# A.N._ Fomin, S.V. Kolmogorov - Measure, Lebesgue Integrals, and Hilbert Space -Academic Press, New York (1961)

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Measure, Lebesgue Integra's, cnd Hilbert Space A. N. KOIMOCOROV AND S. V. FOMEN

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Measure, Lebesgue Integrals, and Hilbert 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 06

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COPYRIGHT C 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 FIFTH AVENUE NEW YORK 3, N. Y. United Kingdom Edition Published by ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY 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

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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 suffcient 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 51, 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. FOMIN

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CONTENTS x Direct Products of Systems of Sets and Measures. 14. Expressing the Plane Measure by the Integral of a 15. Linear Measure and the Geometric Definition of the Lebesgue Integral.. Fubini's Theorem. 16. The Integral as a Set Function 17. CHAPTER IV Functions Which Are Square Integrable 18. The L2 space. ... Mean Convergence. Sets in •1.72 which are Everywhere 19. Complete. L2 Spaces with a Countable Basis.. 20. Orthogonal Systems of Functions. Orthogonalisation . 21. Fourier Series on Orthogonal Systems. 22. Riesz-Fischer Theorem The Isomorphism of the Spaces L2 and 12 .. 23. CHAPTER V The Abstract Hilbert Space. Integral Equations with a Symmetric Kernel Abstract Hilbert Space ..... 24. Subspaces. Orthogonal Complements. Direct Sum. 25. Linear and Bilinear Functionals in Hilbert Space.. 26. Completely Continous Self-Adjoint Operators in H. 27. Linear Equations with Completely Continuous Operators. Integral Equations with a Symmetric Kernel. 29. ADDITIONS AND CORRECTIONS TO VOLUME 1. SUBJECT INDEX. . 78 82 86 90 92 97 100 109 115 118 121 126 129 134 135 138 143

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AcB AGB AAB B(f, g) ö-ring öik II fil x(A) 17 MOM' sn(A) List of Symbols set linear operator adjoint to (the linear operator) A , p. 129. B is a proper subset of A. A is a subset of B. element a is a member of set A. difference, complement of B with respect to A symmetric difference of sets A and B defined by the expression A A B = (A B) u (B A) = bilinear functional, p. 128. Borel algebra containing e, p. 25. P. 24 Kronecker delta, p. 107. arbitrary positive number. null set norm of f, p. 95. scalar product of functions in 112, p. 94. continuation of measure m defined for A, closure of the set M, p. 105. direct sum of spaces M and M', p. 124. system of sets A, p. 20. 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. p. 39.

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xli c-ring {\$1, 32, LIST OF SYMBOLS outer measure of set A, p. 31. inner measure of set A, p. 31. system of sets, p. 25. system of sets, p. 20 direct product of sets XI, X2,• • • quadratic form, p. 129. ring, p. 19. system of sets, p. 19. P. 24 xn, p. 78. set of elements with property f(x) < c, p. 49. union intersection, product. set of elements \$1, \$2, • • •

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CHAPTER I MEASURE THEORY The concept of the measure g(A) of a set A is a natural gener- alisation of the following concepts. 1) the length I(A) of a segment A; 2) the area S(F) of a plane figure F; 3) the volume V (G) of a figure G in space; 4) the increment p (b) — p(a) of a non-decreasing function p (t) on the half open line interval [a, b) ; 5) the integral of a non-negative function taken over some line, plane surface or space region, etc. The concept of the measure of a set, which originated in the theory of functions of a real variable, has subsequently found applications in probability theory, the theory of dynamic systems, functional analysis and other fields of mathematics. In the first section of this chapter we shall introduce the notion of measure for sets in a plane, starting from the concept of the area of a rectangle. The general theory of measures will be given in "3—7. The reader will however easily notice that all arguments used in Sl are of a general character and can be re-phrased for the abstract theory without essential changes. 1. Measure of Plane Sets Let us consider a system e of sets in the (x, y)-plane, each of which is given by one of the inequalities of the form b, b, 1

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2 1. MEASURE THEORY together with one of the inequalities where a, b, c and d are arbitrary numbers. The sets belonging to this system we shall call "rectangles". The closed rectangle, given by the inequalities is the usual rectangle (including the boundary) if a < b and c < d, or a segment if a = b and c < d, or a point if a = b and c = d. The open rectangle is, depending on the relations among a, b, c and d, respectively, a rectangle without boundaries, or the empty set. Each of the rectangles of the other types (let us call them half-open) is either a real rectangle without one, two or three sides, or an inter- val, or a half-interval, or, finally, an empty set. We shall define for each of the rectangles a measure correspond- ing to the concept of area, well known from elementary geometry, in the following way: a) the measure of an empty set is equal to zero; b) the measure of a non-empty rectangle (closed, open or half-open) given by the numbers a, b, c and d is equal to In this way, to each rectangle P there corresponds a number m (P)—the measure of this rectangle; moreover the following conditions are obviously satisfied: 1) the measure m (P) takes on real non-negative values; 2) the measure is additive, i.e., if P = U Pk and Pi n Pk = Ø

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1. MEASURE for i k, then OF PLANE SETS E m(Pk). 3 Our task now is to generalise the measure m (P) , which up to now has been defined only for rectangles, to a wider class of measures, preserving the properties 1) and 2). The first step in this direction consists of generalising the con- cept of measure to so called elementary sets. We shall call a plane set elementary if it can be represented, at least in one way, as a union of a finite number of pairwise non-intersecting rectangles. For what follows we shall use the following Theorem 1. The union, intersection, difference and symmetric difference of two elementary sets is also an elementary set. Proof. It is clear that the intersection of two rectangles is again a rectangle. Therefore, if A = Upk and B = U Qi are two elementary sets, then is also an elementary set. The difference of two rectangles is, as is easily checked, an ele- mentary set. Hence, taking away from the rectangle some ele- mentary set we again obtain an elementary set (as the intersection of elementary sets). Now let A and B be two elementary sets. One can obviously find a rectangle P containing each of them. Then A u B = A) n (PXB)) is an elementary set by what has been said above. From this and the equalities

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1. MEASURE OF PLANE SETS satisfies the conditions 2 5 (For this it is suffcient to replace each of the k rectangles P, which form A by the closed rectangle which is wholly contained in it and having an area larger than m(Pi) —e/2k+1.) Furthermore, for each An one can find an open elementary set Än containing An and satisfying the condition m'(Än) m' (An) + 2n+f It is clear that Ä C U Än. From we can (by the Borel-Lebesgue lemma) select a sys- An which covers Ä. Here moreover, obviously m'(Ä) Em'(ÄnD (since otherwise Ä could be covered by a finite number of rec- tangles having an added area which is less than m'(Ä), which is obviously impossible) . Therefore, m'(A) m'(Ä) 4- —e + + 2 + -e = Em'(An) + e, 2n+1 yielding ( * ) , since e is arbitrary and positive. The set of elementary sets does not exhaust all those sets which were considered in elementary geometry and in classical analysis. Therefore it is natural to ask the question how to generalise the concept of a measure to a class of sets which is wider than the finite combinations of rectangles with sides parallel to the co- ordinate axes, preserving its basic properties.

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6 1. MEASURE THEORY The final solution of this problem was given by H. Lebesgue at the beginning of the twentieth century. In presenting the theory of Lebesgue's measure we will have to consider not only finite but also infinite combinations of rect- angles. In order to avoid dealing with infinite values for measures we shall limit ourselves to sets which fully belong to the square On the set of all these sets we shall define two functions (A) and (A) in the following way. Definition 1. We shall call the number inf m (Pk) A c UP.k the outer measure g* (A) of the set A; the lower bound is taken over all possible coverings of the set A by finite or countable rectangles. Definition 2. We call the number the inner measure (A) of the set A. It is easy to see that always Indeed, suppose that for some A C E i.e., Then, by definition of the exact lower bound, one can find systems of rectangles such that The union of the and E m(Pi) systems covering A and E A, respectively, + Em(Qk) < 1. and we shall denote by

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1. ; we obtain MEASURE and OF PLANE SETS E m(Ri) , 7 which contradicts Theorem 2. Definition 3. We call a set A measurable (in the sense of Lebesgue) , if The common value "(A) of the outer and inner measure for a countable set A is called its Lebesgue measure. Let us find the basic properties of the Lebesgue measure and of countable sets. Theorem 3. If A C U An, where An is a finite or countable system of sets, then Proof. By the definition of the outer measure, we can find for each An a system of rectangles, finite or countable, such that An C U Pnk and Em(Pnk) + ± where > 0 is selected arbitrarily. Then A C U U Pnk, and (A) E Em(Pnk) E (An) e. Since e > 0 is arbitrary, this establishes the theorem. We have already introduced above the concept of measure for sets which we called elementary. The theorem below shows that for elementary sets Definition 3 leads to exactly the same result.

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8 1. MEASURE THEORY Theorem 4. Elementary sets are measurable, and for them the Lebesgue measure coincides with the measure m'(A) constructed above. • Pk are the pair- Proof. If A is an elementary set and Pl, P 2, wise non-intersecting rectangles comprising it, then, by definition, = Em(Pi). Since the rectangles Pi cover all of A, E m(Pi) = m'(A). But, if is an arbitrary finite or countable system of rec- tangles covering A, then, by Theorem 2, m' (A) E m(Qi). Hence m'(A) Therefore, m'(A) = Since E A is also an elementary set, = A). But = 1 — m'(A) and = 1 — yielding Therefore From the result obtained we see that Theorem 2 is a special case of Theorem 3. Theorem 5. In order that the set A be measurable, it is necessary and suffcient that the following condition be satisfied: for any e > 0 there exists an elementary set B, such that, Thus those sets and only those sets are measurable which can be "approximated with an arbitrary degree of accuracy" by elementary sets. For the proof of Theorem 5 we shall need the following Lemma. For two arbitrary sets A and B

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1. MEASURE Proof of the lemma. Since we have OF PLANE SETS 9 This implies the lemma if If then the lemma follows from the inequality which can be established in an analagous manner. Proof of Theorem 5. Suffciency. Let us assume that for any e > 0 there exists an elementary set B, such that Then and since we have analogously that (1) (2) From the inequalities (1) and (2) we have, taking into account that + —II < 2e, and, since e > 0 is arbitrary, i.e., the set A is measurable. Necessity. Let A be measurable, i.e., let

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1. MEASURE OF PLANE SETS But by assumption, Em(Bn) + Em(Cn) + + 3 Subtracting (3) from (4) we have i.e., 3 Therefore, 11 (4) 3' Hence, if A is measurable, there exists for any arbitrary e > 0, an elementary set B such that A B) < e. Theorem 5 is thus established. Theorem 6. The union and intersection of a finite number of measurable sets are measurable sets. Proof. It is clear that it suffces to give the proof for two sets. Let A1 and .42 be measurable sets. Then, for any e > 0, one can find elementary sets Bl and B2 such that, 2 Since (A1 U AD A (BI UB2) C (A1 A BD u (A2 A B2), we have U AD A (BI UB2)] A BI) + A BD < e. (5) Bl u B2 is an elementary set; hence, by Theorem 4, the set A1 u A 2 is measurable. But, just by the definition of measurability, if A is measurable, then EXA is also measurable; hence the fact that the intersection of two sets is measurable follows from the relation A1 n = U (EXA2)].

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1. MEASURE OF PLANE SETS Since 6: can be made arbitrarily small, The reverse inequality 13 is always true for A = A1 u 142, therefore, we finally obtain Since A1, A 2 and A are measurable, we can replace by g. The theorem is thus established. Theorem 8. The union and the intersection of a countable number of measurable sets are measurable sets. Proof. Let A1, .42, • be a countable system of measurable sets and A = U An. Let us set A 'n = An XU Ak. It is clear that A = U A'n, where the sets An' are pairwise non-intersecting. By Theorem 6 and its corollary, all the sets A'n are measurable. According to Theorems 7 and 3, for any n Therefore the series converges and hence for any e > 0 one can find an N such that (11) 2 Since the set C = U A'n is measurable (being a union of a finite

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1. MEASURE THEORY number of measurable sets), we can find an elementary set B such that 2 Since (12) A ABC (C B) U U A'n (11) and (12) yield Because of Theorem 5 this implies that the set A is measurable. Since the complement of a measurable set is itself measurable, the statement of the theorem concerning intersections follows from the equality Theorem 8 is a generalisation of Theorem 6. The following Theorem is a corresponding generalisation of Theorem 7. Theorem 9. If is a sequenæ of pairwise non-intersecting measurable sets, and A = U An, then Proof. By Theorem 7, for any N, Taking the limit as N —+ n , we have On the other hand, according to Theorem 3, (13) Inequalities (13) and (14) yield the assertion of the Theorem.

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1. MEASURE OF PLANE SETS 15 The property of measures established in Theorem 9 is called its countable additivity, or c-additivity. An immediate corollary of c-additivity is the following property of measures, called con- tinuity. Theorem 10. If A1 A 2 is a sequence of measurable sets, contained in each other, and A = n An, then g(A) = lim pan). It suffces, obviously, to consider the case A = Ø, since the general case can be reduced to it by replacing An by An A. Then and Therefore and g(An) = (15) (16) since the series (15) converges, its remainder term (16) tends to Thus, zero as ( n ) for n —Y , which was to be shown. Corollary. If A1 C A 2 C urable sets, and then is an increasing sequence of meas- A =UAn, = lim g(An).

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16 1. MEASURE THEORY For the proof it suffices to go over from the sets An to their complements and to use Theorem 10. Thus we have generalised the concept of a measure from ele- mentary sets to a wider class of sets, called measurable sets, which are closed with respect to the operations of countable unions and intersections. The measure constructed is a-additive on this class of sets. Let us make a few final remarks. 1. The theorems we have derived allow us to obtain an idea of the set of all Lebesgue measurable sets. Since every open set belonging to E can be represented as a union of a finite or countable number of open rectangles, i.e., measurable sets, Theorem 8 implies that all open sets are meas- urable. Closed sets are complements of open sets and consequently are also measurable. According to Theorem 8 also all those sets must be measurable which can be obtained from open or closed sets by a finite or countable number of operations of countable unions and intersections. One can show, however, that these sets do not exhaust the set of all Lebesgue measurable sets. 2. We have considered above only those plane sets which are subsets of the unit square E =. It is not diffcult to remove this restriction, e.g., by the following method. Repre- senting the whole plane as a sum of squares Enm = (m, n integers), we shall say that the — A n Enm with plane set A is measurable if its intersection Anm each of these squares is measurable, and if the series E g(Anm) converges. Here we assume by definition that = P (Anm). All the properties of measures established above can obviously be carried over to this case. 3. In this section we have given the construction of Lebesgue measures for plane sets. Analogously Lebesgue measures may be constructed on a line, in a space of three dimensions or, in general, in a space of n dimensions. In each of these cases the measure is constructed by the same method: proceeding from a measure de-

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2. SYSTEMS OF SETS 2. Systems of Sets 19 Before proceeding to the general theory of sets we shall first give some information concerning systems of sets, which sup- plements the elements of set theory discussed in Chapter I of Volume I. We shall call any set, the elements of which are again certain sets, a system of sets. As a rule we shall consider systems of sets, each of which is a subset of some fixed set X. We shall usually denote systems of sets by Gothic letters. Of basic interest to us will be systems of sets satisfying, with respect to the operations introduced in Chapter I, Sl of Volume I, certain definite condi- tions of closure. Definition 1. A non-empty system of sets is called a ring if it satisfies the conditions that* A G and B G implies that the sets A A B and A n B belong to jt. For any A and B and hence A G and B G also imply that the sets A u B and A B belong to jt. Thus a ring of sets is a system which is in- variant with respect to the operations of union and intersection, subtraction and the formation of a symmetric difference. Ob- viously the ring is also invariant with respect to the formation of any finite number of unions and intersections of the form Any ring contains the empty set zf, since always A \ A = Ø. The system consisting of only the empty set is the smallest possible ring of sets. A set E is called the unit of the system of sets e, if it belongs to e and if, for any A G e, the equation holds.

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20 1. MEASURE THEORY Hence the unit of the system of sets is simply the maximal set of this system, containing all other sets which belong to Z. A ring of sets with a unit is called an algebra of sets. Examples. 1) For any set A the system YJ2(A) of all its subsets is an algebra of sets with the unit E = A. 2) For any non-empty set A the system { Ø, A consisting of the set A and the empty set Ø, forms an algebra of sets with the unit E = A. 3) The system of all finite subsets of an arbitrary set A forms a ring of sets. This ring is then and only then an algebra if the set A itself is finite. 4) The system of all bounded subsets of the line forms a ring of sets which does not contain .a unit. From the definition of a ring of sets there immediately follows Theorem 1. The intersextion = n of any set of rings is also a ring. Let us establish the following simple result which will however be important in the subsequent work. Theorem 2. For any non-empty system of sets there exists one and only one ring containing e and contained in an arbi- trary ring which contains e. Proof. It is easy to see that the ring is uniquely defined by the system e. To prove its existence let us consider the union X = U A of all sets A contained in e and the ring m (X) of all AEE subsets of the set X. Let 2 be the set of all rings of sets contained in (X) and containing e. The intersection of all these rings will obviously be the required ring Yt(€).

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2. SYSTEMS OF SETS 21 Indeed, whatever the m * containing e, the intersection = n (X) is å ring of 2 and, therefore, i.e., really satisfies the requirement of being minimal. m (C) is called the minimal ring over the system Z. The actual construction of the ring Yt(Z) for a given system is, generally speaking, fairly complicated. However, it becomes quite straightforward in the important special case when the system is a "semiring". Definition 2. A system of sets is called a semiring if it contains the empty set, is closed with respect to the operation of intersec- tion, and has the property that if A and A1 C A belong to e, then A can be represented in the form A = U Ak., where the Ak are pairwise non-intersecting sets of e, the first of which is the given set A1. In the following pages we shall call each system of non-inter- secting sets A1, A2, • • An, the union of which is the given set A, a finite decomposition of the set A. Every ring of sets Yt is a semiring since if A and A1 C A belong to m, then the decomposition where takes place. As an example of a semiring which is not a ring of sets we can take the set of all intervals (a, b), segments [a, b] and half seg- ments (a, b] and [a, b) on the.real axis. * In order to find out how the ring of sets which is minimal over a given semiring is constructed, let us establish some properties of semirings of sets. * Here, of course, the intervals include the "empty" interval (a, a) and the segments consisting of one point [a, a].

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3. MEASURES ON SEMIRINGS intersecting and all belong to A, then 27 Proof. Since s g is a semiring there exists, according to Lemma 1, p, the decomposition where the first n sets coincide with the given sets A1, A 2, Since the measure of any set is non-negative, Theorem Q. If A r, A 2, • • An, A belong to Sy and A C U Ak, then Proof. By Lemma 2, p, one can find a system of pairwise non- Bt from Sy, such that each of the intersecting sets Bl, B2, • • • sets A1, A2, • • • An, A can be represented as a union of some of the sets Ba.• 8EBf0 Moreover, each index s G Mo also belongs to some member of Mk. Therefore each term of the sum E g(B8) enters once, or at most a few times, into the double sum

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28 This yields 1. MEASURE THEORY In particular, for n = 1, we have the Corollary. If A C A', then g(A) g(A'). Definition 2. The measure (A) is called the continuation of the measure m(A) if Sm C Sg and if, for every A G Sm, the equality holds. The main aim of the present section is the proof of the follow- ing proposition. Theorem 3. Every measure m(A) has one and only one continua- tion P(A), having as its domain of definition the ring Yt(Sm). Proof. For each set A G m(Sm) there exists a decomposition A=UBk, Bk€sm, (Theorem 3, P). Let us assume by definition = Em(Bk). (1) (2) It is easy to see that the quantity P(A), given by equation (2), does not depend on the selection of the decomposition (1). Indeed, let us consider the two decompositions A = ÜBt- Since all intersections Bi n Ci belong to Sm, the additivity of measures, E m(Bi) = EEm(Bi n CD = we have, because of E m(Ci),

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4. CONTINUATIONS OF JORDAN MEASURES 29 which was to be proved. The fact that the function g(A), given by equation (2) is non-negative and additive is obvious. Hence the existence of a continuation P(A) of the measure m(A) is shown. To show its uniqueness let us note that, by definition of continuation, if A = U Bk, where Bk are non-intersecting sets from S,n, then for any continuation of the measure m onto the ring m (S,n) = = Em(Bk) i.e., the measure coincides with the measure l.' defined by equation (2) . The theorem is proved. The connection between this theorem and the constructions of Sl will be completely clear if we note that the set of rectangles in the plane is a semiring, the area of these rectangles is a measure in the sense of Definition 1, and the elementary plane sets form a minimal ring over the semiring of the rectangles. 4. Continuations of Jordan Measures* In the present section we shall consider the general form of that process which in the case of plane figures allows one to generalise from the definition of areas for a finite union of rec- tangles, with sides parallel to the axes of coordinates, to areas of all those figures for which areas are defined by elementary ge- ometry or classical analysis. This extension was given with complete precision by the French mathematician Jordan around 1880. The basic idea of Jordan goes back, incidentally, to the mathematicians of ancient Greece and consists of approximating from the inside and from the outside the "measurable" set A by sets A' and A" to which a measure has already been prescribed, i.e., in such a way that the inclusions are fulfilled. * The concept of a Jordan measure has a definite historical and methodological interest but is not used in this exposition. The reader may omit this section if he wishes.

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34 1. MEASURE THEORY coincide, it is necessary and suffcient that the following condi- tions be satisfied: G sew mr(A) = on m2(A) = gr(A) on The necessity of the condition is obvious. Let us prove their sufficiency. IRtA e sgr and Then there exist A', A" G kS„u, such that ml(A") — 77h(A') < — 3' By the conditions of the theorem, (A') = (A') and (A") = From the definition of the measure it follows that there exist B' G kSm2 and B" G ST, 2 for which, Here and, obviously, and (A — m2(B') and m2(B ) " — P2(A") m2(B") — m2(W) < e. i' 3 Since e > 0 is arbitrary, A G S , and from the relations = m2(B') m2(B") = gt(B") it follows that P2(A) = The theorem is proved. To establish that the Jordan measure in the plane is inde- pendent of the choice of the system of coordinates, one need only

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5. COUNTABLE ADDITIVITY 35 convince oneself that a set which is obtained from an elementary set by a rotation through some angle a is Jordan measurable. It is suggested that the reader do this for himself. If the initial measure is given, not on a ring, but on a semiring, then it is natural to consider as its Jordan continuation the measure j(m) = obtained as a result of a continuation of m to the ring (Sm) and a subsequent continuation. 5. Countable Additivity. General Problem of Continuation of Measures Often one must consider the union of not only a finite, but of a countable number of sets. In this connection the condition of additivity which we have imposed on measures (Definition I, P) turns out to be insuffcient and it is natural to replace it by the stronger requirement of countable additivity. Definition 1. The measure is called countably additive (or c-additive), if, for any sets A, A1, • • A belonging to its domain of definition and satisfying the conditions — for i the equality holds. The plane Lebesgue measure which we constructed in Sl is CT-additive (Theorem 9). An example of a CT-additive measure of a completely different kind can be constructed in the following way. Let

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be an arbitrary countable set, such that The domain consists of all A C X we æt 36 1. MEASURE THEORY n = 12 and let the numbers pn > 0 be subsets of the set X. For each It is easy to check that g(A) is a a-additive measure, where 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 c-additive. Let X be the set of all rational points of the segment [0, 1 J, and Sy 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 u is a semiring. For each such set we put (Aab) This is an additive measure. It is not o--additive because, for example, g(X) = 1 and at the same time X is the union of a countable number of separate points, the measure of each one of which is zero. In this and the two following sections we shall consider a-addi- tive measures and their different a-additive continuations. Theorem 1. If the measure m, defined on some semiring Sm, is countably additive, then the measure = r (m), obtained from it by continuation to the ring is also countably additive. Proof. Let A mcsm), and mcsm),

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5. COUNTABLE ADDITIVITY 37 where B, n - Sm such that for s r. Then there exist sets Ai and Bnt• from 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, P). Let Cnij Bni n A,. It is easy to see that the sets Cnij are pair- wise non-intersecting, and hence, E Em(c .) nil Em(C ) nii , and, by definition of the measure r(m) on (Sm) = Em(Ai), g(Bn) = Em(Bni). Equations (1 (2), (3) and (4) imply g (A) = Therefore, and because of the additivity of the measure m on Sm, we have m(Ai) = m(Bni) = (1) (2) (3) (4) E "(Bn). (The summations over i and j are finite, the series in n converge. ) One could show that a Jordan continuation of a o--additive measure is always radditive; 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 CT-additive measures, Theorem 2 of 53 may be extended to countable coverings. Theorem 2. If the measure ,u is c-additive, and the sets A, A1, , An, • • • belong to s u, then A C U An

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38 I. MEASURE THEORY implies the inequality Éß(An). 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 g is a ring, the sets belong to s u. Since and since the sets Bn are pairwise non-intersecting, Egan). From now on we shall, without special mention, consider only a-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 Yt(Sm) in 53 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 Si(m)), then, for any measure continuing m and defined on A, the value g(A) coincides with the value J (A) of the Joün continuation J = j(m). One can show that the extension of the measure m beyond the boundaries of the system Si(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 measur- able with respect to the measure m, i.e., with the system of sets S

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6. LEBESGUE CONTINUATION OF MEASURE 39 However, if one considers only (Y-additive measures and their continua- tion (radditive), then the system of sets of uniqueness will be, generally speaking, wider. Since it is the case of CT-additive measures that will interest us in the future let us establish Definition 2. The set A is called the set of a-uniqueness for a raddi- tive measure g, if: 1) there exists a c-additive continuation X of the measure m defined for A (i.e., such that A G SD; 2) for two such CT-additive continuations Xl and X2 the equation = X2(A) holds. If A is a set of cadditivity for the a-additive measure g, then, by our definition, there exists only one possible X(A) for the CT-additive continuation of the measure g, 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 insuffcient 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 æt of points, the coordinates of which are rational and satisfy the condition + 1, is not Jordan measurable. A generalisation of a a-additive measure defined on some semi- ring to a class of sets which is maximal in the well known sense can be obtained with the help of the so-called Lebesgue con- tinuation. In this section we shall consider the Lebesgue con- tinuation of a measure defined on a semiring with a unit. The general case will be considered in 57. The construction given below represents, to a large degree, a in abstract terms, of the construction of the Lebesgue measure for plane sets given in Sl.