly further generalized. Given
a set 23, let us denote, for any set X, by i'? (X) the lower bound of
the values taken by F(E-Y) for the sets 7 measurable Br) that
contain X.
D.7) Theorem. Given a set E, (i) to every set X corresponds
a set X° D X measurable Br) such that Fs (X) = F{23 ¦ X"),
(ii) r(^-liminfX,,Xll(liminfX,!)<liminfr»(X,.) for every se-
quene'e {Xn) of sets, and, in particular, F{23-ixmX,,) ^ i
= limrj(I,,) for every ascending sequence (X,,).
Proof, re (i). For every positive integer n there is a set ,
measurable Br), such that F{23 ¦ 7u)^raE (X) +l/n. Writing
X°=/7 3r,,, we verify at once that the set X° has the required
properties.
re (ii). Taking (i) into account, let us associate with each set X,,
a set X°nZ)X,,, measurable (Sr) and such that F{E-Xl)=F%{Xn).
The set limini X,°Oliininf X,, is measurable (Sr) and, we therefore
n n
have, by Theorem 4.6 (iii)
F(E -liminf X,") s^limi
l{Xn).
The second part of (ii) follows at once from the first part.
* § 5. The operation (A). We shall establish here that
measurability Br) is an invariant of a more general operation
than those of addition and multiplication of sets.
We call determining system, any class of sets 9l={J.i,llllj n/t} in
which with each finite sequence of positive integers nltn2,..., nk there
is associated a set A,,lilh ,,k. The set
where the summation extends over all infinite sequences of indices
nx, w2, ..., nk, ..., is called nucleus of the determining system 31 and
denoted by NC31). The operation leading from a determining system
to its nucleus is often called the operation (A).

48
(JIIAPTKll 1L (lura.l.hcodory
The operation (A) wiw first dofitiwl by M. KoUHliu [l| in 1017. When
applied to Borel sofas, it. leadw to a wide olaMH of hoIh (Following N. lamin, wo call
them analytic) and those play 11.11 important part in l.he theory of h«|,h, in the
theory of real functions, mid even, in Home problomH of ehuwical type. A Hywtem-
atic account of tlio theory of those sets will be found in the troatiHON of Jl. II aim
[II], F. Hausdorff [II], ('. KuraU>wnki |I], N. LuHin |II| and W. Siotpiu-
ski [II].
We mentioned at the beginning of this § that the operation (A) inelndoH thomi
of addition and of multiplication of HotH. This remark must bo undui'stood iih FoIIowk:
If 3J1 is a class of sets such that the, nutikus of every determining xydem. formed of neln (9ft)
itself beloiigs to 9J1, then the sum and the emmnon fart of every mpunioe {Ni} of neln
(931) are aho sete(iR). In fact, writing Pnj,»2 «A-¦¦¦¦= JVh, and (Ji'u't-i "i,' ** we
see at once that the nuclei of the determining HyKtoniH {Ph|,«.>, ,„,„„} and
{Q»i<n2,-,"k} coincide respectively with, tlio mim nnd with the eimnnon part, of
the sequence {Ni}, Thus, Theorem 5,5, now to be proved, will, complete tlio I'lwilt
contained in Theorem 4.5, and in conjunction with. Theorem 7.'I, OMtablmh mea-
surability (S2j<) for analytic, hoIjh in any tmskieiil hjxlc.h (el1, N. IjiiHin and
W. Sierpinaki [1], N. Lusin [3, pp. 2E .-2d|, and W. Mierpii'iHki [12; lfi|).
The proof of this ean bo simplified if we immune regulu.riliy of the miter idouh-
nre I1 (of. §6) (sec C. Kuratownki [I, p, fiK|).
With every determining system <& = {Alii<Hi ,,fr), we Hhall alHo
associate the following seta. Wli<''?¦¦•••''«(?!) will denote,, for each
finite sequence h1} ?i,a, ..., h,H of positive integer, the. mim E.1)
extended over all sequences, %, i\, ...,%/,, ... such that n/'^h/ for
i—l,2,...,s. We see, at once, that the sequence (IP C1)}/, 1,1,...,
together with every sequence {isryii'/fai¦— ''*•7|(91))/, 1,2,,.., in monotone,
ascending and that
E.2) N(<Jl) = limN""Al), W'i-i'-i i>k{<a.) = lim'N''i. ''/,.''C3t).
h h
Further, for every sequence of positive integers hit h,M ..., hi,, ..., we
shall write
We see directly that if the sets ot the. determinins HyHtem 91
belong to a class of sets 2ft, the. seta 'N'y,,,^ /lftC3l) belong to Uio
class 9Waff.
[§5]
The operation (A),
E.3) Lemma. For every determining system 91={A,,,,,,.,,..,,,,ft)
for every sequence of positive integers \, h2, ..,, ]i/c,...
E.4) 3STAl(9l) • ^i^k,^)-...^„,,,,, iA(9l) ¦ ¦
Proof. Let a? be any point belonging to the left-hand side
of E.4). We shall show firstly that a positive integer n^^Ji^ can
be chosen so that, for each Jc^2, the point % belongs to a set
Ani-A,HiH-...-Ani,ni,...,nk for which nx=n\ and m^h for i—2, 3, ..., 7c.
Indeed, if there were no such integer nj, we could associate with
each index n^.hlt a positive integer h,,, such that a? belongs to no
product An-An,n.i-...-An,n.2 ,,A for which n^h when t=2,3,..., hn.
Denote by pi the greatest of the numbers \, Tc.2, ..., Jcih. The point x
thus belongs to none of the sets Ani-Anbn.2--.-Anhnii^n ^ for which
ni^fii when i= 1, 2, ..., plt and therefore is not contained in
their sum N/,,,/^ aPi(9I). This is a contradiction since, by hypo-
hypothesis, x is an element of the left-hand side of E.4).
After the index n°v we can determine afresh an index n^^Ji^,
so that, for each fc^3, the point a? belongs to a aQ>tAItl-Aniin2-...-An1,ni,...,nk
for which % = n°v n% = <n% and nt^ht when i = 3, 4,..., fc. For, if
there were no such index, we could find, as previously, a positive
integer p2~^3 such that x belongs to no product Ani-Anhn.,-..,-A,,lin^,.t,,
for which w1=wj and n^hi when i=2, 3, ..., p%. And this would
contradict the definition of the index n\.
Proceeding in this way, we determine an infinite sequence of
indices [nf, such that n^^-Jh when i = l, 2, ... and such that
oceAnQ-Anw — 'An,°,<n?,...,n0-—- Thus a?e!NCl), and this completes
the proof.
Lemma 5.3 is duo to W. Sierpiiiski [13]. The proof contains a slightly
more precise result than is expressed by the relation E.4) and shows that the
loft-hand side of that relation coincides ¦with the sum E.1), when the latter is
extended only to systems of indices nls n2, ..., nit, ... restricted to satisfy
ftj < /ij, n.z <-; h2, ..., nh -s?- hk, ... .
Let us call degenerate, a determining system {An1,n.i,...,n/t} such that, for
some sequence {hi,} of positive integers, we have An1,n.2,...,nl!'=0 whenever nii'?hk.
Then, for this sequence {Jilt), the relation of inclusion E.4) becomes an identity
and we are led to the following theorem:
If a degenerate determining system consists of sets beloriging to a class 331,
Us nucleus is a net (90lfimi). A similar theorem cannot hold for non-degenerate
systems: in fact, as shown by M. Souslin, the operation (A) applied to Borel
sets (and even to linear segments) may lead to sets that are not Borel seta (of.
?. Hausdorff [II, p. 182—184]).
S. Silks, Tlinory of Lho Inli'ural. 4

50
CUAPTliJJ If. OnratlKiodor.y in(wimiro.
E.5) Theorem. The nualens of any dekmmdnf/ ,s]/nlem <H~~[Anu,,^
consisting of sets measurable (Sir) *« itxdf >meaxurabl<> (Sir).
Proof. Let us write, for short,
N = T${Vl)> JV"i'- ¦••¦"* = N"i-"i' ••••"* C31), JVH),,,2 „,,=-:= N;,,,,,, /(/(
We have to show that, for any sot E
E.6) F(B)^F(B-N) + VtfhQN).
We may assume that F(E)<oot since (fi.(l) is evidently fulfilled
in the opposite case.
Let us denote (as in § 4, p. i.7) by F%(X) tlic lower bound
of the values of F(E-l) for nuts JO A' measurable («/•) and lol, e
be an arbitrary positive number. Taking into account (fi.U) and
Theorem 4.7, we readily define by induction a .sequence <>? positive*
integers {hk} such that 'f?:(N"<)^ P(I<J-N) — s/'2 and
r^(N'1i'i'i.-'i'k)^ri(N"hh"-'i'i, i)_.<?/2* /w fc-U, 3, ... .
Thus the sets -F^,^ nft3JV'i-"*¦•••"* being measurable (i!r) together
with the Ani,ni ,,kt
• N) — p
for each ft, and therefore
E.7)
Nhlhi hh)
N) + F(JE-ON
Sow the sequence of sets {JV/,!,/,^...,;,^. ii2,.,. is deHcending', and
byLemma5.3its limit is a subset of JV. Thesoquence (CW/(|>/,2 /,/(}/, i,2,...
is thus ascending and its limit contains the set (W. Henc(i, making
in E.7), we find, by Theorem 4.(i(ii), fcho inequality
)+r(E.GN)— e, and this impUos E.6) since e'is an
arbitrary positive number.
§6. Regular sets. A set X will bo called regular (with respect
to the outer measure F), if there exists a set A measurable (Sir),
containing X and suoh that F(A) = F(X). Kvory meaciurable not
is evidently regular, and so is also every set X. whose oulier nu^wure (F)
is infinite, since we then have J'(.Z)=/1(iF/)™cx.i. If every mt
of the space considered is regular with respect; to the outer measure F,
this measure is itself called regular; c:f. H. Halm 11, p. -1,'WJ,
0. Carath^odory [1; I, p. 258].
[§7]
Bore! sets.
51
Denoting by Fa(X) the lower bound of the values of F(Y)
for sets 131 measurable Br), we see readily that the relation
F°[X) — F(X) expresses a necessary and sufficient condition for
the set X to be regular. Prom Theorem 4.7(ii), taking for the
set E the whole space, we derive the following:
F.1) Theorem. For any sequence {Xn} of regular sets P(Kminf Xn)^.
n
^ liminf F(X,,), and, if further the sequence {Xn} is ascending
The generality and the importance of this theorem consist in that all
outer measures J1 that occur in applications satisfy the condition of regularity,
and, for these measures, the last relation of Theorem 6.1 therefore holds for
every ascending sequence of sets. Nevertheless, for measures that are not them-
themselves regular, the, restriction concerning regularity of the sets Xn is essential
for the validity of Theorem 6.1 as is shown by an example of irregular measure
due to C. Caratheodory [II, pp. 693—696].
We may observe further that, for any fixed set E, the function of a set J'e(X),
defined in § 4, p. 47, is always a regular outer measure, even if the given measure
-T(X) is not. Conditions (Cj) and (C2) together with that of regularity, are at onee
seen to hold, and (C3) may be derived from Theorem 7.4, according to which
closed sets are measurable
§ 7. Borel sets. We shall show in this § that, independently
of the choice of the outer measure F, the class ?r contains all
Borel sets.
G.1) Lemrna. If Q is any set contained in an open set G, and Qn
denotes the set of the points a of Q for which Q(a, GG)^l/n, then
lim F(Qn) = F(Q).
n
Proof. Since the sequence (Qn) is ascending and Q = lim Qn,
n
it suffices to show that liraF(Qn)^F(Q). For this purpose let us
write Dn=Qn+i—Q,,. We then have ()(i)n+i,Q7I)^l/w(»+l)>0,
provided that D,,+i=f=O and (?„=(= 0. Hence, taking into account
condition @3), p. 43, it is readily verified by induction that
G.2)
for every positive integer n. Writing, for short, an—J]F(Di/!) and

52 CHAPTUH II. OaraUu'odory men
>n,-i), we obtain at once, by condition (U2), n. <\;\,
G.3) "
+a,,
GO
How two possibilities arise: either both norms, }] F(J)U) und
/i 1
CO
^i^Da/t-i) have finite siims, or, one at least ha« it,s .sum infinite.
In the former case a,,—>0 and 6,,—>0, ho that, the required ineqiud-
ity r(QXlimP(Q,,) follows by making '«,->¦ oo in G.3); while, In.
the latter case, the inequality in obvioun, Kineo by G.2) we, have
th li P((?)
then lim JT(Q,,) = oo,
n
G.4) Theorem. Every set measurable (SB) in measurable (ii ).
Proof. Since the class 2/1 is additive, and winoo 'B in the smallest
additive class including the closed sets (cf. § 2, p. 41), iii is enough to
prove that every closed set is measurable (&/¦), i. <»., denoting any
such a set by F, that
G.5) r(P~\-Q) ^ F(P) -|- J'(Q)
holds for every pair of sets PC'F and QCQ-F. Since the set OF
is open, there is, by Lemma 7.1, a sequence {Qn\ of sets hu«1i that
QnCQ, P(Q«, -P) > ljn for w=l, 2, ..., and lim F{Q,,) = i1^). Thus
n, F) > 0, and so, on account of condition (ds), p. 48,
we derive r(P+Q)^Z1(P + (J,,) = r(P) + P(gJ1) for oacli" »,' and,
making w->oo, we obtain G.5).
The arguments of this § depend essentially on property ((!„) oE outer nu-as-
me, and on the metrical character of the spado il/, winch did not onto into
§§ &—6. It is possible however to give to those iu-gnmfiiits a form, independent
of condition (C,), valid for certain topological spacos thiit, iwo not nocosHfixily
metrical (of. N. Bourbaki [1]).
Prom the preceding theorem coupled with Theorem ,'W, we
derive at once the following
G.6) Theorem, (i) Every fimction mmmrabU C33) on a act Ii
is measurable (?r) on E. (ii) Every function thM in mm-eonbmuom
on a set Br) is measurable Bi<) on this net.
[§S]
Length of a set.
53
§ 8. Length of a set. We shall define in this § a class
of functions of a set that are outer Caratheodory measures and
that play an important part in a number of applications.
Let a be an arbitrary positive number. Given a set X, we
shall denote, for each e>0, by Alaf)(Z) the lower bound of the
sums ^[^(Z/)]01, for which (X-),¦=!,•>,... is an arbitrary partition of X
into a sequence of sets that have diameters less than s. When
e->0, the number A^iX) tends, in a monotone non-decreasing
manner, to a unique limit (finite or infinite) which we shall denote
by Aa(X), The function of a set Aa(X) thus defined is an outer
measure in the sense of Caratheodory. For, when e>0, we clearly
have (i) A$(X)*Z,A$(Y) if XC?, (ii) ^'(IX^iX^X,), if{X«}
n n
is any sequence of sets, and (iii) A%\X+Y) = AL?)(Z) + A^(Y),
if q[X, Y)>e. Making e->0, (i), (ii), (iii) become respectively
the three conditions (CJ, (C2), (C3), p. 43, of Caratheodory for Att(X).
We shall prove further that the outer measure Att (for any a>0)
is regular in the sense of § 6, i. e. that every set is regular with
respect to this measure. We shall even establish a more precise
result, namely
(8.1) Theorem. For each, set X there is a set Se<S>s such that
XCS and An(H) = A*(X).
Proof. For each positive integer n, there is a partition of X
into a sequence of sets {Xt^i^ix- such
(8.2) (i(I,w)<l/2» for 1=1,2,..., and f
We can evidently enclose each set Z/n) in an open set <xi"' such that
(8.3) dffl^Xd+l/nJ^Z?0).
Writing R — U E&T\ we see a* once tliat s is a set (®a) and that
XCB. Moreover, for each n, H = J? H-G-\n) and the relations (8.2)
and (8.3) imply that 6{R-Gf])<lln for i=l, 2,... and that
J>)]. Making »->oo,
J[
we find in the limit Aa{H)^Aa(X), and, since the converse in-
inequality is obvious, this completes the proof.

54 CHAPTER II. Oarntheodory muiiHuro.
In Euclidean w-dimensional space It,, (hcc Ohap. .1.11), the sets
whose measure (A,,) is zexo may be identified, with tho.se of meamire
zero in the Lebesgue sense. By analogy, in any metrical space,
sets whose measure (A«) is zero are termed nets having a-dimensional
volume zero, and in particular, when a=l, 2, 3, sets of zero length,
of zero area, of zero volume, respectively. For the same reaHon, sets
of finite measure (An) are termed sets of finite a-dimensional volume
(or of finite length, finite area, finite volume, in the caseh <t~ 1, 2, &),
In particular, in Mv i. e. on the straight line, the outer measure A1
coincides with the Lebesgue measure, and, on this account, we call
the number A^X), in general, outer length of X, and when X is
a set measurable (S.i,), simply, length of X, For short, wo often -write A
instead of Av
We have, mentioned only the more olomonliary proportion of tho tudiumnw
Aa, those, namely, that wesliall have some further occasion to uho. lAor u. doopw
study, the reader should consult I1, llauwdorff [J, |. Among the, rowinrolioH de-
devoted to the notion of length of sets in Kuclidoun spawn, Hpwial mention muni;
be made of the important memoir of A. H. Bo.Riooviteli [I |; of. iiIho W. Wiur-
piiiski [1] and J. Gillis [1].
§ 9. Complete space. A metrical space is termed aomplde,
if a sequence {a,,} of its points converges whenever lini <?(«,„, a,,)=0.
In any metrical space, this is evidently a necessary condition for
convergence of the sequence ¦(«„}, but, as a rule,, not a sufficient
one. The following theorem concerns a characteristic property of
complete spaces:
(9.1) Theorem. In a complete space, when (I?,,) is a descending
sequence of closed and non-empty sets whose diameters tend to zero, the
common part TlFn is not empty.
n
Proof. Let a,, be an arbitrarily chosen point of 1'V For n^m,
we have q{am, an)^d(Fm), and hence lira q[am, a,,)=0. The sequence
771,11-^-CO
{an\ is thus convergent. Now the limit point of this sequence clearly
belongs to all the sets Fm, since aneF,,C^m whenever n^m, and
since the sets Fm are closed by hypothesis.
(9.2) Baire's theorem. In a complete spaee M, every non-empty
set (©j) is of the second category on itself, i. c. if II is a nel (<S,i)
in M and H=2Hn, one at hast of the sets //„ in even/where dense
in a portion of H.
[§9]
Complete space.
'00
Proof. Suppose, accordingly, that H = ff Gn where Q,, are
71=1
open sets, and further that
(9.3) H = EHa
where Hn are non-dense in R. The partial sums of the series
(9.3) are then also non-dense in H (cf. § 2, p. 41), and it is easy to
define inductively a descending sequence of portions Sn of H
n
such that (i) AC^n, (ii) Sa-ESj=0, (iii) <5(^)<1/b. On account
y=i _
of Theorem 9.1 and of (iii), the sets Sn have a common point, which
by (i), belongs to all the sets Gn, and so to R, while at the same
time, by (ii) it belongs to none of the Hn. This contradicts (9.3)
and proves the theorem.
The ease of Theorem 9.2 that occurs most frequently, is that in which E
is a closed set. For closed sets in Euclidean spaces the theorem was established
in 1899 "by E. Baire [1]. To Baire, we owe alBO the fundamental applications
of the theorem, which have brought out the fruitfulness and the importance
of the result for modern real function theory. As regards the theorem by itself
however, it was found, almost at the same time and independently by W. F.
Osgood [1] in connection with some problems concerning fimotions of a complex
variable (ef. in this connection, the interestingfarticle by W. H. Young [7]).
The general form of Theorem 9.2 is due to P. Hausdorff [I, pp. 326—328; II,
pp. 138—145].
If * is a non-isolated point (cf. § 2, p. 40) of a set K, the set (a)
consisting of the single point * is clearly non-dense in M. It there-
therefore follows from Theorem 9.2 that
(9.4) Theorem. In a complete metrical space, every non-empty
set (©j) without isolated points, and in particular every perfect set,
is non-enumerable.
More precisely, "by a theorem of W. H. Young [1], every set that fulfills
the condition of Theorem 9.4 has the power of the continuum; cf. also S1. Hatis-
dorff [II, p. 136].

CHAPTEB III.
Functions of bounded variation and the
Lebesgue-Stieltjes integral.
§ 1. Euclidean spaces. In this chapter, the notions of
measure that we consider undergo a further specialization. Accord-
Accordingly we introduce for Euclidean spaces, a particular class of
outer measures of Caratheodory, determined in a natural way by
non-negative additive functions of an interval. These outer meas-
measures in their turn determine the corresponding classes of meas-
measurable sets and measurable functions, and lead to processes of
integration usually known as those of Lebesgue-Htieltjes.
By Euclidean space of m dimensions It,,,, wo moan the Hot of
all systems of m real numbers (wu %i, ..., wm). The number wk is
termed Mh coordinate of the point (at], xt,..., ,»,„). The point @, 0,..., 0)
will be denoted by 0.
By distance Q(os,y) of two points x=(os\, x%, ...,*„,) and
y—tyhVh —tVin) in the space JRm, we mean the non-negative number
[B/i—*iJ + B/2—x%f + ...+ (ym~xmf\Vs. Distance, thus defined,
evidently fulfills the three conditions of Chap. II, p. 40, yjid hence
Euclidean spaces may be regarded as metrical spaces. All the defi-
definitions adopted in Chap. II therefore apply in particular to spaces
JRm. In § 2 we supplement them by some definitions more exclusively
restricted to Euclidean spaces.
The space J?x is also termed straight line and tho Hpijico /t'a,
plane. Accordingly, the sets in i^ will often bo called linear, and
those in M2 plane sets.
Intervals and figures.
57
§ 2. Intervals and figures. Suppose given a Euclidean
space -Bm.
The set of the points (xh xh ..., xm) of JRm that fulfill a linear
equation 0,1X1+0,1X2+...+amxm=b, where b, ait a>, ..., a.,, are real
numbers and, of these, the coefficients au a>, ..., am do not all vanish
together, is called hyperplane aiXi+aiXi+ ...+amxm=b. For each
fixed &=1, 2, ..., to, the hyperplanes %=6 are said to be orthogonal
to the axis of xk. The term hyperplane by itself, will be applied ex-
cbisively to a hyperplane orthogonal to one of the axes. In Iilf
hyperplanes coincide with points. In i?2 and li$ they are respectively
straight lines and planes.
Given two points a=(a,i, a*, ..., am) and b=[bi, fo, ...,&,„)
such that ajt^bz for ft=l, 2, ..., m, we term closed interval
[«i, b\\ a,i, bi; ...; a,m,bm] the set of all the points (a?i, x>,..., xm) such
that ak^xji^bk for ?=1, 2,..., m. The points a and b are called
principal vertices of this interval. If, in the definition of closed in-
interval, we replace successively the inequality aft^*A^&A by the in-
inequalities A°) ak<xh<.bk, B°) <Sft<3sA<&A and C°) ah<ask^bk, we
obtain the definitions A°) of open interval («i, 61; a2, &2;...; am, bm)s
B°) of interval half open to the right [aL, 61; 0,1, bf,...; a,», hm) and C°) of
interval half open to the left @,1, &i; a^, 62;...; am, bm]. If «*=&* for
at least one index A-, all these intervals are said to be degenerate.
In what follows, an interval, by itself, always means either a closed
non-degenerate interval or an empty set, unless another meaning
is obvious from the context.
We call face of the closed interval I=[0,1, b\; a^, bi\ ...; am, bm]
the common part of I and any one of the 2m hyperplanes xl!=ai1
and xk=bh, where 7<:=1,2,..., m. If J is an open or half open in-
interval we call faces of J those of its closure J. We see at once that
the faces of any non-empty interval 1 are degenerate intervals and
that their sum is the boundary of the interval I.
If 61—at=bi—*2=-=&m—a,n>Q, the interval [*i,&i; ai,bi;...;am,bm]
is termed cube (square in -B2)- We define similarly open cubes and
half open cubes (half open to the right or to the left).
We call net of closed intervals in Xtm any system of closed non-
overlapping intervals that together cover the space .R,,,. Similarly,
by net of half open intervals, we mean a system of intervals half
op mi on the same side, no two of which have common points, and
whose sum covers ltm. A sequence of nets {91*} (of closed or of half
open intervals) is regular, if each interval of 9U+i is contained in

58
CHAPTER TTI. Functions of bounded vuriiition.
an interval of %, and if the characteristic numbers d[%;) (cf.
Chap. II, p- 40) tend to 0 as h->oa. Given a net of half open, inter-
intervals, we clearly change it into a net of closed intervals by replacing
the half open intervals by their closures. The flume operation changes
any regular sequence of nets of half open intervals into a regular
sequence of nets of closed intervals.
B.1) Theorem. Given an enumerable system of hyperplanes xk=aj,
where Jc=l, 2, ..., m, and j = l, 2, ..., we can always construct a
regular sequence {%,) of nets of cubes (closed or half open) none of
which has a face on the given hyperplanes.
To see this, let b denote a positive number not of tho form
qajjp where p and g are integers and j==l, 2, .... Huch a. number b
certainly exists, since the set of the numbers of tho form, qctj/p is
at most enumerable. This being so, for each positive integer h let
us denote by 9U the net consisting of all the cubes half upon to the
left (^6/3*^+1N/2*; pb/2« (+lN/a* b/2" (+ N/*
p
a*; ...; pmb/2", (pm+ 1N/2*J,
left (^6/3*,^+1N/2*; p,b/2«, (pg+)/; ; pm/, (pm+ 1N/2J,
where pi, p-2, ..., pm are arbitrary integers. The sequence of nets
{91*} evidently fulfills the required conditions.
Let us observe that, given a regular sequence {3i/t} of nets of
half open [closed] intervals, every open set G is expressible as the
sum of an enumerable system of intervals [%,) without common points
[non-overlapping]. To see this, let 93^ be the set of intervals of 9?j that
lie in G, and let 2R/;+i, for each fc>l, be the sot of intervals (9}/H i)
that lie in G but not in any of the intervals (J1/,). Hinco A(%,) >()
as ~k->oo, the enumerable system of intervals _V2ft/, covers the set G,
it
and the other conditions required are evidently satisfied also.
On account of Theorem 2.1, we derive at once the following
proposition which will often be useful to us in the course of this
Chapter:
B.2) Theorem, Given a sequence of hyperplanes {Hi\, every open
set G is expressible as the sum of a sequence of half open cubes
without common points [or of closed non-overlapping cubes] tohose
faces do not lie on any of the hyperplanes II).
A set expressible as the sum of a finite number of intervals
will be termed elementary figure, or simply, figure. 10very sum of
a finite number of figures is itself a figure, but thin is not in, tfoueral
the case for the common part, or for the difference, of two figure's.
[§3]
Functions of an interval.
59
We shall therefore define two operations similar to those of multi-
multiplication and subtraction of sets, but which differ from the latter
in that, when we perform them on figures, the result is again a figure.
These operations will be denoted by 0 and 3 and are defined by
the relations
AQB = (AJ}j5 and A 9 B = {A—Bf.
The relation AQB=Q means that the figures A and B do not
overlap (cf. Chap. II, p. 40).
Given an interval I=[ai,b1;a1,bi;...;am,bm], the number
(&i—ai)-(k—a±)-...-[bm—am) will be called volume of I [length for
»i = l, area for m=2), and denoted by L(I) or by \I\. If 1=0, by
L(I)=|I| we mean 0 also. When several spaces Iim are considered
simultaneously, we shall, to prevent any ambiguity, denote the
volume of an interval J in JRm by Lm(I). We see at once that every
figure R can be subdivided into a finite number of non-overlapping
intervals. The sum of the volumes of these intervals is independent
of the way in which we make this subdivision; it is termed volume
[length, area) of R and denoted, just as in the case of intervals,
by L(E) or by |J?L
§ 3. Functions of an interval. We shall say that F[I) is a
function of an interval on a figure i? [or in an open set 6?], if F(I) is a
finite real number uniquely defined for each interval I contained
in R [or in 6?]. To simplify the wording, we shall usually suppose
that functions of an interval are defined in the whole space.
A function of an interval F[I) will be said to be continuous
on a figure R, if to each e>0 there corresponds an ??>0 such that
|I|<«7 implies |.F(I)|<e for every interval IC.R- A function of
an interval will be said to be continuous in an open set G, if it is
continuous on every figure R(^G. Finally, functions continuous in
the whole space will simply be said to be continuous.
The reader will have noticed that we use the terms "on" (or
"over'') and "in" in slightly different senses. We may express the
distinction as follows. Suppose that a certain property (P) of func-
functions of a point, of an interval, or of a set has been defined on fig-
figures. We then say that a function has this property in
an open set G, if it has the property on every figure RC&-
Further, if a function has the property (P) in the whole space, we
say simply that it has the property (P). Thus, for instance, a function

60
CHAPTER III. FuiuitioiiH til' "bmuiilud variation.
of a set <[>{X) is additive C3) ((if. Chap. .1, § 5 and Chap. .II, § 2)
in an open .set G, if it is additive C3) on every figure RQO-
if (i is a measure C3) (el Chap. 1, § 0), a function of a point, de-
defined in the whole space, is said to be integrablo C3, //), if it ;«
integrable C3, fi) on every figure, and no on,
We shall call oscillation ()(/''; F) of a function of an interval F
on a set F,, the upper bound of the values \lP{f.)\ iov intervals I(ZF.
If D is an arbitrary set and R a figure, we shall donol;e by On(F;]))
the lower bound of the numbers O(F; Ji-G), whore 0 is any open
set containing I); the number Oi<{F;D) will, be termed oxeAUatiun
of F on B, at the set D. Finally we shall say simply oNcillation of F
at D, and use the notation o(F; D), for the upper bound of tbe
numbers oR(F;D) where It denotes any figure (or, what amounts
to the same, interval or cube).
In the sequel, D will usually be a hyperplaiie (a point, in. Itu
a straight line in i?8) or else the boundary of ti. figure. .In. l,huso cases
we shall say that the function F is wnMnitoiix, or dmwrdinuoun,
at D on R, according as oR{F;D) = () or oR [I? ;]))>(). Similarly,
we shall say that F is continuous, or discontinuous, at I), according
as o(F; Z>)= 0 or o(F; D) > 0.
C.1) Theorem. In order that a function F of an interval be, con-
continuous on a figure R [or in the whole, space], it is wwsaanj and suf-
sufficient that on{F;D) = Q [or thai o(F; .#)=()] for mmy hypur-
plane D.
Proof. Since the condition is clearly necessary, lot, uh suppose
that the function F is not continuous on li. There is than a number
?0>0 and a sequence of intervals [I{n)^\a{{:), b\"]; ...; al',',\ 1$]}
contained in B and such that for n=l, 2, ..., \F(l{"])\>i0, and that
I(n)j->•(). By the second of these conditions, we can extract from
the sequence {I^), a subsequence {l{n"\ in such a manner that
UmF/1* —aini'])=0 for a positive integer *=?0<m. The so(,|U(iiu'.os
\% ' j\=i, 2,... an(l \^oy'}j(=i,a,,,. ttien have a common limit point a,
and, denoting by D the hyperplane x,^a, we me thai; every
open set GOD contains an infinity of intervals /"", so that;
(FI)
Functions of bounded variation.
61
§ 4. Functions of an Interval that are additive and
of bounded variation. A function of an interval F(I) is said to
be additive on a figure Eo [or in an open set G], if F(I1+Ii)=
=F(,I1)+F(IZ) whenever Iv I.2 and Ix+I2 are intervals contained
in Bo [or in G] and Ix, I2 are non-overlapping. A function additive
in the whole space, is for instance the volume L(I)=|I|. Just as
in the case of the function L(I) (cf. § 2, p. 59), every additive
function of an interval F(I) on a figure Ba [or in an open set (?]
can be continued on all figures in BQ [or in 6?] in such a manner
that JI(i?1+.R2) = F(.S1)+.F(.K!!) for every pair of figures J^C^o and
J?aC-Ro that do not overlap. In the sequel, we shall always sup-
suppose every additive function of an interval continued in this way
on the figures.
If F is an additive function of an interval on a figure -fi'o, we
shall term respectively upper and lower (relative) variations of F
on ff0 the upper and lower bounds of F(R) for figures i?C^o- 'We
denote these variations by W(F; Ro) and W(.F; Ro) respectively.
Since every additive function vanishes on the empty set, we have
W(F; RQ)^0^W(F;B0). The number W(F;B0) + W(F;E0)\, dearly
non-negative, will be called absolute variation of F on Ro and de-
denoted by W(.F; Bo). If W(F; J?0)<+oo, the function F is said
to be of bounded variation on i?0. In accordance with the convention
of § 3, p. 59, an additive function of an interval in the whole space
is of bounded variation, if it is of bounded variation on every figure.
It is obvious that a function of bounded variation on a figure Bo
is equally so on every figure contained in i?0, and also that the sum,
the difference, and, more generally, any linear combination of two
additive functions of an interval that are of bounded variation on
a figure, is itself of bounded variation on the same figure.
An additive function whose values are of constant sign is
termed monotone. A non-negative monotone additive function is
also termed non-decreasing (for the same reason as in the case of
non-negative additive functions of a set, cf. Chap. I, p. 8). Similarly,
non-positive additive functions are also termed non-increasing.
Every monotone additive function F of an interval on a figure J?o
is clearly of bounded variation on Bo.
If a function is of bounded variation on a figure Bo, its relative
variations on -Z?o are evidently finite. Conversely, if for an additive
function F of an interval on a figure Bo, one or other of the two
relative variations is finite, then both are finite, and therefore the

CHAPTER III. Functions of hounded variation.
absolute variation is finite. For, if W(F; 7t!0)<+oo say, then as
W[F;R0) is the lower bound of the numbers ifl^^/y-FI^^/^
where B is any figure contained in. 7^>, we find. W (./''; Ro) >
^F{B0)—W{F;B0)>—oo. Moreover, this last inequality may also bo
•writtenin the form F{B0)^W(F;Ra)+W(F;K0). Here we replace.//'
by—.]p to derive the opposite inequality and then finally the equality
F(R0) = W{F; -ffo)+W(-^; -So)- Hence, any additive function of bounded
variation is the sum of its two relative eariationn on any figure for
which it is defined. This decomposition i»s termed, the Jorilmi de-
decomposition of an additive function of bounded variation, and is
similar to the Jordan decomposition of an additive function of
a set (Chap. I, § 0).
If F is an additive function of bound ml variation of ;ui, inter-
interval on a figure Bn, the three monotone functions defined for every
figure RBB0 by the relations
W(R)=W{F; B),
--W(F;B) and
are likewise additive on B,o and are termed, respectively, absolute,
upper, and lower variations of I'7. The first two are non-negative
and the third non-positive. It therefore follows from the Jordan
decomposition that every additive function of bounded variation
on a figure is, on this figure, the difference of two non-doereasing
functions. The converse is obvious.
We shall now prove some elementary theorems concerning
continuity properties of functions of bounded variation.
D.1) TJieorem. If F is an additive fwndum of an interval,
of bounded variation on a figure Ra, (i) the series _VOnu(F;.D,,) eon-
n
verges for every sequence [Dlt} of hyperplanes distinci from, one another,
and (ii) then is at most an enumerable infinity of hyperplanen I) such
that oRo(F;D)>0.
Proof. In the proof of (i) we may clearly suppose that the
hyperplanes I)n are orthogonal to the same axis. Consider now the
first k of these Dn. We can associate with them, k non-overlapping
intervals Iu h,...,Ik, contained in Ba and such that ol{n(:P;O,,X
^\F(I,,)\+1/W for »=lf2,...,fc. Hence JX(i';Z),,Ki\F(L,)\-H//^
n~-1 ii j
<W(-F; B0) + ljk, and, since k is an arbitrary positive integer,
|o^;J),KW(?;iJG)
co.
Functions of bounded variation.
To establish (ii), suppose that there is a non-enumerable in-
infinity of hyperplanes D such that Oro(F;D)>0. There would then
be a positive number e, such that Ojt0{F',D)>s for an infinity,
(which would even be non-enumerable) of hyperplanes. But this
clearly contradicts part (i) which has just been proved.
We obtain at once from Theorem 4.1 the following
D.2) TJieorem. For each additive function of an interval of
bounded variation, there is at most an enumerable infinity of hyper-
hyperplanes of discontinuity.
D.3) Theorem. If F is an additive function of an interval of
bounded variation on a figure Bo and we write W(I) = W(F; I),
the relations Qr,,{F; D) = 0 and o#n(TF; D) = 0 are equivalent for
every hyperplane D.
Proof. Suppose, if possible, that
D.4) oRil{F;B) = Q and D.5) Or,,(W;D)>s
for a hyperplane D and a number e> 0. We shall show that it is
then possible to define a sequence of figures {i?n}«=i,2,...j non-over-
non-overlapping, contained in Ro, and such that
F(En)\>ej2.
D.6) E;i-D=0 and D.7;
To see this, suppose defined k non-overlapping figures J?i,i?2,.... JS*
contained in BQ, and let D.6) and D.7) hold for )j=l, 2,..., k.
On account of D.5) there is then an interval IC-R0 sucn tnat J--E/i=0
for n=l, 3, ..., k, and such that W(F; I)=W(!)>?. Hence, there
exists a figure RQI such that \F{E)\>W(F; I)j2>ej2. Moreover
since D.4) asserts that F is continuous on Ba at D, we may sup-
suppose that B-D = Q. But if we now choose Bii+t=R, we see that
the figure i?A+i does not overlap any of the figures Bn for
l^n^ik, and that D.6) and D.7) continue to hold for n=k~\-l.
Having obtained our sequence {Rn}, we conclude from D.7)
that W(F;Bo)^ y \F(B,,)\=cc, and this contradicts our hypotheses.
n
The conditions D.4) and D.5) are thus incompatible, i. e. D.4)
implies o^ii(W;D) = 0. And since the converse is obvious, this
completes the proof.
From Theorems 4.3 and 3.1, we obtain at once the following
D.8) Theorem. In order that an additive function of bounded
variation on a figure Bo be continuous on i?0, it in necessary and suf-
sufficient that its three variations be so.

CHAPTER, III. Functions of bouiulpd variation.
§5. Lebesgue-Stieltjes integral. Lefoesgue integral
and measure. We need hardly point oat tho analogy between
additive functions of bounded variation of an interval and additive
functions of sets. This analogy will be made clearer and deeper in
the present §, by associating a function TJ* of a net with each addi-
additive function U of bounded variation of an interval. In order to
simplify the wording, we shall suppose that the functions of an
interval are defined in the, whole space.
Suppose given in the first place, a non-negative additive
function TJ of an interval; we then denote for any set E, by TJ*(J<J)
the lower bound of the sums 21 Pftli whore {//,) in an arbitrary
it
sequence of intervals such that EC. ?11- For an arbitrary additive
function TJ of bounded variation, with tho upper and lower varia-
variations Wi and Tf2, we denote by W* and (—W^)* tho functions
of a set that correspond to the non-negative functions W, and, —W2,
and we write, by definition, U*=Wf — (—WB)*. The function U*
is thus defined for all sets and is finite for bounded nets.
When TJ is non-negative, TJ* is an outer measure in the House
of Carathe'odory, i. e. fulfills the three conditions (Gx), (Oa) and (C3)
of Chap. II, § 4. Condition (C3) is the only one requiring proof, the
other two are obvious. Let therefore A and ? be any two sets whose
distance does not vanish, and let s be a positive number. There
is then a sequence {!„) of intervals such that A-\-BB]^I", aud
II
2U{In)^U*(A+B)-\-t. We may clearly suppose that the intervals
of the sequence have diameters lees than q (A, li), i. e. that none
of them contains both points of A and points of B. We then
have U*{A)+U*{B)<:y;'O{InX'Cf*(A-\~B)+e. This gives the in-
n
equality U*{A)+U*{B)^U*(A+B) and establishes condition (O3).
The function TJ'*, determined by a non-negative function, of
an interval, itself determines, since it is an outer Carathe'odory
measure (cf. Chap. II, § 4, p. 46), the class 2r;» of the sets measurable
with respect to TJ* and the process of integration (U*). To simplify
the notation, we shall omit the asterisk and write simply 2a for
2u*, integral (TJ) for integral A7*), measure (U) of a set in-
instead of measure {TJ*), j fdU instead of jfdlf*, and ho oil.
L§5]
Lebesgue-Stieltjes integral.
This slight change of notation cannot cause any confusion, since
the measure TJ* is uniquely determined by the function cf an in-
interval TJ.
When TJ is a general additive function of an interval, of bounded
variation, we shall under? band by 2a the common part of the classes
?tf, and 2-jk,, where Wx and W2 denote respectively the upper and
lower variations of TJ. A function of a point f(x) will be termed
integrable {TJ) on a set E, if /(as) is integrable (TFX) and (—F2) si-
simultaneously; by its integral (TJ) over E we shall mean the number
ffdW1— Ifd(—TF2), ar±d we write it IfdTJ as in the ease of
E E E
a non-negative function TJ. This integration with respect to an
additive function of bounded variation of an interval
is called Lebesgue-Stieltjes integration. In the ease of the integra-
j
tion over an interval I=[a,b] in i?lt we frequently write I f&TJ
a
for ffdU.
i
When the function TJ is continuous, every indefinite integral (TJ)
vanishes, together with the function TJ*, on the boundary of any
figure. Consequently, an indefinite integral with respect to a
continuous function TJ of bounded variation of an interval is
additive not only as function of a set Bc) out also as function
of an interval.
The most important case is that in which the given function
of an interval 77 is the special function L (cf. § 2, p. 59) that denotes
the volume of an interval. The outer measure (L) is also termed
outer Lebesgue measure, and the integral (L), Lebesgue integral,
while functions integrable (L) are often called, as originally, by
Lebesgue, summable. The class of sets 2l will be denoted simply
by 2. The outer measure (L) of an arbitrary set E is written meaSeJB,
and meas E without the suffix when E is measurable B). We shall
also denote this outer measure by \E\ or by L(i5) (or sometimes
by Iim(E) in Mm), thus extending to arbitrary sets the notation
adopted for all figures R, since for the latter, as we shall see
(cf. Th. 6.2), the measure (L) coincides with the values LB?)=|JB|.
Finally, owing to the special part played by Lebesgue measure
in the theory of integration and derivation, the terms "measure
of a set", "measurable set", "measurable function", and
S. Saks. Theory of the Integral. 5

66
CHAPTER III. Functions of hounded variation.
so on, -will, in the .sequel, be understood in tho Lebcsgue nouho,
whenever another sense lias not been explicitly assigned to liheiri.
We also modify slightly the integral notation for a. Lobosgue
integral; and instead of ff{x)iL(w) wo write j j{x)dx, or else
Ohi,, ...,:b,,,)
... dx,,,, when we wish to indicate the
number of dimensions of the space It,,, under consideration. This
brings us back to the classical notation.
A special part, similar to that of Lebosgue measure in the
theory of the integral, is played by Borel Notn in l;hc theory of ad-
additive classes of sets. In the first place, it follows from Theorem 7.4,
Chap. II, that every class 2u, whero // is an additive funoliiou of
bounded variation of an interval, contains all tho wots (3). Iu the
sequel, we shall agree that additive functions of a ,s«t will
always mean functions additive C3), unless there is oxpli.di.ti reference
to another additive class of sots. Wimilarly, additive functions of
a set that are absolutely continuous (93, L) or singular (93, L), will
simply be called absolutely oontinuow or singular. In noinl; of fact,
Theorem 6.6 below, winch asserts that every net measurable (?)
is the sum of a set (93) and a set of zero measure (L), will show that
every additive function of a set, absolutely continuous (93, L), can
be extended in a unique manner to all sets B) so iiiH to remain ab-
absolutely continuous B, L).
The special role of the measure (L) and of tho note C33) Hhowed itnolf already
during the growth of the theory. LobOBguB mousure wan tho Htarfcing point for
further extensions of the notions of measure and integral, whwean tho Boiel
sets were the origiu of general theories of additive classon and XniiotioiiH. Tho
sets (95) were introduced, with measure (L) defined for thorn, by E. .Born). [I,
p. 46—SO] in 1898. But it was not until some yearn later that J[. Lobonguo [1; I],
by amplifying and extending the definition of measure (L) to nil sots (ii), made
clear the importance of this measure for the theory of integration and especially
for that of derivation of functions. Vide E. Borol [1] and II. Ix»<boHguo [(!].
We have already seen iu § 1 of this book, how, by an apparently very Blight
modification of the classical definition of Rieinann, w<> obtain tho Lclxwgue, in-
integral. A similar remark may be made -with regard to tho rolationHlnp of bo-
besgue measure to the earlier measure of Peauo-Jordan. Tlui outer lnoiuurro
of Peano-Jordan for a hounded sot 14 is tho Jowor bound, of Urn uuiiiIimh V \f,t
n
where {/„} is any finite system of intervals liovoriug ,!'J. IjoIhwkuo'h hn.|>|>y itlou
was to replace m this definition, tlio finite syatoniM of inlorvulH liy oinuiusr-
able ones.
[§6]
Measure defined by a function of an interval.
67
We have given in the text a more general form to Lebesgue's definition,
relative to an arbitrary non-negative function of an interval. This relativizing
of Lebesgue measure is due to J. Radon [1] and to Ch. .T. de la Vallee-Poussin
[1; I]. The parallel extension of the Lebesgue integral is also due to J. Eadon.
In the text we have termed it Lebesgue-Stieltjes integral; it is sometimes also
termed Lebesgue-Radon integral or Eadon integral. For a systematic
exposition of the properties of this integral, vide H. Lebesgue [II, Chap. XI].
A particularly interesting generalization of the Lebesgue integral, of the Stieltjes
type, has been given by X. Bary and D. Menchoff [1]; it differs considerably
from the other generalizations of this type. Finally, for an account of tho
Riemann-Stieltjes integral (which we shall not discuss in this volume) vide
W. H. Young [2], S. Pollard [1], R. C. Young [1], M. Frechet [5] and
G. Fichtenholz [2].
It was again J. Eadon [I, p. 1] who pointed out the importance of the
Lebesgue-Stieltjes integral for certain classical parts of Analysis, particularly
for potential theory. The modern progress of this theory, which is bound up with
the theory of subharmonic functions, has shown up still further the fruitfnlness
of the Lebesgue-Stieltjes integral in this branch of Analysis (cf. the memoirs
of F. Riesz [4] and G. C. Evans [1]).
§ 6. Measure defined by a non-negative additive
function of an interval. In this §, U will denote a fixed non-
negative additive function of an interval. In the preceding § we
made correspond to any such a function V, an outer CaratWo-
dory measure V*. Besides the properties established in Chap. II
for all Caratheodory measures, the function of a set V* possesses a
number of elementary properties of a more special kind which we
shall investigate in this §.
F.1) Lemma. If D denotes a liyperplane or a degenerate interval,
the relation o{V;D) = 0 implies U*{D) = O.
Proof. Since every hyperplane is the sum of a sequence of
degenerate intervals (cf. § 2, p. 57), it is enough to prove the lemma
in the case in which I) is a degenerate interval [ai, &i; «>, b>; ...; am, 6m].
Let E be a cube containing D in its interior, 0 an arbitrary
open set such that D(Z_G, and let
Z>, = [«i—e, b(-]-e; a2—f, h+s; ...; am—e, &„,+?],
whei'e ? is any positive number, sufficiently small to ensure
that DeCR-Gr. Since Dt is then an ordinary closed interval con-
containing D in its interior, we find U*(D)^U(DC)^O{U; B-G),
whence V*(D)^oR(U; D) = 0.

6S
CHAPTEB III. FnnctioiiH of lioundt.'d variation.
F.2) Theorem. For every figure Ii we have
F.3) t/*(in< U(K)< V\R),
and, if the oscillation of 11 at the boundary of li vanishes,
F.4) V*{B,°)=V{K) = U*{E).
In particular therefore, if U is a continuous function, the equal-
equality F.4) holds for every figure R.
Proof. In virtue of Theorems 2.2 and 4.2 the sot It" is ex-
expressible as the sum of a sequence of noii-ovorlappiug cubes {//,)
such that the oscillation of U vanishes at all faces of nil the lh.
Hence, by the preceding lemma U*(.B")—}]U*(ri,), and since
i
~^]ilTl*{It) for Mich positive integer n, we got
U*(R°) < TJ{E).
To establish that U{Ii)^"O*{R), it is enough to nIiow that
C(-fix!?Z7(Ia) for every sequence of intervals {In} such that I&C}Jl°if
k "~ I,
, if (I/;) is such a sequence, we have, by the well-known covering
N
theorem of Borel-Lebesgue,
&>r some sufficiently large
U(Ih).
If
5]
value of N. Hence iji
lt—\ /!"¦ 1
Finally, denoting by B the boundary of R, let us supiio.se that
o{U;B)=0. It then follows from Lemma (U. tiiali U*(H)=~-(), so
that U*(R)=U*{R°), and the equality F.4) follows at onco from @.3).
F.5) Theorem. Given an arbitrary set IS and any positive e, there
is (i) an open set G such that -E?C# and U*{G)^U*(I<J)+s, (ii) a set
-ffeSa such that EQE and U*(H)=U*(E).
Proof, re (i). There exists for each e>0, a, sequence of inter-
intervals {I,,} such that ECZln and that }JU(L,)^U*{It])~\-r. Hence,
writing G=?l?,, we find, on account of condition @2) of (!;irat1u'>odory
(Chap.II, p.43) and Theorem (S.2, that
s; U*(E)+e.
Measure defined bv a function of an interval.
69
re (ii). Let us make correspond to E, for each positive integer n,
an open set Gn containing E and such that JJ*{Gn)^JJ*{E)+ljn;
this is always possible by (i). The set H=[]G,, clearly fulfills our
requirements and this completes the proof.
Every set (©d) is of course measurable (?#). Hence it follows
at once from Theorem 6.5 that every set is regular (of. Chap. II, § 6)
with respect to the outer measure V*, and therefore that this
measure is itself regular.
F.6) Theorem, Bach of the following conditions is necessary
and sufficient for a set E to be measurable (fiF):
(i) for every e > 0 there is an open set
such that
(ii) there is a set (©a) containing E and differing from E at most
by a set of measure (U) zero;
(ill) for every e>0 there is a closed set FQE such that
(iv) there is a set (^n) contained in E and differing from E
at most by a set of measure (U) zero.
Proof. We shall first prove all these conditions necessary.
Let E be a set measurable Bu) and s a positive number. We begin
by representing E as the sum of a sequence {E,,)n==h2,... of sets measur-
measurable (fit/) of finite measure; we may write for instance, E,l=E-B(,Q;n).
This being so, we associate with each set En, in accordance with The-
Theorem 6.5, an open set GnZ)En such that ?"*(<?„)<P*(^)+f/2".
Hence, the sets En being measurable Bf), we have TJ*(Gn—En)^.ej2"
for every n, and if we write G=YGn, we find EB_G and U*{G—?7)<
and this proves condition (i) necessary.
To prove the necessity of condition (ii), we attach to the given
set E measurable Bc) a sequence {Qn} of open sets such that EQQn
and U*(Qn—E)^l/n for each n. Writing E=[jQn, we see that
He<$>6, ECS and U*(H—E) = 0.
Finally, we observe that for any set A, the relation
implies ciC-® and E—CA=A—CE- further, if A is a set (<B)
or (®ri), the set GA is a set E) or (<$„) respectively. Hence every
set E measurable Bc) fulfills conditions (iii) and (iv), since, by the
results just proved, its complement CE fulfills conditions (i) and (ii).

70
CHAPTER, III. Functions o? hounded va.riatio
The sufficiency of conditions (ii) and (iv) is evident, since
sets of measure G7) zero, and sets (©a) or (%„), are always measur-
measurable Bu).
To establish the sufficiency of conditions (i) and (iii), we
need only observe that they imply respectively conditions (ii)
and (iv). Thus, for instance, if (iii) holds, there i,s for each positive
integer n a closed set F,,CE such that <7*(A1—./''„)<!¦/«¦ The set
P=?Fn is therefore a set (<$„) contained in E and such that
u*(E—P)=o.
Prom Theorem 6.6 it follows in particular that the general
form of a set measurable Bu) is B+.N, where ./>' is a set measurable
E8) and N a set of measure (II) zero. In other words, 2a is the
smallest additive class containing the Borel sets and the sets of
measure (P) zero. It follows that iiwD^ whenever the function of
an interval U is absolutely continuous (vide § 112).
F.7) Theorem. For any net E there in a mi ?/eF,i containing E
and such that
F.8) U*(E-X)=U*{E-X) for every sd X measurable BV).
Proof. It is enough to show that there is a set H'Z)E measurable
BU) for which F.8) holds, For, by Theorem 0.6 we can always en-
enclose such a set H in a set @5<s) differing from it by a set of measure
(Z7) zero.
Let us represent E as limit of an ascending sequence [E,,]
of "bounded sets, which, are therefore of finite outer measure ((/);
and let us associate, as we may by Theorem O.fS, with each Elt a set
Ene<S>i such that E^C^.,, and t/*(,7/,,) = f/*(K,,). Then, for every
set X measurable Bu),
from which, writing IT=Urn inf Hn, we deduce by moans of The-
orem 9.1 of Chap. I, that
V*(H-X)^lhnixd U*(En-A')<lim U*[Wn-X) < 7/*( AM'),
« ii
and this implies F.8), since // = lim inf //„ ~J lim K,,^ K.
n n
In the theorems of this § we have supposed the function U ot
an interval to be non-negative. But, by slight changes in lihe wording,
L§6]
Measure defined tiv a function of an interval.
the theorems can easily be extended to arbitrary functions of bounded
variation. As an example we mention the following theorem which
corresponds to Theorem 6.5:
F.9) Theorem. If F is an additive function of bounded variation
of an interval, then for any bounded set E and any e>0 there is an
open set GJE such that \F*(X)—F*{E) <? for every bounded set X
satisfying the condition
Proof. Denoting by TFj and TF2 two functions of an interval
that are respectively the upper and the lower variation of F, we
can, by Theorem 6.5, enclose E in each of two open sets Gx and G2
such that Wf(G1)^!Wf(E)+e and Wf(Gi)^Wi(E)~E. Therefore^
wilting G=GVGZ we have ECG; and for any bounded set X
such that EQXCG, we find 0<W?(X) — Wf(E)^s and
whence by subtraction \F*(X)—
Let us still prove a theorem which allows us to regard all non-
negative additive functions of a set in Bm as determined by non-
negative additive functions of an interval. We recall that, according
to the conventions of § 3, p. 59 and § 5, p. 66, we always mean by
additive functions of a set, functions of a set that are additive C5)
on every figure.
F.10) Theorem. Given any non-negative additive function 9 of
a set, there is alicays a non-negative additive function F of an interval
such that <P(X)=F*(X) for every bounded set X measurable C3).
Proof. Let us denote for each interval J=[ai, &i; ...; am, bm],
by I the interval (al: b\; ...; am, bm] half open to the left, and let
us define the non-negative additive function of an interval by writing
i?(I)='7i(I) for every interval I.
This being so, we observe that any bounded open set G can
be expressed (cf. Theorems 2.2 and 1.2) as the sum of a sequence
¦\In} of non-overlapping intervals at whose faces the oscillation of
F vanishes; and therefore by Theorem6.2, ®(G)=?®(III)=?F(In) =
n n
=?.F*(I,,)=F*(G). Thus the equation 0(X)=F*(X) holds whenever X
n
is a bounded open set, and therefore also whenever X is a bounded
set @3,5), since the latter is expressible as the limit of a descending
sequence of bounded open sets. It follows further that <P(X)=
— F*(X) = 0 for every bounded set X of measure (F) zero, since,

OHAPTEK 111. Functions of bounded variation.
by Theorem 6.6, sucli a set X can be enclosed in a bounded Hot @3,,)
of measure (P) zero. This completes the proof, [Hiiico every bounded
set E8) is, by Theorem 6.6, the difference of a. bounded net {<$>&) and
of a set of measure (F) zero.
The proof of Theorem 6.10 could also hv attached to ilio following gonwal
theorem concerning functions additive ($>) defined on any motrioal Hjmoe, /U:
if two such functions coincide for every open set, tkoy are identical for all hbU («B).
This theorem is easily proved.
§ 7. Theorems of Lushi and Vitali-Carath^odory. We
shall establish in this § two theorems concerning the approximation
to measurable functions by continuous functions and by somi-
continuous functions. As in the preceding §, U will stand for a. non-
negative additive function of an interval, fixed in any manuor tor
the space Rm.
G.1) Lusin's Theorem. In order thai a function f(x), finite,
on a set E, be measurable Bu) on E, it is necessary and sufficient that
for every e>0, there exists a closed set J?C_E nueh that U*(I<J—:f)<,r, and
on which f(x) is continuous.
Proof. To show the condition necessary, we suppose /(a?)
finite and measurable B^) on E, and we deal first with two partic-
particular cases:
(i) f(x) is a simple function on I<J. The Net 1<J in, in this
case, the sum of a finite sequence Hi, th, ..., K,, ol! h«1,h moaauraible
Bu) no two of which have common points, such that f(x) \h con-
constant on each of these sets. By Theorem 0.6 tboro oxi.sliN, for each
set Ei, a closed set FtCEi such that U*(/<Jj—/(',)<f/it. Writing
F=IFi, we then have MQJE and U*(HI—F)<e, and moreover
the function f(x) is clearly continuous on ./''.
(ii) Sis a set of finite measure (U). In this ctwe, by
Theorem 7.4, Chap. I, (applied separately to the non-negative and
to the non-positive parts of f{wj), there in a Hequonco [j„('»¦¦))„ i,a,...
of simple functions, finite and measurable Btl), that converft'OH on M
to /(«;). By Egoroff's Theorem (Chap. I, Th, ().(!), thiw k«(|ikmi<'.«
converges uniformly on a set PQE, mwwurabh, {ii,,) and wucli l,hii,t
U*(E—P)<e/2. This set P juay further be mipixwcd (,o l>o cloHod,
on account of Theorem (i.6. Finally, by (i) w« can attach to oaoh
fiuaction /„(«), a closed set P,,CK Hiuth that //¦(/*'—/'„)< f/2""',
[§']
Theorems of Lusin and Vitali-Caratheodory.
and on which fa[x) is continuous. Hence, writing F=P-[JPn,
we get U*(E—I'X.U*(E—P)+?u*(i:—P!l)^e; and moreover, all
the fn(x)9 and therefore also the function f(%) = ]im fn(x)} are con-
tinuous on F, a set which is evidently closed.
We now come to the general case where E is any set measur-
measurable BU). Let En=E-{Sn—Sa-i), where ?0=0 and #,,=S@;m)
for ra>l. By (ii), there exists, for each n^l, a closed set QnQEn
such that U*(Ea—Qn)<?J2", and on which f{x) is continuous.
Writing F=^Qn, the set P is closed, we have U*(E—I'X
71=1
^?U*(En—QnXe, and f(x) is continuous on F.
n
The proof of the necessity of the condition is thus complete.
Let us now suppose, conversely, that the condition is satisfied.
The set E is then expressible as the sum of a set N of measure (U)
zero and of a sequence {Fn} of closed sets on each of which f(x) is
continuous. The function / is thus measurable {2u) on ^T and on
each of the sets Fn (cf. Chap. II, Th. 7.6), and therefore on the whole
set E.
For the various proofs of Lusin's theorem, vide X. Lusin [1], W. Sier-
pinski [6] and L. W. Cohen [1].
G.2) Lemma. Given a function f(x), measurable Bu) and non-
negative in the space lim> there exists, for each e>0, a lower semi-
eontinuous function h(x) sucli that
G.3)
and
G.4)
at each point x,
j
Urn
(where, in accordance with the convention of Chap. I, p. 6, the
difference h(x)—f(x) is to be understood to vanish at any point x
for which h(x)=f(x) = -\-^o).
Proof, (i) First suppose that f(x) is bounded and vanishes
outside a bounded set E measurable Bu). Let i?=?/[l-f?7*(.E)].
We write jE?A=E|>e-E; (fc—1)??</(.e)<&ij] for Jc=l, 2, ... and we
associate with each set Bk an open set
such that

(IHAl'TiiJR 1H. Funoliimm of bounded variation.
G.5)
Further, denoting by eft(a?) the characteristic function of (Jk} we
write h(x)^=^hi}-ci,(x). Since each function oA(a)) is evidently lower
semi-continuous, the function h{x) is so too. We also observe that
li(x) fulfills condition G.3). On the other hand
*=i
le- 1
whence, by G.5), we obtain
-K..
x) + s.
K,,
«.,
From this, remembering that f{x) is integrablo on It,,,, G.4) follows
at once.
(ii) We now pass to the general case and represent firstly f(x)
CO
in the form f(x)=2fn(x), where the /„(#) are bounded non-nogative
n—1
functions measurable B^), each of which vanishes outside a bounded
set. We may do this, for instance, by writing f,,(x) = s,,(#)—,v,, , (x) whore
sn(x) =
/(as) for <?@,
n for q{0,
0 for q{0,
and
and f{w)>n,
n-0,1,2, ...
Ey what has been proved in (i), there exists for each function f,,(x)
a lower semi-continuous function %n{x) such, that \,(ai) >/„(*¦) at
every point x, and that l\hn(x)—f,,(x)]dV(x)<ej'2". The function
DO
7i{x)—^hn{x) is then evidently lower Hemi-continuouhs and fulHllH con-
ditionG.3).Finally f[h(x)-~f(x)]dU(m)
Iim , «¦ 1 Km
and this .completes the proof.
Theorems of Lusin and VitaJi-Carath&Mlorv.
(O
G.6) Theorem of VitaU-Caratheodory. Given a function f{x)
measurable Bu) in the space JRm, there exist two monotone sequences
of functions {ln(x)) and [un(x)} for which the following conditions are
satisfied:
(i) the functions ln are lower semi-continuous and the functions
<un are upper semi-eontimious,
(ii) each of the functions ln is bounded below and each of the
functions nn is bounded above,
(iii) the sequence {I,,} is non-increasing and the sequence [vn) is
non-decreasing,
(iv) ln{w)^f{x)^un{x) for every x,
(v) Mmln{x)=f(x) = ]imu,,{x) almost everywhere (V),
n ' n
(vi) on every set IS on which f(x) is integrable (U), so are the
functions ln{x) and Unix) and we have
Iim /' ltt {x) dTJ{x) =lim fn
" i " i
= ff(x) dU{x).
Proof, By expressing the function f{x) as the sum of its
non-negative and non-positive parts f(co) and f(x) (Chap. I, § 7),
we may suppose that f{x) is of constant sign, say non-negative.
By the preceding lemma, we can associate with f(x) a sequence
of lower semi-continuous functions {hn(x))ll=1^,., such that lin(x)^f{x)
for every x and
G.7) -Umf[hn(x)—f(x)]dU(x.) = O.
Writing ln(x) = min [fti(a?), h(x), ..., hn{x)] we therefore obtain anon-
increasing sequence of lower semi-continuous functions \ln[x)}
that evidently fulfills conditions (i), (ii), (iii) and (iv); moreover,
it follows from G.7) that Mm / \ln{x)—f(x)]dU(x) = Q, and hence
Jim
that the functions ln{x) fulfill also conditions (v) and (vi).
In order now to define the sequence {ii,,(x)), we attach to the
function ljf(x) a non-increasing sequence of lower semi-continuous
functions [g,,(x)) such that ]xmg,,(x)—llf(x) almost everywhere (U).
n
Such a sequence certainly exists by what, has just been proved.
The functions l/gn(x) then form a non-decreasing sequence of upper
semi-continuous functions, that converges almost everywhere (TJ)
to f(x). If we now write «,,(,r) = l/f/,,(.r) when llgn(x)^.n, and

76
CHAPTER 111. Functions of hounded variation.
nn(x)=n when i/r/,,(*)>»> w« obtain a sequence [u,,{x)) of bounded
functions with the same properties, which therefore isatisdiieH con-
conditions (i—v). Finally, since the functions u,,{ic) are uon-nogativo,
we can apply Lebesgue's Theorem (Chap. I, Th. 12.0) to derive
from (iii) and (v) that lJm / u,,{ai)dU(:c)= j f (ne) d U (ui) on every
" ii is
set U measurable (Cy), and this implies (vi).
Conditions (i) and (v) of Theorem 7.0 imply that, every function measur-
measurable (80) is almost everywhere (If) the limit (it a convorgont Hoquencc. (with
finite or infinite limit) of semi-continuous funotiouH, and tlttiH coincideH almost
everywhere (II) with a function of tlio second clans of Uairo. Thin nwulfc, duo
to 6. Vital) [2] (of. also W. Sierphiski [6J) wa» completed b,y (',. Oarathrtodory
[I, p. 406], who established for every measurable function /(;<;) tlio oxiHtoneo
of two sequences of functions fulfilling conditions (i) -(v). Condition (vi), which
includes, as we shall see Inter, the theorem of do la Va.lloe PounHin. and I'ermn
on the existence., for summable functions, of m n, j o ran t and m in ortm I;
functions, has been added hero because its proof is naturally i/oIhUmI to those
of conditions (i—v).
There is an obvious analogy between the, property of mwwuvablo functions
expressed "by the theorem of Vitali-Uaratlie'odoTy, :uid tlio proportieH of moas-
urable sets stated in conditions (i) and (iii) of Theorem 6.A. By taking into ac-
account the geometrical definition of the integral (cf. below § 10), we might even
base the proof of Theorem 7.6 directly on Theorem <i.(l (vide the first od. of this
book, pp. 88—91).
§ 8. Theorem of Fubini. Given two Euclidean, upaees R,,
aud Mg, if ,B=(ai, a*, ..., a,,) and j/=(*/H-i, «/;+.>, ..., a,, \,,) are two
points situated respectively in these two apaeos, we shall denote
by (x,y) the point (a,, a2, ..., a,,+l,) in the space H,,\.tl, If X and Y
are two sets situated respectively in the spaces It,, aud /»',„ w<i sluilL
denote by 1x7 the set of all points (x,y) in If/,.].,, such that xeX
and yeY. In particular, if X and Y are two interval — closed, open,
or half open on the same side — XxY also is an interval in lt,,\.,,,
which is closed, open, or half open on the same side a« X and Y.
Every interval I—[ah bt; ...; a,)+l/) bp+v] can evidently be expressed —
and in a unique manner — in the form IxxL2 where /-, and l%
are intervals in ltp and liq respectively; we merely haves 1;o write
!,= [«!, &,; ...; ap, b,,] and I2=|>/)+i, b,1+); ...; up+m b,,, „].
Given two additive functions of an interval, (I and V, in, the
spaces Bp and Rq respectively, we determine a function of an inter-
interval T in Mp+l, by writing 2T(./1x/a)==f7(./l)- K(Ja) .for each pair o.f
intervals IiC^,> and hCRr The function T 1,1ms defined, clearly
additive when U and V are, will bo denoted by UV. \n particular,
[§sj
Theorem of Fubini.
we see easily that Lp_Lr/=LpL?, where Lp, hq and
denote
the volume in the spaces lip, Bq and i?c+v respectively (cf. § 2, p. 59).
It is known since Cauchy that, if I1 and I.2 are respectively
two intervals in the spaces Iip and Iiq, integration of any continuous
function over the interval IiXl-idRp+q may be reduced to two
successive integrations over the intervals Ix and I2. By repeating
the process, any integral of a continuous function on an m-dimensional
interval may be reduced to tn successive integrations on linear
intervals in Rx. This classical theorem was extended by H. Lebes-
gue [1] to functions measurable B) that are bounded, and then by
G. Fubini [1] (cf. also L. Tonelli[2]) to aU functions integrable (L),
whether bounded or not. We shall state this result in the follow-
following form:
(8.1) FiibinVs Theorem. Suppose given two non-negative ad-
additive functions U and V of an interval in the spaces Mp and Hq
respectively, and let f(x,y) be a non-negative function measurable Bur)
in Itp+q. Then
(ijj f(x, y) is a function of x, measurable Bp) in lip for every
y e Rq, except at most a- set of measure (F) zero,
(i2) f{x, y) is a function of y, measurable Br) in JRq for every
except at most a set of measure {U) zero,
(ii) //(.r, y) dUY(x, y) = /' [ ff(x, y) dV(x)] dY{y) =
= f[ff(x,y)dV(y)]dU(x).
Proof. Let us write for short, T=VY. By symmetry, it is
enough to show that every non-negative function f(x, y) measurable
Br) in JSp+q fulfills condition (ij) and also the relation
(8.2) \f{x, y) dT(x, y)=f [ff(x, y) dV(x)] dY(y).
For brevity, we shall say that a function f[x, y) in Rp+q has the
property (F), if it is non-negative and measurable (?7) in Rp+q, and
if it fulfills condition (ix) and the relation (8.2). For the sake of
clearness, the reasoning that follows is divided into a several
auxiliary propositions.

78
CHAPTER HI. I'uiHttioim (if bounded vnriiitioii.
(8.3) The sum of two functions with, the property (If1), and the
limit of any non-decreasing sequence of sueh functions, have the, prop-
property (JP). Also, the- difference of two function* with the 'property (/*'),
has the property (F), provided that, it is non-negative and that one at
least of the given functions is finite and integrabk (T) on the space
Mp+q.
For the sum, and for the difference, of two fuu.ctio.iiK, the
statement is obvious. Let therefore {h,,(x,y)) be a non-decreasing
sequence of functions in lt,,+q having the property G'1), and let
Ji(x,y) = ]imh,,{x,y). The definite integrals j h,,(,x,y)dU{,r,) exist,
it,,
and constitute a non-decreasing sequence, for. every y e It,, except
at most those of a set of measure (V) xoro. <!ou,scq neatly, by
Lebesgue's theorem on integration oi!monotone sequences of functions:
fh(x,y)dT{x,y)=\im fhn{x, y) dT(x, y)-lim /" | fh»{w,y) dU(x)
K
"«.,
)=\im fhn{x, y) dT(x, y)
'x, y) dV{x)\ dV(y) -
/ | f
h(x, y) d(J{x) dV(y),
and this establishes the property (F) for the function li(m,y).
(8.4) The characteristic function of any net BC^f? measurable (If)
has the property (F).
We shall establish this, first for very special sots M and then,
by successive stages, for general measurable sets. Suppose in the
first place that
1° E—AxB, where A and B are intervals half open
to the left, situated respectively in .It,,, and Jt.,,, and such
that the oscillations of U and of V vaniKli at the bound-
boundaries of A and B respectively (cf. § 3, p. 60). The oscillation
of the function T=UV therefore vanishes at the boundary of the
interval F=AxB, and we find by Theorem 6.2
(8.5) T*(E)=T(E)=UA)-V(B)=U*{A)- V*(B).
On the other hand, for every y e It,, the function (^(.p, //)
is in x the characteristic function either of the half open, inter-
interval A, or of the empty set, according as y tB or ye tt,,—Ii. This
function is therefore measurable Bu), and indeed inoaHtmiblo C3),
for every yelt(n and, by (8.5)
[§8]
Theorem of Fubini.
79
fas(x, y) cl,T(x, y)=T*W)= U*(A) • V*(B)= f [ [eE(x, y) dU(x)] SV(y).
K K
2° Fl is an open set. We shall begin by showing that, in
this case, FJ is the sum of a sequence of half open intervals [IB) no
two of which have common points, these intervals Ia being of the
form A,,xBn where (a) An and Bn are intervals, half open to the
left, situated in ltp and Rq respectively, and (b) the oscillations of U
and V vanish at the boundaries of An and Bn respectively.
To see this, let [Uik)) be a regular sequence of nets in lip formed
of intervals half open to the left and such that the oscillation of U
vanishes at the boundary of each of these intervals; by 'Theorems
i.2 and 2.1, such a sequence certainly exists. And let {SB**5} be
a sequence of nets similarly constructed for the space liq and for
the function Y. We denote, for each k, by S(ft) the system of all
half open intervals in JSp+? which are of the form A xB where Aellm
and SeSB'*'. The systems of intervals S'*', thus defined, form a reg-
regular sequence of nets of half open intervals in the space Rp+q. The
set Fi being open, we can therefore express it (ef. § 2, p. J58) as the
sum of a sequence of half open intervals (/„} taken from the nets
S( and without points in common to any two. We see at once that
each interval In of this sequence is of the form AnxBn where A,,
and Bn satisfy conditions (a) and (b).
This being so, we have cE{x, y)=^Ci!l{x,y), where on account
of the result established for the case 1°, each of the characteristic
functions C/n (%, y) has the property (F). Therefore, to verify that
the function cE(x, y) also has this property, we need only apply (8.3).
3°^ is a set (©a). First suppose that, besides, the set F is
bounded. F is then the limit of a descending sequence of bounded
open sets {<?„}. The functions c3l(x,y)—eSii(-x,y) constitute a non-
decreasing sequence of non-negative functions which have, by 2°
and (8.3), the property (F). Consequently, again on account of (8.3),
the limit function of this sequence h(x,y) = c3l(x,y)—cE{x,y) itself
has the property (F) and the same is therefore true of the function
cs{x, y) = ca,(x, y) — h{-x, y).
Now if E is an arbitrary set (©a), we can express it as the limit
of an ascending sequence [En\ of bounded sets (©a). Ey what has

so
CHAPTER III. FuiK'.Uoiw of bounded variation.
just been proved, the characteristic functions of the, sets H,, have the
property (F) and, consequently, the function Mtf,i/)=lime,/;i (,¦!?,#)
itself has the property {F).
4° E is a set of measure (?) zero. There is then, by
Theorem 6.5, a set He (Si a containing E and of meaHiirc (T) zero.
By the result established for sets (<S,i), the function a,,(x,y) has
the property (F), and therefore / \ \ a,i{x, y) dU{x)\dV(y) =
= [en(x,y)dT{x,y)=T*(H)=Q. Hence, for every yelt,,, except at
most a set Y of measure (F) zero, j <.'¦// («,?/) r/?7 (*)---(), i.e. cw(«,?/),
as function of a?, vanishes almost everywhere (f/) in />',„ Hence,
a fortiori, Cji(x,y)^Cn(x,y) m function of t», vaiiMieH almost every-
everywhere (P), and is consequently measurable E3//), for all yt>lt,n except
at most for those of the set X of measure ( V) zero, finally, we
clearly have /[ fcs(x,y)dV(x)\dV(y)=0=T*(E) = / e^/.',?/) <T/'(«,?/).
The function cE(x,y) thus has the property (P).
On account of Theorem 6.C every set .E measurable (Cr) is
expressible in the form IS=H—Q, where // m a set ((&,;) and Q
is a set of measure (T) zero contained in //. We thus have cn{x,y) —
= Cg{x,y)—(iQ{x,y), and by (8.3) the proposition (8.<1) reduces to
the special cases 3° and 4° already treated.
The proposition (8.4) being thus established, let /(*',y) be
any non-negative function measurable (&</¦) in the njnice H,l+,r By
Theorem 7.4, Chap. I, the function / is the limit of a non-decreasing
sequence of simple functions, finite, non-negative, and measurable (fir).
Now each of these simple functions is a linear combination, with
positive coefficients, of a finite number of characteristic functions
of sets measurable Br), and therefore has the property (/*') on ac-
account of (8.4). Thus the function / is the limit of a jiou-decroasing
sequence of functions with the property (J?), and so, by [H.'<i), f itself
has the property (F). This completes the proof of Theorem S.I.
Let us make special mention of the particular cum of the
theorem in which f{x,y) is the characteristic function, of a meas-
measurable set:
Theorem of Fubini.
81
(8.6) Theorem, If V and T are two non-negative additive
functions of an interval in the spaces Rp and-ltg respectively, and if Q
is a set measurable {2uv) in the space M-p+q, then
(ij) the set E[(ic, y)eQ] is measurable (QD) for every y e Rq,
X
except at most a set of measure (F) zero,
(i2) the set E[(ap, y) eQ] is measurable (ilv) for every xeltF
u
except at most a set of measure (U) zero, and
(ii) the measure [TJV) of Q is equal to
J'U* IE l(x, y) b q\\ dV(y) =[v* (E [(x, y) eQ]) dV(x).
Fubini's theorem is frequently stated in the following form:
(8.7) Theorem. Let U and V be two additive functions of bounded
variation of an interval in the spaces Mp and JR9 respectively. Then
for every function f{x,y) integrable (VV) on RP±g, the relation, (h) of
theorem, 8.1 holds good and the function f{x,y) is integrable (TJ)
in x on lip for every yeMg, except at most a set of measure (F) zero,
and integrable (F) in y mi Rq for every xeRP) except at most a set
of measure (Z7) zero.
We reduce this statement at once to that of Theorem 8.1 by
expressing the function / as the sum of its non-negative and non-
positive parts, and by applying to the functions of an interval U
and F the Jordan decomposition (§ 4, p. 62).
Further generalizations of Fubini's theorem for the Lebesgue-Stieltjes
integration (in particnlar including the theorems analogous to Theorem 15.1 of
Chap. I) were studied by L. C. Young in ;his Fellowship Dissertation (Cam-
(Cambridge 1931, unpublished). An account of these researches will be given in the book
The theory 0/ Stiettjes integrals and distribution-junctions by L. C. Young (Oxford,
Clarendon Press).
It follows in particular from Theorem 8.6 that for any set 9 measurable
in the sense of Lebesgue in the space Kp+q, its measure (Lp+i/) is given by the
definite integrals /Lp{E [(x.y)eQ]}d'Lv(y)= f li9 {E [(x,y) 6§]}dLp(je). It is never-
theless to be remarked that the existence of these two integrals does not in
general enable us to draw any conclusion as to nieasurability B) of the set Q.
TV. Sierpinski [5] has in fact constructed in the plane a set non-measurable (?)
having exactly one point in common with every parallel to the axes. This con-
construction depends, needless to say, on the axiom of selection of Zermelo.
For an interesting discussiou of Fubini's theorem for Lebesgue integration
of functions of variable sign, vide Or. Fichtenliolz [1].
S. Saks. Theory of the Integra!. 6

82
CHAPTER III. FunctiioiiH of hounded variation.
Wo complete the theorems of this § by the following
(8.8) Theorem,. If Q is a set measurable A3) in the xpaoa lt,,\,,,
the set E[(x,y)f.Q] is measurable (93) in the apace It,, for every ye Hq,
and the*set E[(cc,y)eQ] is measurable, A3) in It,, for every xt-tt,,.
Similarly, if a function f(x,y) in measurable C3) in the apace ltp+ll,
then in lip the function f(x,y) is measurable (93) in x for every yeM,,,
and in Rq the function f{m,y) is measurable (93) in y for every noeltp.
Proof. It will be enough to prove the first; half of the theorem,
since the second half obviously follows from the first. Lot; us denote
by 93O the class of all sets Q in .R,,..\-<, mush that the sots Vi\(x, y)(-Q],
for every yell,,, and the sets R[(as, y)f-,Q\, for every x^R,,, sue
measurable (93) in the spaces Jip and M,, respectively, if a sot QQJt,, u,
is closed, so are the sets E[{x,y)eQ\ and \i\(x, y)aQ\. The class 93O
thus contains all closed sets of the space Jt;,\.,n and on the other
hand we see at once that 33O is additive. It follows that 93O includes
all Borel sets in the space -Mp+q (cf. the definition, Chap. II, p. 41),
and this completes the proof.
* § 9. Fubini's theorem in abstract spaces. We shall
return in this § to the abstract considerations of Chap. I and show
that for abstract spaces, theorems similar fco those of tho preceding §
hold good.
Given any two sets X and Y, we shall denote by X x Y the
set of all pairs of elements [x,y) for which xtX and yeY. The
set X x Y is often called eombinatory product or Cartesian product
(cf. C. Kuratowski [I, p. 7]) of the sets X and Y. The following
identities are obvious
(9.1) (Xxx Yx) • (Z2x Y2) = (Xx -X2) x (Yt ¦ Ya),
(9.2) (i8xia)- (x1xr1)=[(ZJ—x1)xr2iH-[(x2..x1)x(Y.,--y1)]J
the sets Xt, X2, Yv Y2 being quite arbitrary.
If 9E and ^ are additive classes of sets in the spaces A' and 1"
respectively, ?i^j will denote the smallest additive class of >sc(;s
in the space Arx V, containing all jwoducirsets of tho form A'x.V,
where Ze9? and
Tubini's theorem in abstract spaces.
83
For auxiliary purposes, we shall make use in tins § of
the following definition: a class 9t of sets will be termed normal,
if (i) the sum of every sequence of sets Ct) no two of which have
common points is itself a set C1) and (ii) the limit of every
descending sequence of sets C1) is a set (91).
We shall begin by proving the following analogue of Theorem8.8:
(9.3) Theorem. Let S? and ^ be two additive classes of sets in
the spaces X and Y respectively. Then, if Q is a set measurable (?3))
in the space XxY, the set E[(jj, y) e Q] is measurable (S?) for every
X
y eY, and the set E[(j, y)eQ] is measurable (?)) for every xeX.
n
In the same way a function f{x,y) winch is measurable (SPg))
in the space lrx 1', is measurable (9?) in as for every y eY and meas-
measurable C)) in y for every weX.
Proof. It is enough to prove only the first part concerning
sets. To do this, we denote by 9ft the class of all sets Q in Xx Y
such that the set E[{x,y)eQ] is measurable E?) for every y eY,
X
and that the set 'E[{x,y)eQ] is measurable B)) for every xeX. We see
H
at once that the class 9ft is additive in the space Xx Y and that,
besides, it includes all sets Xx Y for which XeSE and Ye?). Hence
$DC !0t) and this completes the proof.
Before proceeding further we shall establish the following
lemma:
(9.4) Lemma. If 'S, and, 3) are two additive classes of sets in
the spaces X and Y respectively, the class J??) coincides tvith the smal-
smallest normal class that includes the sets XxY for which X e 3: and Y e *?).
Proof. For brevity let us term elementary any set XxY
for which XeSt and Ie?), and let 910 denote the smallest normal
class which includes all elementary sets (i. e. the common part
of all the normal classes that include these sets). Clearly 91OC^2)
since K'J) is also a normal class. In order to establish the opposite
inclusion, it is enough to prove that the class 910 is additive, and this
will be an immediate consequence of the following two properties of
the class 910:
6*

o,( CHAPTER HI. FuhcUoiih of bounded variation.
(9.5) The common part of any sequence of sets (9}()) in itself a mi (91,,).
(9.6) The complement (with respect to the. space Xy.Y) of any sd{%)
is again a set Eft0).
To prove (9.5), it is enough, since the class 910 is normal, to
show that the common part of two sets (9?(>) is a net (9?,,).
For this purpose, let % be the class of all the. nv>tn (9}J whose
common parts with every elementary wet belong to %.
From the identity (9.1), it follows that the common part, of two
elementary sets is an elementary set, and hence that 9^ includes
all the elementa,ry sets. On the other hand, we verify at once that
Sftj is a normal class. This gives $R0C9ii> iv.tt.cl since by definition
9^i C 9^>! we obtain 5ft1=5ft0.
Let now % be the class of all the sets (%) whose common
parts with every set (9J0) belong to 9?n. Wince 9ly¦¦¦~<3\a, the class
5ft2 includes all elementary sets. Furthermore, 9L is clearly a> normal
class. We therefore have 5U2=910) tM1<I l'h*N 1/rovcs (9.5).
To establish (9.6), let % be the class of all the nets (9?A) whose
complements are also sets (9t0). On account ot tho identity (9.2)
the complement of any elementary set is the sum of two elementary
sets without common points, and so, a set (9}(>). Therefore the class 9}3
includes all elementary sets and, to conclude that 913=9?0, it suffices
to show that the class 5ft3 is normal.
Let therefore [Xn} be any sequence of. sets ER3) without com-
common points to any two of them, and let X he the sum of the se-
sequence. The set X clearly belongs to the class 910. On the other
hand, the sets OX",, are, by hypothesis, sets (9?0); so that, by (9.5),
the same is true of their product 0X=//0X,,. Thus we have at
the same time, Xe5R0 and CXe5ftfl, and. therefore X«--913.
Again, let {3Tn},,.^i,2,.., be a descending sequence ot sets (9]3),
and Y its limit. The set Y clearly belongs to the class 9t0. On the
other hand, consider the identity
CY =
!„ ¦ CF,,
.|,,
and observe that no two of the sets QYt and Vr,,-OY,, (.| for n~l,ii,...
have common points. Since these sets belong, by (9.5), to the class 9?,,,
so does the set GY. Thus we have both If e9t0 and OY a 9]0, whence
Ye9*3. The class % is therefore normal, and l,htn «stiU)lislies (<».«)
and completes the proof of Lemma 0.4.
[§9]
Puliini's theorem in abstract spaces.
85
We can restate Lemma 9.4 in the following more general form:
@.7) Given in an abstract space T a class Q of nets additive in the weak
sense, then the smallest class that is additive (in the complete sense) and contains Q,
coincides with the smallest normal class containing Q.
The proof is the same as for Lemma 9.4.
If 9: and 3) are additive classes in the spaces _V and 1" respectively, the
finite sums of the sets Xx r for which XeS and Tety, constitute, according to
formulae (9.1) and (9.2), a class that is additive in the weak sense (vide Chap. I,
p. 7) in the space A' x I'. Another example of a class of sets additive in the
weak sense consists of the class of all the sets of an arbitrary metrical space 3T,
that are both sets ((8ii) and Cfto). The smallest class that is additive (in the com-
complete sense) and contains these sets is clearly the class of Borel sets in J/.
The assertion of (9.7) enables us to prove easily the following theorem
due to H. Halin [2, p. 437] and in some respect analogous to Theorem 6.6:
Let Q be a class of seta, additive in the iceah sense in a space T, and let Z be
the smallest class of sets that is additive in th-e complete sense and contains Q. Sup-
pose further that t is a measure (Z) such that the space T either has finite measure (r),
or, more generally, is expressible as the sum of a sequence of sets of finite measure (r).
Then (i) for every set JE measurable (?) and for every ?>O, there exists a setFeQ$,
and a set GeOa, such that F(Z^CG and that t(E — F)<e and zF — JB)-Cej
(ii) for every set E measurable (%) there exist a set (Qla) contained in E, and a set
containing E, which differ from E at most by sets of measure (r) zero.
(9.8) Theorem. Let Ii and 2) be additive classes of sets in the
spaces X and Y respectively¦, and let /< and v be measures defined
respectively for tliese classes. Suppose that ji(X)<ioa and v(Y)<joo,
or, more generally, that
(9.9)
— v
X=2.XB
n n
where X^eSE, Yne^, ft(Xn)<ioo and i'(Y,,)<ioa for n=l, 2, ...
Then, for every set QCXxY measurable {&§), (i) fi'^t[x,y)eQ]},
as function of y, is measurable (D) in the space Y and v[T&[(as,y)eQ]},
g
as function of x, is measurable (9E) in the space X; furthermore
(ii)
x
Proof. We may clearly suppose that no two sets Xn, and
also no two sets Yn, have common points. The same will then be
true of the sets Xa X Ym in the space Jxl".
Let us denote by 5ft the class of all the sets P measurable (9F?))
in the space Xx Y, such that conditions (i) and (ii) of the theorem

86
CHAPTER IH. Function b of bounded variation.
hold good for every set Q^=l'-{X,,x Ym) whore n and m are arbitrary
positive integers. Since Q=?Q-(XnX Yw) for every set; QC^'X V,
and since no two of the sets X,,x Y,,, have common points, it; follows
easily from Lebesguo's Theorem 12.3, Ohap. I, that, every set Q
belonging to the class 91 fulfills conditions (i) and (Ii). We have to
prove that this class includes all sets measurable (96?)).
To do this, we observe that it follows at once from the identity
(9.1) that every set XxY for which A"e96 and Ye%), belongs to 91.
On the other hand, since by hypothesis !<{Xn)<oo and v(Y,,)<oo
for every n, we easily deduce from Lobesgue's Theorems 12.'A and .1.2.11
Ohap. I, that the sum of any sequence of sots {31) no two of which
have common points, and the limit of any descending sequence
of sets (91), are themselves sets (9t). The class 91 is therefore normal,
and by Lemma 9.4, contains all sets (96?)). This proves the theorem.
If we suppose the hypotheses of Theorem !).H HiUiisl'iod, id meas-
measure can be defined for the class 91%) so aw to correspond naturally
to the measures fi and v that are given for the classes 96 aud SE).
We do this by calling measure (fir) of a. sot Q measurable (96?)) the
common value of the integrals (ii) of Theorem 9.8. It is immediate
that we then have jiv(XxY)—fi\X)-v(Y) for every pair of sets
Xe<3l and Ye<§.
This definition enables us to state Theorem 9.<s iu a manner
analogous to Theorem 8.6. But the analogy would be incomplete
if we neglected to extend at the same time the class 96?). Thus,
for instance if 96 and ?J denote respectively the classes of sets meas-
measurable in the Lebesgue sense in Euclidean spaces It,, and It,,, the
class S?2) does not coincide with that of the sets measurable (ii)
in Rp+q, although it is evidently included in the latter. Tin* extension
of the class 3-?), that we require in the general ease, will be defined
as follows.
Given an additive class of sets % and a measure % associated
with this class, we shall call the class % complete with reject lo the
measure % if it includes all subsets of wets (S) of measure (r.) zero.
Thus for instance, if F denotes any moannro of Aarathc'»odory, tlic
class fir is complete with.respect to I1 (cf. (limp. II, p. -II), and in
particular, the class 2 iu a Euclidean Npaee is complete with respect
to Lebesgue measure; whereas the class of sctH moamirablo C33) is
not complete with respect to that measure.
§9]
Fubini's theorem in abstract spaces.
87
Every additive class of sets ? may be completed with respect
to any measure t defined for the class, i. e. there is always an ad-
additive class 6DS such that the function of a set % can be continued
as a measure on all sets (S) and such that S is complete with respect
to the measure r thus continued. Among the classes 6 of this kind,
there is a smallest one that we shall denote by ?r. As is seen di-
directly, tins class consists of all sets of the farm T—2^j_+X2, where
TeS and Nlt Ns are arbitrary subsets of sets (?) of measure (t)
zero. The extension of the measure t to all sets of this form is evident.
We can now state the following theorem which corresponds
to Fubini's Theorem 8.1:
(9.10) Theorem. Under the hypotheses of Theorem 9.8, if f{x,ij)
is a non-negative function measurable (9??)'"') in the space 1'x I)
(ij f{x,y) as function of x is measurable (S?") in X for every
yel', except at most a set of measure (v) zero;
(i2) f(x, y) as function of y is measurable (?)'') in Y for every
xeX, except at most a set of measure («) zero;
x)] dv(y) = f[[f(iB,y) dv(y)]dn(x).
±±
(ii) j'f(x,y) iuv{x,y)=
i
Proof. In the special case in which / is the characteristic
function of a set measurable (9?^), the theorem is an immediate
consequence of Theorem 9.8. The same is true when / is the cha-
characteristic function of a set measurable (96^)-'"') of measure (/w) zero,
and in consequence Theorem 9.8 remains true when / is the cha-
characteristic function of any set (SED'"').
This being so, we pass as usual to the case in which / is a finite
function, simple and measurable (96?)''") and finally, with the help
of Theorems 7.4 and 12.6, Chap. I, to the general ease in which /
is any non-negative function measurable "
Condition (9.9) is essential to the validity of Theorems 9.8 aad 9.10. To
see this, let ns consider some examples for which the condition is not fulfilled.
Put X= V=Ji1, and let ft"=9) be the class of all sets iu R^ that are measurable
in the sense of Lebesgue. We choose for ,h the ordinary Lebesgue measure, and
-we define the. measure v by making *' (T) equal to the number of elements of 1 (so
that »'{in=co if r is an infinite set). Finally, let Q be the set of all the points
(x, x) iu Bs=A'xr such that O«sc<:l. The integrals ooourmg in condition
(ii) of Theorem 9.8 are then respectively 0 and 1 so that condition (ii) does not
hold. (We could also, by a suitable modification of the set Q, choose I'= Bi
and take as measure v the length Ax; cf. Chap. II, § 8.)

88
OHAPTEK III. Functions of bounded variation.
Another example allowing tlin importiuico of condition (().!») in duo to
A. lAndenbiium. Put A'= I'— Iiv let ?=?) 1>« the clam of nil Borol sots in Itu and
let /i(X) = i'(X) denote for every not X tlui number of ita olomontH. By u thoorum
of the theory of analytic sets (of., lor instamso, A. Kunitownki fl, p. 21A]) there
exists in the plane /?a=Xx V a Borol sot y mush Lliafc tho Hot of xu It, for which
E[{^, y)eQ] reduces to a single point, is uot moammiblo in tho kojiho of Borel.
n
In other words, the set of xe.V for which »'{R[(»;, ?/) sy.|J = 1. in not lueamirablo (S?),
.v
Thus condition (i) of Theorem 9.8 does uot hold.
For the results of this §, vide H. Halm [2]; of. al.Ho S. Ulam [2|, rA. -Lorn-
nicki and S. Dlatu [1], and W. Feller [IJ. For a rtinrmnHimi of K'ubiui'ti theorem
applied to fnn.ctio.ns whose values belong to au abnti-act vector njjaec*, ¦vide aluo
S. Bochner [2]. Finally we observe that certain theoremH, ilnalog-ouH bo thoKO
established in this § for measurable nots and nuts of measure ssoro, can bo stated
for the property of Baire and tho Bairc caliegoricK. (IF. on this point (!, K.u.
ratowski and R. Ulaui [1],
§ 10. Geometrical definition of the Lebesgue-Stieltjes
integral. Tlie geometrical definition of tin in.t«K''*iil in iUH])iro<i by
the older and more natural idea of regarding tho iutiogi'ikl hh 1;Iio
measure of an "area", or of a "volume", attached to the function
in a certain way that is well known.
Let us hegiu by fixing our notation. Givon a Inaction. f{x)
defined on a set QC.Hm, we call graph of f(te) on Q, and we denote
by B(/;<3), the set of all points (as,«/) of lin,+\ for which atf-Q and
y=f(x)^-oo. If f(oa) is non-negative on Q, tho sot of all tiho points
(x,y) of lim+1 such that xeQ and 0<;i/</(*•) i« tonnod, iiccording
to G. Caratheodory, ordinata-sei of / on Q and will ho donotod by
MfQ)
;
As in §§ 3—7 we shall suppose tho upauo Hm fixod and a non-
negative additive function V of an interval givon in. It,,,. And. in
accordance with § 2, p. 59 and § 5, p. 65, Lj donolios tho Lehcwgne
measure in i^.
A0.1) Lemma. If QQJim is a net measurable (ijy), «/uj ,ve«
T2i[xeQ;a*^y^.b] in lim+1 is, for every pair of real number* a and
, measurable (S^l,), and its measure (f/Lj) in (b—a)-U*(Q).
Proof. Let us write for short, Q<hh=K[:t:eQ; a<T.y^b\ and
2'=DrL1. We shall begin by showing that if Q hm rnca,Niir<i ((/)
zero, the set Qnfi is of measure G') wo, and ho in certainly meas-
measurable (Sp).
[§10]
Geometrical definition of the Lebesgue-Stieltjes integral.
89
To see this, observe that there is then, for any f>0. a sequence
of intervals {I,,} in Bm such that QG^In and yU(I,,)^.e. Writing
n n
Jn=Inx[a—e, b+e], we obtain a sequence of intervals [J,,\ in Rm+i,
such that Q«,bCZJ°n and JSlT(Jn)=rJ7GJI).(fr-a+2eKF-a+2«)-f.
Thus T*(Qa,b)=0.
Let now Q be any set measurable Bu). By Theorem 6.6, Q is
the sum of a sequence of closed sets and a set of measure (U)
zero. Therefore, by the above, the set §„.& is also the sum of a se-
sequence of closed sets and of a set of measure (T) zero, and is thus
measurable Br). Finally, for every real number y, we have
E[(ic, y)eQn,b]=Q if as^y^b, and E[(j?,y)eQn,6]=0 if y is outside the
.v -v
interval [>,&]. Hence, by Theorem8.6, we have T*(Q,,,b)= / U*{Q)dy=
a
= {b—a)-U*(Q), which completes the proof.
A0.2) Theorem. If f(x) is a function measurable Bu) on a set
Us graph on Q is of measure (?7L1) zero.
Proof. Since any set measurable Bu) can be expressed as
the sum of a sequence of bounded measurable sets, we can restrict
ourselves to the ease in which Q is bounded.
Let us fix an e>0 and write Qa=E[xeQ)ne*^.f{x)<(n+l)e]
X
for every integer n. By Lemma 10.1 the measure (Z7LJ of the graph
of f{x) on Qn does not exceed s-U*{Q,,); therefore, on the whole s&bQ,
it does not exceed e-U*{Q), and so vanishes.
We can now prove the following theorem which includes the
geometrical definition of the Lebesgue integral:
A0.3) Theorem. In order that a function /(*) defined and non-
negative on a set QCl^m measurable (So-) be measurable [2v) on Q, it
is necessary and sufficient that itu ordinate-set A(f; Q) on Q be meas-
measurable (SmJ. When this condition is fulfilled, the definite integral, (Z7)
of f on Q is equal to the measure (DT^) of the. set A(/; Q).
Proof. Write, for short, T=LTL1 and suppose first that f(x)
is a simple function, finite, non-negative, and measurable Ba) on Q,
i.e. that f=[vi,Qi; v>,Q->; ¦•¦; vn,Q,,} where Qi are sets measurable (Sy)

90
CHAPTER III. Functions of bounded variation.
no two of which have common points. By Lemma .10.1., all the
sets A(f;Qt) are measurable (Hr) and T* I .A (/; Q,)\ =<«,.. [J* (Q^
for i—l,2,...,n. Hence, the not A(f;Q)=}jA{f; Q,) Is itself meas-
measurable (fir), and its measure (T) is equal to ^J T*['A(f;Q,)] =
i
Let now / be any non-negative function measurable (ilr) on Q.
There is a non-decreasing sequence [h,,(if,)) of simple functions,
finite, non-negative, and measurable (i3w) on Q such that'/{;»)—Mm h^x).
We then have
A0.4) A (/; Q) = lim A (A,,; Q)¦ -|-B (/; Q).
n
Now, by the above, all the sets A(h»;Q) arc measurable DJy) and
T*[A(hn;Q)~\=\'hndU for w=l,2, ... . On. the other hand, by
Theorem 10.2, the set B(f;Q) has measure G1) zero. It therefore
follows at once from A0.4) that the set A(f;Q) in itself measur-
measurable Bt) and that
T*[A(f; Q)] = lim fhHdU= ffdU.
" Q u
It remains to prove that, if the set A(/; Q) is measurable (iiy),
the function / is measurable (&{/). To do (.his, write for short
A=A(f;Q), and observe that, for every non-negative number y,
the set B[xeQ;f(x)^y] coincides with the Net K[(x,y)eA'\.
Thus, by Theorem 8.6, if A is measurable (iiy), the set .13 [>e<,J ;/(#
.V
is measurable B#) for all numbers y except; at most thone of a set
of measure (Lx) zero. But this suffices for the moannralrility of /
on Q (cf. Chap. I, G.2)) and so completes the proof.
* § 11. Translations ot sets. As an application of Theorem
8.6, we shall prove in this § a theorem on parallel trannlaUons
of sets. As a matter of course, in, what follows, traiiHliiUons could
be replaced by rotations, or by certain other transformations consti-
constituting continuous groups and preserving Lobcs^'iic measure.
Translations of sets.
91
Given two points x—(xu ;ri; ..., xm) and y= (yh y,,..., ym) in the
space Jim, we shall denote by %+y the point {x±+yi, xi+y-,,..., xm+ym)
and by
the number (x~\+a%-\- ...
We shah1 write
as-j-0 when |a?
| —>0. If Q is a set in the space Rn, and a any
point of this space, Q{a) will denote the set of all points x+a where
XfQ. The set Q is termed translation of Q by the vector a. If <B is
an additive function of a set in Rm and aeBm, we shall write
() i) for every set X bounded and measurable C3).
A1.1) Theorem. If Q is a bounded set C3) of measure (L) zero in the
space lim and $ is an additive function of a set C3) in JR,n, the func-
function 0 vanishes for almost all translations of Q, i. e. 0(Q^a))=§l'a)(Q)^=Q
for almost all points a of Jim.
Proof. We may clearly assume # to be a non-negative function,
and Q to be a bounded set (©<>). Hence, by Theorem 6.10, there is
a non-negative additive function TJ of an interval, such that
®(X)=TJ*(X) for every set X bounded and measurable C3).
Denote by M, for any set MC-Km, the set of all points (x, y)
of the space M2m which are such that xeRm, yelim and x+yeM.
The set If is clearly open whenever the set If is open. It follows
at once that if M is a set (G5«s), so is the set M. Finally, observe that
for every point zeRm we have B[(rc, g) ei?] = E[(s, y)eM]=Mi~2)-
x U
Since the given set Q is, by hypothesis, a set (®,i), so is the
setg, and by Theorem 8.6, fu*{Q^)dIjm{g)= j'hm{Q^yiU(s)=0,
' ttm Urn '
because all translations QS~^ of the set Q are of measure (Lm) zero.
Hence 0(Q^-^)—V*(Qt--z))=O for every zeRmf except at most a set
of measure (Lm) zero. Eeplacing —z by a, we obtain the required
statement.
A1.2) Theorem. Given an additive function of a set 5>, each of
the following three conditions is both necessary and sufficient for the
function 0> to be absolutely continuous'.
1° lim <I>(Q{a)) =]hn.<l>("\Q) = (l>(Q) for every bounded set Q meas-
a-H) 17-/0
¦urable C3) and of measure (L) zero;
2° tim${Q{")) = ]im<I>{")(Q)= ®{Q) for every bounded set Q
measurable C3);
3° limWt^' — <7>;I]=0 for every interval I.

OlIAI'TER IIX FuuoMoiih of bounded yiiriiitiou.
Proof. It is evidently Hufficiont to oNlisibliHli the y
of condition 3° and the sufficiency of condition ,1".
Suppose first that (P i« iui absolutely eontinitouK additive
function of a set. In virtue of Theorem M.ll, (!hap. ], (f> in thus
tlie indefinite integral of a. function / measurable C3). Let
I=[ai, b\;...; a,,,, bm] be sin interval in the tspuico conHiderod and l«t,/
be an interval containing / in its interior, for inntauco tho interval
[«,—l, &i+i; -; a.,,—l, bm+l]-
Let e be any powitivo number. Hince the function /(*) in in..
tegrable over J, there exists a number (?>() mich thnt / \f(x)\ div<,e/3,
x
for every set XQ.J measurable C3) and of nxuwuro (L) Iran than ;/.
Therefore
/'\f{x)\ dsa < e/3 and f |/(ji+«)| d*--* f |/0«)| d,r, < f /;{
(U-3) x x ;.,„)
if Xe% XCJ, \X\<7i ami |w| < I.
On the other hand, by Lumu's Theorem 7.1, tliei'e oxiNtu a cloHdd
set 1B1° such that the function. / is coutibiuous on ./*' mul Hiieli.
that \I—F\ <i?/2. Let G<1 be a powitivo Jiumlxu? nucIi that
I^C-T whenever |w|<ff, and such that
A1.4)
'3-1
whenever xe'If, x-\-v,fh\ awl |w|<ff.
Let now a be any point of Itm .such that |«|<(r. Hy (U..'4)
/
/P.'/r(- a)
On the other hand, \I— F.i^
and therefore, by A1.3),
f\f(-x+a) —/(sb)| dse<2f/S.
If we add this inequality to A.1.5) we obtain /"|/(jj-f-u) -/(./;)| dsKs,
i. e. the variation W[<T>W-®; I] = f\f(x+a)^f(x)\dx t«n<lH to 0
with |«|. The function <fi therefore fulfills <iondi|,ion ;j».
It remains to prove the HiiMictoncy ol; condifcum 1". Now if
the function <P fulfOls thin condition, 0 vmMm by TIkmhw,, I i.l for
every bounded set measurable (93) of menHuro (L) zero, n,n<l ho Ih
absolutely continuous.
t§12]
Absolutely continuous funotioas of an interval.
93
It was long known that every absolutely continuous fuuetion 0 fulfills
conditions 1°, 2° and 3° of Theorem 11.2. The converse, however, (i. e. the suf-
sufficiency of these conditions, in order that the function 0 of a set he absolutely
continuous) was established more recently. The sufficiency of condition 3° was
first proved by A. Plessner [1] (with the help of trigonometric series and for
luuctions of a real variable). As regards the other conditions A° and 2"), and
as regards Theorem 11.1, vide H. Milicer-Grrnzewska [1], and N. Wiener
and E. C. Young [1]. In the text we have followed the method nsed by the
latter authors.
§ 12. Absolutely continuous functions of an interval.
An additive function F of an interval will be termed absolutely con-
continuous on a figure Bo, if to each ?> 0 there corresponds a number
??>() such that for every figure BC.B0 the inequality |i2|<»? implies
|JP(-R)[ < f. In conformity with § 3, p. 59, we shall understand by
absolute continuity in an open set G, absolute continuity on every
figure R(ZG> and by absolute continuity, absolute continuity in the
whole space.
Every additive function of an interval, absolutely continuous on
a figure Ra, is of bounded variation on Bo. For, if F is a function
that is absolutely continuous on i?0, there exists a number ??>0
such that, for every figure BC.BOt the inequality |i?|<'/ implies
|JP(JB)|<1. Therefore, if we subdivide i?0 into a finite number of
intervals I\, I?, ..., In of measure less than »;, we obtain W(F; i?0)^
An additive function of an interval F, of bounded variation
on a figure i?0, will be termed singular on Bo, if for each e>0 there
exists a figure RC^o auola tlaat \s\<e alld ~W(F; ?,0QB)<.e.
The reader will observe the analogy between the above definitions and
the criteria given in Theorems 13.2 and 13.3, Chap. I, in order that an additive
function of a set should be absolutely continuous or singular. This analogy could
be pushed further by introducing the notions of absolutely continuous function,
and of singular function, with respect to a non-negative additive func-
function of an interval. But this "relativization", althongh useful iu certain
cases, would not play an essential part in the remainder of this book.
The following theorem is, abnost word for word, a duplicate
of Theorem 13.1 of Chapter I.

CHAPTER III. KunuUouH of bounded variation.
H2.1) Theorem, 1° In order that an additive, function of wn
interval be absolutely oonUmwun [singular] on a figure, ft0, it is neces-
necessary and sufficient that Us two variations, the upper mid the lower,
should both be so. 2° Every linear combination, with constant coefficients,
of two additive functions of an interval which are absolutely continuous
[singular] on a figure Bo is itself absolutely continuous [singular]
on Ro. 3° The limit of a bounded monotone sequence of additive functions
of an interval that are absolutely continuous [singular] on a figure Ro
is also absolutely continuous [singular] on Rv. 4° // an additive function
of an interval is absolutely continuous [singular] on a figure h'(l, the
function is so on every figure ItCJt0. fi" // an additive function of
an interval is absolutely continuous [singular] on each of the figurca
R1 and R2, the function is so on the figure Ky |-/i'2. IS" An additive function
of an interval e<mnot be both absolutely continuous and singular on
a figure R0) without vanishing identically on h'Q.
Part 3°, at most, perhaps requires a. proof. (Id difl!or\s slightly
from the corresponding part of Theorem ltt..l, (Jlniip. I.) Lot therefore
F he the limit of a hounded monotone sequence [fin) of additive
functions of an interval on a figure Ro. Lot ? bo any positive number.
Since the functions F—F,, are monotone on Ro, there exists a pos-
positive integer n0 such that
A3.2) \F(B)—FllJiE)\^!\F(R0)—FaSR<t)\<ef3 for every figure HQRV
This being so, let us suppose that the functions /''„ iwo ab-
absolutely continuous on Bo. There is than an ?;> 0 huc.Ii that, for
every figure R(ZR0, \R\<V implies the inequality \.I<\h(,R)\<e/2
and therefore, by A2.2), the inequality \F(R)\<.s. The function F
is thus absolutely continuous on Bo.
Suppose next that the functions JB1,, are singular on ft0. There
is then a figure -RiQ^o su°h ^nat; |^i|<? an<l Wfi'1,,,,; ,/i'0!~iiVI<E/2.
Hence, by A2.2), W[F; E0QR1]<e, -which shows that the function
F is singular. This completes the proof.
We shall now establish two simple theorems that show ex-
explicitly the connection between the absolutely eonl.inuons or sin#'-
ular functions of an interval and thoHO of. a sot. To avoid misnixlor-
standing, we draw the reader's attention, to |,ho abbreviations
adopted in § n, p. 66, in the terminology of functions ol' ;i. sot.
[§12]
Absolutely continuous functions of an interval.
95
A2.3) Theorem. In order that a non-negatire additive function F
of an interval be absolutely continuous, it is necessary and sufficient
that the corresponding function of a set F* should be so.
Proof. Suppose that the function F is absolutely continuous.
In order to prove that the function F* is so too, it is enough to
show that F* vanishes on every bounded set of measure (L) zero.
Let therefore E be such a set, and let J be an interval that contains
F, in its interior. For any e>0, let t] be a positive number such that
A2.4) \.R\<V implies \F(R)\<e for every figure RQJ-
Since \F\=0, there exists a sequence of intervals {!„} in J such that
A2.5) ECSln and E\I,,\<v.
" n
Denote by Bk the sum of the h first intervals of this sequence.
By Theorem 4.6 (or 6.1) of Chap. II, and Theorem 6.2, the relations
A2.4) and A2.8) give F*(E) ^]imF*(Rfl)^]hn.F(Bk)^e) from
which it follows that F*(JB)=0. * /
Conversely, if F* is an absolutely continuous function of a set,
the absolute continuity of F follows at once from the inequality
F(B)^F*(B) which holds by Theorem 6.2 for every figure B.
A2.6) Theorem. In order that a non-negative additive function
of an interval F be singular, it is necessary and sufficient that the
corresponding function of a set F* should be so.
Proof. Suppose that the function of an intervalJ? is singular,
and let J he any interval. Given any number ?>0, there is then
a figure B,(ZJ such that \B\<Ce and F(JQR)<.e. Consequently,
by Theorem 6.2, we have F*(J°—R)sZ,F(JG%)<e, which shows
on account of Theorem 13.3, Chap. I, that the non-negative function
of a set F* is singular in the interior of every interval J, and there-
therefore in the whole space.
Suppose, conversely, that the function of a set F* is singular,
and let s be any positive number. Given any interval I there is then
a set ECI° such that |J5| = 0 and F*(I°—E) = 0. Consequently,
there is a sequence of intervals {!„} in I such that
A2.7) r—ECUl,, and A2.8) I" JF (!„)<«?.
Denote by Bk the sum of the Ic first intervals of this sequence. Since
|E|=0, we obtain from A2.7) that \RK\ > \I\ — e for a sufficiently
large kB, and writing 'P=lQBk{l, this gives |P|<f. Again, by A2.8),
which proves that the function F is singular.

90
CHAPTER HI, KunctioiiK of hounded variation.
§ 13. Functions of a real variable. The momt important
of the notions and theorems of this chapter wore originally given
a rather different form: they were made to refor, not; to additive
functions of an interval, but to functions of a real variable. It in,
however, easy to establish between funetionH of a real variable and
additive functions of a linear interval, a. correspondence rendering
it immaterial which of these two kinds of fimctiouN is considered.
To do this, let f(x) be a,n arbitrary finite function of a real
variable on an interval /„. Let us term increment of f(x) over any
interval 1=0, b] contained in l0, the difference f(f>)—/(«). Tims
defined the increment is ¦ an additive function of a linoar interval
I Clot anA corresponds in a unique manner to the function /(*),
Conversely, if we are given any udditivo function /''(/) of a linoar
interval I, tliis in itself defineH, except for an additive constant,
a finite function of a real variable f(m) wIiohc increments on tho
intervals I coincide with the corresponding vaJucs of the function /*'(/).
We shall understand by tipper, lower and absolute, variations
of a function of a real variable f(x) on an interval /, the upper,
lower, and absolute, variations of the increment of /(*¦) over /,
To denote these numbers, we shall use symbols similar, to those
adopted for additive functions of an interval, i. e.: W(/; /), W(/; I),
and W (/-,!).
A finite function will be termed of bounded variation on an
interval Io, if its increment is a function of an interval of bound?;!
variation on Io. Similarly the function is abnolutdy continuouti, or
singular, if its increment is absolutely continuous, or Hingnbw. As we
see immediately, in order that a function f(x) bo of bounded vari-
variation on an interval Io, it is necessary and sufficient that there
exists a finite number M such that }J\f{bi)—f(ai)\<.M for every
sequence of non-overlapping intervals {[at, ?>,]} contained iu Ia.
Similarly, in order that f{x) be absolutely continuous, it is necessary
and sufficient that to each s>0 there corresponds an ^>0 such
that S\f{b[)—/(«,)|<e for every sequence of non-overlapping- inter-
intervals {[«;,&,¦]} contained in Io and. for which ?\bi—
[§13]
Functions of a real variable.
97
If f(x) and g(x) are two bounded functions on an interval Io,
and M denotes the upper bound of the absolute values of /(.r) and
g(x) on Zo, we have
\f(b)g(b)—f(a)g(a)\*ZM[\f(b)—f(a)\+\g(b)~g(a)\]
for every interval [a, b]CI0- I* follows at once that
A3.1) The product of two functions of bounded variation [absolutely
continuous] on an interval is itself of bounded variation [absolutely
continuous] on this interval.
Finally we see that if a function of an interval F corresponds
to a finite function of a point / (i. e. is the increment of /), we have
Oj(f; a)= Oi(F; a) for any interval I and any point a el (of. Chap. II,
§3, p. 42, and the present Chapter, §3, p. 60). Thus, in particular,
in order that the function / be continuous at a point a according
to the definition of § 3, Chap. II, it is necessary and sufficient that
the function of an interval F that corresponds to / should be so
according to the definition of § 3 of the present Chapter.
If at a point a a function of a real variable / has a unique,
limit on the right, this limit will be denoted by /(«+); similarly,
f(a—) will stand for a unique limit on the left. If the function /
is defined in a neighbourhood of a point a and both limits /(a+),
f(a—) exist, then the ossillation o(/; a) (vide Chap. II, p. 42) is
equal to the largest of the three numbers \f(a-\-) — f(a—)',
If both limits /(«+) and f(a—¦) exist and are finite, and
/(«)=?[/(«+)+/(«—)] *ne Emotion f(x) is termed regular at the
point a. It is regular if it is regular at every point.
Let / be any function of a real variable, of bounded variation,
and {a,,} a sequence of points. Let us put s(a) = 0 and
slx) =
)—/(»—) for x>a
f(a—)~f(a)+?(a'x)[f(a!l—)—f(an+)]+f(m)—f(w+) for
where the summation Z!(a'x) is extended to all indices n such that
«<««<>, when x>a, and a>an>%, when w<La. The function s
thus defined is termed the saltus-function of f corresponding to the
sequence {a,,} of points. It is continuous everywhere except, perhaps,
S. Saks. Thuovv of the Intesiul. 7

98
CHAPTER III. Functions of bounded variation.
at the points «„, ami by subtracting it from / we obtain a function
of bounded variation, continuous at all points of continuity of / and,
besides, at all the points «„. If {a,,} is the sequence of all points of
discontinuity of /, the corresponding function s is called simply
the saltus-function of f. By varying the fixed point a we get the
various saltus-funetions of / which can obviously differ only by
constants. A function of bounded variation which is its own saltus-
function, is called a saltus-function.
The functions of a real variable whose increments over each
interval I coincide respectively with the variations W(/; I), W(f; I)
and W(/; I) of a function /, are also termed (upper, lower, and ab-
absolute) variations of /. By applying the Jordan decomposition (§ 4,
p. 62), we can express any function of a real variable / of bounded
variation as the sum of two functions that are respectively its upper
and lower variations. Thus any function of bounded variation, is
the difference of two monotone non-decreasing functions, and con-
consequently is measurable (93) and has at every point the two uni-
unilateral limits, on the right and left. Moreover, the set of its points
of discontinuity is at most enumerable, since the sum of its oscil-
oscillations at the points of discontinuity lying in any finite interval is
always finite (this is actually the special case of Theorem 4.1).
In various cases it is more convenient to operate on functions
of a real variable than on additive functions of an interval in lt^
The difference is, of course, only formal, and all the definitions
adopted for functions of an interval can be stated, with obvious
modifications, in terms of functions of a real variable. We need
not state them here explicitly. If F is a function of a real variable,
of bounded variation, the meaning of expressions such as Lebes-
gue-Stieltjes integral with respect to JF, integral (F),
sets (&f), and so on, may be regarded as absolutely clear, in view
of the definitions of § 5. If F is a continuous function and g
a function integrable (F), the integral fgdF, where 1 is a variable
i
interval, is an additive continuous function of an interval I (vide
§ 5, p. 65). There is, consequently, a continuous function of a real
variable whose increment on any interval I coincides with the
definite integral (JF) of g over I. This function, which in determined
uniquely except for an additive constant, is also termed indefinite
integral (F) of g.
[§13]
Functions of a real variable.
99
When there is no ambiguity, the additive function of an interval
that is determined by a finite function of a real variable F, will
be denoted by the same letter F, i. e. F(I) will stand for the in-
increment of F(x) on an interval I. By means of the corresponding
function of an interval, any function of a real variable F of bounded
variation determines an additive function of a set which we denote
by F* (cf. § 5). We see at once that F*(X)—F(b+)—F(a—) when
X=[a,b], and that F*(X)=F(a+)—F(a—) when X=(a), i.e.
when X is the set consisting of a single point a.
If W(x) is the absolute variation of a function Fix) of bounded
variation, we clearly have V?(F*; X)^W*(X) for every set X
bounded and measurable C3). The opposite inequality does not
hold in general. If, for instance, X is a set consisting of one point
only, and F is the characteristic function of X, then W(F*; JT)=
=F*(X)=0, while W*(X)=2. We can, however, state the following
theorem:
A3.2) Theorem. If F(x) is a function of a real variable of bounded
variation, and W(x) is the absolute variation of F(x), then W(F*; Z)=
=W*(X) for every set X bounded and measurable B3) at aM points
of which F(x) is continuous.
Proof. Suppose first that the set X is contained in an open
interval Jo in which the function F(x), and consequently the function
W(x) also, is continuous. Let GCZ^a be an arbitrary open set such
that XBO. Then, expressing 0 as the sum of a sequence of closed
non-overlapping intervals {!„}, we get W*(X)^W*{G)=ZW*{In)=
= ZW(In)^ZW(F*;In)=W(F*; G); whence W*(X)<W(F*;X), and
n n
since the opposite inequality is obvious, W*(X)=W(F*; X).
Let us pass now to the general case. Let Io be an interval
containing X in its interior, and let ?>0. Denote by {a,,} the se-
sequence of points of discontinuity of F{x) interior to Io, and by 8n(a>)
the saltus-function of F (as) corresponding to the points an for n>N.
Let us put G(x)=F{x)—SN(x), where N is a positive integer suf-
sufficiently large in order that W(8N; I0Xe- The points at, a-i,..., aN,
none of which belongs to X, divide Jo into a finite number of sub-
intervals Jo, J\, —,Jn in the interior of which the function G(a>)
is continuous. Hence, denoting by V(a>) the absolute variation of
7*

100
CHAPTER III. Functions o? hounded variation.
G{x), there follows, by what has already boon proved, V*(X-Jk)=
=W(?*; X'Jk) for 7c=0,1, ..., N, whence HI)=W(«*; I). On
the other hand, \W*{X)—V*{X)\ and |W(^*; X)—VV(<7*; Z) are
both at most equal to W(SW; /„)<?. Thus |W*(A')—W^1*; A')|<s»e,
and finally F*(X)=W(.F*; Z).
If Fix) is a finite function of a real variable and E an arbit-
rary set in Ii1} the set of the values of F(x) for ,*fB will be denoted
by F[E].
A3.3) Theorem. If F(x) is a function of a real variable of
bounded variation and ~W{x) is the absolute variation of F(x), then
\F[T$~\\ *^.W*{E) for every set E in i?t; and if further the function
F(os) is non-deereasing, and continuous at all points of 'E, then
\F[E]\=F*(E).
Proof. Let e be a positive number and {/„} a Moquonee of
intervals such that EC2]l,, and W*(E)+e>}]WUn). Then, if m,, and
M» denote the lower and upper bounds, respectively, of F(x,) on 1,,,
the sequence of intervals {[m,,, II,,]} covers tlie set F[E], and con-
consequently \FtE]\^?{M,l—-ma)sz2]W(l»)*ZW*(l!l) + e. Hence,
]\^()
Suppose now F(x) continuous at the points of B and non-
decreasing. By what has already been proved, \F[W\\*^.F*\E). To
establish the opposite inequality, let v be an arbitrary positive
number, and {</„} a sequence of intervals subject to the conditions
F[E\CSJn and \F\W\\+V>E\J »\- Let En denote the sot of the
points xeE such that F(x)ej',i Then JF*(.E,,X|J/(| for each n; and
consequently -F%E)^2V,,|<|.#pr]| + ??; whence F*(E)^ \F[E]\.
The characteristic function of a set consisting of a single point provides
the simplest example of a singular function of a real variable, tliat does not vanish
identically. This function is however discontinuous. It is easy to give examples
of functions of an interval that are additive, singular, continuous, and not idea-
tieally zero, in the spaces Itm for ro>2. For simplicity, consider tlio plane, and
denote, for any interval I, by F(I) the length of the segment of the line y = x
contained m I- the function of an interval F(I) will evidently have the dosired
properties. A similar example for ft, is less trivial. We shall therefore conclude
thisit, with a short description of an elementary method for the contraction of
continuous singular functions of a real variable.
We shall begin with the following remark which frequently provon u^d
[§13]
Functions of a real variable.
101
A3.4) Let E be a linear, bounded, perfect and non-dense -set, und a and /S>o
two arbitrary numbers. Then, ij a and b denote the lower and upper bounds of E,
a junction F(x) may be defined on the interval J,r=la, b] su as to satisfy the following
conditions: (i) F{a) = a, F(b) = ,J, (ii) F(x) is constant on each interval contiguous
to the set E, and (iii) F(x) is continuous and non-deereasing on the interval Jo and
strictly increasing on the set E.
To see this, let {I,,} be the sequence of intervals contiguous to E, and let
us agree to -write In-^J.m whenever the interval 1,, is sitnated on the left of Im.
By induction (cf. e. g. F. Hausdorff [II, p. 50]) we can easily establish a one-
to-one correspondence between the intervals In and the rational numbers of the
open interval (a, /3) so that, denoting by u(In) the number which corresponds
to the interval In, the relation 1,,-^Im implies u(I,,),:u(Im)- Let us now put
F(x) =ii{In) for xeln where n=\,2, ..., and then extend F(x) by continuity
to the whole of the interval Jo. We see at once that the function F(x) thus ob-
obtained satisfies all the required conditions (i), (ii) and (iii) of A3.4).
Xow let us choose for the set E in A3.4) a set of measure zero. Then if
{In} is the sequence of the intervals contiguous to E, we have
y,-(F;y ik)= y
n
for each positive integer n; and since |t70 — _.I*|—>0 as n—>oo, the fune-
*=i
tion F{x) is evidently singular on the interval -/0.
The singular function obtained by the foregoing construction is continuous
and monotone non-decreasing; the, function is not constant on the whole interval Ja,
but is so on certain partial intervals. Now, by the method of condensation
of singularities, it is easy to derive from it a singular continuous function
that increases everywhere.
To do this, suppose in A3.4), |E[ = O, a=a = 0, & = /?=!, and extend the
function F[x) on to the whole axis R1 by stipulating F{z + 1)= 1 +F(x). Write
A3.5)
This series is a uniformly convergent series of singular functions, since F{nx) is
clearly singular with F{x). Xow the functions F(nx) are monotone non-deereasing.
By Theorem 12.1 C°), the function ff(jr) is thus singular. This function is also con-
continuous, as the limit of a uniformly convergent series. To prove that H(x) is strictly
increasing, let x1 and xi.:-jr1 denote an arbitrary pair of points in [0, 1]. For
n' lj(x2 — j-j), we have nx2 — «Xj ¦ 1, and consequently F(nx,) ^(h^i);
while for every n, i?()«2K--iin(Hx1), whence by A3.5), E(xi)>H{x1) as asserted.
Various examples of this kind have been constructed by A. Denjoy [1],
W. Sierpinski [3], H. Hahn [I, p. 538], L. C. Young [1] and G-. Vitali [4];
cf. also O. D. Kellog [1], and E. Hille and J. D. Tamarkin [1].

102
CHAPTER III. Fniw'tiiiiiM of bounded variation.
§ 14. Integration by parts. Aw in the preceding §, we .shall
deal only in this § with functions of a real variable. For the latter,
we shall establish two classical theorems, of importance on account
of their many applications to various branches of Analysis. We shall
first prove them for the Lebesgue-Htieltjes integral and then .spe-
.specialize them for the ordinary Lebesgne integral.
A4.1) Theorem on inter/ration by parts. If U(nn) and V(x)
are two functions of hounded variation, we hart' for erery internal
I0 = [a,b] ¦
YdU=U(b+)V(h+)— U(a—)V{a—),
provided that at each point of l0 either one at leant of the fmieHonx
U and V is continiwux, or both are regular.
Proof. In order to simplify the notation assume «=o and
6=1, and consider the triangle Q=E[0^.r^l;j/^,/-'| on the, piano /»'„.
The set E[(#, y)eQ] is then the interval [;//, 1] or the empty set,
X
according as y belongs, or does not belong, to the interval [0, 1].
Similarly, E [(#, y)eQ] is the interval [((, i»] or the empty set,
according as we have, or do not have, 0<a?^l. Hence, by Pubiiii's
theorem in the form (8.0),
i. e.
i i
A4.2) fu(x-)cW(x) + fV(x
» 6
Interchanging TJ and V and adding the corresponding equation
to A4.2), we get, on dividing by 2,
i i
A4.3) f ±[U(x+)+U{x—)] dV(x)+fi[V(x+)+ V(x—I
— U@—)F@—).
Let M be the set of the points in /0=[0, 1 | at which the
i V{) i h
function V{x) is regular. Then
—)]dV(%)= l'U{a)dV(a>).
A4.4) f ^
Integration by parts.
103
On the other hand, the set Io—M is at most enumerable and, by
hypothesis, the function T{x), and consequently both its relative
variations, are continuous at each point of Io—JI. Thus, the def-
definite integral (T) of any function over the set Io—If is zero, and
i i
it follows from A4.4) that f]iiU(x->-)-^r(x—)]dT(x) = [r{x) dY(*).
Similarly, the second member on the left-hand side of the rela-
i
tion A4.3) is equal to / Y(x) dU(ic), and this relation may be written
o
L 1
[UdY+ [YdU=F(l+) F{1+)—C@—)r@—),
6 6"
-which proves the theorem.
The theorem may be also proved independently of Fnbini's theorem, but
then the proof is slightly longer. The proof given above was eominunieated to
the author by L. C. Young.
A4.5) Second Mean Value Theorem. If U(x) and Y(x) are
two non-decreasing functions and the function Y(x) is continuous,
then in any interval [a, b] there exists a point | such that
A4.6) fUdY=U(a)-[Y(l)-Y(a)l+U(b)-[Y(b)-Y(»)l
a
Proof. Since the values of U(x) outside the interval [«, b]
do not affect A4.6), we may suppose that U{a—)=Z7(«) and,
U(b+)=U(b). Therefore, making use of Theorem 14.1 and of the
first mean value theorem (Chap. I, Th. 11.13), we obtain
A4.7)
= V(b) Y(b) —U{a)Y(a) — ji • [ U(b) — Z7(o)],'
where /t is a number lying between the bounds of the function Y(x)
on [a, V\. But, since this function is by hypothesis continuous, there
exists in [a, b] a point ? such that ^=7A). Substituting this value
for fi in A4.7) we obtain the relation A4.6).

204 CHAPTER III. Functions of bounded variation.
As a special case of Theorem 14.1 we have the following theorem
on integration by parts for the Lebesgue integral:
A4.8) Theorem. If u(x) and v(x) are two .summable functions
on an interval [a,b] and XJ(x) and V(x) are their indefinite in-
integrals (L), then
b b
A4.9) fmx)v(x)dx+ fV(x)u(x)dx = U(b)-V(b)—TJ(a)-V(a).
Proof. Observe first that by writing for instance u(x) = Q
and v(x)=0 outside the interval [a,b], wo may suppose that the
functions u{x) and v{x), and their indefinite integrals 77 (a;) and V{x),
are defined on the whole straight line i^. Also, by altering, if neces-
necessary, the values of the functions u(x) and v{x) on a sot of measure
(L) zero, which does not affect the values of the integrals in A4.9),
we may suppose that these functions, together with the functions
V(x)v(x) and V(x)u(x), are measurable C3) (cf. Theorem 7.6
of Vitali-Carathe'odory or else Lusin's Theorem 7.1). We may,
therefore write, according to Theorem 15.1, Chap. I,
6 6 b b
fU(x)v(x)dx=fu{x)dV(x) and fv(x)u(x)dx= [V(a>)dU(x),
a a a a
and A4.9) follows at once from Theorem 14.1.
Similarly, we derive at once from Theorem 14.5 the second
mean value theorem for ihe Lebesgue integral;
A4.10) Theorem. If TJ{x) is a non-decreasing function on an
interval [a, b] ami v (x) is a summable function on this interval, then
/ U{x) v{x) dx = V[a) fv(x) dx + U(b) j'v[x) dx,
a " i
where | is a point of [«,&].

CHAPTEE IV.
Derivation of additive functions of a set
and of an interval.
§ 1. Introduction. In this chapter we shall study Lebesgue's
theory of derivation of additive functions in a Euclidean space of
any number of dimensions. When other spaces are considered, or
when we specialize our space (to be, say, the straight line JR1 or
the plane JS2), we shall say so explicitly.
In what follows, an essential part is played by Yitali's Covering
Theorem (vide, below, § 3) which is restricted to the case of Lebes-
Lebesgue measure. For this reason, the theorems of the present chapter
have not in general any complete or direct extension to other meas-
measures, not even when the latter are determined by additive functions
of an interval. In accordance with the conventions of § 5, Chap. Ill,
the terms measure, integral, almost everywhere, etc. will
be understood in the Lebesgue sense whenever we do not explicitly
assign another meaning to them. Similarly, by additive functions
of a set we shall always mean functions of a set C3) (some of which
may of course be continued on to wider classes of sets, cf. Chap. Ill, §5).
We have already remarked in § 1, Chap. I, that any additive
function of a set $ in a space Rmj may be regarded as a distribution
of mass. It is then natural to consider the limit of the ratio <P{S)I\8\
where 8 denotes a cube, or a sphere, with a fixed centre a and with
diameter tending to 0, as the density of the mass at the point a.
By the fundamental theorem of Lebesgue {vide, below, Theorem 5.4)
this limit exists almost everywhere. Moreover Lebesgue has shown
that in the above ratio, 8 may be taken to denote much more gene-
general sets than cubes or spheres. Of these, further details will be
given in the next §.

106
CHAPTER IV. Derivation of additive
§ 2. Derivates of functions of a set and of an interval.
Suppose given a Euclidean space i?m. By parameter of regularity r (E)
of a set E lying in this space, we shall mean the lower bound of
the numbers |jE7|/|J1 where J denotes any cube containing E. Thus
when E is an interval, l'"jL'"<:v{E)^:ljL, where I denotes the
smallest and L the largest of the edges of E; in particular, the
parameter of regularity of a cube is equal to 1.
A sequence of sets [E,,} will be termed regular, if there exists
a positive number a such that v(E,,)>a for w—1,2, ....
We shall say that a sequence of sets {A1,,! tends to a point a, if r>(E „)->()
as «-»¦ oo, and the point a belongs to all the sets of the sequence.
Given a function of a set i> (not necessarily additive) we call
general upper derivate of (J> at a point a the upper bound of the
numbers Z such that there exists a regular sequence of closed sets J/i1,,}
tending to a, for which lim $(En)l\E,,\ — l- We shall denote this
n
derivate by ~D<I>{a). Similarly, merely replacing the cloned sets
by intervals, we define the ordinary upper derivate of (I> at a
point a, and we denote it by '3 (a). If we remove the condition
of regularity of the sequences of intervals considered, we obtain
the definition of strong upper derivate. In other words the
strong upper derivate of # at a point a is the upper limit of the
ratio <S(I)/|I|, where I is any interval containing a, whose diameter
tends to zero. This derivate will be denoted by <2Js(a).
The three lower derivates at a point a, general I)@(a), ordinary
#(«), and strong 0s{a), have corresponding definitions, and if at
a point a the numbers B#(«) and D$(a) are equal, their common
value is termed general derivative of (I> at the point a and will be
denoted by DW(a). If further D # (a) =j= °o, the function 0 is said
to be derivable in the general sense at the point a. Similarly we define
the ordinary derivative $'(«) and the strong derivative $'s(<t), as well
as derivability in the ordinary sense, and in the strong sense, of the
function $ at the point a. Sometimes the derivatives D®(«), 0'(a)
and <h(a) are termed unique derivates, while the upper and lower
derivates (general, ordinary and strong) are termed extreme derivates.
At any point a, we clearly have M(a)<^(a)<^(a)<D!P(fl)
and similarly f,(«)^!(flK^)<$,(«); so that the existence
either of a general derivative, or of a strong derivative, always
implies that of an ordinary derivative. On the other hand, no such
relation holds between the general and the strong extreme derivates.
Derivates of functions of a set and of an interval.
107
It may be noted that in order that it be possible to determine
the general derivates of a function of a set #, the latter must be
defined at any rate for all closed sets; whereas in order to determine
the extreme ordinary derivates, or the extreme strong derivates.
we need only have the function <f> defined for the intervals. This
is why the process of general derivation is most frequently applied
to additive functions of a set, and that of ordinary, or of strong,
derivation to functions of an interval. We shall often omit the terms
"ordinary", "in the ordinary sense", in expressions such as "ordinary
derivate", "derivability in the ordinary sense".
We have seen (Chap. III. § 5) that an additive function of
an interval F of bounded variation determines an additive function
of a set F*. Let us mention, in the case in which the function F
is non-negative, an almost evident relation between ordinary deri-
derivates of F and general derivates of F*:
B.1) Theorem. If F is an additive vnn-tiegatire function of an inter-
interval, then, at any point x, which belongs to no liyperplane of discontinuity
of F, we hare BF*{x)<iF(x)^F{x)^DF*(jc).
In particular therefore, the ordinary derivative F'(x)=~DF*{x)
exists at almost every point x at ichich F* has a general derivative.
Proof. Since jP(Z)<P*(J) for any interval I, the inequality
F(x)^DF*(x) is obvious. On the other hand, let I denote any number
exceeding F(x). Then there exists a regular sequence of intervals {!„)
tending to the point x and such that ]im F(In)I\In\<.l. Since x does
n
not belong to any plane of discontinuity of F, we may asssume
that it is an internal point of all the intervals !„. Hence we can
make correspond to each interval I,, an interval J,,(ZIl such
ihabxeJa, \Ja\^{l—lhi)-\In\ and r(Jn) = r(In). We then have
UmsupJ*(Jn)/|J^nm-F(In)/|In|<Z. Now since {Jn} is a regular
n n
sequence of intervals tending to x, it follows that DP(i)<!, and
therefore that BF*{x)^F{x).
Let us note also the following result:
B.2) Theorem. If f is a summable function and <I> is the indefinite
integral of f, then D3>(a?X/(;p) and <Ds{x) ^.f (x) at any point x at which
the function f is upper semi-continuous, and similarly ~DjI> (x)^f [x) and
^_s(x.)^f{x) at any point x at which the function f is lower semi-continuous.
In particular therefore, $'{x)= @'s(x)=D${x)= f(x) at any
point x at ichich the function f is continuous.

108
CHAPTER IV. Derivation of additive functions.
In lf1 there is no difference between ordinary and strong deri-
derivation. If F{x) is a finite function of a real variable, we understand
by its extreme derivates F[x) and F{x), and by its derivative, or unique
derivate, F'(x), tlie corresponding derivates of the function of an
interval that F(x) determines (cf. Chap. Ill, § 13). Besides these
derivates, which we shall often term bilateral, we also define, for
functions of a real variable, unilateral derivatives and derivates.
Thus, if F (x) is a finite function of a real variable defined in the neigh-
neighbourhood of a point x0, the upper limit of [F{%)—F(xo)]j(x—x0)
as x tends to x0 by values of x>x0 is called right-hand upper derivate
of the function F at the point x0 and is denoted by F+(x0). Similarly
we define at the point x0 the right-hand lower derivate F+{sc0) and
the two left-hand, upper and lower, derivates, F~(ss0) and JT~(x0).
These four derivates are called unilateral extreme, or Dini, derivates.
If the two derivates on one side (right or left) are equal, their
common value is called unilateral {right-hand or left-hand) derivative
of the function F at the point in question. Finally, we shall call inter-
intermediate derivate of F(m) at the point x0, any number I such that
there exists a sequence {osn} of points distinct from xQ for which
xn=x0 and ]im[F{xn)—F(xo)]j(xn—a>0) = l.
Let B be a linear set, x0 a point of accumulation of U, and
F{x) a finite function defined on E and at the point J3O. The upper
and lower limit of the ratio [F{a>)—F(xo)]j(x—xa) as a? tends to x0
by values belonging to the set E, are called respectively the upper
and lower derivate at F at x0, relative to the set E. We shall denote them
respectively by FE(xQ) and FE(x0). When they are equal, their
common value is termed derivative of F at x0 relative to the set E,
and is denoted by F'E{x0).
Besides this derivation relative to a set, we define also
derivation relative to a function. Suppose given two finite
functions F{x) and U{x), and let x0 be a point such that the func-
function U is not identically constant in any interval containing a>0.
We then call upper derivate Fu(x0) and lower derivate Fu(ir,()) of the
function F with respect to the junction V at the point ,% the upper
limit and the lower limit of the ratio [F{x)—F(xo)]/[V{x)~- U(nc0)]
as x tends to x0 by values other than those for which F(x)—F(xo) =
= U{x)—U{xo)=O. Similarly, considering unilateral limits of the
same ratio, we_ define four Dini derivates of F with respect to U:
*?() ??() J??() and F~(x).
Titali's Covering Theorem.
109
When all these extreme derivates are equal, their common
value is denoted by F'u{x0) and called derivative of F with respect
to the function U at the point x0. The most usual ease in which this
method of derivation is applied, is when V is a monotone increasing
function; and it is then easy, by cha-nge of variable, to reduce deri-
derivation with respect to U to ordinary derivation.
§ 3. Vitali's Covering Theorem. We shall say that a family (?,
of sets covers a set A in the sense of Yitali, if for every point x of A
there exists a regular sequence of sets ((E) tending to x (cf. p. 106).
C.1) Titali's Covering Theorem. If in the space Bm a family
of closed sets (? covers in the sense of Yitali a, set A, then there exists
in <? a finite or enumerable sequence [En) of sets no two of which have
common points, such that
C.2) \A—2E»\=0.
11
Proof, a) We first prove the theorem in. the special ease in
which (i) the parameters of regularity of all the sets (<?) exceed
a fixed number a>0 and (ii) the set A is bounded i. e. contained
in an open sphere 8. We may clearly assume that, in addition, all
the sets (ffi) are also contained hi 8.
This being so, we shall define the required sequence {En} by
induction in the following manner.
For Ex we choose an arbitrary set (?), and when the first p
sets Ex, E2,..., Ep no two of which have common points, have
been defined, we denote by Sp the upper bound of the diameters
p
of all the sets (?) which have no points in common with 2]Et, and
by Ep+\ any one of these sets with diameter exceeding <5p/2. Such
a set must exist, unless the sets Ev Et,..., Ep already cover the
whole of the set A, in which case they constitute the sequence whose
existence was to be established. We may therefore suppose that
this induction can be continued indefinitely.
To show that the infinite sequence {En} thus defined covers A
almost entirely, let us write
C.3)
B=A—
and suppose, if possible, that |S|>0. On account of condition (i), we
can associate with each set En a cube Jn such that EnrJ* and

110
CHAPTER IV. Derivation of additive functions.
|2Jn|>a-[Jn|. Let Jn denote the cube with the same centre as Jn
and with diameter Dm + l) <5(J»). The series
C.4) ?\Jn\^(im+l)m -ElJnKiim+ir ¦ a .ZlEnl^iim+ir-a"-1
n n n
converges; therefore we can find an integer N such that Jj \Jn\<i\B.
It follows that there exists a point x0 e B not belonging to any Jn
for n>N; and since by C.3) the point x0 does not belong to 2jEn
n
and the sets En are supposed closed, there must exist in <? a set E
containing x0 and such that
C.5)
for n = l,2,.,.,
Hence the set E has common points with at least one of the sets En
for n>N; for otherwise we should have 0 < 6(!?)< 6n^ 26(JEn+i) <
<2E(JPn+i) for every positive integer n, and this is clearly impossible
since by C.4) we have lira d(Jn)=0. Let na be the smallest of the
n
values of n for which •??¦.#„ 4=0- Then on the one hand, E-En=0
for m=1, 2,..., n0—1, so that
C.6) S(E)^dtt<i^;
and on the other hand, by C.5), no~>N, which implies, by definition
of x0, that x0 does not belong to Jn,,. Thus we find that there are
both some points outside Jn,, and some points belonging to the set
EnvC.Jna, which are contained, in the set E; this set must therefore
have diameter exceeding 2<5(Jn,,)>2<5(_©„„)>Enil_i, in contradiction
to C.6). The assumption that \B\ > 0 thus leads to a contradiction
and this proves the theorem, subject to the additional hypotheses
(i) and (ii).
b) Now let (? be any family of closed sets covering the set A
in the sense of Vitali; and let us denote, for any positive integer n,
by Sv the sphere S@; n) and by An the set of the points x e A ¦ 8n
for which there exists a sequence of sets tending to x and consisting
of sets ((?) whose parameters of regularity exceed 1/w. The sets An
constitute an ascending sequence and J.= limJ.n.
n
We can now define by induction a sequence of families of
sets {Sn} subject to the following conditions: 1° each family Zn
consists of a finite number of sets ((?) no two of which have common
[§3]
Vitali's Covering Theorem.
Ill
points and none of which, for »>1, have points in common with
the sets of preceding families %, ?»,..., Sn-,; 2° denoting by Tn
the sum of the sets which belong to Sn,
C-7)
7=1
To see this, suppose that for n^Lp we have determined families Zn
subject to 1° and 2°. Write AP+1=AP+1—z.Ti, and consider the
family of all the sets (?) which are contained in the open set G2. Tt and
1=1
whose parameters of regularity exceed l/(p+l). This family evid-
evidently covers the set J.p+iCA>+iC$p+i m the sense of Vitali, and
by what we have already proved we can extract from it a sequence {Et}
of sets, no two of which have common points, so as to cover Ap+i
almost entirely. Therefore, for a sufficiently large index i0,
*=1 1=1
/=i
and, denoting by Zp+i the family consisting of the sets Elt E.2,..., Eit,
we find that conditions 1° and 2° hold when n=p+l.
Let us now write Z=T!Zn. The family Z consists of a finite
n
number, or of an enumerable infinity, of sets ((?) no two of which
have common points. Denoting the sum of these sets by T, we find,
by C.7), |J.—T|=0 and this completes the proof.
The proof given above is due to S. Banaeh [3] (for other proofs ef. C. Ca-
ratlieodory [II, pp. 299—307] and T. Rad6 [3]).
Theorem 3.1 was proved by G. Vitali [3] in a slightly less
general form; he assumed the family (? to consist of cubes. H. Le-
besgue [5] while retaining the line of argument of Vitali, showed
that the conclusion drawn by Vitali could be generalized as follows:
C.8) Theorem. Given a set A and a family <? of dosed sets, suppose
that with each point xeA we can associate a number a>0, a sequence
{Xn} of sets ((?) and a sequence {J,,} of cubes such that
for ?i=
and
Then <? contains a sequence of sets no tico of which have common
points, that covers the set A almost entirely.

112
CHAPTER IV. Derivation of aflrlitivo functions.
This statement, although apparently more general than
that of Theorem 3.1, easily reduces to the latter. For let us denote
by & the family of all the sets of the form E+(a>), where E is any
set of ? and x any point of the set A. The family (?° clearly covers
the set A in the Vitali sense, and by Theorem 3.1 we can therefore
extract from it a sequence {E%} of sets no two of which have commou
points and which covers the set A almost entirely. Now each set 17"
either already belongs to S, or becomes a set (®) as soon as we
remove from it a suitably chosen point. Therefore, removing where
necessary a point from each El, we obtain a sequence of sets ((?) no
two of which have common points and whose sum, since it differs from
l by an at most enumerable set, covers A almost entirely.
For further generalizations of Vitali'a theorem (which, again, win bo proved
without introducing fresh methods), see B. Jessen, J. Marcinlriuwioz ami
A. Zygmund [1, p. 224].
It is easy to see that the hypothesis that the family ffi covers the set A
in the Vitali sense (and not merely in the ordinary sense) is eRH<mti»il for the
validity of Theorem 3.1. But, as has been shown by S. Banacli [1] and II. Bohr
(vide C. Caratheodory [II, p. 689]), this hypothesis cannot he dispensed with
in the theorem even in the case where ffi is a family of intervals such that to each
point x of the set A there corresponds a sequence {I,,} of intervals belonging to IS,
of centre % and diameter <!(!„) tending to zero as n—>-co.
For covering theorems similar to that of Vitali and which correspond to
linear measure (length, cf. Chap. II, § 9) of sets, vide W. Sierpinski [7],
A. S. Besicovitch [1] and J. Gillis [1].
§ 4. Theorems on measurability of derivates. Of the
two theorems which we shall establish in this §, the first is due to
S. Banach [4, p. 174] (cf. also A. J. Ward [2, p. 177]) and concerns
the extreme derivates of any function of an interval (not necessarily
additive). We begin by proving the following lemma:
D.1) Lemma. Any set expressible as the sum of a family of intervals
is measurable.
Proof. Let 3 be any family of intervals and let S be the sum
of the intervals of 3- Let d denote the family of cubes each of which
is contained in one at least of the intervals C). The set S is clearly
covered by (E in the Vitali sense and by Theorem 3.1, 8 is therefore
expressible as the sum of a sequence of cubes ((?) and of a set of
measure zero. Therefore the set S is measurable, as asserted.
D.2) Theorem. If.F is a function of an interval, its two extreme ordinary
derivates F and F and its extreme strong derivates Fs and Fs are measurable.
[§*]
Theorems on measurabilitv of cleriva-tes.
113
Proof. Let us take first the strong derivates of F, say Fs.
Let a be a finite number and P the set of the points x for which
F*{x)>a. For any pair of positive integers h and A-, let us denote
by Pv, the sum of all the intervals I for which 6(I)^.ifk and
We see at once that P=H[JPhk. Sow the sets
h k
h k
P/,,A are measurable on account of Lemma 4.1 and so is the set P.
This proves F*(%) to be a measurable function.
Consider next the ordinary upper derivate F. As before let a
be any finite number, and Q the set of the points ,r at which F(x)> a.
In order that a point x should belong to Q, it is clearly neces-
necessary and sufficient that there should exist a positive number a
and a sequence {!„} of intervals tending to x such that T(I,,)~^a
and F{In)l\In\^a+a for n=l, 2,.... Hence denoting for any pair
of positive integers h, h by Qn,k the sum of the intervals I such that
t(I)^l/i, ii(I)<l/i and F(I)l\I\^a+ljJi, we find easily that
Q=SF[Qh,k. Thus, since each set Qi, t is measurable by Lemma 4.1,
h k __ *
so is also the set Q. The derivate F{x) is therefore measurable.
It follows in particular from Theorem 4.2 that the bilateral
extreme derivates of any function of a real variable are measurable.
The same is not true of unilateral extreme, derivates. Neverthe-
Nevertheless, as shown by S. Banach [2] (cf. also H. Auerbach [1]), these
derivates are measurable whenever the given function is so. Similarly
by a theorem of W. Sierpinski [8], the Dini derivates of a function
measurable (93) are themselves measurable (93). These two results
are included in the following proposition, from which they are
obtained by choosing the class S? to be either 2 or 93.
D.3) Theorem. If&is an additive class of sets in Iiv which includes the
sets measurable (S13), the Dini derivates of any function of areal variable
wliicli is finite and measurable (SE), are themselves measurable (S?).
Proof. If F is any finite function of a real variable, x any
point and h,h any pair of positive integers subject to t>/i, let us
write Diuk{F; x) for the upper bound of the ratio {Fit)—F(x)]j(t—x)
when x + l/i<it<x + ljfi. At any point x we clearly have .
D.4) F+ (x) = lim lim Bhh{F: x).
h k
Now let a be any finite number and consider the set
D.5)
S, Saks. Theory of the Integral.

114
CHAPTER IV. Derivation of additive functions.
We see at once that if the function F is constant on a set E, the
set of the points x of B at which Bh,h(F;x)>a is open in B (cf.
Chap. II, p. 41). Thus the set D.5), and consequently the expression
Da,a(-P;«) as function of x, is measurable (9E) whenever the fun.ction.F
is finite, measurable (9t) and assumes at most an enumerable infinity
of distinct values.
This being so, let F be any finite function measurable ($?).
We can represent it as the limit of a uniformly convergent sequence
{Fn} of functions measurable ($?) each of which assumes at most
an enumerable infinity of distinct values: for instance we may write
F,,{x)=ijn, when ijn^F{x)<i{i-\-l)jn for i=...—2,—1,0,1,2,....
We then have Dhtk{F;x) =lim!>/,,/,(F,,;x), and since by the above
n
the functions Dhtk(Fn;x) are measurable Bt) in3 so is Di,tll(F;x).
It follows at once from D.4) that the derivate F+(x) is also meas-
measurable (96), and this completes the proof.
§ 5. Lebesgue's Theorem. We shall establish in this § the
fundamental theorem of Lebesgue on derivation of additive functions
of a set and of additive functions of bounded variation of an interval.
E.1) Lemma. If for a non-negative additive function of a set 0 the
inequality Jy&(x)^a holds at every point x of a set A, then
E.2) $(X)>a>\A\
Iwlds for every set X~2)A, bounded and measurable EB).
Proof. Let e be any positive number and b any number less
than a. By Theorems 6.5 and 6.10 of Chap. Ill, there exists a bounded
open set G such that
E.3)
and
Let us denote by (? the family of closed sets EQG for each
of which ^(H)^b-\H\. Since by hypothesis, B® (x)^a>b at any
point xeA, the family (E covers the set A in the Vitali sense, and
by Theorem 3.1, we can extract from it a sequence {&,} of sets no
two of which have common points, so as to cover almost entirely
the set A. Therefore, on account of E.3),
In this we make
and b->a, and E.2) then follows at once.
Lebesgue's Theorem.
115
E.4) Lebesf/ue's Theorem. An additive function of a set is almost
everywhere derivable in the general sense. An additive function of
bounded variation of an interval is almost everywhere derivable in
the ordinary sense.
Proof. On account of Theorem 2.1 we may restrict ourselves
to non-negative additive functions of a set.
Let ® be such a function and suppose that D2>(.3?)>I)(P(a;)
holds at each point x of a set A of positive measure. For any pair
of positive integers h and h let us denote by Ah^k the set of the points
x of A for which D$(x)>(h+l)i7c>hjk>I)_${x). We have A=?Ahk,
and therefore there exists a pair of integers li0 and ft0 such that
|^-a,,,aJ > 0. Let us denote by B any bounded subset of Aj^x of
positive outer measure. Let e be any positive number and G
a bounded open set such that
E.5)
and
\G\<:\B\+e.
Consider the family of all closed sets E(ZG for which
This family covers the set B in the Yitali sense, and therefore con-
contains a sequence {En) of sets no two of which have common points,
which covers the set B almost entirely. Writing Q =X &•> we find, on
n
account of E.5),
E.6)
a.q n
On the other hand, D#(#)^(A0+l)/&0 at each point xeB. Therefore,
since all but a subset of measure zero of the set B is contained in the
h ' 1
set Q, it follows from Lemma 5.1 that W(Q)^~^-—\B\, and therefore,
on account of E.6) that (fto+l)-[B|^Ao-(|5| + e). But this is clearly
a contradiction since |-B|>0 and e is an arbitrary positive number.
Thus the function 3> has almost everywhere a general derivative
D$. It remains to prove that the latter is almost everywhere finite.
Suppose the contrary: there would then exist a sphere S such that
D<P(ic)=+oo at any point x of a subset of $ of positive measure.
We should then have, by Lemma 5.1, #($)=+00, which is impos-
impossible and completes the proof.
8*

116
CHAPTER IV. Derivation of additive functions.
The preceding theorem was proved by H. LobeMguo [I, p. 12BJ fimt
for continuous functions of a real variable and later [5, ji. 408—425] for additive
functions of a set in Rn. Among the many memoirs* devoted to Himplifying the
proof we may mention: G. Faber [1], W. H. and (i. V. Young [1], II. Stoiu-
haus [1], Cli. J. do la Vallee Poussin [1, 1; p. 103], A. Rajclutian and
S. Saks, [1], J. Bidder [3] (of. also the direct proof of Lebesgue'.s theorem for
additive functions of bounded variation of an interval: in the first edition of lliis
book). More recently F. Riesz [6; 7] has given an elegant proof of Ix'beague'n
theorem for functions of a real variable. Finally S. Bauac.h [4, p. 177] baa exten-
extended the theorem in question to a class of functions of an interval which in slightly
wider than that of additive fimctions of bounded variation. The proof given
liere applies also without any essential modification to the theorem of Bamich.
For an extension of the theorem to certain abstract spa oca, see J. A. Clark-
son [1] (cf. also y. Bochuer [3]).
Another application of Lemma 5.1 is the proof oil the fol-
following theorem on term by term derivation of monotone seqvenees
of additive functions:
E.7) Theorem. If an additive function of a set 0 in the limit of a mono-
monotone sequence [$„} of additive functions of a net, then almost everywhere
In the, same way, if an additive function of bounded variation
of an interval, F, is the limit of a monotone sequence [J?n) of additive
junctions of bounded variation of an interval, then almost everywhere
T{x)=lmiF'n{x).
Proof. Suppose, to fix the ideas, that the 8oquwi.ee. {$„} is
non-decreasing and write ®n = ® — $>,¦ To establish the first part
of the theorem, we need only show that
E.8)
limD @,,(x) = 0
almost everywhere.
For this purpose, let A denote the set of the points x at which
E.8) is not satisfied and suppose that \A\ > 0. For any positive inte-
integer lc we write Ak for the set of the points x at which lim Bd,,{x)
Since A=ZAh there is an index k=k0 such that |^
b
0. Let B do-
note any bounded subset of Aku of positive measure aad I an interval
containing B. Since the sequence {&»} is non-increasing, ho i.s the
sequence [T>Sn) and therefore we must have B®,,(«)^ !//<;„ for
W = l, 2, ... at any point xeBC Aki. Hence by Lemma fi.l, we find
[§6]
Derivation of the indefinite integral.
117
) \B\fh fo:r every positive integer n, and this is clearly a con-
contradiction since \B\ > 0 and lim &„(!) = 0A) —lim #„(!) = 0.
n n
We might prove similarly the second part of the theorem,
but actually the latter can be reduced at once to the first part.
In fact if we suppose the given sequence {Fn} non-decreasing and
write Tn = F—Fn, then the functions of an interval, Tn, are non-
negative and the sequence {Tn{Ij} converges to 0 in a non-increasing
manner for every interval I. The sequence [T?} of additive and
non-negative functions of a set is then also non-increasing and con-
converges to 0 (cf. Chap. Ill, Theorem 6.2). Hence, by the first part of
our theorem and by Theorem 2.1, we have ]imK(<Jo)=~\im~DT%(x)=Q>
n n
and therefore lim F'n(x)=F'(x), for almost all x, which completes
n
the proof.
For fimctions of a real variable, Tbeorern 5.7 may be stated in the fol-
following form (vide 6. Fubini [2]; cf. also L. Tonelli [3] and P. Eiesz [6; 7]).-
Ij F(x)=yFn(x) is a convergent series of monotone non-decreasing junctions,
n
then the relation F'{x)=yjFn(x) holds almost everywhere.
§ 6. Derivation of the indefinite integral. Given a set A,
let us write ~LA{X)=\A-X\ for every measurable set X. The function
L.4 of a measurable set, thus defined, is termed measure-function
for the set A. Considered as function of a measurable set, or as
function of an interval, L_4 is additive and absolutely continuous;
and, if further the set A is measurable, we have L.4(JT) = [
for every measurable set X, i. e. the function L.4 is the indefinite
integral of the characteristic function of the set A.
F.1) Theorem. For any set A ice have
F.2) &~La(x) = aA(x)
at almost all x of A; and if further, the set A is measurable, then F.2)
holds almost everywhere in the whole space.
Proof. By Theorem 6.7 of Chap. Ill the set A can be enclosed
in a set He($>d for which the measure-function is the same as for
the set A. Let us write H—[]Gn, where {Gn} is a descending

118
CHAPTER IV. Derivation of additive tuiiefciouH.
sequence of open sets. We clearly have I)Lft,n(aj)=l at suiy point xeGn,
and so a fortiori at any point seeACH-CQn- Hence, remembering
that the sequence {hg} of functions of a set is noa-in.creu.sing and
converges to the function JjA=1Lh7 it follows by Theorem 5.7 that
~DLA{x) = limDL(? (as) =1 == ca{x) almost everywhere in A.
n n
Sow suppose the set A measurable. Then LA{X) -\- ~Lga{X) = |X|
for every measurable set X, and consequently DL,i(a;)^DLia(,t)=l
at any point * at which the two derivatives. DL^as) and I)hcA{x)
exist, i. e. almost everywhere by Lebesgue's Theorem 5.4. Now, by
what has been proved already, DLcu(«) = l almost everywhere
in GA. Therefore DL^(«)= 0 = 0,4@;) at almost all a? of CA. and
this shows that F.2) holds almost everywhere in the whole .space.
F.3) Theorem. If § is the indefinite integral of a smnmable
function f, then
F.4) T>0 (»)=/(*)
at almost all points x of the space.
Proof. We may clearly assume that / is a non-negative function.
If / is the characteristic function of a measurable set, the relation F.4)
holds almost everywhere by the second part of Theorem 6.1. The rela-
relation therefore remains valid when / is a finite simple function, i.e.
the linear combination of a finite numba1 of characteristic functions.
Finally, in the general case, any non-negative smnmable function /
is the limit of a non-decreasing sequence {/„} of finite simple measur-
measurable functions; therefore, denoting by 0,, the indefinite integral of /„,
it follows from Theorem 5.7 that D®(x) = lim D ®,,{x) = lim /„(»)=/(«)
n n
almost everywhere, and this completes the proof.
§ 7. The Lebesgue decomposition. In this § we shall give
for additive functions of a set (93), a more precise form to the
Lebesgue Decomposition Theorem 14.6, Ohap. I. We shall prove in
fact that the absolutely continuous function which occurs in this
theorem is the indefinite integral of the general derivative of the
given function. At the same time we shall establish the corresponding
decomposition for additive functions of bounded variation.
[§7]
The Lebesgue decomposition.
119
G.1) -Lemma. If § is a singular additive function of a set, then
D$(*) = 0 almost everywhere.
Proof. We may assume (cf. Chap. I, Theorem 13.1 (I0)) that
the function ® is non-negative.
The function <? being singular, there exists,a set Eo measur-
measurable (93) and of measure zero, such that
G.2) ${X-CE0) = 0 for every set X bounded and measurable (iB).
Suppose that the set of the points x at which D#(#)>() has
positive measure. Then denoting by Qn the set of the points x e CE0
at which D$(?c)>1/m, there exists a positive integer JT such that
\Q^>0. Consequently, there also, exists an interval I such that
\I ¦ QN\ > 0, and by Lemma 5.1 we find 0A ¦ GE0)^${1-QN)>
which clearly contradicts G.2).
G.3) Theorem. If 0 is an additive function of a set, the derivative
D0 is suminable, and the function & is expressible as the sum of its
function of singularities and of the indefinite integral of its general
derivative.
Proof. By Theorem 14.6, Chap. I, we have 0=0+2*" where &
is a singular additive function of a set and W is the indefinite integral
of a summable function /. Hence, making use of Theorem 6.3 and
of Lemma 7.1 we find almost everywhere B # (x)~T> &{x) +D W(x)=f(x)
and this proves the theorem.
We can extend the theorem to additive functions of bounded"
variation of an interval. We have in fact:
G.4) Theorem. If F is an additive function of bounded variation
of an interval, the derivative F' is summable, and the function F is
the sum of a singular additive function of an interval mid of the in-
indefinite integral of the derivative F'.
Moreover, if the function F is non-negative, ive have for every
interval Io
G.5)
eguality holding only in the case in icliich the function F is absolutely
continuous on Io.

120
CHAPTER IV. Derivation of adilitivfi Junctions.
Proof. We may clearly assume the function F to be non-
negative in both parts of the theorem. The corresponding function
of a set F*, together with its function of singularities, will then also
be non-negative; and on account of Theorems 7.3 and 2.1 we shall have
G.6) F*{X)^-fl>F*(x)dx=fF'{x)dx for every bounded set Xe33.
1 x
Let us write for any interval I
G.7) T(I)=F(I)—[F'(x)dw.
i
The function of an interval thus defined is clearly non-negative,
since by G.6)
F(I)>^*(I°)> [f'(x) dx = fF'(x) dx
i° i
for every interval I. Moreover, if we take the derivative of both
sides of G.7)in accordance with Theorem 6.3, we find DT*(x) = T'(x)=Q
almost everywhere. It therefore follows from Theorem 7.3 that the
function of a set I* — and so by Theorem 12.6, Chap. Ill, the
function of an interval T — are singular. The relation G.7) therefore
provides the required decomposition for the function F.
Finally, since the function T is non-negative, it follows from
G.7) that the inequality G.5) holds for every interval Jo, and re-
reduces to an equality if, and only if, T(I) = 0 for every interval
IQIq. In other words, in order that there be equality in G.5) it
is necessary and sufficient that the function F be on Io the inde-
indefinite integral of its derivative, i. e. be absolutely continuous on Io.
Theorem 7.4 provides a decomposition of an additive function
of bounded variation of an interval into two additive functions
one of-which is absolutely continuous and the other singular. Just
as for functions of a set, this decomposition is termed Lebesgue
decomposition and is uniquely determined for any additive function
of bounded variation. For suppose that Gx+Tj=G2+Ts where 6t
and <?2 are absolutely continuous functions and Tx and ~1\ axe sin-
singular functions; then Gi—G1=T1~Ti and by Theorem 12.1 B°, 6"),
Chap. Ill, this requires <?x=<?2 and T^T2. The absolutely con-
continuous function and the singular function occurring in the Lebesgue
decomposition of a function of bounded variation F are called,
respectively, the absolutely continuous part and the function of sin-
singularities of the function F.
[§8]
Eectifiable curves.
121
As a special case of Theorems 7.3 and 7.4, let us mention the
following result, in which part 2° includes Lemma 7.1 and its
converse.
G.8) Theorem. 1° An additive function of a set, or an additive function
of an interval of bounded variation, is absolutely continuous, if, and
only if, it is the indefinite integral of its derivative.
2° An additive function of a set, or an additive function of an
interval of bounded variation, is singular if, and only if, its derivative
vanishes almost everywhere.
Finally, let us mention also an almost immediate consequence
of Theorems 7.3 and 7.4:
G.9) Theorem. The derivative of the absolute variatimi of am, ad-
additive function of a set, or of an additive function of an interval of
bounded variation, is almost everywhere equal to the absolute value
of the derivative of the given function.
Proof. Consider to fix the ideas, an additive function of bound-
bounded variation of an interval, F. Let T be the function of singular-
singularities of F, and let W and V be the absolute variations of the func-
functions F and T respectively. In virtue of Theorem 7.4 the relation
F{J) = JF'(x)dxJrT(I) holds for every interval I, and hence also
i
i
Now the function TT is singular together with T, so that its deriva-
derivative vanishes almost everywhere by Lemma 7.1. Hence taking the
derivative of G.10), we find on account of Theorem 6.3 that
W'(x) ^ \F'(x)\ almost everywhere, and this completes the proof
since the opposite inequality is obvious.
§ 8. Rectifiable curves. By a curve in a space i?m we shall
mean any system C of m equations xt=XAt) where i=l,2, ...,m and
the Xt(t) are arbitrary finite functions defined on a linear interval
or on the whole straight line Bv The variable t will be termed para-
parameter of the curve. The point (X^t), X,(t), ...,Xm{t)) will be called
point of the curve corresponding to the value t of the parameter,
and denoted by p(C; t). If E is a set in Blt the set of the points
p(C;?) for teil will be called graph of the curve G on E and de-
denoted by B(O; E) (ef. the similar notation for graphs of functions,
Chap. Ill, p. 88).

122
CHAPTER IV. Derivation of additive functions.
For simplicity of wording we consider in the rest of this §,
only curves in the plane jB8; we shall suppose also that the functions
deta'mining these curves are defined in the whole straight line Jtv
But needless to say, these restrictions are not essential for the
validity of the proofs that follow.
Let therefor© G be a curve in the plane, defined by the equa-
equations -x=X{t),y=Y{t). Given any two points a and &>a, a finite
sequence f=[t]ij=a,i,...,n of points such that a=io^i1^...^in=6
will be called chain between the points a and b, and the number
where pj-
will be denoted by c(C; i). (We may regard tins number ax the
length of the polygon inscribed in the curve 0 and whose vertices
correspond to the values a=to,t1,... ,tn=b of the parameter.)
The upper bound of the numbers a(G;v) when r is any chain
between two fixed points a and b, will be called length of arc of the
"curve G on the interval I=[a, 6], and will be denoted by SF"; I)
or S(C; a, b). If S(C;I)=j=oo the curve G is said to be rectifiable on
the interval I; and if this is the case on every interval we say sim-
simply that the curve G is rectifiable.
(8.1) For any curve C we have S\C; a, &)+S@; 6, c) = SF'; a,c)
whenever a < b < c.
It is enough to prove that 8@; a, b)+B(G; b, c)>S(C; a, c),
since the opposite inequality is obvious. Let z={a—l0, tx,..., t,,=c}
be any chain between a and c, and let h be the index for which
*fr-i<6<fa. Writing %=>=*„, t1? ...,«„_,,&} and *„={&, tA, ...,tn=c},
we have o(C; i)^a{G; h)+o{C; t2)^ S(C; a, &)-fS(C; b, c), and so
It follows from (8.1) that if a curve G is rectifiable, the length
of arc S(C;I) is an additive function of the interval I. We shall
call this function length of the curve G. Any function of a real vari-
variable that corresponds to this function of an interval, i. e. any
function 8{t) such that 8{b)—8(a)=>8(C',a,b) for every interval
O, 6], will also be termed length of the curve G.
Rectifiable curves.
123
(8.2) Jordan's Theorem. If C is a curve given by the equations
x=X(t), y=Y{t), we have
W(Z; I) <S(<7; I), W(J; I) <S(C; I),
(8.3)
for any interval I; and therefore, in order that the curve G be rectifiable
on an interval Io, it is necessary and sufficient that the functions X
and Y be of bounded variation on Ia.
¦¦ Proof. Given an interval I=[a,b], we easily find that for
any chain r={a=t0, tt, ..., tn—b) between the points a and b,
\X(tj)
:o{G;
from which the inequalities (8.3) follow at once.
(8.4) Theorem. If C is a rectifiable curve given by the equations
x=X[t), y=Y{i), and the function S(t) is Us length, then
(i) in order that S(t) be continuous at a point [absolutely contin-
continuous on an interval] it is necessary and sufficient that the two func-
functions X{i) and Y(t) should both be so;
(ii) we have A{B{0; E)X.\8[$J]\ for any linear set E (i.e. on
any set E the length of the graph of the curve C does not exceed the
measure of the set of values taken by tlie function 8[t) for teE);
(iii) [_S\t)Y=[X\t)Y + [Y'{t)J for almost every t;
b
(iv) S@; a, b) > f\\_X\t)Y+{Y'{t)J dt for every interval \_a, 6],
a
and the sign of equality holds if, and only if, both the functions X(t)
and Y (t) are absolutely continuous on [a, 6].
Proof. re(i): This part of the theorem is an immediate con-
consequence of the relations (8.3), since by Theorems 4.8 and 12.1 A°)
of Chap. Ill, a function of bounded variation is continuous at a point
[absolutely continuous on an interval] if, and only if, its absolute
variation is so,

.124
CHAPTER IV. Derivation of additive funetiouH.
retji): Let ? be any positive number and let {I,,} be a sequence
of intervals such that
(8.5) SimCSln and (8.6) 2|I»K|flpT)|+fi.
We may clearly assume that no interval In has a diameter exceeding t.
This being so, we -write 2?=B(C; E) and we denote by En
the set of the points p (C; t) of the curve 0 for each of which 8{t)eln.
It is easy to see, on account of (8.5), that BCIjB,,. On the other
n
hand <5(SnX<3(I,1)=|InK? for any n, and it follows from (8.6)
(cf. Chap.II, §8) that A&(E)^\8{E]\ + e, and hence by making e
tend to zero, that A{E) ^ |<S'[-E]|5 as asserted.
re (iii): Let I0=[ao?&o] De any interval. We shall show suc-
successively that both the relations
(8.7) l8'(t)f^ir(t)f+[T{t)f and (8.8) [,
hold almost everywhere in Io; here the derivatives 8'(t),X'(t)
and Y'{t) exist almost everywhere on account of Theorem 8.2 and
of Lebesgue's Theorem 5.4.
We have 8{t+h)—8(t)^{[X(fi+h}-X(t)f+[Y{t+h)—Y(t)f}li2
for any point t and any h > 0, and if we divide both sides by h and
make ft-> 0, this implies the relation (8.7) at any point t for which all
three functions are derivable at the same time, i. e. almost everywhere.
Now let A denote the set of the points telo at which the three
derivatives X'(t), Y'(t) and S'(t) exist without satisfying the inequal-
inequality (8.8); and for any positive integer n, let A,, denote the set of
the points teA for each of which the inequality
holds for all the intervals I containing t, whose diameters are less
than 1/w. Clearly A=J^An.
Keeping n fixed for the moment, let e be any positive number.
There exists a chain %=[a0=t0,ij,... ,tp=b0) such that tk—t*—i<l/«
for fc=l, 2, ...,p, and such that 8{C;I0)^a{G;t)-\-s. Consequently,
writing for brevity J^p^j, tj, pft==p(C;tA), «>A=(i(pft_.15pA), we have
(8.9) S(Ja)>$a4-|Ja|/» /or fc=l, 2,..., p, whenever Jk-j.
and on the other hand
(8.10)
[§ 9]
I)e la Vallee Ponssiu's theorem.
125
Therefore if _V stands for a summation over all the indices 7:
k
for which Jk-An 4= 0, we find on account of (S.9) and (8.10) that
\An
p
Now since e is an arbitrary positive number, it follows that
J.,,| — 0 for any n, and therefore also that |J-| = 0. Thus the in-
inequality (8.8) holds almost everywhere, as well as the inequality (8.7),
and this completes the proof of part (iii) of the theorem.
Finally, since S(<7; a, b)=8(b)—8(a), part (iv) reduces on
account of (i) and (iii) to an immediate consequence of Theorem 7.4.
Theorem 8.4 (iu particular its parts (iii) and (iv)) is due to L. Tonelli
[1: 4]; cf. also P. Kiesz [6; 7].
As regards part (ii) of the theorem, it may be observed that hi the case
in -which the curve C has no multiple points (i. e. when every point of the curve
corresponds to a single value of the parameter) the inequality A[B{C; E)) -^ |Srjj]i
can easily be shown to reduce to an equality.
As proved by T. Wazewski [1], any hounded continuum P of finite
length may be regarded as the set of tho points of a curve G on the in-
interval FO. 1], such that S{0; 0,1) ¦: 2-A(P).
§ 9- De la Vallee Poussin's theorem. With the help of
the results established in the preceding § we can complete further,
for continuous functions of bounded variation of a real variable,
the decomposition formula of Lebesgue.
We shall begin with the following theorem, which itself com-
completes, in part at any rate, the second half of Lebesgue's Theorem 5.4.
(9.1) Theorem. If F(x) ?s a function of bounded-variation and ~W(so)
denotes its absolute variation, then for the set JV of the points at which
the function F(x) i.s continuous but has no derivative finite or infinite,
we have
(9.2)
and
(9.3) A{B(F; N))=0.
Proof. Consider the curve C: x~.r, ?/=F(.r), Let S(x) be
the length of this curve and let E be the set of the values assumed
by the function 8(:c). For any seE, denote by J(s) the value of x
for which 8(jo) = s and write T(s)=F{X{s)). Since (cf. Theorem 8.2)
and |Y(s2)—
^—%| for any pair
of points sl7 s2 of B, the functions X(s) and Y{s) are continuous

126
CHAPTER IV. Derivation of additive functions.
on the set E, and moreover may be continued on to the closure E
of this set by continuity. If now [a, b] is an interval contiguous
to E, the function X(s) assumes equal values at the ends of this
interval, and the point x=X(a)=X(b) is a point of discontinuity
of the function F(x). We shall complete further the definition of
the functions X(s) and Y{s) on the whole straight line Mx so as
to make the former constant and the second linear, on each interval
contiguous to B.
This being so, consider the curve Gx given by the equations
x—X(s), y=Y(s). We verify easily that the parameter s of thia
curve is its length. (Actually we see easily that the graph of the
curve Cx is derived from that of the curve G by adding to the latter
at most an enumerable infinity of segments situated on the lines
x=a where ot are the points of discontinuity of the function F.)
By Theorem 8.4 (iii), we therefore have [X'(s)f-\-[Y'(s)f=l for
almost all s, and therefore the set H of the points s for which either
one of the derivatives X'(s) and Y'(s) <l°es not exist, or both exist
and vanish, is of measure zero.
Now we see at once that if s eE—H, then at the point x—X (s)f
the derivative F'(x) exists (with the value T{s)jX'(s) if X'(s)^=0,
or with the value ±oo if X'(s) = Q and Y'(s)§0). Therefore
N(ZX[E-E], or what amounts to the same, S[N](ZE-H, and
hence !?[J]Kpf|=0.
3?rom this we derive at once with the help of Theorem 8.4 (ii)
the relation (9.3), since ~B(F; N)=B(C; N). Finally, the function
S(x) is continuous (cf. Theorem 8.4 (i)) at any point at which the
function F (x) is continuous, and so at any point of the set N, and
therefore it follows from Theorem 13.3, Chap. Ill, and from Theo-
Theorem 8.2, that
the proof.
= 0; this completes
(9.4) Lemma, If F{x) is a function of bounded variation, then
(i) F*[A)^lc-\A\ for any bounded set A and any finite number 1c
whenever the inequality F'{x)^tJc holds at every point % of A (and
the assertion obtained by changing the direction of both inequalities
is then evidently also true),
(ii) F*(B)=0 for any bounded set B of measure zero throughout
which the derivative F'{x) exists and is finite.
[§9]
De la Vallee Poussin's theorem.
127
Proof, re (i). Let e be any positive number. Denote, for any
positive integer n, by A{'n the set of the points xeA such that
F{I)^(k—e)-\I\ holds whenever I is an interval containing x and
of diameter less than Ijn. Clearly A~]imA{n).
n
Keeping the index n fixed for the moment, let us denote by (?(n)
a bounded open set containing Ain), such that (cf. Theorem 6.9,
Chap. Ill)
(9.5)
\F*(X)-F*{Am)\<,e whenever
and let us represent (?(n> as the sum of a sequence {l?\=Ui... of non-
overlapping intervals. We may clearly suppose that all the inter-
intervals 4° are of diameter less than Ijn and that their extremities
are not points of discontinuity of F {x). So that if ?{n> stands for
p
summation over the indices p for which
$>-A™±0,
and X{
")
denotes the sum of the intervals if corresponding to these indices.
we find
and
p p
hence on account of (9.5), F*{A(n))^(k— t)-\A(n)\ — e. Making e-»0
and n-^-oo, we obtain in the limit F*(A)^Jc-A\.
re (ii). Let B{n) denote the set of the points xeB for which
F'{x) ^s—n. By (i) we have F*(B{n))^— n-\B{n>\ = 0 for any positive
integer n, and so .F*(J3)>0. By symmetry we must have also
JF*(J3)<0, and therefore F*(B)=0.
(9.6) De la Vallee Poussin's Decomposition Theorem. IfF{x)
is a function of bounded variation with W(x) for its absolute varia-
variation! and if E+cx, and E-oo denote the sets in which F(x) lias a deriv-
derivative equal to + oo and to — oo respectively, then
(i) for any bounded set X measurable (93) at each point of which
the function F(x) is continuous, we have the relations:
(9.7) F*(X)=F*(X-E+oo)+F*(X-E-co)+fF'(x)dx,
-a>)\ +i\F'{w)\'dx;
(9.8)
(ii) the tivo derivatives F'(x) and W'{x) exist and fulfil the rela-
relation W\x)=\F'(x)\ at any point x of continuity of F, except at
most at the points of a net N such that W*{N)=\N\—b (i.e. a set
which is at the name time of measure (L) zero and of measure (W) zero).

128
CHAPTER IV, Derivation of additive funrtioiiN.
Proof. re(i). On account of Theorem 7.3 there exists a wot A
of measure zero such that for any set X bounded and measurable («B)
(9.9) F*(Z)=F*(X-A)+JF'(x) ix.
Supposing further that the function F in continuous at every
point of X, we may assume, in virtue of Theorem 9.1, that the function
F{x) lias everywhere in A, a derivative, finite or infinite. Moreover
since by Lemma 9.4 (ii) the function F*(X) vanishes for any bounded
subset of A—(E+oa+E-«,), we may assume that iC^+i^
Finally, we obtain directly from (9.9) that F*(X) vanishes when
XC.0A and |X|=0. We may therefore choose simply A =W+oo-f-B „
and formula (9.9) becomes (9.7).
Formula (9.8) is, by Theorem IX :2, (limp. Ill, an, immediate
consequence of (9.7), since Lemma 9.4 (i) shows that A1 *(.V¦/</+«>)JM)
and F*lX-E-m)%(l for every set X bounded and measurable (93).
re(ii). Let N be the set of the points x at which the. function
F(x) is continuous and either one at least of the derivatives F\x)
and W'{x) does not exist, or both exist but do uot satisfy the re-
relation W'{x)=\F'{x)[. We then have jV-/i'+oo = N-E-ao = 0, since
evidently W'{x) = + °°=\F'(ci;)\ at any point where F'(x)—±oo.
Therefore, since the set N is further, by Theorem 7.9, of measure (L)
zero, it follows from formula (9.8) that W*{N)=Q and this com-
completes the
Let us mention an immediate consequence of Theorem !).(!. hi order that
a continuous function of bounded variation F(x) be absolutely ootM-wuowi, it in [/«*«¦
sary and sufficient that the function of a net F*(X) should wininh identically on the
set of the points at which F(x) has an infinite derivative, hi- particular therefore,
any continuous function of bounded variation which is not- absolutely coM'mttoutt
has an infinite derivative on a non-enumerable set.
Let us remark further that the theorems of this § cannot be extended directly
to additive functions of an interval in the plane. Thus if F(I) denotes the con-
continuous singular function of an interval in /?», which for any interval / equals
the length of the segment of the line x=y contained in I, we have lf(x)—()
for every point x, so that F'{x)=oo does not hold at any point.
§ 10. Points of density for a set. Given a set E in a apace
Iim, the strong upper and lower derivates of the measure-function oiE
{oi. § 6, p. 117) at a point # will be called respectively the onlcr upper
and (niter lower density of E at x. The points at which these two den-
densities are equal to 1 are termed points of outer dennty, and the points
at which they are equal to 0, points of dispersion, for the set K.
[§10]
Points of density for a set.
129
If the set E is measurable we suppress the word "outer" in these
expressions. We see further, that if the set E is measurable, any
point of density for E is a point of dispersion for GE, and vice-versa.
We shall show in tins § (of. below Theorem 10.2) that almost aE points
of any set E axe points of outer density for E, or what amounts to the same that
A0.1) For almost all points x of E, if [In\ is any sequence of intervals tending
to x (in the sense of §2, p. 106), we have |JMn|/|In|—>1.
This proposition presents an obvious analogy to Theorem 6.1 and it is
in the form F.1) that the "density theorem" is often stated and proved either
with the help of Vitali's Covering Theorem or by more or less equivalent means
(vide, for instance, E. W. Hobson [I], Ch. J. de la Vallee Poussin fl, p. 71]
and W. Sierpinski [10]). Theorem 10.2 will be however, so to speak, independent
of Vitali's theorem, because the sequences of intervals occurring in A0.1) are
not supposed regular.
Also, Theorem 10.2 will be more precise than Theorem 6.1, because at
any point x of outer density for a set E we have a fortiori jyLE[x) — \, and iu-
deed, at any x the relations Lg(ai) = l and DL?,(^)= 1 are equivalent. To see
this it is enough to show that the former of these relations implies the latter,
the converse being obvious from Theorem 2.1. We may assume moreover, on
account of Theorem 6.7, Chap. Ill, that the set E is measurable.
Let therefore ssa be a point such that L^(zo)=l and let JXnj be a regular
sequence of measurable sets tending to x0. Then there exists a sequence {Jn\ of
cubes such that XnCV,, and \Xn\J\Jn\>a, where ?i = l,2, ... and a is a fixed
positive number. Since by hypothesis, \E;Jn\l\Jn\-^l, and so [Ci?-J',!]/|Jn|-»0,
it follows that \CF,-X11\j\Xn\~>0, or what amounts to the same that \E-Xn\j\Xn\—> 1.
Therefore 1)LB(*O) = 1.
It is, of course, only for spaces lim of dimension number m 5s-2 that Theo-
Theorem 10.2 -vv-ll differ from Theorem 6.1. The two statements are equivalent for 1{X.
For the various proofs of Theorem 10.2 vide P. Eiesz L8] and H. Buse-
rnann and W. Feller [1]. In the second of these memoirs will be found a
general discussion of the different forms of "density theorems".
It is of interest to observe that the proposition A0.1) ceases to be true
even for closed sets E, if the intervals In are replaced by arbitrary rectangles
with sides not necessarily parallel to the axes of coordinates. Tins remarkable
fact has been established by O. Nikodym and A. Zygmund (vide 0. Nikodym
[1, p. 167]) and by H. Busemann and W. Feller [1, p. 243].
A0.2) Density Theorem. Almost all the points of an arbitrary setE
are points of outer density for E; and if further the set E is measurable,
almost all the points of GE are points of dispersion for E.
Proof. For simplicity of notation, we shall state the proof
for sets which lie in the plane; the corresponding discussion in any
space if,,, is however essentially the same (except that in Mx, as
already remarked, the theorem reduces to Theorem 6.1).
S. Suks, Theory of the Integral. 9

130
CHAPTER IV. Derivation of additive .fimetionn.
By Theorem 6.7, Chap. Ill, any set can be enclosed in a meas-
measurable set having the same measure-function. On, the other hand
any measurable set is the sum. of a set of zero measure and of a
sequence of bounded closed sets. We may therefore suppose that
the set considered is bounded and closed.
Let s be any positive number. We shall begin by defining
a positive number a and a closed subset A of E such that \B—A\<.e
and that, for any point (?,17) in the plane,
(i)
¦whenever (E,t])eA,
—•«)(& —a)
and b— a<a.
To do this, let us write for brevity, when Q is any set in the
plane, $['']=E[(;k, 1/) eQ], and let us denote, for any positive
integer n, by An the set of the points (?, vf) of B such that
IJE^-T]^ A—e)-|I| whenever I is a linear interval containing ?
and of diameter less than Ijn. The sequence {A,,} is evidently
ascending. Let us write
A0.3) N=E — limJL,,.
n
For any 1?, if ? is a point of the set Nlv\ there will then exist,
for each positive integer n, a linear interval I such that ?el,
6{I)<lln and \Nlv]-I\^\Bln]-I\<(l — E).\I\. Therefore the lower
derivate of the measure-function for the linear set JVlvl cannot
exceed 1—e at any point of this set, whence by Theorem 6.1,
A0.4) |JVr';1|=|E[(fl!, rj)eN]\=Q for every real number r).
X
Let us now remark that the sets An are closed. For, keeping
lor a moment an index n fixed, let (f0, j?0) be the limit of a sequence
{(?*j*7ft)}fc=i,2,... °* points of An. Let Io be a linear interval such
that ?0el0 and <5(I0)<J/«,j and let I be any linear interval con-
containing Io in its interior, whose diameter is less thanl/ra. Then for any
ffiil l g ^]
sufficiently large ft, gkel and therefore
h
other hand,
—?)-|I
the set IB being closed, we easily
; so that !#''o]-Ij5?mnsup \E[n^-
On the
see that
ff).|J|,
and therefore also |JBw.J0|^(i_?).|j0|. Hence (lo,Vo)eAn, i.e.
each set An is closed.
It follows, accordingTto A0.3), that the set N is measurable.
[§10]
Points of density for a set.
131
Therefore, applying Fubini's theorem in the form (8.6), Chap. Ill,
we conclude, on account of A0.4), that the plane set JV is of meas-
measure zero. Consequently \B—A^. < e for a sufficiently large index w0,
and writing a=l/n0 and A=Alla we find that the inequality IE—A < e
and condition (i) are both satisfied.
In exactly the same way, but replacing the set M by A and
interchanging the rdle of the coordinates x and y, we determine
now a positive number oi1<o- and a closed subset B of A such that
\A—B\<ie and that, for any point (?,i]) in the plane
A1)
v
whenever (^
A—c) (b—a)
and b — a<ir1.
This being so, let (?0, j?0) be any point of B. Let J==[a,, jS-,; a2, /Sa]
denote any interval such that {Ea,ri0)eJ and d(J)<o-1<(j. By Fubini's
theorem (in the form (8.6), Chap. HI) we have
A0.5)
Since {twt]0)eB and aa^?;0^^2) it follows from (n) that the
set of the y such that {?wy)eA and a^^y^.^ is of measure at
least equal to A—s) (|32—a2). On the other hand, since
it follows from (i) that ^[(x,y)eE; a
i
whenever (gg,y)eA. Hence, formula A0.5) gives
B-J\ > A—e)« (&—«!> {h—a2)={l—ef-\J\,
i. e. the lower density of E is at least equal to A—sJ at any
point (fo,%) of B. Therefore, since B —BI<B — A] + A—B\^2e
and since e is an arbitrary positive number, the lower density of E
is exactly equal to 1 at almost all the points of the set E, i. e.
almost all points of this set are points of density for it.
The second part of the theorem is an immediate consequence
of the first part. In fact, if the set E is measurable, so is the set CE,
and almost all points of GB, since they are points of density for GE,
are points of dispersion for E.
In connection with the definition of points of density, Denjoy
introduced the important notion of approximate continuity of
a function. We call a function of a point f{x) (in any space i?m),
9*

132
CHAPTER IV. Derivation of additive funo.tioiw.
approximately continuous at a point x0, if f(xQ) =(= oo and f(x) ->/(«„)
as x tends to x0 on a measurable set E for which x0 is a point of
density.
A0.6) Theorem* If f is a measurable fimction almost everywhere
finite on a set E, then the function f is approximately continuous at
almost all points of E.
Proof. On account of Lusin's theorem (Chap. Ill, § 7), given
any e > 0, we can represent the set E as the sum of a closed set F
on which the function / is continuous and of a set of measure less
than s. The function / is clearly approximately continuous at any
point of density for the set F, and so, by Theorem 10.2, at almost
all points of F. This implies, as e is an arbitrary positive number,
that the function / is aproximately continuous at almost all points
of the set E itself.
Theorem 10.6 is due to A. Denjoy [5] (of. also W. Sierpit'iNki [0; 9]).
It is easy to see that the converse holds also, i. e. that every junction wlueh is
approximately continuous at almost all the points of a measurable set Ji is measurable
on E (vide W. Stepanoff [2] and E. Kamke [1]).
Let us mention also the following theorem, an almost im-
immediate consequence of Theorem 10.6, which completes, in part,
Theorem 2.2 on derivation of an indefinite integral:
A0.7) Theorem, If 0> is the indefinite integral of a hounded meas-
measurable function f, then ®'B(x)=j(x) at almost all points x, and in
fact, at any point x at which the function f is approximately continuous.
Proof. Let x0 be a point at which the function / is approx-
approximately continuous and let E be a measurable set for which x0 is
a point of density, while /(a;)->/(«„) as x tends to x0 on the set E.
We may suppose (by subtracting, if necessary, a constazit from
f{x)) that f(xQ)=O. Therefore, given any positive number e, we
have for any interval I of sufficiently small diameter con-
containing x0, (i) |Z.OEj<e.|l! and (ii) |/(a»)|<c for every xel-E.
Denoting by M the upper bound of \f{x% conditions (i) and (ii)
imply |0{I)| ^ \®(LCE)\+\$(I.E)\ < Me- \I\+s-\I\ < e-(M+1)-\I\,
whence ®'s(x0)=0=f[x0).
If 0 is the indefinite integral of a function / which is auwmablo but un-
unbounded,*, may happen thatthe relation *;(«) = /(»)« not fulfilled at any point.
In vrrtue of Theorem 6.3, this relation clearly holds at almost all points at
[§H]
Ward's theorems.
133
which the strong derivative 3>s(x) exists; but if the function / is unbounded, its
indefinite integral $ may have no finite strong derivative at any jtoint (cf. on
this point Busemann and Feller [1, p. 256]; the result of Banach and Bohr
mentioned above, p. 112, follows as a particular case).
Nevertheless, the result contained in Theorem 10.7 may be generalized
considerably. In fact, according to a theorem of B. Jessen, J. Marcinkie-
wicz and A. Zygmund [1] (see below § 13) the indefinite integral of a junction f
in a space Bm is almost everywhere derivable in the strong sense whenever the jumtion
|/|-(log+|/|)m~1 is summatle (in a less general form, for functions / of -which the
power p>l is summable, this theorem was established a little earlier by
A. Zygmund [1]). On the other hand however, given an arbitrary junction o(t)
positive for i>0 and such that liminf a(t)=O, there always exists a junction j(x)
in J?m such that ike junction tr(|/|)-|/|-(l°g"
,+m,m—1
is summable and such that the
indefinite integral oj j is not derivable in the strong sense (has the strong upper
derivate -\-co) at any point of Km-
* § 11. Ward's theorems on derivation of additive func-
functions of an interval. In the preceding §§ of this Chapter, we
have treated the Lebesgue theory of derivation of additive func-
functions of an interval of bounded variation. As regards functions
of a real variable, this theory has been extended to arbitrary func-
functions by Montel, Lusin and especially by Denjoy. Becently Denjoy's
theorems, which already belong to the classical results of tie theory,
have been generalized still further. On the one hand they have been
given a geometrical form by which they become theorems on certain
metrical properties of sets, and in this form an account will be given
of them in Chapter IX. On the other hand, recent researches of
Besicovitch and Ward have made it possible to extend an essential
part of the Denjoy results, particularly the relations between the
extreme bilateral derivates, to additive functions of an interval
in a space Sm of any number of dimensions. These researches will
form the subject of the present §.
It was A. S. Besicovitch [5] who started these researches, by establishing
between the extreme strong and ordinary derivates of absolutely continuous
functions of an interval, relations analogous to those proved by Denjoy for
derivates of functions of a single variable. A. J. Ward [2; 5] has extended this
result to quite arbitrary additive functions of an interval. Of the two theorems
of Ward (vide, below, Theorems 11.15 and 11.21) one concerns only ordinary
derivates, while the other applies also to strong derivates. It is the latter that
generalizes the result of Besicovitch; this second theorem is one which can be
proved fairly simply for functions of an interval in the plane; it is rather curious
that it requires much more delicate methods in an arbitrary space Km.

134
CHAPTEJS IV. Derivation of additive functions.
We shall make use in tliis § of some auxiliary notations. If F
is an additive function of an interval and a^l is a positive number,
F(«ix) and JT(a)(a?) will denote at any point x the upper and lower
limit of the ratio F(I)I\I\ where I is any interval containing x,
which is subject to the condition r(I)>a, and which has diameter
tending to zero. We see at once that at any point x, F(K){x) and F{a)(x)
tend to Fix) and F[x) respectively as a->0.
We shall suppose fixed a Euclidean space Mm, and in it we
define a regular sequence of nets of cubes {Q/;}/,=i,2,.,., denoting by
{Qis} the family of all the cubes of the form
[ft 2^,(^+1) 2-*; p22-",(p2+lJ-";...; pm2~", (pm+l) 2'k]
where pvp.v-., pm are arbitrary integers.
A1.1) Lemma. Given an additive function of an interval G, and
positive numbers a^2~m and a, suppose that the inequalities <)<;<?(«)(*)<«
hold at every point x of a set E having positive outer measure; then
there exists for each e > 0, a cube Q which belongs to one of the nets Qk
and for which we have
A1.2) 6[Q)<e, \E-Q\>A—b)-\Q\ and G(Q)<&'" ¦ cTm ¦ a-\Q\.
Proof. By replacing, if necessary, the set E by a suitable
subset of E having positive outer measure, we may suppose that
there exists a positive number a such that for every interval I,
A1.3) <?(J)>0 whenever I-E^O, v{I)^a and 6{I)<ia,
We may further clearly assume that a is less than both s and amj&"\
This being so, let xoeE be a point of outer density for the
set E (cf. §10). Since G(a)(x0)<a, we can determine an interval
t7=[fli, bx; ...; am, bm] containing x0 and such that
A1.4) i(J)<a, x(J)^a, \E-J\>A—afi).\J\ and G(J)<a-\J\.
It follows in particular that
A1.5) \E-I\>A— a)-\I\ for any interval IQJ suchthat \I\>a-\J\.
Let Z be the smallest of the edges of J. Since r (J) ^a, no edge
of J can exceed Ija, and therefore we have \J\^l'"la'". Finally
let ft.be the positive integer given by
[Hi]
Ward's theorems.
135
A1.6)
1/2*
and let Q = [>i,&i; ...;a'm,b'm] be a cube which belongs to the net.Q*
and which contains the centre of the interval J. By A1.6) we find
that QC J and that a\—a^l/'i, bt—b'^lji and b\—«;-=l/2*^Z/8.
It follows easily that the figure J 3Q can be subdivided into a finite
number of non-overlapping intervals with no edge smaller than Ij8.
Isov? any such interval can clearly be further subdivided into
a finite number of non-overlapping subintervals whose edges all lie
between ijS and lj±. We thus obtain a subdivision of the figure
J36 into a finite number of non-overlapping intervals, whose para-
parameters of regularity are greater than, or equal to, 2~™^e, and whose
volumes are greater than, or equal to, 8~OTZm^8~'nara-|Jj>a.|t7|. It
therefore follows from A1.5) and A1.3) that
A1.7) G(J3Q)>Q.
Similarly, it follows from A1.6) that |0j=2~Ain^8""xr>
;>8~ni am-|J|>o--|J|, whence by A1.5) we derive at once the second of the
relations A1.2); at the same time, by the relations A1.7) and A1.4),
G{Q)<G{J)<a-\J\^8mcTma-\Q\, and this gives the third of the
relations A1.2) and completes the proof.
A1.8) Lemma.. Let G be an additive function of an interval in Mm,
E a set in JRm, Q a cube belonging to one of the nets Qft, and a>0,
? > 0 and b arbitrary fixed numbers. Suppose that
(i) IS-QXl —e)-|Q],
(ii) <?(!) > 0 for every interval I such that ICQ, I-E^Q and
(iii) G(a)(x)>b at any point xeE;
then G{Q)>12-m-ccmb-{l — 2ms).\Q\.
Proof. We may clearly assume that the set E is contained
in the interior of Q and that every point of the set is a point
of density.
This being so, we shall begin by establishing the following
result:
A1.9) Given any *7>0, we can associate- with any point xeE
a cube P, belonging to one of the nets Q*, and a cube JJP, such
that (a) G(P)>am-12~m>b-\P\ and (b) xeJ, 6{J)<n and-\J\=3m-\P;.

136
CHAPTER IV. Derivation of additive functions.
Por this purpose, let us associate with the point * an interval 8
such that xeS, <5(#)<l/4, v{8)^a and G(8)> b-\8\. Let h denote
the largest edge of 8, and let 7% be the positive integer satisfying
the inequality 1/2>27i>1/2*1+1. Let 8X be a cube of the net Qki
having points in common with 8, and let J denote the cube formed
by the 3"' cubes of the same net (including the cube 8X itself) which
have points in common with 8V
The cube J clearly contains the interval 8, and since no edge
of 8 can be less than ah, we find that
A1.10) \J\=3'"-2-""" =6™-2-m(*1+1)
". \8\.
On the other hand, since 2~*' — 7i>2~(*>+1) and ah^a-2 ~(*I+2), the
figure JQ8 can be subdivided into a finite number of non-over-
non-overlapping intervals with edges greater than, or equal to, a-2' (*rh2), and
therefore, as in the proof of Lemma 11.1, into a finite number of
non-overlapping intervals whose edges have lengths between a-2 ~(/f|+2)
and a-2~(Al+1). Therefore, denoting by 7 any interval of this sub-
subdivision, we find rG)>2-m and |I|>a'".<r"'(*'+2) = 12~"' a'"- \J\,
Consequently, by supposing the interval S, and a fortiori the
cube J, sufficiently small, we may assume that d(J) < rj and
that each of the intervals of the subdivision in question contains
points of JS. It follows, by condition (ii) of our lemma, that
G(Je8)>0, and so, by A1.10), that G{J)>G(8)>b-\8\^l2~"I-a'"-b-\J\.
Thus among the 3"' cubes of the net Q*, which make Tip the cube J,
there is one at least, P say, such that G(P)>12~'"-a'"-b-\P\,
and the cubes P and JDP, thus defined, clearly satisfy the con-
conditions (a) and (b) of A1.9).
It now follows, on account of A1.9) and condition (i) of the
lemma, that (with the help of Vitali's theorem in the form C.8))
we can determine in Q a finite system of non-overlapping intervals
Pi, P2,..., Pn belonging to the nets Q*, such that:
A1.11) <?(P;)>12-"!.am&.ip,.| and PrJ5={=0 for *=1,2,...,»,
A1.12)
Among the cubes of the nets Q*, we shall consider specially
two classes of cubes. A cube of a net Qk contained in Q will be said
to be of the first class if it is one of the cubes Ph P2j ...,P,,;
and of the second class if it contains points of E and if further
Ward's theorems.
137
among the 2m cubes (Q*+i) composing it, there exists at least
one which does not overlap with any cube P;. Since the number
of cubes Pt is finite, there exists a net Q# such that no cube of this
net contains cubes of the first class. Let 31 be the set of all the cubes
of the first or second class contained in Q and belonging to the nets
Q* for ks^K.
The set of these cubes covers the whole cube Q. Por if not,
there would certainly exist in the net Q^ a cube I0(ZQ not con-
contained in any cube C1). H"ow, since 70 contains no cube of
the first class, 70 would not contain any point of the set E; and
since, by hypothesis, Io is not contained in any cube of the first
or second class, we could, starting with 70, form in Q a finite as-
ascending sequence of cubes without points in common with E and
which belong respectively to the nets QK, Qjr-i,..., ?!*„, where Q*,
is the net containing the cube Q. But the last term of this sequence
of cubes is evidently the cube Q itself, and we arrive at a contra-
contradiction since E(ZQ.
Let us now remark that since all the cubes B1) belong to the
nets of the regular sequence [Q],}, it follows that, of any two over-
overlapping cubes C1), one is always contained in the other. Hence,
we can replace the system of cubes 31 by another system ^C^
which also covers Q, and which, this time, consists of non-over-
non-overlapping cubes. Let A be the sum of the cubes C1]^) of the first class.
On account of A1.11) we have
A1.13) GU)>12~m-am-b-\A\.
Moreover, since the figure Q<~.A is formed of a finite number
of cubes of the second class which do not overlap, it follows from
condition (ii) of the lemma that
Finally, in each cube I of the second class, there is always
n
a cube which is contained in Q~TP, and whose volume is 2~m-[I|.
It therefore follows from A1.12) that \QQA\<Z2m-t-\Q\, and in virtue
of A1.11) and A1.13) we find
G(Q) >G(A)^12~'"-a"-b-\A\>V2~m-anl-b-(l'-2me)-iQ\,
which completes the proof.
A1.15) Theorem. Any additive function of an intervalF is derivable
at almost all the points x at which either F{x) > — oo, or F(x) < + oo.

138
CHAPTER IV. Derivation of additivo functions.
Proof. Consider the set of the points x at which F{x) > —oo
and suppose, if possible, that the set A of the points a; at which
F(x)>F(x)>~~ oa is of positive measure. We could then determine
a number a>0, and a set BQA of positive outer measure, such
that F(a)(x) =f= oo and F(a){x) —F(_a)(x) > a at every point * of B. We
may clearly assume that a<;2~~"!.
Let e be any positive number. Let us denote for any integer p
by Bp the set of the points * of B at which ps < Jf(«)(a>) < (p+1) e, and
let p0 be an integer sucli that |J5j > 0. We can determine a number
<j>0 and a set EQBlh, whose measure is not zero, so as to have
F(I)>pos-\I\ whenever the interval I is subject to the conditions
<5G) O, r(I)>a and JE-I^O.
Now write G(I)=F(I) — p0 e- \I\ (where I denotes any interval).
Thus defined, the function G clearly fulfils the conditions:
1° 0< <?(a) (a)< 2 e and <?(«)(*)>« at any point xeE,
2° <?(!)> 0 for any interval I such that <$(!)< a, rG)^a
and E-I^Q.
By Lemma 11.1 we can therefore determine a cube Q, belonging
to one of the nets Q*, so as to have <$((?)<cf, |JE?-Q|'X1—e)-|Q|
and <?(#)^8ma~'"-2«-|Q|. From the first two of these relations and
from conditions 1° and 2°, it follows, on account of Lemma 11.8, that
Thus 12-'" am+x ¦ A—2'" e)
. 2e
G{Q)>12~mam+1-(l— 2me
for every e > 0, and this is clearly impossible. We arrive at a
contradiction and this shows that |4|=0, i. e. that F(x)=F(oo) for
almost all x for which F(x)~>—oo.
It remains to be shown that the set of the points x at which
the derivative F'(x) is infinite, is of measure zero. Suppose then,
if possible, that F\x)=-\-oo at each point a; of a set M of positive
measure. We may clearly assume that there exists a number ij > 0
such that _F(I)>0 whenever I is an interval containing points
of M and subject to the conditions 6 (I)<ij and r(I)^2~™. There-
Therefore, denoting by B any cube which belongs to one of the nets
Q* and which satisfies the relations \M-B\ > A—2~{m+1))-\B\ and
6{B)<t], we find easily from Lemma 11.8 that F(B) >l2~1-12~~"l-b-\B\
for every finite number b. We thus again arrive at a contra-
contradiction and this completes the proof.
Ward's theorems.
139
It should be remarked that for the validity of Lemma 11.1 it is enough
to suppose merely that a<l (instead of a^2~m). Similarly in condition (ii) of
L'emma 31.8, the inequality r(J)>2""°* may be replaced by r(I)>o. The proofs
of the lemmas remain essentially the same; we need only observe that if I is an
interval whose parameter of regularity is greater than, or equal to 2~m, and if
a<l is a positive number, the interval I can always be subdivided into a-finite
number of non-overlapping subintervals I., where j = l, 2,..., such that r(I.)^o
and |i-|^-fcre-ir|, where ka is a constant depending only on a.
We can now easily see that Theorem 11.15 may be stated in a slightly
more general form as follows: any additive junction of an interval F is derivable
at almost all ihe points x at which either F,aJx)>—^co or F,aJx)<-Jroo, ichere a
is any positive number less than 1. The question whether the condition a < 1 is
necessary here, does not seem to have been solved yet completely. It may how-
however easily be proved (by the method of nets used in the proof of Lemma 11.8}
that for any additive function F, ihe set of the points x for which, either F^(x) = —co
or F^(x)=+oo is of measure zero. For a discussion of these questions, vide the
memoir of A. J. Ward [5],
We shall now proceed to prove the second theorem of Ward,
in which the ordinary extreme derivates F(x) and F{x) of Theoremll.lo
are replaced by the strong derivates Fs{x) and Fs{x). It should be
remarked however, that we cannot at the same time replace, in
the assertion of Theorem 11.15, derivability in the ordinary sense
by derivability in the strong sense: in fact, in general, a non-negative
function, even when it is absolutely continuous, may yet have a
strong upper derivate which is everywhere infinite (see p. 133 above).
We shall begin, by j>roving the following lemma which is sim-
similar to Lemma 11.1.
A1.16) Lemma. If G is an additive function of an interval in Km
and if for some fixed number a we lime 0 < (?3(j?) < a at every point
of a set Fi of positive outer measure, then given any e>0 there exists
an interval Q such that
A1.17) <(())<«, v(Q)>2
and G(Q)<3m-a-\Q\.
Proof. Let us write for brevity y=l/3'". We may suppose
(by replacing, if necessary, the set E by a subset of positive outer
measure) that G{I)>0 for every interval I containing points of E,
which has diameter less, than a positive number a<e. Let xoeE be
a point of outer density for E and let J=[a1,bl;a.2,bi;...;a!,,,bm]
be an interval containing x9 such that

140
CHAPTER IV. Derivation of additive functions.
A1.18) d(J)<0, \E-J\>[l—y)-\J\ and, G(J)<a-\J\.
Let us denote by I the smallest edge of J and by % the positive
integer satisfying the inequality
A1.19) WjZ^&j— a1<(%+l)?.
Writing d1=(b1—%)/%, let us subdivide the interval J into %
equal non-overlapping subintervals
where i=l,2,..., %. We shall call an interval J; of the first kind
if |JJ-J,-|>A—3y)-\Ji\, and of the second kind in the opposite
ease. Denoting by ^" a summation over the indices i corresponding
i
to intervals of the second kind, we see easily, on account of the
"'
second of the relations A1.18), that S'Sy-\Ji\<y-\J\=Zy-\Ji\' Hence,
if p and q are the number of intervals Jt of the first and second
kind respectively, we find 3#<%=p+g. Now let us subdivide the
interval J into a finite number of non-overlapping subintervals, in
such a manner that each of these is the sum of a certain number
of intervals J; among which exactly one is of the first kind. Since
2q<p, the intervals of this subdivision include some which coincide
with certain intervals Jt of the first kind, and their number is at
least equal to p— gr>w1/3. Thus if we denote their sum by A,
we find
A1.20) \A\ > | J|/3.
On the other hand, the figure JQA consists of a finite number
of non-overlapping intervals each of which contains an interval Jt
of the first kind, and therefore points oiH. Consequently, G{JQA)>0,
and, on account of A1.18) and A1.20)
G(A) <G(J)<a-\J\<3a-\A\.
It follows that among the intervals J,- of the first kind of which
the figure A is formed, there exists one at least, </,¦„ say, such that
Let us write, for brevity, a«=a1+(i0—l)d1( fi^^+i,^ and
J4=[46»;ai,6.2;...;am,&J.Bytheabove,JA)CJr, G(j'J))<3a.\J{1)\
and, since Jm coincides with an interval J", of the first kind,
J3jW()|j«j by (n.i9)} l^—a^d^Zl.
ill]
Ward's theorems.
141
If we now operate on J"'" just as we formerly did on J (except
that we replace y by 3y, a by 3a and the linear interval [a±, 6X] by
[a.2Na])' ve obtain an interval J"l'2>=[a5,6J;a°,&°;a3,63;...;am,6m]CJ<l)
such that <?(/2>)<32-a-|JC2)|, jJJ-J<2)Xl—32y)-|J"B>: and J<6»—o°<2Z
for ?=1,2.
Proceeding in this way m times, we obtain after m oper-
operations an interval J('n)= [a", 6°; < &j>;...; a°m, b°J C J such that
())<3ma-|J-(m)|, \E-Jlm)\>(l—3my)-\Jim)\ and
for j=l,2,...,m. It follows that r(J"
(m))
2~'", and if we write
j,,, (),
Q=Jlm) and substitute y=3~m, we find at once that the interval Q
fulfils the conditions A1.17).
A1.21) Theorem. If F is an additive function of an interval,
we have F'(x)=Fs(x)^°o [F'{x) =Fs{x) 4= <=o] of almost all tlte points
at which Fs{x) > — oo [Fs{x) < + oo].
Tims, in particular, the function F is derivable in the strong
sense at almost all the points at which both the extreme stnmg derivates
Fs{x) and Fs{x) are finite.
Proof. Since F(x)^Fs{x) holds for all a?, the function F is,
by Theorem 11.15, derivable (in the ordinary sense) at almost all
the points x for which Fs(x)> — oo, and we have only to show
further that at almost all these points F'(x)=Ff(;c). Suppose there-
therefore that the set of the points x for which F'(x)~>Fs{x)>—oo is of
positive measure. We could then determine a number a >> 0 and a set
B of positive measure such that F'(x)—JT6(*)> a at every point o?eB.
For brevity, write e=a-3~"(m+I), and let Bp denote the set of
the points xeB for which ps <Fs(x)^(p+l)-e. Let p0 be an
integer such that iBpj>0, and write G(I)=F{I)—po?-\I\
(where I is any interval). Since <?'(») > Gs(x)-\-a > a at every point
xeBPa, we can determine a positive number a and a set UQBpit of
positive measure, such that G{Q)>a-\Q\ whenever Q is an interval
satisfying the conditions
A1.22
6{Q)]<a, v(Q)>2"m and
But since 0<6>(a;)<2? at eveiy point xeEC.BPi1, there
exists by Lemma 11.16, an interval Q subject to the conditions
A1.22) and such that G(Q)<3m-2s-\Q\<a-\Q\. We thus arrive at
¦a contradiction and this proves the theorem.

112
CHAPTER IV. Derivation of additive functions.
*§ 12. A theorem of Hardy-Little wood. The theorem of
Jessen, Marcinkiewiez and Zygmund concerning strong derivation
of indefinite integrals, which was mentioned in § 10, p. 133, is
connected with an important inequality due to G. H. Hardy and
J. B. Littlewood [2]. This inequality, which was established in
connection with certain problems of the theory of trigonometrical
series, thus obtains a new and interesting application.
We reproduce in this § the elegant proof given by F. Biesz [5] (of. also
A. Zygniund [I, pp. 241—245]) for this inequality. Although simpler than the
other proofs, it requires nevertheless some rather delicate considerations. Certain
parts of the argument have been touched up in accordance with suggestions
communicated to the author by Zygmund.
The reasonings of this § concern functions of a real variable.
A2.1) F. Riesz's lemma. Let F(x) be a continuous function on
an interval [a, ?»] and & a finite number. Let E be the set of the points x,
interior to the interval [a, 6], for each of wMoh the inequality
F{x)—F{u)>~k-{ae—u) is fulfilled by at least one point u subject to
a<u<x-.
Then the set E is either empty, or else expressible as the sum of
a sequence {(a^, bn)} of open non-overlapping intervals such that
Proof. By subtracting from F{oo) the linear function lex, we
may suppose that fe=0. Then E is the set of all the points * of the
open interval (a, 5), for each of which there exists a point u such
that F(u) <F{x) and «< u < x. Since the function F is continuous,
the set E is clearly open, — and, unless empty, it is therefore ex-
expressible as the sum of a sequence {{an, b,,)) of non-overlapping
open intervals. We have to prove F(a,,)^.F{bn) for each n.
To see this, let us fix an index n and suppose that F (a,,) > F (&„).
Let h be any number such that
A2.2)
F(an)>h>F(bn),
and let x0 be the lower bound of the points x of the interval [a,,, &„]
for which F{x)=h. By A2.2) the point *0 belongs to the open
interval [a,,, bn), and so to the set E; thus there exists a point y
such that F(yXF{a>0)—h<F(a,,) and a<2/<a?0. This last relation
implies a<y<an, since, by A2.2) and by the definition of the point
[§12]
A theorem of Hardy-Littlewood.
143
xm the inequality F(y)<~h cannot hold for any y of the interval
[an,a]- Tiras F{yXF{an) and a<y<an, and consequently arieE;
but this is clearly contradictory, since an is an end-point of one
of the non-overlapping open intervals which constitute the set E.
Besides the results treated in this §, many other applications of Lemma 12.1
are given by P. Biesz [6; 7; 8], particularly in the theory of derivation of func-
functions of a real variable. Cf. also S. Izumi [1]. The lemma might also have
been used in the considerations of § 9 (instead of appealing to the theorems of
§ 8 on rectifiable curves).
To shorten our notations we shall restrict ourselves in the rest
of this § to functions defined in the open interval @,1); and we
shall agree to write E[/ > a] for E [/ (x) > a; 0 < x < 1]. The symbols
X
E[/^a], B[6>/>a] and so on, will have similar meanings.
Two measurable functions g and h in @,1) will be called (in
accordance with the terminology of P. Eiesz) equi-measurable if
|E[<7> a]|=|E[7i> a]\ for every finite number a. We see at once that
we then also have
|B[flf>o]|=jE[A>a]|, Blb>g^d]\=\E[b>h>a]\, etc.
A2.3) If tico non-negative measurable functions g and h in the in-
interval @,1) are equi-measurable, their definite integrals over this in-
interval are equal.
To see this, let us associate with the function a a non-decreasing
sequence {gj of simple functions by writing gn{x)=(k—1)/2" when
(k—l)/2"<5r(*)<fc/2'i and h=l,2,...,2"-n, and gn{x)—nwheng{w)^zn.
Similarly with g replaced by h, we define the sequence {hn} con-
converging to the function h. If we calculate directly the integrals
of the functions gn and hn over @,1) by formula 10.1 of Chap. I,
p. 20, we see at once from the fact that the given functions g and h
i i
are equi-measurable that jgn(x)dx=jh:i{x)dx. Making >i->oor
i i * 6
this gives /g{x)dx= jli{x)dx as asserted.
6 o
If / is a continuous function in @,1) which is not constant
on any set of positive measure, and if m and M denote the lower
and upper bound of / respectively, the function ?(!/)=|E[/>2/]| is
evidently continuous and decreases from 1 to 0 in the open interval

144
CHAPTER IV. Derivation of additive functions.
(m, M). Its inverse function is therefore continuous and decreasing in
@,1) and, as we easily verify, equi-measurable with the given function.
"We shall extend this process with suitable modifications, to
arbitrary measurable functions finite almost everywhere in @,1).
"With any such a function / (x), we associate the function /"(a?) de-
defined for each x of @,1) as the upper bound of the numbers y for
which |E[/>2/]|>». The function f{x) is clearly finite and non-
increasing in @,1). To show that this function is equi-rneasurable
with /(as), let y0 be any finite number, and let »„ denote the upper
bound of the set E[/re>2/0], or else oj0=0 if this set is empty. Then
since |E[/">?/o]|=flJoj & uas to be P^ved that |B[/> #0]|=*¦(,.
We have, in the first place, /"(a5o+eX?/o for every e>0
(provided, of course, tliat *0+?<l), so that |E[/> j/o+^lKavK
and therefore |B [/ > ?/0]K x0 • On the other hand, f"{w0—e)>?/0
for every e>0 (provided that x0—e>0), so that |E[/>j/0]|>-a;0—«,
whence |E[/>i/0]|^a;0, and finally |E[/> ?/o]|=«o = B[f > yDl
We shall further define, in connection with any summable
function f{x)} three functions /3l(as), f°{x) and f{x). At any point x
of @,1) we shall denote by f'{x) the upper bound of the mean
values of / on the intervals (w, as) contained in @,*), i.e. the
upper bound of the numbers ff(t)dt for 0<w<as. Simi-
it
, i ;
larly, f"{x) will denote the upper bound of the means f(t) dt
X
for *<»<1. Finally, f{x) will denote the larger of the two num-
numbers f'{x) and f-[%), or what comes to the same, the upper bound
of the means ff(t) dt where u and v are subject to the
condition 0<«<><»<1.
A2.4) Lemma, If f(x) is a non-negative measurable, function in
the open interval @,1) and if -E is a set contained in this interval, then
\f\
Jj
[§12]
A theorem of Hardv-Little-wood
145
Proof. Let fx be the function equal to / on the set _E and to 0
elsewhere. We evidently have f"(x)^f'(.t) at each point x of the in-
interval @,1). Furthermore /"(#) = (.> as soon as x>\E\. Therefore on
account of A2.3) we find
f
A2.5) Lemma. If f{x) is a 'non-negative summable function in the
interval @,1), tlien for each point x of the interval, we have
Proof. Let x0 be any point in @,1), let yo=fia(xo), and
let ? denote an arbitrary positive number. Wxe write A = "&\j^a^>ya—s]
and B=E[/^'>i/0—e]. Then since the function fl" is non-increasing,
we have \A\^-x0 and therefore, remembering that the functions fl"
and fl are equi-measurable, \B\=\A\~^x0.
B is the set of the points x for each of which there exists
X
a point « subject to the conditions jf(t)dt>{y0—e)-(x—u) and
u
0<«<a;. Therefore, applying F. Eiesz's Lemma 12.1 to the in-
indefinite integral of /, we find easily that B is an open set and that
ff(t)dt^{yo—e)-\B\. It follows by Lemma 12.4 that
B
A2.6)
Now since \B\ > x0 and since tie function f' is non-increasing, the
last term of A2.6) cannot exceed — jf'{t)dt; and since e is an
arbitrary positive number, we must have f'a(x(s)=y<s^.— I f[t)dt.
This completes the proof.
A2.7) Theorem of Hardy-Littiettood. If /.(.as) u a non-negative
summable function in @,1) and s is a positive number, then
i i i
A2.8) ff(x) dx^A-ff(x)-log+f{x) dx+B-ff{oo) dx+e,
0 0 0
where A and B are constants depending on e, but not on f.
S. Saks, Theory of the Integral. 10

146
CHAPTEE IV. Derivation of additive, functions.
Proof. We first evaluate the integral of /'"' over @,1). Ac-
According to A2.3) and Lemma 12.5 we have
jf(w)ta= ff'"(x)dm</[ / tM
o 6 6 hi
In virtue of Fuhini's theorem the last member of this inequality
is the surface integral of the function f'{y)jx over the triangle
'O^ic^l, 0^2/<; x. Therefore inverting the order of integration in
this member, we find
A2.9) ff'(x) aaxzf \ f~]fw %== [ftMogyl %¦
o o U) -1 h
>j _
Let now 77 <1 be a positive number such that /|log y\/Vydy < f/2.
0
Let us denote by J3X the set of the points y of the interval @, ij) at
which f(y)<~lj\'y, and by E2, the set of the remaining points of
this interval. We find
A2.10)
f
b1
^ dy+2 ff(y)-logf"(y)dy+
V 4,
1
og+ f(y) dy+\log 4 ff(y) dy+e/2.
Further, since the functions / and /" are equi-nieasnrable, so
are the functions f-log+f and f-log^f, and it therefore follows
from A2.9) and A2.10) that
1 1 1
A2.11) ff(x) dx ^2 ff (x)-log+ f(x) da>+ |log^j- ff(x) dw+ f/2.
0 0 0
A similar inequality clearly holds when on tie left-hand side
of A2.11) f' is replaced by f% and on adding the two inequalities
•we find jf{x) dm <//¦'(*) dx+f/<(oo) dx^i j'f(cc) ¦ log+ f(cc) dx +
+\21ogi]\- ff{a>)dx+s; this gives A2.8) with J. = 4an<l witli i?=2|logi?|.
[§13]
Strong derivation of the indefinite integral.
147
*§ 13. Strong derivation of the indefinite integral.
We proceed to prove the theorem of Jessen, Mareinkiewiez and
Zygmund. We shall give the proof for the case of the plane;
its extension to spaces Sm of any number of dimensions (cf. §10,
p. 133) presents no fresh difficulties and is effected by means of
the well-known inequality of Jensen.
We shall begin with some auxiliary remarks. Suppose given
a non-negative function f{x,y) summable over the open square
J0=@,1; 0,1). By Fubini's theorem, the function f(jc,y) is sum-
summable in x over @,1) for almost all y of @,1), Denote by E the
set of these values of y. For any y e H and for any x of the interval
@, 1), we shall denote (cf. §12, p. 144) by f(x, y) the upper bound
of the mean jf(t,y)dt for 0<M<t»<y<l; and whenever
yeCS, we shall write, for definiteness, f(x,y)=0 identically in x.
We shall prove that the function f(x, y) thus associated with any
function /(as, y) which is summable over the open square J"o= @,1; 0,1),
is measurable.
For this purpose, let a and b denote two positive numbers,
x-rb
and write ga,b(x, y)= j f{t,y)dt when y eE and 0<!.r—»<a!-i-6<l,
and ga,b(x, y) = 0 elsewhere in Jo. We shall begin by showing that
each of the functions ga,b{x,y) is measurable. By Lusin's theorem,
or more directly by the theorem of Yitali-Caratheodory (Chap. Ill,
§ 7), the function / is equal almost everywhere to the limit of a non-
decreasing sequence (/(n)) of non-negative, bounded, upper semi-contin-
x+b
uous functions. Now, let us put g%\{ia>, y) = ffinHt,y)dt when
X—(I
0^x — a <icH-6< 1, and g(fb(x,y)=0 elsewhere. As is easy to show
(e. g. by means of Theorem 12.11, Chap. I), each of the functions
g^\{x, y) is then also upper semi-continuous, and since, as we readily
see, ga h(x,y)=Ym.^b(x,y) almost everywhere, the function ga,b(x,y)
n
is measurable.
Finally, with the same, notation as above, if [up) is the
sequence of rational numbers of the interval @,1) we have
f(x, y) = upper bound gWo,uo

148
CIIAPTEB IV. Derivation (if additive* functions.
at any point {x, y) of Jo. Thus the function f(x, y) is also measurable,
and this proves our assertion.
A3.1) Theorem. If f(x, y) is a measurable function in the plane
J?8 and if the function f log+1/| is summable, then the indefinite integral
of f is almost everywhere derivable in the strong sense.
Proof. Clearly we need only consider the function / iti the
open square Jo=@,1; 0, 1); and we may also suppose that this
function is non-negative.
We write gn(x,y)=f{sc,y) wherever f(x,y)^n, and yn(x,y)=n
wherever f(x,y)^>n; we write further hn(x, y) — f(x, y)—g (a),y)
and we denote by a an arbitrary positive number. The functions
hi{x, y) are measurable and non-negative; ho that by Theorem 12.7
of Hardy-Iittlewood,
I l~h{(x,y)dxd'nK,
A3.2) , ./',
/ fh,,{x,y)-log+hn(x,y) dxdy+B f jh,,{x, y
where A and B are finite constants depending only on a. And since
the integrals on the right-hand side of A3.2) tend to 0 as n-><x>,
there exists a positive integer N such that the left-hand side of A3.2)
becomes less than o2 for n=$f. Therefore, writing for brevity
h{%, y)=hN{x, y) and g{co, y)=gN(a>, y), we have
A3.3) J' fh\x,y)dxdy<(fi,
j
so that hi particular the function h?{x, y), besides being measurable
and non-negative, is summable on Jo.
^ow denote by E the set of the points (x0, y0) of Jo such that
1° jJi?{ie0,t)dt<+oo, and 2° the indefinite integral j'li?(xo,t)dt
0 o
has at the point y=y0 the derivative h\xa,ya) with respect to y.
Since, by Theorem 6.3, condition 2° is fulfilled for almost all ?/„
of @,1) provided that condition 1° is satisfied, it follows at once
from Fubini's theorem in the form (8.6), Chap. Ill (cf. also Theo-
Theorem 6.7, Chap. Ill) that |JS|=|J0|=l.
Let us write F, E and G for the indefinite) integrals of the
functions /, I and g, respectively, in Jo. Let (*¦„, j/0) be si point of the
set E and I=|>0—ult «0+m2; ?/o_ Vl, yo+v.2\ 'any interval wm-
tamrng (x0, y0) and contained in Jo. We have
[§14]
Syniffletrieai derivates.
149
= ^TT / L . „ }H-»:o+u,yo+v)du\dt-
—D, —H,
"^ «2 + »1 J "'
mi)
\i\
whence making <5(I)->0 we obtain JBh(j:0, yo)^h:'(xo, ;/0). Thus,
smee (x0, y0) is an arbitrary point of the set EQJo of outer measure 1,
and since the extreme derivate Ss(x0, y0) is measurable (cf. Theo-
Theorem i.2), it follows from A3.3) that 0^Ee(x,y)^Hs(x,y)^a
at every point {x, y) of Jo, except at most a set of measure less than a.
On the other hand, since the function g = gN is bounded, its in-
indefinite integral G is, by Theorem 10.7, derivable in the strong
sense almost everywhere. Therefore F?{x, y)—Fs{x, y)^a at all but
a subset of measure a of the points of Jo; and so finally, smee a
is an arbitrary positive number, Fs(x, y)=Fs{x, y) almost every-
everywhere in Jo, which completes the proof.
By Ward's Theorem 11.21, to prove that the non-negative function F is
almost everywhere derivable in the strong sense, it is enongh to show that
P5 {;?)<+oo almost everywhere. Hence by using Theorem 11.21, the proof of
Theorem 13.1 might be slightly shortened.
*§14. Symmetrical derivates. If <P is an additive function
of a set in a space i?m, we shall denote by D?ym @{x) the upper, and
by paym(P(j) the lower, symmetrical derivate of <P at a point x,
these being defined respectively as the upper, and as the lower,
limit of the ratio 0{S)j\S\ where S represents a closed sphere of
centre x and of radius tending_to zero. It is obvious that, for any
point x whatsoever j5&(x)^Dsrmt>(x)p!l>s,m&{x)^T)<P{x).
Following A. J. Ward [5], we shall establish a decomposition
theorem in terms of symmetrical derivates, which is similar to
Theorem 9.6. We shall begin by the following "covering theorem":
A4.1) Theorem. If 0 is an additive function of a set in Rm and E
a bounded set measurable (93), contained in an open set G, then for
any e > 0 there exists in G an enumerable sequence of closed sphe-
spheres {Si} such that (i) the centre of each Sk belongs to 2? and the ra-
radius is less than e, (ii) SrSj=0 whenever i^j, and {Hi) the spheres
Si cover together the whole of the set E, with the possible exception
of a subset on which the function $ vanishes identically.

150
CHAPTER IV. Derivation of additive functions.
Proof. We can clearly assume (by replacing, if necessary,
the function $ by its absolute variation) that the function # is
monotone non-negative.
a) We shall first prove that, "with the hypotheses of the
theorem, there always exists in G a finite system of equal sphe-
spheres {Sk} which sasisfy the conditions (i) and (ii) and cover 'the
set E except perhaps for a set TBE such that
A4.2) ®(T)^(l-llim+1 m"')-0(E).
To see this, let A be a subset of E, measurable (95), such
that §(A)^h;${E) and q{A, C(?)>0. Let n0 be a positive in-
integer such that m/na<Q(A, CG) and m/wo<?.
Denote by '¦p the net in the space J?m, which consists of tlie cubes
of the form [pjno, (px+l)X; p.J%, (p.f\-l)ln0;...,pjn0, (v,,,+l) ra0]
where pv p2, —,pm are arbitrary integers. We can clearly sub-
subdivide the net Sp into Dm)"! families of cubes, 'P1,'^.<j,.")^(,lm)/« say,
such that the distance between any two cubes belonging to tlie same
family is not less than Dm—l)jn0. Denote, for each ft=l, 2,..., Dm)"'
by Ak the part of the set A covered by the cubes of the family %,.
Then there exists a positive integer ^^.(im)'" such that
A4.3) 0(A)^ 3s(^)/Dm)">^(-B)/4m+1m.
Now let Pi, P2,..., Pr be those cubes of ^ which contain
points of A^. With each Pt we can associate a closed sphere Si,
of radius mj%, whose centre belongs to Akl-Pt. The system of spheres
8\, 82,..., Sr thus defined is contained in G and clearly satisfies
the conditions (i) and (ii) of the theorem. Again, since PtC^i for
every i=l, 2, ..., r, the spheres St cover the whole of the set E
with the possible exception of the points of the set T=E—Ak,
which, in virtue of A4.3), fulfils the condition A4.2)
b) We now pass on to the proof of the assertion of the
theorem. By what has already been proved, we can define by
induction a sequence {Sn}n=12 of finite systems of closed spheres with
centres in E and radii less' than e, subject to the following two
conditions: 1° If Bo denotes the empty set and Bn, for »>l, the
sum of the spheres belonging to 6x+S2+... + Sn, then the system
6n-j.i, where »>0, consists of a finite number of closed spheres
contained in the open set G-Bn, no two of which have common
points; 2^0(E-Bn+1)^(i-hm).0(E-Bn) where fcm=l/4m+1 m'"
and w=0,1,... Now, arranging the spheres belonging to the family
Symmetrical derivates.
151
"- in a sequence {Sjj, we see at once that the latter
fulfils conditions (i) and. (ii) of the theorem. On the other hand, by 2°
we have &(E—BB)<i(l—IimY'-&(E) for each n; whence, denoting
by B the sum of all the spheres 8h it follows that &(E—B)=Q,
which establishes condition (ill) and completes the proof.
Theorem 14.1 may be established iu a sliglitly more general form:
Given a bounded set E measurable C3), a sequence of positive numbers {rn}
converging to 0 and a fawily of dosed sets 91, suppose that with each point x of B
there are associated tiro finite numbers a=c(x), Jf=3T{E), and a sequence {An(x)\
of sets (?I) sueh, that $>(x; r,i)(Z^-n{x)(Z^{x; arn) for ?i5>5"(.r).
Then, for any sequence (<Z>n) of additive functions of a set, we can extract
from 91 a sequence of sets [Ai=Ani{xl)} such that (i) XieE for i=l,2,..., (ii) Ai-Aj=0
whenever i±j, and (in) the sets Ai cover the, whole of the set E, with the exception,
at most of a set of measure zero on which all the functions 2>n vanish identically.
A4.4) Lemma. If 0 u an additive function of a set in JRm, and if
psym^(*)>0 at each point ,v of a bounded set X measurable (93),
then 0(X)>O.
Proof. Let us denote, for every positive integers, by Xn the set
of the points xeX such that §(8)^0 whenever #is a closed sphere
of centre x and radius less than l/«. Each set Xn is evidently meas-
measurable C3), in fact closed in X. Hence, for any e> 0, we can associate
with each Xn an open set fiJI,, such that Y\r($;Gn—Xa)^s
(cf. Theorems 6.9 and 6.10, Chap. III). Next, keeping n fixed for
the moment, we can (on account of Theorem 14.1) define in <?„
a sequence {8k} of closed spheres with centres in Xn and radii less
than 1/ra, such that (i) SrSj—0 whenever ?#=?', and (ii) the sphe-
spheres Sm cover the whole of the set Xn with the exception at most
of a set T on which the function 0 vanishes identically. Since
for every Jc, we find by (i) and (ii) that ®{Xn) >
^T)+W($;Gn—Xn)J^—e. Hence, as X = limXn and ?
n
is an arbitrary positive number, it follows that E(X)^0, which
completes the proof.
A4.5). Theorem. If <P is an- additive function of a set' in Mm,
and if A^ denotes the set of x at which, one at least of the derivates
Dsym<P(*) and D5ym #(¦*¦) is infinite, then for any bounded set X
measurable E3), ice have
A4.6) &(X)=<P{X-Aco) + f~D<P(a))dx.
x
Consequently, if D4ym<2>(.r) >—<=o at every point x and D<Z>(a*)^0
at almost every point x of a bounded set X measurable B3), then

152
CHAPTER IV. Derivation of additive functions.
Proof. We firstly remark that if —oo<DBym0(x) at each point a;
of a bounded set Q measurable C3) and of measure zero, then
#($p*0. In fact, denoting for each positive integer n by Q,, the
get of the points x of Q at which —n<psym<35(a;), and writing
0n(X)=0(X)+n-\X\, we obtain Dsym0,,(a?)>O at every point xeQn.
Hence, by Lemma 14.4, we must have &(Q,,)=0,,(Qn)^Q, and making
w->oo we find 0{Q)^O. By symmetry we also have 0(Q)^O when-
whenever Q is a bounded set measurable (93) of measure zero, such
that D$(a;)< + oo at each point x of Q,
We pass on to the proof of formula A4.6). By Theorem 7.3
there exists a set Af measurable (93) and of measure zero, such that
the relation
A4.7;
0(X)=0{X-A) + fl>0(x)dx
holds whenever X is a bounded set measurable E8). Since the set
Aca is of measure zero, we see at once from the equation A4.7) that
the function $ must vanish identically for all the subsets of A^—A,
which are bounded and measurable (93). On the other hand, by what
has just been proved, the function vanishes also for all subsets of
A —A<x. Hence the set Ax may be taken in place of tie set A in
A4.7) and this gives A4.6). Finally, if Daym<25(a;)>—oo at every
point a) of a bounded set X measurable C3), then 0{X-AX)^O
and the second part of the theorem follows at once from the first.
Let us mention the following ooiisequence of Theorem 14.5: If at each
point x both the symmetrical derivates of a given additive ¦fwiwiion of a set are 'finite,
the latter is absolutely continuous. For ordinary derivates the corresponding pro-
proposition has long been known (cf. H. Lebesgue [5, p. 423]) and in moreover
included in Theorem 15.7 of this chapter, as well as in Theorem 2.1 of Ohap. VI.
*§15. Derivation in abstract spaces. With certain hypo-
hypotheses, a process of derivation may be defined for additive func-
functions of a set in any separable metrical space, and for such a process,
theorems similar to those of §§ 7 and 9 may be established.
A5.1) Zemma. If 0 is an additive function of a set (93) on a metrical
space M, then given any set X measurable (93) and any e>0, there
exists an open set G such that
A5.2) W{0;G—X)<s and W@; Z— <?)<?.
[§15]
Derivation in abstract spaces.
153
Proof. Let 93O denote the class of the sets X measurable (93)
for each of which there exists, however we choose e>0, an open
set G satisfying the relations A5.2). Since any closed set F is the
limit of a descending sequence of open sets, we observe easily
(cf. Theorems 5.1 and 6.4, Chap. I) that there exists for each ?>0
an open set GZ)F such that W{0\ G—F)<e. The class 93O thus
includes all closed sets; to prove that 93=93O, it suffices, therefore,
to show that the class 93O is additive.
To do this, we choose s>0 and denote by X the sum of a se-
sequence !Xi)n=i.2,... of sets B3O). To each set Xn there corresponds an
open set Gn such that W(<2>; Gn~Xn)<sf2n and W(<Z>; X,—Ga)<sj2\
Writing G=?Gn, we clearly find that the inequalities A5.2) are
satisfied. Therefore XeS80.
Again, suppose that. e>0 and that X=CY, where re©0.
There will then exist an open set H such that W((P; F—H)<sj2
and W@; H—T)<e. Consequently writing P=CH, we find that
A5.3) W{0;P—X)<ej2 and W@; X—P)<e.
But since the set P is closed, there exists an open set <? such that
G^)P and W@; G—P)<e/2; and this implies, on account of A5.3),
the inequalities A5.2) and so completes the proof.
We shall call net in a metrical space -If any finite or enumer-
enumerable family of sets measurable (93) no two of which have common
points and which together cover the space M. The sets constitut-
constituting a net will be called its meshes. A sequence [3Ra] of nets will
be termed regular, if each mesh of 9Kn+i (where ?i>0) is contained
in a mesh of 9Jln and if further A{3Rn)->0 as ?i->oo (where 4BJl«)
denotes the characteristic number of %\n; cf. Chap. II, p. 40). It is
easy to see that in order that there exist a regular sequence of nets
in a metrical space, it is necessary and sufficient that this space
be separable.
In the rest of this § we shall keep fixed a separable metrical
space ilf and we shall suppose given in JUT a regular sequence 9ft=[3Rn}
of nets and a measure /; which is defined for the sets measurable
(93) and which is subject to the condition i«(jyr)<+oo. Let 0 be
an additive function of a set (93) on M. For x e M, where M is any
mesh of a net 2Rn, let us write
I 0B1I/j{M) Klien /i(JtfL=0,
dn(x)-\ +oo u-hen /<(!!)= 0 and ${M)S?0,
— oo when ,«(J1/)=O and 0(M)<O.

154
CHAPTEK IV. Derivation of additive functions.
The functions dn{os) are thus defined on_the whole space M
and are measurable C3). Let us write {ft, W)D&(cc) = lim sup dn(x).
n
The number (//, 9K)D$(a;) thus defined will "be called upper derivate
of the function 0 at the point x with respect to the measure p and the
regular sequence of nets 3R. Considered as a function of x, this upper
derivate is clearly measurable (93). Similarly we define the_lower
derivate {/j., %l)IH{x). If at a point x the two numbers (p, Sffl)I)<P(x)
and (/{, <3R)D0{x) are equal, their common value will be written
{n, 3R)D0(x) and called derivative of the function 0 at x with respect
to the measure p, and the regular sequence of nets 3R. For the rest
of this §, a measure p and a regular sequence of nets SOI will be
kept fixed in the space M.
A5.4) Lemma. Let 0 be an additive function of a set (93) on the
space M. Then
(i) if the inequality ({il!3R)T>0(x)^k, where k is a finite number,
holds at every point x of a set A measurable (93), we have &(A)^Jc-/,i(A);
(ii) if at each point xof a set B measurable (93) and of measure (/<)
zero, the derivative (/i,<3}\)D0{x) either does not exist or else exists and
is finite, &(B) = 0.
Proof, re (i). By subtracting from the function 0 the function
hp,, we may assume that h—0. Let e be any positive number. By
"Lemma 15.1 there exists an open set G such that
A5.5) W{0;G—A)<e and W{0;A—&)<s.
Let %\ be the set of the meshes M of the net 9^ such that
A5.6) MQG and
aad generally, for w>l, let 3Jtn+1 be the set of the meshes M of the
net mn+i which fulfil the conditions A5.6) and are not contained in any
of the meshes of^St1+§t2+...+«mn. By arranging the sets belonging
to9K1+9K2+...+2Rn+... in a sequence {Mk}, we have 0{Mk)>—s-f.i{Ml[)
for fc=l, 2,..., and A-GCZMk. Since the sets Mh are measurable (93)
and no two of them have common points, it therefore follows
from A5.5) that 0(A) = ${A-G) +<P(A — G)>d>(ZMlt) — 2?^
>~e-Sft{X*)—2e^ — e-[/j,(G)+2l <™d so that
[§15]
Derivation in abstract spaces.
155
re (ii). Denote, for any positive integer n, by Bn the set of
the points x of B at which D0(x)^—n. On account of (i) we have
0[Bn)^—n-n(Bn)=0 for each n, and, since B = MmBn, this gives
0(J3)>O. By symmetry &(B)s^0, and so finally 0(?)=O.
A5.7) Theorem. If 0 is a function of a set (93), which is additive
on the space 31, the derivative (,«,!3R)D0{jj) exists almost everywhere
and is integrable (93, fi) on M; moreover, if E+ca and H-^ denote the
sets of the points at which {u, SR)'D0(x) = -\-oo and (p, 3Sl)I>0(x)~—oo
respectively, we have
A5.8)
and
A5.9)
for every set X measurable (93).
Proof. By Theorem 14.6, Chap. I, there exist a function of
a point / integrable (93, ft) on M and an additive function of a set 8
singular (93, /<) on M such that
A5.10)
for every set X
Let E be a set measTxrable (93) such that fi(E)=0 and that
the function 6 vanishes identically on CE. Writing, for brevity,
D, Uandl? in place of (/<,3R)D, (,«, 901)D and (/t,9!ll)D respectively,
let us denote for any parr of integers »i>0 and fc, by Pn,A the set of
the points a? at which B0(x)^(]c+1) ln>~kjn^f(x). If we substitute
Pn,k-GE for X in A5.10), we find on account of Lemma 15.4 (i) that
n Pn/'CE
and so that [i{PBjt)=fi{Pn^CE)=0. Therefore D0(x)<,f(x) at
almost all points x. By symmetry JH{x)^f(x) must also hold
almost everywhere in M. Therefore the derivative I>0(x) exists and
equals f{x) at ahnost all the points % of M, and the identity A5.10)
takes the form
A5.11) 0{X) = @{X)-rfl>0dii=0(E-X) + [D0dfi for every set Xe93.
X
Moreover, since D<P(x)=f(x)^co ahnost every where, the set E
is of measure (,«) zero, and it follows directly from A5.11) that the
function 0 vanishes identically on the set (-E+^+E—x,)—JE.

156
CHAPTER IV. Derivation of additive functions.
On the other hand, by Lemma 15.4 (ii), 0 vanishes identically
on E—{E+ao+E-cc). Therefore in A5.11) the set E may be replaced by
the set E+oo+E-cc, and the relation A5.11) becomes the required
formula A5.8). Finally, since by Lemma 15.4 (i) the function <t>
is non-negative for the subsets C3) of E+oa and non-positive for
the subsets C3) of E. „, formula A5.9) follows at once from for-
formula A5.8).
Let us mention specially the following corollary of Theorem 15.7:
A5.12) Theorem. Suppose given in the space ftl two regular sequences of nets
3t and ?, and, as before, a measure fi defined for the sets C3) and subject to ihe
condition fiCJ)<+oo. Then for every function <t>of a set (93), which is uddiUve on M,
roe have almost everywhere (,«;, 3l)D<2> (?) = (,«, ^)D'S(x); moreover, if Ii denotes
the set of the points x at which either one at least of ihe derivatives (//, 3I)D'/'(x)
and {,u, ^)D^{x) does not exist, or else both exist but have different values, then
the function 0 vanishes identically on E, i. e. Vf('I';E) = 0.
In fact, if we write, for brevity, Dx and B2 in place of (,», 9t)D iukI {/r, Sf,)l>
respectively, and if we denote by 8 the function of singularities of <1>, wo have
by the previous theorem
for every set X measurable (93). Equating the two integrals which occur in this
relation, we obtain almost everywhere D1&(o;)=D20(x).
Now the set E of the points at wliioh this relation does not hold, may be
expressed as the sum of three sets Ax, A% and As, where AL is the set of the points
ieE at which one at least of the derivatives DjCffa) and D2tf'(x) does not exist,
or else exists and is finite, A% the set of the points x at which 'D10(x) = +oo and
D2$(x)=— co, and Aa the set of thepoints x at which DjfiPfx) = - oo and D2 ?&(.«) =H-oo.
It follows directly from Lemma 15.4 (ii) that the function <I> vanishes identically
on ?v In the same way, it follows from part (i) of this lemma that wo have
simultaneously 0(X)^Q and 0(X)^O, and so $(X) = 0; for every subset X
measurable (93) of A2 or of A2- Consequently W(#;-ft) = 0, and this completes
the proof.
Theorem 15.7, which corresponds, to a certain extent, to Theorem 9.6,
was first proved by Ch. J. de la Vallee Poussin [1; of. also I, p. 103] for
derivation with respect to the Lebesgue measure, and with respect to the regular
sequences of nets of half open intervals in Euclidean spaces. Strictly, the Lebes-
Lebesgue measure does not fulfil the condition which we laid down for the measure fi,
since Euclidean space has infinite Lebesgue measure. Nevertheless it is easy
to see that for the validity of Theorem 15.7 (as well as for that of the other pro-
propositions of this §) it suffices to suppose only that the meshos of the nets considered
have finite measure.
For the derivation of additive functions of a set in abstract Hpaces, see
also E. de Posse! .[1].
[518]
Torus space.
157
* § 16. Torus space. As an. example and an application of the
results of the preceding §, we shall discuss in this § a metrical space
which, from the point of view of the theory of measure and inte-
integration, may be considered as one of the nearest generalizations
of Euclidean spaces. This space, called torus space of an in-
infinite number of dimensions, occurs in a more or less ex-
explicit form in the important researches of H. Steinhaus [2], of
P. J. Daniell [2; 3], and of other authors, in connection with
certain problems of probability; but the first systematic study of
this space is due to B. Jess en [2].
Following Jessen, we shall call torus space Qa the metrical
space whose elements are the infinite sequences of real numbers
?=(%l,xs,...,%n,...) where 0<a3y,<l for «=1,2,..., the distance
q{$, tj) of two points ?=(xv x.2, ..., x»,...) and t?=(.Vi, Vi, -. .?«> ••¦)
in Qa being defined by the formula g(f, fj)—^l{yn—xn\j2n. By Qm,
n
where m is any positive integer, we shall denote the half ojien cube
[0,1; 0,1;...; 0, 1) in the Euclidean space JRm. If g—fa, x2, ...,xn,...)
is a point of Qa> we shall denote, for any positive integer in, by t-'m
the point (.%, x2, ..., xm) of Qm, and by %'m the point (xm+i, xm+-i,—)
of Qoi, and we shall write l=(l^, O- According to this notation,
A,7?) is a point of Qa whenever ?eQm (where m is any positive integer)
and yeQo. So that,if AQQm andB<ZQ<*, the set AxB (ef. Chap.Ill,
§§ 8, 9) lies in the space Qa>; and in particular QmxQt,f=Qa>.
We shall call closed interval, or simply interval, in the space
Qa, any set of the form I x Qa, where I is a closed subinterval of Qm
for some value of m=l, 2, .... Similarly, taking I to be an interval
which is half open (on the left or on the right) in Qm, we define in
the space Qa the half open intervals (on the left or on the right).
Every (closed) interval J in Q,,, has only one expression of
the form IxQa where I is an interval in a space Qm. (It is to be re-
remarked that the space Qa itself is not a closed interval in the sense
of the definitions given above.) By the volume of the interval J=I x Q,»
we shall mean the volume of the interval I in Qm(Z^m (cf. Chap. Ill,
§ 2). Just as in Euclidean spaces, the volume of an interval J in Q«
will be denoted by \J\ or LM(J). Again, as in Euclidean spaces
(cf. Chap. Ill, § 5). we shall extend the notion of volume in the space
Qm by defining for every set E in this space the outer measure L,*(JE7)
of the set E as the lower bound of the sums j?|J*| where [Jk] is
k
any sequence of intervals such that ECZJi- Thus defined, the
b

158
CHAPTER IV. Derivation of additive f
outer measure evidently fulfils the three conditions of Caratheodory
(cf. Chap. II, § 4) and determines, first the class of sets measurable
Bf), and then the class of functions measurable Bi,*). For brevity,
the" sets and the functions belonging respectively to those classes,
¦will simply be termed measurable. Also by the measure of a set E
in the space Qa we shall always mean its measure (LIS).
It is easily shown, with the help of Borel's Covering Theorem,
that the measure of any closed interval coincides with its volume
(cf. Chap. Ill, § 5, p. 65), so that we can, without ambiguity, write
\E\ or L4-E) (omitting the asterisk) to denote the outer measure
of any set E in Qa. We also see that the boundary of any closed
interval is of measure zero. Finally, we remark that the whole space Qa
is of measure 1.
We shall now define in Q,,, a regular sequence of nets (ef. § 15,
p. 153) of intervals half open on the right. We shall, in fact, denote
for any positive integer m, by Q1'"' the finite system of li'"" intervals
half open on the right
where the h are arbitrary non-negative integers less than 2"'.
We see at once that each system Q(m) is a net in Q,,,. To see
that the sequence of these nets is regular, we observe in the first
place that each interval of Q(m+1) jg contained in one of the intervals
of Qlm). On the other hand, no interval of the net Q("° can have
a diameter exceeding the number J|l/2+/'-|-2"'l/2/''^l/^"' "', so that
te=l ft—rn-fl
the characteristic number zl(Q("°) of the net Q(";) tends to zero as m->oo.
If * and y are two real numbers, x+y will denote the number
x+y—[®+y], where, as usual, [x+y] stands for the largest integer
not exceeding x+y. If ?=(x1} xK ..., aB,...) and ri=(yv yv ..., ?/„, ...)
are two points of Qa, we shall write t+n for {xx+yv x^+y,,, -, *'„+ yn, •••)•
The point f-f»7 clearly belongs to Qa.
We shall call translation by the vector a, where a is a point of Q,c,
the transformation which makes correspond to each point f of Qa
the point -?+a. The translation by the vector a will be termed of
order m, if all, except at most the first m, coordinates of a vanish.
A function / in Qa will be termed cylindrical of order m, if /(?)
does not depend on the first m coordinates of the point, ?, i. e. if
Torus apace.
159
/(?)=/(?+<*) identically in f, for every point a whose coordinates,
except perhaps the first m, all vanish. A set E in Q,o will be termed
cylindrical of order m, if its characteristic function is so, or, what
amounts to the same, if E=QmxA where A is a set in Qa.
A6.1) Theorem. A function which is measurable on ?>,., mid cylindrical
of every finite order, is constant almost everywhere, i. e. /(|) = c for
almost all points ? of Qa, where c is a constant.
Proof. Suppose first that the function / is bounded, and
therefore integrable, on Qa. Denoting by 0 the indefinite integral
of /, let us define, for each value of m and for each mesh Q of the
net Q(m), f'"\?)=<I>{Q)IL4Q) whenever ?eQ. For every pair of
meshes Qt and Q2 of the same net Q(m), there always exists a trans-
translation of order m which transforms Q% into Q2; therefore, since the
function / is cylindrical of order m, it follows that <!>{Qi)=&{Qi);
and since further LU(Q1) = LU(Q2), each of the functions fm)(?) is
constant on Qa. On the other hand, we deduce from Theorem 15.7
that /(|) = lim fm)(?) almost everywhere in Q,,,, i. e. that the function
m
is almost everywhere identical with a constant.
S"ow let / be any measurable function which is cylindrical of
every finite order. Let us write /j,(f)=/(l) when !/(?)[^»i and
f(g)=n when \f(tj)\>n. Each of the functions /„(?) is bounded and
cylindrical of every finite order, so that by what has just been proved,
each of these functions is constant almost everywhere. Therefore
the same is true of the function /(?) = lini fB{?) and this completes
the proof. "
The fundamental properties of our measure in the space Qa
may be established by methods similar to those used in Euclidean
spaces. To illustrate this, let us enumerate some of these properties.
Given any measurable set E and any e>Q, there exists a closed
set F and an open set G such that FQEQG and suck that \G—23;<e
and \E—.F\<e (cf. Theorem 6.6, Clap. III).
Prom this we may deduce next Lusin's theorem (cf. Theorem 7.1,
Cha,p. Ill): If f is a finite junction measurable on a set E, there exists
for each e>0, a closed set F(ZE such that the function f is continuous
on F and that \E—F\<s; and its immediate corollary: any function
which is measurable in Q,,,, is equal almost everywhere in Qa to a function
measurable (©). Finally Fubini's theorem (cf. Chap. Ill, § 8) may
be stated as follows for the space ?><„:

160
CHAPTER IV. Derivation of uddditive fuuctiotw.
A6.2) Theorem. If f is a non-negative measurable function in the
space Q,o, then for any positive integer m,
(i) the definite integral ff(?, ^)dLw(»?) exists for every ? 6 Qm,
except (A most for those of a set of measure (L,,.
zero.
(ii) the definite integral jf{?,
Qm
exists for every r;
except at most for those of a set of measure (Lw) zero,
(iii)
Q»
Qu Qm
Proof. We begin by verifying this directly when / in the
characteristic function of a dosed interval, or of a half open interval,
and then successively when / is the characteristic function of an
open set, of a set (©a), of a set of measure zero, and finally of any
measurable set. It follows at once that the theorem is valid in the
case where / is a simple function, and then, by passage to the limit,
in the general case where / is any non-negative measurable function.
The line of argument that we have sketched, does not differ substantially
in any way from the proof of Fubini's theorem for Euclidean spaces, and is oven
in a sense simpler than the latter, since in proving Theorem 8.1 of Chap. Ill we
had to allow for the possibility of there being liyperplanes of discontinuity oE
the functions V and 7.
In the space Qa there is, however, as shown by B. Jossen [2, p. 273],
another theorem of the Fubini type, whose proof requires new methods. This
theorem allows integration over the space Qa to be, so to speak, reduced to
integrations over the cubes Qm iu Euclidean spaces, whereas each of the three
members of the relation (iii) of Theorem 16.2 contains an integration extended over
the space Q,,,.
A6.3) Jesseris theorem. If f is a non-negative measurable function
in the space Qa, the integral
A6.4) U0
Q»
exists, and we have
for almost all ? in Qu.
), where m=l,2, ...,
q
[§16]
Torus space.
161
Proof. Let us first remark that if Q is a set of measure
in Qa, it follows from Theorem 16.2, applied to the characteristic
function of Q, that for any m whatever, the set E[(f, t])eQ; ?e Qm~]
is of measure (Lm) zero for almost all t\ of Q,,,. Hence (-with the not-
notation adopted p. 157) we also have Lm(E[{gt?m)eQ]) — 0 for almost all
? of Qa. It follows that if g and h are two non-negative measurable
functions which are almost everywhere equal in Qa, the integrals
[g{S, O dhm (?) and / *ft(f, Q dLJS) are equal for almost aU the ?
Qm Qm
of Qw, whatever m may be. We may therefore, without loss of gen-
generality, assume in the proof of Theorem 16.3 that the given func-
function / is measurable (SB); for any measurable function is almost
everywhere equal to a function measurable (93).
The integral in the formula A6.4) then clearly exists for every ?,
and moreover it follows directly from this formula that the function
/„,(?) is cylindrical of order m. The upper and lower limits of
the sequence {f„,(?)) are thus cylindrical of every finite order and
by Theorem 16.1 we may write almost everywhere in QM
limsup/nj(?)=J5
where A and B are constants. It remains to be proved that A=M=B,
where 1/ denotes the integral on the right-hand side of A6.5).
We shall prove in the first place that A >M. For this purpose,
let A' be any number exceeding A (if A = -\- oo our assertion is
obvious), and write
m
A6.6) Pt=E[/4(f)<l'], 8n=ZPk and 8=limSm.
The set $ coincides, except for a set of measure zero, with the whole
space Qa. Keeping an index m fixed, let us evaluate the integral of
fm over 8m. Writing i?ra=Pm,Rm-i=P,»-vGPm,...,2?I=P1-CP2-...-0Pnij
we have
A6.7) Sm=
and A6.8) i?,-E;=0 whenever
On the other hand, since every function fk is cylindrical of order fe,
so are the sets P* and CP* and therefore the sets Rk for fc=l, 2,..., m.
We may thus write (cf. above p. 159) Ra=QkXRk where RtClQu.
According to A6.6), we have /*(?)<!' for every ?6E*CP*
where fc=l, 2,..., m, or, what amounts to the same by formula A6.4),
S- Saka. Theory of the Integral.
11

162 CHAPTER IV. Derivation of additive functions.
i)^A' for every yeRk. Therefore, on account of
Theorem 16.2, we obtain for h=l, 2,..., m,
whence it follows by A6.7) and A6.8) that ff{C)dh@(C)^A'-Jjlo(/Sm)^A'.
Making m->oo, we obtain in the limit M=jf(?)dli(i,(?)^A' and
so M <;J., Q@
By symmetry M^B and, since it is clear that A <;J5, this
requires .A=Jf=.B and completes the proof.

CHAPTER V.
Area of a surface z=F(x,y).
§ 1. Preliminary remarks. We saw (cf. Chap. IV, §8) that
the Lebesgue theory enables us to solve completely the elementary
problems concerning the length of a curved line and the expression
of this length by an integral. However, similar problems concerning
curved surfaces involve difficulties of a much more serious Mnd.
Certain classical treatises on the differential and integral calculus,
even in the second half of the XIX-th century, contain an inaccurate
definition of the area of a surface. By analogy with the definition
of length of a curve, the authors attempted to define the area of
a surface as the limit of the areas of polyhedra inscribed in the
surface and tending to it. H. A. Schwarz [I, p. 309] (cf. also
M. Frechet [3]) was the first to remark that such a limit may
not exist and that it is possible to choose a sequence of inscribed
polyhedra whose areas tend to any number not less than the
actual area of the surface. About the same time Peano and Her-
mite subjected the old definition to similar criticisms and proposed
new definitions based on quite different ideas. It was H. Lebesgue
who first returned in Ms Thesis [1] to the old method, in a modified
form that may be roughly described as follows: the area of a surface
is the lower limit of the areas of polyhedra tending uniformly to
the surface in question (without, however, being necessarily in-
inscribed in the latter).
Nevertheless, in the more general case in which the surface
is given parametrically, this definition requires various additional
notions and considerations (cf. T. Bad 6 [I; 1; 4]) and the results
obtained are far from being as complete as those available
for curves. The,' difficulties that arise belong to Geometry and
ll*

164
CHAPTER V. Area of a Burfae-e s—l
Topology rather than to the Theory of functions oJf a real vari-
variable. (For the special case in which the functions* x='X(u,v),
y—Y{u,v) and z—Z(u,v) which define the .surface parametrieally
fulfil the Lipschitz condition vide T. Eado [4] and I-I. Radema-
cher [4]).
We shall therefore restrict ourselves to the case of continuous
surfaces of the form z—F{x, y). The most elegant and the most
complete results concerning these surfaces are due to L. Tonelli
[5; 6; 7]; they will be givenin § 8 and are the principal object of this
chapter.
Tonelli's theory is based on the definition of area proponed by Lebotigue.
As regards the modem work on area of surfaces based on other definitions, wo
should mention: W. H. Young [4], J. C. Burkill [3], S. Banaeh [f>|, A. Kol-
mogoroff [3] and J. Schauder []].
T. Eado [1, pp. 154—169; 2] lias developed further lihe
methods of Tonelli by means of older ideas due to do Georae
and with the help of certain functional introduced by the latter.
The principal result of Ead6 {vide Theorem 7.3), applications of which
will be discussed below, enables us to define the area of a surface
as the limit of certain simple expressions, whereas the Lebesgue
definition only enables us to obtain it as a lower limit. An-
Another expression is due to L. 0. Young [vide below § 8) and
constitutes a direct generalization of the classical formula for the
area of a surface.
Except where the contrary is expressly stated, the reasoning
of this chapter will be formulated for functions of two real variables.
The extension to spaces of any number of dimensions offers no
difficulty.
§ 2. Area of a surface. By a continuous surface on a plane
interval Io, we shall mean any equation of the form g=T?{x,y),
where F is a continuous function on Jo.
A continuous surface z=P{x,y) on an interval Io is termed
polyhedron if there exists a decomposition of Io into a finite
number of non-overlapping triangles Tlt T2!..., Tn such that the
function P is linear on each of these triangles, i. e. such that
P{x, y)=atx+biy+ei for (as, y) e Tt, where i—1,2,..., n arid «,-, bh Oi
are constant coefficients. We shall call, respectively, faces and ver-
vertices of the polyhedron z=P{x) y), the parts and the points of the
R3J
The Burkill integral.
165
graph (cf. Chap. Ill, § 10) of the function P, which correspond to
the triangles .TV, and to the vertices of the T,. The sum of the areas
of the faces in the sense of elementary Geometry, i. e. the number
SlTiHal+bj+l)^ff[{9I>j9xf+(9PISyf+l^dxdy, will be called
'' ' >„
elementary area of the polyhedron z=P(x, y) on Io and denoted
by S0(P; Io).
Given any continuous surface s=F («,;//) on an interval 10, we
shall term its area an. Io, and denote by SA";IO), the lower limit of
the elementary areas of polyhedra tending uniformly to this sur-
surface, i.e. the lower bound of all the numbers s for each of which
there exists, given any e>0, a polyhedron s=P(x, ;//) on Io such
that 1° \P(x,y)—F(w,ij)\<s at every point (;«,(/) elo and
2«S0(P;I0)<*.
We might verify hero that for polyhedra the elementary area agrees with
the area according to the general definition just given. As, however, this is an
easy consequence of this theoreuiH given further ou (vide p. 181), ft special proof
is unnecessary at thin point. It Hhould be remarked tliat, in accordance with the
definition adopted, the urea, of a surface may be either finite or infinite.
The following theorem in an immediate consequence of the
definition.
B.1) Theorem. For any sequence of continuous functions {F,,}
which con-verges uniformly on an interval Lo to a function F, we have
§ 3. The Burkill integral. Instead of treating the theory
of area of surfaces by itself, it is more convenient to associate it
with certain differential properties of functions of an interval. How-
However, the functions of an interval occurring in the theory of area
are not in general additive, and in consequence we shall have
to complete in some minor points the theory of functions of an
interval, developed in the two preceding chapters.
We shall begin with some subsidiary definitions. To simplify
the wording we shall understand in the sequel by subdivision of
a figure J?o any finite system of non-overlapping intervals Iv I2) ••¦, In
such that B0—?li,. Given any function of an interval U and given
a finite system of intervals 3=(I*), we shall write, for brevity, 17C)
in place of ?U(Ji,)- 'In particular therefore, LC) will denote the
sum of thc aroas of the intervals belonging to the system 3-

166
CHAPTER V. Area of a surface s=[i'{x,y).
We call upper and lower integral in the sense of Burkill of
a function of an interval 17A) over a figure Hw and we denote by
~~ , /¦ „ ,. , ,, .. . -. ,, , ...
of Burkill of
terval 17A) over a figure Kw and we denote by
I'U and I'U respectively, the upper and the lower limit of
the numbers !7C) for arbitrary subdivisions 3 of JK0, whose eharac-
teristic numbers 4C) tend to zero (cf. Chap. II, p. 40). When
these integrals are equal, their common value is called the Burlcill
definite integral (or simply the integral) of the function U over Ko
and is denoted by JZJ. If this integral exists and is finite, the fune-
tion V is said to be integrable on Ro (in the sense of Burlcill). If the
function U is integrable on every figure li (in the whole plane or
in a figure i?0) its integral / U considered as a function of the figure
M is called indefinite integral of V (in the whole plane or on #„).
C.1) Theorem 1° If U is a function of an interval and li^ R2
are non-overlapping figures, we have
and
C.2)
provided that both integrals of U over i?!+jBa are finite.
2° Any fmction of an interval U which is integrable on a
figure Bo, is equally so on every figure BQBa and its indefinite
integral on 2?0 is an additive function of a figure.
Proof. Part 1° of the theorem is a direct consequence of the
definition of the Burkill integrals, and part 2° follows at once from
the formulae C.2) when we subtract the second of these formulae
from the first.
If U is a function of an interval on a figure B, we shall call
variation of U on B at a set D the upper limit of | J7C)| as 4C) -*0,
¦where 3 denotes any finite system of non-overlapping intervals
contained in R and possessing common points with D. The follow-
following analogue of Theorem 4.1, Chap. Ill, may be noted.
C.3) Theorem. Given on a figure Ba a function of an interval U
such that j\U\<+oa, there can be at -most an enumerable infinity of
straight lines D, which are parallel to the coordinate axes and at
which the variation of V on Ro is not zero.
t§3.1
The Burkill intogriil.
167
In fact, the number of straight lines which are parallel to the
axis of a? or of y, and at which the variation of U on Ro exceeds
a positive number e, cannot; be greater than 2e~1- I\U\
oo.
C.4) Lemma. Given a function of an interval U integrable on
a figure Bo, there exists, for each e > 0, an rj>0 suck that for every
system 3 = Ui> ^s>— > I/>) °f non-overlapping intervals situated in Rm
the inequality A C) < V implies the inequality
C.5)
Proof. Let ^>0 be a number such that, for every subdivision
% of Ii0, A{Z)<r) implies \U{Z)--[[J\ O/2, and let 3={Ii, I*,..., I,}
be any finite Hys1;em of non-overlapping intervals situated in Iim
l>
such that A{%)<n. Lot, K^lt^ ?lk- By Theorem 3.1, the function
V is integrable on I\\. It follows that there exists a subdivision 3i
of R-l such that
C.6)
)-[U\<ej2.
i
Now 3 + 3i clearly constitutes a subdivision of Rq such that
<»7. We therefore have |?7C+3i)—/'f|<e/2, and we need
H,,
only subtract the second of the relations C.6) from it to obtain C.5).
If jK0 is a fixed figure, then to any ?j>0 there corresponds
a positive integer p such that every interval IC-^o m&J ^e su^"
divided in p subintervals of diameter less than v\. Hence applying
Lemma 3.4, we obtain at once the following
C.7) Theorem. If a function of an interval 17A) which is in-
integrable on a figure jB0, is continuous, then the same is true of its in-
indefinite integral B(E)= I U.
h
C.8) Theorem. If If is a function of an interval which is in-
integrable on a figure R,o and if B is its indefinite integral, then
B{x,y) = U{x,y) and B{x,y) = U{x,y) at almost all points {x,y)elio.
In particular therefore, if one of the functions U and B is almost
everywhere derivable in li0, the same is true of the other and the deriv-
derivatives of U qnd of B are almost everywhere equal.

168
CHAPTEE V. Area of a surface s=F(x,y).
Proof. Suppose that the set of the points {x, y) at which
V{x,y)>B{x,y) has positive measure. We could then determine
a set .EC^o of positive outer measure and a number a> 0, such
that V{x,y)—B(x,y)>a at each point (x,y) of JS. Therefore, on
account of Vitali's Covering Theorem (Chap. IV, Theorem 3.1),
¦we could determine in Rm for any v\ > 0, a finite system of non-
overlapping intervals 3=(Ja}*=i,2,...,„ such that zlC)<??, L C) >|i?|/2,
and U{Ik)—B(Ik)>a-\Ik\ for ft=l, 2,..., n. Now it follows from
the last two relations that C/C)—BC)>a-|J5|/2, which contradicts
Lemma 3.4. Hence, U(x,y)^B(x,y) almost everywhere in i?0.
In the same way we prove that, the opposite inequality holds also
almost everywhere in Bo, and this completes the proof.
C.9) Theorem. Suppose that U is a eontinuuun fnndion of an
interval on a figure Bo and that (i) /|J7|< + °° and (ii) U[I)^OF)
for every interval IC^o aw* 6myU subdivision 6 0/ /. Then the. func-
function V is integrable on Ro.
Proof. Given a number e>0, let ? = {t/,-)M,;> „ be a Kub-
division of Ba such, that
C.10)
V(X)>[TJ — e.
Let us denote by D,, D2,..., D,. the sides of the intervals (%) which
do not belong to the boundary of i?0. By Theorem 3.3 it may
be assumed, in view of the continuity of the function U and of
condition (ii), that the variation of U on ?0 vanishes at each aide I),.
It follows that there exists an j? > 0 such that, given any finite
system 6 of non-overlapping intervals situated in 22O and'having
points in common with the sides JD,, the inequality 4F)<»; im-
implies |tf F)| <e. We can clearly assume that v does not exceed
the length of any side of the intervals {%).
This being so, consider an arbitrary subdivision 3={Iu I2,..., I,,}
of 50 such that dC)<?7. By numbering the intervals of 3 suitably,
we may evidently suppose that !„!„... ,1, are those having
pouts rn common with the sides JD,, while the remaining intervals
of 0 (if any) have none. Finally, let us agree to write U{ JtQh) = 0
when J.Qi^o. Then \%V{I>)\<, and |^ I^(J,f.)J*)|<*, eo
t§4]
Bounded variation for functions of two variables.
169
that, by C.10) and by condition (ii) of the theorem, we have
'/ l> q __
~ " ' e. It
follows that f'u<.fu+3e, and so, that fu— fu.
111 connection with this §, vide J. 0. Bin-kill [2; 3; 4], E. C. Young [2]
mid F. Riesz [fi; 7].
§ 4. Bounded variation and absolute continuity for
functions of two variables. Given a function F(x, y) contin-
continuous on an interval I = [«i, b^, a2, &a], let us denote for any value x
subject to fi^O^^, by W^F; x; «a, f»2) the absolute variation
of the function F(x, y) with respect, to the variable y on the interval
[«a> &a!) mK^ '-01' iUiy value,?/ nubject to «2^2/<&2, by W2{F;y; a1}bx)
that of the function /''(*',?/) with respect to x on [ax, 6J. Denoting
by Jj and J2 respectively the linear intervals [<h>K] aud [«a, &2]
we shall also write W^i'1; x; Ja) for W^i'1;,*; «2)^a) and W2(J';y;t71)
for W2(.F; ;(/; au bx).
By eoniiinxiity of the function F, the non-negative expressions
W^jF; .b; J2) and. W2(F; y; Jt) are, as is easily seen, lower semi-
continuous functions of the variables x and y respectively. When
the. integrals i~S\\(F)x;J%)(lx and j^\(F;y; J^dy are both finite,
the function .F is said, to be of bounded variation on 1 in the Tonelli
sense. It follows at once that any function of bounded vari-
variation of two variables x, y is of bounded variation with respect
to *¦ for almost every value of y and with respect to y for almost
every value of *.
A continuous function F(x,y) will he termed absolutely con-
continuous on an interval I=[ax, 6X; ait 62] in the Tonelli sense, ii
it is of bounded variation on I and moreover, absolutely continuous
with respect to x for almost every value of y in [a2,62], and absolute-
absolutely continuous with respect to y for almost every value of x in [a1} \].
We say that a function F(x, y) fulfils the Zipschitz condition on I,
if there exists a finite constant V such that \F(x',y')-F(x",y")\<.
<JV.Ca!'—,'c"|-|- y'—!l"\) im <'very pair of points (x',y') and
(x",y") of I.
Any function which fulfils the Lipschitz condition on an
interval / is evid(uitly absolutely continuous on I. In particular

170
CHAPTER V. Area of a surface z=F(x,y),
polyhedra and also functions of two variables with continuous
partial derivatives, are always absolutely continuous functions.
A function F which is continuous and of bounded variation
[absolutely continuous, or subject to the Lipschitz condition] on an
interval Ia=[avhl a2,h] ean easily be continued, even so as to
be periodic, over the whole plane in such a manner as to remain
continuous and of bounded variation [absolutely continuous, or
subject to the Lipschitz condition] on every interval. In fact, de-
denoting by Ii one of the intervals congruent to Io with a, common
side parallel to the axis of x, let us continue the function F from
the interval Io on to the interval Ij by symmetry relative to the
common side of these intervals. Let us further continue similarly
the function F from the interval I<f\-Ix on to an interval J2 con-
congruent to Ia+Ii which has with the latter a common side, parallel
to the axis of y. The function F is then defined on the interval
la-\-lxJrli whose sides are respectively of lengths 2-(fr1—ax) and
2-(&2—«2). Writing u=2-(b1~a1) and v=2-(b2—a2), and continuing
the function F from the interval I^I^I^ on to the rest of the
plane by the periodicity condition F(x+u, y)=F(x, y-\-v)—F(?, y),
we see easily that the continuation obtained for the function F
has the properties required.
Besides the definition of Tonelli several other definitions have been given
of conditions under -which a function of two variables is said to be of bounded
variation. For a discussion of these definitions see C. R. Ada in» and
J. A. Clarkson [1;2]. Throughout this chapter, use is niado of Tonelli's
concept only.
We shall subsequently make use of the following theorem
concerning the partial derivates of any continuous function:
D.1) Theorem. Given a continuous function F(x,y), its -partial
Dini derivates, F?,F7,Ft,F~ and F^,F^,F^,F~, are func-
functions measurable (93).
Proof. It will suffice to prove this for any one of these deriv-
derivates, say Ft.
Given an arbitrary real number a, consider the set
[
and denote by En the set of all the points (as, y) such that for every h
the inequality 0<*^i/n implies [F(x+h,y)~F{x, y)]jh^a—1/n.
The expressions of de Ueiicze.
171
We find that E=?EI, and, since by continuity of the function F
I!
each of the sets F,, in closed, E is a set ($„), so that the derivate f?
is a function measurable (93).
Theorem 4.1 may be compared with Theorems 4.2 and 4.3 of Chap. IV
concerning moasuralrility of the derivates of functions of one real variable. Never-
Nevertheless it is to be remarked that contrary to what occurs for functions of one
variable, the partial Dini derivates of a function measurable (¦?) need not in
general be measurable ($5), although they are still measurable B) (the proof of
this requires, however, the tlieory of analytic sets; vide F. Hausdorff [II, p. 274],
M. Neubauer [1] and A. E. Currier [1]). On the other hand, a function of two
variables may be measurable (i!) without its partial Dini derivates being so.
§ 5. The expressions of de Geocze. We shall make
correspond to a,uy function F(x, y) which is continuous on an interval
!=[«!, b±; «2. '>g]j the following expressions introduced by Z. de
Geocze [1] into the theory of areas of surfaces:
While studying the fundamental properties of these expressions,
we shall often find the following two inequalities useful:
(i
(=1
for any three sequences {xi), {yi} and fa) of real numbers;
E.2)
is e is b
for my measurable set B in a space B,» and any three non-negative
functions x(t), y(t) and z(t), measurable on E.
The inequality E.1) is easily deduced by induction from the
case n = 2 which can be verified directly. The inequality E.2), in
the special case in, which the functions x(t), y{t), z{t) are simple-,
is an obvious consequence of E.1); and we pass at once to the
general case with the help of Theorems 7.4 and 12.6 of Chapter I.

172
CHAPTER V. Area of a surface ts=F(x,y).
E.3) Theorem. The expressions of de Geocssc G1{I) = GX{F;II
#2(I) = G2(JT;I) and G{I)=O{F;I), associated with a continuous
function F (%, y), are continuous functions of the interval 1 and their
integrals over any interval exist (finite or infinite); these integrals
over any interval !„=[«,, bx; ait Js] fulfil the following relations:
E.4)
E.6)
= / WX(JP; x; a2, b2) dx and
and
toliare i=l,2.
Proof. Given an arbitrary e>0, let ?;<e be a positive number
such that, for any pair of the points (a?,?/,) and {x,y.,) in I().
E.7)
'y*—!li\ <V implies \F {no, y^—F(x, 7/x)| <e.
Let If denote the upper bound of F(oo, y) on Io and consider
in Io an interval !=[%, §x; a2, /Sa] such that \I\<if. We then
have either /5X—ax<»; or /92—a2<7j. In the former case, we find
G(IX(Pi—a1)-'2M<2Mrj<:2Ms, and in the latter we derive
7) fffJX^)<F
from E.7),
^—a1)-e<F1—ax) • e, so that in both cases
jX^1)<Ax) , so that in both cases
(?1(IXBJi+&1—%)•«. The function G-^I) is therefore continuous.
The same is of course true of (?2(I) and the continuity of these
two functions at once implies that of G(I).
This being so, we shall show that the functions Gx and (?2 are
integrable and, at the same time, we shall deduce the formulae E.4).
Let {3n} be a sequence of subdivisions of Io such that
limzlCJt)=0 and lim 0C )= [ Gt; and denote, for any positive
integer n and any ?e\av 6J, by Fn(f) the sum of the absolute
increments of the function F{?, y) on the linear intervals cut off
on the line x=S by the rectangles of the subdivision 3,,. We then
have, on the one hand,
E.8)
for »=1,2,...,
and on the other hand, on account of continuity of the. function F,
lim7^)=W1(J?I;f;a2, 6a) for any |e[a,, 6,]. Therefore, in virtue of
n
The expressions of de G-eocize. 173
Fatou's Lemma (Chap. I, Theorem 12.10) and by E.8), we obtain
'?. But since G1{^)^jW1{F;t,ai,bi)dS for
every subdivision 3 of Io, we have also
; «
2,
Therefore the function G1 has a unique integral over Jo and this
integral fulfils the first of the relations E.4). The existence of the
integral j G% and the validity of the second of these relations are
i»
deduced by symmetry.
Let us pass on now to the function G. We first remark that
the integral j G clearly exists in the case in which one at least of
the integrals j Gx and I Gs in infinite, and is then also infinite
on account of the relations
E.9) (?!(!)<(?(I) and (?a(J)^(?(J) for any internal I.
In the remaining case, the two integrals in question being finite,
the evident inequality G{I)^G1{I) + G^{I) + \I\ yields
4 /„ /„
and on the other hand, for every subdivision 3 of any interval I
the equally obvious relations
@.11) ff1(/)<(?1C) and 02(J)<e2C)
lead, in view of the inequality E.1), to
E.12) 0A)^0C).
Fow, continuity of the function (? being already established,
the formulae E.10) and E.12) imply, by Theorem 3.9, that this
function has over Io a unique integral.
To complete the proof we need only remark that the formulae
E.11) and E.12) imply at once the formulae E.5) and finally that
formula E.0) follows directly from E.9) and E.10).
As a corollary of Theorem 5.3, and more particularly as a con-
consequence of the formulae E.4) and E.6), we have:

174
CHAPTER V. Area of a surface z = F[x.y).
E.13) Theorem. In order thatthe function of an interval G{I)=G(F;I),
associated with a continuous function F(x, y), be integrable on an in-
interval Io {i. e. in order that Jg<-\-oo), it is necessary and sufficient
u
that the function F(x, y) be of bounded variation on Io.
§6. Integrals of the expressions of de Geocze. Given
a continuous function F (x,y), we shall denote, for any interval Io,
by H^JFjIo), Ha(F;I0) and R{F; Io) respectively, the integrals of
the functions of an interval G1{I)=G1{F; I), GZ(I) = Q2(F; I) and
G{I) = Q(F; I) over the interval IO. All these integrals exist on
account of Theorem 5.3 and their importance in the theory of area
of surfaces is due to the fact that, as will be shown in the next §,
the number H{F; Io) coincides with the area of the surface z=F{x, y)
on Io.
F.1) Theorem. For any function F(x, y) which is continuous and
of hounded variatio-n, the expressions i?1(I) = H1(i?I;I), E2(I) = 'Hi(F; I)
and E(I)=H(F;I) are additive, continuous, and non-negative func-
functions of the interval I, and we have at almost all points (x,y) of the
plane
E:(x,y)=\F'B(x,y)\, E'^y^F'Mvl
E'(x, y) = {[Fx{x,y)f + [F'B(x, y)? + l}\
Proof. Additivity and continuity of the functions in question
follow at once from Theorems 3.1 and 3.7 on account of Theorem 5.3.
We have therefore only to establish the relations F.2). B"ow, for
any interval !=[«!, bx; a2, 6J we have according to Theorem 5.3
and Theorem 7.4 of Chap. IV, the following relation (in which the
transformation is effected in accordance with Fubini's Theorem 8.1,
Chap. Ill, rendered applicable to the partial derivates of the func-
function F{x, y) by Theorem 4.1):
x;
a2, 68)dx^J [[{F'^x, y)\ dy] dx= f f\F'0(x, y)\ dxdy;
whence
F.3) E'ix, i/)> \F'g(x, y)\ for almost every point (x,y).
Let us now denote by {Jn) the sequence of the linear intervals
with rational extremities. In view of Theorem 7.4, Chap. IV, we
[§(i| Integrals of the oxpresHionu of de Ge.oesse. 175
have for »=1, 2, ... and for every linear interval J,
'WjiF;x;J»)dx^M^JxJ,,M? / JE\(x,y)dxdy = f [\'e\{x, y)dy]dx,
J J,,
and consequently, for each positive integer n, the inequality
W^F; x; Jn)^ I H\(x, y)dy holds at every point x, except at most
those of a set E,, of linear measure zero. Therefore, writing E=ZF»,
n
we obtain the inequality W^-F; «;</);> fs[(x, y) dy, whenever J has
rational extremities and x lies outside the set E of linear measure
zero. If we now regard the two sides of this inequality, for a given
value of x outside the set E, as functions of the linear interval J,
we obtain by derivation with respect to this interval (on account
of Theorem 7.9, Chap. IV) for almost all y, the inequality
Therefore, since the derivatives E[(x,y) and JP|;(a?,i/) are meas-
measurable (cf. Theorem 4.1), it follows from Fubini's theorem (in the
form (8.6), Chap. Ill) that the set of the points {x, y) at which the
relation F.4) is not fulfilled, is of plane measure zero. By F.3) we
therefore have almost everywhere the first of the relations F.2).
The proof of the second relation now follows by symmetry,
and that of the third from the remark that if we write G1(I)=G1(F; I),
G2{I)=G2{F; I) and G(I) = G(F; I), we have by Theorem 3.8,
TT'f/v, ,,\ nit™ n.\_ /r/V/rv. ,,\n2 i r/>'/•« ,i<\n2 i i \-_
= ![jri(«,
at almost every point (x, y) of the plane. This completes the proof.
F.5) Theorem. In order that the function of an interval H{I)=IL(F;I),
corresponding to a continuous function F(x, y) of bounded variation
on an interval I0=[a1,b1;ai,b2], be absolutety continuous on this
interval, it is necessary and sufficient that the function F{x,y) itself
be absolutely continuous; and when this is the case, we have
F.6) H(I0) = [[{[F'Ax, y)f + [F'B(x, y)f + if dxdy.

176
CHAPTEB V. Area of a, surface »=V(x,y).
Proof. By Theorem 5.3, absolute continuity of the function
S(I) is equivalent to absolute continuity of the functions H^I)
and E%{I) together.
Therefore if the function E is absolutely continuous on Io,
we have, by Theorem 6.1, for any interval i|= [«!, g; a2, b2], whore
!, the relation
^F; x; a2, b2) Ab=H'1(Ij)=/*j\F',,{x, y)\ dx dy=j [j \F'y{x, y)\dy] dx,
and, taking the derivative with respect to |, we obtain for almost
every value of x,
F.7) W^F; x; a2, &2) = /V>, y)\ dy.
Now, for any given value of x (for which F{x, rj) is of bounded
V
variation in rj) the difference W^J1; x; a2, rj)— f\F',i{x, y)\ dy in a non-
"j
negative and non-decreasing function of the variable rj (cf. Theo-
Theorem 7.4, Chap. IV). It therefore follows from F.7) that we have
for ahnost every value of x, and for any r\ e [a2, Z>2],
r
i. e. that the function W^F; x; a2, rj), and consequently also F{x, rj),
is absolutely continuous with respect to rj on [a2, 62] for almost
every value of x. By the symmetry of the variables, we conclude
also that the function F(?,y) is at the same time absolutely contin-
continuous with respect to f on [a1? b{\ for almost every value of y.
The function F, which is by hypothesis of bounded variation on Io, is
therefore absolutely continuous in the Tonelli sense on this interval.
Conversely, if the function F is absolutely continuous on Jo,
we have by Theorems 7.8 and 7.9, Ohap. IV, for every suMnterval
!=[>!, &} a2, j82] of Io, the relations:
a2, /9a)
e= f f\Fy(x, y)\dxdy,
'x(x, y)\dwdy,
so that the two functions of an interval S1 and /f2, and therefore
also B", are absolutely continuous.
[§7]
Eado's Theorem.
177
Finally, since the function E is absolutely continuous, the for-
formula F.6) is a direct consequence of the third of the relations F.2),
the latter being valid almost everywhere on account of Theorem 6.1.
Up to the present we have regarded the expression H(iin; J)
as a function of an interval I. If we treat this expression as a func-
functional depending on the function F, we obtain the following theorem,
whose geometrical interpretation will appear in § 8 when the theorem
appears to be a generalization of Theorem 2.1.
F.8) Theorem. Given any sequence of continuous functions {F,}f
which converges to a continuous function F, we have for every interval I
F.9)
lim inf K(Fn; I) > H(F; I).
Proof. Denoting by 3P the subdivision of I into p2 equal
intervals, similar to I, we have by Theorem 5.3 for any pair of in-
integers p and n, ~K(F,,; I)^G(Fri; %), and by Fatou'sLemma (Chap. I,
Theorem 12.10), for every integer p, lim inf G(Fj,; %)^G(F; %).
We therefore have lim inf H(.Fn; I)^G{F; %), and this leads to
F.9) when p—>oo.
§ 7. Rado's Theorem. Before passing to the proof of the result
of Eado, according to which the area of any surface z=F(x, y)
on an interval I is equal to H(F;I), we shall prove the following
G.1) Theorem. If a continuous function F(x,y) has on an
interval !<)—[%> ^ii au ^al continuous partial derivatives, there exists
a sequence of polyhedra {z=Pn(x, y)} inscribed inthe surface z=F(x,y),
such that the sequence converges uniformly to this surface and such that
G.2) lim S0(P,,; Io)=
'x(x, y)?+[F'g(x,
Proof. Let 5n—{In,i, In,i, ..., InA denote the subdivision of Io
into w2 equal intervals similar to Io, and (»„,,-, yn,t), where i=l, 2,..., n2,
the lower left-hand vertex of In,t- Let us divide any interval In,i
into two right-angled triangles T'a,i and F^t by a diagonal, in such
a way that, the vertex (a?n>i, yn,i) is that of the right angle of T'nti.
Consider for any n the polyhedron z=Pn{x, y) inscribed in the surface
z=F(x,y) and corresponding to the net formed on Jo by the 2«a
triangles T'nJ and Tn',; where i=l, 2,..., n%.
S. Saks. Thoory of the Intejral. 1"

178
CHAPTER V. Area of a surface z=F(x,y).
For brevity let hn=(b1—a1)ln and fe=Fa—aa)/n; and let
fin denote the upper hound of the differences \F'x(x", y")~F'x{x', y')\
and \FB(a>",y")—F'g(x',y')\ for all points {x',y') and (x",y") of Jo
such that \x"—x'\^Jht and |y"—2/'|<fo.-
Now if s'n,i and si,',,- denote respectively the elementary areas
of the faces of the polyhedron z=Pn{x, y) which correspond to the
triangles T'n,i and T',',,i, we notice at once that the areas of the pro-
projections of the former of these faces on the planes xz and yz are
respectively equal to
\h'hk\F'{nj, y'n,i)\
n,b yn,,)\
and
iTen-\F{Xn,t+K yn,<)—F(xn,t, ynii)\=^
where .-Kn.,-<av;<;%,<+7!« and yn,i^y'n,i
We therefore have s'n,i^[[^'M,b y»,i)f+[Ftsl{x,uh y'n
and so, by the inequality E.1), p. 171,
}^ ¦ \I,,,
Since the partial derivatives Fx and F'u are by hypothesis continuous,
it follows by making w->oo that
lim
and the same limit is clearly obtained for the sum of the s'!,,i.
By addition, together with an appeal to Theorem 6.5, we now
derive the formula G.2) and this complete;-; the proof.
In what follows we shall apply the method of mean value
integrals. Given in the plane a summable function F(x,y), the
l/n l,-'n
sequence of functions Fn{x,y)=n2J(F{x+u,y+v)d%dv where
o 6
n=l, 2,..., will be called sequence of mean value integrals of the
function F(x, y). It is clear that if the function F is continuous,
(i) the sequence of its mean value integrals {Fu{x,y)\ converges to F(x, y)
at every ¦point (x, y) of the plane, and uniformly on any interval, and
(ii) the partial derivatives 5Fnj3'x and 2Fnj5y exist everywhere and are
[§7]
Radd's Theorem.
179
continuous. In fact, at any point (x, y) a direct calculation gives
9F,,(x, y)j3x=n2 I [j? (#+!/», y + v)—F{x, y+v)]dv
and
SFn{x,
l/n
, yfltin.
It was T. Rado [21 who first applied in the theory of area of surfaces the
method of mean value integrals. The role of these mean values is due to the fact
that in the case in which the given function JF is continuous, the sequence of
areas of the surfaces z=Fn(x,y) on any interval tends to the area of the surface
«=J?(o!, y) (vide, below, Theorem 7.3).
In tlie definition given above, the functions J?n are defined at each point
(x, y) as "mean valnes" of the function F over squares of which (x, y) is a vertex;
it goes without saying that we could also make use of mean values taken over
squares, or circles, having (x, y) as their centres. These mean values over circles
are used for instance in potential theory (cf. F. Riesz [4] and G. C. Evans [1]).
G.3) Badd's Theorem. If F{x,y) is a continuous .function and
{Fn{x, y)} is the sequence of mean value integrals of F(x, y), then
G.4) H(_F; I0) = 8{F; I0)=lim S(.Fn; Io)
n
for every interval Io.
Proof. Let {s=Pn{x, y)) be a sequence of polyhedra converging
uniformly to the surface z—F(x, y), such that
II
Since the functions Pn(x, y) are absolutely continuous, it follows
from Theorem 6.5 (cf. also §2, p. 165) that S0(Pn;I0)=H(Pn; Io)
for every n. Consequently, since the sequences of functions {Fn}
and {Pn} converge uniformly to the function F, it follows by using
successively Theorem 2.1, the formula G.5) and Theorem 6.8, that
G.6) lim inf S(J?"n; Io) > S (F; I0)=lim
^ H(J?;
Now if the function F is not of bounded variation on Io, it
follows from Theorem 5.13 that K(F; 10) = + oo and consequently
the formula G.4) follows at once from G.6). We may therefore as-
assume that the function 1" is of bounded variation on Jo, and further
(cf. § 4, p. 170) that F is continuous and of bounded variation on
each interval of the plane.
12*

180
CHAPTER V. Area of a surface z=F(«,y).
Let us agree to denote, for any set Flva the plane, by -B('!lU) the
parallel translation, of E by the vector {u, v) (cf. Chap. Ill, § 11);
similarly, for a family of sets (? in the plane, (?("'"' mil denote the
family of all the sets obtained from sets (?) by subjecting them
to this translation. For any subinterval I=[«i, h; a2, &2] we then
obtain
C^(J?n; I)=f\Fn(a>, bt)—Fn{x,
6, 1/n 1/n
'f\ f f\F(x-}-u, b%+v)—F{x+u, a^-\-v)\dudvjdx=
1,'nl/n
f [
o o
and a similar formula for G2. Hence by the inequality E.2), p. 171,
l.n 1/n
0 0
i 1/n
=n2
0 0
Denoting by 3P the subdivision of Io into p2 equal intervs
similar to Jo, we obtain therefore, for every p,
1/n 1/n
and since by Lemma 3.4, G(-F;3ji"'o)) tends to H(J"; Jo"') as p-
uniformly in m and v, we obtain in the limit
1/n Vn
G.7)
H(Fn; Io)<m2/ JH(J; I?'u)
o o
MnaUy the areas of the figures I^'v)QI0 and
^'v)
tend
to 0 with u and ¦» and each of these figures is a sum of two intervals.
Hence since the expression H(F;I) is by Theorem 6.1 a contin-
continuous function of the interval I, we have lim H(_F;Io")) = H.(.F;Io).
On the other hand, since the functions Fn(x, y) have continuous
partial derivatives, we have, by Theorem 7.1, S(J?n,-IoXH(JP,1;Io)
for each w. Therefore matingw^oo iu G.7)jwefind lim sup B{Fn;I0)^.
n
<H(F',I0), which in conjunction with G.6) gives the required
relation G.4).
[§8]
Tonelli's Theorem.
181
§ 8. Tonelli's Theorem. The theorem of Eado just established,
enables us to replace in all the theorems of this chapter the ex-
expression H(JF;I) by the surface area Q(F;I).
Thus for instance, Theorem 6.5 (formula F.6)) expresses the
fact that the elementary area of a polyhedron coincides with its area
according to the general definition of area of a surface.
Theorem 6.8 contains a generalization of Theorem 2.1; it enables
us to replace in its statement uniform convergence by ordinary
convergence: we thus obtain a theorem similar to Lemma of Fatou
(Chap. I, Theorem 12.10). It follows that the uniform convergence
of the inscribed polygons, required in the definition of area, may
be replaced by the ordinary convergence, so that the area of a contin-
continuous surface z—F (x, y) is the lower limit of the areas of polyhedra
tending to this surface. Further, by Theorem 7.1, if a function F(x, y)
has continuous partial derivatives, there exists a sequence of polyhedra
inscribed in the surface z—F(x, y), tending uniformly to the latter
and having areas which converge to the area of this surface. (For further
generalizations vide S. Kempisty [1]. Cf. also on this subject
H. Eademacher [3], W. H. Young [5], M. Fre'chet [2] and
T. Eado [5].) Finally, we obtain the following theorem, which
sums up the most essential considerations of this chapter:
(8.1) Tonelll's Theorem, a) In order that a continuous surface
z=F{x,y) have a finite area on an interval IQ, it is necessary and
sufficient that the function F{x, y) be of bounded variation on Io.
b) When this is the case, we have
3(F;I0)
Sx)
w)+l^dxdy;
the expression 8(I)=S{F; I) is then an additive continuous function
of the interval ICZI0 and we have for almost every point (x,y)elo
,Sxj'\9y
c) In order that we should have
(8.2
it is necessary and sufficient that the function F{x,y) be absolutely
continuous on Io; and in order that this be the case it is necessary and
sufficient that the area S(F;I) be an absolutely continuous function
of the interval I(T-?o-

182
CHAPTEE V. Area of a surface s--=F(x,y).
[§8]
ToneUi's Theorem.
183
Proof. The assertion a) follows directly from Theorem 5.13;
b) and c) follows from Theorems 6.1 and 6.5 on account of Theo-
Theorem 7.4, Chap. IV. ¦
With regard to Theorem 8.1 vide L. Tonelli [5; 6; 7]. The necessity of con-
ditioa a) was "established a little earlier by G. Lampariello [1].
According to Tonelli's theorem, the relation of equality (8.2)
can hold for a continuous surface s=F[x, y) only in the case in
which the function F is absolutely continuous. Nevertheless, as
proved by L. C. Young, this relation will remain valid for arbitrary
continuous surfaces, as soon as we replace on the right-hand side
the partial derivatives by ratios of finite differences and transpose
the passage to the limit outside the integral sign. In fact:
(8.3) Theorem. For any continuous surface z=F{x,y) and any
interval Io we hceoe
and in order that the function F be of bounded variation on Io, it is
necessary and sufficient that
(8.5)
F(x,y)\dxdy
Proof. Let {Fn} be the sequence of mean value integrals
(cf. § 7, p. 178) of the functionF. Denote, for brevity, by B (x, y; a, /?)
the expression under the integral sign on the right-hand side of (8.4),
and by Bn(x, y; a, /?), for each positive integer n, the expression
obtained from R(x, y; a, /?) by replacing F by Fn. Finally let us
write for n=l, 2,...
Bn(x, y)=lim Bn(x, y; a, P)={[SFn(x, y)l9xf+[9Fn(x, ff\
In order to establish the identity (8.4), it evidently suffices
to show that
(8.6)
and
(8.7)
S(.F;I0Xliminf f fB(x,y; a
«,|3-W> J j
im sup f fB(x,y; a,P)dxdy.
«/S-H) r
f
For this purpose, let I be any interval in the interior of Io.
By means of the inequality E.2), p. 171, we easily find that
In In
(8.8)
Bn{x,y; a,fi)^n2 f fR(x+u,y+v; a,P)dudv.
6 o
Now let n be a positive integer, sufficiently large in order that
(x,y)el, \u\<l/n and
should imply {x+u, y+v) eIo.
We then have jrfB{x+u,y+v; a, P)dxdy^j JR{x,y;a,P)dxdy and
' i , , ' h
consequently, by (8.8), / jBn{x,y; a,P)i!edy<. f [B(x,y; a, §)dosdy.
i ' f
Making a->0 and /S-»0, we obtain in the limit
<;liminf / f B(x,y; a,C)dxdy. This relation being thus established
«,ii-M> J ;'
for each interval I(ZIS, we may replace, on its left-hand side, I
by Io, and making still n^oo we obtain the relation (8.6).
In order to prove the relation (8.7), let) us first observe
that the latter is obvious in the case in which S(JI;I0) = + oo.
We may therefore assume that the function F(x,y) is of bounded
variation on Io and moreover (cf. § i, p. 170) of bounded variation
on every interval in the plane and periodic with respect to each
variable. We can therefore determine an interval Joz=\-ai>Wi a%i ^al
containing Io in its interior, such that its sides bt—ax and b%—«2
are the periods of F{x,y) with respect to x and y respectively.
This being so, we find easily, on account of the inequality E.1),
p. 171, that, for any pair of positive integers n and h,
B,,(x,y; a,p)<lj-
By integrating the two sides of this inequality over Jo, and taking
account of the periodicity of the function F, we obtain
/ / Bn(x,y; a,
j
and hence, passing to the limit, making first
we find by Eado's Theorem 7.3,
j fRn{x, y; a/Te, pjty
oo, and then «->oo,

184 CHAPTER V. Area of a surface z=F(x, y).
f [B(x, y; a, p) dxdy ^7dm f J'Bn{a:, y)dxdy=lim &(F,,;J0)=&{F; Jo)
¦h
and so
(8.9)
lim sup / /.
, by the result already established in the inequality (8.6),
¦we have lim inf f fs{x,y; a, j8) dxdy > S {F; I) for every interval I.
a, f?-x> •> y
It follows at once that (8.9) remains valid when we replace the
interval Jo by any subinterval of Jw and in particular by the
interval Io. We thus obtain the relation (8.7).
Finally let us remark that on account of the relation (8.4),
in order that the area of the surface z=F(x, y) on Io be finite,
it is necessary and sufficient that
lim sup— / / \F{x+a, y)—F(co, y)\dxdy
«^o a J J
u
<+oo
and
limsup-rsr / I \F{x, y+P)—F(x, y)\dxdy <+oo.
^o \p\JJ
Now this pair of relations is easily seen to be equivalent to the
relation (8.5) which therefore expresses a condition necessary and
sufficient in order that the function F should be of bounded
variation on Io. This completes the proof.
A statement analogous to Theorem 8.3 oan be made for curves (cf. Chap. IV,
§ 8). If 0 is a continuous curve defined by the equations x=X (i), y=Y(t), its ',
on any interval I0=[a, fi] is given by the formula
(8.10) 3@.,
¦;!,) = lim f{\
X(t+h)—X(t)] \T(t+h)-Y(t)
h J + L h
2,1/2
dt.
In particular therefore, in order that a continuous functions G(t) be of bounded
variation on an interval [a, 6], it is necessary and sufficient thai
(8.11)
limrap ^
[§8]
Tonelli's Theorem.
185
This assertion can he proved by the method of mean value integrals in a man-
manner quite similar to that m made use of in the theory of areas of surfaces z=F(x, y),
but for curves this method can be very much simplified. Let us observe further
that the relation (8.11) rnay lie interpreted in a more general sense. In fact, given
any summable function G(i), the relation (8.11) is the necessary and sufficient
condition in order that the function O be almost everywhere on [a, 6] equal to
a function of bounded variation (vide G-. H. Hardy and J. E. Littlewood [1];
cf. also A. Zygmund [I, p. 106]).
Finally the relation (8.10) holds for any rectifiablo curve given by the
equations x—X{t), y = Y(t), where the functions X(t) and Y(t) are not necess-
necessarily continuous, provided however that for each t the point (X(t), Y{t)) lies
on the segment joining the points (X(t—), Y(i—)) and (X{t+), Y(t+)).

CHAPTEE VI.
Major and minor functions.
§1. Introduction. Major and minor functions (defined in § 3
of this chapter) were first introduced by Ch. J. de la Valise Poussiu
in his study of the properties of the Lebesgue integral and those
of additive functions of a set. Entirely equivalent notions (of
"Ober"- and "Unterfunktionen") were introduced independently
by O. Perron [1], who based on them a new definition of integral,
which does not require the theory of measure. Although, in its original
form, this definition concerned only integration of bounded functions,
its extension to unbounded functions was easy and led, as shown
by O. Bauer [1], to a process of integration more general than
that of Lebesgue. Moreover, as we shall see in § 6, the Perron integral
may be regarded as a synthesis of two fundamental conceptions of
integration: one corresponding to the idea of definite integral as
limit of certain approximating sums, and the other to that of
indefinite integral understood as a primitive function.
It is usual to associate these two conceptions of integration
with the names of Leibniz and Newton. In accordance with this
distinction (which is largely a matter of convention) we shall caE
a function of a real variable F indefinite integral, or primitive, of
Newton for a function /, if F has everywhere its derivative finite and
equal to /. The function / will then be termed integrable in the sense
of Newton, and the increment of the function F on an interval Jo,
will be called definite integral of Newton of / on Io. As is seen im-
immediately, this definition implies that any function which is integrable
in the sense of Newton is everywhere finite. This restriction is
essential (cf. the example of § 7, p. 206) for the unicity of integration
in the sense of Newton, which then follows from classical theorems of
Analysis, or, if we like, from Theorem 3.1, or from Theorem 7.1
of this chapter.
The theory of the integral was first developed on Newtonian lines.
This is easily accounted for if we think how much simpler the inverse
Introduction.
187
of the operation of derivation must have seemed than the notion of de-
definite integral as defined by Leibniz. It was A. Cauehy [I, t. i, p. 122]
who returned to the idea of Leibniz in order to apply it to integi'ation
of continuous functions, for which the methods of Cauehy and
Newton are actually completely equivalent. This equivalence dis-
disappears, however, as soon as we pass on, with Eieniann, to inte-
integration of discontinuous functions. In fact, even in the domain
of bounded functions to which the Eiemann process applies, there
exist on the one hand (as we see at once) functions which are inte-
integrable in the sense of Eiemann but have no primitive, and on the
other hand (as shown by V. Volterra [1]; cf. also H. Lebesgue
[II, p. 100]) functions which have a primitive but are not integrable
in the Eiemann sense. Also the Lebesgue process of integration
does not include the integral of Newton, not even when the func-
functions to be integrated are everywhere finite.
Thus, the function. F(x) = x2 sin (njx-) for x^O, completed by -writing
F @)= 0, has in the whole interval [0,1] a finite derivative -which vanishes for x=0
and which is bounded on every interval [e, 1], -where 0<e<l. On every interval
[e, 1] the function F(x) is therefore absolutely continuous. On the other hand,
on the whole interval [0,1] the function is not even of bounded variation. Hence
F'(x) is not sumniable ou [0,1], since its indefinite Lebesgue integral could
then differ only by an additive constant from F(x) on [0,1], and this is im-
impossible.
We have thus been led to the problem of determining a process
of integration which includes both that of Lebesgue and that of
Newton. As an application of the method of major and minor func-
functions, we shall consider in this chapter (§§ 6 and 7) the solution of this
problem constituted by the Perron integral. Another solution, the
Denjoy integrals, will be treated in Chapter VIII.
The notions of major and minor functions, and their applications to Le-
Lebesgue integration, will be discussed here for arbitrary spaces Rm. In defining
the Perron integral however, we shall limit ourselves to functions of one real
variable. Although recently various authors have treated the extension of this
integral to Euclidean spaces of any number of dimensions, the present state of
the theory does not allow us to decide as to the importance of this genera-
generalisation. On the contrary, in the domain of functions of a real variable, tlie
method of major and minor functions as a means of generalizing the notion of
integral has already repeatedly shown its fruitfulness. In the memoir of
J. JIarcinkiewicz and A. Zygmund [1], the reader will find new applications
of this method in connection with certain fundamental problems of the theory
of trigonometrical sei'ies (cf. also J. Bidder [II])-

188
CHAPTER VI. - Major and minor functions.
§ 2. Derivation with respect to normal sequences
of nets. Given a regular sequence 3t=CU} of nets of intervals
{vide Chap. Ill, § 2) in a space Bm and a function of an interval F
in i?m, we shall call upper derivate of F at a point x with respect to
the sequence of nets 31 the npper limit of the ratio F{Q)j\Q\ as
S^Qj-tO^ where Q denotes any interval containing x and belonging
to one of the nets of the sequence 31. By symmetry we define
similarly the lower derivate of F at x with respect to the sequence of
nets 31. We shall denote these two derivates by C1) F(x) and C1) F(x).
When they are equal at a point x, their common value will be
denoted by C1) F'(x) and called 'derivative of F at so with respect
to the sequence of nets 31.
These definitions are similar to those given in § 15, Chap. IV, in connection
with derivation of additive functions of a set C3) in a metrical space. It should
be observed, however, that additive functions of a set (93) correspond to ad-
additive functions of an interval of bounded variation, whereas in the present § we
treat derivation of additive functions of an interval without supposing them
a priori of hounded variation. Tor tins reason it will be necessary to impose
certain restrictions on the nets considered in this §, and to distinguish a class
of nets which we shall call, for brevity, normal nets. The latter are, in point of
fact, the nets occurring most frequently in applications (of., for instance, Chap. Ill,
p. 58).
A system of intervals will be called a normal net in the space Mm,
when it consists of the closed intervals [a^\ «4+i; af\ «4+i; ¦¦•; a*"', «l+\]
for Ic—O, +1, +2,..., which are determined by systems of num-
numbers a® subject to the conditions af <.af+1 for j=l, 2, ..., m and
? = ...,— 1, 0, +1, ..., and lira af = ± oo. A regular sequence of
normal nets will be termed normal sequence.
B.1) Theorem. Let 3t={9U} be a normal sequence of nets, g{x)
a function which is summable in the space Hm and F a continuous
additive function of an interval such that (i) C1)J1(.'K)>—oo at every
point x, except at most those of an enumerable set, and (ii) F'(x)^g(x)
at almost all the points % at which the function F is derivable in the
ordinary sense.
Then for every interval I, we have
B.2)
i. e. F is a function of bounded variation, whose function of singul-
singularities is monotone non-negative.
[§2]
Derivation with respect to normal sequences of nets.
189
Proof. Consider the points in every neighbourhood of which
there exist intervals I for which the inequality B.2) is false, and let
P denote the set of these points. The set P is evidently closed,
and we see easily that the relation B.2) must hold for every interval
I such that 1° C CP. For if this were not the case, we could deter-
determine first an interval IQCP such that F{I)<zj g{x)dx, and then,
by the method of successive subdivisions, a descending sequence {!„}
of subintervals of I such that 6(In)—>0 as n->oo and that
F(In)<l g{x)dx for n=l, 2,.... Therefore, denoting by a the com-
4
mon point of the intervals In, we should have a eP, which is clearly
impossible.
It follows that in order to establish the validity of the in-
inequality B.2) for all intervals I, we need only prove that P = 0.
Suppose therefore, if possible, that P4=0. Let us denote, for any
pair of positive integers & and h, by 3Tj>,ft the sum of all the intervals I
oo
of the net 9^ for which F{I)>—h-\I\. Therefore by writing JVr/,=/7-^r*,A,
we obtain a sequence {Ata} of closed sets whose sum, according to
condition (i), covers the whole space except for an at most enumerable
set. Consequently, on account of Baire's Theorem (Chap. II, Theo-
Theorem 9.2), the set P contains a portion which either consists of
a single point, or else is contained in a set Nh. The former case is
excluded since it is evident from the continuity and additivity of
the function F that the set P contains no isolated points. Therefore
there exists a positive integer 7i0 and an open sphere S such that
0=j=P-#C-ftv Let us write H(I)=F(I)+ho-\I\ + [\g[as)\dx where I
is any interval. We shall have HA)^0 for any interval I such
that J°CCP, as well as for any interval I belonging to a net 3U
of index Tc^hQ and having points of the set Nho in its interior.
Therefore H{I)^0 for any interval IQ? belonging to a net 31*
of index Tc^h0, and consequently, by additivity and continuity of E,
we have H{I)^0, i.e. F(I)^—ho-\I\— j\g(x)\dx, for any inter-
i
val I(Z& whatsoever. It follows at once that the function F
is of bounded variation in 8 and that the function of singularities
of F (cf. Chap. IV, p. 120) is monotone non-negative in 8.

190
CHAPTER VI. Major and roiaor functions.
Hence, by condition (ii), F(I)^ j'F'(x)dx^fg(x)dx for every inter-
i i
valIC#- But since P-'S' + O we thus arrive at a contradiction and
this completes the proof.
As an immediate corollary of Theorem 2.1, we have
B.3) Theorem. If 91 is a normal sequence of nets in the space JSm
and if F is a contin%ous additive function of an interval such that:
(i) —oo<z('Sl)F(x)^:{<Sl)F{x)< + oo for each point x except at most
the points of a% enumerable set, (ii) the (ordinary) derivative F\so) is
summable on each portion of the set of the points at which this deriva-
derivative exists; then the function F is almost everywhere derivable and is
the indefinite integral of its derivative.
For Theorems 2.1 and 2.3 of. J. Bidder [2]. Let u& remark that in the
case where the function F is of bounded variation, these theorems are included
in Theorem 15.7, Chap. IV, which concerns derivation of additive functions of
a set in an abstract metrical space.
It follows easily from Theorem 15.12, Chap. IV, that F'{x) = {$l)F'[%)
almost everywhere for any regular sequence of nets of intervals 9? and for any
additive function of an interval F which is continuous and of bounded variation.
This remark enables us to replace condition (ii) of Theorem 2.1 by the following:
(ii-te) F'{x)=('3l)F'(x)'^g {x) at almost all the points x at which the two derivatives
F'(x) and (Sl)F'(x) exist, are finite and equal. Similarly we may modify condi-
condition (ii) of Theorem 2.3.
As it follows from an example due to A. J. Ward [7], the jnequality
(SSl)F(x)> —oo in condition (i) of Theorem 2.1 cannot be replaced by (<&)F(x)>-co.
§ 3. Major and minor functions. Before introducing the
fundamental definitions of the theory of the Perron integral, we
shall prove
C.1) Theorem. If an additive function of an interval F (not neces-
necessarily continuous) has a non-negative lower derivate at each point x
of an interval Io, then F(IO)^O.
Proof. Let she any positive number and write G(I)—F(I)-\-e-\I\
for every interval I. Then 6r(a?)>e>0 at each point xelo. Suppose
that (?(I0)^0. By the method of successive subdivisions, we could
then determine a descending sequence \In\ of intervals similar to Io,
such that <?(InXO for n=0,1,2, ... and that d(!„)->¦ 0 as «->oo.
Therefore, denoting by x0 the common point of the intervals I,,,
we should have 0{os0)^;0 which is impossible. Hence #(I0)>0, and
this gives F{I0)>—e-\I0\ for each ?>0, and finally .F(I0)>0.
Derivation with respect to binary sequences of nets.
191
An additive function of an interval F is termed major [minor]
function of a function of a point / on a figure Bo if, at every point x
of this figure, ~°°^Fd?)>1(x) l+°°^Fs(x)^f{x)]. It follows at
once from Theorem 3.1 that if the functions of an interval U and 7
are respectively a major and a minor function of a function / on
a figure Ro, their difference U—Y is monotone non-negative on Bo.
C.2) Theorem. If f is a summable function, then, for each e>0, the
function f has an absolutely continuous major function V and an
absolutely continuous minor function Y such that, for each interval I,
C.3)
and
Proof. On account of the theorem of Vitali-Carathiodory
(Chap. Ill, Theorem 7.6) we can associate with the function / two
summable functions, one a lower semi-continuous function g
and the other an upper semi-continuous function h, such that
(i) — oa^p g(x)^f(x)^h{scLz + °° at every point x and that
(ii) l[g(x)—f(x)]dx<e and J[f{x)—h(x)]dx<e for every interval I.
Therefore, if we denote by U and V the indefinite integrals of the
functions g a,nd h respectively, we find by Theorem 2.2, Chap. IV,
that TJs{x)^g{x)^li{x)^Vli{x), and so, on account of (i), that
—°°^=Us{x)^f(x) and +oo=)= F«(a;X/(<E) at each, point x. Finally,
(ii) then implies the relations C.3) and this completes the proof.
Theorem 3.2 can easily be inverted. Thus: in order that a function of a point f
be sumviablr., ii is necessary and sufficient that for each e>0 there, exist two absolutely
continuous functions of an interval TJ and F, the one a major and the other a minor
function of f, which fulfil the condition U(I)—V(I)<e for every interval I. (These
absolutely continuous functions may clearly be replaced by functions of bounded
variation, and if the function / is supposed measurable, then, of course, for its
summability there suffices the existence of two functions of bounded variation,
one of which is a major and the other a minor function of /.)
* § 4. Derivation with respect to binary sequences of
nets. The theorems of §2 concerned derivation of additive functions
with respect to any normal sequence of nets of intervals. For certain
purposes however, more special sequences of nets are required.
We shall say that a normal sequence {9U}/t=i,2,... of nets in the space
Hm is binary, if the net 9L+i (where &=1, 2, ...) is obtained by
subdividing each interval JV of the net 5R* into 2'" equal intervals
similar to N.

192
CHAPTEE VI. Major and minor functions.
An application of this notion may be found in the following theorem which
is proved similarly to Lemma 11.8 of Chap. IV: If 31 is a binary sequence of nets,
any additive function of an interval F is derivable with respeol to 91 at almost all
the points at which either (91) P(x)> — co or (91).F(a;)<+oo.
Another application, of particular interest, is due to A. S. Besicovitoli [3]
who, by using derivation with respect to a binary sequence of nets, established
a theorem on complex functions (vide below § 5). The substance of Besicovitch's
result is contained in Theorem 4.4 below. We must first, however, give some
subsidiary definitions.
For definiteness, just as in § 11, Chap. IV, we shall fix in the
space B,m a binary sequence of nets Q={Q*}, where Q* denotes, for
=1, 2,
the net formed by the cubes
[Pl/2*,
*; pj2"
where pv pv ..., pm are arbitrary integers; it goes without saying
that in Theorem 4.4 this sequence may be replaced by any binary
sequence whatsoever.
Given a non-negative number, a, we shall say that a function
of an interval F fulfils the condition (it) [condition A,7)] at a point x,
if liminf .F (I)/[<5 {!)]"> 0 [lim supl'(J)/[(S(J)]"^0], where I is any
interval containing a). If a function / fulfils the condition A7) [A,,)]
at every point of a figure R, we shall say simply that / fulfils this
condition on R. Finally, we shall say that a function fulfils the
condition (la) at a point, or on a figure, if it fulfils simultaneously
the conditions A7) and A7).
We recall further the notation Aa(E) for the a-dimensional
measure of a set E (cf. Chap. II, p. 53).
D.1) Lemma. Given a set E in the space Mm, together with a positive
integer Tc0 and a non-negative number a<m, there exists for each s> 0
a sequence {Qn} of intervals belonging to the nets QA for fc^&o, which
fulfils the following conditions:
(i) 2 [d (Qn)]a ^ [im)m-[Aa(E)-\-e]-
n
(ii) to each point x of E there corresponds a positive integer h^Je0
such that all the intervals of the net Qft which contain the point x belong
to the sequence {Qn}.
Proof. Let us cover E by a sequence {.E,},=i 2 of sets such
that O«5(j;,)<i/2A«+1 for i=l, 2,... and such that'
Derivation with respect to binary sequences of nets.
193
Let us denote by A-,-, for each i=l, 2, ..., a positive integer
such that
D.3) l/2*'><5(S,)S?=l/2*rH.
We easily see that fc( > &o for every i, and that each net Qi/}
for i=l,2,..., can contain at most 2'" intervals having points in
common with Et. Let {<?n}n=12 De the sequence of all the intervals
belonging to the nets Q*,, ?U2, ..., ?h,;, ... and having points in com-
common with the sets E1,E,,...,Ei,... respectively. The sequence (Qj
clearly fulfils the condition (ii). Moreover, we find on account of D.3),
and this by D.2) gives at once the condition (i).
D.4) Theorem. Suppose that F is a continuous additive function of
an interval in the space JRm and fulfils the condition (L) where 0<a<m,
andthat g is a summable function. Suppose further that (i) (Q) F(x) >—oo
at every point x except at most those of a set jB expressible as the sum
of an enumerable infinity of sets of finite measure (Aa), and that
(ii) [Q)F(x)^g{x) at almost all points x; then
D.5)
for every interval Io.
Proof. Since the function F is continuous, it will suffice to
prove D.5) in the case in which the interval Io belongs to one of
the nets Q*, to the net ?}*„, say. Further by changing, if necessary,
the values of g on a set of measure zero, we can assume that
the inequality (Q)Jf(as) ^g{x) holds at every point x.
Let e be a positive number and let V be a minor function of g
(cf. § 3, particularly Theorem 3.2) such that
D.6)
V(I0)>fg(x)dx~B.
Let us write <?(I)=F(I)—7(I)+?-|J|, where I denotes any interval.
We shaH have {?i)G(x)^{Q)F{x) — f («)+?>s>0_at every point
x except at most at the points of E. Finally, since 7s(x)<Z + °° at
every point x, the function V fulfils the condition A7) and the func-
function G therefore fulfils the condition [it)- - .
S. Saks. Theory of the Integral.
13

194 CHAPTEE VI. Major and minor functions.
Let us now represent the set E as the sum of a sequence
{JE,}i=i,2,... of sets of finite measure (Aa), and denote, for each pair of
positive integers % and n, by BUn the set of the points x such that the
inequality G{I)>—e-[d(I)fl2i[Aa[Ei) + l'] holds whenever I is an
interval containing x and belonging to one of the nets Q/( for h^n.
The sets Bt,n are evidently measurable B3) (they are actually sets (®<s)).
Moreover, since the function G fulfils the condition (it), the sum
?Bt,n must, for each integer i, cover the whole space R,,,. Hence,
writing Ei,n=Er(Biin
find that
D.7)
B,,n-i) for n>l, and Eui=ErBi,i, we
for 4=1,2,....
This being so, it follows from Lemma 4.1 that for each pair
of positive integers i and n, we can determine a sequence {<2^=] 2
of cubes which belong to the nets Q* for 1c ~^n, and fulfil the fol-
following conditions:
D.9) to each point xeE^n there corresponds am, integer 7c^=/c0 such
that each cube of the net Q*, containing x, belongs to the se-
sequence {Q[i)n}J=i 2 ;
D.10) each cube Q^n has points in common with the set Jy. n and therefore
fulfils the inequality
For brevity, let us agree to say that an interval has the prop-
property (A), when it is representable as the sum of a finite number
of non-overlapping intervals I each of which either fulfils the in-
inequality t?(I)>0, or else coincides with one of.the cubes Q\J\ We
remark that on account of D.10), D.8) and D.7), the inequality
i.n.J
is valid whenever B is a figure consisting of any finite number
of non-ova-lapping cubes Q^, and therefore that the inequality
G(I)>—(imf-e must hold for every interval I having the prop-
property (A),
Applications to functions of a complex variable.
195
We shall now show that the interval Io itself has the prop-
property (A), so that <x(I0)^ — Dm)me. Let us suppose the contrary.
We could then, starting with Io, construct a decreasing sequence
{Ip} of cubes belonging to the nets Q* and none of which has the
property (A). Let x0 be the common point of these cubes. Then
either x^eE, and consequently, by D.9), the sequence contains
cubes Qfn; or xoeCE, so that (Q)<?(a;o)>0, a^d therefore G(Ip)>0
for each sufficiently large p. Thus in both eases, the sequence {Ip}
would contain intervals with the property [A) and we arrive at
a contradiction. It follows that G(Ia)~^ — (itn)me, and therefore,
by D.6), that
since e is an arbitrary positive number, this gives the relation D.5).
* § 5. Applications to functions of a complex variable.
We now interpret the points of the plane JS2 as complex numbers
and, as usual, we call complex function of a complex variable every
function of the form u-\-iy where u and v are real functions in
the whole plane, or in an open set. The functions u and v are
termed real part and imaginary part of the function /. A complex
function is said to be continuous (at a point, or in an open set), if
its real and imaginary parts are both continuous.
Given a complex function /, continuous in an open set G, and
having the real and imaginary parts u and v respectively, we shall
write for every interval I=[a1, 6X; aa, 5a] contained in G:
E.1)
and
/;-?) = — /[«(», b2)—u(x, ffla)] dx—f lv(bv y)—v(au y)] Ay,
b., 6t
/;^)= J{v>(buy)—n{a1,y)'\&y—ji'o(.x,ll)—v{x,al)\dx,
The expression J(/;I), which will also be denoted by ifdz, will
be called curvilinear integral of the function / along the boundary
of the interval I. The function / will be termed holomorphic ii1 an
open set G, if J(/;I)=0 for every interval ICG. (The equivalence
13*

196
CHAPTER VI. Major and minor functions.
of this definition of the term. ,,holomorphie" — used here in place
of terms such as "regular", "analytic", etc. — with the more familiar
definitions of the theory of complex functions, follows from the
well-known theorem of Morera [1].) We verify at once that this
relation holds when f{z) = az-\-b where a and b are any complex
constants.
If / is a complex function, continuous in an open set 0, the
expressions J^f; I) and J2(/; I) are continuous additive functions
of the interval I in G. Moreover
|J,(/;I)|<|J(/;I)| and |J,(/; I)|<|J(/; I)\
for each interval I in G. On account of Theorem 2.3, we there-
therefore obtain at once the following theorem due to J. Wolff [1]
(cf. also H. Looman [2] and J. Bidder [1; 2]):
E.2) Theorem. A complex function f, continuous in an open set G,
is Jiolomorphie in G if at almost all points z of G,
liminf-
w\fm*lr°>
Q)
Iv!
and if at all points « of G, except at most those of an enumerable set,
1
<S«?HO IV I
Q)
z)dz
where Q denotes any square containing st.
A complex function is called derivable at a point z0, if the ratio
[/(»)—/(ao)]/(a—«o) tends to a finite limit when s tends to z0 in any
manner. This limit is called derivative of / at z0 and is denoted by f{z0).
Let / be any complex function, defined in the neighbourhood
of a point a0. If we have lim sup |[/(ao+7i)— /B0)Pi< + 00, we
can write f{s>)=f(so)+M{ss)-(g—mo), where M(ss) is a function of s
which is bounded in the neighbourhood of s0; and we then easily
find that the ratio \J{f;Q)\l\Q\, and a fortiori the ratios |J1(/;#)|/|y'|
a&d \^z(flQ)\l\Q\, must remain bounded when Q denotes any suf-
sufficiently small square containing «0. If, further, the function / is
derivable at *0, we have /(*)=/(*0)+/'(«„)¦(»—*„)+*(«).(«—*„),
where |«(«)|->0 as «->*:„, and the ratios in question tend to zero as
<5(Q)->-0. Finally, let us observe that if the function / is continuous,
the expressions J^/; I) and Ja(/; I), considered as functions of the
[§5]
Applications to functions of a complex variable.
197
interval J, both fulfil the condition (LJ of § 4. Therefore, if we apply
Theorem 4.4, we obtain the following theorem due to A. S. Besi-
covitch [3] (ef. also S. Saks and A. Zygmund [2]):
E.3) Theorem. A complex function f, continuous in an open set G,
is holomorphic inGif it is derivable at almost all the points of G and if
further limsup \{f[z-\-h) — f(z)]jh\<. + oa at each point z of G except at
A-M>
most those of a set which is the sum of a sequence of sets of finite length.
The theorem of Besieoviteh may be regarded as a generalization of the
classical theorem of E. Goursat [1]: A complex function /, continuous in an open
set G, is holomorpMc. in 6 if it is everywhere derivable in G.T. Pompeift [1] showed
that it is enough to suppose / derivable almost everywhere, provided that
we restrict the expression lim sup \[f(z+h)—/(«)]/ft| to be bounded in Q. Finally,
H. Looman [3] (cf. also J. Eidder [2]) replaced the condition that the expres-
expression limsup [/(z+7s)—f{z)]/h\ is bounded bv the condition that this expres-
sioa is finite at each point of G. Theorem 5.3 evidently includes all these
generalizations.
The theorems of Morera and of Goursat, and their generali-
generalizations furnished by Theorems 5.2 and 5.3, contain criteria for
holomorphism which are based on the notion of curvilinear integral
and of derivation in the complex domain. The classical theorem of
Oauchy is an instance of a criterion of a different kind, expressible
in terms of real variable conditions on the real and imaginary parts
of a complex function; we have in fact, according to this theorem:
in order that a continuous function of a complex vari-
variable /(a) = u(cs,y) + iv{x,y) be holomorphie in an open
set G, it is necessary and sufficient that the partial
derivatives u'x, u\, v'x, v't should all exist in G and be
continuous, and that they everywhere fulfil the Oauchy-
Eiiemann equations u'x=v , and u=—v'x.
A series of researches begun by P. Mont el [1] has been de-
devoted to the reduction of these conditions, particularly that of the
continuity of the partial derivatives. The problem was finally solved
by H. Looman [2] and D. Menchoff (vide the first ed. of this book,
p. 243, and D. Menchoff [I]) who succeeded in removing completely
the condition in question without replacing it by any other. It is
remarkable that a classical problem of such an elementary aspect
should only have been solved by a quite essential use of methods
of the theory of real functions.

198
CHAPTEE VI. Major and minor functions.
E.4) Lemma. Let w he a real f%nction of one variable, derivable
almost everywhere in an interval [a, b]; let F be a closed non-empty
subset of this interval, and let N be a finite constant such that
\w{x2) — w
whenever x^eF and xse[a,?\. Then
E.5)
w(b)—w(a) — j
F
-a-\F\).
Proof. Let us denote by Fx the set obtained by adding the
points a and b to the set F. The function w, equal to w on Fx and
linear on the intervals contiguous to Fx, is evidently absolutely
continuous on [as, 6] (and even fulfils the Lipschitz condition). Hence
E.6)
o
w{b)—w{a)—w(b) — w(a)= jw'{x)dx.
How w'[x)=w'[x) at almost all the points x of F and | w'[x) |^ N
at each point x outside F. The relation E.5) therefore follows at
once from E.6).
E.7) Lemma. Let w(x,y) be a real function whose partial derivatives
with respect to the Mm variables x and y exist at every point of
a square Q, except at most at the points of an enumerable set; and
let F be a closed non-empty subset of Q, and N a finite constant
such that
h—^il and
whenever (x^y^eF, (x2,yj)eQ, and (as1)yi)eQ.
Then if [%, \; a2,62] denotes the smallest interval (which, may
be degenerate) containing F, we have
![w[x,b2)—w(x, a2)~\dx— f fw'g(x, y)dxd,y <^oN-\Q—F\
F
,y)—w{aliy)]dy—ffw'x(so,y)da>dy\^!5N-\Q — F\.
F
Proof. Let us choose arbitrarily two points (as', a2) and (a)") ^2)
belonging to the set F and situated on the two sides of the interval
1"/^, &x; ffl2, 52] parallel to the s-asis. For any point f of [%, /;,] we
E.8) Ol
(,il)|<|(|,a)w(a!,6a)| + l«'(a!»6a)-«'(ai, M +
|to(a;',62)— w(x',ao)\-{-\w{x',a2)—td(§,a2)|; and hence, denoting by I
[§5]
Applications to functions of a complex variable.
199
the length of the side of the square Q, we obtain
,.Q> |«>(?, &a)—w&aE)K
We now denote for any point ? of [alt b^, by IPs tlie set of
all the points y of [a2, 62] such that (?,y)eF. Let A be the set of
the points f of the interval [alt 6J for each of which -P|=j=O, and
let B denote the set of the remaining points of [%, 6J. On account
of Lemma 5.4 we have
whenever ^e A, and if we integrate the two sides of this inequality
with respect to § on the set A, we find
E.10)
On the other hand if we integrate E.9) with respect to % on the
set B, we obtain
|, &2)—
— F\,
and by adding this to E.10) we obtain the first of the inequalities E.8).
The second inequality follows by symmetry.
E.11) Theorem of JOooman-Menchoff. If the functions u{x,y)
and v(x,y), continuous in an open set 6, are derivable with respect
to x and with respect to y at each point of G except at most at the points
of an enumerable set, and if u'x{x,y)=v'g(x,y) and u'y{x,y) = —v'x{x,y)
at almost all the points [x, y) of G, then the complex function f=u-\-iv
is holomorphic in G.
Proof. Let us denote by F the set of the points [x,y) of G
such that the function / is not holomorphic in any neighbourhood
of (*, y). The set F is evidently closed in G and the function / is
holomorphic in G—F. It thus has to be proved that F is empty.
Suppose therefore, if possible, that F 4= 0 and let Fn denote, for
each positive integer n, the set of the points (x,y) of G such that,
whenever |A|^l/?i, none of the four differences u{x-\-h,,y)—u(a>>y)
ii(x,y-\-h)—u(x,y), v[x-\-h,y)—v{x,y), v{x,y-\-h)-—v{x,y) exceeds \nh\
in absolute value. By continuity of the functions u and v, each
of the sets Fn is closed in G. On the other hand, the sets Fn cover
the whole set (?, except at most an enumerable set consisting of

200
CHAPTER VI. Major and minor functions.
the points at which the functions u and v are not both derivable
with respect to x and with respect to y simultaneously. Therefore,
on account of Baire's Theorem (Chap, II, Theorem 9.1), the set FQ &
contains a portion which either reduces to a single point, or else is
contained entirely in one of the Fn. The former possibility is ruled out,
since, as we easily see on account of the continuity of /, the set F
cannot contain any isolated points. There must therefore exist a
positive integer N and an open sphere 8 such that Q=\=
Let Q be any square contained in 8, such that
and Q-F=^0. We denote by !=[»!, bx; as, 62] the smallest interval con-
containing Q-F. By applying the evaluations of Lemma 5.7 to the
integrals on the left-hand sides of the formulae E.1) and by taking
into account the relations u'x(x,y)=v'!J{x,y) and u's{x,y) =—v'x(x,y)
which are, by hypothesis, fulfilled almost everywhere, we find
iJitfjIJl^lO-W-K?— F\ and |J3(/; I)\^1QN-\Q— F\, and therefore
|J(/; I)|^20iV-|<3— F\. This last inequality may also be written
|J(/;G)l<20if-|#— F\, since the figure QQI contains no points
of F in its interior, and since therefore J{f;R) = Q for each interval
B contained in QQI.
Sow let so= (o;o, y0) be any point of 8, and let Q be any square
Containing z0. By what has just been shown, if z0 e F we have
\J(f;Q)\l\Q\^20N-\Q^F\l\Q\ as soon as <J@Xl/2r; the ratio
J(/j Q)I\Q\ therefore remains bounded as <S(Q)-»O and tends to zero
whenever s0 is a point of density of F. Further \J{f;Q)\j\Q\->0 as
^(§)->-0, whenever zoe8—F, since J(/;<2)=0 for every square Q
which does not contain points of F. Therefore by Theorem 2.3,
the function / must be holomorphic in 8, This is, however, excluded
since 8-F^=0. We thus arrive at a contradiction and this completes
the proof.
Theorem 5.11 was stated (even in a more general form) by P. Montel [2]
as early as 1913, but -without proof. The proof supplied by H. Looman [2] in 1923
¦was found to contain a serious gap which was only finally filled in by D. Men-
ohoff (of. D. Menchoff [I] and the first edition of this book, p. 243).
By making use of general theorems on derivates (vide, below, Chap. IX)
it is possible to weaken slightly the hypotheses of the theorem. Thus instead
of assuming partial derivability of the function u and v, it is sufficient to sup-
suppose that at each point of <? (except at most those of an enumerable
set) these functions have with respect to each variable, x and y,
their partial Dini derivates finite. This condition implies (of. Chap. VII,
§ 10. p. 236, or Chap. IX, §4) partial derivability of the functions n and v with
The Perron integral.
201
respect to each variable at almost all points of G (this generalization of the
theorem of Looman-ilenehoff does not require any alteration of the proof;
for other and much deeper generalizations, vide the memoirs of D. Menchoff
[1; 2)).
The extension of Theorem 5.11 which we have indicated, includes in par-
particular the theorem of Looman mentioned above, p. 197, but not however the
theorem of Besicovitch E.3). It would be interesting to establish a theorem which
would include both the theorem of Besicoviteh and that of Loomau-Menchoff.
§ 6. The Perron integral. For functions of one real vari-
variable, as announced in § 1, the method of major and minor func-
functions leads to an important generalization of the Lebesgue integral.
A function of a real variable, /, is termed integrable in the sense
of Perron, or oP-integmble, on a figure Ro in Rx, if 1° / has both major
and minor functions on Ro, and if 2° the lower bound of the numbers
U(E0), where U is any major function of / on Ro, and the upper
bound of the numbers V{RQ), where V is any minor function of /,
are equal. The common value of the two bounds is then called
definite Perron integral, or definite ^-integral, of / on JR0, and de-
denoted by (df) I f(x)dx. It is evident that for S'-integrability of afunc-
tion f on a figure RQ it is necessary and sufficient that for each e > 0
there should exist a major function U and a minor function V of f on Ro
such that U(B0)—V(BQ)<€.
Since (cf. §3, p. 190) the function U—V is monotone non-
decreasing for every major function U and every minor function V
of /, it follows that every function which is ff'-integrable on a figure Rw
is so also on every figure R(Z.Ro- The function of an interval
P(I)=(&) ff(x)dx, thus defined for every interval JC^oj ig called
i
indefinite Perron integral, or indefinite ^-integral, of / on Ro. As. we
see at once, P (I) is an additive function of the interval I. Moreover,
given any positive number e, there exist always a major function U
and a minor function V of/, such that O^U(I) — P(I)^.e and
O^P(I)—V{I)^e for every interval IC-Roi and since TJ{x)> — oo
and V(x) <+ oo at each point x of Ra, it follows at once that the
function P is continuous. Just as in the case of the Lebesgue integral,
a function of a real variable is termed indefinite if-integral [major
function, minor function] of a function /, if this is the case for the
function of an interval determined by it (ef. Chap. Ill, § 13).

202
CHAPTER VI. Major and minor functions.
As we see at once from Theorem 3.2, every function which
is integmble in the sense of Lebesgue on a figure Bo, is so in the sense
of Perron, and Us definite Lebesgue mid Perron integrals over Ro are
equal. On the other hand, if F is the primitive of Newton (cf. § 1)
of a function /, tie function F is at the same time a major and
a minor function of /, and therefore is the indefinite ^-integral of /.
It follows that Perron's process of integration includes both that of
Lebesgue and that of Newton.
We shall establish some fundamental properties of the Pen-on
integral.
F.1) Theorem. Every oP-integrable function is measurable, and is almost
everywhere finite and equal to the derivative of its indefinite integral.
Proof. Let / be a function of a real variable, <f-integrable
on an interval Io, and let P be its indefinite ^-integral on Io. It has
to be proved that the function P has at almost all points x, a finite
derivative equal to f(so).
For this purpose, let e be any positive number and U a major
function of / such that
F.2) U(I0)-P(I0)<e>.
Let us write H= U—P. The function H, as monotone non-decreasing,
is almost everywhere derivable, and if we denote by E the set of
the points x of Io at which H'{x)^e, we find, by F,2) and Theo-
Theorem 7.4, Ohap. IV, that |.E| < e.
H"ow at each point coelo where the function H is derivable,
U{x)=H'(x)+P(x); hence P(x)>— °o and P{x)>U{x)—e^f(x)--e
at almost all the points x of Jo—E. Therefore, since |JB|^c, s being
an arbitrary positive number, it follows that — °o=|=P (;*;);> /(#) at
almost all the points x of Io. By symmetry this gives also
+ oo^zP(x)^f(so), and finally oo^P'{x)—f{x) almost everywhere
in Io.
F.3) Theorem. If two functions f and g are almost everywhere equal
on a figure Bo and one of them is S'-integrable on JRW so is the other
and the definite ^-integrals of f and g over Bo are equal.
Proof. Suppose that the function/ is ^-integrable and denote
by A the value of its definite integral over RQ. Let e be any positive
number and let U and V be two functions of an interval, which
are respectively a major and a minor function of / on Bo and which
fulfil the inequalities
[§7]
Derivates of functions of a real variable.
203
F.4)
and
Let us denote by E the set of the points x at which f{x) =3pg(x).
The function equal to +oo at all the points of E and to 0 every-
everywhere else is therefore almost everywhere zero, and by Theorem 3.2
has a major function G such that 0^ff(E0)^f/3. We have G {x) = + oo
at each point xeE and writing XJl = XJ-\-G, V1=V—Cf, we see that
the functions of an interval J7i and Vt thus defined are respectively
a major and a minor function of the function g on Bo. Moreover
by F.4), ?71(-R0)>.iL>F1(E0) and
Therefore
the function g is cT'-integrable on i?0 and A = (S') I g[x)dx, which
completes the proof. *«
F.5) Theorem. Every function f which is S'-integrable and- almost
everywhere non-negative on a figure Bo, is summable on this figure.
Proof. We may assume, by Theorem 6.3, that the function /
is everywhere non-negative on BQ. Therefore if U is any major
function of /, we have U(x}^ f(x)^O at every point xeBa, and
consequently, by Theorem 3.1, the function U is monotone non-
decreasing. Its derivative U'[x) is therefore summable on Bo, and,
since U'[x)^ f(x)^Q almost everywhere, the function/is also sum-
summable on Bo.
Theorem 6.5 shows that, although Perron integration is more general
than Lebesgue integration, the two processes are completely equivalent in the
case of integration of functions of constant sign.
§ 7. Derivates of functions of a real variable. Certain
of the theorems of §§ 2 and 3 can be given a more complete statement
when we deal with functions of one real variable. We shall begin
with the following proposition which is due to Zygmund:
G.1) Theorem. If F{x) is a finite function of a real variable sueh
that (i) limsup_F(a'—h)^.F(x)^]imsu-pF(x-\-h) at every point x, and
h0+ frM)+
+ M+
(ii) the set of the values assumed by F(x) at the points x where F()
contains no non-degenerate interval, then the function F is monotone
non-decreasing.
Proof. Suppose, if possible, that there exist two points a
and b sueh that a<6 and that F{b)<.F{a). Then, denoting by E
the set of the points x at which F+(x}^.0, we can determine a value yo
not belonging to the set F[E] and such that F{b)<y0<F(a). Let

304
CHAPTER VI. Major and minor functions.
xu be the upper bound of the points x of [a, b], for which
We shall obviously have ffl<a?0<&, F{xo) = yo, and F{x)^y0 for
each point x of the interval [xB,b]. Therefore F+[x0)^0, although x0
does not belong to B. This is in contradiction with the definition
of the set F.
Let us mention the following consequence of Theorem 7.1:
Dint's Tlieorem. Given on an interval I=[a, b] a continuous junction F(x),
the upper and lower bounds of each of its four Dini derivates are respectively equal
^
where x^ and x* are any
to ths upper and lower bounds of the ratio
points of I.
Let, for instance, m be the lower bound of the derivate P+(x) on the inter-
interval/, and suppose first that m>—co. Then, if m' denotes any finite number
less tlian to, the function F{x)—m'-x has everywhere on [a, b] its upper right -
hand derivate positive; and so by Theorem 7.1, F(xi)—F{x1)'^-'m,'-{x,i—x{),
and therefore also IFfa)—F(x1)}/(x1—xj^m, for every pair of points n, and xs
of I such that x^-c^x^. Since the inequality juBt obtained is trivial in the case
m=—od, the theorem follows.
An immediate consequence is the following theorem:
If any one of the four Dini derivaies of a continuous fun-ation is continuous
at a jwint, so are the three others, and all four derivates in question are. equal, so
that the function considered is derivable at this point.
These two propositions were proved by U. Dini [I] in 1878.
G.2) TJieorem. If E is a finite function of one variable such that
(i) lim sup if(a:—h)-^E{x)^1imsTxpE{x+h) at every point x, and
1H4+ h-Hl+
(ii) E' (x)^-Q at every point x except at most at those of an enumer-
enumerable set, then the function H is monotone non-decreasing.
Proof. Let e be a positive number and write F{x)=E(x) + ex.
We have F+(x)^e>Q at each point x except at most at those of
a finite or enumerable set E. The set F[D] being, with E, at most
enumerable, it follows from Theorem 7.1 that the function
F[w)=E(x) + ex is non-decreasing for each e>0; and by making
e->-0 we obtain the assertion of the theorem.
G.3) Theorem. Suppose that F is a continuous function and g
a P-integrable function of a real variable, and that, further, we
have (i) F+{x)^g{x) at almost all points x and (ii) F+[x)> — oo
at every point x, except at most at those of an enumerable set; then
b
for every pair of points a and b such that a<b.
Derivates of functions of a real variable.
205
If, in addition, (ix) F+{x)^g(x)^F'( (x) at almost all points x
and (iij F+{x) >—co and F~[x) <-j-oa «J every point x except at most
at those of an enumerable set, then the function F is an indefinite
iP-integral of g.
Proof. We may obviously assume that F+(x)^g{x) at every
point x. Therefore, denoting by V any minor function of g, and
writing E=F—V, we shall have S+(x)^F+[x)— V+{x)^0 at every
point x, except at most at those of a finite or enumerable set where
F+(x)=—oo. Further, since the function F is continuous, the
inequality V(x)<+ oo, which holds at every point x, implies that the
function S satisfies the condition (i) of Theorem 7.2. Consequently,
by Theorem 7.2, H{b)—H(a)^0, i.e. F{b)—F(a)^Y{b)—V(a), and
since V is any minor function of g, we obtain the inequality G.4).
The second part of the assertion is an immediate consequence
of the first part.
As we easily see, the condition of continuity of the function F in the
first part of Theorem 7.3 may be replaced by the condition (i) of Theorem 7.1.
Theorem 7.3 constitutes, on account of Theorem 7.4, Chap. IV, a general-
generalization of the following theorem of Lebesgue [I, p. 122; 2; 3; 4; II, p. 183]:
in order that one of ihe derivates of a continuous function, supposed finite, be surn-
mable, it is necessary and sufficient that this function be of bounded variation; its
absolute variation is the integral of the absolute value of ihe derioate in question.
Let us add that in the case in which the function F is assumed to be of bounded
variation, Theorem 7.3 is included in Theorem 9.6 of Chap. IV.
The condition (ii) of the first part of Theorem 7.3, as well as the con-
condition (iij) of the second, is quite essential for the validity of the theorem.. It is
possible, in fact, to give an example of a continuous function whose derivative
exists everywhere and is summable, without the fuuetion being the indefinite
integral of its derivative, and this because the latter assumes infinite values.
To see tbjs, we shall first show that given any closed set B of measure zero in
an interval Jo= [a, 6], there exist* a function Gf, absolutely continuous and in-
increasing in /„, which has a derivative everywhere in Ja and fulfils the conditions
G.5) ff'(a;) = +co for xeE and G'{x) + +oa for xeJ0—E*
Let us suppose for simplicity that E contains the end-points a and b
of Ja and let us denote by j[an,6n]} the sequence of the intervals contiguous
to JB. Let {hn) be a sequence of positive numbers such that
G.6) ]im1inl(bn—an)=+co
and
G.7)
(it suffices to write, for instance, hn = \rn—\'rn+l, where rn= -j \ F,-—a/)).
Let us wnte
,_ c, ,i\\ hn/(x—an)v- (bn—xI'1, when an<x<bn,
I +co, when xeE.

206 CHAPTEK VI. Major and minor functions.
Thus defined the function g(x) is non-negative on Jo and summablo on Jo,
bn b
since [g(x)dx=7ihn, so that by G.7) we have jg(x)dx^--r. Let G be the indefi-
an a
nite integral of g on Jo. In order to verify that the function G fulfils the con-
conditions G.5), we observe that the function g(x) is continuous for every xeJB—E;
on the other hand,-if we denote by mnthe lower bound of g(x) in \an, brt] we derive
from G.6) that limmn=lim 2ft*/(&n—a,,)=+oo, from which it follows that
lim g(a:) = +CQ=g(xa). for every 'XoeM. Consequently <?'(&) = j (as) for every x,
and therefore, by G.S), the conditions G.5) hold.
Now let (of. Chap. Ill, A3.4)), H(x) be a continuous non-decreasing sin-
singular function on Jo, which is constant on each interval contiguous to the, set JS,
and such that H(o) +HF). Let us put F=6+H. As we verify easily from G.5),
we have F'(x)=&'(x)=g(x) at every point x of Jo. The function F therefore
has everywhere a derivative which is suininahle on Jo. But, since II in the
fuuetion of singularities of the function F, the latter is certainly not absolutely
contiunoua, let alone the. indefinite integral of its derivative. (The functions
67 and F provide at the same time an example of two functions whose deriv-
derivatives, finite or infinite, exist and are everywhere equal, without thn difference
6—F being a constant; cf. H. Hahn [1] and" S. Ruziewic/ [1].)
In connection with these examples, it may be interesting to mention the
following theorem (vide G-. G-oldowsky [1] and L. Tonelli [8]):
G.9) Theorem. If a continuous junction F has a (finite or infinite) derivative
at each point of JR1 except perhaps at the points of an enumerable set, and if this
derivative is almost everywhere non-negative, the function F is monotone non-de-
non-decreasing.
Proof, Let E be the set of the points x such that the function F is not
monotone in auy neighbourhood of x. The set B is evidently closed, and the
function F is non-decreasing on every interval contained in CB. It 'therefore
has to be proved that the set B is empty.
Suppose, if possible, that E r?0, and denote for every positive integer n
by Pn the set of the points x for which the inequality 0<x'—as<l/ra implies
F(x'}—F(&)<! — (x'—x) however we choose x'. Similarly let Qn be the set of the
points x for which the same inequality implies F(x')—F[x)^—2(x'—x). We
see easily that the sets Pn and Qa are closed, and that they cover the whole straight
line 'JS, except at most the finite or enumerable set of the points at which the
function F is without a derivative. Consequently, by Baire's Theorem (Chap. II,
Theorem 9.2) the set E must contain a portion .which either 1° reduces to a single,
point, or else 2° is contained in one of the. sets Pn, or finally 3° is contained in
one of the sets Qn. The first case is obviously impossible, since the sot 14 has no
isolated points. Let us therefore consider case 2°, and suppose that there exists
a positive integer n0 and an open interval I such that 0=f2iMC-Pn,,. We may
clearly suppose that <5(I)< l/».o. Since by hypothesis, .F'(x)»0 almost everywhere,
the set Pn,, is- certainly non-dense. Let [a, &] denote any interval contiguouH
to E-I. The function F is then non-decreasing on [a, b] and this contradicts Uio
fact that, since a aud b belong to P,h, and b—a--:,l/n0, we have
[58]
The Perron-Stieltjes integral.
207
There now remains only ease 3°. In this case there exists an open interval I
such that the set E-I is non-empty and is contained in one of the sets Qn. But
then F^(x)^—2 everywhere in I, and F'(x)^0 almost everywhere, in I. Therefore,
by Theorem 7.3, the function F is non-decreasing in J, and this again is impos-
impossible since the interval I contains points of E in its interior.
We thus arrive at a contradiction in each of the three cases, and this
proves our assertion.
Let us mention a corollary of Theorem 7.9:
If'F is a continuous function having a derivative at every point, except per-
perhaps af those of an enumerable set, and if there exists a finite constant II such that
\F'(x)\^2I at almost all points x, then the function F is the indefinite integral of
its derivative.
*§ 8. The Perron-Stieltjes integral. Among the various gen-
generalizations of the Stieltjes type for the Perron integral (vide for in-
instance E. L. Jef f ery [2; 3], J. Bidder [9] and A. J. Ward [3]), that
. due to Ward has the advantage of including the others and of defining
the process of Stieltjes integration with respect to any finite func:
tion whatsoever. In. this § we shall give the fundamental defi-
definitions and results of the theory of Ward. For a deeper analysis,
in the case in which the function with respect to which we integrate
is of generalized bounded variation in .the restricted
sense [vide below, Chap. VII) the reader should consult the memoir
of Ward referred to.
As in the two preceding §§ we shall consider only functions
defined in JR^ i. e. functions of a linear interval or of a real variable.
We shall, moreover, restrict ourselves to integration of finite func-
functions. This restriction is essential for the methods which we shall
employ.
Given two finite functions / and Q-, an additive function of an
interval V will be termed major function of / with respect to G on
an interval Jo, if to each point x there corresponds a number e> 0
such1 that U[I)^f(x)G(I) for every interval I containing x and
of length less than s. The definition of minor function with respect
to G is symmetrical, and by following the method of § 6, p. 201, with
the help of the notions of major and minor functions with respect
to G, we define Perran-Stieltjes integration, or S'S-integration with
respect to any finite function G whatever. The ^-integral of a func-
function / with respect to a function G on an interval Io— [a, b~\ will
b
be denoted by (i?S)[f(x)<lG[x), or by [&•§) ff(x)dG[x). ,

208
CHAPTER VI. Major and minor functions.
If U and V are respectively a major and a minor function
of the same function / with respect to the same function G, their
difference 17-7 is evidently monotone non-decreasing. The criterion
for #S-integrability of a function is entirely similar to that for
c?-integrability given in § 6, p. 201, and it foUows that every func-
function which is c^-integrable on an interval Io, is so equally on each
subinterval of Io. We are thus led to the notion of indefinite
S'S-integral with respect to any finite function G. TMs indefinite inte-
integral is an additive function of an interval, and is continuous at each
point of continuity of the function G. Finally we observe that the
^-integral possesses the distributive property which we may express
as follows: If each of the two finite functions fx and /2 is iPS-integrable
on an interval Io with respect to each of the two functions Gx and <?2,
then eaeh linear combination of the functions ft and /2 is i7S-integrable
with respect to each linear combination of the functions Gx and G2,
and we have
for all numbers a1, «2, \ and 52.
If G{x)=x for every point x (or, what amounts practically
to the same, if G(I)=\I\ for each interval I) ^-integration with
respect to G coincides with ^-integration. In fact, if / is any finite
function, each major [minor] function of / with respect to the
function G[x)=x in the sense of Ward, is at the same time a major
[minor] function of / in the sense of the definition of § 3; the con-
converse is not true in general, but we see at once that if U is a major
function of / in the sense of § 3, the function U(cs) + ex is for eaeh
e>0 a major function of / with respect to G{x) = x. Thus the
Perron-Stieltjes integral includes the ordinary Perron integral, at
any rate as regards integration of finite functions. On the other
hand, the Perron-Stieltjes integral includes also the Lebesgue-
Stieltjes integral. We have in fact
(8.1) Theorem,, A finite function f integrable in the Lebesgue-Stieltjes
sense on an interval io=[fflc» VI w^1 respect to a function of bounded
variation G, is so also in the Perron-Stieltjes sense and we, have
(8.2)
The Perron-Stieltjes integral.
209
Proof. Let us denote for brevity, by A the right-hand side
of the relation (8.2). We may evidently assume that the function /
is non-negative and it is enough to consider only the following
two eases:
1° G is a continuous non-decreasing function. The
proof is then just as in Theorem 3.2. Let e be any positive number.
Since the function / is finite, there exists by the theorem of Vjtali-
Oarathe"odory (Chap. Ill, § 7) a lower semi-continuous function g,
integrable (G) in the Lebesgue-Stieltjes sense, such that g{x)>f(x) at
each point x and such that f[g{x)—f(x)]dG{x)<:?. Denoting by U
the indefinite integral (G) of the function g, and taking account
of the lower semi-continuity of g, we see easily that V is a major
function of / with respect to G on Io. Moreover, the function G being
continuous by hypothesis, the number A is equal to the integral
Jo
JfdG and we find 0^U(Io)—J.<s. By symmetry we determine
also a minor function V oif with respect to G on Io in such a manner
that 0=^ A—V(I0)<e, and this establishes ^S-integrability of /
on Io and at the same time the validity of the formula (8.2).
2° G is a non-decreasing saltus-funetion. Let us denote
by W»=u,,. the sequence of the points of discontinuity of G
which are in the ulterior of the interval Io; and let e be any positive
number and {&„}«=!,2,..- a sequence of positive numbers such that
(8.3)
and
Let us define a function h in Mlt by writing: h(x)=f(x) for all
the points x of Jo which are distinct from the points xn; h{xn) =/(*„)+hn
for n=l, 2, ...; and h{x)=f(a0) for a?<«0, and h(x)=f(b0) for a?>50.
Finally let us write, for each interval !=[«, 6],
b
U (J) = fji{x)dG{x)—{h{a)[G(a)—G{a—)]+h(b)lG(b+)—G(b)]}.
The function of an interval U thus defined is evidently ad-
additive, and as we easily verify, is a major function of / with respect
to G on Io. Moreover, it follows at once from (8.3) that 0 <?7(I0)—A^e.
Similarly we determine a minor function V of / with respect to G
so as to have 0^^. — V[I0)^.s; hence A = ($'S) I fdG, and this
completes the proof. '^
S. Saks. Theory of the Integral.
14

210
CHAPTEE VI. Major and minor functions.
Formula (8.2) brings out tlie faet that the definite Perron-Stioltjes and
Lebesgue-Stieltjes integrals are not always equal, even for a function / integrable
in both senses. This is due to the fact that the indefinite integral of Lobesgne-
Stieltjes is not in general an additive function of an interval. We could, of course,
modify the definitiou of this integral so as to ensure its additivity as a function
of an interval. The term in brackets { } would then disappear from the
formula (8.2), hut it would then be necessary to give up the additivity of the
indefinite Lebesgue-Stieltjes integral considered as a function of a set (of.
Chap. VIII, §2).
Let us mention further the following generalization of Theo-
Theorem 6.1 on derivation of the indefinite Perron integral:
(8.4) Theorem. If P is an indefinite i'S-integral of a finite function f
with respect to a function G, then, at almost all points x, the ratio
(8.5) [P(I)-f(x)G{I)]j\I\
tends to 0 as d (I)->-(), where I denotes any interval containing x.
Hence at almost all points x, P(x)—f{x)G{x) and P(oa)=f(x) G{x)
or else P(x)—f(x)G{x) and P{x) = j[x)G[x) according as f[x)^0 or
f(x)^.O; in particular P'(x) = Q at almost att points x where f(x) = O.
Proof. The proof is quite similar to that of Theorem 6.1.
Let Io be an interval, e a positive number, and U a major function
of / with respect to G on Jo such that U(I0)—P{I0)< s2. We write
E=JJ—P. The function H is monotorje non-decreasing, and we
have H'{x)<e at every point xelo except at most those of a set E
of measure less than e. Sow, since ?7(I)—/(»)<? (I) > 0 for every
point x and for every sufficiently small interval I containing x, the
lower limit of the ratio (8.5), as <5(I)-+0, exceeds —e at each point %
except at most at those of E. Therefore, e being any positive number,
this limit is non-negative for almost all points x. Combining this
with the symmetrical result for the upper limit of the same ratio,
we complete the proof.
Another generalization of Theorem 6.1, also due to Ward,
uses the following definition of relative derivation, which is slightly
different from that given in Chap. IV, § 2 (cf. A. J. Ward [3] and
A. Eoussel [1]).
Given two finite functions of a real variable F and G, we shall say
that a number a is the Boussel derivative of the function F with
respect to G at a point x0, if when I denotes any interval containing x0,
we have (i) F{I)—a-G(I)->0 and (ii) \F{I)—a-G(I)\IO(G;I)->0,
as <5(J)—>0 (the ratio in (ii) is to be interpreted to mean 0 whenever
its numerator and denominator vanish together; O(G;I) denotes,
in accordance with Ohap. m, p. 60, the oscillation of G on I).
[§S]
The Perron-Stieltjes integral.
211
When the oscillation of the function 0 at x0 is finite, the condition (ii)
evidently implies (i); however, when o((r; .?0) = +co, the condition (i) plays an
essential part, whereas (ii) is then satisfied independently of F and of a.
It is also to be observed that when o (&; x0) < +oo, and when Pis a function
which has the relative derivative F'o(x0) (cf. Chap. IV, § 2, p. 109), the latter is
also the Eoussel derivative of F with respect to 6. Finally, in the case of deri-
derivation with respect to monotoue functions, the two methods are completely
equivalent. In particular therefore, when G(x) = x, Eoussel derivation with
respect to 0 it equivalent to ordinary derivation.
The proof of the theorem on Boussel derivability of the in-
indefinite e^-integral is much the same as that of Theorems 6.1 and 8.4;
it depends, however, on the following lemma which may be regarded
as a generalization of a result of W. Sierpinski [4].
(8.6) Lemma. Let Gbe a finite function of a real variable, E a bounded
set in R-y, and 3 a system of intervals such that eaeJi point of E is a
(right- or left-hand) end-point of an interval C) of arbitrarily small
length.
Then, given any number /j,<C\G\_E]\, we can select from 3 & finite
system [Ik] of non-overlapping intervals such that
Proof. Suppose, for simplicity, that the set E lies in the open
interval @,1). For each positive integer n, let An and Bn denote
respectively the sets of the points of E each of which is respectively
a left- or right-hand end-point of an interval C) contained in @,1)
and of length exceeding 1/w. We evidently have E—lim{An-{-Bn) and
n
there therefore exists a positive integer n0 such that ^[J.^ -j-5nJ|>,«.
Suppose, for definiteness, that \G[AnJ\>^/x.
Xow, it is easily seen that, if |(?[J.,,J|=+oo, there exists apoint x0
such that \G[Ana- J]|=-|-oo for any interval J containing x0 in its in-
interior. Hence, from the family of intervals C) whose left-hand end-
points belong to Ana and whose lengths exceed ljnw we can ob-
obviously select an interval J so as to have |<?[I]| > \G[Ani[-I~\\^>\fi.
Suppose now that |ff[J.^,]l<+oo. Then, by induction, we can
extract from 3 a finite sequence of intervals {I*=(tf/(, &*)}*=i,2,...,P
in such a manner that, writing for symmetry 50=0 and fflp+i=l,
we have: (i) bk—aA > 1/m0 for k—1, 2, ..., p, (ii) bn-i < ak and
,,,,]|— ip)/n0 f°r ?=1,2,..., p, and (iii) the
14*

212 CHAPTER VI, Major and minor functions.
interval {bp, ap+1)={bp, 1) contains no points of Afln. Since, on
account of (i), we certainly have p < n0, it follows from (ii) and
(ill) that il|ff[I*3|>|ff[i.JIJ|-p-(|Q[[i.iJ|--W/»o>^i i- -e. that
the system of intervals {h) fulfils the required conditions.
(8.7) Theorem. Every finite function f which is tfS-integrable with
respect to a function G on an interval Io, is the JRoussel derivative with
respect to G of its indefinite ^-integral at each point x of Io except
at most those of a set E such that \G[E]\ = 0.
Proof. Let e be any positive number and U a major function
of / with respect to G such that U{I0)— P(I0)<?2, where P de-
denotes the indefinite ^-integral of /. Let us write JS—U—P, and
denote by Et the set of the points x of Io for which there exist
intervals I of arbitrarily small lengths, such that x el and that
H(I)^e-\G[I\\. It follows that each point of EE is an end-point of
intervals I, as small as we please, which fulfil the inequality
E(I)^^e-\G[I]\. Therefore, denoting by fi any number less than
|(x[J5y| and applying Lemma 8.6, we can determine in Io a finite
system of non-overlapping intervals {Ik} such that E(Ik)^~$?-\G[Ik]\
for h=l, 2,..., -p and that j?|6r[I*]| > ^p. Consequently, since E
is non-decreasing, ?2>?T(ro)^e^/4; and therefore |tt<4e, and hence
isTow let x be any point of Io. We have for every sufficiently
small interval I containing x,
and, unless x belongs to the set Es, we also have
Combining this with the similar npper evaluations of P(I) —f{x)G{I)
obtained by symmetry, we see, since ? is an arbitrary positive
number, that / is the Eoussel derivative of the function P with
respect to G, at every point x of Zo except at most those of a set E
such that \G[E]\ = 0.

CHAPTEB VII.
Functions of generalized bounded variation.
§ 1. Introduction. The definition adopted in Chap. I (§ 10)
as starting point of our exposition of the Lebesgue integral, con-
connects the latter with the conception of definite integral due to Leibniz,
Cauchy and Biemann (cf. Chap. I, § 1 and Chap. VI, § 1). On ac-
account of the results of § 7, Chap. IV, we may, however, also regard
the Lebesgue integral as a special modification of that of Newton
(cf. Chap. VI, § 1) and define it as follows:
(L) A function of a real variable f is integrable if there exists
a function F such that (i) F'(x) = f(x) at almost all points $ and
(ii) F is absolutely continuous.
The function F (then uniquely determined apart from an ad-
additive constant) is the indefinite integral of the function f.
A definition of integral is usually called descriptive when it is based
on differential properties of the indefinite integral and therefore connected with
the Newtonian notion of primitive; this is the case of the definition (JD) of the
Lebesgue integral. In the note of F. Riesz [9] the reader will find an elementary
and elegant account of the fundamental properties of the Lebesgue integral based
on a descriptive definition differing slightly from the one given above (an ac-
account based directly on the definition (L) is given in the first edition of this book).
By contrast to the descriptive definitions, we call constructive the
definitions of integral which are based on the conception of definite integral
of Leibniz-Cauchy, i. e. on approximation by the usual finite sums. Thus for
instance, the classical definition given by H. Lebesgue [1] in his Thesis may
be regarded as constructive (the reader will find a very suggestive explanation
of this definition in the note by H. Lebesgue [8]); cf. also the definitions of
Lebesgue integral given in the following memoirs: W. H. Young [3], T. H. Hil-
debrandt [1], F. Riesz [1] and A. Denjoy [7; 8].
As is readily seen, the definition {L) constitutes a modification
of that of the integral of Newton, in two directions: firstly, a gen-
generalization which enables us to disregard sets of measure zero

214 CHAPTER VII. Functions of generalized bounded variation.
in the fundamental relation F'(x)=f(x); and secondly, an essential
restriction, which excludes all but the absolutely continuous
functions from the domain of continuous primitive functions con-
considered. Some such restriction is, in fact, indispensable, unless we
give up the principle of unicity for the integral: to see this it is enough
to consider, for instance, singular functions which are continuous
and not constant^ and whose derivatives vanish almost everywhere
(cf. Ohap. in, § 13, p. 101).
But although the condition (ii) cannot be wholly removed
from the definition (L), it is possible to replace it by much weaker
conditions, and the corresponding generalizations of the notion of
absolute continuity give rise to extensions of the Lebesgue integral,
known as the integrals ¦?>„ and S> of Eenjoy. . ' .
We shall treat in this Chapter two generalizations of absolutely
continuous functions: the functions which are generalized abso-
absolutely continuous in the restricted sense or ACG,,,, and
those which are generalized absolutely continuous in the
wide sense or ACG. If, in the definition (L), we replace the con-
condition (ii) by the conditions that the function F is ACG* or ACG
respectively, we obtain the descriptive definitions of the integrals S\,
and S>. It must be added however that the second of these defini-
definitions requires a simultaneous generalization of the notion of deriv-
derivative, to which is assigned the name of approximate derivative
(or asymptotic derivative) and which corresponds to approximate
continuity (vide Ohap. IV, § 10). A function which is ACG (unlike
those which are absolutely continuous or which are ACG*) may in
fact fail, at each point of a,.set of positive measure, to be derivable
in the ordinary sense, and yet be almost everywhere derivable in
the approximate sense. Therefore, in order to obtain the definition
of the integral S> from the definition (L), it is necessary not only to
modify the condition (ii) as explained above, but also to replace
in the condition (i) the ordinary by the approximate derivative.
The integrals % and 3> -will T>e studied in the next chapter; the preliminary
discussion of their definitions just given, is intended to emphasize the important
part played by the generalizations of the notion of absolute continuity which are
treated in this chapter. The results of which an account is given in the following ss
are essentially due to Denjoy, Lusin and Kliintehine. The first definition" of
the integral % was given in notes dating from 1912 T>y A. Denjoy [2- 31 who
employed the constructive method baaed on a transfinite process (vide Chap VIII
15). These notes at once attracted the attention of N. Lusiu [2] who originated
the descriptive theory of .this, integral. Finally, A. Khintohine [I- 2] ami
L§2]
A theorem of Lusin.
215
A. Denjoy [4] defined, independently and almost at the same time, the process
of integration S1 as a generalization of the integral 'i1*. A systematic account
of these researches may be found in the memoir of A. Denjoy [6].
As shown by W. H. Young [6] the generalization of the Denjoy inte-
integrals can be carried still further if we give up, partially at least, the continuity
of the indefinite integral. For subsequent researches in this direction, vide J. C.Bur-
kill [5; 6; 7], J. Bidder [6;7], M. D. Kennedy and 3. Pollard [1], S.Ver-
blunsky[l], and J. Marcinkiewicz and A. Zygmund [1],
Except in a few general definitions in § 3, we shall consider
in this chapter only functions of a real variable. As therefore we
shall be employing in Hx notions established in the preceding chapters
for arbitrary spaces JRm, it will be convenient to add a few com-
complementary definitions.
We shall say that a point a is a right-hand point of accumulation
for a linear set E, if each interval [a, a-\-Jt], where A>0, contains
an infinity of points of E. A point of E which is not a right-hand
point of accumulation for the set E is termed isolated on the right
of this set. The definitions of left-hand points of accumulation and
of points isolated on the left are obtained by symmetry.
Similarly, for each linear set E, in addition to the densities
defined in § 10, Chap. IV, we define at each point x four unilateral
densities: two outer right-hand, upper and lower, and two outer left-
hand, upper and lower, densities of E. We shall understand by these
four numbers the values of four corresponding Dini derivates of
the measure-function (cf. Chap. IV, § 6) of IS at the point x. If at
a point x, two of these densities on the same side (right or left) are
equal to 1, the point x is termed unilateral (right- or left-hand) point
of outer density for the set E. The term "outer" is omitted from
these expressions if the set E is measurable.
Finally, we shall extend the notation of linear interval and
denote, for each point a of JEtlt by (— °o, a), (— oo, a],- (a,-f-co)
and [a, -4-°°) the half-lines x<_a, x^a, x>a and x^a respectively.
*§ 2. A theorem of Lusin. While discussing the significance
of the condition (ii) in the definition (L) of an integral, we remarked
that a continuous function which is almost everywhere derivable
is by no means determined (apart from the additive constant) when
we are given its derivative almost everywhere. It is, however, of
greater interest that, for a function /, the property of being almost
everywhere the derivative of a continuous function, itself represents ¦
no restriction at all, except, of course, in so far as it implies that

316 CHAPTEfi VII. Functions of generalized bounded variation.
the function / is measurable and almost everywhere finite (this last,
assertion follows, for instance, from the corollaries to Theorem 10.1,
p. 236). We shall prove this result; which is due to F. Lusin [I; 4]
(of. also E. W. Hob son [II, p. 284]), by means of two lemmas.
B.1) Lemma. If g is a function summable on an interval [a-,b], there
exists, for each «>0, a continuous fimotion 0 such that (i) G'{x) — g{x)
almost everywhere on [a, 6], (ii) G{a) = G{b)—0, and (iii) |ff(a;)|^e
at every point % of [a, b].
Proof. Let H(x) be the indefinite integral of g{x). We insert
in [a, b] a finite sequence of points a—ao<iai<i...<ian = b such
that the oscillation of H is less than e on each of the intervals
[ah ai+l] where i=0,1, ..., to—1. Let F (cf. A3.4), Chap. Ill, p. 101)
be a function which is continuous and singular on [a, b], monotone
on each interval [a,-, al+1] and coincides with the function H at the
end-points of these intervals. Writing G=3—F, we shall have
(i) G'{x)=E'{x)—F'{x)=S'{x)=g{x) at almost all the points x of
[a,b], (ii)G(a)=G(b)=0, and finally (iii) \G(so)\=\H(x)~-F(x)\^e
on each interval [ah ai+{\, and therefore on the whole interval [a, b].
B.2) Lemma. If g is a function which is summable on an interval
J=[a, b] and if P is a closed set in J, there exists for each. (>0o con-
continuous function G such thai (i) G'{x)—g{x) at almost all the points x
of J—P, (ii) G{x) = 0 and G'{x)=0 at aU the points x of P and
¦{iii) |(?(a! + A)|<e-|^| for every x of P and every h.
Proof- Let us represent the open set J° —P as the sum of a se-
sequence {Ik={ak, i*)}ft=i,2,... of non-overlapping open intervals, and in-
insert in each interval Ik an increasing sequence of points {af}t_
infinite in both directions and tending to ak or bk according as;'!?->—oo
or i-^+oo. Let us further denote, for each &=1,2, ..., and
*=0,- ±1, ±2,..., by e$ the smaller of the numbers ^(a^—a^lCk+lil)
and «•(&*—ajf+^/fJi+lii). Lemma2.1 enables us to determine in
each open interval Ik a continuous function Gk such that G'k[x)=g{x)
almost everywhere on It, <?(a«>)=0 for t=0, ±1, ±2, ..., and
iWKsj? vhen atf^x^a<i+ll. If we now write G(x)=G,,{x) for
xelk and k=l, 2,..., and G{x)= 0 elsewhere on J^, we see at once
that the function G is continuous and fulfils the required conditions
(i), (ii) 'and (iii).
[§2]
A theorem of Lusin.
217
B.3) Lusin's Theorem. If f is a function which is measurable and
almost everywhere finite on an interval J=[a, b], there always exists
a continuous function F such that F'{x)=f(x) almost everywhere on J.
Proof. We shall define by induction a sequence of continuous
functions {ffn}n=o,i,...> each of these functions being almost everywhere
derivable, and a sequence of closed sets {Pn{n=o,i,... in J, such that,
n n
writing Qn=?-E'k and Fn = J]Gi, the following conditions will be
satisfied for n=l, 2, ....
(a) F'n{x) = f{x) for xeQn,
(b) Gn(x)=0 for xeQn_v
(c) |6rn(a;+7i,)|^|7!,|/2" for every xeQn_1 and every h,
For this purpose, we choose G0(x) = 0 identically and P0=0,
and we suppose that for ra = 0,1, ...,r the closed sets Pn and the
continuous functions Gn, almost everywhere derivable, have been
defined so as to satisfy the conditions (a), (b), (c) and (d) for each
n^r. Since the function / is measurable and almost everywhere-
finite, and since the function Fr is almost everywhere derivable,
we can determine a measurable subset _Er of J—Qr such that
B.4) \J-Qr-Er\<ll(r+1),
and such that the derivative F'r(x) exists at each point x of the
set -Er and is bounded, together with the function f(x), on this set.
Hence by Lemma 2.2, we can determine a continuous function Gr+i,
almost everywhere derivable, in such a manner that (i) G>+i(a;) =
=f(x)—F'r{x) at almost all points of ErCJ—Qr, (ii) Gr+i(x)=G'r+1(x)=0
at all points of Qn. and (iii) |(?r+1(a; + 7i)|<:|7i|/2/"fI for every xeQr
and every h.
Now it follows from the first of these conditions and from B.4),
that there exists a closed set Pr+i(ZF<r such that:
B.5) \J-Q-PrJ<ll(r+l), B.6) G'r^(x)=f(x)-F'r(x) for xePr+i,
and we easily verify, on account of B.6), (ii), (iii) and B.5), that the
conditions (a), (b), (c) "and (d), still remain valid for n—r-\-l.
Let us now write:
B.7) F(x)=]imFk(x)=SGk(x),
k k
B.8) Q=:limQk=
k A

218 CHAPTEfi VII. Functions of generalized bounded variation.
In view of the condition (c), the series occurring in. B.7) con-
converges uniformly, and the function F is therefore continuous. Let xg
be any point of Q. Then for every sufficiently large integer n we1
have xot-Qn, and since
o + h)-F{xo) FB[x0 + h)—Fn[x0)
It=--n+i
h h
we find, on account of the conditions (a), (b) and (c), that
F[xo+h)-F[xo)
lim sup
A-W)
h
and so, that F'[xo)=f(xo). Now it follows from the condition (d)
that \J—Q\=0; we therefore have F'{x)=f{x) at almost all the
points x of J, and this completes the proof.
Theorem 2.3 remains valid for any space Um:
If f is a measurable function which is almost everywhere finite in a space fim,
there exists an additive continuous function of an interval F such lliat F'(x) = j{x)
almost everywhere in'JSm.
The proof is almost the same as that of Theorem 2.3. We may also, in the
foregoing statement, replace the ordinary derivative F'(x) by the strong de-
derivative (vide Chap. IV, § 2, p. 106),.but the proof is then more elaborate.
It may be remarked further that Lusin's theorem in tlie form B,3), is
obvious if the function / is summablej for / is then almost everywhere the derivative
of its indefinite integral. But this is no longer so when we wish to determine
a function F with a strong derivative almost everywhere equal to / (of. Chap. IV,
p. 132). Nevertheless, it can be shown that given in a space It,,, any suminable
function of a point f, there always exists an additive continuous junction of an inter-
interval, of bounded variation, F, such that I?'s(x) = f(x) almost everywhere in lim.
Lusin's method is applicable in several other arguments. It has been used,
for instance,' by J. Marcinkiewicz [1J, to derive the theorem:
There exists a continuous function of a real variable F which lias the following
-property: with each measurable function f, almost everywhere finite, there can be
associated a sequence of positive -numbers {hn} tending to 0 such that
lim [F(x+hn)—F(x)yhn=f(x)
n
at almost aU tlie points x.
§ 3. Approximate limits and derivatives. Given any
function F defined in the neighbourhood of a point soa of a space It,,,,
we shall call approximate upper limit of F at x0 the lower bound
of all the numbers y (-f-oo included) for which, the set E[?(*•)> y]
has x0 as a point of dispersion (cf. Chap. IV, § 10). Similarly, the
approximate lower limit of the function F at the point a>0 is the
§3]
Approximate limits and derivatives.
219
upper bound of the numbers y for which the set ~E[F[x)<y] has x0
X
as a point of dispersion. These two approximate limits of F at x0 are
called also extreme approximate limits and denoted by lim sup a,-pF[x)
and lim inf ap F [x) respectively. When they are equal, their com-
x^-x0
nion value is termed approximate limit of F at a?0 and denoted by
lim ap F[x).
It is easily, seen that if E is a measurable set for which x0 is
a point of density, then, in the preceding definitions of extreme
approximate limits, the sets ~E[F[x)>y] and "E[F [ot>) <iy] may
be replaced by the sets ~E[F[x) >y; xeE] and "E[F[x)<iy\ xeE]
X X
respectively. Hence
C.1) Theorem. If two functions coincide on a measurable set E,
their approximate extreme limits coincide at almost all points of E,
a%d in fact at every point of density of E.
We see further that if x0 is a point of density for a measurable
set E and if the limit of F[x) exists as x tends to x0 on E, then this
limit is at the same time the approximate .limit of F at the point xQ.
Therefore, if a function F is approximately continuous (cf. Chap. IV,
p. 131) at a point x0, we must have F(a;0)=lim ap F[x).
If x0 is a point of density for a measurable set E and if, further,
the function F is measurable on E, it is easily seen that the ap-
approximate upper limit of F at x0 is the lower bound of the num-
numbers y for which the set J&[F(x)^y; xeE] has x0 as a point of
X
density. It follows, by the definition of approximate lower limit,
that with the same hypotheses on the set E and on the function F, in
order that J = lim ap F[x), it is necessary and sufficient that for each
e>0 the set E[Z—e^F(x)^l+s; xeE] should have, the point x0 as
X
a point of density.
Let us remark finally that the following inequalities hold be-
between approximate and ordinary extreme limits:
C.2) b'rainfJ1(j))^liminfapi'(a;XlimsupapJ1(a3)^limsupJ'(a:i);
.V-f.V,, i'-M'o X+Xa x~^xo
and hence the approximate limit exists and is equal to the ordinary
limit, wherever the latter exists.

220 CHAPTER VII. Functions of generalized bounded variation.
In order to understand better the meaning of the definitions of ap-
approximate limits, it may be remarked that the definitions of the ordinary
limits are expressible in a very similar form. Thus the upper limit of F(x) at x0
may be defined as the lower bound of all the numbers y for which x0 is not a point
of accumulation ?or the set ~E[F(x)>y]. The inequality C.2) then becomes obvious.
X
For functions of a real variable, in addition to the approximate
limits defined above, and which in this case we call bilateral, we
introduce also four unilateral approximate limits. The approximate
wpper rigM-Mnd limit of a function F at a point x0 is the lower bound
of the numbers y for which the set ~E,[F(x)~>y; x>as0] has x0 as
a point of dispersion. This limit is written lim sup ap F{x). The
three other approximate extreme unilateral limits are defined
and denoted similarly.
These generalizations of the notion of limit lead very naturally
to parallel generalizations of derivates. Thus, given a finite func-
function of a real variable F, we define at each point xQ the approximate
right-hand upper derivate FIp(w0) and lower derivaie Ftp{x0), the
approximate left-hand upper derivate F^p(x0) and lower derivate
F^(x0), and the approximate bilateral upper derivate Fap(xa) and
lower derivate _FnV[x0), as the corresponding approximate extreme
limits of the ratio [F{x)—F(xo)]l(x—'Xo) as x~>x0. When all these
derivates are equal (or, what comes to the same, when S'^ix^^F^Xg)),
their common value is called approximate derivative of F at x0
and is'denoted by Xp(a?o)j if further, this derivative is finite, the
function F is said to be approximately derivable at oo0.
For some further generalizations, such as "preponderant derivates"
("nombres derives preponderants"), and for a deeper study of the prop-
properties of approximate derivates, the reader should consult A. Denjoy [6]
and A. Khintchine [5].
The properties of bilateral approximate limits, discussed
above, can be taken over, with the obvious formal modifications,
so as to apply to unilateral approximate limits. In particular,
Theorem 3.1 may be completed as follows:
C.3) Theorem. If two functions of a real variable coincide on a mea-
measurable set F, their approximate extreme limits and their approximate
derivates coincide respectively at almost all points of F, and in fact
at every point of density of F.
Also, if a function F is measurable on a set F, we have F'ult{x)=F'K{x)
at almost all the points x of F at which the function F has a derivative
with respect to tie set F.
L§4]
Functions VB and VBG.
221
§ 4. Functions VB and VBG. We shall denote by Y{F; F),
and call weak variation of a finite function F(x) on a set F, the upper
bound of the numbers ]}\F {h;)—F(ai)\ where {[a,, b{]} is any se-
i
quence of non-overlapping intervals whose end-points belong to F.
If Y(F;F)<.+oo, the function F is said to be of bounded variation
in the wide sense on the set F, or, simply, of bounded variation on F,
or VB on F.
In the special case in which the set E is a closed interval, we clearly have
V (F; E)=W (-P; E), i.e. the weak variation of the function F on E then coincides
with its absolute variation in the sense of Chap. Ill, §13.
The definition of functions of bounded variation in the wide sense on a set
thus constitutes a generalization (for functions of a real variable) of that of func-
functions of bounded variation on an interval. If E is a linear figure formed of
disconnected intervals we only get the inequality Y(F;E)^-"W(F;E), but
it is easy to see that even then the relation "W(i*';.E)<+O0 always implies
V(F;E)<+co. ¦ - '
Plainly, every function which is VB on a. set F is bounded
on F and is VB on each subset of F. Again, any fimction F which is
continuous on a set F and VB on a set AC1F everywhere dense in F
(cf. Chap. II, § 2) is VB on the whole set F (for then Y{F;F)=Y{F;A)).
Finally, if F and <? are two functions which are "bounded on a set F
and Jf denotes the upper "bound of the absolute values of these
functions on F, we have Y{aF+bG;F)^\a\-Y{F;F)+\b\-Y{O;F) for
each pair of constants a and b, and Y(F-G; F)^M-[Y(F;F)+Y{G;F)]
(cf. Chap. Ill, p. 97). Hence every linear combination, with constant
coefficients, of two functions which are VB on a set, and the pro-
product of the two functions, are themselves VB on this set.
A function F{po) is said to be of generalized bounded variation
in the wide sense on a set F, or simply, of generalized bounded variation
on E, or again, for short, VBG on F, if F is the sum of a finite
or enumerable sequence of sets on each of which F(x) is VB.
From what has just been proved for functions which are VB we
see at once that every linear combination of two functions which are
VBG on a set, and the product of the two functions, are themselves VBG
on this set.
D.1) Lemma. In order that a function F be bounded and non-
decreasing [of bounded variation] on a set F, it is necessary and
sufficient that F coincide on F with a function which is bounded
and non-decreasing {of bounded variation] on the whole straight
line jRx.

222 CHAPTEE VII. Functions of generalized bounded variation.
Proof. Let us denote for each x, by Eix) the set of the points
of E which belong to the interval (—00, a;]. We shall consider two
cases separately.
1° The function F is bounded and non-decreasing
on E. For each x, let G[x) denote the upper bound of the function
F on the set E{x), or else the lower "bound of the function F on JQ,
according as $(*>=# 0 or JJW=O. The function 6 thus defined is evid-
evidently bounded and non-decreasing on the whole straight line ttx
and coincides with the function F on E.
2° The function F is VB on E. For each point x, let
Y{x)=Y{I;E{x)) if #w=|=0, and V{x)=0 if JEw=O. We see at once
that the function T{x) is monotone and bounded on the whole
straight line M1 and that 7(x)—F(x) is non-decreasing and bounded
on E. Hence, by what has just been proved in 1°, there exists a func-
function Q(x) which is bounded and non-decreasing on Jtx and which
coincides on E with V (a?) —F[x). We have therefore F (x)=V(%) —G[x)
for every xeE, and since the function V(x)—Q(x), as difference
of two bounded monotone functions, is clearly of bounded variation
on JRy this completes the proof.
D.2) Theorem. Let F be a function which is measurable on a set E
and which is VB on a set E±dE. Then (i) F is approximately deriv-
derivable at almost all points of E± and (ii) there exists a measurable
set E2 such that F,±(ZE2CF and that F is VB on J?a.
Proof. By Lemma 4.1, there exists a function O which coin-
coincides with F on the set Ex and which is of bounded variation on
the whole straight line Mv Let E2 be the set of the points x of E
at which F{x)=G{x). Then since F is, by hypothesis, measurable
on E, the set E% must be measurable. Moreover, as E-^C'E^GE,
the function F is, with G, of bounded variation on E2, and by
Lebesgue's Theorem 5.4, Chap. IV, and Theorem 3.3, the finite
approximate derivative F'^{x)=Q'{x) exists at almost all the points
x of Es.
Theorem 4.2 leads at once to the following theorem, which
for the Denjoy integral takes the place of Lebesgue's Theorem on
derivability of functions of bounded variation:
D.3) Theorem of Denjoy-Khintchine. ¦ A function which is
•measurable and VBG on a set is approximately derivable at almost
all points of this set.
Functions AC and ACG.
223
Finally, if we make use of Theorem 9.1, Chap. IV, and
Lemma 4.1, we may complete Theorem 4.2 as follows:
D.4) Theorem. A function F which is VB on a set E, is derivable
with respect to the set E at almost all points of E. Moreover, if N de-
denotes the set of the points at which the derivative F'e{x) (finite or in-
infinite) does not exist, then the graph of the function F on N is of length
zero and consequently the set of the values taken by F on N is of meas-
measure zero; in symbols A(B{F; N))=\F[N]\ = 0.
For an extension of Theorem 4.3 to functions of two variables, vide
V. G. Celidze [1].
§ 5. Functions AC and ACG. A finite function F will be
termed absolutely continuous in the wide sense on a set E, or absolutely
continuous on E, or simply AC on E, if given any e~>0 there exists
an ??>0 such that for every sequence of non-overlapping intervals
{[at,, bk]} whose end-points belong to E, the inequality ?(h—a>k) <V
implies 2\F{bk)-F{ak)\<e. k
h
A function F will be termed generalized absolutely continuous
function in the wide sense on a set E, or generalized absolutely con-
continuous function on E, or finally AOG- on E, if F is continuous on E
and if E is the sum of a finite or enumerable sequence of sets En
on each of which F is AC.
These definitions generalize that of functions absolutely con-
continuous on a linear interval (cf. Chap. Ill, §§ 12, 13) and allow us
to generalize certain fundamental properties of the latter. We see
at once, by the arguments of the preceding §, that every linear combina-
combination of two functions which are AC [AOG] on a bounded set, and the
product of such functions, are themselves AO [AOG] on this set. Further,
every function which is AC on a bounded set E is VB on E. In fact, if F
is such a function, there exists an %>° suc]a *na't V[F; E-I)<*1 for
each interval I of length <^0. It follows that F is bounded on E.
Let M be the upper bound of the absolute values of F on E, and
let J be an interval containing E; then, J is the sum of a finite num-
number of non-overlapping intervals Ji,</2, ••¦ ,JP each of which is of
length <t]0, and we find Y(F;E)^SV{F; E-Jk)+2pM<+oo.
It follows at once that any function which is AOG on a set E
(bounded or unbounded) is VBG on E, and therefore, by the theorem
of Denjoy-Khintchine given in the preceding §, every function
¦which is AOG on a measurable set is approximately derivable at
almost all points of this set.

224 CHAPTEE VII. Functions of generalized bounded variation.
Nevertheless we can construct an example of a function wldch is ACG
on an interval and which is not derivable in the ordinary sense at the points
of a set of positive measure.
For this purpose, let H denote a bounded, perfect, non-dense set of
positive measure, with the bounds a and6. Let !=[«,&] and lot {!„= [a^bj
be the sequence of the intervals contiguous to E. We denote further by <?„ the
length of the largest subinterval of [a, b] which does not overlap the first n in-
intervals Iv I2 In of this sequence. Plainly
E.1) lim|IJ = 0 and liin ?>„=().
n i
Now let cn denote for each n=l, 2,..., the centre of the interval In, and
let F be the function defined on the interval I by the following conditions:
1° F(x)-Q for xeH; 2° F(cB) = \In\+en for ri=l> 2' -> 3° the fu^iction F is linear
hi each of the intervals [an, oj and [cn, bn] where n=l, 2,.... Thus defined, the
function F is continuous on I by E.1) and is AC on H and on each In; since
ln, it follows that F is ACG- on I.
We shall show that F is not derivable at any point aseJX. In fact, since 2?
vanishes on S, we have
E.2) V_(x)^.Q-^.F(x) jor every xeH.
If therefore a point x0 is a left-hand end-point of an In, there omi.be no derivative
F'(x0) since it is clear that F~(xa)=F+(!t>0)> 0 and therefore, by E.2), that
F{_xo)±F(xa). Similarly, JJUxJ < 0*?F(x0) if xa is a right-hand end-point of an
interval Io.
If, on the other hand, xoeM, x0Jpan and xo--^bn for n=l,2,..., denote byin
the suffix of that interval of the systBm I±,I2,..., In which is nearest to xa. Then
Km in=+oo and 0<|c/n—xo\ < |I/n|+Pn. and so, by the definition of JP(aj), we have
Fiain)-F(x<1)^\I(n\+Sin^\Iin\+en> |c/n-«0|- Since Mm oin=x0, it follows that either
n
F(xo)^g>\ or F(x0)^.—1, which by E.2), proves that F is not derivable at x0.
Let us remark, in conclusion., that a function F which is con-
continuous on a set E and which is AC on a subset of E everywhere dense
in JS, is AC on the whole set JB.
• . § 6. Lusin's condition (N). A finite function F is said to
fulfil the condition (N) on a set E, if \F[E]\ = 0 for every set EC_E
of measure zero (for the notation cf. Ohap. Ill, p. 100). Clearly,
a function which fulfils the condition (N) on each of the sets of a finite
or enumerable sequence, also fulfils this condition on the sum of these sets.
,Tke-;- condition (N) was introduced by N. Lusin [I, p. 109], who
was the first to recognize the importance of this condition in the theory of the
integral. It is easy to see that in the domain of continuous functions the con-
condition (N) is necessary and sufficient in order that the function should transform
every measurable set into ameasurable set (cl H. Kademackor [1] and II. Halm
[I, p. 586]). Among the more recent researches devoted to the condition (N) and to
other similar conditions (cf., below, Chap. IX) the reader should consult above
all N. Bary [3].
[§6]
Lusin's condition (X),
225
F.1) Theorem. A function whieh is ACG- on a set necessarily fulfils
the condition (N) on this set.
Proof. Since each set on which a function is ACG is the sum
of a sequence of sets on which the function is AC, it will suffice
to prove that \F[E]\=0 whenever E is a set of measure zero and F
a function AC on E.
For this purpose, let e be any positive number. We denote,
for brevity, by M{~E) and m{B) respectively the upper and lower
bounds of F on J8, when E is any subset of E, and we write
M{E)=m{B) = d in the case in which 25—0. Since the function F is AC
onB", there exists a number ^> 0 such that y[M(E-Ik)~m[E- J*)]<e
k
for every sequence of non-overlapping intervals {I/,} which sat-
satisfies the condition J^I*] < tj. Now since the set E is of measure
zero, we can determine a sequence of non-overlapping intervals
{It} which satisfies this last condition and which covers, at the same-
time, the whole set E. Therefore, since \FlH-Ik]\^M(H-Ik)—m[E-Ik)
for each fe, it follows that ^[BH^e. Hence, e being arbitrary,
|JF[JET]|=O.
It follows from Theorem 7.8 A°), Ohap. IV, that every func-
function which is absolutely continuous on an interval and whose deriv-
derivative is almost everywhere non-negative, is monotone non-decreasing.
With the help of Theorem 6.1, this result can be extended to functions
which are ACG and we have:
F.2) Theorem. Every function F (x) which is ACG o% an interval I
and for which we have almost everywhere in this interval F's.v(%)~^-(},
or more generally, JP+(aj) ^ 0, is monotone non-decreasing.
In particular therefore, if the app'oximate derivative of a func-
function which is AOG on an interval vanishes almost everywhere on this
interval, then the function is a constant.
Proof. Let e be any positive number and let G{x)~F{x) + ex.
The function G is then ACG- on the interval I (together with the
function F), and moreover, we have G+[oc)=F (x)Jrs^:e>0 at
almost all the points x of I. Hence, denoting by 21 the set of the
points x at which G+{x)^.0, we have |2?| = 0, and this implies, by
Theorem 6.1, that |<?[2I]| = 0. Thus the set G[E] cannot contain
any non-degenerate, interval, and by Theorem 7.1, Chap. VI, the
function G (x)=F{x)-j-ex is non-decreasing on I. It follows at once,
by making e->0, that the function F is itself non-decreasing.
S. Saks, Theory of the Integral.
15

226
CHAPTER VII. Functions of generalized bounded varinliiou.
If we analyze the preceding proof, wo notice that the hypothecs of
generalized absolute continuity of F(x) has been used only to nhow that
every function of the form F(x) + ax, where e> 0, fulfils tho condition (N). It is
remarkable that the condition (N) need not remain satisfied whon wo add a
linear function to a function fulfilling the condition, oven whon thin last
function is restricted to be continuous (vide S. Majuirkiow.ioz [1]). For
this reason it is not enough to suppose in the preceding proof that the function
F(x) merely fulfils the condition (N).
Nevertheless, Theorem 6.2 itself does remain true for arbitrary funutions
which fulfil the condition (N.). The theorems which will bo proved in Cliap. IX,§7,
include a more general result, namely that every continuous junction which jviljils
the condition (N) and wlwse derivative is nmi-neqative at almost all tlie points at
loMeh it exists, is monotone non-deareasini/.
We shall show {vide, below, Theorem 6.8) that for continuous
functions of generalized bounded variation on closed sots, the con-
converse of Theorem 6.1 is true, i. e. that in this case tho condition
(S) is equivalent to generalized absolute continuity. Similarly, for
continuous functions of bounded variation the condition (E") is
equivalent to absolute continuity in the ordinary sense.
We shall begin with a lemma which will also prove useful
elsewhere.
F.3)- Lemma. If, for a finite function F, the inequalities F+{x) ^ M
and F~{x)~^—M, where M is a finite non-negative number, hold at
each point x of a set B, then \F[B]\^M-\B\.
Proof. Let e be any positive number. Let Bn denote for each
positive integer n the set of the points x of B for which we liave
F{t)—F{x)^(M+e)-\t—x\ whenever \t— aij<l/». The 'sets Du evid-
evidently constitute an ascending sequence and we see easily that
B = lim Bn.
t With each 2>n we can associate a sequence of intervals {JT/">}/,=i,Si...
which covers Bn and fulfils the condition
F-4) J?|i*n)|<|25,,|4-?,
k
and in which, further, no J$ has length greater than ljn. By defini-
definition of Bn, we therefore have, for every pair %,, xz of points of
B,,-Ik\ the inequality \F[xz) —F^xJl<(M + e)¦ \x, —xx\<(M + e)• \I{^\,
so that \F[Bn-I^]\^(M+s)'\I^\. In view of "the inequality F.4)
it therefore follows that, for every n,
and, by making first w-voo and then e->0, we derive \F[D]\ ^M-
[§6]
Lusin's condition (JJ).
227
F.5) Theorem. If a function F is derivable at every point of a meas-
measurable set B, then
F.6) ;JF[D]j </>'(«)! (ta-
(tail
Proof. We may clearly assume that the set B is bounded.
Given any e>0, let Br, denote, for each positive integer n, the set
of the points xeB, at which (n—l)e< F'(x)\<%-s, We then have,
by the preceding lemma,
and hence, s being arbitrary, the inequality F.6).
The formula F.6) remains true when we replace in it the derivative H"(x)
by any Dini deiivate, provided however that we restrict the latter to be finite
in D. The proof then becomes rather more elaborate and requires certain general
theorems on derivates which will be established later (vide Chap. IX, §4).
F.7) Theorem. In order that a function F{x) which is continuous
and VB on a bounded closed set 25, be AC on E, it is necessary and
sufficient that F[x) fulfil the condition (S) on this set.
Proof. In view of Theorem 6.1, it remains to be shown that
the condition is sufficient.
Suppose then that F fulfils the condition (N) on 23. Let aQ
and b0 be the bounds of E, and let 0 denote the function which
coincides with F at the points of F and is linear in the intervals con-
contiguous to E. The function (? is evidently continuous and of bound-
bounded variation, and fulfils the condition (N) on the whole interval
I>o, &o]-
Given any subinterval I=[a,b] of [a0, i0], let us denote by B
the set of the points of I, at which the function Q is derivable, and
write S=I—B. Plainly [J3T| = 0, and therefore also \G[H]\ = 0.
On the other hand, since the interval with the end-points G{a)
and <?(&) is contained in (?[!], we have by Theorem 6.5
|ff (J) -
Since this inequality is valid for every subinterval I=[a,b]
of [a0, be] and since by Theorem 7.4, Chap. IV, the derivative Q'{x)
is sumniable on [ao,fio], it follows that the function G is AC on [a0, &0],
and therefore that F is AC on the set E, where F and G coincide.
15*

228 CHAPTEE VII. Functions of generalized bounded variation.
It is easy to see that the same argument loads to a more general theorem:
in order that a continuous function F which is continuous on cm interval Jo bu ab-
absolutely continuous on this interval, it is necessary and sufficient ihat .F fulfil the
condition (8) on Io and that Us derivative exist almost, everywhere on Ia and be sum-
viable on Io. This theorem will again be generalized in Chap. IX, §7.
F.8) Theorem. In order that a function F which is continuous
and VBG on a closed set E be AOG on E, it is necessary and sufficient
that F fulfil the condition (S) on this set.
Proof. In view of Theorem 6.1, we need only prove the con-
condition (8) sufficient. Now, since F is VBG on the set E, this set
is expressible as the sum of a sequence of bounded sets {En} such
that the function F is VB on each Ea. By continuity of F on the
closed set E, we may suppose (cf. § 4, p. 221) that each set En is
closed. Since further F fulfils the condition (N) on E, it follows
from Theorem 6.7 that the function F is AC on each E,,, and there-
therefore ACG on E.
§ 7. Functions VB* and VBG*. We shall denote by Y*(F;E)
and term strong variation of a finite function I on a set E, the
upper bound of the sums ?O(F;Ik) where {Ia} is any sequence of
k
non-overlapping intervals whose end-points belong to E (in accord-
accordance with Chap. Ill, p. 60, O(F;Ik) denotes the oscillation of F
on the interval I*). If V*(.F, .E)<-f-oo, the function F will be said
to be of bounded variation in the restricted sense on the set E, or
VB* on E.
Following the order of the definitions of §4, we shall say
further that a finite function is of generalized bounded variation in
the restricted sense, or simply, is VBG* on a set E, if E is the sum
of a finite or enumerable sequence of sets on each of which, the
function is VB*.
In the special case in which the set E is a closed interval, we
clearly have Y*(F;E)=Y{F;F,)=W{F;E). It is easy to see that
we always have Y{F;E)^Y^(F;E); so that every function which
is VB* on a set, is VB on this set, and consequently, every function
which is VBG* on a set, is VBG on this set. We next observe (by
using trivial inequalities for the VB* case, and then.ee passing on
to the VBG* case) that every linear combination, with constant coef-
coefficients, of two functions which, are VB* [VBG*], and also the product
of two such functions, are themselves VB* [VBG,].
Functions VB* and VBG*.
229
Let us observe that, for a function, the property of being VB, VBG-, AC,
or ACG, on a set B depends solely on the behaviour of the function on B; whereas
the property of being VB* or VBG* on B depends on the behaviour of the func-
function on the whole of an interval containing the set B. In other words, of two
functions which coincide on a set B, one may he VB* or VBG* on E and the
other not. The same remark applies to the property of being AC* or ACG* with
which we shall be concerned in the nest §.
We have remarked in § 4, p. 221, that a function which is
continuous on a set E and which is VB on an everywhere dense-
subset of E, is necessarily VB on E. A similar result is true for func-
functions which are VB*, the assumption of continuity of the given
function being now superfluous. We have in fact:
G.1) Theorem. Every finite function F which is VB* on a bounded
set E is equally so on the closure E of this set.
Proof. Let a and b denote the bounds of E and therefore
also of E. Let a=ao<Ca1<C...<lan=b be any finite sequence of
points of E; we write I=[a,b] and IA=[a,*_i, «*] for lc—1,2,...,n.
We shall say that an interval Ik is of the first class if it contains
points of E, and otherwise of the second class. The intervals Ix
and In are clearly of the first class, and we see easily that, if an
interval Ik is of the second class, then both the adjacent intervals
Ik-x and Ik+i are certainly of the first class.
Let us denote by l = i0<%<...<«',=«¦ the suffixes of the
intervals Ik of the first class and by jo<j1<...<js those of the
second. With each interval Ii of the first class we associate a point
&&eZj -E and we write Jh — [b!,-i,bh] for h=l,2,... ,r. It is easy
to see that
and
h=0
Hence, ?O(JF;I*)< 3-ZO(F; Jh)+2-O(F;IX3-[Y*(F;E)+O(F;I)],
and therefore Ysg{F;E)^3-[Y:,{F;E)+0{F;I)]<+x. This completes
the proof.

230 CHAPTER VII. Functions of generalized bounded variation.
G.2) Theorem. If a function F is VBG* on a set E, then F is
derivable at almost all points of this set; and further if N denotes the
set of the points x of E at which the function has no derivative, finite
or infinite, then \F[N]\=A{B{F; AT)}=0.
Proof. We may clearly suppose that the set E is bounded
and that the function F is VB* on E. Moreover, by Theorem 7.1,
we may suppose that the set E is closed.
Let therefore a and b denote the bounds of E on the left and
on the right, and {In}n=i,2,... the sequence of the intervals contiguous
to E. Writing mn and Mn respectively for the lower and upper
bounds of F on I,,, we define two functions m(x) and M(x) on
[a,b] making m{x) = mn and M{x) = MH for a;el,0, where «=1, 2,...,
and m{x)=M{x)=F{x) for xeE. The two functions m{x) and M{x)
thus defined are plainly of bounded variation on the whole interval
[a,b] and coincide with F[x) on the set E. Therefore, denoting
by -A70 the set of the points xeE at which either one at least
of the (finite or infinite) derivatives M'(x) and m'(x) does not
exist, or both exist without being equal, we find by Theorem 9.1,
Chap. IV, that
G-3) \F[No]\ = \A{B(F;N0))\ = 0.
On the other hand, m{x)=F[x) = M{x) at every point x of E, while
m{x)^J?{x)<s.M{x) on the whole interval [a,b]. It follows that
the derivative F'(x)=m'(x)= M'(x) exists at each point x of E,
except at most those of the set No which is subject to the rela-
relation G.3). Finally, since the functions m(x) and M(x) are deriv-
derivable almost everywhere on the interval [a, b], tie function F
must be derivable at almost all points of E, and this completes
the proof.
Theorem 7.2 (for continuous functions and in a slightly less complete form)
was first proved by Denjoy and "by Lusin, independently. It plays in the theory
of the Denjoy-Perron integral (vide, "below, Chap.VIII) apart similar to that of
Lebesgue's Theorem (Chap.IV, § S) in the theory of the Lebesgue integral. A cor-
corresponding part is played in the theory of the' Denjoy-Khintchine integral by
Theorem 4.3. But the latter is stated in terms of approximate derivation (of.
the example of p. 224) whereas Theorem 7.2, which requires uo modification
of the notion of derivative, is, for functions of a real variable, a direct generaliza-
generalization of Lebeague's Theorem.
[§8]
Functions AC* and AC'G*.
231
§ 8. Functions AC* and ACG*. A finite function F is said
to be absolutely continuous in the restricted, sense on a bounded set E,
or to be AO* on E, if F is bounded on an interval containing E and
if to each e> 0 there corresponds an. rp> 0 such that, for evexy finite
sequence of non-overlapping intervals {I*} whose end-points belong
to E, the inequality _S|I*|<»7 implies y,O[F;IJl)<.s.
k k
A function will be termed generalized absolutely continuous on
a set E, or AOG* on E, if the function is continuous on E and if
the set E is expressible as the sum of a sequence of bounded sets
on each of which the function is AO*.
In the case in which the set E is an interval, the class of func-
functions AO* on E coincides with that of the functions which are ab-
absolutely continuous on E in the ordinary sense. Every function
which is AC* on an arbitrary set E is AO on 25, and every function
which is A0G% on E is ACG on E. On the other hand, anvy function
which is AC* on a bounded set is VB* on this set, and therefore, any
function which is ACG* on a set is VBG* on this set. To see this,
let F be AC* on a bounded set E. We can then determine a positive
number rj0 such that Y^F;E-1)^.1 for every interval I of length
less than t]0. Let J be the smallest interval containing E, let M be
the upper bound of \F(x)\ on J, and suppose J expressed as the
sum of a finite number of non-overlapping intervals J1,J2,...,JP
each of length less than ??0. We shall then have
and this shows that the function F is VB* on E.
Thus a function which is AC* on a bounded set E is both AO
and VB* on this set, and similarly a function which is ACG* on H.
is both ACG and VBG* on E. The converse also is true, provided
that the set E is restricted to be closed {vide, below, Theorem 8.8).
Instead of giving a special proof of this result, we shall establish
some more general theorems about the relations between the notions
VB, AO, VB*, AC*, VBG, AOG, VBG* and ACG*.
(8.1) Lemma. Let E denote a bounded closed set, {Jt} the sequence
of the intervals contiguous to F, and Io the smallest interval containing E.
Then, for any function F which is finite on Io, we have
(8.2)

232 CHAPTER VII. Functions of generalized bounded, variation.
Proof. Let M,m and M0,m0 be the bounds (upper, lower)
of F, on E and on Io respectively. Let M'o be any finite number
less than Mo, and cc0 a point of Io such that M'0^F(x0). If we have
aro"e JZ?, this inequality implies M'0<M, while if sc0 belongs to an
interval, Jha say, of the sequence {J*}, M'o^M+OiF-jJ^). Hence
(8.3)
and similarly
(8.4) ma^m —
R
On subtracting (8.4) from (8.3), we obtain, since M—m^.Y{F;E),
the relation (8.2).
(8.5) Theorem. In order that a function F which is VB [AC]
on a bounded closed set E, he VB* [AC*] on E, it is necessary and suf-
sufficient that the series of its oscillations on the intervals contiguous
to E be convergent.
Proof. The necessity of these conditions is obvious (cf.
above, p. 231); we have therefore only to prove them sufficient.
Let then {Jj} denote the sequence of the intervals contiguous
to E, and suppose that
(8.6)
We shall consider the two cases separately:
1° The function F is VB on E, i.e. Y(F;E)< + oo. Then
by Lemma 8.1, we have for every sequence {!„} of non-overlapping
intervals whose end-points belong to E,
It follows by (8.6) that V*(_F; E)< + °o, i. e. that the function F
is VB* on E.
2° The function F is AC on E. Then,, given any e>0,
there exists a number r\ > 0 such that, for every sequence of non-
overlapping intervals {!„} whose end-points belong to E, the in-
inequality ?\In\<r] implies ^(J^.^x^. Now by (8.6), there
exists a positive integer 7i;0 such that
CO
(8.7)
'*„[>
Denote by % the smallest of the Tco+1 numbers th IJJ, |J8|, ..., \JW
and let {!„} be any sequence of non-overlapping intervals "with end-
Li 9]
Definitions of Denjoy-Lusin.
233
points in E, the^sum of whose lengths is less than i]a. None
of these intervals In can contain one of the first fc0 intervals of the
sequence {Jlt}, and it follows from (8.7) and from Lemma 8.1, that
?Z Ze. Therefore the function F is AC*
on E, and this completes the proof.
(8.8) Theorem. In order that a function F be AC* [ACG*] on a bound-
bounded closed set E, it is necessary and sufficient that F be both VB* and
AC [VBG* and ACG] on E.
Proof. The necessity of these conditions is obvious, so that
we have only to prove them sufficient.
Now, if the function F is both VB* and AC on 22, it follows
at once from Theorem 8.5 that F is AC* on E. If on the other hand,
F is VBG* and ACG on E, we can express the set E as the sum of
a sequence of sets {En} on each of which F is both VB* and AC.
Since F is AOG, and so continuous, on the set E, which is by hypo-
hypothesis closed, F is AC on the closure En of each En. Similarly, by
Theorem 7.1, F is VB* on each Ett. Therefore by what has just been
proved, F is AC* on each of the sets En and so, AOG* on the set E.
Theorem 8.8 ceases to hold if -we remove the restriction that the set & is
closed. Let B be the set of irrational points, and {ajn!=12 *ne sequence of ra-
rational points, of the interval [0, 1]; and let _F(k)=0 ior'xejB, and F(an)=l/2"
for m=l,2 The function F thus defined is evidently VB* and AC on B. To
show that F is not AC*, nor even ACG*, on B, suppose that the set B is the sum
of a sequence of sets {En} on each of which F is AC*. By Baire's Theorem (Chap. II,
Theorem 9.2), one at least of the sets Bn would be everywhere dense in a (non-
degenerate) snbinterval of [0, 1]. But this is plainly impossible, since every snb-
interval of [0, 1] contains, in its interior, points of discontinuity of the function F.
§ 9. Definitions ol Denjoy-Lusin. The definitions which
we have adopted in this chapter for the classes of functions VBG,
AOG, VBG* and ACG* are based on the ideas of A. Khintchine [3].
Eather different definitions were given by N. Lusin [I] and
A. Denjoy [6], which are equivalent to those of Khintchine when
we restrict ourselves to continuous functions. We give them here,
in the form of necessary and sufficient conditions, in the following
theorem.
(9.1) Theorem. In order that a function which is continuous on
a closed set E, be VBG [VBG*, ACG, AOG*] on E, it is necessary
and sufficient that every closed subset of E contain a portion on which
the function is VB [VB*, AC, AC*].

234 CHAPTER VII. Functions of generalized bounded variation.
Proof. We shall deal only with the VBG case, the proof for
the other three cases being quite similar.
1° The condition is necessary. Let F be a function which
is continuous and VBG on E. We can then express the set E as
the sum of a sequence of sets {En} on each of which the function F
is VB and, by continuity of F, the sets En may be supposed, closed.
Then by Baire's Theorem (Chap. II, § 9), every closed subset of E
has a portion P contained wholly in one of the sets En. The func-
function F, which is VB on each En, is thus certainly VB on P.
2° The condition is sufficient. Suppose that F is a continuous
function on E and that every closed subset of E contains a portion
on which F is VB. Let {I,,} be the sequence of all the open intervals
I with rational end-points such that F is VBG- onE-I. Let Q ==]>]E-In
n
and E—E—Q. Plainly F is VBG- on Q and we need only prove
that the set E is empty.
Suppose therefore that E^O. Since if is clearly a closed set,
there exists, by hypothesis, an open interval J sucli that II-J -|-0
and that the function F is VB on H-J. We may evidently assume
that the end-points of J are rational. Therefore, the function F,
which is VBG on the set Q, is also VBG on the set E-JC.S-J+Q.
This requires J to belong to the sequence of intervals {I,,} and we
have a contradiction, since the set if, by definition, has no points
in common with any of the intervals I,,.
Theorem 9.1 shows in particular that every continuous function which
is VBG- on an interval I is at the same time "VB on some sulrinterval of I. It fol-
follows that for every continuous function which is VBG- on an interval /, there
exists an everywhere dense system of snbintervals on each of which the function
is almost everywhere derivable, although this function may, as shown in § 5,
have no derivative at the points of a set of positive measure.
§ 10. Criteria for the classes of functions VBG*, ACG*,
VBG and ACG. A series of theorems enabling us to distinguish
certain types of functions of generalized bounded variation and cer-
certain types of generalized absolutely continuous functions, are due
to A. Denjoy [6].
A0.1) Theorem. If F(x) is a function which fulfils at all points
of a set, except at most those of an enumerable subset, one of Ike
inequalities
A0.2) F{x)< + oo or #(fl3)>—oo,
then the function F(x) is VBG* on this set.
[§ 10] Criteria for the classes of functions VBG*, ACG*, VBG and ACG. 235
Proof. It is enough to show that the set E of the points at
which we have, say, F{x)<+oo, is the sum of an enumerable in-
infinity of sets on each of which F is VB,,,.
For any positive integer n, let En denote the set of the points x
of E such that for every t,
A0.3) 0<|i— *|<l/w implies [F{t)—F{x)~\l(t—x)^n.
Further, for each integer i, let Eln denote the part of En sit-
situated in the interval [ijn, (i+l)/ra], and a'n, b'n the lower and
upper bounds of those of the E'n which are not empty. We have
clearly E=2jISn=2j
/[=1 ,,=1 t=—co
Let now F,,(x)=F(x)—n%. For every point xeEn and for
every point t which fulfils the first of the inequalities A0.3), we
then have [Fn{t)— F,,(x)]l(t— «)<0. In particular, given any pair
of points xlt %2 (where x1^.x2) of En, we obtain
A0.4) F,,{an) ^ Fn(x{) J? Fn(ccz) > F,^),
and for every t such that ^^i^^a we find that F^x-^^Fniij^FJjOo)-
This last relation implies that, for every interval I={a,§~\ whose
end-points belong to the set E'n, we have 0(Fn;I)=Fn[a)—Fri{^),
and therefore by A0.4), for every sequence (i;-= [a;-, ft]j of such
intervals (which do not overlap),
The function Fn{x), and therefore also the function F{x)=Fn(x)+nx,
is thus VB* on every set E'n and this completes the proof.
A0.5) Tlieorem. If F(sc) is a function which fulfils at all points
of a set E, except, perhaps, at those of an enumerable set, one at least
of the conditions
or —o
A0.6) — o
then the set E is the sum of an at most enumerable infinity of sets
on each of which the function F is AC*.
If, therefore, we are given further that F{x) is continuous on E,
then F(x) is ACG* on E.

236 CHAPTER VII. Functions of generalized bounded variation.
Proof. It is enough, to show that if at every point x of a set A
the two extreme right-hand derivates PH"(») and F+{x) arc finite,
then A is expressible as the sum of an at most enumerable infinity
of sets on each of which the function F is AC*.
Let An denote, for each positive integer n, the set of the points
soeA such, that, for every*,
A0.7) 0<i—%<.lln implies \F[t)—F{x)\<.n-{t—x)\
and, for each integer *, let A'n denote the common part of A,, and
CO OO -{-CO
of the interval [¦?/«., (*+!)/«]• Plainly A—EAn=? SX,.
Now, if 1= [>!, *2] is any interval whose end-points belong to
An, we have, for every tel, the inequality O^i—x^l/n, and so,
on account of A0.7), \F{t)— F^l^nit — x^^n-ll]. This gives us
0{F;I)<!2n-\I\; and therefore for any finite sequence \Ij) of such
intervals, ?O(F;IJ)<;2n-S\Ij\. It follows that the function F
i i
is AC* on each of the sets A'n, and this completes the proof.
Theorem 10.5 shows, in particular, that every function which is continuous
and everywhere derivable (even only unilaterally) is ACG*. Nevertheless as we
saw in Chap. VI, p- 187, such a function need not be absolutely continuous.
In view of Theorem 7.2, we may state also the following corollary of Theo-
Theorems 10.1 and 10.5: A function F which fulfils at each point of a set E one at least
of the inequalities A0.2) or A0.6), is derivable at almost all points of JS. In part-
particular therefore, the set of the points at which a junction has (on one side at least)
its derivative infinite, is of measure zero. These statements will be generalized
in Chap. IX, §4.
Theorems 10.1 and 10.5 contain sufficient conditions in order that a func-
function be VBG* or ACG^, but these conditions are clearly not necessary. Never-
Nevertheless, by employing the notion of derivates relative to a, function (of. Chap. IV,
p. 108), it is easy to establish conditions similar to those of the preceding theorems,
the conditions being this time both sufficient and necessary. Thus, as shown
by A. J. Ward [3]:
In order that a finite function F be VBG-* on a set E, it is necessary and suf-
sufficient that there exist a bounded increasing function TJ such that the extreme deriv-
derivates of F with respect to TJ are finite at each point of E except, perhaps, those of
an mumerable set.
1° In order to establish the necessity of the condition, let us suppose first
that the function F is VB* on JS. In view of Theorem 7.1 we may assume that
the set E is bounded and closed. Let [a,i] be the smallest interval' i;ontmning E,
and, for each point x of the interval [a, b], let V^x) and 72(x) donate the strong
[§ 10] Criteria for the classes of functions VBG*. ACG*, VBG and ACG. 237
variations of F (of. §7) on the parts of B contained in the intervals [a, a:] and
[x,b] respectively. Filially for each x of [as, 6], let T(x)=V1(x)—V^{x)+x. The
function V thus defined is increasing and finite on [a, &], and can therefore be
continued as a bounded increasing function on the whole straight line -ft,. We
see at once that throughout the set E, except at most at the points a and b, the
derivates of the function F with respect to V aTe finite and indeed cannot ex-
exceed in absolute value the number 1.
Suppose now given any function F which is VBG* on E. The set E is then
expressible as the sum of a sequence {En) of sets on each of -which the function
F is VB*. Consequently, by what has just been proved, there exists for each n
a bounded increasing function Vn with respect to which the function F possesses
finite derivates at each point of the set En except at most at the bounds of
this set. Therefore, denoting by Mn the upper bonnd of |rn(i)| and wilting
U(x)—S7n(x)/2"Jiln, we see at once that the function 17 thus defined is in-
n
creasing and bounded and that at each point of E, except perhaps those of an
enumerable set, the function F possesses finite derivates with respect to TJ.
2° The condition is sufficient. Let J1 be a finite function having at each
point of E, except perhaps at those of an enumerable subset, finite derivates
with respect to a hounded increasing function TJ. For each positive integer n,
let En denote the set of the points x of E for which the inequality t—x\^.l/n
implies \F{t)—J?(ii)|<:ni|D'(t) — TJ(x)\; and let each En be expressed as the smn
of a sequence {E'n}t=-,2,.., of sets of diameter less than 1/n. We see easily (as in
the proof of Theorem 10.1) that the function F is VB* on each set E'n, and
since the sets E1,, plainly cover all hut an enumerable subset of JS, it follows at
once that the function F is VBG* on E. This completes the proof.
If we analyze the first part of the above argument, we see that if the func-
function F is VBG-* on E and moreover bounded on an interval containing the set E
in its interior, there exists an increasing hounded function TJ with respect to
which the function F has its derivates finite at each point of E. Moreover, if
the function F is continuous on an interval containing E in its interior, the func-
function TJ may be defined in such a way as to be itself continuous (cf. the proof
of Lemma 3.4, Chap. VIII). Finally, it can be shown that in order that a func-
function F be ACG* on an open interval I, it is necessary and sufficient that there exist
an increasing and absolutely continuous function with respect to which the function
F has its derivates finite at every point of I.
A0.8) Theorem. If at every point so of a set E, except perhaps at
the points of an enumerable subset, a function F fulfils any one of
the inequalities
A0.9) F+{x)<+^, F+(x)>—oo, F~(x)<+oo, F~(x)>—oa,
A0.10) Fa ]p{)>
then F is VBG on E.

238 CHAPTER VII- Functions of generalized bounded variation.
Proof. We need only consider the case of the first of the
inequalities A0.9) and that of the first of the inequalities
A0.10). It is therefore sufficient to show that each of the sets
A=B[F+(x)< + co] and B=E[Jap(a3)< + oo] is expressible as
X X
the sum of an enumerable infinity of sets on each of which F is
of bounded variation.
Consider first the set A. Given any positive integer n, let A,,
denote the set of all the points xeA such that, for every t,
A0.11)
implies F{t)—F{x)^n-(t—x),
and by A'n, for each integer *, the part of A,, contained in the
interval [i/w,(i+l)/w]. Let Fn{x)=F[x)—nx.
For every pair xuxz of points of Aln, where a;^;^, we have
0^»2—x^ljn, and so, by A0.11), F(x2)—F(x1)^.n-(x2—,•%), i.e.
Fn[x&)—Fn(xl)^.0. The function Fn(x) is thus monotone non-in-
non-increasing on each set Aln, and it follows that A',, is expressible as the
sum of a sequence of sets {ij'j^y,... on each which Fn{x) is mono-
monotone and bounded. The function F{x)=Fn(x)-\-nx is then plainly of
bounded variation on each of the sets A'n, aad moreover we have
-f-oo co
Consider now the set B. Prom the definitions of approximate
upper limit and approximate upper derivate (ef. §3, p. 220), it
follows at once that to each point xeB we can make correspond
J
a positive integer n such that the set M
I
,J h,as tjle
I L fe—a? J
point ? as a point of dispersion. Therefore, denoting by Btl the set
of the points xeB such that the inequality Os^/i^l/w implies both
the inequalities
A0.12)
and
A0.13)
co
we have B=?Bn- We denote further, for every integer i, by B\,
«=i
the part of Bn contained in the interval [ijn, {i+l)jn] and we write
as before, Fn{x)=F[x)—nx.
[§ 10] Criteria for the classes of functions VBG*, ACG>, VBG and ACG. 239
The main part of the proof consists in showing that, for every i,
the function Fn(x) is monotone on B*a.
For this purpose, let xx, x2 be any pair of points of a B'n, and
let xx<ca2. We plainly have 0<<r2—x^l/n, so that (by writing
x—xx and Ji=co2—xl in A0.12), we obtain
)-*1 (fli) 33= «¦(§-»!);
Similaaiy, from A0.13) with x—x% and A=a;2—%, we derive
The two inequalities thus obtained show that the interval
%!, x2] contains a point ?0 such that
-JPKXn-tlo— ih) and
By adding these two inequah'ties term by term, we obtain
F(%2)—F{x1)<n-{xz—Xj), and so finally Fn{x2)—J'n(ir1)<0.
We have thus shown that the function Fn(x) is monotone de-
decreasing on each B'n. It follows that B'n is expressible as the sum of a se-
sequence of sets {.B«'/}i/=i,2,... on each of which Fn{x) is monotone and
bounded, and on which, the function F{x)=Fn[x)Jrnx is therefore
CO OO -{-DO CO
of bounded variation. Moreover, we have B=E Bn—E 2 UB1^1.
,,=1 »=1 j=—oo ]=l
This completes the proof.
On account of Theorem 4.2, it follows immediately from Theorem 10.8
that any measurable junction which satisfies one of the inequalities A0.9) or A0.10)
at each point of a set E, is approximately derivable ai almost all points of E. This
proposition will be generalized and completed in Chap. IX (§§ 9 and 10).
A0.14) Theorem,. If two extreme approximate derivates on the
same side are finite for a function F{x) at every point of a set IS,
except at most in an enumerable subset, then the set E is tlie sum of
a sequence of sets on each of which F{x) is absolutely continuous.
Consequently, if the function F(x) is further given to be contin-
continuous on JO, then F[x) is AGO- on E.
Proof. It is clearly enough to show, for instance, that the
set A^K[—oo<J5n,"J|l(*'XP+(*)<+00] is tne sam of an at most
enumerable infinity of .sets on each of which F is AC.

fan
240 CHAPTER VII. Functions of generalized bounded variation.
Now we can make correspond, to each point we A, a positive
integer n such that x is a point of dispersion for the set
E[\F{g)—F(x)\^n-($—(B)] (cf. § 3, p. 220). Hence, denoting by An
the set of the points xeA such that, for every h, the inequality
0 ^ h <J 2/n implies
A0.15) E[|JF(|)—F{so)\^n-[i—%)', x^^^x-\-K\ ^.h/i,
we have A=?An; and, denoting as before by A,, (for each integer i)
71=1
711
the part of J.,, contained in the interval [ijn, (i+l)/w], we obtain
72=1 /—~OO
Consider now any two points xx and x2 of J.',, where a^ < x2,
and let x3=2xi—sc1.
We have, oh the one hand, 0<a%—a;1=2-(a;3—a;2)^2/n,, so
that by writing x=xx and li=xi~-x1 in A0.15), we obtain the
inequality
and « fortiori
A0.16)
On the other hand, we have 0^a?3—a;2=a;2—x^ljn, and so
by A0.15) with x=x2 and A=ic3—as2, we find
A0.17)
The inequalities A0.16) and A0.17) show that there exists
a point e0 in [«2,%] such that we have at the same time
and this requires \F(x2)—F(x1)\<in-\x2—sc1\. This last inequality
is thus established for every pair of points %, x% of any one of the
sets A'n, and it follows at once that F is AC on each of the sets A1,,.
This completes the proof.
Theorem 10.8 shows in particular that a continuous function which is
everywhere approximately derivable, even unilaterally, is necessarily ACG.
In Chap. IX, § 9, we shall give two tether criteria for a function to
be ACG* or ACG.

CHAPTEB VIII.
Denjoy integrals.
§1. Descriptive definition of the Denjoy integrals.
We shall base the study of the Denjoy integrals on their descriptive
definition. The essential ideas have already been sketched in
Chap. VII, § 1. We now complete them further as follows.
A function of a real variable / will be termed S-integrable
on an interval I—[a,bJ if there exists a function F which is ACG-
on I and which has / for its approximate derivative almost every-
everywhere. The function F is then called indefinite S>-integral of / on I.
Its increment F (I) — F (b) — F {a) over the interval I is termed
definite ^-integral of f over I and is denoted by
b
{2>)jf(x)dx or {S))ff{x)dx.
I a
Similarly, a function / will be termed Sf^-integrable on an
interval I=[a, 6], if there exists a function F which is AOG* on I
and which has / for its ordinary derivative almost everywhere.
The function F is then called indefinite 3^-integral of / on I; the
difference F(I)=F(b)—F(a) is termed definite B^-integrcd of / over I
b
and denoted by (%).jf{x)dx or by Eyjf{x) dx.
For uniformity of notation, the Lebesgue integral will fre-
frequently be called ?-integral.
The, integrals S> and % are often given the names of Denjoy
integrals in ilie wide sense, and in the restricted sense, respectively.
The first of these is also termed Denjoy-KMntcMne integral (cf.
Chap. VII, § 1), and the second, Denjoy-Perron integral (for the
latter, as we shall see below in § 3, is equivalent to the Perron in-
integral considered in Chap. VI).
S. Saks. Theory of the Integral. 16

242
CHAPTEE VIII. Donjoy integrals.
It is immediate, by Theorem 6.2, Chap. VII, that when a func-
function is 3- or SVintegrable on an interval, its definite Denjoy
integrals are uniquely determined on this interval (its indefinite
integrals being determined except for an additive constant). More
generally, if two functions me equal almost everywhere and the one
is integrable in the Denjoy sense (wide or restricted) on an interval Io,
then so is the other and the two functions have the same definite in-
integral over Io. Another immediate consequence of the preceding
definitions is the distributive property for Denjoy integrals. Thus,
if two functions g and h are S>- or 2>v-integrablc on an interval I, the
same is true of any linear combination ag + l)h of these functions, <md
we have
{2>)J[a-g{x)+b-h(x)} dx=a- B>)fg{x)dx+b-{5>) jh(x) dx.
It foEows from Theorem 10.14, Chap. VII, that a continuous
function which is approximately derivable at all points except, perhaps,
at those of an enumerable set, is necessarily an indefinite ^-integral
of its approximate derivative. Similarly, by Theorem 10.5. Chap. VII,
a continuous function which is derivable (in the ordinary sense) at
all but an enumerable set of points, is an indefinite P^-integral of
its derivative. The process of integration S+ therefore includes that
of Xewton (cf. Chap. VI, § 1). The fundamental relations between
the Denjoy and Lebesgue processes are given in the following
A.1) Theorem. 1° A function f which is S^-intograble on an
interval I is necessarily also 9-integrable on I and we have
[ f
i i
2° A function f which is S-integrable on an interval I is
necessarily Si^-integrable on I and we have (?>„) / fdx= I f dx.
i i
3° A function which is 9-mtegrable and almost everywhere non-
negative on an interval I is necessarily S-integrable on I.
Proof. 1° and 2° follow at once from the definitions of the
Denjoy integrals and from the descriptive definition of the Lebes-
Lebesgue integral (Chap. VII, § 1). As regards 3°, it is sufficient to recall
the fact that, in view of Theorem 6.2, Chap. VII, a, function which
is ACG and whose approximate derivative is almost everywhere
non-negative, is necessarily monotone non-decreasing, and therefore
its derivative is summable.
Descriptive definition of the Denjoy integrals.
243
Part 3° of Theorem 1.1 shows that for functions of constant
sign the Denjoy processes are equivalent to that of Lebesgue (cf.
Theorem 6.5, Chap. VI, for the corresponding result concerning the
Perron integral). Hence, we derive the following further extension
of Lebesgue !s theorem on term by term integration of monotone
sequences of functions (Chap. I, Theorem 12.6).
A.2) Theorem. Given a non-decreasing sequence {/J- of functions
tcMoh are 9-integrable on an interval I and whose 2>-integrals over I
constitute a sequence bounded above, the function f(x) = lhafn(x) is
itself, necessarily, 9-integrable on I and we have
(S>) ff(x)dco = Urn (S)/>»<&!.
i " i
Exactly the same is true with 2>^ in place of Si in the hypothesis
and conclusion.
Proof. This theorem reduces at once to the theorem of Le-
Lebesgue just referred to, for we need only consider in place of the
functions fn, the functions fn—fv which are integrable in the Denjoy
sense and non-negative, and which are therefore integrable in the
Lebesgue sense on account of Theorem 1.1 C°).
We shall show later on (Chap. IX, § 11) that the extreme
approximate derivates of any measurable function are themselves
measurable functions. This includes the result that any function
which is 3-integrable is measurable. In the meantime we give an
independent proof of this last assertion.
A.3) Theorem. A function which is 9-integrable is necessarily
measurable and almost everywhere finite.
Proof. Let / be ©-integrable on an interval I and let F be
its indefinite integral. The function F is therefore ACG- on I, so
that I is the sum of a sequence {En} of closed sets on. each of which
F is AC. By Lemma 4.1, Chap. VII, there exists for each % a func-
function Fn of bounded variation on I, which coincides with F on JSn. We
therefore have almost everywhere on En the relation f(x)=F'sv(x)=F'n(x)-.
and since the derivative of a function of bounded variation is meas-
measurable and almost everywhere finite, it follows that / is meas-
measurable and almost everywhere finite on each Hn and consequently
on the whole interval I.
16*

244
CHAPTEE VIII. Denjoy integrals.
Finally, let us mention aa an immediate consequence of The-
Theorem 9.1, Chap. VII,
A.4) Theorem. If a function f is Q-integrable on an interval Io,
then every closed subset of Io contains a portion Q such that the function f
is summable on Q and such that the series of the definite 3-integrals
of f over the intervals contiguous to Q is absolutely convergent.
Similarly, if the function f is ^-integrable on Io, then every
closed subset of Io contains a portion Q such that the function f is sum-
summable on Q and sueh that the series of the oscillations of the indefinite
S>*-integrals of f over the intervals contiguous to Q is convergent.
§ 2. Integration by parts. We have already observed
(Chap. VI, p. 210) that a slight modification of the definition of
Lebesgue-Stieltjes integral leads to an indefinite integral -which is
an additive function of an interval. As this modification will
be useful to us in the present §, we now formulate it explicitly.
Given a finite function g integrable in the Lebesgue-Stieltjes
sense with respect to a function of bounded variation F on an
interval I=[a,6], we shall write
4 b
(S) JgdF= f gdF-{g(a)-[F(a)-F(a-)]+g(b)-[F(b+)-F(b)]).
a a b
The number ($)JgdF will be called definite S-integral of g with
a
respect to F over I. As we see at once, this number (unlike tlxe
Lebesgue-Stieltjes integral) does not depend on the values taken
by the function F outside the interval I, and for each point c
of [a,?] we have
ebb
(S) f g dF + (S) / g dF = {8) f g dF.
a c a
B.1) Theorem. Let g be a bounded function integrable with respect
to a monotone non-decreasing function F on an interval [a, 6]. Then:
b
(i) (?) / gdF=/u-[F{h)—F(aI, where /j, is a number between
a
the bounds of the function g on [a,b~];
X
(ii) writing 8(x) = {§) fg dF for
, we have S'{cc) =
a
=g(x)-F'(ai) at almost all points of continuity of the -junction g, and
in fact at every point x where g is continuous and F derivable.
[§2]
Integration by parts.
215
Proof. Clearly (i) follows at once from the obvious inequality
m-[F(b)—F(a)]^(S)fgdF^3f-[F(b)~-F(a)], where m and M are
a
the lower and upper bounds of g on [«,&]. In order to establish (ii),
consider a point xv at which g is continuous and F is derivable.
We may suppose, by subtracting a constant from g if necessary,
that g{xo)—O. Denoting, for each interval J, by e(J) the upper
bound of \g{x)\ on J, we have |?(Jr)|/|J|<s(J)-_F(J)/|J| and taking J
to be an interval containing x0 and of length tending to zero, we
find S'[xo)=O=g{xo). This completes the proof.
C.2) Lenima. Let F be a function of bounded variation on an interval
I0=[a,b], G a continuous function on Io, and E the function defin-
defined on Io by the formula
X
C.3) E{x)=F{x)G{x)—{S)JQ{t)dF[t) for
a
Then, if the function G is ACG- [AOG*] on Io, so is the function S.
Proof. We may clearly assume F to be monotone non-de-
non-decreasing. Denoting by Mo the upper bound of \F(x)\ on Io, we shall
begin by proving that for every interval ICI0 we must have
B.4) \H(I)\^M0-\G(I)\ + O(G;I)-F(I) and O(E; I)<3JT0-O((?,-1).
In fact, by Theorem 2.1 (i), we have, for every subinterval J=[a, ff\ of I,
where /i is a number between the bounds of G on J. Consequently,
\H(J)\^MO-\G(J)\ + O(G; J)-F(J), and the first of the relations B.4)
follows by choosing J = J. On the other hand, we derive
\H(J)\^3M0-O{G;I) for every interval JCI, and hence the second
relation B.4).
Hence, since the function G is continuous, the function H is
continuous also. Further, if {I*} is any finite sequence of non-
overlapping intervals and if oj denotes the largest of the numbers
O(Cr; Ik), we obtain from the relations B.4)
EZ and

246
CHAPTEE VIII. Denjoy integrate.
The first of these inequalities implies that if the function G is AC
on a set JS, so is the function H, and consequently, that if the func-
function G is AOG on the whole interval Io, then the function H is also
ACG on Io. Similarly, the second of the above inequalities shows
tlxat if G is ACG* ou Io then so is F, and this completes the proof.
We can now complete Theorem 14.8, Chap. Ill, which concerned
integration by parts for the Lebesgue integral, by establishing
a similar theorem for the Denjoy integrals:
B.5) Theorem, If F{x) is a function of bounded variation and g(x)
a function ?>- or ^-integrable on an interval I0—[a,b], then the function
F[x)g(x) is integrable on Io in the same sense, and moreover denoting
by G the indefinite integral of g, we have
ft b
{&)]'F{x)g{x)dx=G(b)F{b)—G{a)F(a)-(S) j'6{x) dF{x).
'a a
Proof. We shall establish the theorem for the 2>-integral.
The proof for the % -integral is quite similar.
By Lemma 2.2, the function R defined by the formula B.3)
is ACG- on Io. Moreover, by Theorem 2.1 (ii), if we form the ap-
approximate derivative of both sides of B.3), we obtain almost every-
everywhere, the relation E'&v(x)=F(x)G's.li(x)=F{x)g{x). It follows that
the function F[x)g{x) is 9-integrable ou the interval IQ and that
{&) JF{x) g(x) d<u=H(b)—H{a). This last relation is equivalent to
a
the one to be proved.
The idea of the above proof, which is directly based on the descriptive defi-
definition of the Denjoy integrals, is due to Zygmund. For another proof, depending
on the constructive definition of these integrals, of. for instance E. W. Hobson
[I, p. 711]. For an interesting generalization of the theorem on integration by
parts to the ^-integral (cf. Chap. VI, § 8) vide A. J. Ward [3].
From Theorem 2.5, there follows easily the second mean
value theorem for the Denjoy integral, which may be regarded
as a generalization of Theorem 14.10, Chap. III.
B.6) Theorem. Given a non-decreasing function F on an interval
Io=[«,&] and a function g which is S>-integrable on Io, there must
exist a point f in Io such that
(S1)/ g(x)F(x)dx=F{a)-{S) [g(x) dx + F{b)-{?>) ('g(x)dx.
[§3]
Theorem of Hake-Alexandroff-Looinan.
247
Proof. Writing G{x)=(S>)jg(t)dt, we have by Theorems 2.5
and 2.1 (i) the relation
(x) F{x) dx = G(b)F(b) - (S) / G(x) dF(x)=
where p is a number between the bounds of G (x) on I. It follows
that there exists a point i in Io such that /{=?(?), and the rela-
relation just obtained becomes
(S>)fg(x)F(x)dx=F(a)-G(?)+F(b)-tG(b)~G(!;)],
n
which, by definition of G[x), reduces to the required formula.
§ 3. Theorem ol Hake-Alexandroff-Looman. Therelations
between the Denjoy integrals and those of Lebesgue and of Newton
having already been obtained in § 1, we now proceed to establish
an important result of Hake, Alexandroff and Looman, which asserts
the equivalence of integration in the restricted Denjoy sense with
Perron integration.
At the same time we shall show that in the definition of Perron
integral (Chap. VI, § 6) we need only take account of thecontinuous
major and minor functions. In order to make this assertion, quite
precise, let us agree to say that a function / is ^-integrable on an
interval Io if 1° the function has continuous major and minor func-
functions on Io and 2° denoting by V any continuous major function
and by V any continuous minor function of / on Io, the lower bound
of the numbers U{I0) is equal to the upper bound of the numbers
7(I0). The function / is then plainly c?-integrable on I0) the definite
^-integral of / on Io being equal to this common bound. We have
to prove the converse, i. e. that every function which is c/Mnte-
grable is also c?0-integrable.
C.1) Lemma. If a function f is ^-integrable on each interval interior
to an interval [a, b] and if the definite c?-integral over the interval
[a + e,b — ij] tends to a finite limit as e->0+ and ??->-0+, then the
function f is ^-integrable on the whole interval [a,6].

248
CHAPTER VIII. Denjoy integrals.
Proof. It is clearly sufficient (by halving the given interval)
to consider the case of a function / which is ^0-integrable on each
X
interval of the form [a,b —e] where 0<s<6—«. Let P{x)=(&) j fdx
a
for a<Jc<& and p=P(b—).
We choose any positive number a. Writing for symmetry a0— a,
we consider any increasing sequence of points {aA}te=Oi li#i, which con-
converges to b. The function / being ^-integrable on each interval
[a*, ajH-i], we easily define, on the half open interval [a,b), a con-
continuous function F such that F is a major function of / on each
of the intervals [a*, «i+1] and that [F(x) —J?(aA)]—[P(a?) -P {ak)]<oj2k
for «a<«<«a+i and ft=0,1,.... By the second of these conditions
the oscillation of the function F on the interval [«*,&) tends to 0
as fc->oo, and therefore J1 has a finite limit F{b—) at the point 6.
Writing _F(a;)=P(a)+(a;—aI;3 for %<a, and F{x)=F(b —) for o!>6,
we extend the definition of F to make this function continuous
on the whole straight line JSj, and the following conditions are then
satisfied:
C.2) —
for
and C.3) F{b)—F{a)^p+2o.
Now let c be an interior point of [a, 6] such that the oscillation
of Jp oh [c',6] is less than <r. For each point x of [e,6], let O(x) denote
the oscillation of F on [*,&]. The function 0(as) is continuous and
non-increasing on the interval [c,6], and we extend its definition
on to the whole straight line JS^ by making 0{x) = 0(o) for sc<c
and 0(as)=0(&)=O for «>&. We now write G{x)=F{x) — O(ni)
and TJ{x)=G[x) + a-[x — 5I/3/(& — aI/3. Since the fanction
ct-(x—6I3/F—aI'3—0{as) is non-decreasing, it follows at once from
C.2) that -oo4=J7(«)>/(as) for «<tc<6. Moreover, since <?(&) —(?(«)
is non-negative for each point x of the interval [e,&], and 0 for
x^b, we find (?FJsO, and therefore R(b)=-+co, Hence C is a con-
continuous major function of / on the interval [a,b] and fulfils, by C.3),
the inequality U(b)-U(a)^F{b)—F{a)+2ct^p+4o. Similarly we
define a function 7 which is a continuous minor function of / on
[a,6] and fulfils the condition V(b)—V{a)^p—ia. It follows that
the function / is <P0-integrable on 0,6]. TMb completes the proof.
[§3]
Theorem of Hake-Alexandroff-Looruan.
249
C.4) Lemma. Let Q be a closed and bounded set, a, b its bounds,
[Ih=[ak, 6a]} the sequence of intervals contiguous to Q, and f a function
which is summable on Q and t^-integrable on each interval contiguous to Q.
Then, if the series of the oscillations of the indefinite ^-integrals
of the function f on the intervals I/, converges, the function f is
tPintegrable on the whole interval [a, 6] and we have
C.5)
Proof. Let s be a positive number and let K be a positive
integer such that
oo
C.6) EOk< e,
where Ok denotes the oscillation of the indefinite ^-integral of /
on the interval IA. Denote by /x the function which agrees with /
on the set Q and on the intervals Ik for h^.K, and which is 0
elsewhere. By Theorem 3.2, Chap. VI, and by the hypotheses
of the lemma, the function fx has a continuous major function Z7X
and a continuous minor function 7X such that
C.7) TJ-Jfi) — TJx{a)~ si
We shall now define a continuous major function for f—fx.
Let Fk be, for each h, a continuous major function of / on
the interval I* such that Fk[aIl)=Q and 0{Fn; Ia)^20a, and let Ak(x)
and Bi,(os) denote, for any point x e I*, the oscillations of the func-
function Fk on the intervals [«a,oj] and [as, 6*] respectively. We write
G(x) = Ff,{x)+Ak{x)—[Bk{x)—Bk(ak)-\ when xel% and fc>Z, and
G(x) = 0 elsewhere. Finally, for each x, we write
where the summation
which
is extended over the indices h for
. Since, for every k, we have G(ak-{-)=Gr(ak) = O and
?A; I/,X6-0a, the function V2 is continuous on the
straight line li^ and since the function G vanishes identically on
each interval Ii, for fe<if, we have by C.6),
C.8)

250
CHAPTER VIII. Denjoy integrals.
Now, for each /fc, we have G(-x)^d and G{h—)—G(a>)^0 for
every point xelk. Therefore the increment of the function ?72 is
non-negative on each interval containing points of the set Q, and
consequently ]72(tf)>0=/(a?)— h(x) at each point x of this set.
Again, since the function (J vanishes on each interval Ih for its^R,
we have ?2(j)) = 0=/(a!)— f^x), whenever xel% for ic^K. Finally,
since the function Ak{x)—Bk{x) is non-decreasing on each I/e, we
see that -oa^U2{x)^Fk{x)^f{x)=f{x)—f1{x) at each point xel%
for k>K. Thus Vo, is a continuous major function of f—fx on [a,b].
Similarly, we determine a continuous minor function F2 of /—/1;
subject to the condition 7a(&)—Fs(a)>—-6s which corresponds to
C.8). Therefore, writing Z7= Z7r + Ca and 7 — Fi+Fg, we obtain
a continuous major function J7 and a continuous minor function V
for / on [a,b], and if we denote by p the right-hand side of C.5),
we obtain from C.6) and C.7), U(b)~U(a)—Se^p^V(b)—V(a)+86.
The function / is thus c^-integrable on the interval [a, 6] and its
definite ^-integral over this interval is given by the formula C.5).
C.9) Theorem. A function f which is %-integrable on an interval Io
is necessarily (?0-integrable on IQt and we have
Proof. Let F be an indefinite S^-integral of / on Io. We call
an interval IC-^o regular, if the function / is c/g-integrable on I
and if the function F is on I an indefinite ^-integral of /. Farther,
we call a point xeIQ regular, if each sufficiently small interval
I(ZIo containing x is regular. Let P be the set of the non-regular
points of Io. We see at once that the set P is closed and that every
subinterval of Io which contains no points of this set is regular.
We have to prove that the set P is empty.
Suppose, if possible, that P=j=O. By Lemma 3.1 we see easily
that every interval contiguous to P is regular and that the set P
therefore has no isolated points. On the other hand, by Theorem 9.1,
Chap. VII, the set P contains a portion Po on which the function F
is AC*. Let JQ be the smallest interval containing Po. Since the
set P has no isolated points, the same is true of any portion of P,
and therefore P.Jg=^o. It follows that in order to obtain a con-
contradiction, which will justify our assertion, we need only prove that
the interval Jo is regular.
[§3]
Theorem of Plake-Alexandroff-Looman.
251
To show this, let J be any subinterval of Jo and let Q be the
set consisting of the points of the set P-J and of the end-points of J.
We denote by ¦{!„) the sequence of the intervals contiguous to Q
and by G the function which coincides with F on Q and is linear
on the intervals I,,. Plainly the function 0 is absolutely continuous
on J. Therefore, since G'{x)=F'(x)=f{x) at almost all points x of Q,
and since G(I,,)=F{In) for each n, we obtain
C.10)
Now the function / is summable on Q and ciVhrtegrable on
each interval I,, and moreover, F is an indefinite ^-integral of / on
each of these intervals. The series of the oscillations of F on the
intervals ln being convergent, it follows, by Lemma 3.4, that the
function / is -j^-integrable on J and that, on account of C.10),
F(J) — {S')fdx. Therefore, since J is any subinterval of JOt the
j
interval Jo is regular and this completes the proof.
C.11) Theorem. A function wliieli is ff-integrable on an interval Io
is necessarily ?)%-integrable on Io.
Proof. Let / be a function ^"-integrable on an interval Ia
and let P be its indefinite ^-integral. We shall show that the func-
function P is an indefinite SVintegral of /. Since P'(x)=f{x) almost
everywhere (cf. Theorem 6.1, Chap. VI), it is enough to show that
the function P is ACG* on Io, i. e. that any closed set QQIo con-
tains a portion on which, the function P is AC*.
Let E be any major function of /. Since E_(x)>—oc at each
point x of Io, the function E is by Theorem 10.1, Chap. VII, VBG»
on l0, and hence Io is expressible as the sum of a sequence of closed
sets (cf. Theorem 7.1, Chap. VII) on each of which the function E
is VB,,. It follows, by Baire's Theorem (Theorem 9.2, Chap. II) that
the set Q contains a portion Qo on which the function E is VB*.
Since the difference P—H is a monotone function, the function P
is actually VB,, on Qo, We shall show that P is further AC_ on QQ.
For this purpose, we denote by J0=[a,b] the smallest interval
containing Qo. Let ? be any positive number and V a major func-
function of / on Io such that
C.12)

CHAPTEB VIII. Deujoy integrals.
Let Px and U^ denote the functions which coincide on Qo with the
functions P and 0 respectively, and which are linear on the intervals
contiguous to Qo and constant on the half-lines (— oo, a] and [b,+oo).
The function P1 is clearly of bounded variation. On the other hand,
we see easily that U1[x}>—oo at every point, and that U^x^P^x)
at almost all points, of the interval Jo. Therefore, writing Mai)=P'i{x)
wherever the second of the above inequalities holds, and f1[x)=—oo
elsewhere, we see at once that the function f^x) is summable on Jo
and has Z7X for a major function. It follows that V1{I)> f h{x) dx
I
for each interval irjB, and therefore that the function of singularities
(cf. Chap. IV, p. 120) of ZJj is monotone non-decreasing on Jo. Let Tx
be the function of singularities of Pv Since the function P1 — TJ1
is monotone non-increasing on Jo and since, by C.12), we have
0>P1(Jr0)-Cr1(Jr0)=-P(J'o)-^o)^-e) it foUows that Tx{I)^-b
for each interval IC^oi ail(i e being any positive number, this
requires 1^1)^0 for every interval I(Vo- Similarly, by considering
minor functions of / m place of major functions, we find Ti(I)<0,
and therefore, finally, T1(I)=0, for each interval IC^o- The func-
function Px is thus absolutely continuous on Jo. This requires the func-
function P to be AC on the set Qo as well as VB*, and therefore AC*
on this set on account of Theorem 8.8, Chap. VII. Thus every clos-
closed set Q(ZIo contains a portion Qo on which the function P is
AC,, and this completes the proof.
The first of the theorems proved in this §, which together establish the
equivalence of the processes of %-, <?„- and ^-integration, was derived
in 1921 by H. Hake [1] from the constructive definition of the integral ?>*
{vide below, § 5). The second theorem was obtained some years later by P. Alex-
androff [1; 2] and H. Looman [4] independently. For an interesting extension
of these results to Perron-Stieltjes integral, vide A. J. Ward [3].
It should, perhaps, be added that in their original definitions 0. Per-
Perron [1] and 0. Bauer [1] employed only continuous major and minor functions.
The equivalence of the original Perron-Bauer definition with that of Chap. VI,
§ 6, has therefore been established here as a consequence of Theorems 3.9 et 3.11.
Let us remark further that in the definition of Perron integral, ordinary
major and minor functions may be replaced by generalized continuous major
and minor functions defined as follows. A function IT is a generalised continuous
major function of a function / on an interval I if 1° IT is continuous and VBG*
on I, 2° the set of the values assumed by U at the points at which U'(x) = — oo,
is of measure zero, and 3° U(x)^f(x) at almost all points x. The definition of
generalized continuous minor functions is obtained by symmetry.
[§3]
Theorem of Hake-Alexandroff-Looman.
253
We shall conclude this § with the following result, due to
Marcinkiewicz:
C.13) Theorem. A measurable function f loiiieh has on IQ at least
one continuous major function and at least one continuous minor
function, is necessarily i?-integrable on IQ.
Proof. Let TJ and V be respectively a continuous major func-
function and a continuous minor function of / on Io. We shall call
a point xelo regular if the function / is cF-integrable on each suf-
sufficiently small interval ICJa which contains x. Let Q be the set
of the points x of IQ which are not regular. The set Q is plainly closed
and we see at once that the function / is ^-integrable on each sub-
interval of Io which contains no points of Q. Thus it has to be proved
that Q=0.
Suppose, if possible, that (H=0. For every interval I on which
the function / is (f-iutegrable, we have TJ[I)^{&) ff(x)dos^V{I).
i
Therefore, if [a,b] is an interval contiguous to Q, the definite
o?-integral of / on the interval [a+e, 6 —»;] interior to [a, b] tends
to a finite limit as e-»-0 and tj-vO. By Lemma 3.1, the function /
is thus c?-integrable on each interval contiguous to Q. It follows,
in particular, that Q can have no isolated points.
Now let Qo be a portion of Q on. which the functions V and V
are both VB*. Such a portion exists by Theorem 9.1, Chap: VII,
since the functions V and Fare VBG* on Io on account of Theorem 10.1,
Chap. VII. Let JQ be the smallest interval containing Qo. Since
U.(x)^f(x)^V(x) everywhere on Io, the function / is gummable
on Qo together with the two derivatives V_[x) and V(x). On the
other hand, denoting by {I,,} the sequence of the intervals contiguous
to Qo and by On the oscillation on I,, of the indefinite ^-integral
of /, we shan have 0,,<0(Z7;In) + 0(F;In) for every n, and so
ZO,:<+oo. It follows by Lemma 3.4, that the function / is cf-inte-
n
grable on the whole interval Jo. But this is clearly impossible, for
since the set Q has no isolated points, the interval Jo contains in
its interior some points of Q. We thus arrive at a contradiction
which completes the proof.
Just as in the definition of the Perron integral, we may replace, in The-
Theorem 3.13, ordinary major and minor functions by generalized continuous ones
(cf. above, p. 252). Nevertheless, the conditions of Theorem 3.13 differ from those
of the definition of Chap. VI, § 6, in that continuity is essential. In fact, if
•we write /(^) = 0 for sc^.0 and f(x) = —ljxi for x>0, U(x)=0 identically in ifx
and V(x) — 0 for x~<-0 and Y {x) = Ijx for x >0, we see at once that V and V are
respectively a major and a minor function of /: and yet / is evidently not
cV0-mtegTable on [0, 1].

25-i
CHAPTEE VIII. Denjoy integrals.
* § 4. General notion of integral. We shall deal in this §
with some notions of a more abstract kind which we shall employ,
in the next §, aa a basis for the constructive definition of the
Denjoy integrals.
Let c be a functional operation by winch there corresponds
to each interval I=[a,b] a class of functions defined on I, and to
each, function / of this class a finite real number. This class of func-
functions will be called domain of the operation t on the interval I, and
the number associated with / will be denoted by ?(/;!).
An operation C will be termed an integral, if the following three
conditions are fulfilled:
(i) If a function / belongs to the domain of the operation Z
on an interval IOf the function belongs also to the domain of C on
any interval IC.I0, and ?(/;!) is a continuous additive function
of the interval IO0-
(ii) If a function / belongs to the domain of the operation &
on two abutting intervals Ix and I2, the function belongs also to
the domain of t on the interval Ix+I2-
(iii) A function / which vanishes identically on an interval I
belongs to the domain of ? on I, and we have ?(/;!)=0.
If ST is an integral, any function / which belongs to the domain
of € on an interval Io will be termed G-integrable on Io and the
number ?(/;I0) will be called definite ^-integral of / on Io. The
function of an interval IQI0, ?"(/; I), which is additive and continuous
on account of (i), will then be called indefinite ^-integral of / on Io
and its oscillation on Io (i. e. the upper bound of the numbers ]?(/; I)\,
where I denotes any subinterval of Io) will be denoted by O(€; f; Io).
Two integrals ?i and ?g will be termed compatible, if
h(f>J-) = ^(f\I) for every interval I and for every function /
which is both ?i- and ?2-integrable on I.
We shall say that the integral ? includes the integral ?„
if the two integrals are compatible and if every function which is
trintegrable is also ?2-integrable. When this is so we shall write
Given an integral t and a function g which vanishes out-
outside a bounded set 13, it is evident that if g is ?-integrable on an
interval Io which contains B in its interior, then g is so also on
any interval I which contains E, and we have €(g;l) = ?{g;l0).
General notion of integral.
255
This fact justifies the following definition: we shall say that
a function / is t-integrable on a bounded set E, if the function g which
coincides with / on E and is 0 elsewhere, is ?-integrable on each
interval IZ)E. The number €{g;I) is then independent of the choice
of the interval 10)%', we shall call this number definite ^-integral
of the function / on the set E and we shall denote it by ?"(/; IS).
Of the known processes of integration, all those which give rise to a con-
continuous indefinite integral (for instance those of Lebesgue, Newton, Denjoy, etc.)
are easily seen to be integrals according to the above definition. If, however
we wished to include also discontinuous integrals (e. g. that of W. H. Young
of. Chap. VII, p. 215) we should have to modify some details of the definition
Given a function / on an interval Io and given an integral ?,
we shall say that a point aelo is a ^-singular point of / in Io if there
exist arbitrarily small intervals IC-^o containing a on each of which
the function / is not ?"-integrable. Denoting by S the set of these
points, we see at once that the set S is closed and that the function /
is c-integrable on every subinterval of Io which contains no points of 8.
With each integral € we now associate three "generalized"
integrals €L, ?H and €H*, defined as follows.
Given any interval Io, the domain of the operation ?c on Io
is the class of all the functions / which fulfil the following two
conditions:
(c1) the set of the ^-singular points of / in Io is finite (or empty);
(c2) there exists a continuous additive function of an interval F
on Io such that F(!)=&(/; I) whenever I is a subinterval of Io
which contains no ^-singular point of /.
Since such a function F (if existent) is uniquely determined
by the conditions (c1) and (c2), we can write ? '(f;I0)=F(IQ).
The domain of the operation ?H on Io is defined as the class
of the functions / which fulfil the following conditions:
(h1) if S denotes the set of all €¦ singular points of / in Io, the
function / is c-integrable on the set 8 and on each of the intervals lk
contiguous to the set consisting of the points of 8 and of the end-
points of Io;
(h2) ?\?{f; !/,•)[< +°° an(i, in the case in which the sequence {I*}
is infinite,' lim 0{€;f; IA) = 0.
k

256 CHAPTEK VIII. Deujoy integrals.
For any such function /, we write by definition:
Finally, we obtain the definition of the operation <TH* by re-
replacing in the definition of the operation ?H the condition (li2) by
the more restrictive condition:
04)
00.
We verify at once that the operations ?c, ?H and ?H* all
fulfil the conditions (i), (ii) and (iii), p. 254. These operations are there-
therefore integrals according to the definition, p. 254, and we evidently
have ?C?C and cTC^C^- For brevity, we shall write ?CH and
?CH* in place of (CC)H and (?"C)H* respectively.
The integral Ec and the integrals EH and <TH* may be regarded respectively
as the Cauchy and the Harnaok generalizations of the integral <T. They correspond,
in fact, to the classical processes employed by Cauchy and Harnack to extend
integration from bouaded to unbounded functions of certain classes. The original
process of Harnack actually corresponds to the operation tH* rather than to the
operation cH. Cf. A. Harnack [1], E. W. Hobson [I, Chap. VIII] and A. Kosen-
thal [I, p. 1053].
If we were to add to the conditions (h1) and (h2) which characterize the
generalized integral <La, the condition that limO(C; /;/*)/?(»> !*)= 0 for almost
h
all x e S, we should arrive at a generalized integral CH' intermediate between
CH and EH*. By applying the process ?H' in the constructive definitions of Den-
joy integrals of the next §, we should then obtain an integral .'?', intermediate
between ¦? and '!>„. Its descriptive definition is very simple: a function f is 'i1'-
integrable if it is tf-integrable and if its indefinite ^-integral is almost everywhere
derivable (in the ordinary sense). This integral has been discussed by A. Khin-
tchine [1]; cf. also J. C. Eurtill [1].
*§5. Constructive definition of the Denjoy integrals.
With the notation of the preceding §, we see at once that for each
integral ?"C®) we have also ?CC% similarly the relation
implies 6°C%- It is not quite so obvious that the relations
and ?C% imply respectively ?HC@ and CH*CS*- This last as-
assertion is a consequence of the following theorem which is anal-
analogous to Lemma 3.4.
[§5]
Constructive definition of the Denjoy integrals.
257
E.1) Theorem. Let Q be a bounded dosed set with the bounds
a and b, and let {Ik} be the sequence of intervals contiguous to Q;
and suppose that f is a function §)-integrable on the set Q as well as
on each of the intervals Ik, and that (in the case in which the sequence
{Ik} is infinite]
k
C>)lfdx<+oa and lim0C;
Then the function f is &-integrable on the whole interval
I=[a,b~\ and we have
E-2) B>)[f dx= @) // dx+2(9) ff da.
1 Q " i'k
If we suppose, further, that the function f is %-integrable on Q
as well as on each of the intervals Ik and that X (?»»;/; Ik)< +00,
then the function f is Q^-integrable on I. *
Proof. We shall prove the theorem for the 3>-integral. The
case of the ?>„ -integral is similar.
Let I [x) denote the interval [a, x] where we suppose x e [a, 6],
and let
E.3) F(x)=2(®) ff(t)dt.
h ik:m
We shall show that the function F, thus defined, is ACG on
the interval I. For this purpose, it will suffice to show that F is
AC on the set Q, the function being evidently continuous on I and
ACG on each of the intervals Ik-
Let g{x) be the function equal to 0 for xeQ and to T7-7 ¦ (®) ff{t)dt
for xel% where &=1, 2, .... The function g is summable on I and
X
if G{x)— j g(t) dt, the function F clearly coincides with G on Q;
a
F is thus AC on Q and therefore ACG on I.
This being so, we have F'^[x)=G'{x) = g{x)=Q at almost all
points x of Q, while it follows at once from E.3) that Fi?(x)=f(x) at
almost all points x of I—Q. Hence,- F being AOG on I, it follows
that the function equal to / on I—Q and to 0 on Q has F for an
indefinite ©-integral. On the other hand, the function equal to /
on Q and to 0 elsewhere is, by hypothesis, 3-integrable on I.
It follows that the function / itself is 3-integrable on I, and
that {&) ff(x)dx=F[b)~F{a) + (S>) \fdx, which, on account of E.3),
/ Q
is equivalent to E.2). This completes the proof.
S. Saks. Theory of the Integral.
17

258
CHAPTEE VIII. Denjoy integrals.
We now pass on. to the constructive definition of the Denjoy
integrals. We begin by introducing the following notation.
Let (S'1} be a sequence of integrals, in general transfinite, such
' ?& i
(}
that ?S'C ?'' whenever ?<tj. We then denote by ?& the operation €
whose domain on each interval I is the sum. of the domains of the
operations €l for ?<a, and which is defined for every function /
of its domain by the relation ?(/;J)=s:l(l(/; I), where ?0 is the least
of the indices |< a such that / is fi^-integrable on I. It then follows,
of course, that €(f; !)=?*(/; I) for every f >f0, since by hypothesis €'
then includes ?*°.
This being so, let {?"} and {?*} be two transfinite sequences
defined, by an induction starting with the Lebesgue integral ?, as
follows:
?o = ?% = ?,
,=\CH
and
K«
for a>0.
Denoting by Q the smallest ordinal number of the third class
(cf. for instance, W. Sierpiriski [I, p. 235]) we shall show that
and
V
We shall restrict ourselves to the case of the ?>-integral (that of
the S1,-integral being quite similar).
Since ?(Z3>, we find at once by induction (cf. above, p. 256)
that for every f, ?*(Z®, and so, obviously, 27 ?S(Z®- In order to
change this last relation into one of identity, it is enough to show
that every function / which is 2>-integrable on an interval 2"o = [<M]>
is ?;-integrable on Io for some index ?<Q.
Let 55 denote the set of the .^-singular points of / in Io. The
sequence [8% as a descending sequence of closed sets, is sta-
stationary, i.e. there exists an index v<Q such that Sv=Sv+i-
(For if not, there would exist for every ?<Q a point
scteS'—S*^, and therefore also an interval It with rational end-
points, containing the point x* of $; but without points in common
with the closed set 8i+i, nor therefore, with any of the sets
8;+1, Ss+S,.... We should thus obtain a transfinite sequence of type
Q of distinct intervals with rational end-points, and this is impossible.)
We shall prove that 8"=0,
[SB]
Constructive definition of the Denjoy integrals.
259
Suppose, if possible, that ?"+0. We see at once that the func-
function / is -if-integrable on each interval I(ZI0 which contains no
points of S". It follows that the function / is (^")G-integrable, and
a fortiori r''+1-integrable, on each interval contiguous to 8". Since
$"=#"+I) it follows, in particular, that the set 8" contains no iso-
isolated points.
The function / being, by hypothesis, ?>-integrable on Jo, the
set 8" (cf. Theorem 1.4) must contain a portion Q such that the
function / is summable on Q and such that the series of the definite
^-integrals of / over the intervals contiguous to Q converges absolu-
absolutely. Since i'C(~")GC?); it follows at once that the function / is
(-C")CH-integrable, i. e. -Cw+1-integrable, on some interval Jo contain-
containing Q. But this is clearly impossible, since, in view of the fact that
the set 8" has no isolated points, the interval Jo certainly contains
points of the set S"=S''+1 in its interior.
We thus have 8"= 0, which establishes the ^"-integrability of
/ on Io an(l completes the proof.
Various definitions, constructive and descriptive, of Denjoy integrals mil
be found in the papers mentioned in Chap. VI, p. 207, and Chap. VII, pp. 214-215,
as well as in the following treatises and memoirs: N. Lusin [I;4], T. H. Hil-
debrandt [1], P. Nalli [I], E. Kamke [I], A. Kolmogoroff [2],
H. Lebesgue [7; II, Chap. X], A. Eosenthal [1] and P. Eomanowski [1].
For further extensions to functions of two or more variables, see also
H. Looman [1] and M. Krzyzaiiski [1].
17*

CHAPTEE IX.
Derivates of functions of one or two real variables.
§ 1. Some elementary theorems. The first part of this
chapter (§§ 1—10) is devoted to studying the various relations bet-ween
the derivates of a function of a real variable. With the help of the no-
notion of extreme differentials introduced by Haslam-Jones, certain of
these relations -will subsequently be extended, in the second part of
the chapter (§§ 11—14), to functions of two variables.
Accordingly, the term "function" will be restricted in the first
part of this chapter to mean function of one real variable.
Before proceeding to the theorems directly connected with the
Lebesgue theory, we shall establish in this § some elementary results.
We first observe that a linear set E contains at most a finite num-
number, or an enumerable infinity, of points which are isolated on one side at
least. To fix the ideas, let A be the set of the points of E which are
isolated points of E on the right. For each integer n, let AB denote the
set of the points x of A such that the interval [xtx+ljn] contains no
point of E other than x. Then it is plain that, for each integer 1c, the in-
interval [fc/w, (fc+l)/»] can have at most one point in common with An.
Hence each set An is at most enumerable, and the same is true of
the set A = J!,An.
n
We say that a finite function F assumes at a point x0 a strict
maximum if there exists an open interval I containing x0 such that
F(x)<F(xQ) for every point soeJ other than xa. By symmetry we define
a strict minimum.
Some elementary theorems.
261
A.1) Theorem. Given a finite function of a real variable F, each of
the following sets is at most enumerable:
(i) the set of the points at which the function F assumes a strict
maximum or minimum;
(ii) the set of the points x at which
lira sup F(t) >lim sup F(t)
t t+
or
lirninf J1(i)<liminf F{t);
(iii) the set of the points x at which
F+(x)<F~(x) or F~
Proof, re (i). Consider the set A of the points at which, for in-
instance, the function F assumes a strict maximum, and let An denote,
for each positive integer n, the set of the points x such that F(t)<F(x)
holds for each point t^=x of the interval (x—l[n, x-\-ljn). We see at
once that each set An is isolated, and therefore at most enumerable.
Since A—J^An, it follows that the set A is at most enumerable.
n
re (ii). Let us consider, for definiteness, the set B of the points
¦x at which lira sup F{t) >lim suy F(t). We denote, for each pair of in-
tegers p and q, by BM the set of the points x such that
h'm sup F(t) >plq> lira sup F{t).
Olearly each point of a set Bp,q is, for that set, an isolated point on
the right. Each of the sets BM is thus at most enumerable, and, since
B=2BPiq, the same is true of the whole set B.
p>i _,
re (iii). Consider the set G of the points x at which F^(x)<JT(x),
and denote, for each pair of integers q>Q and p, by GM the set of the
points jo at which F+{x)<plq<Jf'{x). Write FM{x)=F{x)-pxlq. We
find F+q{x)<Q<F^,q{x) at each point xeGp,v, and this shows that the
function FM assumes a strict maximum at each point of Gp,q. By the
result just established, each set Gp,q is at most enumerable, and con-
consequently, the same is true of the whole set G.
It is sometimes convenient (vide, below, § 5) to appeal to a
slightly more general form of the last part of Theorem 1.1, which
concerns relative derivates (cf. Chap. IV, p. 108) and which reads thus:
A.2) Theorem. If V andF are two finite functions of areal variable,
the set of the points t at which the derivative U'(t)>0 (finite or infinite)
exists and at which F%{t)<Fu(t), is at 'most enumerable.

262
CHAPTER IX. Derivates of functions of one or two real variables.
This is proved in the same way as the corresponding part of
Theoreml.l. In fact, if we denote, for every pair of .integers q> 0 and p,
by Cp,q the set of the points x at which t
Cp,q
we see at once that the function F[se)—(plq)-U(x) assumes at each
point of Cp,q a strict maximum. Therefore each set GM is at most
enumerable.
For Theorem 1.1 and its various generalizations, vide: A. D enjoy [1, p. 147],
B. Levi [1], A. Rosenthal [1], A. Schonflies [I, p. 158], W. Sierpinski [1; 2]
and G. G. Young [J]. As regards the enurnerability of the set of the points at which
the function assumes a strict maximum or minimum, it is easily seen that this
result remains valid for functions in any separable metrical space (cf. F. Haus-
dorf f [I, p. 363]). Mention should be made also of the elegant generalizations of
Theorem 1.1, obtained successively by H.Blumberg [1], M. Sckmeiserfl] and
V. Jarnlk [3].
§ 2. Contingent of a set. Ws have mentioned earlier (in
Chapter IV, p. 133), that certain theorems on derivates of functions
may be stated as propositions concerning metrical properties of
sets in Euclidean spaces. In connection with these results, we shall
state in this § some definitions which begin with some well-known
notions of Analytical Geometry.
By the direction of a half-line I in a space ttm (where m^2) we
shall mean the system of the m direction cosines of I. The half-line
issuing from a point a and having the direction d will be denoted by
ad. The half-line issuing from a point a and containing a point b^=a
—*-
will be denoted by ab.
If we interpret the system of the m direction cosines of a half-
line as a point in Bm (situated on the surface of a unit sphere), we may
regard the set of all directions in a Euclidean space as a complete,
separable, metrical space (cf. Chap. II, § 2). It is then clear what is
to be understood by the terms: convergence and limit of a se-
sequence of directions, everywhere dense set of directions, etc. We
shall say further that a sequence of half-lines {ln) issuing from the
same point a converges to a half-line I issuing from a, if the sequence
of the directions of the half-lines I,, converges to the direction of I.
Given a set E in a space lim a half-line I issuing from a point aeE
will be called an intermediate half-tangent of JS at a, if there exists
a sequence {an) of points of E distinct from a, converging to a and such
that the sequence of half-lines {^n} converges to Z. The set of all inter-
intermediate half-tangents of a set E at a point a is termed, following
[§2]
Contingent of a set.
263
G. Bouligand [I], the contingent of E at a and denoted by contgAa
(by the contingent of E at an isolated point of E, we shall understand
the empty set). A straight line passing through a which is formed of
two intermediate half-tangents of E at a is called intermediate tangent
of E at a. Similarly a hyperplane 7i passing through the point «is called
intermediate tangent hyperplane of E at a, if each half-line issuing from
a and situated in h is an intermediate half-tangent of E at a. In Mz
the notions of intermediate tangent hyperplane and inter-
intermediate tangent are plainly equivalent.
Given in the space JRm a hyperplane h, a1x1+a2x2Jr...+amxm=b,
(cf. Chapter III, §2) the two half-spaces {half-planes if m=2)
a1x1+a2%g-\-...-\-amxm'^b and a1a;1+a2a;2+»-+»mflJm<;&, into which 7i
divides Mm, will be termed sides of the hyperplane h. In the case in
which h is an intermediate tangent hyperplane of a set E at a point a
and in which, further, the contingent contgg a is wholly situated on
one side of 7i, the side opposite to the latter is called empty side of Ji
and the hyperplane h is termed extreme tangent hyperplane of E at a.
The two sides of h may, of course, both be empty at the same time,
and this occurs if the contingent of E at a coincides with the set of
all half-lines issuing from a which lie in the hyperplane Ji itself.
The hyperplane h is then termed unique tangent hyperplane, or simply,
tangent hyperplane, of E at a.
For simplicity of wording, we shall restrict ourselves in the sequel
to the case of sets situated either in the plane JSa or in the space Mz.
Needless to say, the extension to any space Rm presents no essential
difficulty (an elegant statement, which sums up the results of §§ 3
and 13 of tins chapter and which is valid for an arbitrary space Rm,
will be found in the note of P. Eoger [2]).
As usual, the hyperplanes in i?2 and JRZ are termed straight
lines and planes respectively. Moreover, in the case of plane sets we
shall speak of tangent (intermediate, extreme, unique) in place of
tangent straight line (intermediate, extreme, unique).
We shall discuss the case of the plane (§3) and that of the space
(§13) separately, although the proofs of the fundamental theorems
3.6 and 13.7 which correspond to these two cases, are wholly analogous.
The proof of the former is, however, more elementary, whereas the
latter requires some subsidiary considerations connected with the
notion of total differential (cf. below §13).

264 CHAPTER IX. Derivates of functions of one or Wo real variables.
§ 3. Fundamental theorems on the contingents of plane
sets. For brevity, we snail say that the contingent of a plane set
B at a point a is the whole plane, if it includes all half-lines issuing
from this point. Similarly the contingent of E at a point a 'will be
said to be a half-plane, if B has at this point an extreme tangent I
and if contgua consists of all the half-lines issuing from a- and
situated on one side of I.
We shall see in this § that, given any plane set B, at each point a of B except
at most in a subset of zero length, either l'the contingent of JSis the whole plane, or
' 2° it is a half-plane, or finally 3° the set JB has a unique tangent. This result (together
with the more precise result contained in Theorem 3.6) was first stated by
A.Kolmogoroff andJ.Verfienko [1;2]. It was rediscovered independently, and
generalized to sets situated in any space Jim, by F.Eoger [2]. The proofs, together
with some interesting applications of the theorem of Kolmogoroff and Vereenko,
will be found in the notes of IT. S. Haslam-Jones [2;3]. (For the. first part
of Theorem 3.6 cf. also A. S. Besicoviteh [4].)
A finite function of a real variable F, defined on a linear set B,
is said to fulfil the Lipschitz condition on B, if there exists a finite
number JST such that \F[x2)—F(x1)\^.N-\x2—o!i1\ whenever x1 and xz
are points of B. As we verify at once, we then have A{B(F; B)) <
<(A7+l)-|J5?j (for the notation, cf. Chap. II, § 8, and Chap. Ill, § 10).
Thus, if a function F fulfils the Lipschitz condition on a set E of finite
{zero] outer measure, its graph B(F; B) on E is of finite [zero]
length.
It is also easy to see that any function which fulfils the Lipschitz
condition on a linear set B, can be continued outside B so as to fulfil
the Lipschitz condition on the whole straight line M1 and so as to be
linear on each interval contiguous to B.
C.1) Lemma. Let B be a plane set, 0 a fixed direction and P the set
of the points a of B at which contg^ a contains no half-line of direction 6.
Then (i) tie set P is the sum of a sequence of sets of finite length,
and (ii) at each point a of P, except at most at those of a subset of length
zero, the set B has an extreme tangent such that the side of the tangent
containing the half-line ad is its empty side.
In the particular case in which 6 is the direction of the positive semi-
axis of y, the set P is expressible as the sum of an enumerable infinity of
sets each of which is the graph of a function on a set on which the
function fulfils the LipseUte condition.
[§ 3] Fundamental theorems on the contingents of plane sets.
265
Proof. By changing, if necessary, the coordinate system, we
may suppose in both parts of the theorem that d is the direction of
the positive semi-axis of y. Let us denote, for every positive inte-
integer n, by P,, the set of the points {x, y) of P such that the inequali-
inequalities \x'—as|<l/m and \y'—y\^.ljn imply y'—-y^.n-\se'—x\ for every
point (x',y') ofB. Since there is no point a of P at which the contingent
of B contains the half-line with the direction of the positive semi-axis
of y, it is clear that P=%Pn. Let us now express each Pn as the sum of
n
a sequence {P,Uk)k=\,i,... of sets with diameters less than ljn. We shall
then have \y2—yi\^.n-\x2-—x1\ for every pair of points, (#,,#,) and
(aJ2, y2), belonging to the same set P,,,A. Let Qn,k be the orthogonal
projection of Pn,k on the axis of x. We easily see that each point of
Q,hk is the projection of a single point of P^k. Consequently, the set
Pn,k may be considered as the graph of a function Fnj, on Qn,k. Moreover
we have \Fn>k[x-i)—Fn,k{xi)\^n-\x%—x\\ for each pair of points xt and
x2 of C,,,A, i.e. the functionFn,i, fulfils the Lipschitz condition on Q,,ik
and therefore (cf. above p. 264) each set Pn,h='B(FnX, Qajt) is of
finite length. Thus, since P=?Pn *, we obtain the required expres-
sion of the set P as the sum of an at most enumerable infinity of sets
of finite length, which are at the same time graphs of functions
fulfilling the Lipschitz condition on sets situated on the #-axis.
It remains to examine the existence of an extreme tangent to
the set B. at the points of P. For this purpose, let us keep fixed for the
moment a pair of positive integers n and h, and let Qnik be the set of
the points of Qn,k which are points of outer density for Qn<k and at
which the function Fn,b is derivable with respect to the set Qn<k. Since
the set Qn,k—Qn,k is of measure zero (cf. Theorem 4.4, Chap. VII) and
since the function Fn,k fulfils the Lipschitz condition on Qn>k, it follows
that A[B(Fn-k;Q,l,k-Qn,k)] = Q.
We need, therefore, only prove that B has an extreme tangent
at each point of the set B(F,,x, Qn,k) and that, further, the side of
this tangent which contains a half-line in the direction of the
positive semi-axis of y is its empty side.
Let (?„, va) be any point of B{Fny,Qn,k), and J_othe derivative
of F,,ik at So with respect to the set Qn,k- Let e be a positive number less
than 1. Since f0 is a point of outer density for the set Qn,k, we can

266
CHAPTEE IX. Derivates of functions of one or two real variables.
associate with each point (?,??), sufficiently close to (f0, %), a point
f'eQn,* such that
C.2)
and
C.3)
(for otherwise, the outer lower density of Qn,k at ?0 would not ex-
exceed 1—e).
Eemembering now that %—F^a)) let us write for brevity
We shall have
Now suppose that the point (f, v) belongs to B and that
|f—?0!«?1/2k2 and (97—??0|<l/2m2. By C.3), we have |f—f|<l/w,
and, by C.2), 1-^A')—j^ra-lf—fj<w-|f—?0|<l/2ra, so that
|-Pn.*(f')—);|<l/«i. Since the point {?',Fn,k{?')) belongs to PnJl(^Pn, it
follows from the definition of the set Pn that i)—Fn,k{i'XM-\?—?'\,
and using C.3) again, we derive from C.4) that
3.5
Now as i, and therefore f, tends to f0, the ratio Dn<h{? ')/(?'—f0)
tends to zero; the same is therefore true, on account of C.3), of the ratio
DnJ.?')l{?—fo)- Consequently, since sis an arbitrary positive number
less than 1, it follows from C.5) that the upper limit of the ratio
0?—%~~JV(?~?o)]/|?~fol> as the point (?,w)eB. tends to (?„,??„),
is non-positive. Further, since the line 1/—rjo=Ao-(x—10) is plainly
an intermediate tangent, of the set B(Fny, QnM) QR at the point (f 0, jj0),
we see that this line is an extreme tangent of the set B at this point
and that the half-plane y—t]q^A0-{x—f0), which contains the half-line
issuing from (f0, rja) in the direction of the positive semi-axis of y,
is an empty side of this tangent.
This completes the proof.
C.6) Theorem. Given a plane set B, let P be a subset of B at no point
of which the contingent of B is the whole plane. Then (i) the set P is the
sum of an enumerable infinity of sets of finite length and (ii) at every
point of P, except at those of a set of length zero, either the set B hat a imigne
tangent or else the contingent of B is a half-plane.
[§3]
Fundamental theorems on the contingents of plane sets.
267
Proof. Let {9,,} be an everywhere dense sequence of directions
in the plane and, for each positive integer n, let Pn denote the set of
the points of P at which the contingent of B does not contain the half-
line of direction 8,,. We clearly have P=y,Pn, and by the preceding
II
lemma each set P,,, and therefore the whole set P, is the sum of a se-
qiisnee of sets of finite length. Further, the same lemma shows that
the set B has an extreme tangent at every point of P, except at most
in a set of length zero.
Now let Q be the set of the points of P at which 1° there exists
an extreme tangent which is not a unique tangent and 2° the con-
contingent of B is not a half-plane. For each positive integer n. let Qn
denote the set of the points 6 of Q such that the half-line b8n is situated
on the non-empty side of the extreme tangent of B at b, but does
not belong to eontg^J. Plainly Q=?Qn. Now, by the preceding
n
lemma, for every point b e Qn, except at most those of a set of length
zero, the half-line bdn is situated on the empty side of the extreme
tangent at 6. It follows that all the sets Qn, and therefore also the
whole set Q, are of length zero. Hence, at every point of P, except
perhaps those of a subset of length zero, either there is a unique
tangent or the contingent at this point is a half-plane.
C.7) Theorem. Given a plane set B, let P be a subset of B at every
point of which the set B has an extreme tangent parallel to a fixed
straight line I). Then the orthogonal projection of P on the line at right
angles to D is of linear measure zero.
Proof. We may clearly assume that the line D coincides with
the axis of x. Let $ and T denote, respectively, the sets of the points
(?,ri) of P for which the half-planes y^rt and y^t] are respectively
the empty sides of the extreme tangents. Consider the former of
these sets. By Lemma 3.1, the set 8 is the sum of a sequence of
the sets. B(Fn;Qn), where the Qn are sets on the .r-axis and the Fn
functions fulfilling the Lipschitz condition on these sets, respectively.
We may suppose (cf. p. 264) that each function Fn is defined, and
fulfils the Lipschitz condition, on the whole .r-axis and is linear
on the intervals contiguous to the set Qn.
This being so, we easily see that, for every n, the relation
J?7(.v)^Q^F?(.r) holds at each point x of Qn -which is not an isolated
point on any side for Qn, i.e. (cf. §1, p.260) at all the points of Qn,

268
CHAPTER IX. Derivates of functions of one or two real variables.
except at most those of an enumerable set. Thus by Lemma 6.3,
Chap. VII, we have \Fa[Qj = Q for every positive integer n, and
since the projection of 8 on the y-axis coincides -with the sum of
the sets Fn[Qn], this projection is itself of measure zero. By sym-
symmetry, the same is true of the projection of the set T on the
(/-axis, and this completes the proof.
As an immediate corollary, we derive from Theorem 3.6 the following prop-
proposition:
C.S) Given a plane set B, let P be a subset of B at each point a of which there exists
a straight line through a which contains no half-line of oontg^fl. Then the set B has
a unique tangent at all the -points of P except at most those of a subset of length zero.
This result can "be easily extended to the space (cf. F. Roger[2]) as follows:
C.9) Given a set B in the space i?3, let P be a subset of R at each point a of which there
exists a plane through a which contains no half-line of contgff a. Then (i) the set P is
the mm of an enumerable infinity of sets of finite length atid (ii) the set B has a unique
tangent at all the points of P except at most those of a subset of length zero.
Proof. Let (9nf be an everywhere dense sequence of directions in tlie space
i?8. For each positive integer h, let Pn Sl denote the set of the points a of P such
->
that |cos(a&, «n)|>l/7i for every point b of B distant less than I/A from a. We
express each set Pn A as the sum of a sequence [Pn ^ /l}/l=:-i •> of sets of diameter
less than 1/h. We then have
n,h,k
Keeping, for the moment, the indices n, h, k fixed, we choose a new system of
rectangular coordinates, taking for the positive semi-axis of s the half-line of
direction 0n. Let a, ft and y be, respectively, the three positive semi-axes of the new
coordinate-system. For any set, or any point, Q, we denote by Q^a\ Q^ and Q^ the
orthogonal projections of Q on the planes fiy, ya and a/3, normal to the axes a, p
and y respectively. _^
We have |cos(ai, y)[>l/i whenever aePnhl;, JeEand 0<g(a, b)<_\jh. It
follows at once that there is no point P^h k at which the contingent of the plane set
B'-"i contains a half-line at right-angles to the semi-axis y. Hence, by C.8),
the set J?j,"j^ is the sum of an at most enumerable infinity of sets of finite
length, and the set E(K) has a unique tangent at all the points of P("\ k except at most
those of a set Mnhjl of length zero. Similarly, the set iJ® has a linique tangent at
all the points of P^'AA except at most those of a set Nn h k of length zero. It follows
that the set if has a unique tangent at each point a of Pn h k, except perhaps when
«<E) e Mnhj or when afc3) <¦ Nn^k. Now we easily see' that the two ratios
p(a, &)/?(a(a), b(a)) and (>(«, b)ls(al-s\ 6(-3)) remain bounded (by h) when a and 6
belong to the set PnJlk- It follows that the set of the exceptional points of Pn h k
at which the set S, has no unique tangent is, with the sets 2?nll k and Sfn h ,,,' of
length zero. For the same reason, since the set P^'\ k is the sum of an at most
enumerable infinity of sets of finite length, so is also the set Pnhk- This com-
completes the proof, on account of the relation C.10).
Denjoy's theorems.
269
§ 4. Denjoy's theorems. We shall apply the results of the
preceding § to establish certain important relations, valid almost every-
everywhere, which connect the DM derivates of any function what-
whatsoever, and which are known by the name of the Denjoy relations.
For simplicity of wording, we agree to call opposite derivates of
a function F at a point x0 the Dini derivates F+[xa) and F_~(x0),
or else F+(x0) and F~~(%0).
We shall begin with some preliminary remarks.
Let F be a finite function defined in a neighbourhood J of
a point x0 and let B denote the graph of F on J. It is clear that if
the function F is derivable at the point x0, the set B has at (osO)F(xo))
a unique tangent not parallel to the axis of y. Similarly, if two op-
opposite derivates of F are finite and equal at xw the set B has at
(x^F(xQ)) an extreme tangent y—F(oiQ)=k-(x—x0), whose angular
coefficient h is equal to the common value of these derivates. Con-
Conversely, if at the point (ao,F(xo)) the set B has the extreme tangent
y —F(xo)~h-(x—x0) where fc=f=oo, then 1° F+(x0)=F~(xa)=k in the
case in which the half-plane y—y^lt-[x—x0) is an empty side of
this tangent and limsup F{x)^F{x0), and 2° F+(x0)=F'~(cc0)='k in
the case in which the half-plane y—yQ^k-(x—xQ) is an empty side
and ]iminiF{x)^F{x0).
-v->.v,,
In the enunciations of the theorems which follow, we shall
frequently be concerned with exceptional sets E, connected with
a function F and subject to the condition A{B(F;E-)) = (). This
condition evidently implies both |J?| = 0 and \F[E]'=Q, since the
sets U and F[E~\ are merely the orthogonal projections of the set
B(F;E) on the x- and y-axes, respectively.
D.1) Theorem,. If at each point of a set JE one of the extreme uni-
unilateral derivates of a function F is finite, this derivate is equal to its
opposite derivate at every point of E except perhaps at the points of
a set J?! of measure zero such that yl{B(ir';-E!1)} = 0.
Pr o o f. We may clearly suppose that the same derivate, F+[x) say,
is the one whichis finite throughout E. We thus have lim sup F{x) ^F{x0)
at every point xoeE and, on account of Theorem 1.1 (ii), we may
even suppose that limsupF(x)^.F(xQ) at every point x0 of E.
Now, when xoeE, the contingent of B(F;E) at the point (xo,F{xB))
contains no half-line situated in the half-plane x^x0 and having

270 CHAPTER IX. Derivates of functions of one or two real variables.
angular coefficient exceeding F+(x0). Therefore, by Theorem 3.6,
the set B {F;E) has an extreme tangent at each of its points {%0,F{x0)),
except for those of a subset Bx of length zero, and this tangent has the
half-plane y—y0>F+{xQ)-(x—x0) for its empty side. Hence, denot-
denoting by Et the orthogonal projection of -B3 on the a;-axis, we see, from
the remarks made at the beginning of this §, that at every point x
of the set E —Ex the derivates F+{x) and F~(x) are equal. This com-
completes the proof since A(B(-F;-B1))=A(-B,)=0.
D.2) Theorem. If at each point of a set E a finite function F has
either two finite D-ini derivates on the same side, or else a finite extreme
bilateral derivate (F{x) or F{x)), then the function F is almost every-
everywhere derivable in E; moreover, denoting by Ea the set of the points x
of E at which the function IP is not derivable, we have A{B{F;E0)) = Q.
Proof. It will suffice to consider separately the following
two cases:
1° The function F has two Dini derivates on the same
side finite at each point of E. We then have, by Theorem 4.1,
D.3) F+{x)=F~{a)) and F+{x)=F~{x)
at each point x of E, except perhaps those of a set Eo such that
A{B(F;E0)} — 0. But the relations D.3) imply the equality of all four
Dini derivates at the point x, and since two of them are finite, by
hypothesis, at each point x of E, the function F is derivable through-
throughout E—Eo.
2° The function F has an extreme bilateral derivate
finite at each point of E. By applying twice over Theo-
Theorem 4.1, and making use of the obvious relations F+{x)^F_+{x) and
F~{x)^F~{x), we see that the four Dini derivates are finite and
equal at each point of E, except perhaps at those of a set on which
the graph of F is of zero length. This completes the proof.
Theorem 4.2 (in a slightly less complete form, it is true) has
already been mentioned in Chap. VII, p. 236, as a corolla.ry of
Theorems 10.1 and lO.n, Chap. VII. We have also stated that (as
a consequence of these same theorems) the set of the points at which
a function has a unique derivative (even a unilateral derivative)
infinite, is necessarily of measure zero. We can now extend tins
result by taking the modulus, as follows:
D.4) Theorem. For any finite function F, the set of the points x
at which Mm\F(x+h)~-F(x)\jh=+oo, is of measure zero.
[Ml
Denjoy's theorems.
271
Proof. Denoting the set of the points in question by A, we
see at once that the graph of the function F has at every point of
the set B(F;A), except perhaps at those of a set of length zero,
an extreme tangent parallel to the ?/-axis. Thus, by Theorem 3.7,
the set A, which is the projection of the set B{F;A) on the a-axis,
is of measure zero and this completes the proof.
It results, in particular, from Theorems 4.1, 4.2 and 4.4 that the Dini
derivates of any finite function F satisfy one of the following four relations at almost
every -point x: laj!'+(x)=F~{x)=*+co, J?+(x)=J?~{x)=~oq; 2aF+(x)=F^(:x)^co,
F+(p) = —oo, F~^x)=+oo, 3° J'+(a;)=F-(a;)±oo; F+(x)=+oo, J1~(x)=—co;
4° F+(x)=F+(x)=F~(x)=F^(x)JFoo. For direct proofs of this theorem, which was
established first by Denjoy for continuous functions and then generalized to
arbitrary functions, vide: A. Denjoy [1], G-. C. Young [2], F. Eiesz [7], J. Rid-
der [4], J. C. Burkill and U. 8. Haslam-Jones [1], and H. Bluraberg [2] (cf.
also A. N. Singh [1]). A further discussion of the Denjoy relations ¦will be found
in the notes of V. Jarnik [1] (for functions of one variable) and of A. 8. Besi-
covitch [6] and A. J. Ward [4] (for functions of two variables). For Theo-
Theorem 4.4 see S. Saks and A. Zygmund [1] (cf. also S. Banach [I]).
A part of the Denjoy relations has recently been generalized to differential
coefficients of higher orders; see the important memoirs of A. Denjoy [9],
J. Marcinkiewicz and A. Zygmund [1], and J. Marcinkiewiez [2].
We may now supplement Lemma 6.3, Chap. VII, by the fol-
following result:
D.5) Theorem. Let II be a finite number and F a finite function
such that |-F+(*)K M at every point x of a set E. Then \F[E]\ < M-\E\.
Proof. Let Ex denote the set of the points $ of E at which
F'{x)drF+{x). By Theorem 4.1, we have .A{B(JI;.E1)}=0 and there-
therefore, |P[^J|=0. On the other hand, since \F {x)\ = \F+(x)\^3? at
each point %eE—Eu it follows from Lemma 6.3, Chap. VII, that
\F[E— -BJK M-\E\, and this completes the proof.
An immediate consequence is the following criterion for a func-
function to fulfil Lusin's condition (N) (Chap. VII, § 6):
D.6) Theorem., If a finite function F has at each point x of a set E
a finite Dini derivate, the function necessarily fulfils the condition
CS) on E.
Proof. It is enough to show that if at each point x of a_set E
of measure zero the function F has one of its Dini derivates, F+ say,
finite, then \F[H]\=0. For this purpose, let Bn be the set of the
points xfH at which \F+{x)\^n. We have, by Theorem 4.5.
|-F[H,,]|<M-|-Er,,|=0 for each positive integer n, and hence |Z[H]|0

272 CHAPTER IX. Derivates of functions of one or two real variables.
It is easy to see that the hypotheses of Theorem 4.5 imply that
A{B(F; E))^(M+1)-\E\. This remark enables us to complete Theorem 6.fl of
Chap. VII, as follows: If the derivate F+ of a finite junction F is finite at every point
of a measurable set E, except at those of an enumerable subset, then the function F,
together with its derivate F+, is measurable on E and we have
\F[E)K f\F+(x)\dx and A{B(F; B))=j {l+[f+(x)f)'!:dx.
E B
We may note also the following consequence of Theorem 4.5: If one of the
four Dini derivates of a function F vanishes at every point of a set B, then \F[E}\—0.
For functions F(x) which are continuous, or more generally continuous in
the Darboux sense (i. e. assume in each interval [a, b] all the values between F(a)
and F(b)), we deduce at once the following result:
D.7) Theorem. If F is a finite function, continuous in the Darboux sense 011 an
interval I, and if at each point of this interval, except those of an enumerable set, one
at least of the four Dini derivates is equal to aero, then the function F is constant on I.
* § 5. Relative derivates. The Denjoy relations can be
extended in various ways to relative derivates of a function
with respect to another function. Let us remark that, in accordance
•with the definition given in Chap. IV, p. 108, the extreme derivates
of any function with respect to a finite function V are determined
at each point which belongs to no interval of constancy of the func-
function U; consequently, the set of values taken by the function U
at the points at which the extreme derivates with respect to U
remain indeterminate is at most enumerable.
In the sequel it will he useful to employ the notation adopted
in Chap. IV, § 8. Let us recall in particular, that if C is a curve given
by the equations x=Z(t), y = J(t), its graph on a linear set E (i.e.
the set of the points (X{t), Y(t)) for teE) is denoted by B(G;E).
E.1) Lemma. If G is a curve given by the equations x— U(t), y = F{t),
the set E of the points t at which Fu(t)<+oo, may be expressed as
the sum of a sequence of sets {En} such that
(V) for every n and for every open interval I of length less than ljn,
the set E(O;I) has a unique tangent at every point of B(C;I-E,,)
except those of a set of length zero.
Proof. Let us denote, for each positive integer n, by E,, the
set of the points t such that, provided that ths differences F(t')—F(t)
and U(t')—U{t) do not vanish simultaneously, the inequality \V—1\ <l/n
implies [F(t')-F(t)-]l[U(t')-U(t)]^n. We see at once that, for any
[§5]
Relative derivates.
273
open interval I of length less than 1/w, the contingent of B(G;I)
at a point of B(C;I-En) cannot contain half-lines of angular coef-
coefficient greater than n, and can be, therefore, neither a whole plane,
nor a half-plane (cf. §3, p. 264). The property (y) of the sequence
{En} thus appears as a direct consequence of Theorem 3.6.
E.2) Theorem. If U and F are continuous functions and we hare
F[,(t)<+oo at every point t of a set E, then there is a finite derivative
F'u(t) at each point t of F, except at the points of a set E such that
\U[H]\ = 0.
Proof. Let G denote the curve given by the equations a;= TJ(t),
y=F(t). On account of Lemma 5.1, the set E is expressible as the
sum of a sequence of sets {En} which fulfil the condition (v) of this
lemma. Keeping fixed, for the moment, a positive integer n, let us
consider an open interval I of length less than Ifn. Let Bn(I) denote
the set of the points of B{C;I-En) at which the graph of the curve
G on I either has no unique tangent, or else has a unique tangent
parallel to the ?/-axis. Further, let Bn(I) be the projection of the
set Bn(I) on the a;-axis. On account of the condition (y) and
Theorem 3.7, we have \Bn(I)\ = 0. ^"ow since V andF are continuous,
it is clear that the derivative F'v{t) exists and is finite at each point
t el-En, provided that U(t) does not belong to the set Bn(I). Hence,
I being any open interval of length less than Ijn, this derivative
exists and is finite at each point teEn, except at most at the points
of a set Hn such that \U[EnW^O. This completes the proof, since
E.3) Theorem. If U is a continuous function mid F any finite
function for which Fu{t) = Q at each point t of a set E, then \F[E]\=0.
Proof. Let G denote, as in the proof of the preceding theorem,
the curve x= U{t), y=F(t), and let E be expressed as the sum of
a sequence of sets {E,,\ subject to the condition (¦%) of Lemma 5.1.
Keeping fixed, for a moment, a positive integer n, let us consider
any open interval I of length less than lln. At each point of ~B{C;LEn),
except those of a set of length zero, the set B(C;I) then has a unique
tangent, and since the function V is continuous and Fa(t) — Q at
each point tell, this tangent is parallel to the a?-axis. It follows,
by Theorem 3.7, that the set F[I-En], which coincides with the
projection of the set B[G;I-En) on the y-axis, is of measure zero.
Since I is any interval of length less than Ifn, it follows that j.F[ J?n]|=0
for each positive integer v, and finally that \F[E]\ = 0.
S. Sakn. Theory ol1 tin- I li
18

274 CHAPTER IX. Derivates of functions of one or two real variables.
The hypothesis of continuity of the function U is essential for the validity
of Theorems 5.2 and 5.3 (the hypothesis of continuity of the function F, which is
not required in Theorem 5.3, may, however, be removed also from Theorem 5.2).
Let F(t)=—t identically, and let TJ(t)—t for irrational values .and Z7(<)=*+1 for
rational values of t. Denoting by E the set of irrational points of the interval @,1),
we shall have at each point t of this set
0 and J?,@
Nevertheless |P[.E1| = |.F[S]|=1. On the other hand, the hypothesis of continuity
of the function U may be removed from Theorem 5.3, if we replace the con-
condition Jj,(i)=0 by the more restrictive condition F'u (i)=0. To see this, we shall first
establish an elementary lemma.
E.4) Lemma. If U is a finite function on a set B, there exists a set
TCE such that the set Z7[T] is at most enumerable and sueh that each
point xeE—I is the limit of a sequence of points {tt} of E which
fulfils the conditions (i) to% and U(<,¦)#= U{r) for each ?=1,2,...
and (ii) lim U(h)=V(r).
i
Proof. Let T be the set of the points xeE none of which is
the limit of a sequence {«,} of points of B subject to the conditions
(i) and (ii) of the lemma. Let us denote, for each positive integer k,
by Tk the set of the points t of T for which there is no point t e E
such that both 0<f—t<1/7c and 0<\U{t)—U{r)\<l/k. We have
T=^Tk. Plainly the function U cannot, on any portion of Tt! of
diameter less than Ijk, assume Wo distinct values differing by less
than Ijk. It follows that each set 17[T*] is at most enumerable,
and the same is therefore true of the whole set U[T].
E.5) Theorem. If U and F are any finite functions and Fu{t) — Q
or, more generally Ft(t)=F~u(t) = Q, at each point t of a set E, then
\F[E]\=0.
Proof. Let G be the curve x= U{t), y=F{t), and let Bn denote,
for each positive integer n, the set of the points t of E such that
the inequality 0<f—t<l/ra implies \F{t')~F(t)\^.\U(t') — U(t)\
whatever be the point V. We can express each set E,, as the sum
of a sequence {En,^^,... of sets of diameter less than Ijn.
Let us keep n and k fixed for the moment. It is clear that
the contingent of the set B(C;Bn,k) cannot, at any point of this
set, contain a half-line whose angular coefficient exceeds tlie num-
number 1. Consequently, denoting by Bn,k the set of the points of B (C; E,,j<)
at which the set B{C;Bn,k) has no unique tangent, we see from
Theorem 3.6 that A{Bn,k) = Q.
[§5]
Relative derivates.
275
JSTow the set En,k contains, by Lemma 5.4, a subset Tn,k such
that U[Tnik] is at most enumerable and that each point reBn,k—Tn,k
is the limit of a sequence {ty of points of En,k which fulfils the con-
conditions (i) and (ii) of this lemma. Hence, the relations Fn(t)~Ft{t)=O
being satisfied, by hypothesis, at each point teEn,k, the set B(C;En,k)
has a unique tangent parallel to the a?-axis at each point of the
set B(G;E,hk— T,,,k)— B,,,k. Since A(Bn,k)=Q, it thus follows from
Theorem 3.7 that the set F[En<k— Tn,A], which coincides with the
projection of the set B (C;En,k—In,k) on the y-axis, is of measure zero.
The same is therefore true of the set F[En,k], for the set F[Tn,k]
is, with U[Tn,k], at most enumerable. It follows at once that
\F[E]\ = Q, since E=^En,k.
k
We may mention an application of Theorems 5.3 and 5.5, which is connected
with the following theorem of H. Lebesgue [II, p. 299]: Ifthe derivative of a con-
continuous function F, with, respect to a function XJ of bounded variation, is identically
zero, then the junction F is a constant. J. Petrovski [1] and R. Caccioppoli [1]
extended tliis theorem, in the ease when the function U is continuous, by removing
the hypothesis of bounded variation for 77. At the same time, Petrovsti remarked
that it was sufficient for the validity of the theorem to suppose that the relation
_p'?/(t)=O holds everywhere except in an enumerable set.
It is easy to see that this result is contained in each of the separate theorems
5.3 and 5.5. These theorems actually enable us to state the result of Petrovski
and Caccioppoli in two slightly more general forms. Thus:
1" Suppose that U and F are continuous functions and that at eaoh point t,
except at most those of an enumerable set, one at least of the four relations
Fu(t)=O, Fa{t)=O, F%(t)-=F+{t)=0 or F~(t)^F~(t)=0 is fulfilled. Then the
funetio7i F is a constant.
2° Suppose tliat U is any finite funotio-n and F a continuous function, and let
one of the relations Fy(t)=F'ij(t)=0 or F'jj(t)=F~[;{t)=Q hold at each point t except
at most those of an enumerable set. Then the function F is a constant.
We observe further that, in both the statements 1" and 2°, we may replace
the hypothesis of continuity of F by the hypothesis that F is continuous in the
Darboux sense (cf. § 4, p. 272); moreover the condition that the exceptional set be
at most enumerable may be replaced by the condition that the set of values assumed
by the function F at the points of this set be of measure zero.
The Denjoy relations have a more complete extension to rela-
relative derivates when the function U of Theorem 5.2 is subjected
to certain restrictions. Thus:
E.6) Theorem. Let V and F be. finite functions, and suppose that,
at each point t of a set E, the derivative U'(t) (finite or infinite) exists
and that Fj(i)<+oo. Then FT;(t)=F~u(t) 4= °° «< ea^ Voini { °f -®
except at most the points of a set H such that \U[H]\ = 0.
18*

276 CHAPTER IS. Derivates of functions of one or two real variables.
Proof. We may clearly restrict ourselves to the case in which,
the derivative U'(t) is non-negative throughout E, and even, by
Theorem 4.5, to the ease in which A°) U'(r)>0 at each point x
of E. We then have limsup Z7(<X?7(T)<;]iminf U{t) at each point
fr»r— H«+
tell, and consequently, on account of Theorem 1.1 (ii), we may
suppose that B°) the function U is continuous at each
point of E. This implies that we then have also Om su-p F {t) z^F (x)
at each point x of E, and hence, appealing again to Theorem 1.1 (ii),
we may suppose further that C°) limsup J(*)<J(t) at each point
xeE. Finally by Theorem 1.2, we may suppose D°) F_u(
at each point xeE.
Let now G be the curve defined by the equations x=U(t),
y=F{t). We denote, for each positive integer n, by En the set
of the points teE such that, for every point V, (i) the inequal-
inequality 0<t'—t<l[n implies the two inequalities U(t')>U(t) and
F(t')—F{t)<n-lU{t') — U{t)~], and (ii) the inequality 0<t—t'<l[n
implies U{t)>TJ(t').
Since, by hypothesis, Fu{t) <+oo and since, by A°), ?7'(t)>0
at each point t of E, we see that E=]^
n
Keeping a positive integer n fixed for tbe moment, let I be
any open inteival of length less than Ijn. Whenever (?„??) is a point
of B{C;I-En), the contingent of the set B(G;I) at (?, 97) clearly con-
contains no half-line which, is situated in the half-plane as>| and which
has an angular coefficient exceeding n. Let B{I) denote the set
of the points of the set B(C;I) at which this set has an extreme
tangent, non-parallel to th.fr 2/-axis, with an empty side containing
the half-hue in the direction of the positive semi-axis of y. Further,
let Bn{I) be the set of the points of B{G;I-Efl) which, do not belong
to D[I), and let B,,(I) be the projection of Bn(I) on the o>axis.
By Theorems 3.6 and 3.7, the set Bn{I) is of measure zero.
Tins being so, let i0 be any point of the set I-En such that
TJ(t0) does not belong to the set Bn(I). Let us denote by fr0 the
angular coefficient of the extreme tangent to the set B(C;I) at
the point (U(tQ),F{t0)). It follows easily from the hypotheses A°),
B°) and C°) that i^(i)>?>i4@), and this, in view of (-1°), leads
B°) and C°) that i_
to the relation Fu{tQ)=F^(t0)^= 00.
[§6]
The Banach conditions
and (T2).
277
Thus, since I is any open interval of length less than Ijn, we
find that the last relation holds at each point t0 of En other than
those belonging to a set Hn such that \TJ[Hn]\=Q. This completes
the proof, since we have seen that E=J]En,
n
In view of Theorem 7.2, Chap.VII, we derive from Theorem 5.6 the following
theorem which has been established in a different way by A. J. Ward [3]:
E.7) Suppose that the -junction TJ is VBG-* and let F be any finite function. Let 25
be a set at each point t of which we have either i^(i)<-f oo or -P^ (i)>—co. Then the
derivates Fjj and F~^ are finite and equal at all points of 25 except at most those of a set
H such that |Z7[ff]|=0.
It will result from the considerations of § 6 (see, in particular, Theorem 6.2)
that Theorem 5.7 remains valid for all continuous functions V which fulfil the
condition (Tx). Nevertheless, its conclusion ceases to hold if we allow U to be any
function which is VBG- or even ACS. To see this, let & be a non-negative contmuous
function which is ACG- on the interval [0,1] and for which G(i)=0 and ff~(t)<—1
at any point t of a perfect set B of positive measure (for the construction of such
a function cf. Chap. VII, § 5, p. 224). Let us choose U{t)=t+G{t) and F(t)=t We
shall then have at _every point t of E, U(t)=t, P+(<)>1 and ZZ~(*)<0<P~D
so that 0^F?(t)s$F~fr(t)*Cl, while F^(t)=— oo and i?p(i)=+oo. Nevertheless
|Z7[2S]|=|JE|>0. (This example is due to Ward.)
* § 6. The Banach conditions (Tj) and (Tj). A finite func-
function of a real variable F is said to fulfil the condition (Tx) on an
interval I if almost every one of its values is assumed at most
a finite number of times onl. A finite function F is said to fulfil the
condition (T2) on an interval I if almost every one of its values
is assumed at most an enumerable infinity of times on I.
These two conditions were formulated by S. Banach [6]. We shall begin
by studying the condition (Tt) and we shall establish a differential property which
is equivalent to this condition in the case when F is continuous (vide below
Theorem 6.2). Another equivalent condition, due to Nina Bary, will be estab-
established in § 8 (Theorem 8.3).
F.1) Lemma. Suppose that F is a continuous function and that E
is a set at no point of which the function F has a derivative (finite
or infinite). Suppose further that each point x of E is an isolated
point of tie set E[F(t)=F{x)l Then A{B{F;E)} = 0, and conse-
t
qiienthf \E\ = \F[E~\\ = 0.

278 CHAPTEE IX. Derivates of functions of one or two real variables.
Proof. For each xsF, there exists a neighbourhood, I such
that, -when tel, the difference F(t)~F(x) remains of constant sign
as long as t remains on the same side of .as; this difference then ehanges
sign as t passes from one side of*, to. the .other, except in the
case in which the function F assumes a strict maximum or mini-
minimum at x. Therefore, if we'denote by Eo the set of the points at
¦which the function assumes a strict maximum or minimum, we
see at once that the four Dini derivates oi-F have the same sign
at any point x of F — Eo. In other words, since, by hypothesis, the
function F has no finite or infinite derivative, at any point of $,
we shall have at each point x of E—BQ either-~\-oo>F(as)>0 or
else — oo<F(x)^Q. Hence, by Theorem 4.2,- A{B(F;F — -#„)}= 0,
and, since the set Eo is at most enumerable (cf. Theorem 1.1), it
follows that A{B(F;E)}=0.
F.2) Theorem. In order that a function F which is continuous
on an interval I, fulfil the condition (Tx) on this interval, it is neces-
necessary and sufficient that the set of the values assumed by F at the points
at which the function has no derivative (finite or infinite) be of meas-
measure zero.
Proof. Denoting by Y the set of the values assumed an in-
infinity of times by the function F on I, and denoting by H the set
of the points of I at which F has no derivative, we have to prove
that the relations |Y| = 0 and \F[W]\ = Q are equivalent.
1° |r|==0 implies \F[E]\ = Q. Let X be the set of the points
xel such that F(x)eY. Then F[X]=Y, whence \F[X]\ = 0.
On the other hand, for each xoeE—X, the set of the points x
such that F{x)=F(x0), is finite, and consequently an isolated set.
It follows from Lemma 6.1 that \F[E—X]\=0, and hence finally
that \F[E]\^\F[X]}+\F[F-X]\ = 0. .
2° \F[E]\=Q implies |r|=0. Let R denote the set of the
points x at which F\x)=Q. By Theorem 4.5, we have \F[S]\ = 0.
Now let y0 be any point of Y~-F[E], and let EB denote the
set of the points x at which_F(a:) = 2/0. The set Fa being infinite and
closed, let x0 be a point of accumulation of Ea. Since the function F
has a derivative at each point of Ew we find that F'{xo)=O; thus
xoeH and therefore yQeF[E]. It follows that Y-F[E]CF[E],
and hence that \Y—F[F,]\=0. Thus \F[E]\ = 0 implies |r| = 0
and the proof is complete.
[§«]
The Banach conditions (TL) and (T2).
279
F.3) Theorem, 1° A continuous function which is VBG* (in par-
particular, one of bounded variation) on mi interval I, necessarily fulfils
the condition (Tx) on I.
2° A continuous function which is VBG on an interval I neces-
necessarily fulfils the condition (T2) on I.
Proof. On account of Theorem 7.2, Chap. YII, the first part
of the theorem is an immediate corollary of Theorem 6.2. To
establish 2°, let us suppose that F is continuous and VBG on an in-
interval I. The interval I is then expressible as the sum of a sequence
{Bn} of closed sets on each of which the function F is VB. We may
clearly suppose that each En contains the end-points of the interval I.
Let us denote, for each positive integer n, by Fn, the function which
is equal to F on En and which is linear on the intervals contiguous
to Hn. The functions Fn are plainly, of bounded variation on I, and
therefore, by 1°, they fulfil the condition (TJ. It follows at once
that the function F fulfils the condition (T2) on I.
In the part of Theorem 6.3A") that applies to functions which are
the continuity hypothesis for the function F is not a superfluous one (thus, the
function F(x)=&m(ljx) forx=f=0 and F[0)=0 is VBG* and does not fulfil the condi-
condition (Tt) on [0, 1]). This hypothesis may however he replaced by a weaker one,
which consists in supposing that the function F has no points of discontinuity
other than of the first kind (i. e. that, at each point x, both the unilateral limits
F(x+) and F{x—) exist). In particular, functions of bounded variation, -whether
continuous or not, all fulfil the condition (Tj) (and from this it follows easily that
the continuity hypothesis may be removed altogether from the second part B°)
of the theorem).
For functions of bounded variation, the condition (T,) may also be deduced
from the following general property of plane sets, established by W. Gross [I]
(cf. J. G-illis [1]): If E is a plane set and En denotes the set of the values of ?; such that
the line y=>i contains at least n distinct points of the set E, then A(E)^-ii-\En\.
In connection with part 2° of Theorem 6.3, it may be noted further that
functions which are VBG-, or even ACG, need not fulfil the condition (Tt). An
example is furnished by the function V considered in § 5, p. 277. The latter is also,
as -will follo-w from results to be established in § 7 (cf. in particular, Theorem 7.4),
an example of a continuous function -which is ACG, and consequently fulfils the
condition (N), without fulfilling the condition (8) of Banach.
For continuous functions of bounded variation, the condition
(Tx) is also a consequence of the following theorem of S. Banach [5]
(ef. also N. Bary [3, p. 631]), which contains at the same time an
important criterion for a continuous function to be of bounded
variation:

280 CHAPTEE IX, Derivates of functions of one or two real variables.
F.4) Theorem. Let F be a continuous function on an interval
Io=[»,&] and let s{y) denote for each y the number (finite or infinite)
of the points of Io at which F assumes the value y. Then the function
s(y) is measurable B3) and we have
F.5)
fs{y)dy=W[F;I0).
Proof.For eachpositiveinteger n,let us put I^=[a,a+ (b—a)j2"]
and I^=={a+(k— I){b—a)l2",a+lc(b-a)l2n], when 7c=2,3,...,2".
This defines a subdivision 3(n> of the interval Io into 2" subintervals,
of which the first is closed aud the others are half-open on the
left. For ft=l,2,...,2n, let sfi denote the characteristic function of
the set J[lf], and let s(n\y)=?s?\y).
We see at ouce that the functions s(">(«/) constitute a non-
decreasing sequence which converges at each point y to s(y). Hence,
the functions s<-n)(y) being clearly measurable B3), so is also the func-
function s(y). +oa
On the other hand, fs^\y)dy = \F[I^]\=,O(F;I^). Therefore,
denoting by W{!l) the sum of the oscillations of the function F on the
intervals of the subdivision 3(n), we obtain fs{n)(y) = W{n\ and the
relation F.5) follows by making n-+oa.
F.6) Theorem. If F{x) is a continuous function which fulfils the
condition (T2) on an interval IM the set D of the points at which the
derivative F'(x) (finite or infinite) exists, is non-enumerably infinite.
Moreover, if we write
and N
then, for each interval I=[a,b]CI0, we have
Proof. We may, plainly, suppose that
since the other case may be discussed by changing the sign of the
function F.
Let Y be the set of those values of F on I which are assumed
by the function F at most an enumerable number of times on I
Denoting, for each y, by F!s the set of the points xel such that
[§6]
The Banaoh conditions (TL) and (T2).
281
F(x) = y, we shall show that with each point jelwe can associate
a point xveF,g, in such a manner that (i) F(xg)^0 and (ii) xg is
an isolated point of the set Hg.
For this purpose, we remark first that if the set Fs reduces
to a single point, the latter may "be chosen for our ccs. For, in that
case, the condition (ii) is clearly fulfilled, while the condition (i)
holds on account of the hypothesis F.8).
Let us therefore consider the other case, in which the set JES
contains more than one point. Then, since the function F is con-
continuous by hypothesis and yeT, the set UB is closed, and at
most enumerable. This set, therefore, contains a pair of isolated
points a, /? between which it has no further points. (This is obvious,
if the set Eg is finite. If EB is infinite, its derived set (cf. Chap. LT,
p. 40) is itself closed, non-empty, and at most enumerable; the
latter, therefore, contains an isolated point x0. Thus near xa there
are only isolated points of Eg. It will, therefore, suffice to choose,
among the latter, any two consecutive points as our points a and 0.)
Consequently, at one at least of the points a and /?, the upper deri-
vate of F(x) is non-negative. We choose this point as our xg. We
then see at once that the conditions (i) and (ii) are fulfilled.
This being established, let X denote the set of all the points
x,, which are thus associated with the points yeY. It follows
from the conditions (i) and (ii) and from Lemma 6.1, that
\F[X—P]\ = \F[X—D]\={), and so, by definition of the set X,
that \J\ = \F[X]\ = \F[X-P]\^\F[PJ. Since the condition (T2) im-
implies that |^[J]| =1^1, we obtain, in view of F.8), the inequality
— \F[N]\^0<F{b)— F{a)^\F[I]\^\F[P]\, i.e. the inequality F.7).
Finally, since this relation holds for every subinterval [a,b]
of Io, we see that, unless the function F is a constant, one at least
of the sets F[N~] and F[P] is of positive measure. The set Z^F-l-P
is thus non-enumerably infinite, and this completes the proof.
F.9) Theorem. Let F be a continuous function which fulfils the
condition (T2) and let g be a finite summable function. Suppose further
that F'(x)z^g{x) at each point x at which the derivative F'(x) exists,
except perhaps those of an enumerable set or, more generally, those of
a set E such that \F[E]\ = 0. Then the function F is of bounded vari-
variation and, for each interval [a,b], we have
b
F.10) [d

282 CHAPTER IX. Derivates of functions of one or two real variables.
Proof. Let P be the set of the points x of [a,b] at which the
derivative F'{x) exists and is non-negative. Then, since at each point
¦x eP—E we have 0<F'(x)^g(x)< +00, it follows from Theorem 6.5,
h
Chap.VII, that \F[P—E]\^fF'(x)dx^f\g(x)\dx. On the other
P a
hand, by hypothesis, \F[E~\\ = Q. Hence, on account of Theorem 6.6,
b
F{b)—F(a)^\\g(x)\dx. for each interval [a,b], and, in conse-
a
quence, F is a function of bounded variation whose function of
singularities is monotone non-increasing. The inequality ('6.10) fol-
follows at once.
In view of Theorem 6.3 B°), we may apply Theorem 6.9, in particular,
to continuous functions _F -which are VBG-. We also observe that Theorem 8.9,
when F is of bounded variation, may be deduced from de In, Vallee Poussiu's
Decomposition Theorem (Chap. IV, ^ 9).
Theorem 6.9 may be generalized further, iiy replacing the condition that
the function g is summable, by the condition that the latter is ^-integrable
(the function F then shows itself to be VBG*). We thus obtain a proposition similar
to Theorem 7.3, Chap. VI. The proof of Theorem 6.9 in this generalized.form is,
however, more complicated.
*§ 7. Three theorems <rf Banach. We have repeatedly
emphasized the importance of Lusin's condition BST) in the theory
of the Denjoy integrals. We shall show in.this §, that every con-
continuous function which fulfils the condition (S), also fulfils the con-
condition (T2). This result due to S. Banach [6] (cf. also N. Bary
[3, p. 195]) renders Theorems 6.6 and 6.9 applicable to functions
which fulfil the condition (N).
We shall also study another condition, introduced by
S. Banach [6] and termed condition (8). We say that a finite function
F fulfils the condition (S) on an interval Io, if to each number e>0
there corresponds an ?j>0 such that, for each measurable set
Ed0, the inequality \E\<rj implies \F[E]\<s. (This condition is
essentially more restrictive than the condition (K); cf. the remarks,
p. 279, also G. Pichtenholz [4].)
G.1) Lemma. Given a function F which is continuous on an in-
interval I, every closed set ECI contains a measurable set A on which
the function F assumes each value yeF[E] exactly once.
[§7]
Three theorems of Banach.
283
Proof. With each y sF[E] we associate the lower bound xv
of the set. of the points x of E at which F{x) = y, and we denote
by A the set of all the points xff which correspond in this way to
the values y e F[E]. Since the set E is closed, we plainly have ACE
and F assumes on A each of the values y eP[E] exactly once.
In order to establish the 'measurability of A, let us denote,
for each positive integer n, by En, the set of the points x e E such
that E contains at least one point t which is subject to the conditions
F{t) = F{x) and so — f>l/ra. We have A=E~2X, where I?is closed
n
by hypothesis, and where each En is closed by continuity of F.
The set A is thus measurable and this completes the proof.
G.2) L&mrna. Let F be a continuous function which fulfils the con-
condition (8) on an interval I. Then
(i) every measurable set EC I contains, for each s>0, a meas-
measurable subset Q, such that \F[E~\—F[Q]\<e, and on which the func-
function F assumes each of its values at most once;
(ii) every measurable set Eel contains a measurable subset B,
such that \F[E] — F[R]\=Q, and on which the function F assumes
each of its values at most an enumerable infinity of times.
Proof, re (i). As a measurable set, E is the sum of a set H of
measure zero and an ascending sequence of closed sets {En\. Since
the function F fulfils the condition (K), we have |.F[-H"]|=0, and
hence, the sets F\_E,^ being measurable, there exists a positive in-
integer n0 such that \F[E]—F[E,,J\<e. Now, by Lemma 7.1, there
exists a closed set QCE,,a such that each value y eF[EnJ is as-
assumed exactly once by F on Q. This set Q plainly fulfils the condi-
conditions stated.
re (ii). In view of (i), there exists for each positive integer %
a measurable set QnCE, such that \F[E]— J[Qn]|<l/%, and on which
the function F assumes each of its values at most once. Therefore,
writing R=EQn, we see immediately that \F[E]— F[B] =0 and
that on B the function F assumes each of its values at most an
enumerable infinity of times. This completes the proof.
We shall establish in tbis § three theorems due to Banach on
functions which fulfil the conditions (N) or (S). The first of these
theorems, which concerns functions fulfilling the condition (F),
is aa follows:

284 CHAPTER IX. Derivates of functions of one or two real variables.
G.3) Theorem. Any continuous function F which fulfils the condition
(N) on an interval I, necessarily fulfils also the condition (T2) on I.
Proof. Let us denote, for each measurable set Ed, by SRg
the class of all measurable sets BCE which are subject to the
following two conditions: (i) \F[E]—F[E]\ = 0, and (ii) each
value yeF[E] is assumed by the function F at most enumerably
often on B. By Lemma 7.2r the class 9Rg is non-empty, however we
choose the measurable set ECI- We shaE denote, for any such set JB,
by pE the upper bound of the measures of the sets (9t«).
Consider, in particular, a sequence {Hn} of sets (9ti) such that
]hn.\En\=fir Let H=HHn and let U be a set (%-&)• We verify
n n
at once that |J7| = 0, whence on account of the condition (N),
\F[U]\ = 0. Therefore, \F[I—H]\=\F[U]\=0, so that almost every
value y eF\I] is assumed by F only on the set H, and therefore at
most enunierably often.
The second of the theorems of Banach concerns functions
which fulfil the condition (S).
G.4) Theorem. In order that a continuous function F be subject
to the condition (8) on an interval I, it is necessary and sufficient
that F be subject on I to both the conditions (N) and (Tx).
Proof. 1° Suppose that the function F fulfils the
condition (S) on I. Since this condition clearly implies the con-
condition (N), we need only prove that F fulfils the condition (T2).
Suppose then, if possible, that the set of the values assumed
infinitely often on I by the function F, is of positive outer measure.
Since this set is measurable by Theorem 6.4, it contains a closed
subset T of positive measure. Let X denote the set of all the points
xel such that F{x) e T. The set X, plainly, is also closed.
We shall now define by induction a sequence of measurable
sets {Xi\ subject to the following conditions: (i) Xt-Xj = Q whenever
*+J» (») |-F[X,]|>|r|/2 for *=1,2,..., and (iii) the function F
assumes each of its values at most once on each set Xt.
For this purpose, suppose defined the first ft sets X, for which
the conditions (i), (ii) and (iii) are satisfied. Let Ek=I—Zx,. Since,
on 2X-, the function F assumes each of its values at most a finite
number of times, it follows that, each value y e 7 is necessarily
assumed on the set Fk. By Lemma 7.2, this set therefore contains
[§7]
Three theorems of Banaah.
285
measurable subset Xk+i such that iJ7[XA+i]|>|J][EA]|/2>|r|/2 and
that each value is assumed by F at most once on Xk+1. The sets
Xi,Xz,...,Xk+\ clearly fulfil the conditions (i), (ii) and (iii).
The sequence {Xt} being thus defined, it follows from (i) that
lim \X( | = 0, and hence, remembering that the function F fulfils
the condition (S), we have 'also lim |.F[X,]| = 0. But this clearly
i
contradicts (ii), since |3T| >0.
2° Suppose now that the function F fulfils the conditions (N)
and (Tx), but not the condition (S). We could then determine a po-
positive number a and a sequence, of sets ¦
G.5) \Ek\<lj2k, and
; in I so that for ?=1,2,...,
G.6) \F[Ek]\ > a.
Let us write E = lim sup Ek and A = lim sup F[Ei,]. We easily see
A k
that, if y e A, then either ye FIE], or else the value y is assumed
by F on I infinitely often (in fact there exists an increasing sequence
of positive integers {hi) and a sequence of points {*,-}, every two
of which are distinct, such that x^
and F(xt) = y for i'=l,2,...).
Now, on account of G.5), we have \F\ = Q, and therefore
also \F[E]\ = 0. On the other hand, by G.6), \A\^a>0. Thus
\A—F[H]\ > a, and since, as we have just seen, each value
y e A — F[M\ is assumed by F infinitely often on I, this contradicts
the hypothesis that the function F fulfils the condition (Tx).
We shall establish, nest a "differentiability theorem" for
the functions which fulfil the condition (N):
G.7) Theorem,. In order that a continuous function F be absolutely
continuous on an interval Io, it is necessary and sufficient that the
function F fulfil simultaneously the condition (N) and the condition
G.8)
/ F'(x) dec
at which the function F has
where P denotes the net of the
a finite non-negative- derivative.
Proof. Since the conditions of the theorem are obviously
necoBHary (of. Theorem6.7, Chap.VII), let us suppose that the
function F fulfils the condition AST) and the inequality G.8). Let g
be the function equal to F'(x) for w e P and to 0 elsewhere. Then,
it' E denotes the set of the points x at which F'{x)=+°°, vre shall
have F'(x)K(!(¦>:) "t every point * of Io-E at which the derivative
F'{x) exists.

286 CHAPTER IS. Derivates of functions of one or two real variables.
On the other hand, since |J?|=0 (cf. Theorem 4.4, or
Chap. VII, §10, p. 236), we have \F[E]\ = Q, and, since the func-
function F fulfils, by Theorem 7.3, the condition (T2), it follows from
Theorem 6.9 that F is of bounded variation on Jo. This completes
the proof, since, by Theorem 6.7, Chap. VII, every continuous
function of bounded variation, which fulfils the condition {?) is
absolutely continuous.
Theorem 7.7 (in a slightly less general form) was first proved by N, Bary
[2; 3, p. 199]. It shows in particular that every continuous function F(x), which is
subject to the condition (X) and whose derivative is non-negative at almost every point
where F(x) is derivable, is monotone noii-decreasviig. This proposition contains an
essential generalization of Theorem 6.2, Chap. VII.
Theorem 7.7 may, moreover, be generalized still further. If a continuous
function F(x) fulfils the aondition (N) and if the function g(x), equal to F'(a>) wherever
F(x) is derivable and to 0 elsewhere, has a major function (in the Perron sense),
then the function F(x) is ACG# i. e. mi indefinite tf-integral.
For the part played by the conditions (N), (Tx) and (Ta) in the theory of
Denjoy integrals, cf. also J. Bidder [8].
Prom Theorem 7.7 we obtain the third theorem of Banach:
G.9) Theorem. Any function which is continuous and subject to the
condition (?) on an interval, is derivable at every point of a set of
positive measure.
* § 8. Superpositions of absolutely continuous functions.
Suppose given a bounded function G on an interval [«,&], and a func-
function E defined on the interval [a,/?] where a and /? denote respectively
the lower and the upper bound of G on [a, &]. We call superposition
of the functions Gt and E on [a,b], the function S{6{x)). The func-
function G is termed inner function and the function E outer function
of this superposition.
If a function F is continuous and increasing on an interval
[a,b], the continuous increasing function G defined on the interval
[F{a),F{b)] so as to satisfy the identity G(F(m)) = <x on [«,&], will,
as usual, be termed inverse, function of F and denoted by F~\
• It has long been known that the superposition of two abso-
absolutely continuous functions is not, in general, an absolutely con-
continuous function. By means of the conditions discussed in the
preceding §§, particularly the condition (8), Mna Bary and D. Men-
ehoff succeeded in characterizing completely the class of functions
expressible as superpositions of absolutely continuous functions.
(Cf. also G. Fichtenholz [3].)
Superpositions of absolutely continuous functions.
2S7
(8.1) Theorem. Any function F which is continuous mid subject
to the condition (Tx) on an interval [a,b] is expressible on this interval
as a superposition of two continuous functions, of which the inner
function is of bounded variation and the outer function is increasing
and absolutely continuous.
If, further, the function F fulfils the condition ($), the inner
function of this superposition is necessarihj absolutely continuous also.
Proof. Let a and /? denote respectively the lower and upper
bounds of F on [a,b], and let sF(y) denote, for each y, the number
(finite or infinite) of the points of the interval [a, b] at which F
assumes the value y. Since, by hypothesis, the function F is contin-
continuous and subject to the. condition (T^, we shall have 0<ljsF(;//)< 1
for almost all the values y of the interval [a,jf\. Let us denote by U
an indefinite integral of the function which is equal to 1/sf(#) on
[a,(J\ and to 1 elsewhere. We now write G(x) = U[F(x)] for x e [a,b].
We thus have F(x) = U~1[G[x)'], and in order to establish the first
part of the. theorem, it is enough to show that (i) the function U
is absolutely continuous and (ii) the function 0 is of bounded variation.
Suppose, if possible, that the function V~x (which is continuous
and increasing together with V) is not absolutely continuous. Then
(cf. Theorem 6.7, Chap. VII), there exists a set F, of measure zero
^). Writing Q=JT\B\ we thus have
and'
We may, plainly, suppose that the set E, and therefore the
set Q, are sets [<&»). Thus (cf. Theorem 13.3, Chap, ni)
which renders the relations (8.2) contradictory, since almost every-
everywhere U'{y) = ljnF{y)>0 for ye[a,^1 and U'[y) = l outside the in-
interval [a, /?].
In order to establish (ii), we shall make use of the criterion
of Theorem 6.4. Denote, for each z by Se{z) the number of the points
of the interval [a, b] at which the function G assumes the value s.
Since the function U is increasing, we clearly harve 8o(U(y)) = 8p(y)
for each y, and s(l{a) = Q for each z outside the interval[U(a), U'(/?¦)].
Hence, remembering that the function U is absolutely continuous,
we obtain (of. Theorem 15.1, Chap. I)

288 CHAPTER SI. Derivates of functions of one or two real variables.
?/(«)
f
=fsF
which shows, by Theorem 6.4, that the function G is of bounded
variation.
Finally, if the given function F fulfils the condition (N), so
does the function G{x) = TJ{F{%)), and the latter, since it is of
bounded variation, is absolutely continuous by Theorem 6.7, Ghap.VII.
This completes the proof.
(8.3) Theorem. 1° In order that a continuous function F be expres-
expressible as a superposition of two continuous functions of which the inner
function is of bounded variation and the outer function is absolutely
continuous, it is necessary and sufficient that F fulfil the condition (Tt).
2° In order that a continuous function be representable as a super-
superposition of two absolutely continuous functions, it is necessary and
sufficient that the function fulfil both the conditions (T^ and (N), or
what amounts to the same, the condition (8).
Proof. Since it follows at once from Theorem8.1 that these
conditions are sufficient, we need only prove them necessary.
Let therefore F{x)—E{G{%)) on an interval [a,b], where G
is a function of bounded variation and E an absolutely continuous
function. Let a and /S be respectively the lower and the upper bound
of G on [a, 6]. Let EG and FH denote the sets of the values which the
functions G and E assume infinitely often on the intervals [a,b]
and [a,/S], respectively. Since the functions G and E fulfil the con-
condition (T2), we have \Ea\—\F,H\=§, and since the function H is,
moreover, absolutely continuous, we have also \H[Eo]\=Q. Now
we see at once that each value which is assumed infinitely often
on [«,&] by the function F, belongs either to EH, or to H[EG]. The
set of these values is thus of measure zero, and the function F ful-
fulfils the condition (Tj).
If, further, the function G is absolutely continuous (as well
as E), then the function F is a superposition of two functions which
fulfil the condition (N), and, thei fore, itself fulfils this condition.
This completes the proof.
[§8]
Superpositions of absolutely continuous functions.
289
Theorems 8.1 and 8.3 are due to Nina Bary [1; 3, pp. 208, 633]
(cf. also S. Banach and S. Saks [1]). Paxt 2° of Theorem 8.3 was
established a little earlier in a note of N. Bary and D. Menchoff [1]
(cf. also N. Bary [3, p. 203]) in a form analogous to Theorem 6.2. Thus:
(8.4) Theorem. In order that a function F which is continuous on
an interval [a, 6] be on this interval a superposition of two absolutely
continuous functions, it is necessary and sufficient that the set of the
values assumed by the function F at the points of [a,b] at which F is
not derivable, be of measure zero.
Proof. Let QF be the set of the points of [a,6] at which F
is not derivable. Suppose first that
l°F(x) = H(G[x)) on [a,b], where E and G are absolutely
continuous functions. Let Qg and QH be'respectively the sets
of the points at which the, functions G and E are not derivable.
We have \QQ\=\QH\=O mi, consequently, \F[Qa]\ = \E[QH]\=0.
Now, we see a/fc once that if the function F is not derivable at a point x,
then either xeQe, or G{x)eQH. Therefore F[QF] CF[Qa]+ E[QH],
and hence, \F[QF]\=Q.
Conversely, suppose that
2° I^W^hO. By Theorem 6.2, the function then fulfils the
condition (Tx). To show that F also fulfils the condition (N), con-
consider any set of measure zero, E say, in [«,&]. Since the function F
is derivable at each point of E—QF, we have, by Theorem 6.5,
Chap. VII, \F[E-QF}\ = (), and since, by hypothesis, {FlQ^l^Q,
we obtain |-F[K]| =0. The function F thus fulfils both the conditions
(Tx) and (N), and is, therefore, by Theorem 8.3 B°), a superposition
of two absolutely continuous functions.
It follows from Theorem 8.3 B°) that a superposition, of any finite number
of absolutely continuous functions is expressible as a superposition of two absolutely
continuous functions. For the superposition of any finite number of functions
which fulfil the condition (S), itself fulfils this condition.
The results exposed in this § have been the starting point of the important
researches of Nina Bary [3] on the representation of continuous functions
"by means of superpositions of absolutely continuous functions. Let us cite two of
her fundamental theorems: 1™ Every continuous function is the sum of three super-
superpositions of absolutely continuous functions, and there exist continuous functions
which cannot be expressed as the sum of two such superpositions. 2" Every continuous
function-which fulfil* the condition (N) — or, more generally, every continuous function
which is derivable at every point of a set which has positive measure in each interval—
is the sum. of turn superpositions of absolutely continuous functions, and there exist
contiinious functions which fulfil the condition (N), but are not expressible as one
nuperpoHilion of alwuhikiy continuous functions (the function V(x) discussed above
in § 6, p. 279, is an example of such a function).
For further researches, Me K.Bary [4] and J. Todd [1; 2].
S. Silks. TJicnrv or tin- IntL'irnll.
19

290 CHAPTER IX. Derivates of functions of one or two real variables.
§ 9. The condition (D). We shall now establish, for the
extreme approximate derivates, a theorem, analogous to Theo-
Theorem 4.6, but whose proof depends on a different idea. It is con-
convenient to formulate it, from the beginning, in a slightly more ge-
general manner.
Given two positive numbers N and s, we shall say that a func-
function F fulfils at a point x0 the condition (Dn,*), if there exist positive
numbers h, as small as we please, such that the difference between the
outer measure of the set ~E[F(x)— F{x0) > N ¦ (x—x0
x0);
and that of the set ~E[F(x)—F{x0)^—N-{x-~a?0); 0<re
X
exceeds the number he in absolute value. By symmetry, merely
replacing F{x)—F{xa) and x—x0 by F{xo)—F{x) and xo—x respec-
respectively, we define the condition (Dj7,?).
If, for a point x0, there exists a pair of finite positive numbers N
and ? such that the function F fulfils at this point the condition (D^,,.),
or the condition (D]v;t)> we say that F fulfils at x0 the condition (D).
For measurable functions the condition (D) may be formulated
more simply: a measurable function F fulfils at a point x0, the con-
condition (D), if there exists a finite positive number N such that x0
is not a point of dispersion for the set of the points x at which
\F(x)~-F{xo)^N-\x—xo\.
(9.1) Theorem. If any one of the four approximate extreme derivates
of a function F is finite at a point x0, then the function fulfils the con-
condition (D) at this point.
JProof. Suppose, to fix the ideas, that \FJv(x0)\<-\-oo and write
JV"=|JFj(a;0)|+l. Let Ex and E% be the sets of the points x which are
situated on the right of the point x0 and which fulfil respectively the
inequalities F{x)—F(xo)^N-(x— x0) and F(x)—F(xo)^—N-{x—xo).
It follows at once from the definitions of approximate derivates
(Chap. VII, § 3) that x0 is a point of dispersion for the set E1} while
E2 has at x0 a positive upper outer density. Denoting the latter
by 6, we see at once that the function F fulfils at sc0 the condition
(Dn,b) whatever be the positive number g<<5.
(9.2) Lemma, Let N and e be finite positive numbers and suppose
that a finite function F fulfils the condition (Djv,?) at each point of
a set B. Then \F[E]\^{2N/e)-\Il\.
l§9] The condition (D). 291
Proof. We shall first show that for every interval /=[«,*],
(9.3) \I\~^{el2N)-\F[E-I]\.
For this purpose, we write, for every y,
The function H thus defined is non-increasing and bounded on the
whole straight line (— oo, + oo); we have, in fact, for every y,
(9-5)
Given an arbitrary point y0 of F[E-I], which is distinct from
and at which the function E is derivable, let us consider a point
xoeE-I such that F[xQ) = y0. Plainly a;0=)=6. Let us write, for brevity,
and
For every subinterval [os0,x0+h] of I, we then have the re-
relation B(h,N)CA(h,N)CA{h,~N)CB(h,—N), whence it follows
easily, on account of (9.4), that
Now, since F fulfils, by hypothesis, the condition (J)t,e) at x0, there
exist positive values h, as small as we please, such that
\A(h,-N)\~\A(h,N)\^hs,
and therefore E[y0— Nh)—H{yo+ NJi)^he. Hence, H'{yo)^—sl2N
for every point yo4=F{b) of F[E-I] at which the function H is
derivable. Therefore, denoting, for each positive integer n., by Qn
the part of the set F[E-I] contained in the interval [—n,n\, we
find, on account of (9.5), \I\^\H(«,)—E{— n)\^e-\Qn\j2N, from which
the inequality (9.3) follows by making «->-oo.
This being established, let r\ be a positive number and (I*}
a sequence of intervals such that
(9.6)
and
Since (9.3) holds for each interval I, it follows from (9.6) that
|-B| + »y >{BJ'2N)-^,\F[E-Ih]\^{elM)-\F[E]\, whence, remembering
that r\ is an arbitrary positive number, we see that \F[E]\^BNje)-\E\.
19*

292 CHAPTER IX. Derivates of functions of one or two real, variable*.
(9.7) Theorem. If at each point of a set E, a finite function F ful-
fulfils the condition (D) (and so, in particular, if at each point of E the
function F has] one of its', extreme approximate derivates finite),
then the function F fulfils the condition AST) on IS.
Proof. Let H be any subset of E of measure zero, and let Rn
denote, for each positive integer n, the set of the points x of E at
which the function F fulfils the condition (D,ti/«) or (D~,/;,). We
clearly have H=]jEn, and since, by Lemma 9.2, |F[Ra]\^.&n2-\lIn\= 0,
n
we obtain |JP[S"]|=^0.
Theorem 9.7 enables us to complete Theorems 10.5 and 10.14
of Chap. VII, as follows:
(9.8) Theorem. 1° Every finite function F which is continuous on
a closed set E and which has at each point of E, except perhaps those
of an. enumerable subset, either two finite Dini derivates on the same
side, or one finite extreme bilateral derivate, is ACG* on E.
2° Every finite function F which is continuous on a closed, set E
and which has at each point of E, except perhaps those of an enumerable
subset, either one finite Dini derivate, or one finite extreme approximate
bilateral derivate, or finally two finite extreme approximate unilateral
derivates on the same side, is ACG on E.
Proof. By Theorems 10.1, 10.5, 10.8 and 10.14 of Chap. VII,
the function F is VBG* on E in case 1° and VBG on E in case 2°.
On the other hand, by Theorems 4.6 and 9.7, this function fulfils,
in both cases, the condition (N) on E. Hence, by Theorems 6.8
and 8.8 of Chap. VII, the function is ACG* on E in case 1°, and
ACG on E in case 2°.
In the most important case in which the closed set E is an
interval, Theorem 9.8 may further be stated in terms of Denjoy
integrals. For this purpose, let us begin by noting the following
proposition (cf. A. 8. Besicovitch [2], and J. 0. Burkill and
U. S. Haslam-Jones [1]):
(9.9) Theorem. If a finite function F is measurable on a set E and
has at each point of this set one of its Dini derivates finite, then this
derivate is, at almost all points of E, an approximate derivative of F.
Proof. It follows from Theorem 10.8, Chap. VII, that the
function F is VBG on E, and so, approximately derivable tit almost
all the points of E. Let us denote by E1 the set of the points of E
[§9]
The condition (D).
293
at which one at least of the opposite DM derivates I+ and IT~ is
finite. Plainly, F'{x)^F'a.p(x)^F+{x) at each point x at which
the approximate derivative F'^{x) exists, and therefore by
Theorem 4.1, F~{x) =F+{x) = F'ni{x)__ at almost all points x
of E,_. Similarly, we show that F+(x)=F~(x)=F'aL]i(x) at almost all
the points x of E at which one of the derivates F+ and F~ is finite.
This completes the proof.
(9.10) Theorem. 1° If f is a finite function which, at each point of
an interval Io, except those of an enumerable set, is equal to an extreme
bilateral derivate of a continuous function F, then the function f is S^-in-
tegrable on Io and the function F is an indefinite ^-integral of f.
2° If f is a finite function which, at each point of an interval Io,
except those of an enumerable set, is equal either to a Dini derivate, or to
an extreme approximate bilateral derivate of a continuous function F,
then the function f is ?>-integrable on Io and the function F is an in-
indefinite ^-integral of f.
Proof. In view of Theorem 9.8, the function F is ACG* in
case 1°, and ACG in case 2°. Moreover, at almost all the points x
of E, we have F'{x)—f(x) in case 1°, and by Theorem 9.9, F's_p(x)=f(x)
in case 2°. This proves the theorem.
Although Theorem 9.8 presents a formal analogy with Theorems 10.5 and
10.14 of Chap. VII, there is an essential difference between the result of this § and
those of § 10, Chap. VII. We see, in the first place, that the criteria of Theorems
10.5 and 10.14 of Chap. VII concern functions which are given on quite arbitrary
sets, whereas those of Theorem 9.8 are established only for closed sets. In the
second place, if the derivates of a quite arbitrary function satisfy on a set E the
conditions of Theorem 10.5, or of Theorem 10.14, of Chap. VII, then the set E can,
by these theorems, be decomposed into a sequence of sets on which the function
is absolutely continuous. On the contrary, Theorem 9.8 of this § does not enable
us to draw any conclusion as to a similar decomposition of the set E (even when
this set is an interval), unless the function considered is continuous.
Two examples will now be given to show that this feature of
Theorem 9.8, [which represents a restriction as compared with the results of § 10,
Chap. VII, is essential for the validity of the theorem.
(i) Consider the function F(x)=TJ[2nai]/5n, where [2"k] denotes, as usual,
n
the largest integer not exceeding 2nx. This function is increasing. Its lower right-
hand derivate is finite everywhere, and even, as we easily see, vanishes identically.
Nevertheless, there is no decomposition of the interval Jo=[0,1] into a sequence
of sets on which F is absolutely continuous, or even only uniformly continuous.
In fact, no such decomposition can exist for a monotone function F whose points
of discontinuity form a set everywhere dense in Ja.

294 CHAPTEE IX. Derivates of functions of one or two real variables.
For, if such a decomposition {S1, E.2, ..., En,—} existed, one at least of the
sets En -would, by Baire's Theorem (Chap. II. § 9), be everywhere dense in an interval
ICJo- Tllis is plainly impossible since the function F, monotone by hypothesis,
is uniformly continuous on each set En and has points of discontinuity in the in-
interior of I.
(ii) Let us now consider an example of a continuous function
F{x), increasing on the interval J0=[0,l], and which has its lower
right-hand derivate zero at every point of a set E, without being
ACG on E.
For this purpose, let us agree to call, for brevity, function attached to
an interval I=[a, 6], any function S(x), which is continuous and non-decreasing
on I, and which fulfils the conditions:
(a) H(z) is constant on each of the intervals It of a sequence {Ik} of non-
overlapping sub-iiitervals of I such that |i1=,S|Ift|; the length of any sub-
• interval of I on which R(x) is constant does not exceed \I\/2;
(b) H(x)—H(a)jgx—a and E(b)—H(x)^.b—x for every xel.
Sueh a function is easily obtained, by slightly modifying the construction
of the function j(x), considered in Chap. Ill, § 13, p. 101.
This being so, we shall define by induction a sequence {Fn(x)} of functions
attached to the interval Jo, beginning with an. arbitrary function F^x) attached
to this interval. Given the function Fa attached to =/„, let {I^'^a^, bf]}
be a sequence of the intervals of constancy of FB in the interval Jo. (By an
interval of constancy of a function in Ja we mean here any interval I(ZJ0
snch that the function is constant on I without being constant on any sub-
interval of t/0 which contains I and is distinct from I.) For -each fe=l,2,...,
we determine a function B^\x) attached to the interval /W and we write
L n-f-11
for
for
the sum T^ being extended over all the values i such that ftp ^S x.
The sequence {Fn{x)) being thus defined, let
(9.11) F(x)=EFn(x)/2".
n
The function F(x) is clearly continuous, increasing, and singular on Jo-
Consider the set E=f]J!(I^H, and let xa be any point of E. Then there
n Jt
exists a sequence {I$}n__1 2 of intervals each of which contains x0 in its
interior. Plainly, for each positive integer n, FJfi^jf1)—Fj(sea) = Q if Kra, and
Fjlbty—Fj(xo)<.bM—x0 if j>n. Hence, by (9.11),.
for each n, and therefore F+{x0)=Q.
Nevertheless, the function F is not ACG on E. To see this, suppose, if pos-
possible, that E is the sum of a sequence of seta En on each of which the function
F is AC. Since the set Jo—E is the sum of a sequence of non-dense closed sets,
one at least of the sets En is everywhere dense in a sub-interval I of Jo, and,
since the function F is absolutely continuous on each En, this function would
be so also on the whole interval I. This is clearly impossible, for the function F
is singular and increasing.
[§ 10] A theorem of Denjoy-Khintchine on approximate derivates. 295
§ 10. A theorem of Denjoy-Khintchine on approximate
derivates. The considerations of the preceding § -will now be
completed by a theorem which establishes, for the extreme ap-
approximate derivates, relations similar to those which hold for
Dini derivates (cl § 4). This theorem was proved independently
by A. Denjoy [6, p. 209] and by A. Khintchine [4; 5, p. 212]
(cf. also J. C. Burkill and TJ. S. Haslam- Jones [1; 3]).
A0.1) Theorem. If a finite function F is measurable on a set E
and if, to each point x of E, Here corresponds a measurable set Q(x)
sueh that (i) the lower unilateral density of Q(x) at x is positive on at
.least one side of the point x and., (ii) J'QW(*)<+oo or FQ^}(x)>—oo, then
the function F is approximately derivable at almost all the points of E.
Consequently, if a finite function F is measurable on
a set E, then at almost every point of E either the function F
is approximately derivable, or else FfJ)(x)=F'^(x)=+-oo and
Proof. In view of Lusin's Theorem (Chap.HE, §7), we may
suppose that the set E is closed and that the function F is con-
continuous on E. To fix the ideas, consider the set A of the points x e E
such that (ix) the lower right-hand density of Q(x) at x is positive
and (iix) Fq1x)(x)<-\- oo. We shall show that the function F is ap-
approximately derivable at almost all the points of A. By symmetry,
this assertion will remain valid for each of the other three subsets
of E, defined by a similar specification of the conditions (i) and (ii)
of the theorem.
Let us denote by P the set of the points of A at which the
function F is not approximately derivable, and suppose, if pos-
possible, that |P|>0. For each positive integer n, let An be the set
of the points x of E such that the inequality 0<A<l/w implies
A0.2) \E[F{t)-F{x)^n-(t-x);teE;x^t^:x+h'\\>hln.
t
The sets An are closed. To see this, let us keep an index n fixed
for the moment, and let {a;,}i=i,2,... be a sequence of points of the
set An converging to a point ir0. Let h^ljn be a non-negative number,
and, for brevity, let Fi=B[F(t)—F{xi)^;n-{t—xi);teE; x^t^Xi+h]
where 4=0,1,2,.... We obtain \E^lijn for i=l,2,..., and since,
by continuity of F on E, we have Eo D lim sup Bb it follows
(cf. Chap. I, Theorem 9.1) that \E0\^Ii[n, which shows that xoeAn,
i.e. that An is a closed set.

296 CHAPTER IX. Derivates of functions of one or two real variables.
Let us now denote, for every pair of positive integers n and k,
by An,k the set of the points x of An such that the inequality
0<A<l/fc implies
A0.3) \E[teAn;x—A<i<*]|>(l— ^nr^-h.
t
We observe easily that the sets An, and therefore the sets A,hi
also, cover the set A almost entirely. Hence, there exists a pair of
positive integers n0 and Tc0 such that \A^ko-P\>Q. Let R denote
a portion of the set Anm/,tl-P such that
A0.4) |-R|>0, A0.5) S{B)<ljna and A0.6) 8(li)<ljlc0.
Writing &{x)=F(x) — (na+l)-x, we shall show that the func-
function G is monotone non-increasing on B,. Suppose therefore, if pos-
possible, that there exist two points a and b in R, where a<b, such that
A0.7)
G{a)<G{b).
Let </=[>,&]. Since the set A,h is closed and the function G
continuous on A,h, the function G attains, at a point e of the set
Ani,-J, the lower bound of its values on this set, In virtue of A0.7)
we have c<b. Since ceA^, and since, by A0.5), 0<&—c^l/w0,
we may put n=n0, x—o and h=b—c in the relation A0.2). We
thus obtain
A0.8) \E[G{t)-G{c)^ — {t—c);teE;o^t^b-]\^(b-e)fno.
i
Again, since beBCA^^ and since, by A0.6), O<6—o<l/fco,
we may put n=n0, x=b and h=b—c in A0.3). This gives
A0.9) |EP6^,,; c^t^bUMl-bwMb-e).
Now the sets which occur in the relations A0.8) and A0.9)
are both measurable; it therefore follows from these relations that
there exist, in the open • interval (c, &), points teA,,t) for wliich
G[t)—G{c)^.—(t~c)<0. This is plainly impossible, since the
function <? attains its minimum on the set Ana-J at the point o.
The function G is thus monotone on the set B, and since it
is, moreover, measurable (indeed continuous) on the closed set
JBDB, it follows that G is approximately derivable at almost all
the points of B. On the other hand, however, since BcP the
function F is approximately derivable at no point of B, and, in view
of A0.4), we arrive at a contradiction. This completes the proof.
[§11] Approximate partial derivates of functions of two variables. 297
By a slight modification of the proof, we may extend Theorem 10.1, in
a certain way, to functions which need not be measurable. Let us agree to under-
understand by approximate derivability of a finite function F at a point x0, the existence
of a set for which k0 is a point of outer density and with respect to which the
function F is derivable (if the function F is measurable, this notion of approxi-
approximate derivability clearly agrees with the definition of Chap. VII, § 3). When
approximate derivability is interpreted thus in the statement of Theorem 10.1,
this theorem remains valid without the hypothesis that the function F be meas-
measurable on the set M (although the hypothesis concerning the measurability of
the sets Q(x) remains essential).
Prom Theorem 10.1, we may deduce the following proposition: If, for
a finite function F, we can make correspond to each -point x of a set E, a measurable
set Q(x) whose lower right-hand density at x is positive, and with respect to which
the function has an infinite derivative at x, then the set E is of measure zero. This
theorem is similar to Theorem 4.4, but only partially generalizes the latter. It is
not actually possible to replace, in Theorem 4.4, the ordinary, by the approx-
approximate, limit, without also removing the modulus sign in the expression
\F (x+h) -F (x)\. This rather unexpected fact was brought to light by V. J ar n i k [2],
who showed that there exist continuous functions F for which the relation
lim. ap \F(x+h)—F(x)\/h=+co holds at almost all points x,
A-XH-
Pinally, let us note that Theorem 10.1 is frequently stated in the following
form:
If a finite function F is measurable on a set E, then at almost every point x
of -Z? either (i) the function F is approximately derivable, or else (ii) there exists a meas-
measurable set B(x) whose right-hand and left-hand upper densities are both equal to
1 at x, and with respect to which the two upper unilateral derivates of F at x are +00
and tlie two lower derivates —00.
It has been shown by A. Khintchine [4] (cf. also V. Jarnik [2]) that
there exist continuous functions for which the case (ii) holds at almost every point x.
§ 11. Approximate partial derivates of functions of
two variables. The §§ which follow will be devoted to generali-
generalizations of the results of § i for functions of two real variables (their
extension to any number of variables presents, as already said,
no fresh difficulty). In this § we shall establish some subsidiary
results.
Given a plane set Q and a number rj, we shall understand
by the outer linear measure of Q on the line y—r), the measure of
the linear set E[(J, rj) eQ]. Similarly, we define the outer linear measure
of Q on a line x = %, where f is any number. It follows from Fubini's
Theorem in the form (8.6), Chap. Ill, that if Q is a measurable set
¦whoselinear measure on almost all the lines y=i] (i.e. on the lines
y=rj for almost all values of rj) is zero, then the set Q is of plane
measure zero.

298 CHAPTEE IX. Derivates of functions of one or two real variables.
A point (?co?2/o) w111 ^ termed point of linear density of a plane
set Q in the direction of the x-axis, if as0 is a point of density of the
linear set V[{t,yo)eQ]. We define similarly the points of linear den-
density of Q in the direction of the y-axis.
A1.1) Theorem,. Almost all points of any measurable plane set Q
are points of linear density for it both in the direction of the co-axis
and hi that of the y-axis.
Proof. We may clearly assume that the set Q is closed. Con-
Consider, to fix the ideas, the set D of the points of Q which are points
of density of Q in the direction of the rc-axis. Since the set Q—D
is of linear measure zero on each line y= t\, the proof of the relation
\Q— D\ = Q reduces to showing that the set I) is measurable.
In order to do this, we write, for each point {x,y) and each
pair of numh-ers a and b,
E(x,y; a,b) =
eQ;
and we denote, for each pair of positive integers n and k, by QnJl the
set of the points (x,y) of Q such that the inequalities a<x<b and
b—a^l/h imply \E{%,y;a,b)\-^{l-in>-^-{b-a). Plainly D=WZQ,ltk.
n k
We now remark that all the sets Qn,k are closed. To see this,
we keep the indices n and k fixed for the moment, and consider
an arbitrary sequence {(*,,2/,-)},-=i,2,... of points of Q,hk which, converges
to a point {xQ,y0). Let a and b denote real numbers such that a<a;0<6
and b—a^ljlc. For every sufficiently large index i, we then have
a<%t<b, and so \E{xpyt; a,
•(&—«). Now it is easy to
see that lim sup E(xp yt; a,b)CE(xQ,yQ; a,b); it therefore follows from
Theorem 9.1, Chap. I, that \E(xo,yo; a,b)\^(l— w~')-(&— a), and so,
that (xo,yo)eQn,k.
Since the sets Qn,t are closed, D is a set C^at) and this com-
completes the proof.
If F is a finite function of two variables, the extreme approx-
approximate partial derivates of F{x,y) with respect to x will be denoted
by FtPx, FtVx, -Pap,, and F^,x. If these derivates are equal at a point
(x,y), their common value, i. e. the approximate partial derivative
of F with respect to x, will be denoted by KVx{x,y). Analogous
symbols will be used with respect to y. For the partial Dini deri-
derivates, we shall retain the notation of Chap. V, namely Ft, Ft, etc.
[§1.1] Approximate partial derivates of functions of two variables.
299
A1.2) Theorem. If a finite function of two variables F is meas-
measurable, on a set Q, its extreme approximate partial derivates are them-
themselves measurable on Q.
Proof. In view of Lusin's Theorem (Chap. Ill, § 7), we may
suppose that the set Q is closed and that the function F is con-
continuous on Q. Consider, to fix the ideas, the derivate J^Pi, Let a be
any finite number and let P be the set of the points {x,y)
of Q at which F~tPx(x,y) z^ %. We have to prove that the set P is
measurable.
For this purpose, let D denote the set of the points of the
set Q which are its points of linear density in the direction of the
iB-axis. Further, for every point {x,y) and every positive integer n,
let En{x,y) denote the set of the points t such that
t^x, (t,y)eQ and F{t,y)—F{x,y)^{a+n~1)-{t—x).
We easily observe (cf. Chap. VII, § 3) that, in order that
FtPx[xOlyo) < a at a point {xo,yo) e B, it is necessary and suffi-
sufficient that the point [xQ,y0) be a point of right-hand density for
every set En{x0,y()), where m = l,2,... Hence, denoting for every
system of three positive integers n, k and p, by Qn,k,P the set of
the points (x, y) of Q such that the inequality O^h^ljp implies
\En{x,y) ¦ \_x,x-\-1h\\ > A— krl) ¦ h, we have
l-1~1-<5) r-u=[i n /jV,,,*,^.
n k p
Now the set Q is closed and the function F is continuous on Q,
and by means of Theorem 9.1, Chap. I (cf. the proofs of Theorems 10.1
and 11.1) we easily prove that all the sets Qn,h,P are closed. Hence,
by A1.3), the set F-D is measurable, and since, by Theorem 11.1,
\Q— J>| = 0, we see that the set P is measurable also. This completes
the proof.
It follows, in particular, from Theorem 11.2 that the extreme approximate
derivates of any finite measurable function of one real variable are themselves mea-
measurable functions. We thus obtain a result analogous to Theorem4.3, Chap. IV,
whieli ooiioenied the measurability of Dini derivates (cf. also Theorem 4.1, Chap. V,
and the remark p. 171).

300 CHAPTEE IX. Derivntes of functions of one or two real variables.
§ 12. Total and approximate differentials. A finite func-
function of two real variables F is termed totally differentiable, or simply
differentidble, at a point (xwy0) if there exist two finite numbers
A and B such that the ratio
A2.1) \F{x)y)—F(xa,y0)--A-{x—xa)—B-{y—yfs)'\j[\!is~x<i\-+\y—y^\
tends to zero as {x,y)-+{xo,yo). The pair of numbers {A,B} is then
termed total differential of the function F at the point (a?0,?/0) and
we see at once that A and B are the partial derivatives of F at
{(ca,y0) with respect to x and to y respectively.
If, for a finite function of two variables F and for a point
(xo,yo), there exist two finite numbers A and B such that the ratio
A2.1) tends approximately to 0 as {«i,y)-^{xo,yo), the function F
is termed approximately differentiable at {xa,ya) and the pair of
numbers {A,B} is called approximate differential of F at {xo,yo).
The numbers A and B will be called coefficients of this differential.
We see at once that no function can have at a given point
more than one differential, whether total or approximate.
The existence of a total differential of a function F(x,y) at a. point may
be interpreted as the existence of a plane, tangent at this point to the surface
z=F{x,y) and non-perpendicular to the sy-plane. In this way the notion of total
differentiability of functions of Wo variables corresponds exactly to the similar
notion of derivability of functions of one variable. Nevertheless, whereas every func-
function of bounded variation of one variable is almost everywhere derivable, a.
function of bounded variation (in the Tonelli sense), and even an absolutely
continuous function, of two variables may be nowhere totally difforontiable
(of. W. Stepanoff [3, p. 515]).
The coefficients of an approximate differentia] of a function
at a point are not, in general, approximate partial derivatives of
this function. Nevertheless they coincide with the latter almost
everywhere, as results from the following theorem:
A2.2) Theorem. In order that a finite function of two variables F,
which is measurable on a set Q, be approximately differentiable at
almost all the points of this set, it is necessary and sufficient that F
be, almost everywhere in Q, approximately derivable with respect to
each variable.
When this is the case, the approximate partial derivaten !<%(.«,?/)
and F'aVy(x,y) are, at almost all the points (x,y) of Q, the coefficients of
the approximate differential of F.
J12]
Total and approximate differentials.
301
Proof. 1° Suppose that the function F is approximately
differentiable at almost all the points of Q. We denote
for each positive integer n, by Rn the set of the points (?,v) of Q
such that, for every square J containing (f, q), we have
A2.3)
-^'l //)
whenever <5(<7)<2/ra. Writing 5=2X, we clearly have \Q-S\=0.
Let us now denote, for a general plane set E and any number t],
by #;J the linear set of the points f such that (f, j?)ej3. Keeping
fixed, for the moment, a positive integer w0 and a real number %,
we consider any two points ? and ?8 of ifj^for which 0<f2 — ^ljn0,
and we denote by <70 the square [?„?.,; %,%+f*—fi]. We then
have <5(<70)<2/n0, and so, putting n=n0, J=J0, ??=•>;„ in A2.3),
and choosing f = ^ and f=fa successively, we see at once that the
square Jo contains points (x,y) for which we have at the same time
and
Hence |J(f2,%) —jF^,^)! <4«0-|f2 —?x|, which shows that,
for any fixed rj, F{x,7]), as a function of x, is AC on each set B^1,
and so VBG on the whole set R['n (cf. Chap. VII, § 5). Now R is
(with Q) a plane measurable set, so that the linear set Bl>li is meas-
measurable for almost every ?;. Hence (cf. Theorem 4.3, Chap. YII);
for almost all v, the function F{x,v) is approximately derivable
with respect to w at almost all the points of i?'1'1. Since further,
by Theorem 11.2, the set of the points of i? at which the function F
is approximately derivable with respect to one variable, is meas-
measurable, it follows at once that the function F is approximately
derivable with respect to x at almost all the points of E, and so,
at the same time, at almost all the points of Q. Similarly, we establish
the corresponding result concerning approximate derivability of F
with respect to y.
'2° Suppose that the function F is approximately
derivable, at almost all the points of Q, with respect
to x and with respect to y. We shall show that the function F
then has, at almost all the points of Q, an approximate differential
with eoeirieieuts F'll]lx{a:,y) and F'^x^). On account of Theorem 11.2
and of Lusin's Theorem (Ohap. III. § 7), we may suppose that

302 CHAPTER IX. Derivates of functions of one or two real variables.
(a) the set Q is bounded and closed, (b) the function F is approximate-
approximately derivable with respect to each variable at all the points of Q, and
(c) the function F, and both its approximate partial derivatives,
are continuous on Q.
This being so, we write, for each point (?, r/) of Q and each
point [x,y) of the plane,
A2.4)
tf, t?; x) = \F(x, V)—
g
Let e and r be any positive numbers. We shall begin by de-
defining a positive number a and a closed subset A of Q such that
\Q— A\<b and such that, for any point A,^),
whenever
b—a<a.
For this purpose, let us denote, for each positive integer n,
by An the set of the points (f, 97) of § such that the inequality in the
first line of (i) is fulfilled whenever a<t<b and b—a<l/n. Since
the set Q is closed and since the function F and its derivatives F'aVx
and F'ap are continuous on Q, it is easily seen that all the sets A,,
are closed. On the other hand, the sets An form an ascending se-
sequence and we immediately see that the set Q — ]imAn is of
measure zero on each line y=r). Hence, this set being measurable,
we have \Q—]imA,,\ — 0. Consequently \Q—A,,a\<e for a suffi-
n
ciently large index nw and writing <j=1/w0 and A = Am we find
that the inequality \Q— A\<s and the condition (i) are both satisfied.
In exactly the same way, but replacing the set Q by A and
interchanging the role of the coordinates x and y, we determine
now a positive number aj<a and a closed subset B of A such that
|^L —-B| < s and that for any point (?,ji)
whenever (?,ri)eB, a^rj^b and b—a<ov
Finally, let a2<a1 be a positive number such that
\K9x{x2, ;y2)—Kp^, ?a)|<t
for any pair of points {xlly1) and .(asa,i/8) of Q subject to the con-
conditions Ja;a—o^l <: «ra and |y2—^|<ff
[§12]
Total and approximate differeutials.
303
This being so, let (?0, ??0) be any point of B. Let </=!>!,&.; a2!j32]
denote any interval such that {io,Vo)^J an<i 8{J)<ai<a1<a. We
write:
|2/— Vo\\ i€o,y) e A; a2
and, for each ?/,
Then any point (x,y) such that yeEt and
set Q-J and, for such a point, we have
^y) belongs to the
On the other hand, it follows at once from (ii) and (i) respectively,
that |J02|>A—e).(/3a— a2), and lE^JlXl-e)-^-^) whenever
jyei72. Hence, D(f0,i?0; x,y) being a measurable (indeed contin-
continuous) function of the point {x,y) on Q-J, it follows that the set
of the points {x,y)eQ-J such that D(?0,7]0; o;,^X2t-[|;i5— ?o| + I^—^ol]
is of measure at least equal to A— ef (ft— at) (/?2— aa)=(l — sf-\J\.
The point (f0, ?j0) here denotes any point of the set B, and J any
interval, containing (f0, ??0), whose diameter is sufficiently small.
Therefore, since |<J— B|<|Q— A|+|-4— -B|^2e, where s is at our
disposal, we see that, for every positive number t, almost every
point (?,??) of Q is a point of density for the set of the points (x,y)
of Q which fulfil the inequality D(f,»j; x,y) /[|*—f| + |j/—»/|]<2t; and
in view of A2.4), this completes the proof.
We notice a similarity TDefrween the preceding proof and that of the "Den-
"Density Theorem" (Chap. IV, § 10). Actually the result just established constitutes
a direct generalization of the Density Theorem. To see this, we need only inter-
interpret, in the statement of Theorem 12.2, the function F as the characteristic
function of the set Q (of. the first edition of this book, p. 231).
The notion of approximate differential, together with Theorem 12.2, are
due to W. Stepanoff [3]. There is, however, a slight difference between the defi-
definition adopted here and that of Stepanoff, so that, in its original form, as proved
by Stepanoff, Theorem 12.2 generalizes Theorem 6.1, Chap. IV, rather than the
Density Theorem of § 10, Chap. IV.
We conclude this § by mentioning the following theorem,
t
which, in view of Theorems 9.9 and 11.2
sequence of Theorem 12.2:
is an immediate con-

con304 CHAPTER IX. Derivates of functions of one or two real variables.
A2.5) Theorem. Suppose that a finit& function of two variables F
which is measurable on a set Q, has at each point of Q at least one finite
Dim derivate with respect to % and at least one finite Dini derivate
with respect to y.
Then the function F is approximately differentiate, at almost
every point of Q.
§ 13. Fundamental theorems on the contingent of
a set in space. Following F. Boger [2], we shall now extend
to sets in the space jB3, the results obtained in § 3. The proofs ¦will
he largely a repetition of those of §3 with the obvious verbal changes.
We shall therefore present them in a slightly more condensed form.
Generalizing the definitions of §3, p. 264, to functions of two
variables, we shall say that a function F(x,y) fulfils the Lipschitz
condition on a plane set E, if there exists a finite constant N Hiich
that \F(x2, y2) - F{xu yj\ < N- [\x2 — x^ + \y.2 — ?/x|] for every two
points (a^) and (x^Vi) of E. We verify at once that the
graph of the function F on E is then of finite area whenever
|J5|<+oo, and of area zero whenever, in particular, \E\=0
(cf. Chap. II, §8; more precisely, we have, for every set E,
A[B(FE))<Ml+^)\m
In the sequel we shall make use of the following notation
for limits relative to a set. If E is a set (in any space) and t0 is a point
of accumulation for E, the lower and upper limits of a function
F(t) as t tends to t0 on E will be written liminfsJ*1^) and lim sup#7'1(//)
respectively. Their common value, when they are equal, will be
written ]xmEF[t).
'-Hi
A3.1) Lemma. Let E be a set in the space Ms, 8 a fixed direction
in this space and P the set of the points a of R at which contgja con-
contains no half-line of direction d. Then (i) the set P is the sum of a
sequence of sets of finite area and (ii) at each point a of P, except
at most at those of a subset of area zero, the set R has an extreme tan-
tangent plane, for which the side containing the half-line ad is its empty side.
In the particular case in which 8 is the direction of the positive
semi-axis of 2, the set P is expressible as the sum of an enumerable
infinity of sets each of which is the graph of a function on a plane
set on which the function fulfils the UpseMtz condition.
[§ 13] Fundamental theorems on the contingent of a set in space.
305
Proof. We may clearly suppose (in the first part of the theo-
theorem also) that 8 is the direction of the positive semi-axis of z. We
denote, for every positive integer n, by P,( the set of the points
{x,y,s) of P such that the inequalities \x'—x\^lln, \y'—y\^l[n
and \z'—z\^.l/n imply z'—z^n-[\x'--x\-\-\y' — y\] for every
point {x',y',z') of R. We express, further, each Pn as the sum of
a sequence {P,a}*=i a... of sets with diameters less than Ijn. For every
pair of points (x1Jy1,sl) and (aia,^8,«a) of the same set Pn>k, we thus
have |s2—%|<%-[|*2—a>]|+ \y%—?/i|], and if we denote by Qn>k the
orthogonal projection of P,hk on the a^-plane, we easily see that
the set PnJi may be regarded as the graph of a function Fnik on Qn>k.
Plainly \Fn,k{ooz,ys)— Fn^ix^y^n-ljx^— ^[+1^—yx\] for every two
points (^,7/j) and (ic2,j/a) of Q,uh. Thus Fn,k fulfils the Lipschitz con-
condition on Qj,,k and hence (el p. 304) A2{Pn,k)=^i{B{Fn,k;Qa,k)}<+oo.
Thus P=?P,i ji i« the required expression of the set P.
n,k
It remains to discuss the existence of an extreme tangent
plane to R at the points of P. For a fixed pair of positive integers n
and k, the function Fn,h, which fulfils the Lipschitz condition
on the set Q,,,/,, can be continued at once, by continuity, on to the
closure Qn<k of this set, and then on to the whole plane by writing
3?n,k{%,y) = 0 outside Qn<k. On account of Theorem 12.2, the function
Fn,k is approximately differentiate at almost all the points of Qn%k.
Hence, denoting by Qn,k the subset of Qn,k consisting of the points
of density of Qnik at which F,,,k is approximately differentiable, we
see that \Q,,,i,— Qn,ji\ — 0 and hence, that A^BiF^kiQ^n—Q^kY^O.
We need, therefore, only show that R has an extreme tangent plane
at each point of B(F,,,k; Qn,k), and that, further, the half-line -with
the direction of the positive semi-axis of z is contained in the
empty side of this plane. ^
Let (fo,»7(»Co) b« any Point ol B^,*;*^,*) and let \A^Bo)
be the approximate differential of Ftt,k at the point ($0,y<>)- Let
e<l be any positive number, and let Ef be the set of the points
(x,y)eQnJt such that
\Fn,k(x,y) ¦Fn,l,(SB,rl0)-A0-(x-h)-B0iy-r]0)\^ei\x-i0\ + \!i-7lB\l
Since the function i'1,,,* is measurable, (f0, %) is (cf. Chap. VII, § 3)
a point, of outer density for the set EE. Hence we can make cor-
correspond to each point (f,»;,s), sufficiently uear to (fo>%i?o)» a Point
(st(, rj')e. E, .such that:
S. Saks. Tlmory of Ilif IiiU-|trol.
20

306 CHAPTER IX. Derivates of functionn of one or two real variable*.
A3.2)
A3.3)
and
and
Bemembering that to=IV/e(V?o), we now write for brevity
nn],(?,ri')=FnM'>y')-Zo—Ao'(?'—fo)—#o-('?'—%)• WA thus llave
This being so, let (f, ??, 0 be a point of iJ such that each of
the differences |f—fo|, l'?-^?ol and IC—Col is l^s than, or equal
to, l/4w2. Then by A3.3), we have |f—?|<l/n and \t)'—-»?|<1/w,
while, by A3.2), [2fn,*(f',V)-Col<»-[|f'-fol+l'7'-'?o|]<l/2»: and so
l^,^',!?')—C|<1K Since the point (?',»?', Ftt,k^',ri')) belongs to
B(JP;^f)CPii, it follows that ^—Fn,H[i',v')^n-[\^'~^'\+\rl—rl'\]>
and, again mating use of A3.3), we deduce from A3.4) that
A3.5)
We now observe that, since
, A3.2) implies
Hence, e being an arbitrary positive number, we derive from A3.5)
A3.6) limsupK
Moreover, since {A0,.B0} is the approximate differential of the func-
function Fn,k at (fo, ??o) an(i since the point (f0, »7o) *s a point of outer
density for the set QnJl, the plane z—?0—A0-(x—-f0)—JS0-(^-—t?o)^O
is certainly an intermediate tangent plane (cf. § 2, p. 263) of It at
the point (?0, %, ?0). It is therefore, by A3.6), an extreme tangent
plane at this point, with an empty side consisting of the half-space
z—fo>J.o-(j—?a) + Ba-(y—rj0). This completes the proof.
We shall employ in space a terminology similar to that of the
plane (cf. § 3, p. 264) and agree to say that the contingent of a set
EcJR3 at a point a of B is the whole space if it includes all the
half-lines issuing from the point a; and again, that the contingent
of E at a point a of E is a half-space, if E has an extreme tangent
plane at a and if contg^a consists of all the half-lines issuing from a
which are situated on one side of this. plane. We make use of
these terms to state the analogue of Theorem 3.6:
L§13] FundaiWHital thtsoremn on the contingent of a set in space. 307
A3.7) Theorem. Given a set It in Ba, let P be a subset of R at
no point of which the contingent of R is the whole space. Then (i) the
set P is the sum of an enumerable infinity of sets of finite area and
(ii) at every point of P, except at those of a set of area zero, either the
set R has a unique tangent plane, or else the contingent of R is a half-spaee.
The proof of this statement, which follows directly from
Lemma 13.1, is quite similar to that of Theorem 3.6. We need only
replace, in the proof of the latter, the terms length, tangent and
half-plane by area, tangent plane and half-space, respectively.
It only remains to extend to space, Theorem 3.7. This ex-
extension, in the form A3.11) in which we shall establish it, is essen-
essentially little more than an immediate, and almost trivial, conse-
consequence of Theorem 3.7. Its proof requires however some subsidiary
considerations of the rueasurability of certain sets.
(.13.8) Lemma. If Q is a set {^a6) in Ms, its orthogonal projection
on the xy-plane is a measurable, net.
Proof. Let us denote generally, for every set E situated in i?3,
by I\E) its projection on the ay-plane. In order to establish the
measurability of the set F{Q), it will suffice to show that for each
e>0 there exists a closed set PCF{Q) such that |P|>|r(Q)| — e.
We express Q as the product of a sequence {Qn}n=i,2,... of
sets ($„)• It may clearly be assumed that the set Q is bounded and
that, moreover, all the sets Qn are situated in a fixed closed sphere So.
We shall define in jB3, by induction, a sequence {F„}„=<>,i,...,
of closed sets subject to the following conditions for u,= l,2,...:
(i) F,,CF,,_,, (ii) FnCQn and (iii) \r(Fn-Q)\>\r{Fn-rQ)\-el<?.
For this purpose, we choose ^0=^0 and we suppose that the
next r—1 sets F,, have b«en defined. We have QCQn and so
F,-1-Qr-Q=F,-vQ, and since Fr r#ris, with Qr. a set E,), there
exists a closed set FrCFr-vQr such that \r{Fr-Q)\>\r{Fr-VQ)\-el2r.
This closed set Fr clearly fulfils (i), (ii) and (iii) for n=r.
Now let F=nFn = \imFn. It follows from (ii) that FCQ, and
therefore that r{F)"cr{Q). Further, r(F) is a dosed set, for, since [Fn\
is a descending sequence of closed and bounded sets, we easily see
that r(F)—]iiQr{Fn). Finally this last relation coupled with (iii) shows
that \r[F)\^lim\r(Fa-Q)\>\r{F0-Q)\-e=\T(Q)\-e, which com-
/(
pletes the proof.
20*

308 CHAPTER IX. Derivates of functions of one or two real variables.
It would lie easy to prove that the projection oJ a Hot C$m)) it) the,
of a determining system formed of closed sets and tluw to deduce Lemma 13.N
from Theorem 5.5, Chap. II. We have preferred, liowovor, to giviv a direct ele-
elementary proof, based on a method due to N. Lusin [3]. Tlie same argument
shows that any continuous image of a set (<fta() is measurable.
It has been proved more generally (vide, for instance, W. Sierpiriski
[II, p. 149], or P. Hausdorff [II, p. 212]) that any continuous image of an
analytic set (in particular, of a set measurable E8)) situated in Jtt is an analytic
and, therefore, measurable set.
A3.9) Zeniina. Given a set R in Ba, let Q be the set of the, points
(f, r),?) of R which fulfil the condition:
(A) the part of the contingent of R at the point (f, ?j, ?), which
is situated in the plane #=?, is wholly contained in one or other of
the, two half-spacen y^rf and y^rj.
Then the orthogonal projection of the net Q on the, wy-plane is
of plane measure zero.
Proof. We may clearly suppose, that the set R is dosed (for
the contingent of any set R coincides, at all points of It, with that
of the closure of 22).
Let us denote generally, for any set F in _Ka and any number f,
by Em the set E[{g,y,s)eE]. It follows from Theorem 3.6 that,
(ff.2)
for every f, the plane set R[~] has an extreme tangent, parallel to
the s-axis at every point of Qlii except those of a set of length zero.
Hence, by Theorem 3.7, the projection of Q on the awy-plane is of
linear measure zero on each line x=? of this plane, and, in order
to prove that this projection is of plane measure zero, we need
only show that the latter is measurable.
Let us denote, for each pair of positive integers k and n, by
Ak,n the set of the points (?,??,?) of 22 such that the inequalities
A3.10) \x-?\+\y-v\+-\z-?\<l[n and \x~-^\<[\y—r]\ + |s—C|]/n
imply, for any point (oo,y,z)eR, the inequality y—r}^[\x— f| + |a
Similarly, we shall denote by Bh<n the set of the points (,?,)
of R for which the inequalities A3.10) imply, for every point
(x,y,s) of R, the inequality y-..-v^—[\X—||-f |s—f|]/fc. Writing
¦A.=UZAk,n and B=[ISBkl.,
k n in'
we find that Q = A + B. On the other hand, since the set R i.s, by
hypothesis, closed, we observe at once that each set AktB, and like-
likewise each set Bk,n, is closed. The sets A and B, and so the set Q
also, are thus sets EJ, and in view of Lemma 13.8, the projection
of Q on the a$-p!ane is a measurable set.
[§14]
Extreme differentials.
309
A3.11) Theorem. Given a set R in H3, let P be a subset of R at every
point of which the set R has an extreme tangent plane parallel to a fixed
straight line D. Then the orthogonal projection of P on the plane per-
perpendicular to D is of plane measure zero.
Proof. We may clearly suppose that the straight line D is
the 2-axis. Let us denote by P2 the set of the points of P at which
the extreme tangent plane, parallel, by hypothesis, to the a-axis
is not, however, parallel to the j/a-plane. Similarly, P2 will denote
the set of the points of P at which the extreme tangent plane is
not parallel to the res-plane. We then have P=P1+-Pi.
Now we observe at once that each point (f, rj, f) of P1 fulfils the
condition (A) of Lemma 13.9. It therefore follows from this lemma,
that the projection of P1 on the *y-plane is of plane measure zero.
By symmetry, the same is true of the projection of the set P2.
The proof is thus complete.
§ 14. Extreme differentials. Let F be a finite function
of two real variables. A pair of finite numbers {A,B} -will be called
upper differential of F at a point (xo,yo) if, when we write zo=F{os0,y0),
(i) the plane s—ao=A-(x—xo)-\-B-{y—y0) is an intermediate tan-
tangent plane of the graph of the function F at the point (soB,y0,s0) and
F(x,y)—F(xa, yo)—A-(x—xo)—B-(y—yn) «
¦ *¦"" v ¦"' J0> -.—p 2i y—^2Z=0
\\\
lim sup
...
(n
These conditions may clearly be replaced by the following: (i2) the plane
z—zo=A-(x—xQ)-\-B-{y—yg) is an extreme tangent plane of the graph
of F at (d5Oj^o?2o) ^th the empty side e—eu^A-(x—xo)+B-(y—ya),
and (ii2) lim sup F(x,y)^F{xQ,y0).
(.v,i/)->(.v,,,?;,,)
The definition of lower differential is similar, and the two
differentials, upper and lower, will be called extreme differentials.
If a function F has a total differential (cf. § 13, p. 300) at a point,
this differential is both an upper and a lower differential of F at
the point considered. Conversely, if a function F has at a point
(^ok'/o) k°t:tl an upper and a lower differential, these are identical
and then reduce to a total differential of F at {xo,yo).
For a finite function of one real variable F, the existence of an upper dif-
differential at a point x0 is_to be interpreted to mean tliat F+(xa)=F~(xB) = co (in
which ease the number F+{xl))=F~(x()) may be regarded as the upper differential
of F at .'«„). There is a similar interpretation for the lower differential of functions
of owe variable. This interpretation brings to light the relationship between the
theorems of this § and those of § 4.

310 CHAPTER IX. Derivates of functions of one or two real variables.
We propose to give an account of researches concerning the existence
almost everywhere of total, approximate, or extreme differentials. These re-
researches were begun by H. Kademacher [3], who established the first general
sufficient condition in order that a continuous function be almost every-
everywhere, differentiable. W. Stepanoff fl;3] later removed from Bademaoher's
reasoning certain superfluous hypotheses, and obtained, a more complete result,
valid for any measurable function: In order that a function F which is meas-
measurable on a setB, slwuld be differentiable almost everywhere in E, it is necessary and
sufficient that the relation lim sup \F(x,y)~F(g,y)\l[\iii—i\+\V~'i\l<+°° should
hold at almost all the points (#,*?) of E. (Certain details of Stepanoff's proof, par-
particularly those concerning measurability of the Dini partial derivates, have been
subjected to criticism (cf. J. C. Burkill and U. S. Haslam-Jones [1].) U. S.
Haslam-Jones [1] extended further the result of Stepanoff, and by introducing
the notion of extreme differentials (which he called upper and lower derivate
planes), obtained theorems analogous to those of Denjoy for functions of one
variable. The researches of Haslam-Jones have been continued and completed
by A. J. Ward [Is 4] who, in particular, removed the hypothesis of me.asur-
ability in certain of Haslam-Jones's theorems.
We shall derive the results of Haslain-Jones from the theorems of the
preceding § (cf. F. Roger [3]; direct proofs will be found in the memoirs of
Haslam-Jones and Ward referred to, and in the.first edition of this book).
In what follows, we shall make use of some subsidiary con-
conventions of notation. If J1 is a function of two real variables and t
denotes a point (x,y) of the plane JS2, we shall frequently write
F(t) for F(x,y). If t1=[x1,y1) and t2=(xi,yi) are two points of the
plane, |t2—*i| will denote the number \x2—asj + \y2—yx.
Given in the plane two distinct half-lines issuing from a point t0,
each of the two closed regions into which these half-lines divide
the plane will he called angle. The point t0 will be termed vertex
of each of these angles.
We shall begin by proving a theorem somewhat analogous
to Theorem 1.1 (ii).
A-i.l) Theorem. Let F be a finite function in the plane JS2 and
let E be a plane set, each point r of which is the vertex of an angle A{x)
such that limsup4(r)JT(i)<lim sup.F(i). Then the set E is of plane
measure zero.
Proof. Let us denote, for each pair of integers p and q, hy Ep<!,
the set of the points t of E at which hmsupAr)JP(?)<p/4<limsup F{t).
For fixed p and q, we observe that no point xeEM is a point of
accumulation for the part of the set EM contained in the interior of
the corresponding angle A(r). Hence, no point of the set EM can
be a point of outer density for this set. Each of the sets EIhq is
thus of plane measure zero, and the same is therefore true of the
whole set E.
Extreme differentials.
311
Ah wo wiHily hoc, in virtue of Theorem 3.6, each set EM, and consequently
the whole. Hot E, m the sum of a sequence of sets of finite length (this of
course, implies that Ii is of piano measure zero). Cf. A. Kolmogoroff and
J. Verfionko [_!/].
A4.2) Theorem. Let F be a finite function in the plane. Then
(i) if P is a plane set each point r of which is the vertex of an
angle A(r) such that
A43)
f-H
= +OO,
the set P is necessarily of plane measure zero;
(ii) if Q in a plane set each point r of which is the vertex of an
angle Aa(r) such thai
A4.4)
UmaupiW0[J»(<)-JI(T)]/|*-T|<+oo,
the function F necessarily has an upper differential at almost all the-
points of Q;
(iii) if Ii is a plane set each point r of which is the vertex of two
angles A^r) and A^r) such that
lim sapAM [F{t)—F{r)']j\t—r\<+ oo
and '">T
lim inf ^ [F(t)~F{r)] /|<—r|> — oo,
the function F is totally differentiable at almost all the points of ?.
Proof, re (i). By Theorem 13.7 the set B(F;P) has, at each
of its points except those of a subset of area zero, an extreme tan-
tangent plane. The latter is seen to be necessarily parallel to the s-axis.
Hence, by Theorem 13.11, the set P, as the projection of B{F;P)
on the ay-plane, is of plane measure zero.
re (ii). It clearly follows from A4.4) that, at each point r of Q,
we have limsup^()WjP(i)<J?1(T). Hence, by Theorem 14.1, we have
lira sup F{t)^F(r) at all the points r of Q, except at most those
of a set Qa of measure zero.
Let us now denote by B the graph of the function F (on the
whole plane). Let Bl be the set of the points of B(F;Q) at which
the set B has no extreme tangent plane, and -Ba the set of the points
of B(F;Q) at which such a tangent plane exists, but is parallel to
the z-axis. Finally, let Qt and §3 be the projections of the sets Bt
and B2 respectively, on the on/-plane. On account of Theorem 13.7,

312 CHAPTER IX. Derivates of functions of one or two real variables.
we easily verify that A^B^O, and so, that |&J = 0. Similarly,
it follows at once from Theorem 13.11 that |<93| = 0. H"ow, if (?,»?)
is any .point of Q— (ft + <22), the set B has at {?,rj,F(?,y)) an ex-
extreme tangent plane of the form s—?=M{?, rj)-{x—?)+N{?, ii)-{y—rj),
where M{?,rj) and N{S,rj) are finite numbers. We observe further
•without difficulty that the half-space
is an empty side of this plane. Hence (cf. p. 309), at each point
{S,rj) of the set Q~{Q0 + Qi + Qz), the pair of numbers {M(?,ri),N(?,r))}
is an upper differential of the function F. This completes the
proof, since \Qo+Qi+Qz\ = Q-
Finally, (iii) is an immediate consequence of (ii).
In the case in which the function J? is measurable, we can complete
part (i) of Theorem 14.2 (which itself generalizes Theorem 4.4). Thus, if F
is any measurable function of two variables, the set of the points (x,y) at which
lim \P(x-\-h,y)—F{x,y)\fh=-\-co, is of plane measure zero.
This proposition plainly follows from Theorem 4.4, except for meas-
nxalnlity considerations, essential to the proof, which seem to require general
theorems on the measurability of the projections of sets (93) (of. p. 308).
We conclude with the following theorem (ci. A. J. Ward [1]
and the first edition of this book, p. 234) which, in view of Theo-
Theorem 14,2 (i), (ii), may be regarded as an extension of Theorem 9.9
to the functions of two variables:
A4.5) Theorem. If F is a finite function of two variables, which
¦is measurable an a set E and which has mi extreme differential at each
point of a set QCB, then this differential is, at the same time, an
approximate differential of F at almost all the points of Q..
Proof. On account of Lusin's theorem (Chap. Ill, § 7) we
may clearly suppose that the set F, is closed and that the function
F is continuous on B. Let us suppose further, for definiteness, that
the function F has an upper differential at each point of Q, and
let us denote, for each positive integer n, by Qn the set of the points
t of Q such that, for every point V, \V—1\ <l/«. implies the
inequality F[t')~F{t)<n-\t'—t\. Finally, let each set Qn be ex-
expressed as the sum of a sequence {Qn,k}t=i,2,... of sets with diameters
less than 1/n. We shall have Q=S
n.k
Extreme differentials.
313
We see at once that the function F fulfils the Lipschitz con-
condition on otich set Q,,il{, and therefore also on each set Q,uk. Hence,
by Theorem 12.2, the function F has the approximate differential
{FLVx{x,y), KP//(»,;>/))¦ at almost every point (x,y) of each set Qn,h
and therefore at almost every point (x,y) of the set Q.
Let us, on the other hand, denote, for each point (as,y) of Q,
by {A{x,y), B(x,y)} the upper differential of F at this point. It
follows at once from the definition of upper differentia], p. 309, that
F7{x,y)>A{%,y)^Ft(x,y), and similarly F~(x,y)^B(x,y)^F~t{x,y),
at each point (x,y) of Q. Hence, at each point (x,y) of Q at which
the approximate partial derivates F^Px(ix),y) and F^ (x,y) exist,
we have A(x,y)=F:iPx(x,y) and B(as,y)=F^v{x,y). The upper
differential \A(x,y), B{x,y)} of the function F thus coincides at
almost all points {x,y) of Q with the approximate differential of F.

NOTE I.
On Haar's measure
by
Stefan Banach.
§1. This Note is devoted to the theory of measure due to Al-
Alfred Haar [1], Haar's beautiful and important theory deals with
measure in those locally compact separable spaces for which, the
notion of congruent sets is defined. His measure fulfils the usual
conditions of ordinary Lebesgue measure: congruent sets are of
equal measure and all Borel sets (more generally, all analytic sets)
are measurable. The theory has important applications in that
of continuous groups.
To complete the definitions of Chap. II, § 2, we shall say that
a set situated in a metrical space is compact, if every infinite subset
of the set in question has at least one point of accumulation. A met-
metrical space is termed locally compact if each point of this space
has a neighbourhood which is compact.
§ 2. In what follows we shall denote by JE a fixed metrical
space, separable and locally compact, and we shall suppose that,
for the sets situated in E, the notion of congruence = is defined
so as to fulfil the following conditions:
iv A = B implies B = A; A'stB and B=G imply A=C;
12. If A is a compact open set and A=B, then the set B is itself
open and compact;
13. If A = B and {An} is a (finite or infinite) sequence of open
compact sets such that Ac?An, then there exists o- sequence of sets
{Bn} such that BcSBn and such that An=Bn for n=l,2,...;
n
14. Whatever be the compact open set A, the class of the sets con-
congruent to A covers the whole space E;
15. If {$„} is a sequence of compact concentric spheres with radii
tending to 0, and {(?„} is a sequence of sets such that G,,^8U, then the
relations «=lim«n and 6 = lim&,,, where a,,eG,, and b,,eGn, imply a=b.
Oh Haar's measure.
315
§3. Given two compact open sets A and B, the class of the
sets congruent to A covers, by i4, the set B, It therefore follows
from the theorem of Borel-Lebesgue that there exists a finite system
of sets congruent to A which covers B. Let h(B,A) denote the least
number of sets which constitute such a system.
It is easy to show by means of ix—i5 that, for any three com-
compact open sets A, B and G, the following propositions are valid:
ii-L. GC_B implies h{G,A)^]x{B,A);
BG
ii3. B^G implies h{B,A) = h.{C,A);
iiB. If q(A,B)>0 and if {8n} is a sequence of compact con-
concentric spheres with radii tending to 0, then there exists a positive
integer N such that, for every n>N,
C.1) h(A + B, ?„)=h (A, 8n) + hE, Sn).
All these propositions are obvious, except perhaps ii6. To
prove the latter, let us suppose, if possible, that there exists an
increasing sequence of positive integers {n,) such that C.1) does
not hold for any of the values n=ni. There would then exist a se-
sequence of sets {Qt) such that G^ = 8n, while A-Gt=$=0 and B-G^O.
Consider now arbitrary points ateA-Gi and bieB-Gi. Since the
sets A and B are compact, the sequences {as,-} and {&,) contain re-
respectively convergent subsequences {«;} and {bi}. Let a = lima,-.
1 j j j
and b — limbi.. By i6 we must have a—b, and this is impossible
since, by hypothesis, y(A,B) = 0.
We shall now suppose given a fixed compact open set G and
a sequence {&„} of concentric spheres, with radii tending to 0, which
are situated in G and therefore clearly compact. For every compact
open set A, "we write
h(A,8n)
ln(A-> h(G,Sn)'
We then have, by ii4,
and
and hence for each n=l,2,..., ljli{G,A)^ln{)^{,)
Thus, l?,,(A)} is a bounded sequence whose terms exceed a fixed
ponitioe number.

316
Stefan Bauach:
§4. We now make use of the following theorem (of. 8. Ba-
naeh [I, p. 34] and 8. Mazur [1]), in which {?„} and {rjn} denote
arbitrary bounded sequences of real numbers, a and b denote real
numbers, and the symbols lim, lim sup and Mm inf have their usual
meaning:
To every bounded sequence {!„) we can make correspond a num-
number Lim ?„, termed generalised limit, in such a manner as to fulfil
n
the following conditions:
1) Lim (a^n+brjn) = a-Lim ?n -\-b-Lim T)n,
it n n
2) lim inf ? s
3)
The last condition implies that the generalized limit remains unal-
unaltered, when we remove from a sequence a finite number of its terms.
Let us now write, for every compact open set A,
= Limln{A).
We then have, for any compact open sets A and B:
1112. ACB implies
1113. A=B implies l(A) — l(B);
iii6. q(A,B)>Q implies l(A + B)=l{A)-\-l(B).
§5. This being so, we denote, for an arbitrary set JC-E, by
F(X) the lower bound of all the numbers ?l(An) where {An} is any
n
sequence of compact open sets such that XC^An. We shall show
n
that the function of a set P, thus defined, fulfils the following con-
conditions:
1° We ham always 0</1(X) and there exist sets X for which
we have 0<r(Z)<-foo; this is, in particular, the case of all compact
of en sets X;
2" XtCX2 implies
3° XcEJ* implies
±« o(J:1,X2)>0 implies
5° X! = Ja implies r(Z1)=/1(X2).
On I-Iaar's measure.
317
Proof. 1°. Let X l)t> a compact open set. We have, by defi-
definition, r(A')<?(A')<+oo.
On the otlior hand, there clearly exists for each e>0 a (finite
or infinite) sequence of compact open sets (An) such that XclX
V^T II
and r(X)+e^2jl(A»). Let S be any sphere contained in X. Since
n
the set 8 is closed and compact, this set, and a fortiori the set S,
is already covered by a finite subsequence {An) of [An). In view
of iii2 and iii4, we thus have
Hence, s being arbitrary, it follows that l(S)^.r(X), and finally,
by iiii, that 0<F{X).
2° and 3° are obvious.
i°. e(A,A)>0 implies that there exist two open sets G1
and <?3 such that A^c^i, AC(?2 and g(G!1,6(a)>0. On the other
hand, there exists for each s>0 a sequence of compact open sets [An]
such that
E.1)
and
Write A7=An-G1 and A)f=An-G2. Since the sets A? and A?
are open and compact, and since their distance, like that of G1 and Gt:
is positive, we have, on account of iii5 and iii2,
E.2)
But, since on the other hand X^cEaW and X^C^-Af, we
n . i
have the inequaHties /1(A)<2^(^«)) and r{Xt)^Zl(^n), so that,
n "
by E.2), r(X1) + r(Xe)^El{An)- Hence by E.1), e being arbi-
trary, we obtain r(J1) + JT(Z2)<r(A'1-{-Xz), and finally by 3°,
r(x1)+r(xg)=r(xx+xt).
5" follows at once from i3 and iii3.
8 6. It follows from the properties l°-40 of the function J1
that the latter is an outer measure in the sense of Oaratheodory
(cf. Chap. Ill, § -1) and therefore determines iu-Ea class of sets
measurable Br), that we shall call, simply, measurable sets. We
see at once that for each set X in E the number r(X) is the lower

318
Stefan Banaoh:
bound of the measures [F) of the open sets containing X. It follows,
in particular, that /"is a regular outer measure (cf. Chap.II, § 6).
Finally, since the space B can be covered by a sequence of
measurable sets of finite measure (e. g. by a sequence of compact
spheres), we easily establish, for the measure F, conditions of meas-
urability (8j) similar to those of Theorem 6.6, Chap. III. In part-
particular, we shall have:
F.1) In order that a set JE be measurable, it is necessary and
sufficient that there exist a set (©a) containing E and differing from
¦it by at most a set of measure zero.
§ 7. "We conclude this note by giving two examples of spaces E
with the notion of congruence subject to the conditions of § 2.
Example 1. Let E be a metrical space which is separable
and locally compact, and suppose that, among the one to one trans-
transformations, continuous both ways, by which the whole space E
is transformed into the whole space JE, there exists a class S>li of
transformations subject to the conditions:
1) Tedli implies T^1 e Sli;
2) If T1eSll and T^effi, then T^^eM;
3) For every pair a, b of points of E, there exists a transformation
Te8n such that T(a)=b;
4t) If {an\ and {bn} are two convergent sequences of points of E
such that liraan = Urnb,,, and if {T,,} is a sequence of transformations
n n
belonging to d/l such that the sequences {T,,(a,,)) and {Tn(b,,)) are con-
convergent also, then we have lim Tn(an) = lim Tn{bn).
n n
Two sets AcB and BCE will be termed congruent, if there
exists a transformation Te3/i such that T(A) = B (where T{A)
denotes the set into which A is transformed, i. e. the set of all the
points T{a) for which aeA).
It is easy to verify that the conditions i,—i6 are fulfilled.
As special cases of such spaces E we may mention: Euclidean
^dimensional space with d!i interpreted as the class of all trans-
translations and rotations; the 3-dimensional sphere with SUl interpreted
as the class of all rotations.
Let us observe that, in the space considered, the sets which
are congruent to open sets are themselves open. On the other hand,
Ou Haar's measure.
319
on account of 5", p. 316, the sets congruent to sets of measure (f)
zero are themselves of measure zero. It follows therefore from F.1)
that in the space JE considered the sets which are congruent to meas-
measurable sets, are themselves measurable.
Example 2. Suppose that a metrical space JE, separable and
locally compact, constitutes a group, i. e. that with each pair a,b
of elements of E there is associated an element ab of JE, called pro-
dud, in such a manner that the following conditions are fulfilled:
1) (ab)c = a(be) (whatever be the elements a, b mid o of E);
2) there exists in E a unit-element 1 such that we have l-a=a-l=a
for every aeJE;
3) to each element aeE there corresponds an inverse element
a i e IS which fulfils the equation aa~l = l.
Suppose further that B fulfils the conditions:
4) if lima,,= ft and Mm.bn=b, then ]imanbn=ab;
n n n
5) if limfls,,= a, then lim a^1 = aTl.
n n
Given any element eeE and any set BCE, we denote by cB
the set of all the elements aeE such that a=eb where beB.
Given an element a of E, we write, for every element oseJE,
Tn(x) = ax. Thus each element a of JE? determines a transformation Ta,
clearly one to one and continuous? both ways, of the space E into
itself. Denoting the class of all these transformations by SIT, we
see at once that the conditions 1)—4), p. 318, are fulfilled. In ac-
accordance with the definition of congruence employed in Exam-
Example 1, two sets A and B in the space in question are congruent if
there exists an element c such that B — cA.

NOTE II.
The Lebesgue integral in abstract spaces
by
Stefan Banach.
Introduction.
In this note we intend to establish some general theorems
concerning the Lebesgue integral in abstract spaces. This subject
has been discussed by several authors (for the references see this
volume, pp i, 88, 116, 156 and 157). Our considerations differ from
those of other writers in that they are not based on the notion
of measure.
Let us fix a set of arbitrary elements H as an abstract space.
We shall denote real functions (i. e. functions which admit real
values) defined in H by x{t), y(t), z(t),... where teH, or simply by
*,y,z,.... A set ? of real functions defined in H will be called linear
if any linear combination, with constant coefficients, of two ele-
elements of 2, also belongs to 8.
Let 2 be a linear set of functions defined in H. A functional F
defined in 2 is termed additive if for any pair of elements $ and y
of 2 and any real number a, we have F{xJry)—F{x)-\rF{y) and
F{ax) = a'F(x). The functional F is non-negative if I?(j?)>0 for any
non-negative function x e 8.
We say that a functional F defined in 2 is a Lebesgue integral
{^-integral) in 8 if the following condition* are satisfied:
A) The set 2 is linear;
B) the functional F is additive and nou-negativc>;
C) if l°{s,,lC? and M<??, 2
-<~M(t) for n=--\,2,... and
teH, and
) = 2(i) for
, then s
and
The LolKisfiuo integral in abstract spaces. 321
DMf ?efi, *•«,,«0 and |y(,)|<irW for (ejHj thm yfS ^
.15) if
{s,,)C2, zn(t)^z,1+l(t) for n=l,2,..., 2° timz&)=*(t)
for tell', and a»Iira I?(s,,)<+oo, thense2 md]imF(gn)l-.F(&).
The Lebesgue integrals considered in this note will moreover
satisfy the condition:
R) If seQ, then \s\e2.
In Part I, a condition is. established under which an additive
and non-negative functional defined in a linear set of functions (?,
may be extended to an ^'-integral on a certain set 2 containing (?.
The f-integral and the set ? will be explicitly defined.
fa Tart II we admit that // is a metrical and compact space.
Wo consider an ^-integral defined in sets containing all functions
which are bounded and measurable in the Borel sense. It is shown
that each f-integral of this kind is determined by the values which it
admits for continuous functions. Conversely, any additive and non-
negative functional defined for all continuous functions may be
extended as an .^-integral to the class of functions measurable E8).
We thus obtain the most general f-integral defined for all functions
bounded and measurable C3).
In Part III we deal with an analogous problem supposing
that H is the unit sphere of the Hilbert space. In particular, the
integral of a continuous function is expressed by explicit formulae.
I. Abstract sets.
§ 1. We shall employ the following notations:
1. x^zy if x{t)^zy{t) for every teH; in particular a>0 means
that ,i;(i)>0 for teH;
2. |a;|=|a>(i)| is the modulus of x{t) in the ordinary seuse;
3. max {ai,y)=h(is+y+\x—y\), mm{<c,y)=i(x+y—\x—y\);
i. limiP,, = a! means that ttnix11(t) = x(t) for teH; the relations
n «
Mm sup x,, = x, lirninf a;,, = a; are defined similarly;
n n
fl. .'?=.V(*+I4l, ?=K«-M) (cf- oliaP- T> p- 13)- -
S. Snks. Theory nf ttau Integral.
21

322
Stefan Bauach:
§ 2. For the rest of Part I of this note we shall fix a set (g of
real functions defined in H, and a functional /(*) defined for ,<»e(?,
subject to the following conditions:
(ij) The set g is linear;
(ia) if %e(?, then \cc\e(?;
A11) the functional / is additive;
A12) the functional / is non-negative;
A13) if 1° WCS and Me% 2° \x,,\^M for w=l,2,...,
C) W
and 3° lima;n=0, then lim/(a;n) = O.
n «
It follows immediately from the conditions (i) that for any pair
of elements x and y of fi, max {x,y), min (x,y), & and f also be-
belong to (?. It follows further that the condition (ii8) is equivalent to
the following condition:
(%) If 1° {«n)C(? and me?, 2° x^m for w=l,2,...,
and 3° litninf o^O, then ]xmxnff(xn)^0.
§ 3. We shall establish the following
Theorem 1. If the set ? and the functional f satisfy the con-
conditions (i) and (ii), then there exists an ^-integral F, defined in a set 2
containing g, such that F(x)=f(x) whenever xe<&; moreover, this
integral satisfies the condition B).
The proof will result from several lemmas.
§4. We denote by 2* the set of all functional z(t) defined
in H for each of which there exist two sequences {,*,,}C(?, {i/,i}C?
such that
A)
liminf
lim sup yn.
It is easily seen that the "set fi* is linear and that ?C?*.
Given a function g e ?*, we shall term upper ^-integral of z
the lower bound of all (finite or infinite) numbers g for each of which
there exist a function m e ? and a sequence of functions [asn} be-
belonging to ? such that xn^m for n=l,2,..., liminfa'n>s and
gr=liminf/(a5n).
n
The definition of the lower ?-inte.grnl is analogous to that of
the upper X-integral, The upper and lower xMntegrals of a function
ze2* will be denoted by p(s) and q(z) respectively. We obviously
have q(z) = — p{— z).
Tim
Kgiin intojufrul in abstract spaces.
323
§5. Tin* HoqtiMuw {{@n)) iu the above definition of the upper
may obviously be supposed convergent (to a finite limit
or +00). Further, if [wB}CQ, me% s^O, »„>»» for n=l,2,... and
liminf</,•„>s, t;h<Jii limi»,,= 0 and consequently, by the condition
(ii8), §2, ]im/(,c,,) = o. Hence, if sffi*?
and p{z)<P<-\-oc,,
there, always exist* a mjuence of non-negative functions {#„} belonging
to (? such that liminJ!,¦»„>!? and f{xn)<P for 71= 1,2,...
Leuiinal, For any function xe<$, we have p(x)—f(x).
Proof. Writing ,«„=« and tn=x, we have
A) Ian iufi!/;„>;« and x,,^in for w=l,2,...,
(x). ()u the other hand, if xltxz,... and m are
any fimctioiiH which belong to (? and satisfy the relations A), then
liminf(;/)„--¦¦¦/'')^0 and x,,—x^m—x for n=l,2,.... It follows from
(iia), § 2, that. limmif{xn—x)^z0, i.e. lirninff(xn)^f(as). Thus
f{x), and finally p(x)=f(x).
Lemma 2. If s1e2*, «2f2* and if, moreover, p(
Proof. Lot Px and Pg be arbitrary numbers such that pi)i
and p(a,i)<l\. There exist two sequences {x^), {xf} of functions
belonging to (? an<l two functions % e S and »i2 e ? such that
liminf xJ/'^s; and lim/(.x!/')<Py for ; = 1,2 and such that x{i>>tn1
n n
for j = l,2 and n—1,2,.... Therefore, writing xn=Ja+sS) and
m=%-fmj, we have liminf i^^Xi+^a ^D-d xn~^m for w=l,2,.... Oon-
sequently p(
22)<lim/{xn) = lim/(a;^) +lim/(asf )<PX + Pa, whence
f
Lemma 3. Far any function z<=2*, we have p
Proof. Since 5@) = — p(-2) (cf. §4), the inequality
is obvious if one of the, numbers p(s) or p(—s) is +°o; while, if
p(s)<+oo and p{—s)<-[-oo, it follows immediately from Lemma2.
21*

324 Stefan Banaeb:
Lemma4. If ze&*, p{s)<+™, then also p{z)<+oo and
Proof. Given an arbitrary finite number P>p(s), there exist
a function »uG and a sequence {xn) of functions belonging to S
such that xn^miorn=l,2,..., liminfa;,,>s and lixnf(xtl)<P. Note
n n
that Xn^-tn, and consequently f{a:n)^f{xn)—f{m), for n—1,2,...,
whence p(l)<lirninf/D)<+°°- Again
and therefore
); whence, in virtue of Lemma 2,
Finally, we mention two propositions which are directly obvious:
Lemma5. If 2^2*, z^e2* and Si^zi} then p(^i)^p{~i); in
particular, if ze2* and z^O, then p(«)>0.
Lemma6. If sefi*, then p(Xz)=X-p(z) for any non-negative
number A.
§ 6. We shall now denote by ii the set of all functions z e id*
for which p(z) = q(z)^<x>. The folkwing proposition is an immediate
consequence of Lemmas 2 and 6:
Lemma 7. If g16 2 mid zz e 2, then (A^ + A2«2) e 2
pB1s1+A2^2)=A1p(s1) +A2p(«2) for any pair of finite numbers
Lemma 8. If ze&, then \s\e2.
Proof. Since |s|=a—f, it is enough to prove that se2 and
z e 2. To this end, let us remark that, in virtue of Lemma -k,
I p(|)> — oo and p{z) = p{z) + p(s); by symmetry,
, 2B)<+oo and 2(s) = g(s)+g'(f). &ince, by hypothesis,
it follows that [p(«)— ?(«)]+[p(f)— 2(f)]=0, and so
by Lemma3, p(s)=g(!)H=°° and p(f) = g(s)H=c>o-
Lemma 9. If z is the limit of a non-decreasing sequence {&„}
of functions belonging to 2 and ]imp{zn)<-}-c>o1 then St-2 and
Proof. We can clearly assume (by subtracting, if necessary,
the function zx from all functions of the sequence {s,,)) that a1=0.
Writing wn=zn+i—zn for n=l,2,..., we shall now follow an argu-
argument similar to that of Theorem 12.3, Chap. I. First, we have s>s,,
and p{zn)=q{zn) for every n, and so
A)
zn) = limp (zn).
Tllfi U:
integral in abstract spaces.
325
To establish the opposite inequality, let e be an arbitrary
positive integer and let us associate (cf. the remark at the begin-
beginning of § f>) with each function wn a sequence {a#V=i,2,... of non-negative
functions belonging to (? such that
B) Iimmf4'()>w,, and C) f{x?))^p(wa) + el&1.
it
Let us write ;//,,== JX'1'. The functions y^ clearly belong to Q, and,
by B), wo have liminf^^Vs^- On the other hand, in virtue
k It
k
of C), wo find /(jfi)<^(i»J+s<}i(«lffI) + e<liniy(«!)+« for
fc=l,'2,.... Therefore, p(»)<liminf f{yk)^ lim.p{z.) +s, and
since « ia an arbitrary positive number, this combined with A) gives
which completes the proof.
/<
Lemma 10. If MeQ and, {s,,} is a sequence of functions belonging
to 2 xuflh thai \zn\^.M for n=l,2,..., then, putting g~liminfzn and
h = limmij)sn, wa huve </efi, he2, and, "
n
. p(g)^.lirn inf p(sn)^liin sup p (zn)z?.p{h).
n 7i
Consequently, if the sequence {z,,} is convergent and z = ]ssa.zn,
then p(s) = linip(z,,). "
tt
Proof. The lemma corresponds to Theorem 32.11, Chap. I,
and its proof is analogous to that of the latter. Let us write, for
each pair of integers i and j^i, ;iij = min(sj,zt+1,...,sj). The se-
sequence (f/..], t l+J is non-increasing, and consequently the sequence
{M — (.ii})..j+i is non-decreasing. Let gi = 1imgy. Since the func-
functions yr clearly belong to ?, it follows from Lemma 9 that
M~gte2 and p{M—gl)=Mmp{M—g..), i.e. g,e2 and p(ff)^
Hence, applying again Lemma 9 to the non-decreasing sequence {g,}
which converges to g, we obtain #e? and
p (g) = lim p (gt) <lim Mp(zt).
i i
By symmetry we have the analogous result for h and the
proof is complete.
We shall conclude this § by mentioning the following lemma
winch is an immediate consequence, of Lemma 5:
Lemma, 11. If ze2, s^O and p(z) = 0, then any function, a?
such that \n\^a belongs to 2 and for any such function x we have p {x)= 0.

326
Stefan Banacli:
§7. Let F{x) = p{x) for .re 2. The lemmas of the preceding-
sections show that the set 2 and the functional F(z) satisfy the
theorem stated in §3. {Theorem 1 is thus proved.
It is easily seen that if an f-integral Fx defined in a linear
set SjDfi satisfies the condition f(x)=F1{x) for xe<S, then
F{x)=Fl{x) for all xe2. Consequently the functional / determines
completely an ^-integral in the set ?.
n. Metrical compact sets.
§ 8. Let now H be a complete and compact metrical space.
We shall specify S as the set of functions continuous in 2?.
The set (f satisfies evidently the conditions (i), § 2. It may
be shown that any additive and non-negative functional / defined
in 2 satisfies the condition (ii3I).
Theorem 1 permits to define a Lebesgue integral F{x) for all
functions x belonging to a certain set CDS, in such a manner that
the condition E), p. 321, is satisfied and that F{x) = f(x) for ajeffi.
Evidently, every function x (t) which is constant on H belongs
to (?. It follows by condition C), p. 320, that eArery bounded function
measurable in the sense of Borel belongs to (E.
We have thus proved the following
Theorem 2. Finely additive and non-negative functional, defined
for all functions which are continuous in a complete compact space 3,
may by extended to an ^-integral defined in a certain linear set (con-
(containing all bounded functions measurable in the sense of Borel) so
that the condition E) be satisfied.
The values of this X-integral for functions bounded and meas-
measurable C3) are, of course, determined by the given functional /.
Hence the most general ^-integral defined for this class of functions
may be obtained by choosing an arbitrary additive non-negative
functional defined for all functions which are continuous in M and
by extending this functional by means of the method described in
Part I of this note.
1) A functional of this kind is necessarily linear. Every linear functional
defined in ffi satisfies the condition (ii3). See S. Banaeh [I, p.224].
The Lebesgue integral in abstract spaces.
327
Any linear functional f{x) defined in the set E is the difference
of two additive non-negative functional f^x) and f%(x) (cf. S. Ba-
Banach [I, p, 217]). Extending these functional by means of Theorem 1
over two sets, 2X and 22 say, respectively, we see that it is possible
to extend the functional f(x) over the linear set 2=2].-fig. This set
will contain all bounded functions measurable (SB). The extended ad-
additive functional F{x) evidently satisfies the conditions C) and E),
p. 321, and is non-negative.
m. The Hilbert space.
§9. We shall now understand by H the unit sphere of
the Hilbert. space, i. e. the set of all sequences {#,¦) for which
l. The distance of two points «={#,-} and t'=[d't} is defined,
as usually, by the formula
With regard to this definition of distance the space H is
not compact and therefore we cannot apply Theorem 2 directly.
Let (?„ be the set of functions x=x(t) = x(&1,^...) which are
continuous in H and whose values depend only on the first n co-
coordinates &j, so that a;@i,iV..) = a?@i, #>,...,#,,,0,0,...) for any
t={&i) e H." Clearly <E,,C gn+i.
DO
It is easily seen that the set (? =?'(?« satisfies the conditions
(i), § 2. Any functional / defined in G for which the conditions (ii)
hold may be extended to an xMntegral defined in a certain set 2
containing (?.
Lemma 12. The set 2 contains all bounded functions mesurable
(93) defined in H.
Proof. Let nbea bounded continuous function defined in H.
For any point «=@i,04> •¦¦>*«>•••) and any positive integer n, we
write .t,,(/,) = .<?(#>,...,#,,,0,0,...). Evidently xaed and lini.^=x. If
M is the upper bound of \x{t)\ for teJf, then \xn\^M. Since the
constant function s=2f certainly belongs to (E, it follows from the
condition 0), p. 320, that sre2.

328
Stefan Banach:
Consequently every bounded and continuous function belongs
to 2 and by the condition C) the same is true of any bounded func-
function measurable (93).
Lemma 13. Every additive and non-negative functional. f(x)
defined in S satisfies the condition (ii3), § 2.
Proof. We define in H a distance o^t,!/) of two points
t = {&!,&*,...), ?' = {#;,4-} by
A)
2'1
We easily verify that with regard to this distance the set //
is complete and compact.
Let ffi" be the set of all functions defined in H which are
continuous according to the distance defined by the. formula A).
Evidently (ECS.
Let / be an additive non-negative functional defined in (?.
Let xn(,t) = x{&1; ...,&„, 0,0,...) for xett and t=(&1, &?,...)e H.
With regard to the distance A), B is a complete and compact
space, and hence the function x(t) e(? is uniformly continuous. It fol-
follows that the sequence [xn) uniformly converges to x. This implies
the convergence of the sequence {/(a?*)I). Let f{x) = lim /(,«„).
If ,r^0, then x,,^0 for each'M, and consequently f(x)^(). The
functional f(x), clearly additive, is therefore non-negative. The set i3"
being compact, it follows, by what has been established in Part TI,
that / satisfies in H the condition (ii3) (with (? and / replaced by (S
and J respectively). Since (ECtt and J{x)=f{x) for xe% the
functional / satisfies the condition (ii3) in (E.
§10. Now consider an additive non-negative functional f(x)
defined in (E. Let fn(x) denote the functional defined in g,, by the
formula
B) fn{x)=f{x) for XeQn.
We obviously have
C) fn(x) = fn+i(x) for ??<?„.
1) Indeed if e > 0, there exists a positive integer N suuli that — s -^Xp—^n-1 > *
whenever p>N, q>N. Since the constant function x—1 belongs to K, we have,
for fc=/(l), the inequality —ke^.f(x )—f(j;t)<lce which proves the convergence
of \1K)}.
LdbuHguo integral in abstract spaces.
329
y, if we choose any sequence (/„(.«)} of additive non-
negative- functionate, the functional /„ being defined in (?„ (where
n~l,2,...) HubJBot to the condition C), then the formula B) deter-
mincM an additive non-negative functional f(x) in S. We thus obtain
the most general additive non-negative functional f(x) defined in 2,
and by what has been established in the preceding §, the most
general Lehosguc integral for all functions bounded and measurable (93).
The set (?„ may be interpreted as the set of all function of %
variables ¦&\,...,&n which are defined and continuous in the sphere
#?+ ...+^,^.1. It is known that the most general additive and non-
negative functional defined in (?„ may be represented by a Stjeltjes
integral.
These general considerations will now be illustrated by the
following example. Suppose thatj the functionals /„ are given by
the formula
D)
for so e (? , where <pn denotes a fixed non-negative function integrable
in the sphere $f+...+^,<;i. The condition C) maybe written in
the form
n+l"
To satisfy this condition, we may put, for instance, 95,=1/2
and Vn vj = <Pn121'l=«f-.• • ~&l for n>l. We thus obtain
Let x be-an arbitrary function bounded and continuous in H.
We write again .^=^,...,#,,,0,0,...). If \x\^M, where M is
a constant, then \\Kixn = ?

330 Stefau Banach.
let F be an JMntegral which for functions belonging
to (? coincides with the functional / subject to B). We then have
xn} = ]imfn{a..n). If further /„ is represented by the
formula D), then
and, in particular, if <pa is given by E),
This formula defines explicitly a certain ^-integral for all func-
functions bounded and continuous in H.
The above considerations may be extended to certain spaces
of the type [B] (cf. S. Banach [I, Chap. V]), e.g. the spaces /-(/)),
L{p) with