ELEMENTS OF THETHEORY OF

FUNCTIONS ANDFUNCTIONAL

ANALYSIS

Volume 2Measure

The Lebesgue IntegralHilbert Space

All. Kolmogorov and S. V. Fomin

ELEMENTS OF THE THEORY OF FUNCTIONS

AND FUNCTIONAL ANALYSIS

VOLUME

MEASURE. THE LEBESGUE INTEGRAL. HILBERT SPACE

OTHER GRAYLOCK PUBLICATIONS

KHINCHIN: Three Pearls of Number TheoryMathematical Foundations of Quantum Statistics

PONTRYAGIN: Foundations of Combinatorial Topology

NOVOZHILOV: Foundations of the Nonlinear Theory ofElasticity

KOLMOGOROV and FOMIN: Elements of the Theory of Functions andFunctional Analysis. Vol. 1: Metric andNormed Spaces

PETROVSKIT: Lectures on the Theory of Integral Equations

ALEKSANDROV: Combinatorial TopologyVol. 1: Introduction. Complexes. Coverings.DimensionVol. The Betti GroupsVol. 3: Homological Manifolds. The DualityTheorems. Cohomology Groups of Compacta.Continuous Mappings of Polyhedra

Elements of the Theoryof Functions

and Functional Analysis

VOLUME 2

MEASURE. THE LEBESGLTE INTEGRAL.HILBERT SPACE

BY

A. N. KOLMOGOROV AND S. V. FOMIN

TRANSLATED FROM THE FIRST (1960) RUSSIAN EDITIONby

HYMAN KAMEL AND HORACE KOMMDepartment of Mathematics

Rensselaer Polytechnic Institute

GI?A YLOCKALBANY, N. Y.

1961

PRESS

Copyright 1961by

GRAYLOCK PRESSAlbany, N. Y.

Second PrintingJanuary 1963

All rights reserved. This book, or partsthereof, may not be reproduced in anyform, or translated, without permis-sion in writing from the publishers.

Library of Congress Catalog Card Number 574134

Manufactured in the United States of America

CONTENTSPreface viiTranslators' Note ix

CHAPTER V

MEASURE THEORY

33. The measure of plane sets 134. Collections of sets 1535. Measures on semi-rings. Extension of a measure on a semi-ring to

the minimal ring over the semi-ring 2036. Extension of Jordan measure 2337. Complete additivity. The general problem of the extension of

measures 2838. The Lebesgue extension of a measure defined on a semi-ring with

unity 3139. Extension of Lebesgue measures in the general case 36

CHAPTER VI

MEASURABLE FUNCTIONS

40. Definition and fundamental properties of measurable functions.. 3841. Sequences of measurable functions. Various types of convergence. 42

CHAPTER VII

THE LEBESGUE INTEGRAL

42. The Lebesgue integral of simple functions 4843. The general definition and fundamental properties of the Lebesgue

integral 5144. Passage to the limit under the Lebesgue integral 5645. Comparison of the Lebesgue and Riemann integrals 6246. Products of sets and measures 6547. The representation of plane measure in terms of the linear meas-

ure of sections and the geometric definition of the Lebesgue in-tegral 68

48. Fubini's theorem 7249. The integral as a set function 77

V

CONTENTS

CHAPTER VIIISQUARE INTEGRABLE FUNCTIONS

50. The spaceL2 7951. Mean convergence. Dense subsets of L2 8452. L2 spaces with countable bases 8853. Orthogonal sets of functions. Orthogonalization 9154. Fourier series over orthogonal sets. The Riesz-Fisher theorem. ... 9655. Isomorphism of the spaces L2 and 12 101

CHAPTER IXABSTRACT HILBERT SPACE. INTEGRAL EQUATIONS

WITH SYMMETRIC KERNEL

56. Abstract Hilbert space 10357. Subspaces. Orthogonal complements. Direct sums 10658. Linear and bilinear functionals in Hilbert space 11059. Completely continuous self adjoint-operators in H 11560. Linear operator equations with completely continuous operators.. 11961. Integral equations with symmetric kernel 120SUPPLEMENT AND CORRECTIONS TO VOLUME 1 123INDEX 127

PREFACEThis book is the second volume of Elements of the Theory of Functions

and Functional Analysis (the first volume was Metric and Normed Spaces,Graylock Press, 1957). Most of the second volume is devoted to an ex-position of measure theory and the Lebesgue integral. These concepts,particularly the concept of measure, are discussed with some degree ofgenerality. However, in order to achieve greater intuitive insight, we beginwith the definition of plane Lebesgue measure. The reader who wishes todo so may, after reading 33, go on at once to Ch. VI and then to theLebesgue integral, if he understands the measure relative to which thisintegral is taken to be the usual linear or plane Lebesgue measure.

The exposition of measure theory and the Lebesgue integral in thisvolume is based on the lectures given for many years by A. N. Kolmogorovin the Department of Mathematics and Mechanics at the University ofMoscow. The final draft of the text of this volume was prepared for pub-lication by S. V. Fomin.

The content of Volumes 1 and 2 is approximately that of the courseAnalysis III given by A. N. Komogorov for students in the Departmentof Mathematics.

For convenience in cross-reference, the numbering of chapters and sec-tions in the second volume is a continuation of that in the first.

Corrections to Volume 1 have been listed in a supplement at the end ofVolume 2.

A. N. KOLMOGOROYS. V. F0MIN

January 1958

vi'

TRANSLATORS' NOTEIn order to enhance the usefulness of this book as a text, a complete

set of exercises (listed at the end of each section) has been prepared byH. Kamel. It is hoped that the exercises will not only test the reader'sunderstanding of the text, but will also introduce or extend certain topicswhich were either not mentioned or briefly alluded to in the original.

The material which appeared in the original in small print has been en-closed by stars (*) in this translation.

ix

Chapter VMEASURE THEORY

The measure /1(A) of a set A is a natural generalization of the followingconcepts:

1) The length of a segment2) The area 8(F) of a plane figure F.3) The volume V(G) of a three-dimensional figure G.4) The increment c(b) c(a) of a nondecreasing function cc(t) on a

half-open interval [a, b).5) The integral of a nonnegative function over a one-, two-, or three-

dimensional region, etc.The concept of the measure of a set, which originated in the theory of

functions of a real variable, has subsequently found numerous applicationsin the theory of probability, the theory of dynamical systems, functionalanalysis and other branches of mathematics.

Tn 33 we discuss the concept of measure for plane sets, based on thearea of a rectangle. The general theory of measure is taken up inThe reader will easily notice, however, that all the arguments and resultsof 33 are general in character and are repeated with no essential changesin the abstract theory.

33. The measure of plane setsWe consider the collection of sets in the plane (x, y), each of which is

defined by an inequality of the forma x a

MEASURE THEORY [OH. V

where a, b, c and d are arbitrary real numbers. We call the sets of rec-tangles. A closed rectangle defined by the inequalities

axb;is a rectangle in the usual sense (together with its boundary) if a < b andc < d, or a segment (if a = b and c < d or a < b and c = d), or a point(if a = c = d), or, finally, the empty set (if a > b or c > d). An openrectangle

a

33] THE MEASURE OF PLANE SETS 3

A n B = (Pk nis also an elementary set.

It is easily verified that the difference of two rectangles is an elementaryset. Consequently, subtraction of an elementary set from a rectangle yieldsan elementary set (as the intersection of elementary sets). Now let A andB be two elementary sets. There is clearly a rectangle P containing bothsets. Then

A uB = P\{(P\A) n (P\B)}is an elementary set. Since

A\B = An (P\B),(AuB)\(AnB),

it follows that the difference and the symmetric difference of two elementarysets are elementary sets. This proves the theorem.

We now define the measure m' (A) of an elementary set A as follows: IfA UkPk,

where the Pk are pairwise disjoint rectangles, thenm'(A) =

We shall prove that m'(A) is independent of the way in which A is repre-sented as a union of rectangles. Let

A = UkPk = U5Q3,where Pk and Q are rectangles, and n Pk = 0, n Qk = 0 for i kSince Pk n Q3 is a rectangle, in virtue of the additivity of the measure forrectangles we have

= n Q3) =It is easily seen that the measure of elementary sets defined in this way isnonnegative and additive.

A property of the measure of elementary sets important for the sequel isgiven by

THEOREM 2. If A is an elementary set and { A is a countable (finite ordenumerable) collection of elementary sets such that

A

then

(1) m'(A)

/1*(A)

MEASURE THEORY {CH. V

0 there is a countable collection of rectangles such that

A

and/1*(A) Ekm(Pflk)

+This completes the proof of the theorem.

Theorem 4 below shows that the measure m' introduced for elementarysets coincides with the Lebesgue measure of such sets.

THEOREM 4. Every elementary set A is measurable, and /1(A) = m' (A).Proof. If A is an elementary set and P1, , are rectangles whose

union is A, then by definitionm'(A) =

Since the rectangles cover A,/1*(A) = m'(A).

But if is an arbitrary countable set of rectangles covering A, then, byTheorem 2, m'(A) m(Q,). Consequently, m'(A) /1*(A). Hence,m'(A) =

Since E \ A is also an elementary set, m'(E \ A) = /1*(E\ A). Butm'(E\A) = 1 m'(A), = 1

Hence,

m'(A) =Therefore,

m'(A) = = /1*(A) = /1(A).Theorem 4 implies that Theorem 2 is a special case of Theorem 3.THEOREM 5. In order that a set A be measurable it is necessary and sufficient

that it have the following property: for every 0 there exists an elementaryset B such that

/1*(A

33] THE MEASURE OF PLANE SETS 7

LEMMA. For arbitrary sets A and B,

B).Proof of the Lemma. Since

Ac B U (Ait follows that

+Hence the lemma follows if

< the lemmafollows from the inequality

+which is proved in the same way as the inequality above.

Proof of Theorem 5.Sufficiency. Suppose that for arbitrary e > 0 there exists an elementary

set B such that,1*(A

8 MEASURE THEORY [cii. V

and such that

+ /3,

+ /3.Since < there is an N such that

< /3;set

B=It is clear that the set

P =contains A \ B, while the set

Q = (B ncontains B \ A. Consequently, A B P u Q. Also

33] THE MEASURE OF PLANE SETS 9

THEOREM 6. The union and intersection of a finite number of measurablesets are measurable sets.

Proof. It is clearly enough to prove the assertion for two sets. Supposethat A1 and A2 are measurable sets. Then for arbitrary 0 there areelementary sets B1 and B2 such that

B1) < /2, B2) < /2.Since

(A1 u A2) (B1 u B2) c (A1 B1) u (A2 B2),it follows that(5) u A2) (B1 U B2)] B1) + !1*(A2 B2) ji(A),m(A"\A') = m(A") m(A')

h >0for arbitrary A', A" E such that A' A A". Hence A

Conversely, if

M(A)then for arbitrary > 0 there exist A', A" E such that

A' A A",M(A) m(A') 0,that is, the continuous functions f and g cannot be equivalent if they differeven at a single point.

Obviously, the equivalence of two arbitrary measurable (that is, ingeneral, discontinuous) functions does not imply their equality; for in-stance, the function equal to 1 at the rational points and 0 at the irrationalpoints is equivalent to the function identically zero on the real line.

THEOREM 9. A function f(x) defined on a measurable set E and equivalenton E to a measurable function g(x) is measurable.

In fact, it follows from the definition of equivalence that the sets

{x:f(x) > a}, {x:g(x) > a}may differ only on a set of measure zero; consequently, if the second set ismeasurable, so is the first.

* The above definition of a measurable function is quite formal. In 1913Luzin proved the following theorem, which shows that a measurable func-tion is a function which in a certain sense can be approximated by a con-tinuous function.

LUzIN'S THEOREM. In order that a function f(x) be measurable on a closedinterval [a, bJ it is necessary and sufficient that for every > 0 there exist a

42 MEASURABLE FUNCTIONS [CII. VI

function co(x) continuous on [a, b] such that,u{x:f(x) co(x)}

.

In other words, a measurable function can be made into a continuousfunction by changing its values on a set of arbitrarily small measure. Thisproperty, called by Luzin the C-property, may be taken as the definition ofa measurable function.*

EXERCISES1. For A X let XA be the characteristic function of A defined by

XA(X) = lifx E A,XA(x) = Oifx E X\A.a) XAnB(X) = XA(X)XB(X),

XAUB(X) = XA(X) + xn(x) XA(X)XB(X),= I XB(X) ,

x0(x) 0, Xx(X) = 1,XA(X) XB(X) (x E X) if, and only if, A B.

b) XA(X) is if, and only if, A E2. Suppose f(x) is a real-valued function of a real variable. If f(x) is

nondecreasing, then f(x) is Borel measurable.3. Let X = [a, bJ be a closed interval on the real line. If f(x) is defined

on X and X = where each is a subinterval of X, ii = 0and f a step function.

Suppose that f is nondecreasing (or nonincreasing) on X. Show thatall the functions of the approximating sequence of simple functions ofTheorem 4 of this section are step functions.

4. Assume that X = [a, b] contains a non-Lebesgue measurable set A.Define a function f(x) on X such that I f(x) is Lebesgue measurable, butf(x) is not.

5. Two real functions f(x) and g(x) defined on a set X are both ji-meas-urable. Show that {x:f(x) = g(x)} is

6. Let X be a set containing two or more points. Suppose that= {O, X}. Describe all measurable functions.

7. Let f(x) be a ji-measurable function defined on X. For t real defineco(t) = ,u({x:f(x) < t}). Show that o is monotone nondecreasing, continu-ous on the left, co(t) = 0, and co(t) = o is called thedistribution function of f(x).

41. Sequences of measurable functions. Various types of convergenceTheorems 5 and 7 of the preceding section show that the arithmetical

operations applied to measurable functions again yield measurable func-

SEQUENCES OF FUNCTIONS. TYPES OF CONVERGENCE 43

tions. According to Theorem 2 of 40, the class of measurable functions,unlike the class of continuous functions, is also closed under passage to alimit. In addition to the usual pointwise convergence, it is expedient todefine certain other types of convergence for measurable functions. In thissection we shall consider these definitions of convergence, their basicproperties and the relations between them.

DEFINITIoN 1. A sequence of functions defined on a measure spaceX (that is, a space with a measure defined in it) is said to converge to afunction F(x) a.e. if(1) = F(x)for almost all x E X [that is, the set of x for which (1) does not hold is ofmeasure zero].

EXAMPLE. The sequence of functions (x) = ( x) converges to thefunction F(x) = 0 a.e. on the closed interval [0, 1] (indeed, everywhereexcept at the point x = 1).

Theorem 2 of 40 admits of the following generalization.THEOREM 1. If a sequence of /2-measurable functions converges to a

function F(x) a.e., then F(x) is measurable.Proof. Let A be the set on which

= F(x).By hypothesis, ji(E \ A) = 0. The function F(x) is measurable on A byTheorem 2 of 40. Since every function is obviously measurable on a set ofmeasure zero, F(x) is measurable on (E \ A); consequently, it is measur-able on E.

EXERCISE. Suppose that a sequence of measurable functions con-verges a.e. to a limit function f(x). Prove that the sequence convergesa.e. to g(x) if, and only if, g(x) is equivalent to f(x).

The following theorem, known as Egorov's theorem, relates the notionsof convergence a.e. and uniform convergence.

THEOREM 2. Suppose that a sequence of measurable functions f,1(x) con-verges to f(x) a.e. on E. Then for every > 0 there exists a measurable set

E such that1) >

2) the sequence converges to f(x) uniformly on E6.Proof. According to Theorem 1, f(x) is measurable. Set

= f(x) I

44 MEASURABLE FUNCTIONS [cH. VI

Em =

It is clear from the definition of the sets that

for fixed m. Therefore, since a o--additive measure is continuous (see 38),for arbitrary m and > 0 there exists an n(m) such that

,u(Em \En(m)m)We set

= flmEn(m)mand prove that E6 is the required set.

We shall prove first that the sequence converges uniformly tof(x) on E6. This follows at once from the fact that if x E then

f(x)I 0

F(x)

= 0.Theorems 3 and 4 below relate the concepts of convergence a.e. and

convergence in measure.

41] SEQUENCES OF FUNCTIONS. TYPES OF CONVERGENCE 45

THEOREM 3. If a sequence of measurable functions converges a.e. to afunction F(x), then it converges in measure to F(x).

Proof. Theorem 1 implies that the limit function F(x) is measurable.Let A be the set (of measure zero) on which does not converge toF(x). Furthermore, let

Ek(cT) = {x:Ifk(x) F(x) I cT}, Rn(cT)M =

It is clear that all these sets are measurable. SinceD

and because of the continuity of the measure,*ji(M) (np oo).

We now verify that(2) MA.In fact, if x0 A, that is, if

=

then for every o- > 0 there is an n such thatfn(xo) F(xo) I

that i x0 hence xo M.But since ,u(A) = 0, it follows from (2) that M(M) = 0. Consequently,

+0 (n*Since this proves the theorem.

It is easy to see by an example that convergence in measure does notimply convergence a.e. For each natural number k define k functions

(k) (k)

on the half-open interval (0, 1] as follows:(k) Ii (i 1)/k

46 MEASURABLE FUNCTIONS [cii. viAlthough the above example shows that the full converse of Theorem 3

is not true, nevertheless we have the followingTHEOREM 4. Suppose that a sequence of measurable functions con-

verges in measure to f(x). Then the sequence contains a subsequencefnk(x)} which converges a.e. to f(x).

Proof. Let , be a sequence of positive numbers such thatlimno = 0,

and suppose that the positive numbers , are such that theseries

171 + 172 +

converges. We construct a sequence of indicesn1 n1).In general, nk is a natural number such that

,2{x:(fflk(x) f(x)I kl < 17/c (n/c > nk_1).

We shall show that the subsequence converges to f(x) a.e. Infact, let

= f(x)I

k}, Q =Since

R1 R3

and the measure is continuous, it follows that >On the other hand, it is clear that < whence ,u(Rj) > 0

as i * Since ,u(Rj) * 0,= 0.

It remains to verify thatfnk(x) *f(x)

for all x E E \ Q. Suppose that x0 E E \ Q. Then there is an io such thatThen

{x:Ifflk(x) f(x)

41] SEQUENCES OF FUNCTIONS. TYPES OF CONVERGENCE 47

for all k io, i.e.,fnk(xo) f(xo) I 0, fm(X)

I> = 0.

Show that if is fundamental in measure, then there exists a measur-able function f(x) such that converges in measure to f(x). Hint:Use Theorem 4.

5. Let be a sequence of measurable sets and let Xn be the characteris-tic function of Show that the sequence is fundamental in measureif, and only if, Am) = 0.

6. If {gn(x)} converge in measure to f(x) and g(x), respectively.then + gn(x)} converges in measure to f(x) + g(x).

Chapter VIITHE LEBESGUE INTEGRAL

In the preceding chapter we considered the fundamental properties ofmeasurable functions, which are a very broad generalization of continuousfunctions. The classical definition of the integral, the Riemann integral, is,in general, not applicable to the class of measurable functions. For instance,the well known Dirichlet function (equal to zero at the irrational pointsand one at the rational points) is obviously measurable, but not Riemannintegrable. Therefore, the Riemann integral is not suitable for measurablefunctions.

The reason for this is perfectly clear. For simplicity, let us consider func-tions on a closed interval. To define the Riemann integral we divide theinterval on which a function f(x) is defined into small subintervals and,choosing a point in each of these subintervals, form the sum

What we do, essentially, is to replace the value of f(x) at each point ofthe closed interval = [Xk, Xk+1J by its value at an arbitrarily chosenpoint of this interval. But this, of course, can be done only if the valuesof f(x) at points which are close together are also close together, i.e., iff(x) is continuous or if its set of discontinuities is "not too large." (Abounded function is Riemann integrable if, and only if, its set of discon-tinuities has measure zero.)

The basic idea of the Lebesgue integral, in contrast to the Riemann in-tegral, is to group the points x not according to their nearness to each otheron the x-axis, but according to the nearness of the values of the functionat these points. This at once makes it possible to extend the notion of in-tegral to a very general class of functions.

In addition, a single definition of the Lebesgue integral serves for func-tions defined on arbitrary measure spaces, while the Riemann integral isintroduced first for functions of one variable, and is then generalized, withappropriate changes, to the case of several variables.

In the sequel, without explicit mention, we consider a o--additive measure/2(A) defined on a Borel algebra with unit X. The sets A X of the al-gebra are ji-measurable, and the functions f(x)defined for all x E Xare also ji-measurable.

42. The Lebesgue integral of simple functionsWe introduce the Lebesgue integral first for the simple functions, that is,

for measurable functions whose set of values is countable.48

42] LEBESGIJE INTEGRAL OF SIMPLE FUNCTIONS 49

Let f(x) be a simple function with values, , y5 for i j).

It is natural to define the integral of f(x) over (on) a set A as(1) E A,f(x) =

We therefore arrive at the following definition.DEFINITION. A simple function f(x) is over A if the series

(1) is absolutely convergent. If f(x) is integrable, the sum of the series(1) is called the integral of f(x) over A.

In this definition it is assumed that all the are distinct. However, it ispossible to represent the value of the integral of a simple function as a sumof products Ck,u(Bk) without assuming that all the Ck are distinct. This canbe done by means of the

LEMMA. Suppose that A = Uk Bk, ii B, = 0 (i j) and that f(x)assumes only one value on each set Bk. Then

(2) =

where the function f(x) is integrable over A if, and only if, the series (2) isabsolutely convergent.

Proof. It is easy to see that each set

An = tx:x E A,f(x) =is the union of all the sets Bk for which Ck = Therefore,

= Yn (Bk) = Ck/2(Bk).Since the measure is nonnegative,

I Yn ii(An) = En Yn I = I Cic Ithat is, the series Yn,u(An) and Ek ck,u(Bk) are either both absolutelyconvergent or both divergent.

We shall now derive some properties of the Lebesgue integral of simplefunctions.

A) + fg(x)= L +

where the existence of the integrals on the left side implies the existence ofthe integral on the right side.

To prove A) we assume that f(x) assumes the values f1 on the sets

50 THE LEBESGIJE INTEGRAL [CII. VII

A, and that g(x) assumes the values gj on the sets G A; hence(3)

= Lfx =(4) J2 = =Then, by the lemma,

(5)=

L {f(x) + g(x)} = + n Gd).But

=

=

so that the absolute convergence of the series (3) and (4) implies the ab-solute convergence of the series (5). Hence

J = J1 + J2.B) For every constant k,

k ff(x)=

L {kf(x)}where the existence of the integral on the left implies the existence of theintegral on the right. (The proof is immediate.)

C) A simple function f(x) bounded on a set A is integrable over A, and

Lfx dH M on A. (The proof is immediate.)

EXERCISES1. If A, B are measurable subsets of X, then

fIxA(X)2. If the simple function f(x) is integrable over A and B A, then

f(x) is integrable over B.3. Let F0 = [0, 1]. Define the simple function f(x) on F0 as follows: On

the open intervals deleted in the nth stage of the construction of theCantor set F let f(x) = n. On F let f(x) = 0. Compute f f(x) wherej1 is linear Lebesgue measure.

43] DEFINITION AND PROPERTIES OF THE LEBESGUE INTEGRAL 51

43. The general definition and fundamentalproperties of the Lebesgue integral

DEFINITION. We shall say that a function f(x) is integrable over a set Aif there exists a sequence of simple functions integrable over A anduniformly convergent to f(x). The limit(1) J

IA

is denoted by

IA

and is called the integral of f(x) over A.This definition is correct if the following conditions are satisfied:1. The limit (1) for an arbitrary uniformly convergent sequence of

simple functions integrable over A exists.2. This limit, for fixed f(x), is independent of the choice of the sequence

3. For simple functions this definition of integrability and of the integralis equivalent to that of 42.

All these conditions are indeed satisfied.To prove the first it is enough to note that because of Properties A),

B) and C) of integrals of simple functions,

ffn(x) f fm(X)dH sup fm(x) I;x E A).

To prove the second condition it is necessary to consider two sequencesand {f *(x)) and to use the fact that

IA ff*(x) dH

f(z) 1; x E A] + sup [Ifn*(x) f(x) I; x EFinally, to prove the third condition it is sufficient to consider the se-

= f(x).We shall derive the fundamental properties of the Lebesgue integral.THEOREM 1.

IA=

Proof. This is an immediate consequence of the definition.THEOREM 2. For every constant k,

52 THE LEBESGUE INTEGRAL [cH. VII

kff(x)= IA {kf(x)J

where the existence of the integral on the left implies the existence of the integralon the right.

Proof. To prove this take the limit in Property B) for simple functions.THEOREM 3.

IA + f g(x) IA {f(x) + g(x)}where the existence of the integrals on the left implies the existence of the in-tegral on the right.

The proof is obtained by passing to the limit in Property A) of integralsof simple functions.

THEOREM 4. A function f(x) bounded on a set A is integrable over A.The proof is carried out by passing to the limit in Property C).THEOREM 5. If f(x) 0, then

IA 0,

on the assumption that the integral exists.Proof. For simple functions the theorem follows immediately from the

definition of the integral. In the general case, the proof is based on thepossibility of approximating a nonnegative function by simple functions(in the way indicated in the proof of Theorem 4, 40).

COROLLARY 1. If f(x) g(x), then

IA

COROLLARY 2. If m f(x) M on A, then

IA

THEOREM 6. If

IA fAfldii,

where the existence of the integral on the left implies the existence of the inte-grals and the absolute convergence of the series on the right.

43] DEFINITION AND PROPERTIES OF THE LEBESGUE INTEGRAL 53

Proof. We first verify the theorem for a simple function 1(x) whichassumes the values

Yi, ,Yk,Let

Bk = {x:x E A,f(x) YkLBflk = {x: x E , f(x) = Yk}.

Then

f f(x) = = Yk(1) A

= fAfldii.

Since the series is absolutely convergent if f(x) is integrable,and the measures are nonnegative, all the other series in (1) also convergeabsolutely.

If f(x) is an arbitrary function, its integrability over A implies that forevery 0 there exists a simple function g(x) integrable over A such that(2) 11(x) g(x) I

54 THE LEBESGUE INTEGRAL [CH. VII

Since 0 is arbitrary,

IA

COROLLARY. If f(x) is integrable over A, then f(x) is integrable over anarbitrary A' A.

THEOREM 7. If a function is integrable over A, and I f(x)I

then f(x) is also integrable over A.Proof. If f(x) and are simple functions, then A can be written as

the union of a countable number of sets on each of which f(x) andare constant:

f(x) = ( an).The integrability of implies that

I

= IA

Therefore f(x) is also integrable, and

IA = I

IA'

Passage to the limit proves the theorem in the general case.TRANS. NOTE. The proof is as follows: For 0, choose an n0 > 1/.

Let : n no) be a sequence of integrable simple functions converginguniformly to the function + , and let no} be a sequence ofsimple functions converging uniformly to f(x). These sequences are chosenso that they satisfy the inequalities

0,I

+ ] I

43] DEFINITION AND PROPERTIES OF THE LEBESGUE INTEGRAL 55

THEOREM 8. The integrals

= IA j2 = L If(x) Ieither both exist or both do not exist.

Proof. The existence of J2 implies the existence of J1 by Theorem 7.For a simple function the converse follows from the definition of the

integral. The general case is proved by passing to the limit and noting thathal - Ibtl Ia - bt.

THEoREM 9 (THE CHEBYSHEV INEQUALITY). If cc(x)

0 on A, then

E A,

c} (1/c) L

Proof. Setting

A' = Ix:x E cj,we have

IA = IA' +

COROLLARY. If

fAtf(x)f = 0,

thenf(x) = 0 a.e.For, by the Chebyshev inequality,

E A,If(x)l 1/ni 0for all n. Therefore,

b4x:x E A,f(x) O} E A, f(x) I

1/n) = 0.

EXERCISES1. Suppose f(x) is integrable over E, and that F is a measurable subset

of E. Then XFf is integrable over E and

L XF(x)f(x) IF f(x)

2 (FIRST MEAN VALUE THEOREM). Let f(x) be measurable,m

56 THE LEBESGUE INTEGRAL [CH. VII

on A, and suppose that g(x) 0 is integrable over A. Then there existsa real number a such that m < a

M and f f(x)g(x) d/2 = a L g(x)3. Suppose that f(x) is integrable over the set E = [a, b] and that is

linear Lebesgue measure. Then F(x)= f f(x) is defined for[ax]

a < x < b.a) Show that

{F(x2) F(x1)]/(x2 = [1/(x2 xi)] f f(x)[x1 ,X2]for a x1 < x2 b.

b) For any point x0, a < xo < b, at which f(x) is continuous showthat F'(xo) = f(xo).

4. Let f, g be integrable over E.a) If f f(x)

= f g(x) for every measurable A E, thenf(x) = g(x) a.e. on E.

b) If f f(x) dji = 0, for every measurable A E, then f(x) =0 a.e. on E.

5. Suppose E = [a, b], is Lebesgue measure and f is integrable over E.Show that f f(x) = 0 for a c b implies that f(x) = 0 a.e. on

[ac]

E. Hint: Consider the class of A E for which f f(x) dji = 0 and applythe preceding exercise.

44. Passage to the limit under the Lebesgue integralThe question of taking the limit under the integral sign, or, equivalently,

the possibility of termwise integration of a convergent series often arisesin various problems.

It is proved in classical analysis that a sufficient condition for interchang-ing limits in this fashion is the uniform convergence of the sequence (orseries) involved.

In this section we shall prove a far-reaching generalization of the cor-responding theorem of classical analysis.

THEOREM 1. If a sequence converges to f(x) on A andI

cc(x)for all n, where c(x) is integrable over A., then the limit function f(x) is in-

44] PASSAGE TO THE LIMIT UNDER THE INTEGRAL 57

tegrable over A and

fAfnxProof. It easily follows from the conditions of the theorem that

If(x) ILetAk= {x:k 1

cc(x) Ukm+l Ak

m}.By Theorem 6 of 43,

(*)IA

=fAk

and the series (*) converges absolutely.Hence

Lmc(x) d/2

'[Akdji.

The convergence of the series (*) implies that there exists an m such that

IBm

The inequality cc(x)

58 THE LEBESGUE INTEGRAL [OH. VII

THEOREM 2. Suppose that

fi(x) f2(x)

on a set A, where the functions are integrable and their integrals arebounded from above:

IA K.

(1) f(x)exists a.e. on A, f(x) is integrable on A and

LfnxClearly, the theorem also holds for a monotone descending sequence of

integrable functions whose integrals are bounded from below.On the set on which the limit (1) does not exist, f(x) can be defined

arbitrarily; for instance, we may set f(x) = 0 on this set.Proof. We assume that f(x) 0, since the general case is easily reduced

to this case by writing= fi(x).

We consider the set= {x:x E

It is easy to see that = flr whereIx:x E > r}.

By the Chebyshev inequality (Theorem 9, 43),

K/r.Since , it follows that

K/r.Further, since

cfor every r, K/r. Since r is arbitrary,

= 0.

This also proves that the monotone sequence f(x)a.e. on A.

44] PASSAGE TO THE LIMIT UNDER THE INTEGRAL 59

Now let co(x) = r for all x such thatr 1

f(x) < r (r = 1, 2, ).If we prove that co(x) is integrable on A, the theorem will follow imme-

diately from Theorem 1.We denote by Ar the set of all points x E A for which cc(x) = r and set

B8 = Ar.Since the functionsfn(x) andf(x) are bounded on B8 and co(x) f(x) + 1,

it follows that

+

= +

K +

But

lBs=

Since the partial sums in the above equation are bounded, the series

= IA

converges. Hence co(x) is integrable on A.COROLLARY. If

L 0 there is a 5 > 0 such that p(f(xi),f(x2)) 0 such that

< (1 i n)

p(x1 , x2) < 5.If f E D, there exists an such that

Thenp(f(xi),f(x2))

p(f(xi),f2(xi)) ++ + + =

if p(xi, x2) < But this means that the set of all f D is equicontinuous.We shall now prove the sufficiency."

(9) p. 72, 1. 3* Replace "max" by "sup".(10) p. 77, 1. 9. Replace "continuous" by "continuous at a point x0".

123

124 SUPPLEMENT AND CORRECTIONS TO VOL. 1

p. 77, 1. 13. Replace II by II x p. 77, 1. 11. Replace I f(xi) f(x2) I by I f(x) f(xo) I.p. 77, 1. 20. Replace "continuous" by "uniformly continuous".

Delete "everywhere in R".(11) p. 80, 1. 9* Replace "x0 L1" by "x0 is a fixed element of the

complement of Lf".(12) p. 84, 1. 12. Replace "supn by "5UPn I(13) p. 92, 1. 14*, 13*. The assertion that the functionals generate a

dense subset of C is not true. Replace "satisfies the conditions of Theorem1, i.e. linear combinations of these functionals are everywhere dense inC[a,bl" by "has the property that if a sequence is bounded and

p for all E then is weakly convergent to x(t)".(14) p. 94, 1. 9* if. The metric introduced here leads to a convergence

which is equivalent to the weak convergence of functionals in every boundedsubset of (but not in all of In 1. 6*, after "so that" insert "in everybounded subset of On p. 95, 1. 10, after "that" insert "for boundedsequences of 1?'.

(15) p. 116. The proof of Theorem 5 contains an error. It should bereplaced by the following:

Proof. 10. We note first that every nonvanishing eigenvalue of a com-pletely continuous operator has finite multiplicity. In fact, the set 14 of alleigenvectors corresponding to an eigenvalue X is a linear subspace whosedimension is equal to the multiplicity of the eigenvalue. If this subspacewere infinite-dimensional for some X 0, the operator A would not becompletely continuous in 14, and hence would not be completely con-tinuous in the whole space.

2. Now to complete the proof of the theorem it remains to show that ifXnl is a sequence of distinct eigenvalues of a completely continuous op-

erator A, then p 0 as n p Let be an eigenvector of A correspond-ing to the eigenvalue The vectors are linearly independent. Let(n = 1, 2, ...) be the subspace of all the elements of the form

ny

For each y E1 n n 1 ni 1y Ay

j=iajXjXn Xi = i1 (1 XiXnwhence it is clear that y Xn1Ay E En_i.

Choose a sequence { such thatE En

, II yn Ii = 1, p(yn , >(The existence of such a sequence was proved on p. 118, 1. 6 if.)

SUPPLEMENT AND CORRECTIONS TO VOL. 1 125

We flow suppose that the sequence is bounded. Then the set{A is compact. But this is impossible, since

II A(yq/Xq) II= II + A(yq/Xq)] II > (p> q),

inasmuch as + A (yq/Xq) E This contradiction provesthe assertion.

(16) p. 119, 1. 12. The assertion that G0 is a subspace is true, but notobvious. Therefore the sentence "Let G0 be the subspace consisting of allelements of the form x Ax" should be replaced by the following: "LetG0 be the linear manifold consisting of all elements of the form x Ax.We shall show that G0 is closed. Let be a one-to-one mapping of thequotient space E/N (where N is the subspace of the elements satisfyingthe condition x Ax 0) onto G0. (For the definition of quotient spacesee Ex. 5, 57.) We must show that the inverse mapping T' is continuous.It is sufficient to show that it is continuous at y 0. Suppose that this isnot so; then there exists a sequence p 0 such that p > 0, where

= . Setting fln Sill and = we obtain a sequencesatisfying the conditions:

II fln 11 = 1, 0.

If we choose in each class a representative Xn such that 2, weobtain a bounded sequence, and = =

A is completely continuous, contains a fundamental subse-quence The sequence = + (where = andis also fundamental and therefore converges to an element x0. Hence

= * Tx0, so that Tx0 = 0, that is, x0 E N. But then II I! x0 * 0, which contradicts the condition = 1. This contra-

diction proves the continuity of and shows that G0 is closed. Hence G0is a subspace".

Absolutely continuous measure 13abstract Lebesgue measure 32additive measure 20algebra of sets 16almost everywhere (a.e.) 41B-algebra 19B-measurable function 38B-sets 19Bessel inequality 97bilinear functional 112Boolean ring 20Borel algebra 19Borel closure 19Borel measurable function 38Borel sets 19

INDEX

Cantor function 14characteristic function 42Chebyshev inequality 55closed linear hull 92closed orthonormal set of functions 98closed set of functions 92complete measure 37complete set of functions 92complete set of orthonormal functions 98completely additive measure 11, 28completely continuous operator 115continuous measure 11convergence a.e. 43convergence in measure 44convergence in the mean 84convolution 75countable base for a measure 88

o-algebra 19-ring 19direct sum of orthogonal subspaces 108Dirichlet function 48discrete measure 13distribution function 42

Egorov's theorem 43eigenvalues 115eigenvectors 115elementary set 2equivalent functions 41

essentially bounded function 83essential upper bound 83Euclidean space 80extension of a measure 20, 22, 28, 31, 36

Fatou's theorem 59finite partition of a set 17first mean value theorem 55Fourier coefficients 97Fourier series 97fractional integral 76Fubini's theorem 73function integrable over a set 51fundamental in measure 47

Hilbert space 79, 103Holder inequality 83

Infinite-dimensionalspace L2 92inner measure 5, 24, 32inner product 80integral as a set function 77integration by parts 76invariant subset 20isomorphism of 12 and L2 101

Jordan extension of a measure 26, 27Jordan measurable set 23Jordan measure 23

Lebesgue criterion for measurability 15Lebesgue extension of a measure 31, 34Lebesgue integral 48, 51Lebesgue integral of simple functions 48Lebesgue integral as a measure 71Lebesgue measurable set 5, 32Lebesgue measure 8, 32Lebesgue-Stieltjes measures 13Legendre polynomials 93, 96linear functional in Hilbert space 110linear hull 92linear manifold 106linear manifold generated by {cokI 92linearly dependent set of functions 91linearly independent set of functions 91Luzin's theorem 41

127

128 INDEX

Mean convergence 84measurability criterion of Caratho-

dory 15measurable set 5, 23, 32, 36measure of an elementary set 3measure in Euclidean n-space 12measure 13measure of a plane set 12measure of a rectangle 2measure on a semi-ring 20minimal B-algebra 19minimal ring 16Minkowski inequality 84,u-integrability for simple functions 49.i-measurable function 38,4F-measurable set 13

n-dimensional space L2 92nonmeasurable sets 13nonseparable Hilbert space 106normalized set of functions 93

Orthogonal complement 107, 110orthogonal set of functions 93orthogonalization process 95orthonormal set of functions 93

Parseval's equality 98, 100plane Lebesgue measure 5product measure 68product of sets 65properties of Lebesgue integral 51ff.

Regular measure 15relation between types of convergence 87Riemann integral 48, 6264Riesz-Fisher theorem 98ring generated by a collection of sets 16ring of sets 15

function 38scalar product 80Schwarz inequality 105self -adjoint operator 113semi-ring of sets 17set of unicity of a measure 30set of i-unicity 30ti-additive measure 11, 28ti-algebra 19ti-ring 19simple function 39singular measure 13space 12 79space 79, 80space L9 83space 83space of square integrable functions 79square integrable function 79square summable function 79step function 42subspace 106suhspace generated by 92symmetric bilinear functional 112

Unit of a collection of sets 16Quadratic functional 113quotient space 114 Young's inequality 72

CoverS TitleOTHER GRAYLOCK PUBLICATIONSTitle: Elements of the Theory of Functionsand Functional Analysis, VOLUME 2, MEASURE. THE LEBESGLTE INTEGRAL. HILBERT SPACECopyright 1961 GRAYLOCK PRESSLCCN 5704134

CONTENTSPREFACETRANSLATORS' NOTEChapter V: MEASURE THEORY33. The measure of plane sets34. Collections of setsEXERCISES

35. Measures on semi-rings. Extension of a measure on a semi-ring to the minimal ring over the semi-ringEXERCISES

36. Extension of the Jordan measureEXERCISES

37. Complete additivity. The general problem of the extension of measuresEXERCISES

38. The Lebesgue extension of a measure defined on a semi-ring with unityEXERCISES

39. Extension of Lebesgue measures in the general caseEXERCISES

Chapter VI: MEASURABLE FUNCTIONS40. Definition and fundamental properties of measurable functionsEXERCISES

41. Sequences of measurable functions. Various types of convergenceEXERCISES

Chapter VII: THE LEBESGUE INTEGRAL42. The Lebesgue integral of simple functionsEXERCISES

43. The general definition and fundamental properties of the Lebesgue integralEXERCISES

44. Passage to the limit under the Lebesgue integralEXERCISES

45. Comparison of the Lebesgue and Riemann integralsEXERCISES

46. Products of sets and measuresEXERCISES

47. The representation of plane measure in terms of the linear measure of sections, and the geometric definition of the Lebesgue integralEXERCISES

48. Fubini's theoremEXERCISES

49. The integral as a set functionEXERCISES

Chapter VIII: SQUARE INTEGRABLE FUNCTIONS50. The space L2EXERCISES

51. Mean convergence. Dense subsets of L2EXERCISES

52. L2 spaces with countable basesEXERCISES

53. Orthogonal sets of functions. OrthogonalizationEXERCISES

54. Fourier series over orthogonal sets. The Riesz-Fisher theoremEXERCISES

55. Isomorphism of the spaces L2 and 12EXERCISES

Chapter IX: SPACE. INTEGRAL EQUATIONS WITH SYMMETRIC KERNEL56. Abstract Hubert spaceEXERCISES

57. Subspaces. Orthogonal complements. Direct sumsEXERCISES

58. Linear and bilinear functionals in Hubert spaceEXERCISES

59. Completely continuous seif-adjoint operators in HEXERCISES

60. Linear equations in completely continuous operators61. Integral equations with symmetric kernelEXERCISES

SUPPLEMENT AND CORRECTIONS TO VOLUME 1INDEX