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Measure, Lebesgue Integrals, and Hubert Space A. N. KOLMOGOROV AND S. V. FOMIN Moscow Stote University Moscow, U.S.S.R.
TRANSLATED BY
NATASCHA ARTIN BRUNSWICK ond
ALAN JEFFREY Institute of Mothemoticol Sciences New York University New York, New York
ACADEMIC PRESS
New York and London
COPYRIGHT © 1960, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED
NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.
ACADEMIC PRESS INC. 111 Fw'rH AVENUE
NEW YORK 3, N. Y.
United Kingdom Edition Published by ACADEMIC PRESS INC. (LoNDoN) Lm. I3ERKELEY SQUARE HOUSE, LONDON W. I
Library of Congress Catalog Card Number 61-12279
First Printing, 1960 Second Printing, 1962
PRINTED IN THE UNITED STATES OF AMERICA
Translator's Note This book is a translation of A. N. Kolmogorov and S. V. Fomin's book
"Elementy Teorii Funktsii i Funktsional'nogo Analiza., II. Mera, Integral Lebega i Prostranstvo Hilberta" (1960). An English translation of the first part of this work was prepared by Leo F. Boron and published by Graylock Press in 1957 and is mentioned in [A] of the suggested reading matter added at the end of the present book.
The approach adopted by the Russian authors should be of great interest to many students since the concept of a semiring is introduced early on in the book and is made to play a fundamental role in the subsequent development of the notions of measure and integral. Of particular value to the student is the initial chapter in which all the ideas of measure are introduced in a geometrical way in terms of simple rectangles in the unit square. Subsequently the concept of measure is introduced in complete generality, but frequent back references to the simpler intro— duction do much to clarify the more sophisticated treatment of later chapters.
A number of errors and inadequacies of treatment noted by the Russian authors in their first volume are listed at the back of their second book and have been incorporated into our translation. The only change we have made in this addenda is to re-reference it in terms of the English Translation [A]. In this edition the chapters and sections have been renumbered to make them independent of the numbering of the first part of the book and to emphasize the self-contained character of the work. New York, January 1961 NATASCHA ARTIN BRUNSWICK ALAN JEFFREY
V
Foreword This publication is the second book of the "Elements of the Theory of Functions and Functional Analysis," the first book of which ("Metric and Normed Spaces") appeared in 1954. In this second book the main role is played by measure theory and the Lebesgue integral. These con-
cepts, in particular the concept of measure, are discussed with a sufficient
degree of generality; however, for greater clarity we start with the concept of a Lebesgue measure for plane sets. If the reader so desires he
can, having read §1, proceed immediately to Chapter II and then to the Lebesgue integral, taking as the measure, with respect to which the integral is being taken, the usual Lebesgue measure on the line or on the plane.
The theory of measure and of the Lebesgue integral as set forth in this book is based on lectures by A. N. Kolmogorov given by him repeatedly in the Mechanics-Mathematics Faculty of the Moscow State University. The final preparation of the text for publication was carried out by S. V. Fomin. The two books correspond to the program of the course "Analysis III" which was given for the mathematics students by A. N. Kolmogorov. At the end of this volume the reader will find corrections pertaining to the text of the first volume. A. N. KoLMoGoRov S. V. F0MIN
vi'
NOTE. v Contents
TRANSLATOR'S
vii
FOREWORD LIST OF SYMBOLS
xi CHAPTER
I
Measure Theory 1. Measure of Plane Sets 2. Systems of Sets 3. Measures on Semirings. Continuation of a Measure from a Semiring to the Minimal Ring over it.. 4. Continuations of Jordan Measures 5. Countable Additivity. General Problem of Continuation of Measures 6. Lebesgue Continuation of Measure, Defined on a Semiring with a Unit 7. Lebesgue Continuation of Measures in the General Case....
1
19
26 29 35 39 45
CHAPTER II
Measurable Functions 8. Definition and Basic Properties of Measurable Functions.... 9. Sequences of Measurable Functions. Different Types of Convergence
48 54
CHAPTER III
The Lebesgue Integral 10. The Lebesgue Integral for Simple Functions 11. General Definition and Basic Properties of the Lebesgue Integral 12. Limiting Processes Under the Lebesgue Integral Sign 13. Comparison of the Lebesgue Integral and the Riemann Integral
ix
61
63 69
75
CONTENTS
14. Direct Products of Systems of Sets and Measures 15. Expressing the Plane Measure by the Integral of a Linear Measure and the Geometric Definition of the Lebesgue Integral 16. Fubini's Theorem 17. The Integral as a Set Function
78
82 86 .
90
CHAPTER IV
Functions Which Are Square Integrable
The L2 Space. . 19. Mean Convergence. Sets in L2 which are Everywhere Complete 20; L2 Spaces with a Countable Basis 21. Orthogonal Systems of Functions. Orthogonalisation 22. Fourier Series on Orthogonal Systems. Riesz-Fischer Theorem 23. The Isomorphism of the Spaces L2 and 12 18.
92
97 100 104 109 115
CHAPTER V
The Abstract Hubert Space. Integral Equations with a Symmetric Kernel
Abstract Hilbert Space 25. Subspaces. Orthogonal Complements. Direct Sum 26. Linear and Bilinear Functionals in Hilbert Space 27. Completely Continous Self-Adjoint Operators in H 28. Linear Equations with Completely Continuous Operators.... 29. Integral Equations with a Symmetric Kernel
118
ADDITIoNs
138
24.
SUBJECT INDEX
CORRECTIONS TO VOLUME I
121
126 129 134 135
143
List of Symbols A
set
A*
linear operator adjoint to (the linear operator) A , p. 129.
A CB
B is a proper subset of A. A is a subset of B.
aEA
element a is a member of set A.
A\B
difference, complement of B with respect to A
A
B
symmetric difference of sets A and B defined by the expression A B (A\B) U (B\A) = (A
B(f, g)
U
B)\(A
n B).
bilinear functional, p. 128. Borel algebra containing
5-ring
p. 25.
p. 24
Kronecker delta, p. 107.
arbitrary positive number.
0
null set
11111
norm of f, p. 95.
(f, g)
scalar product of functions in L2, p. 94.
X(A)
continuation of measure m defined for A, p. 39.
M
closure of the set M, p. 105.
M®M'
direct sum of spaces M and M', p. 124.
9Q(A)
system of sets A, p. 20.
m(P)
measure of rectangular set P, p. 2. Lebesgue measure of set A, p. 7.
outer measure of set A, p. 6. inner measure of set A, p. 6. x
LIST OF SYMBOLS
Xli
outer measure of Set A, p. 31. inner measure of Set A, p. 31.
system of sets, p. 25. system of sets, p. 20. Xk
Q(f)
direct product of sets X1, X2,. •,
p. 78.
quadratic form, p. 129. ring, p. 19.
system of sets, p. 19. a-ring
p. 24
tx:f(x)
0
—
1J < 2E,
is arbitrary,
+ ,z*(E\A) i.e.,
1,
the set A is measurable.
Necessity. Let A be measurable, i.e., let
+ iz*(E\A) =
1.
I. MEASURE THEORY
10
>
Selecting
0
arbitrarily, we shall seek coverings
A and E\A for and
an N such that
Set N
B=
U
It is clear that the set P=
U n>N
contains A\B, and that the set
Q= contains
B\A, and therefore that A
BçPu
Q. Hence,
Let us evaluate ,z*( Q). For this let us note that
(u
U
(u
=
and hence
+
1.
(3)
1. MEASURE OF PLANE SETS
11
But by assumption,
+
=
(4)
1
Subtracting (3) from (4) we have
=
—
fl B)
Therefore,
B)
0, one can
Since (A1
u
A2)
(B1 u
B2) c
(A1
B1) u (A2
B2),
we have UA2) B1 is is
(B1 UB2)]
B1) +,.L*(A2
(5)
B2)
0 one can find an N
31
is arbitrary, and the sets A' U B', A" u B", A'\B"
0
and A"\B' belong to
(1), (2), (3) and
(4) imply that
and A \B belong to Let 9)1 be a system of those sets A for which the set B
AU B
A
exists. For any A from Sill we set, by definition,
of
= inf m(B), B?A
= supm(B). The functions p(A) and are called, respectively, the "outer" and the "inner" measure of the set A.
Obviously, always Theorem 2. The ring A E 9)1 for which A)
Proof.
=
coincides with the system of those sets p( A).
If
then —
and for any A' and A" from m(A')
h> 0, for which A' ç A ç A", m(A") =
m(A"\A') = m(A") — m(A') i.e., A cannot Conversely,
h> 0,
belong to
if
= then, for any
> 0, there exist A' and A" from
A' ç A ç A", —m(A')
m(A") —p(A)
m(A"\A')
= m(A") — m(A')
for which
I. MEASURE THEORY
32
AE
The following theorems hold for sets from Theorem 3.
If A C LI
Ak,
then jz(A) < k=1
k=l
Proof. Let us select Ak' such that Ak
and let us form A'
Ak',
and since
Ak'. Then,
=
m(A')
9 is arbitrary,
with the domain of definition
Let us now define the function S,1 = as
the common value of the inner and outer measures: /.L(A) = /.L(A) = Theorems
3 and 4, and the obvious fact that for A E
p(A) =
= m(A),
imply
A) is a measure and a continuation of
Iheorem 5. The function the measure m.
The construction given can be used for any measure m defined on a ring. of elementary sets in a plane is essentially The system Sm2 =
connected with the coordinate system: the sets of the system
consist of rectangles with sides which are parallel to the coordinate axes. In going over to the Jordan measure J(2) = j(m2)
this dependence on the choice of a coordinate system disappears: starting from an arbitrary system of coordinates connected with the initial system tx1, x2} by the orthogonal transformation =
cos
=
+ sin
+
+ C05 O.'X2 + a2,
we obtain the same Jordan measure
J=
j(m2) =
denotes the measure constructed with the help of rec(here tangles with sides parallel to the axes This fact can be proved with the help of the following general theorem: = j(m1) and 6. In order that the Jordan continuations = j(m2) of the measures m1 and m2 defined on the rings and
Theorem
34
I. MEASURE THEORY
coincide, it is necessary and sufficient that the following conditions be satisfied:
m1(A) = /.L2(A) m2(A) = p1(A)
E E
S,11,
on on
The necessity of the condition is obvious. Let us prove their sufficiency.
Then there exist A', A" E Sm1, such that
Let A E A'
A",
A
m1(A") — m1(A')
0 be such that =L
The domain S,, consists of all subsets of the set X. For each
A çXweset
/.L(A) = €A
It is easy to check that A) is a or-additive measure, where X) = 1. This example occurs naturally in connection with many questions of probability theory. Let us give an example of a measure which is additive but not or-additive. Let X be the set of all rational points of the segment [0,
1], and S,, consist of the intersections of the set X with
arbitrary intervals (a, b), segments [a, b] and half segments (a, b], [a, b). It is easy to see that S,, is a semiring. For each such set we put l.L(Aab) = h —
a.
This is an additive measure. It is not or-additive because, for X) = 1 and at the same time X is the union of a example,
countable number of separate points, the measure of each one of which is zero. In this and the two following sections we shall consider or-additive measures and their different or-additive continuations. Theorem 1.
If the measure m, defined on some semir'ing Sm, 15
countably additive, then the measure = r (m), obtained from it by continuation to the ring 9? (Sm), is also countably additive. Proof. Let
A E 9?(Sm),
fl = 1,
E
and
A=
U
2,
37
COUNTABLE ADDITIVITY
where B8 11 B,.
Sm such that
Then there exist sets A1 and
= 0 for s
from
A=UA1, where the sets on the right-hand sides of each of these equations are pairwise non-intersecting and the union over i and j is finite. (Theorem 3, §2). are pair— n A2. It is easy to see that the sets Let wise non-intersecting, and hence, A2 = U U = U Therefore,
and because of the additivity of the measure m on
Sm, we have
and,
m(A2) =
(1)
=
(2)
by definition of the measure r(m) on 9? (Sm), /.L(A) =
(3)
=
(4)
Equations (1), (2), (3) and (4) imply ,z (A) = summations over i and j
are
(Ba). (The
finite, the series in n convergeS)
One could show that a Jordan continuation of a ci-additive measure is always ci-additive; there is however no need to do this in this special case since it will follow from the theory of Lebesgue continuations which will be given in the next section.
Let us now show that, for the case of or-additive measures, Theorem 2 of §3 may be extended to countable coverings. Theorem 2.
A2, •..,
If the measure belong to
is or-addItIve, and the sets A, A1, then
AçU
38
MEASURE THEORY
implies the inequality /.L(A) Proof. By Theorem 1, it is enough to give the proof for measures defined on a ring, since from the validity of Theorem 2 for =
r( m) it immediately follows that it can be applied also to the
measure m. If S,, is a ring, the sets = (A
belong to
Since
and since the sets
are pairwise non-intersecting,
/.L(A)
From now on we shall, without special mention, consider only c-additive measures. We have already considered above two methods of continuation of measures. In connection with the continuation of the measure m to the ring 9?(Sm) in §3 we noted the uniqueness of this continuation. The
case of a Jordan continuation j(m) of an arbitrary measure m is analogous. If the set A is Jordan measurable with respect to the measure m (belongs to Sj(m)), then, for any measure continuing m and defined on A, the value coincides with the value J(A) of the Jordan continuation J = j(m). One can show that the extension of the measure m beyond th boundaries of the system Sj(m) is not unique. More precisely this means the following. Let us call the set A the set of uniqueness for the measure m, if:
1) there exists a measure which is a continuation of the measure m, defined for the set A; 2) for any two measures of this kind and = The following theorem holds: The system of sets of uniqueness for the measure m coincides with the system of sets which are Jordan measurable with respect to the measure m, with the system of sets
LEBESGUE CONTINUATION OF MEASURE
39
However, if one considers only c-additive measures and their continuathen the system of sets of uniqueness will be, generally tion speaking, wider.
Since it is the case of o—additive measures that will interest us in the future let us establish 2.
The set A is called the set of c-uniqueness for a measure m defined
a
forA
such that A E Sx); 2) for two such c-additive continuations X1 and X2 the equation X1(A) = X2(A)
holds. If A is a set of c-additivity for the c-additive measure then, by our definition, there exists only one possible X(A) for the c-additive continuation of the measure defined on A. 6. Lebesgue Continuation oF Measure, DeFined on a Semiring
with a Unit Even though the Jordan continuation allows one to generalise
the concept of measure to quite a wide class of sets, it still remains
insufficient in many cases. Thus, for example, if we take as the initial measure the area, and as the domain of its definition the semiring of rectangles and consider the Jordan continuation of this measure, then even such a comparatively simple set as the set of points, the coordinates of which are rational and satisfy the condition x2 + y2 1, is not Jordan measurable. A generalisation of a a-additive measure defined on some semiring to a class of sets which is maximal in the well known sense can be obtained with the help of the so-called Lebesgue continuation. In this section we shall consider the Lebesgue continuation of a measure defined on a semiring with a unit. The general case will be considered in §7.
The construction given below represents, to a large degree, a repetition, in abstract terms, of the construction of the Lebesgue measure for plane sets given in §1.
40
I. MEASURE THEORY
Let a cr-additive measure m be given on some semiring of sets of all subsets Sm with unit E. We shall define on the system of the set E the functions A) and in the following way. 1. The number
= inf n
where the lower bound is taken over all coverings of the set A by finite or countable systems of sets E Sm, is called the ouler
measure of the set A ç E. 2.
The number
=m(E) is called the inner measure of the set A E. From Theorem 2, §3 it follows that always 3.
(A) ,z*(A).
The set A ç E is called measurable (Lebesgue), if =
If A is measurable, then we shall denote the common value
A) =
,z*( A) by A) and call it the (Lebesgue) measure of the set A. It is obvious that, if A is measurable, then its complement is (
alSo measurable.
Theorem 2, §5 immediately implies that for any cr-additive of the measure m the inequality
continuation
0 there exists a
B E 9?(Sm) such that B)
In §1 these statements were proved for the plane Lebesgue measure (Theorems 3—5, §1). The proofs given there can be carried over word for word to the general case considered here,
therefore we shall not repeat them.
L MEASURE THEORY
42
Iheorem 4. The system 9)1 of all measurable sets is a ring. ProofS Since
it is always true that A1 Ii A2 = A1\(A1\A2)
and A1 U A2
= E\[(E\A1) n (E\A2)],
it suffices to show the following. If A1 E 9)1 and A2 E 9)1, then also
A = A1\A2 E Let A1 and A2 be measurable; then there exist B1 E 9? (Sm) and B2 E 9?(Sm) such that and
B1)
B2)
Setting B = B1\B2 E 9? (Sm) and using the relation (A1'\A2)
(B1\B2) c
(A1
B1) u (A2
B2),
we obtain Since
B) <E. > 0 is arbitrary, this implies that the set A is meas-
urable. Remark. Obviously
E is the unit of the ring 9)1 which, therefore,
is an algebra of sets. Theorem 5.
On the system 9)1 of measurable sets, the function
is additive.
The proof of this theorem is a word for word repetition of the proof of Theorem 7, §1. Theorem 6.
On the system
9)1
of measurable sets, the function
A) is i-additive. Proof.
Let
Ac9)1,
for
iJ.
43
6. LEBESGUE CONTINUATION OF MEASURE
By Theorem 1, (1)
and
by Theorem 5, for any N >
=
implying (2)
Inequalities (1) and (2) yield the assertion of the theorem.
Thus we have established that the function defined on possesses all the properties of a or-additive measure. the system Hence the following definition is verified: 4. One calls the function defined on a system of measurable sets, and coinciding on this system with S,1 = 9)1 *( the outer measure A), the Lebesgue continuation = L( m) of the measure m(A). In §1, considering the plane Lebesgue measure, we have shown that not only the finite but also the countable unions and intersections of measurable sets are also measurable sets. This is true also in the general case i.e., the following theorem holds.
Theorem 7. The system 9)1 of Lebesgue measurable sets is a Borel algebra with unit E. ProofS Since
flAg = and since the complement of a measurable set is measurable, it suffices to show the following. If A1, A2,
then A =
U
•,
belong to 9)1,
also belongs to 9)1. The proof of this statement
given in Theorem 8, §1, for plane sets, is literally preserved also in the general case.
44
I. MEASURE THEORY
Exactly as in the case of a plane Lebesgue measure its or-addiif is a or-additive measure, A2 defined on a B-algebra, A1 A. is a decreasing chain of measurable sets and
tivity implies its continuity, i.e.,
A
=
fl
then
= lim
measurable
•••
A2
and if A1
is an increasing chain of
sets and
A = U then /.L(A)
The
=
proof given in §1 for a plane
measure
(Theorem
10) can be
carried over to the general case.
From the results of and 6 it is easy to deduce that every set A is Jordan measurable is Lebesgue measurable; moreover its Jordan and Lebesgue measures are equal. This immediately implies that the Jordan continuation of a or-additive measure is or-additive. 2) Every set A which is Lebesgue measurable is a set of uniqueness for the initial measure m. Indeed, for any t > 0 there exists for A a 1)
which
B E 9? such that
B)
Whatever the extension X of the
measure m may be,
since
X(B) = m'(B), the continuation of the measure m to 9? =
9?(Sm)
is unique.
Furthermore, X(A
B)
B)
and therefore X(A) — m'(B)I
0 there exist a function W(x) which is continuous on [a, b], and such that
w(x)) < E. In other words, a measurable function can be made continuous, if
one
excludes from consideration its values on a set of arbitrarily small measure.
This property which
Luzin called the C-property
as a definition of a measurable function.
can be taken
II. MEASURABLE FUNCTIONS
54
9. Sequences oF Measurable Functions. Different lypes oF Convergence. Theorems 5 and 7 of the preceding Section show that arithmetic
operations on measurable functions again lead to measurable functions. According to Theorem 2 of §8, the class of measurable
functions, in contrast to the set of continuous functions, is also closed with respect to the operation of going to the limit. For measurable functions it is expedient to introduce, aside from the usual convergence at every point, several other definitions of convergence. These definitions of convergence, their basic properties and the connections between them will be investigated in the present section.
The sequence of functions
defined on some
with measure X, is said to converge
almost everywhere to the
1.
space
function F(x),
if
= F(x)
for
almost all x E X
(1)
(i.e., the set of those points at which (1)
does not hold has measure zero).
Example. The sequence of functions = defined on the segment [0, 1], converges as n —p to the function F( x) o almost everywhere (precisely, everywhere with the exception of
thepointx =
1).
Theorem 2 of §8 admits of the following generalisation. Theorem 1. If the sequence of /.L-measurable functions x) converges to the function F( x) almost everywhere, then F( x) is also
measurable. ProofS
Let A
be that set on which
F(x). By the condition,
on
A by
E\A)
Theorem 2,
= 0. The function F(x) is measurable
§8. Since
on a set of measure zero every
is obviously measurable, F( x) is measurable on E\A; therefore, it is also measurable on the set E. function
55
9. SEQUENCES OF MEASURABLE FUNCTIONS
Exercise. Let the sequence of measurable functions converge almost everywhere to some limit function f( x). Show that x) converges to g( x) if and only if g(x) is equivathe sequence
lent tof(x). The following important theorem, proved by D. F. Egorov, establishes the connection between the concept of convergence almost everywhere and uniform convergence. x) conTheorem 2. Let the sequence of measurable functions verge on E almost everywhere to f(x). Then there exists for any ô > 0 a measurable set Ea C E such that 1) /.L(Ea) > 2) on the set Ea the sequence ProofS
f( x).
By Theorem 1, the function f( x) is measurable. Set =
fl {x:Iff(x) —f(x)I
m
Thus E, for fixed m and n, denotes the set of those points x for which
—f(x)I for
all i n. Let Em
= U
E it is clear that for fixed m,
therefore, because of the fact that the continuous, one can find for any m and any ô such that
Let
us set Ea
fl
measure is > 0 an n0(m)
56
II. MEASURABLE FUNCTIONS
and Show that the the theorem.
so constructed satisfies the requirements of
Let us first prove that on x)) converges the sequence uniformly to the function f(x). This follows immediately from the fact that if x E Ea, then for any m f(x)
i
for
Let us now evaluate the measure of the set E\Ea. For this let us note that for every m, = 0. Indeed, if Xo E E\Em, then there exist arbitrarily large values of i for which f(xo)
i.e., the sequence x0. Since by assumption
not converge to f(x) at the point x)) converges to f(x) almost every-
does
where, 0.
This implies = Hence
=
=
The theorem is thus proved. Definition 2. One says that the sequence of measurable functions converges in measure to the function F(x), if for any a> 0 lim
— F(x)
>
= 0.
Theorems 3 and 4 given below establish the connection between convergence almost everywhere and convergence in measure.
SEQUENCES OF MEASURABLE FUNCTIONS
57
Theorem 3. If a sequence of measurable functions converges almost everywhere to some function F(x), then it converges to the same limit function
F(x) in measure.
1 implies that the limit function F(x) is measurable. does not converge to
ProofS Theorem
Let A be that set (of measure zero) on which F(x). Let, moreover,
= {x:Ifk(x) — F(x)I >
=
M =
It is clear that all these sets are measurable. Since Ri(a)
J
J
we have, because the measure is continuous,
for n—p Let us now check
Indeed, if xo
that
A1,
McA.
(2)
if
= F(xo),
then, for a given a>
0,
we can find an n such that
— F(xo) and hence x0
i.e., xo
But
=
0,
M.
therefore (2) implies that
(a)) since
n1. < '72, In general, let nk be a number such that Ifflk(x) — f(x)
> We shall show that the sequence we have constructed converges to < '7kg
1(x) almost everywhere. Indeed, let
=
{x:
f(x)
Ekl,
Q
=
59
9. SEQUENCES OF MEASURABLE FUNCTIONS
Since
the continuity of the measure implies On
—+
the other hand, it is clear that 0
for i
Since
—3
io holds.
Xo
{x:
flRk(x)
— f(x)
fflk(xO) — f(xo)
Since,
by assumption,
0 there exists a simple function g(x)
ii. GENERAL DEFINITION AND BASIC PROPERTIES
67
which is integrable over A and satisfies the condition f(x) — g(x)
(2)
For g(x),
fg(x)
=
(3)
moreover, g( x) is integrable Over every set
and the Series (3)
converges absolutely. This last fact and the estimate (2) imply that f( x) is also integrable over every and f(x)
—
f
ff(x) which
g(x)
—
f
g(x)
together with (3) yields the absolute convergence of the
series
f(
and
the estimate
— Since
>
0
is arbitrary,
ff(x) Corollary.
f(x)
= IA f(x)
If f(x) is integrable over A, then f( x)
is also integrable
ç A. Theorem 7. If the function p(x) is integrable over A and f( x)
over any A'
x), then f( x) is also integrable over A. Proof.
If f(x) and
simple functions, then A can be
are
represented as a union of a finite or countable number of sets, on are constant: each of whichf(x) and
1(x) =
=
where
III. LEBESGIJE INTEGRAL
68
Since
x) is integrable, we have
m}.
=
By Theorem 6 of the preceding section
(*)
= kfAk and the series (*) converges absolutely.
Here
f
=
Bm
f
km Ak
From the convergence of the series (* ) there follows the existence of an m such that d/2
N, on the set C {
}
—f(x)I
Then
f
—f(x))
+
=
ff(x)
—
f
—f(x))
=
12. LIMITING PROCESSES UNDER LEBESGUE INTEGRAL SIGN
Corollary.
If
M and
ffn(x)
71
then
1(x)
Remark In as much as the values which the function takes on
on the set of measure zero do not influence the value of the integral, it suffices to assume in Theorem 1 that converges to f( x) almost everywhere. }
Theorem 2. Let, on the set A,
where
greater
are integrable and their integrals are not
the functions
than a certain constant:
K. Then almost everywhere on A the limit
f(x) = f(
tim
x) is integrable on A, and 1(x)
On a set on which the limit (1) does not exist, the function f(x) can be given arbitrarily, for example assuming on this set
f(x) = 0. Proof. Let us assume f'( x) 0, since the general case can be easily reduced to this by going over to the functions —fi(x).
Let
us consider the set
= {x:x It is easy to see that
= fl
—p
E
where
U
= {x:x
oc}.
E
> r}.
III. LEBESGUE INTEGRAL
72
By Tchebischev's inequalities (Theorem 9, §11), < Since
•..,
C we
have
K/r, but from the fact that for any r
(U
cU K/r. Since r is arbitrary
it follows that
=
0.
This proves that the monotone sequence
has a finite limit almost everywhere on A. Let us now set x) = r for those x for which r—
1
f(x)
have (Ba)
=
0.
Since
UAnUUBm, f2}
Our assertion
is proved.
=
0.
0.
91
CHAPTER IV
FUNCTIONS WHICH ARE SQUARE INTEGRABLE One
of the most important spaces among the various linear
normed spaces which are encountered in functional analysis is the
Hilbert Space. Its name is due to the German mathematician D. Hilbert who introduced this space in connection with investigations in the theory of integral equations. It is a natural infinite analogue of the n-dimensional Euclidean space. We have already made acquaintance with one of the possible generalisations of Hilbert space in Chapter III of Volume I, it is the space 12 whose elements are sequences of numbers
•''),
x = (x1, x2,
which satisfy the relation
The concept of the Lebesgue integral allows us to introduce another, in some cases more convenient, realisation of the same space—the space of square integrable functions. In this chapter we shall consider the definition and the basic properties of the space of square integrable functions and establish that it is isometric (with corresponding assumptions about the measure with respect to which the integration is performed) to the space 12. In the next chapter we shall give an axiomatic definition of Hilbert space. 18. The L2 Space Below we shall consider functions f(x), defined on some set R
for which a measure /2(E)
is
given satisfying the condition
92
18. THE L2 SPACE
93
R) < oc. The functions 1(x) are assumed to be measurable and defined almost everywhere on R. We shall not distinguish be-
tween functions which are equivalent on R. Instead of f we shall, for the sake of conciseness, write f. Definition 1. One
says that the function f(x) is square integrable
(or summable) over R if the integral
112(x) exists (i.e., is finite). The set of all functions which are
square
integrable over R we shall denote by L2. We shall now establish the basic properties of such functions. Theorem 1.
The product of two square integrable functions is an
integrable function.
The proof follows immediately from the inequality 12(x) +g2(x) 2
-
integral.
and the properties of the Lebesgue
Corollary. Every
square integrable function f(x) is integrable.
Indeed, it suffices to set g( x)
1
in Theorem 1.
Theorem 2. The sum of two L2 functions
is also in L2.
Proof. Indeed, (f(x)
and,
+ g(x))2
12(x)
+
+ g2(x),
by Theorem 1, each of the three functions on the right is
summable. Theorem 3. If
af(x) E
L2.
f(x)
E
L2
and a
is
an arbitrary number, then
94
IV. FUNCTIONS WHICH ARE SQUARE INTEGRABLE
Proof.
1ff E L2,
then
=
f
a2ff2(x)
0. If n is sufficiently large, then f(x)
f This
inequality immediately yields the assertion of the theorem.
From Theorem 1 it follows that if one can approximate any function f E L2 to an arbitrary degree of accuracy by functions L2 in the sense of uniform convergence, then one can E M use them to approximate any function from L2 also in the sense of convergence in the mean. Hence one can approximate any function f E L2 to any degree of accuracy by simple functions belonging to L2.
98
IV. FUNCTIONS WHICH ARE SQUARE INTEGRABLE
We Shall Show that any Simple function f E L2, and therefore,
also in general any function from L2, can be approximated as closely as desired by simple functions which take on only a finite number of values.
•'', Yn,
Letf(x) take on the values Since
on the sets E1, •'',
f2 is integrable, the series
ff2(x) converges. Let us select a number N in such a way that < E, n>N
and set f(x)
for x E
i < N,
for
i>N.
fN(x) = 0
Then we have
f (f(x) the functions IN, which take on a finite number of values, approximate the function f as closely as required. Let R be a metric space having a measure which satisfies the following condition (satisfied in all cases of practical interest): i.e.,
all open and all closed sets in R are measurable and for any
MCR,
= inf MCG
(*)
the lower bound is taken over all open sets G containing M. Then the following theorem holds. where
Theorem 2. The set of all continuous functions is complete in L2. Proof. By what has been said above it suffices to show that every simple function which takes on a finite number of values is a limit, in the sense of mean convergence, of continuous functions. Furthermore, since every simple function which takes on a
19
99
MEAN CONVERGENCE
finite number of values is a linear combination of characteristic functions XM(X) of measurable sets, it suffices to give the proof
for these latter ones Let M be a measurable set in the metric space R. Then, from condition (*) it immediately follows that for any > 0 one can find a closed set FM and an open set Gftf for which
CMC
and
—
/2(FM)
0,
0.
we get c1
in the space L2 there exists an every••', then any orthonormal where countable dense set f', f2, •, Let us further note that, if
system of functions {
Indeed, let a
}
is at most countable.
$, then
-
=
For each a let us select from our everywhere dense set an such a way that
if a It is clear that countable, the set of the
in
is j3, and, since the set of all the themselves is also not more than
countable. In studying finite-dimensional spaces an important role is played by the concept of the orthogonal normalised basis, i.e., the orthogonal system of unit vectors, the linear closure of which coincIdes with the whole space. In the infinite case the analogue of such a basis is the complete orthonormal system of functions,
i.e., a system (p1, (P2,
such 1)
2)
,
(Pa,
that ((pj,
(P/c)
(P2,
= •,
= 142.
Above we have given examples of complete orthonormal systems of functions on the segments [— and [—1, 1]. The existence of a complete orthonormal system of functions in any
separable space L2 follows from the following theorem:
108
IV. FUNCTIONS WHICH ARE SQUARE INTEGRABLE
Theorem. Let the system of functions
(6)
11,12,
be linearly independent. Then there exists a system of functions (7)
•, 'Pn,
'P1, 'P2,
satisfying the following conditions:
1) the system (7) is orthonormal; 2) every function p,, is a linear combination of functions , fi, f2, = + + + 0;
where
3) every function 'P2,
,
is a linear combination of functions
(pa:
= where
+
+
+
0.
Every function of the system (2) is uniquely determined (up to the sign) by conditions 1 )—3).
is uniquely determined (up to Sign) by the conditions of the theorem. Indeed, Proof. The
function
(p1
((p1,
= aiifi,
(p1) = a112(fi,fi) =
1,
which yields b11 =
= V(11,f')
=
Let the functions (pk (k < n) satisfying conditions 1 )—3) be
already found. Then = where (ha, 'Pk) = 0
can be represented in the form
+ for k < n.
+
+
_____ 109
22. FOURIER SERIES ON ORTHOGONAL SYSTEMS
= 0 would > 0 (the assumption (ha, Obviously (ha, have resulted in a contradiction to the linear independence of the system (6)). Let us set = Then we have
i
Ck2,
where
Ck
= (g,
23. THE ISOMORPHISM OF SPACES L2 AND 12
115
then, by the Riesz-Fischer theorem, there exists a function I E '12, Such that (1, c',c) = The function f —
g
Ck,
(f,
=
Ck2. k=1
is orthogonal to all
(1'f) = it cannot be equivalent to 'I' (x)
By the inequality
< (g, g),
0. The theorem is proved.
23. The Isomorphism oF the Spaces L2 and 12 From
the Riesz-Fischer theorem immediately follows the
following important Theorem. The space L2 is isomorphic* to the space
12.
Proof. Let us select in '12 an arbitrary complete orthonormal system { } and let us put into correspondence with every function f E L2 a sequence c1, c2, ..•, •.. of its Fourier coefficients
< oc, (ci, c2, •, on this system. Since ) is therefore some element of 12. Conversely, by virtue of the RieszFischer theorem, there corresponds to every element (c1, c2, •••) of 12 some function f of L2, which has the numbers c1, c2, •••, as its Fourier coefficients. The correspondence between the elements of 12 and L2 which we have established is one * Two Euclidean spaces R and R' are said to be isomorphic if one can establish a one to one correspondence between their elements in such a way that if
X4—+X',
then 1)
y4—+y',
X+y4—4X'+y',
2)
ax
3)
(x, y) = (x', y').
4—* ax',
It is clear that two isomorphic Euclidean spaces, considered just as metric spaces, are measurable.
IV. FUNCTIONS WHICH ARE SQLARE
to
one. Moreover, if f
(1)
•,
and f (2) 4—+ (ci(2),
),
,
then f
(1)
..f f (2) 4—* (c1(')
—4—
•,
—4—
—4—
and •,
kf
i.e., the Sum goes over into a sum, and the product by a number
goes over into the product of a corresponding element by the same number. Finally, the Parseval equation yields that (f (l),f (2)) = Indeed,
(1)
from the fact that (f(l),
f(l)) =
(f(2), f(2)) =
and (1
+ f (2), f =
+ f (2)) = (f (i), f + +
+ 2(1
+
2
f (2)) +
(f
f (2))
+
(1) follows. Thus the correspondence which we have established between the elements of the space L2 and 12 is indeed an isomorphism. The theorem is proved.
This theorem means that one can consider 12 to be a "coordinate notation" of the space L2. It allows us to carry over facts we have established earlier for 12 to L2. For example, we have shown in
Chapter III of Volume I that every linear functional in 12
has
the form
= (x,y), where y is some element of 12, uniquely defined by the functional From this, and from the isomorphism we have established be-
23. THE ISOMORPHISM OF SPACES L2 AND
tween '12 and
12,
it
117
follows that every linear functional in 12
has
the form = (1, g) =
f f(x)g(x)
where g(x) is some fixed function from L2. In §24, Volume I it was established that 12 = 12. This, and the theorem just obtained, yield L2 = '12. The isomorphism established between L2 and 12 is closely related to certain problems in quantum mechanics. At first, quantum mechanics appeared in the form of two externally different theories: Heisenberg's "matrix mechanics" and "wave mechanics". Later, established the equivalence of the two theories. From the purely mathematical point of view the difference between the two theories is reduced to the
fact that in constructing the corresponding mathematical apparatus Heisenberg used the space 12 and
the space L2.
CHAPTER V
THE ABSTRACT HILBERT SPACE. INTEGRAL EQUATIONS WITH A SYMMETRIC KERNEL. In the preceding chapter we have established that the spaces
the case of a separable measure) and 12 are isomorphic, i.e., in essence represent two different realisations of one and the same space. This space, usually called Hilbert space, plays an important part in analysis and its applications. Often it is useful not to tie oneself down ahead of time by one or another realisation of this space, but to define it axiomatically, as is done for example in linear algebra with respect to the n-dimensional L2 (in
Euclidean space. 24. Abstract Hilbert Space Delinition 1. The set H of elements f, g, h, of an arbitrary nature is called (abstract) Hilbert space, if the following con-
ditipns are satisfied:
I. H is a linear space.
II. A scalar product of elements is defined in H, i.e., to each pair of elements f and g there corresponds a number U', g) such that, 1)
(f,g) = (g,f),
2)
(af,
3)
(1' +12, g) = (1',
4) In
g)
= a(f, g),
(f,f)>o
g) + (12, g),
if
other words, conditions I and II mean that H is a Euclidean 118
24. ABSTRACT HILBERT SPACE
space.
The number
119
= V' (1, f) is called the norm of the
f
element f.
III. The space H is complete in the sense of the p (f, g) = Hf — gil metric. IV. The space H is infinite-dimensional, i.e., for any natural n, one can find in H, n linearly independent vectors. V. The space H is separable*, i.e., there exists in it a countable everywhere dense set.
It is easy to give examples of spaces which satisfy all the enumerated axioms. Such is the space /2 which we considered in Chapter II of Volume I. Indeed, it is a Euclidean space, which is infinitely dimensional since, for example, its elements = (1, 0, 0, ••, 0, e2 = (0, 1,0, e.3 = (0, 0, 1,
•, 0,
•••),
0,
),
,
are linearly independent; the fact that it is complete and separable was proved in Chapter II, §9 and §13 of Volume I. The space L2 of functions which are square integrable with respect to some separable measure which is equivalent to 12 also satisfies the same axioms. The following statement holds: all Hubert spaces are isomorphic.
To prove this fact it obviously suffices to establish that every Hilbert space is isomorphic to the coordinate space 12. This last assertion is proved in essence by the same considerations as the isomorphism of the spaces L2 and 12, namely: 1) One can carry over to Hilbert space without any changes those definitions of orthogonality, closure and completeness which were introduced for the elements of the space L2 in §21.
2) Selecting in the Hilbert space H a countable everywhere dense set and applying to it the process of orthogonalisation described (for L2) in §21, we construct in H a complete ortho-
normal system of elements, i.e., the system h1, h2, *
.
,
•,
Condition V is often omitted, i.e., one considers nonseparable Hilbert spaces.
(1)
120
V. THE ABSTRACT HILBERT SPACE. INTEGRAL EQUATIONS
which satisfies the conditions for
fo
a)
(hi, hk) =
for
i=k,
b) the linear combinations of elements of the system (1) are everywhere dense in H.
3) Let F be an arbitrary element in H. Set Ck = Then, the series converges and f) for any complete = (J, orthonormal system { } and any element f E H. 4) Let, again, { hk } be some complete orthonormal system of elements in H. Whatever the sequence of numbers c1,c2,
may be which satisfy the condition < OC,
there exists in H an element f E H such that Ck
= (f,
hk),
and
= (1,1). 5) From our statements we can see that one can realise an isomorphic mapping of H onto 12, by setting f 4—+ (c1, c2,
where Ck
= (f,
hk),
and h1,h2,
is an arbitrary complete orthonormal system in H. It is recommended that the details of the proof be completed by the reader himself using the methods of
25. SUBSPACES. ORTHOGONAL COMPLEMENTS
121
25. Subspaces. Orthogonal Complements. Direct Sum.
In correspondence with the general definition of §21, Chapter III, of Volume I we shall call a set L of elements in H having the property: if f, g E L, then af + /3g E L for any numbers a and /3, a linear manifold. A closed linear manifold is called a subs pace. Let us give a few examples of subspaces of Hilbert space.
1. Let h be an arbitrary element in H. The set of all elements f E H which are orthogonal to h forms a subspace in H. 2. Let H be realised as 12, i.e., let its elements be sequences (xi, x2, ..., < oc. The ele...) of numbers such that ments which satisfy the condition x1 = x2 form a subspace. 3. Let H be realised as the space L2 of all square integrable functions on some segment [a, b] and let a < c < b. Let us denote the set of all functions from H which are identically equal to H. zero on the segment [a, c] by D moreover, Ha = H, Hb = (0). Thus If c1 < c2, then we obtain a continuum of subsets of H which are contained in each other. Each of these subspaces (of course with the exception of Hb) is infinite-dimensional and isomorphic to the whole space H.
It is recommended that the reader check that the sets of vectors given in examples 1—3 are indeed subspaces.
Every subspace of a Hilbert space is either a finite-dimensional Euclidean space or itself a Hilbert space. Indeed, it is obvious that axioms 1—111 are fulfilled for each such subspace, and the following lemma shows that axiom IV also holds. Lemma. From the existence of a countable everywhere dense set in a metric space R there follows the existence of a countable everywhere dense set in an arbitrary subs pace R' of R. Proof. Let E1,E2,
be
a countable everywhere dense set in R. Let, furthermore, = inf
n),
122
V. THE ABSTRACT HILBERT SPACE. INTEGRAL EQUATIONS
E R' be Such that
and
0, one can find a ö > 0 such that p (f(xi), 1(x2)) < for all x1, x2 for which p(xi, x2) < & The following assertion holds: Every continuous mapping of a corn pactum into a corn pactum is uniformly
convergenL It is proved by the same method as the uniform continuity of a function which is continuous on a segment." * The additions and corrections listed here are in terms of the page numbering of the English translation of Volume I, published by Graylock, Pre8s.
138
139
ADDITIONS AND CORRECTIONS
(9) Page 61. After line 17 from above, instead of the word "Proof" it Should read: "Let us first prove the necessity. If D is compact, then there Since each of the mappings
exists in D a finite E/3 net 11,12,
is continuous, it is uniformly continuous, therefore one can find ö > 0 such that
i= only
1,2, .. ., n,
if
p(xi,
x2)