Young Measures and Compactness in Measure Spaces 9783110280517, 9783110276404

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Liviu C. Florescu · Christiane Godet-Thobie Young Measures and Compactness in Measure Spaces

Liviu C. Florescu Christiane Godet-Thobie

Young Measures and Compactness in Measure Spaces

De Gruyter

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ISBN 978-3-11-027640-4 e-ISBN 978-3-11-028051-7

/LEUDU\RI&RQJUHVV&DWDORJLQJLQ3XEOLFDWLRQ'DWD A CIP catalog record for this book has been applied for at the Library of Congress. %LEOLRJUDSKLFLQIRUPDWLRQSXEOLVKHGE\WKH'HXWVFKH1DWLRQDOELEOLRWKHN 7KH'HXWVFKH1DWLRQDOELEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ˆ Printed on acid-free paper Printed in Germany www.degruyter.com

This book is dedicated to our wonderful spouses, Cristina and Roger

Preface

In recent years, technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such optimal solution can not be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain a such relaxation by means of Young measures. The Young measures generalize measurable functions. Thus, a Young measure is herself a measurable application that, to every point t of , associates a probability  t on a topological space X; for all Borel set A  X,  t .A/ may be interpreted as the probability that the value in t of the “function” : belongs to A. In the particular case, a measurable application u W  ! X is a Young measure, where, for all t 2 ,  t D ıu.t / (ıu.t / indicates the mass of Dirac in u.t /). Young measures’ theory has a long history; it begins with the work of L. C. Young which, in 1937, introduces the so-called “generalized curves” in order to provide extended solutions for some non-convex problems in variational calculus. A milestone in this history is the appearance of the monograph of J. Warga, “Optimal Control of Differential and Functional Equations” (Academic Press, 1972); here is systematically developed a theory of relaxed control in compact metric spaces. The extension of theory on locally compact metric spaces was made by H. Berliocchi and J. M. Lasry in 1973. The study of Young measures was extended to Polish and Suslin spaces by the works of E. J. Balder (since 1984) and M. Valadier (1990). Lately, Young measures were the object of an intense research due to their applications in obtaining relaxed solutions; here are some of the areas in which these relaxed solutions find applications: non-convex variational problems and differential inclusions, non-linear homogenization problems, micro-magnetic phenomena in ferro-magnetic materials, Nash equilibrium in games theory, Gammaconvergence, different phenomena in continuum mechanics (as elasticity, microstructures’ theory), optimal design and shape optimization problems.

viii

Preface

On this subject, recent monographs appeared: .i/ Roubi˘cek, T.—Relaxation in optimization theory and variational calculus, Walter de Gruyter, Berlin. New York, 1997. .ii/ Pedregal, P.—Parametrized Measures and Variational Principles, Birkhäuser Verlag, Basel. Boston. Berlin, 1997. .iii/ Castaing, Ch., Raynaud de Fitte, P. and Valadier, M.—Young measures on topological spaces. With applications in control theory and probability theory, Kluwer Academic Publ. Dordrecht. Boston. London, 2004. The focus of the first two books is mainly on the applications; therefore, Young measures are used as generalized solutions to non-convex problems of variational calculus, optimization theory, or game theory. The last monograph considers theoretical aspects of the theory of Young measures as well as the applications in control theory and probability theory. Many of the results presented here make reference to a wide bibliography; thus, the work is difficult to use for beginners. The literature on the applications of Young measures in various areas (lower semicontinuity, optimal relaxed control, Gamma-convergence and homogenization, differential games, elasticity, hysteresis, etc.) is extremely rich and the existing monographs main focus on applications rather than on theoretical aspects. We found difficult for a young researcher who wants to clarify the theoretical aspects, to go through the extensive bibliography which is usually referred. Thus, our goal was to write a book where to be gathered all the theoretical aspects related to defining of Young measures (measurability, disintegration, stable convergence, compactness), book which to be a useful tool for those interested in theoretical foundations of the theory: the postgraduate students, the students in the doctoral study, but also to all those interested in measure theory and relaxed control. The developing of Young measures’ theory involves some compactness results for measures on abstract spaces and topological spaces. Hence, to achieve our goal, we considered useful to provide a complete set of classical and recent compactness results in measure and function spaces. The book is organized in three chapters (Weak compactness in measure spaces, Bounded measures on topological spaces, Young measures). For a good comprehension of the subject, we developed in the first two chapters the results used in the third (biting lemma in the abstract measure theory and Prohorov’s theorem in the measure theory on topological spaces). The first chapter covers background material on measure theory in abstract frame. Therefore, we present some results of duality and weakly compactness in ca.A/ and L1 ./. However, here we prove some extensions of Dunford–Pettis

Preface

ix

theorem like biting lemma of Brooks–Chacon or subsequence splitting lemma of H. P. Rosenthal. In Chapter two, we treat the measure theory on topological spaces. The framework is offered by Suslin spaces; on the one hand, these spaces are Radon and on the other hand, they cover the particular case of a separable Banach space provided with his weak topology. We introduce the narrow topology and then we prove the Prohorov’s compactness theorem. In the particular case of Polish spaces, the narrow topology is metrizable; we present the compatible metrics of Dudley and of Lévy–Prohorov. As an application of Prohorov’s theorem, we prove in the last paragraph the existence of Wiener’s measure on C Œ0; 1. With some exceptions, in Chapters 1 and 2 are presented classical compactness results for measures on abstract spaces, or on topological spaces. The originality consists in the selection and ordering of these results and the accompanying remarks and examples. However, we note some approaches and new results, such as: the modulus of -continuity (1.79) and theorems 1.80 and 1.81, a-convergence of nets in L1 and the extension of Dunford–Pettis theorem (1.93), a new proof for Rosenthal’s Subsequence Splitting Lemma using Biting Lemma, the modulus of narrow compactness (2.65), a-convergence of nets in ca.B.T //, theorem 2.69 and obtaining, as corollary of this theorem, a new proof of Prohorov’s compactness theorem. Finally, in the last section of 2 we give a simple and self-contained presentation of Wiener measure (2.6). Compactness results from the first two chapters are used to study Young measures in Chapter three. We prove the disintegration theorem for product measures and we use it to present Young measures as parametrized measures; the frame is that of a regular Suslin space. We remark that the space of Young measures contains the space of measurable mappings as dense subspace and that the narrow topology is an extension of the topology of convergence in measure. Prohorov’s theorem in the case of Young measures highlights the role played by tightness in compactness results. We present a vector version for biting lemma and an extension of this result to some special non-bounded sets of measurable mappings: finite-tight sets. In the seventh paragraph, we will study the two types of products for the Young measures and will give the fiber product lemma. In the last three sections of the book are presented some applications; thus, Prokhorov’s theorem for Young measures was used in the ninth paragraph in the study of strong compactness in Lp .; E/. We obtain, as corollaries, the theorems of Visintin–Balder, Rossi–Savaré, Lions–Aubin and Gutman; in the scalar case, the compactness criterion of Riesz–Fréchet–Kolmogorov is obtained. In the tenth paragraph, we consider some applications of quasiconvexity to the study of gradient Young measures and to the lower semicontinuity. Are studied the Young measures generated by sequences and particularly the gradient Young

x

Preface

measures. We pay special attention to quasiconvexity and its various equivalent definitions. The quasiconvexity is essentially used in the Kinderlehrer–Pedregal’s characterization of gradient Young measures, but also in the study of lower semicontinuity of energy functional that appears in variational calculus. Finally, in paragraph eleven, we present some results of existence of solutions in relaxed variational calculus. There are also, in this chapter, some new concepts and results among which: new proofs for theorems 3.30, 3.32 and 3.33, the density result 3.49 and the proof of theorem 3.50, theorems 3.51, 3.66, 3.67 and propositions 3.54 and 3.56, introduction of finite-tight sets (3.75) and use them to obtain extensions of biting lemma (3.84) and Saadoune–Valadier’s theorem (3.85), Jordan finite-tight sets (3.91) and their utility in obtain of a compactness result in Sobolev spaces (3.102) and an alternative to Rellich–Kondrachov theorem (3.105). All results are accompanied by full demonstrations; for many of these results, are given different proofs from those referred in the literature. The bibliography gives the main references relevant to the content of the book; it is no exhaustive. Understanding the text requires basic knowledge of general topology, functional analysis and Lebesgue integration that may be found in any textbook on the subject. In rest, all the statements are fully justified and proved. To conclude, this text is intended as a postgraduate textbook as well as a reference for more experienced researchers. The book was written over several years of collaboration between authors, with the occasion of stages that the first author has made, as a visiting professor, at the University of Brest. An important role in setting the ideas and in the organization of book’s material was played by discussions with various mathematicians met under these occasions. First, we mention the authors of monograph “Young measures on topological spaces”, C. Castaing, M. Valadier and P. Raynaud de Fitte, that supported and inspired us in writing the last chapter. Also, we have had useful discussions with E. Balder and T. Roubi˘cek at the international conference “Mesures de Young et Contrôle Stochastique” (Brest, 2002); in this way, we thank them all. Ia¸si/Brest, December 2011

Liviu C. Florescu, Christiane Godet-Thobie

Contents

Preface 1

vii

Weak Compactness in Measure Spaces

1

1.1 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Radon–Nikodym Theorem. The Dual of L1 . . . . . . . . . . . . . . . . .

28

and ca.A/ . . . . . . . . . . . . . . . . . . . . . . . .

38

1.4 Weak Compactness in ca.A/ and L1 ./ . . . . . . . . . . . . . . . . . . .

47

1.3 Convergences in

L1 ./

...................................

53

1.6 Extensions of Dunford–Pettis’ Theorem . . . . . . . . . . . . . . . . . . . .

64

Bounded Measures on Topological Spaces

90

2.1 Regular Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

2.2 Polish Spaces. Suslin Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

1.5 The Bidual of 2

L1 ./

2.3 Narrow Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.4 Compactness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.5 Metrics on the Space .RcaC .BT /; T/ . . . . . . . . . . . . . . . . . . . . . 136 2.5.1 Dudley’s Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.5.2 Lévy–Prohorov’s Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2.6 Wiener Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3

Young Measures

175

3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 3.1.1 Disintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.1.2 Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.2 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.2.1 Young Measure Associated to a Probability . . . . . . . . . . . 189 3.2.2 Young Measure Associated to a Measurable Mapping . . . 191 3.3 The Stable Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3.4 The Subspace M.S /  Y.S / . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3.5 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

xii

Contents

3.6 Biting Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3.7 Product of Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.7.1 Fiber Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.7.2 Tensor Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 3.8 Jordan Finite Tight Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 3.9 Strong Compactness in Lp .; E/ . . . . . . . . . . . . . . . . . . . . . . . . . 271 3.9.1 Visintin–Balder’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 271 3.9.2 Rossi–Savaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 274 3.10 Gradient Young Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Young Measures Generated by Sequences . . . . . . . . . . . . 3.10.2 Quasiconvex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.3 Lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288 289 296 310

3.11 Relaxed Solutions in Variational Calculus . . . . . . . . . . . . . . . . . . 314 Bibliography

325

Index

336

Chapter 1

Weak Compactness in Measure Spaces

We will present in this first chapter the main properties of the measure spaces and of the space of integrable functions. We recall the classic results of weak compactness (like Vitali–Hahn–Saks, Radon–Nikodym and Dunford–Pettis theorems) but we will also mention more recent results such as Brooks–Chacon biting lemma or Rosenthal’s lemma.

1.1

Measure Spaces

In this introductory section, we recall the definitions and classic properties of the additive and -additive measures. We finish this section by the Saks’ theorem, the Vitali–Hahn–Saks and Nikodym theorems that we will use in the following sections for a study of weak compactness on ca.A/. For beginning, we will specify the definitions and the notations to be employed henceforward. We consider as known the theory of integration relating to a positive, -additive and -finite measure. We designate by X an arbitrary set and by A a –algebra of subsets of X; an A-partition of A 2 A is a partition of A with the elements in A. According to the usual notations, if  is a positive, -additive and -finite measure, we shall denote by L1 ./ D L1 .X; A; / the set of all real mappings f defined on X with the property that f is A-measurable and -integrable and by L1 ./ D L1 .X; A; / the quotient space L1 ./=DP , where D P is equality  – almost everywhere. In the following, we recall the definition of the signed measures. N D Œ1; C1 is a finitely additive Definition 1.1. A set function  W A ! R measure, or shortly an additive measure, if .i/ .;/ D 0; .ii/ .A [ B/ D .A/ C .B/; for every A; B 2 A with A \ B D ;; .iii/  assumes at most one of the values C 1 and  1: An additive measure  on A is a -additive measure or a countably additive measure if, for every sequence of pairwise disjoint sets .An /n2N  A (i.e.

2

Chapter 1 Weak Compactness in Measure Spaces

An \ Am D ;, for every n ¤ m), 

1 [ nD0

! An

D

C1 X

.An /:

nD0

A -additive measure  is finite or real valued if its range is contained in R.  is -finite if, for every A 2 A, there exists a sequence .An /n2N such that S A D n2N An and .An / 2 R, for every n 2 N. We will designate by ba.A/—the set of all real valued bounded additive measures on A, ca.A/—the set of all real valued -additive measures on A. ba.A/ and ca.A/ are vector spaces under the usual addition and scalar multiplication operations. caC .A/ (baC .A/)—the subsets of all positive measures of ca.A/ (ba.A/). The following properties are easy to demonstrate. N be an additive measure and let A; B 2 A with Proposition 1.2. Let  W A ! R B  A. .i/ If j.A/j < C1, then j.B/j < C1. .ii/ If j.B/j < C1, then .A n B/ D .A/  .B/. Proof. A D B [ .A n B/ and so .A/ D .B/ C .A n B/. (i) If .B/ D C1.1/, then .A/ D C1.1/, what contradicts hypothesis. Therefore, .B/ is finite. (ii) If j.B/j < C1, then .A/  .B/ D .A n B/. Proposition 1.3. Let  be a -additive measure and let .An /n2N  A. .i/ If .An /n is an increasing sequence, then .[1 nD0 An / D limn!1 .An /. .ii/ If .An /n is a decreasing sequence and j.A0 /j < C1, then .\1 nD0 An / D limn!1 .An /. S Proof. (i) Let A D n2N An 2 A. Firstly, we suppose that there is n0 2 N such that j.An0 /j D C1. According to (i) of Proposition 1.2, j.A/j D C1 D j.An /j, for every n  n0 ; since  assumes at most one of the values C1 and 1, .A/ D limn .An /. If, for every n 2 N; j.An /j < C1, then we define the pairwise disjoint sequence .Bn /n  A letting: B0 D A0 ; Bn D An n An1 ; 8n  1; then A D

Section 1.1 Measure Spaces

3

P [1 Bn and, using (ii) of Proposition 1.2, we obtain .A/ D 1 nD0 .Bn / D nD0P limn nkD0 .Bk / D limn .An /. (ii) If .An /n2N is decreasing and j.A0 /j < C1, then the sequence .Bn /n , where Bn D A0 n An , is increasing and the result follows from the first part of the proof. N be a -additive measure and let A 2 A; Definition 1.4. Let  W A ! R A is called -positive if, for every B 2 A; .A \ B/  0. A is called -negative if, for every B 2 A; .A \ B/  0. A is called -null if it is -positive and -negative. A is -null set if and only if, for every measurable set B  A, .B/ D 0. For the following two results, see Theorem A, p. 121 in [93]. Proposition 1.5. Let  W A ! .1; C1 be a -additive measure and let A 2 A with .A/ < 0. There exists a -negative set B  A such that .B/ < 0. Proof. If A is -negative, then B D A. Otherwise, there exists C 2 A such that .C \ A/ > 0. Let n1 be the smallest positive integer for which there exists A1 2 A; A1  A with .A1 / > n11 . If A n A1 is -negative, then B D A n A1 ; since .A/ < 0; .A/ 2 R and then, by Theorem 1.2, .A1 / 2 R and .B/ D .A/  .A1 / < .A/ < 0. If A n A1 is not -negative, let n2 be the smallest positive integer for which there exists A2 2 A; A2  A n A1 with .A2 / > n12 ; obviously, n2 > n1 . If the above construct does not produce a solution of problem after a finite number of steps, then we obtain a sequence of pairwise disjoint sets .Ak /k1  1 k  1 and nk " C1. A; Ak  A n [k1 iS D1 Ai , with .Ak / > nk , for every P1 1 Let B D A n kD1 Ak ; then .B/ D .A/  kD1 .Ak / < 0. For every C 2 A; C  B and for every k 2 N  , C  A n [k1 i D1 Ai so 1 that .C /  nk 1 (nk is the smallest positive integer for which there is Ak  1 An[k1 i D1 Ai with .Ak / > nk ). Then .C /  0 and therefore B is -negative. N be a -additive Theorem 1.6 (Hahn decomposition theorem). Let  W A ! R c measure; there exists a -positive set H 2 A such that H D X nH is -negative. For any other pair ¹H1 ; H1c º  A with H1 -positive and H1c -negative, HH1 is a -null set. Proof. First, let us suppose that .A/  .1; C1. Let a D inf¹.A/ W A 2 A; A D -negative º, let .A /n be a sequence of Sn1 -negative sets such that .An / ! a and let H D X n nD1 An ; then H c D S 1 nD1 An .

4

Chapter 1 Weak Compactness in Measure Spaces

letting B1 D A1 and, for every n  2,P Bn D An n S 1 2 A, .C \H c / D .C \ 1 B / D n nD1 nD1 .C \ -negative. Moreover, for every n 2 N  ,

If we define .Bn /n2N  n1 [i D1 Ai , then, for every C Bn /  0. Therefore H c is

.H c / D .An / C .H c n An /  .An / so that .H c / D a. If we suppose that H is not -positive, then there exists C 2 A; C  H such that .C / < 0. The previous proposition assures us on the existence of a -negative set B  C with .B/ < 0. Then H c [ B is -negative and .H c [ B/ D .H c / C .B/ D a C .B/ < a and this contradicts the definition of a. In the case where .A/  Œ1; C1/;  W A ! .1; C1 is a -additive measure. Let H be a ./-positive set and H c be a ./-negative set; then H c is -positive set and H is -negative set. Let now H1 -positive and H1c -negative an other pair. For every B 2 A with B  H nH1 D H \H1c , .B/  0 and .B/  0, hence .B/ D 0. Then H nH1 is a -null set. Similarly, H1 nH is -null and then HH1 D .H nH1 /[.H1 nH / is a -null set. Definition 1.7. Every pair of sets ¹H; H c º  A, with the property that H is positive and H c is -negative, is called a Hahn decomposition of X relatively to the measure . Remark 1.8. .i/ Hahn’s decomposition of X relatively to a measure  is not unique (we can replace H by H [ N where N is a -null set). .ii/ The Hahn decomposition theorem says that, for every -additive measure , there exists a Hahn decomposition of X relatively to . N be a -additive measure, let ¹H; H c º be a Proposition 1.9. Let  W A ! R N C defined by Hahn decomposition of X relatively to  and let C ,  W A ! R C   .A/ D .A \ H /,  .A/ D .A n H /, for every A 2 A. Then C ;  are two -additive positive measures (one of them finite),  D C    and C .H c / D  .H / D 0.  If C 1 , 1 are two other -additive positive measures (one of them finite) such C  c  that  D C 1  1 and if 1 .H1 / D 1 .H1 / D 0 for a set H1 2 A, then C  C   D 1 and  D 1 .

5

Section 1.1 Measure Spaces

Proof. The first part of the proposition is obvious. We shall prove only the uniqueness of decomposition of  as difference of positive measures. The pair ¹H1 ; H1c º is again a Hahn decomposition of X relatively to . Indeed, C  for every A 2 A; .A \ H1 / D C 1 .A \ H1 /  1 .A \ H1 / D 1 .A \ H1 /  0   and .A n H1 / D C 1 .A n H1 /  1 .A n H1 / D 1 .A n H1 /  0. According to Hahn decomposition theorem, HH1 is a -null set. Therefore, for every C D C . A 2 A, C .A/ D .A \ H / D .A \ H1 / D C 1 .A/ so that  1   Similarly,  D 1 . Definition 1.10. We say that the unique pair ¹C ;  º of -additive positive measures (one of them finite) with  D C   and C .H c / D  .H / D 0 for a set H 2 A, is the Jordan decomposition of . Definition 1.11. A positive measure  on A is concentrated on the set D 2 A if .D/ D .X/. Remark 1.12. If  is a -additive measure on A, if ¹H; H c º is a Hahn decomposition of X relatively to  and if ¹C ;  º is the Jordan decomposition of , then C is concentrated on the set H and  is concentrated on H c . Theorem 1.13. Let  be a -additive measure on A; there exist Am ; AM 2 A such that .Am / D inf ¹.A/ W A 2 Aº  0  sup¹.A/ W A 2 Aº D .AM /: Every -additive measure is bounded either from below or from above. Proof. Let ¹H; H c º a Hahn decomposition of X relatively to  and let ¹C ;  º the Jordan decomposition of ; for every A 2 A, C .A/ D .A \ H /  .H / and  .A/ D .A n H /   .H c / D .H c /. Then .H c /   .A/  C .A/   .A/ D .A/  C .A/  .H /: Therefore we can take Am D H c and AM D H . The following result is a corollary of Theorem 1.13. Corollary 1.14. Every measure  2 ca.A/ is bounded; therefore ca.A/  ba.A/:

6

Chapter 1 Weak Compactness in Measure Spaces

Remark 1.15. If  W A ! R is only an additive measure, it is not compulsory that  should be bounded, as it is shown in the following example. P1 Example 1.16. Let P1 nD0 an be a conditionally convergent series (a convergent series for which nD0 jan j D C1), let A be the algebra of all sets A  N such that A or N n A is a finite and let  W A ! R defined by 8 X < an ; A ¤ ;; .A/ D n2A : 0; A D ;: Then  is an additive measure, but it is not bounded on the algebra A. We notice that A is not a -algebra, but, since all additive function on an algebra can be extended to an additive function on the generated -algebra – in our case P .N/ – (see [29], p.185 and [175],1.8.), it is evident that the extension itself is not bounded. The total variation defined below is introduced in order to define a complete norm on ba.A/ or on its subspace ca.A/ of -additive measures (see Definition III.1.4 and Lemma III.1.6 of [62]). Theorem 1.17. For every additive measure , let jj W A ! RC defined by ´ n μ X  jj.A/ D sup j.Ai /j W n 2 N ; ¹A1 ; : : : ; An º D A  partition of A : i D1

Then: .i/ supB2A;BA j.B/j  jj.A/  2 supB2A;BA j.B/j. .ii/ jj is additive. .iii/ If  2 ba.A/, then jj 2 baC .A/; moreover jj is the smallest element of the set M D ¹ 2 baC .A/ W j.A/j  .A/; 8A 2 Aº. .iv/ If  is -additive, then jj is -additive and jj.H / D .H / and jj.H c / D .H c /, where ¹H; H c º is a Hahn decomposition of X. If  2 ca.A/, then jj 2 caC .A/. Proof. (i) For every B 2 A with B  A; ¹B; A n Bº is an A-partition of A and so jj.A/  j.B/j C j.A n B/j  j.B/j, from where sup¹j.B/j W B 2 A; B  Aº  jj.A/:

7

Section 1.1 Measure Spaces

Now, let n 2 N  and let ¹A1 ; : : : ; An º be an A-partition of the set A. We can suppose that .A1 /; : : : ; .Ap /  0 and .ApC1 /; : : : ; .An / < 0: Then 0 1 ! p p n n n X X X [ [ j.Ai /j D .Ai /  .Aj / D  Ai   @ Aj A i D1

i D1

j DpC1

i D1

j DpC1

1ˇ ˇ !ˇ ˇˇ 0 n ˇ p ˇ ˇ ˇ [ [ ˇ ˇ ˇ @ ˇ A Ai ˇ C ˇ Aj ˇˇ D ˇ ˇ ˇ ˇ ˇ i D1

j DpC1

 2 sup¹j.B/j W B 2 A; B  Aº: (ii) Let A; B 2 A with A \ B D ; and let C D A [ B. If jj.A/ D C1, then C1 D sup¹j.D/j W D 2 A; D  Aº  sup¹j.D/j W D 2 A; D  C º  jj.C / and so jj.C / D C1 D jj.A/ C jj.B/: In the same way, if jj.B/ D C1, we have jj.C / D C1 D jj.A/Cjj.B/: Now suppose that jj.A/ < C1 and jj.B/ < C1. Then, for every " > 0, there exists an A-partition ¹A1 ; : : : ; An º of A and an A-partition of B, ¹B1 ; : : : ; Bm º, such that " X j.Ai /j < 2 n

jj.A/ 

and

jj.B/ 

i D1

m " X j.Bj /j: < 2 j D1

Then ¹A1 ; : : : ; An ; B1 ; : : : ; Bm º is an A-partition of C and therefore jj.A/ C jj.B/  "
0, there exists an A-partition of X , ¹A1 ; : : : ; An º, such that k C k  " D j C j.X/  "
n0 such that, for every n  n1 ˇ ˇ n ˇ ˇ X ˇ ˇ .A/   .A /  ˇ n0 n0 k ˇ < ": ˇ ˇ

(1)

(2)

kD1

Then, for all n  n1 ; ˇ ˇ ˇ !ˇ ˇ ! !ˇ n n n n ˇ ˇ ˇ ˇ ˇ ˇ X [ [ [ ˇ ˇ ˇ ˇ ˇ ˇ .A / A A A  An D  .A/   An  An ˇ ˇ n0 k ˇ k ˇ ˇ k k ˇ ˇ ˇ ˇ ˇ ˇ ˇ kD1 kD1 kD1 kD1 ˇ ˇ ˇ ! ! n n ˇ ˇ ˇ [ [ ˇ ˇ ˇ Ak ˇ D ˇ An Ak  C ˇn0 An ˇ ˇ ˇ kD1 kD1 ˇ !ˇ ˇ n n ˇ ˇ ˇ [ X ˇ ˇ ˇ Ak ˇ C ˇn0 .A/  n0 .Ak /ˇ : n0 An ˇ ˇ ˇ kD1

From (1) and (2), we obtain ˇ ˇ n ˇ ˇ X ˇ ˇ .Ak /ˇ < 2"; ˇ.A/  ˇ ˇ kD1

so that  2 ca.A/.

kD1

8n  n1 ;

(3)

13

Section 1.1 Measure Spaces

Before mentioning the definition of the integral in relation to a signed measure, we need to clarify a number of notations and properties of the integral relatively to a positive measure. Let  be a positive -additive measure on A and let f W X ! R be an AC measurable mapping. We recall that f is R-integrable 0º and R if fC D sup¹f; R  f D sup¹f; 0º are -integrable. Then, X f d D X f d  X f  d; let L1 ./ be the set of all -integrable mappings and let L1 ./ D L1 .X; A; / be the quotient space L1 ./=DP , where D P is equality -almost everywhere. the difference If at least one of the two functions f C and f  is -integrable, R of the integrals is always defined and will be marked by X f d. Definition 1.24. Let  2 ca.A/ and let f W X ! R be an A-measurable mapping; we say that f is -integrable if f 2 L1 .C / \ L1 . /. Let us denote Z

L1 ./ D L1 .C / \ L1 . /; L1 ./ D L1 .C / \ L1 . / and Z Z f d D f dC  f d ; for every A 2 A;

A

A

A

where f marks, according to the context, the function f or the equivalence class R of a function f . It is clear that L1 ./ is a vector space and that A is a linear operator on L1 ./. Proposition 1.25. Let  2 caC .A/ and f W X ! R be an A-measurable mapping such that at least one of mappings f C and f  is -integrable. Let R R C  .A/ D R A f d  A f d. R Then,  is a -additive measure on A and C .A/ D A f C d,  .A/ D A f  d. R If f 2 L1 ./, then  2 ca.A/ and kk D jj.X/ D X jf jd. Proof. Let H D ¹x 2 X W f .x/  0º; H 2 A, f C H D f C and f  H D 0. R R R R R C C C   A f d D A f H d D A\H f d and A\H f d D A f H d D 0. R R .A \ H / D A f C d. If B  RA, then .B/  B f C d  R Therefore C C A f d. Then, according to Corollary 1.22, A f d R Dsup¹.B/ W B 2 C A; B  Aº D  .A/ D .A \ H /, which leads to A f d D  .A/ D .A \ H c /. R If f 2 L1 ./, then .A/ D A f d, for every A 2 A, hence  R 2 ca.A/. C  According to RDefinition 1.19, kk D jj.X/ D  .X/C .X/ D X f C dC R  X f d D X jf jd:

14

Chapter 1 Weak Compactness in Measure Spaces

Proposition 1.26.

R R .i/ L1 ./ D L1 .jj/ and j A f dj  A jf jd jj; 8A 2 A; 8f 2 L1 ./. R .ii/ jj.A/ D sup¹j A f dj W f 2 L1 ./; jf j  1º. R .iii/ The mapping k  k1 W L1 ./ ! RC ; kf k1 D X jf jd jj is a norm on L1 ./ and .L1 ./; k  k1 / is a Banach space. 1 then and f 2 L1 .C / \ L1 . /. R f is A-measurable RProof. (i) Let fR 2 L ./; C  1 X jf jd jj D X jf jd C X jf R jd 0 such that arctan .B/ <  implies .B/ < ı and so jj.B/ < ".

18

Chapter 1 Weak Compactness in Measure Spaces

Let now A 2 A and let B 2 A with d .A; B/ < ; then .A 4 B/ < ı and so jj.A 4 B/ < ". Therefore j.A/  .B/j D j.A/  .A \ B/ C .A \ B/  .B/j D j.A n B/  .B n A/j  j.A n B/j C j.B n A/j  jj.A 4 B/ < " and so  is d -continuous in A. (iii) H) (i). Since  is continuous on A, it is continuous at ; 2 A. Then, for every " > 0, there exists ı 20; 1Œ such that, for every A 2 A satisfying d .A; ;/ D .A/ < ı; j.A/j < ". Let A 2 A with .A/ D 0; then d .A; ;/ < ı and hence j.A/j < " and, as " is arbitrary, .A/ D 0. Therefore  . Proposition 1.31. Let  W A ! RC be a -additive measure and let  W A ! R be an additive measure. If  is d -continuous, then  2 ca.A/. Proof. S1 Let .An /n2N  A be a sequence of pairwise disjoint sets and let A D nD1 An 2 A. Since  is d -continuous, for every " > 0, there exists ı > 0 such that, for every B and C of A satisfying .B 4 C / < ı; j.B/  .C /j < ". Since  is -additive, there exists n0 2 N such that, for every n  n0 ; ˇ ˇ ! ! n n n ˇ ˇ X [ [ ˇ ˇ .Ak /ˇ D  A n Ak D  A 4 Ak < ı: ˇ.A/  ˇ ˇ kD1

Then j.A/ 

kD1

Pn 1

kD1

.Ak /j D j.A/  .[n1 Ak /j < " and so  2 ca.A/.

Remark 1.32. The result of Proposition 1.30 asserts that, if  is - additive, then the absolute continuity of  with respect to  is equivalent to the d -continuity of ; this result is no longer valid if  is only additive. In fact, let  be as in the example of Remark 1.16;  is additive, it is not bounded and, according to Corollary 1.14, its extension to P .N/, still noted , is not  - additive. P Let  W P .N/ ! RC ; .A/ D n2A ın .A/, where ın is the Dirac measure that gives to singleton set ¹nº the measure 1. Then  is a -additive measure and, as  is not -additive, according to Proposition 1.31,  is not d - continuous. However  is absolutely continuous with respect to . Indeed, let A 2 P .N/ with .A/ D 0; then A D ; and therefore .A/ D 0. In the case where  2 ba.A/ n ca.A/, we have the following implications among the conditions of Proposition 1.30: (ii)”(iii) H)(i).

Section 1.1 Measure Spaces

19

In relation to Proposition 1.30, if  is -additive, then in order to avoid the use of the difficult formulation of “d -continuity” or the longer one “absolutely continuous with respect to ”, we will give the following definition: Definition 1.33. A measure  2 ca.A/, continuous on the space .A; d /, is called -continuous. We will note by ca .A/ the subset of all -continuous measure of ca.A/. We can find the following theorem in [57] (see Theorem 9, p. 87). Theorem 1.34. Let  W A ! RC be a -additive measure, let d be the pseudometric associated and let K be a family of measures of ca.A/; then the following properties are equivalent: .i/ the family K is d -equicontinuous at some E 2 A. .ii/ the family K is d -equicontinuous at the point ; 2 A. .iii/ the family K is uniformly d -equicontinuous on A. Each of these conditions entails the following: .iv/ the family K is uniformly -additive. Proof. (i)H)(ii). Let K be d -equicontinuous at E 2 A; then, for every " > 0, there exists ı > 0 such that, for every A 2 A with d .A; E/ < ı and for every  2 K; j.A/  .E/j < ". Let A 2 A such that d .A; ;/ < ı, that is .A/ <  D tan.ı/; then D .A [ E; E/ D  Œ.A [ E/ 4 E D .A n E/  .A/ <  and D .E n A; E/ D  Œ.E n A/ 4 E D .A \ E/  .A/ < : Therefore, for every  2 K, j.A [ E/  .E/j < " and j.E/  .E n A/j < ". We have then: j.A/j D j.A [ E/  .E n A/j  j.A [ E/  .E/j C j.E/  .E n A/j < 2": K is therefore d -equicontinuous at ;. (ii) H)(iii). K being d -equicontinuous at ;, for every " > 0, there exists ı > 0 such that, for every E 2 A satisfying d .E; ;/ D arctan .E/ < ı, we have j.E/j < ", for every  2 K. If d .C; D/ D arctan .C 4 D/ < ı then d .C n D; ;/ < ı and d .D n C; ;/ < ı.

20

Chapter 1 Weak Compactness in Measure Spaces

Then, for every  2 K, j.C /  .D/j D j.C n D/ C .C \ D/  .C \ D/  .D n C /j  j.C n D/j C j.D n C /j < 2": Therefore K is uniformly d -equicontinuous on A. Obviously, (iii) H)(i). (ii) H)(iv). Let K  ca.A/ be a family satisfying (ii), let .An /n2N  A be a sequence of pairwise disjoint sets and let A D [1 1 An . Then for every " > 0, there exists ı > 0 such that, for all E 2 A with .E/ < ı, we have j.E/j < ", for everyP  2 K.  being -additive and positive, there exists n0 2 N such that j.A/  nkD1 .Ak /j D .A n [nkD1 P Ak / < ı, for every n  n0 . Then, for every  2 K; j.A/  nkD1 .Ak /j D j.A n [nkD1 Ak /j < " from where it results that K is uniformly -additive. In a consistent manner with Theorem 1.33, we give the following definition: Definition 1.35. A family of measures K  ca.A/; d -equicontinuous at ; (and therefore on A) is called -equicontinuous. We need to emphasize that the definitions of -continuity and -equicontinuity refer only to the real -additive measures, meaning that they do not refer to the -additive measures taking at most one of the values C1 or 1. However, the previous results can be extended by replacing the -additive and positive measure  by a -additive measure  of finite total variation jj and the measure  2 ca.A/ by a -additive measure with values in a Banach space. We will now give a very important result of equicontinuity (Vitali–Hahn–Saks theorem) which allows us to establish the analogue of the uniform boundedness principle from Functional Analysis for the Measure Theory (see Theorem III.7.2 and Corollary III.7.3 in [62] or Theorem 2.53 of [85]). Theorem 1.36 (Vitali–Hahn–Saks). Let  W A ! RC be a -additive measure and let .n /n2N  ca.A/ be a sequence of -continuous measures. Assume that limn n .A/ D .A/ 2 R exists, for every A 2 A; then: .i/ ¹n W n 2 Nº is -equicontinuous, .ii/  2 ca.A/, .iii/  is -continuous. Proof. According to Proposition 1.30, n W .A; d / ! R is a continuous function, for every n 2 N.

21

Section 1.1 Measure Spaces

For every " > 0 and for all couple .n; m/ 2 N  N, let’s note An;m ."/ D ¹A 2 A W jn .A/  m .A/j  "º: An;m ."/ are closed sets in the complete space .A; d /; then, for every p 2 N, \ An;m ."/ Ap ."/ D m;np

is a closed set in .A; d /: Since limn n .A/ 2 R, for every A 2 A, AD

1 [

Ap ."/:

pD1

According to Baire theorem, there exists p0 2 N such that Ap0 ."/ has nonempty interior in .A; d /. Therefore, there exists A0 2 A, there exists r > 0 such that the ball S.A0 ; arctan r/  Ap0 ."/, i.e., jn .A/  m .A/j < ";

8A 2 A with

.A 4 A0 / < r;

8m; n  p0 : (1)

Since the set ¹1 ; : : : ; p0 º is -equicontinuous, there exists ı 20; rŒ such that jn .B/j < ";

for all

B2A

with .B/ < ı;

8n D 1; : : : ; p0 :

(2)

Let A 2 A with .A/ < ı; then ..A [ A0 / 4 A0 / D .A n A0 /  .A/ < ı < r

and

..A0 n A/ 4 A0 / D .A0 \ A/  .A/ < ı < r: By (1), we have jn .A [ A0 /  p0 .A [ A0 /j < ";

8n  p0

(3)

jn .A0 n A/  p0 .A0 n A/j < ";

8n  p0 :

(4)

and

By (2), (3) and (4), we deduct that, for every n  p0 , jn .A/j D jp0 .A/ C Œn .A/  p0 .A/j  jp0 .A/jCjn .A [ A0 /p0 .A [ A0 /Cp0 .A0 n A/n .A0 n A/j  jp0 .A/jCjn .A [ A0 /p0 .A [ A0 /jCjp0 .A0 n A/n .A0 n A/j < 3":

22

Chapter 1 Weak Compactness in Measure Spaces

Therefore, according to (2), jn .A/j < 3", for every n 2 N. ¹n W n 2 Nº is therefore d -equicontinuous at ; and so it is -equicontinuous. Since .n / converges punctually to ,  is additive on A. We still need to show that  is -continuous. According to Theorem 1.34, since ¹n º is -equicontinuous, it is uniformly d -equicontinuous on A. Therefore, for every " > 0, there exists ı > 0 such that, for all A and B of A with .A 4 B/ < ı, for any n 2 N; jn .A/  n .B/j < ". Now let n tend to 1; so we obtain j.A/  .B/j  ", from where it results that  is uniformly - d -continuous and so it is d -continuous. From Proposition 1.31, it results that  2 ca.A/ and, according to Proposition 1.30,  is -continuous. The previous theorem accepts as corollary the following result (see Corollary III.7.4 and Theorem IV.9.8 of [62] and [57], p. 90): Theorem 1.37 (Nikodym). Let .n /  ca.A/ be a sequence of measures such that, for every E 2 A, there exists limn n .E/ D .E/ 2 R. Then: .i/  2 ca.A/, .ii/ ¹n W n 2 Nº is uniformly -additive and .iii/ ¹n W n 2 Nº is bounded in the space (ca.A/; k  k). Proof. (i) + (ii) Let  W A ! RC be defined by 1 X jn j.A/ 1  .A/ D ; n 2 1 C kn k

8A 2 A:

nD1

P 1 It is clear that, for all A 2 A, .A/ < 1 nD1 2n D 1. S1 Let ¹Ep W p 2 Nº  A be a family of disjoint sets and let E D pD1 Ep . For 1 all " > 0 let n0 2 N such that 2n0 2 < ". Since n 2 ca.A/, jn j 2 ca.A/ and then there exists k0 2 N such that, for every k  k0 and every n D 1; : : : ; n0 , ˇ ˇ ˇ ˇ k X ˇ " ˇ ˇjn j.E/  jn j.Ep /ˇˇ < .1 C kn k/: ˇ ˇ 2 ˇ pD1

23

Section 1.1 Measure Spaces

Then ˇ ˇ ˇ 1ˇ 0 ˇ ˇ ˇX ˇ k k X X ˇ ˇ ˇ1 ˇ 1 j.E/ 1 j.E / j j n n p ˇ.E/  Aˇ @   .Ep /ˇˇ D ˇˇ  ˇ ˇ n n ˇ ˇnD1 2 1 C kn k pD1 2 1 C kn k ˇ ˇ pD1 ˇ ˇ ˇ ˇ n0 k 1 X X X 1 ˇˇ jn j.E/ jn j.Ep / ˇˇ 1  2  C ˇ ˇ n 2 ˇ 1 C kn k 1 C kn k ˇ 2n nD1


†1 A2A

and therefore there exists A1 2 A such that jn1 .A1 /j > †1 . jn1 .X n A1 /j D jn1 .X/  n1 .A1 /j  jn1 .A1 /j  jn1 .X/j  jn1 .A1 /j  sup jk .X/j > 1: k

Let’s note B1 D X n A1 . We have obtained an A- partition .A1 ; B1 / of X such that jn1 .A1 /j  1;

jn1 .B1 /j  1:

(3)

24

Chapter 1 Weak Compactness in Measure Spaces

By (2), we have that sup .sup jn .A \ A1 /j/ D C1:

(4)

n2N A2A

or sup .sup jn .A \ B1 /j/ D C1:

(5)

n2N A2A

If (4) is satisfied, then we note C1 D B1 (otherwise C1 D A1 ). Because all finite subset of ca.A/ is uniformly bounded on A (this is an immediate consequence of Corollary 1.14), we have sup .sup jn .A \ A1 /j/ D C1:

n>n1 A2A

Can one restart this procedure by applying (1) to A1 . Let †2 D supk jk .A1 /jC2; there exist n2 > n1 and an A - partition .A2 ; B2 / of A1 such that jn2 .A2 /j  2;

jn2 .B2 /j  2

and sup .sup jn .A \ A2 /j/ D C1:

(6)

sup .sup jn .A \ B2 /j/ D C1:

(7)

n>n2 A2A

or n>n2 A2A

If (6) is satisfied, we note C2 D B2 (otherwise C2 D A2 ). C2 D B2  X n C1 . Continuing in this fashion, we define a strictly increasing sequence of integers .np /p2N tending to infinity and a sequence of pairwise disjoint sets .Cp /p2N  A such that, jnp .Cp /j  p;

for every p 2 N:

(8)

S Let C D 1 1 Cp 2 A; by (ii), .n / are uniformly -additive. Therefore, for " D 1, there exists k0 2 N such that jn .C / 

k X i D1

n .Ci /j < 1;

for every k  k0

and for all

n 2 N:

25

Section 1.1 Measure Spaces

Then, for p > k0 and for all n 2 N; ˇ p ˇ p1 ˇX ˇ X ˇ ˇ n .Ci /  n .Ci /ˇ jn .Cp /j D ˇ ˇ ˇ i D1 i D1 ˇ ˇ ˇ ˇ p p1 ˇ ˇ ˇ ˇX X ˇ ˇ ˇ ˇ n .Ci /  n .C /ˇ C ˇn .C /  n .Ci /ˇ < 2; ˇ ˇ ˇ ˇ ˇ i D1

i D1

which is incompatible with (8). Therefore, supn2N kn k < C1 and (iii) is proved. We will now give Nikodym’s theorem on uniform boundedness; the proof follows that of [57], p. 80. Theorem 1.38 (Uniform boundedness Nikodym’s theorem). Suppose that .n /  ca.A/ is a sequence of measures such that supn jn .A/j < C1;

for every

A 2 A:

Then ¹n W n 2 Nº is bounded in the space .ca.A/; k  k/. Proof. We suppose that ¹n W n 2 Nº is unbounded in .ca.A/; k  k/. Since the norms k  k and k  k1 are equivalent on ca.A/ (cf. Theorem 1.23), this means that sup .sup jn .A/j/ D C1:

(1)

n2N A2A

By hypothesis sup jn .X/j < C1:

(2)

n

Then, for all > 0, there exist n 2 N and A 2 A such that jn .A /j > . We also have jn .X n A /j D jn .X/  n .A /j  jn .A /j  jn .X/j  jn .A /j  sup jk .X/j: k

Therefore, if D supn jn .X/j C ˛, then we have an A-partition of X : .A ; B D X n A / such that jn .A /j  ˛;

jn .B /j  ˛:

(3)

26

Chapter 1 Weak Compactness in Measure Spaces

Let us take ˛ D 2: According to (1), we have either sup .sup jn .A \ A /j/ D C1;

or

(4)

n2N A2A

sup .sup jn .A \ B /j/ D C1:

(5)

n2N A2A

Suppose that (4) were achieved and let us note n1 D n , A1 D A and B1 D B (otherwise, we note A1 D B and B1 D A ). We restart the operation by replacing (1) with (4), X by A1 , and ˛ by 2 D sup jn .A1 /j C ˛2 where ˛2 D 3 C supn jn .B1 /j. Then, there exist n2 > n1 and A2 2 A, A2  A1 such that jn2 .A2 /j  2 . We have then jn2 .A1 nA2 /j  jn2 .A2 /jjn2 .A1 /j  2 sup jn .A1 /j D ˛2 : Since ˛2  3 C jn2 .B1 /j, we obtain an A-partition .A2 ; A1 n A2 / of A1 such that jn2 .A2 /j  3 C jn2 .B1 /j

and

jn2 .A1 n A2 /j  3 C jn2 .B1 /j:

(6)

As previously, we note A2 D A2 if supn2N .supA2A jn .A \ A2 /j/ D C1 and B2 D A1 n A2 (otherwise, we note A2 D A1 n A2 ). Repeating, we thus define a strictly increasing sequence .np /p2N   N and a sequence of pairwise disjoint sets .Bp /p2N   A such that jnp .Bp /j >

p1 X

jnp .Bk /j C p C 1;

for every p  2; and

(7)

kD1

jn1 .B1 /j > 2 D 1 C 1: Let now ¹Nk ; k 2 N  º be a countable partition of infinite subsets of N  . Because 0 0 1 1 1 X [ [ jn1 j @ Bp A  jn1 j @ Bp A  jn1 j.X/ < C1; kD1

p2Nk

p2N 

S there exists k1 such that jn1 j. p2Nk Bp / < 1; if p1 is the smallest integer 1 .Bp /p2Nk1 ; of Nk1 then np1 > n1 . We note C1 the first set ofSthe sequence S  we have C1 D Bp1 . By replacing N with Nk1 , p2N  Bp by p2Nk Bp 1 and n1 by np1 , we define a second subsequence .B / extracted from the p p2N k 2 S sequence .Bp /p2Nk1 such that jnp1 j. p2Nk Bp / < 1 and, if we note by C2 2

27

Section 1.1 Measure Spaces

the first set of the subsequence .Bp /p2Nk2 , by p2 the smallest integer of Nk2 then np2 > np1 > n1 and C2 D Bp2 . By reiterating the process, we define a decreasing sequence of infinite subsets of N  , .Nkj /j 2N  , a strictly increasing sequence of integers .pj /j 2N  and a sequence of pairwise disjoint sets of A, .Cj /j 2N  , such that, for all j > 2, 0 1 [ B C (8) jnpj 1 j @ Bp A < 1: p2Nkj

If we note C D 0 jnpj j @

S

j 2N 

1 [

Cj , then we have from (8), 1

0

B Ch A  jnpj j @

hDj C1

1 [

C Bp A < 1;

8j 2 N

p2Nkj C1

and by (7) and by inequality pj 1  pj  1, pj 1

jnpj .Cj /j D jnpj .Bpj /j >

X

kD1

jX 1

jnpj .Bk /j C pj C1 >

jnpj .Ch /jCpj C1

hD0

from where jnpj .Cj /j 

jX 1

jnpj .Ch /j > pj C 1 > j C 1:

hD0

Therefore, for every j 2 N, ˇ 0 1ˇ ˇ 0 1ˇ ˇ ˇ ˇ ˇ j[ 1 1 [ ˇ ˇ ˇ ˇ ˇ ˇ ˇ @ A @ A Ch ˇ  ˇnpj Ch ˇˇ > j; jnpj .C /j  jnpj .Cj /j  ˇnpj ˇ ˇ ˇ ˇ hD0 hDj C1 what implies that supj 2N  jnpj .C /j D C1 and this is incompatible with the hypothesis supn2N  jn .C /j < C1. Remark 1.39. .i/ In [62], we can find a different proof of this theorem based on the use of Baire’s theorem.

28

Chapter 1 Weak Compactness in Measure Spaces

.ii/ Nikodym’s Theorem 1.38 is not true if A is only an algebra and not a algebra of subsets of X, as shown by the example given in Remark 1.16: let A D ¹A  N W A finite or N n A finite º, let ın be the Dirac measure supported on the singleton ¹nº and let n .A/ D nŒınC1 .A/  ın .A/, for every A 2 A. Then, for every A 2 A, supn jn .A/j < 1 and supn supA2A jn .A/j D C1.

1.2

Radon–Nikodym Theorem. The Dual of L1

Firstly, we will present the Radon–Nikodym’s theorem to identify the absolutely continuous measures with respect to a positive measure and then we will give the Lebesgue’s decomposition theorem. We will conclude this part with the theorem of isomorphism between L1 ./ and the dual space of L1 ./. Let A be a -algebra on X, let  W A ! RC be a -additive and -finite positive measure and let .L1 ./; k  k1 / be the Banach space of R all integrable mappings with respect to  provided with norm k  k1 W kf k1 D X jf jd. Let W L1 ./ ! Rca.A/ be the function defined by .f / D f , where, for all E 2 A; f .E/ D E f d: Then is a linear application. For every f 2 L1 ./, let A D ¹x 2 X W f .x/ < 0º

and

AC D ¹x 2 X W f .x/  0º: Then A ; AC 2 A and, for every E 2 A; Z Z Z f d C f d  f .E/ D E \A

E \AC

AC

f d D f .AC /:

Therefore f .AC / D sup¹f .E/ W E 2 Aº. Similarly, f .A / D inf¹f .E/ W E 2 Aº. According to Theorem 1.13, Corollary 1.22 or Proposition 1.25, Z Z C f d D f C d and f .E/ D f .E \ AC / D E \AC

f .E/

E

Z D f .E \ A / D 

E \A

where f C D sup¹f; 0º and f  D sup¹f; 0º.

Z f d D

E

f  d;

Section 1.2 Radon–Nikodym Theorem. The Dual of L1

It results that jf j.E/ D

fC .E/

C

f .E/

from where

Z D

.f E

C

29



Z

C f /d D

E

jf jd

Z

k .f /k D kf k D jf j.X/ D

X

jf jd D kf k1 ;

for all

f 2 L1 ./:

Due to the absolute continuity of integral, for all f 2 L1 ./, forR every " > 0, there exists ı > 0 such that, for all E 2 A with .E/ < ı, we have E jf jd < " or jf j.E/ < ". According to Proposition 1.30 (ii), it results that f . Let f1 ; f2 2 Œf , where Œf  is the equivalence class of function f 2 L1 ./; it is easy to see that, for all E 2 A, f1 .E/ D f2 .E/, fC1 .E/ D fC2 .E/ D R R C C and jf1 j.E/ D fC1 .E/ C f1 .E/ D fC2 .E/ C E f1 d D E f2 d R f2 .E/ D jf2 j.E/ D E jf jd. Therefore, we can define ‰ W L1 ./ ! ca.A/ by ‰.Œf / D .f / D f . ‰ embeds the space L1 ./ into the subspace ca .A/  ca.A/ of all -additive measures, absolutely continuous with respect to . The following theorem, allows us to show that ‰.L1 .// D ca .A/; so that ‰ is an isomorphism between .L1 ./; k  k1 / and .ca .A/; k  k/, where k  k is the norm defined in Theorem 1.23 (for Radon–Nikodym theorem, may consult references [62], Theorem III.10.2 and [85], Theorems 1.101 and 1.111). Theorem 1.40 (Radon–Nikodym’s theorem). Let  W A ! RC be a -additive and -finite positive measure and let  be a -additive, -finite and -continuous measure on A. There exists an A-measurable application f W X ! R such that Z .A/ D f d; for every A 2 A: A

The function f is unique modulo the equality -almost everywhere. In this statement, to remark that, according to notations in R R R it is convenient Proposition 1.25, A f d D A f C d  A f  d shows that at least one of the functions f C or f  is integrable on A. The proof from below follows the ideas of [93] (see Theorems A and B, sect. 31, p. 128); it will be achieved in several stages and require the following lemma: Lemma 1.41. Let  and  be two -additive positive measures defined on A; let us suppose that  is finite, that  is -continuous and .X/ > 0. There exist " > 0 and A 2 A such that .A/ > 0 and ."/C is concentrated on the set A.

30

Chapter 1 Weak Compactness in Measure Spaces

Proof. Suppose that, for S every n 2 N  , .  n1  /C is concentrated on the set An 2 A and let D D n2N An . For every n 2 N  , D c  Acn ; therefore, .  n1  /C .D c / D 0 and then .  n1  /.D c /  0. Therefore 0  .D c / 

1  .D c /; n

for every n 2 N  :

Because .D c / < C1, .D c / D 0. Since .X/ > 0, .D/ > 0 and, as  , .D/ > 0. So, there is at least one n 2 N  such that .An / > 0. " D n1 and An satisfy then the conditions of lemma. Proof of theorem. (a) Firstly, we suppose that  and  are positive and finite. Let ³ ² Z 1 f d  .A/; for every A 2 A : F D f W X ! RC W f 2 L ./ and A

¯ Let ˛ D sup X f d W f 2 F . Because  is finite, ˛  .X/ < C1. R Let .fn /n2N   F such that supn X fn d D ˛ and let fN D supn fn . For every n 2 N  , let hn D sup¹f1 ; f2 ;    ; fn º 2 L1 ./; then hn " fN. According to Lebesgue monotone convergence theorem, Z Z N f d D lim hn d; for every A 2 A and therefore ®R

n

A

(1)

A

Z X

fNd  sup n

Z X

fn d D ˛ < C1:

(2)

Now, we show that fN 2 F . Let A be arbitrary in A and let n 2 N  ; we note A1 D ¹x 2 A W hn .x/ D f1 .x/º A2 D ¹x 2 A W hn .x/ D f2 .x/ > f1 .x/º :: : An D ¹x 2 A W hn .x/ D fn .x/ > fi .x/ for every i < nº The sets .Ai /i n  A are disjoint and [i n Ai D A. So we have: Z Z Z Z hn d D f1 d C f2 d C    C fn d A

A1

A2

An

 .A1 / C .A2 / C    C .An /  .A/: Then hn 2 F , for every n 2 N  . According to (1) Z Z N hn d  .A/: f d D lim A

n!1 A

(3)

Section 1.2 Radon–Nikodym Theorem. The Dual of L1

31

R By (3), X fNd  .X/ < C1: Therefore fN is -integrable and, using again (3), fN 2 F . Then Z fNd  ˛: (4) X

By (2) and (4), Z X

fNd D ˛:

(5)

Now, we will show that fN is the solution to the stated problem. Let D   fN  R, where fN   indicates the positive measure which, to any E 2 A, associates E fNd. Since fN 2 F , is a measure positive, finite and -continuous. Therefore, according to the previous lemma, if .X/ > 0, then there exist " > 0 and A 2 A such that .A/ > 0 and .  "/C is concentrated on A. Since .  "/ is concentrated on Ac , for every B 2 A, 0  .  "/C .B/ D .  "/C .B \ A/ D .  "/.A \ B/: Therefore we have

Z

".A \ B/  .A \ B/ D .A \ B/ 

A\B

fNd;

for every B 2 A: (6)

The function g D fN C " A is -integrable and, by (6), Z Z Z Z N N gd D f d C ".A \ B/ D f d C ".A \ B/ C B

B

A\B

fNd

BnA

 .A \ B/ C .B n A/ D .B/; for every B 2 A: R Consequently, g 2 F and then X gd  ˛. On the other hand, according to (5), Z Z gd D fNd C ".A/ D ˛ C ".A/ > ˛ X

X

which is impossible with the choice of fN. Therefore .X/ D 0 and so  D fN  . The uniquenessR of f is easy R to demonstrate. Let us suppose that, for every f d D A 2 A, .A/ D 1 A A f2 d and let B D ¹x 2 X W f1 .x/ > f2 .x/º. R Then B .f1  f2 /d D 0 and so .B/ D 0. It means that f1  f2 -a.e. Similarly, f2  f1 a.e. and then f1 D f2 ; -almost everywhere. The theorem is therefore demonstrated in the case (a).

32

Chapter 1 Weak Compactness in Measure Spaces

(b) Let now  and  be positive and -finite. There exists an A-partition .Xn /n2N of X such that .Xn / < C1 and .Xn / < C1, for every n 2 N. For every n, let An D ¹A \ Xn W A 2 Aº, n .A \ Xn / D .A \ Xn / and n .A \ Xn / D .A \ Xn /. Then n n , for every n 2 N. Therefore, by (a), there exists fn 2 L1C .Xn ; An ; n / such that n D fn  n . We define f W X ! RC letting f .x/ D fn .x/, if x 2 Xn . Then, f is A-measurable and positive and, for every A 2 A, Z XZ XZ X f d D f d D fn dn D n .An / D .A/: A

n2N

A\Xn

n2N

A\Xn

n2N

The proof of the uniqueness of f is identical to that of (a). (c) Now let  be a positive -finite measure and let  be -finite signed measure with  . Then  D C   , where C and  are -finite. According to Remark 1.29, C and  satisfy the hypothesis of (b). Therefore, there exist two A-measurable mappings g and h such that Z Z C  gd and  .A/ D hd; for every A 2 A:  .A/ D A

A

Since  can take only one of the values C1 or 1, at least one of the mappings g or h is -integrable. If f D g  h, then we have Z f d; for every A 2 A: .A/ D A

R Definition 1.42. The unique function f for which .E/ D E f d, for every E 2 A is called Radon–Nikodym derivative of  with respect to  and it is denoted by f D d . One will also write  D f  . d In the sequel, in this section, we will give some important consequences of Radon–Nikodym theorem. If we restrict ourselves to integrable functions, we have the already announced isomorphism theorem: Theorem 1.43. Let  be a -additive, -finite and positive measure. Then, the mapping ‰ W L1 ./ ! ca.A/, defined by ‰.f / D  D f  , for every f 2 L1 ./, is an isometric isomorphism between the normed spaces L1 ./ and ca .A/. Proof. Obviously, ‰ is a bijective linear map. According to Radon–Nikodym theorem, ‰.L1 .// D ca .A/. Moreover, according to Proposition 1.25, Z jf jd D jj.X/ D kk D k‰.f /k; for every f 2 L1 ./: kf k1 D X

Section 1.2 Radon–Nikodym Theorem. The Dual of L1

33

P Example 1.44. Let  W P .N/ ! RC ; .A/ D n2A ın .A/, be the measure given in Remark 1.32. Then  is -additive and positive. We can easily see that  , for every  2 ba.P .N//. The space of -integrable functions, L1 ./, is the space of all absolutely summable sequences, i.e. ´ μ X jxn j < C1 : L1 ./ D `1 D .xn /n2N  R W n2N

If x D .xn /n2N 2 `1 , then ‰.x/ D x where Z X xd D xn : x .E/ D E

n2E

For every  2 ca.P .N// D ca .P .N// and every n 2 N, let’s define xn by xn D .¹nº/. Then x D .xn /n2N 2 `1 and ‰.x/ D x . Definition 1.45. Two signed measures ;  2 ca.A/ are called singular or orthogonal if there exists a set D 2 A such that jj is concentrated on D and jj is concentrated on D c D X n D . This is denoted by  ? . We remark that, if ¹C ;  º is the Jordan decomposition of , then C ?  . Proposition 1.46. The measures ;  2 ca.A/ are singular if and only if, for every " > 0, there exists A" 2 A such that jj.A" / < " and jj.X n A" / < ". Proof. The implication “H)” is obvious. (H : For every n 2 N, let " D 21n ; there exists An 2 A such that jj.An / < and jj.X n An / < 21n . Let A D lim sup An D n

1 [ 1 \

1 2n

Ak :

nD1 kDn 1 [

jj.A/ D lim jj n

kDn

!

Ak

 lim

1 X

n

jj.Ak /  lim n

kDn

1 2n1

D 0:

  jj.X n A/ D jj lim inf .X n An /  lim inf jj.X n An / D 0: n

n

Theorem 1.47 (Lebesgue’s decomposition theorem; see Theorem III.4.14 in [62] and 1.115 in [85]). Let  be a -additive, -finite and positive measure on A and let  be a -additive and -finite measure on A. Then there exist two -additive and -finite signed measures ; on A, such that

34

Chapter 1 Weak Compactness in Measure Spaces

.i/  D C . .ii/ is -continuous. .iii/ and  are singular measures. These two measures are uniquely determined. Proof. (a) Firstly, let ;  2 caC .A/. If  D  C , then it is clear that  . Therefore, from Theorem 1.40, there exists g 2 L1 ./ such that Z Z Z gd  D gd C gd: .A/ D A

A

A

For all A 2 A, 0  .A/  .A/; so that, -a.e., 0  g  1 which means that ¹x 2 X W g.x/ < 0º [ ¹x 2 X W g.x/ > 1º is a set of -measure zero and consequently of -measure zero. 0 Let now R B D ¹x R2 X W g.x/ D 1º and B D ¹x 2 X W 0 < g.x/ < 1º. Then .B/ D B gd C B gd D .B/ C .B/; since  is finite, .B/ D 0. Let’s put, for every A 2 A, .A/ D .A \ B/ and .A/ D .A \ B 0 /. Then is concentrated on B and .B/ D 0; therefore and  are singular measures. Since .B [ B 0 /c is a set of -measure zero, .A/ D .A \ B 0 / C .A \ B/ D .A/ C .A/: Now let us show that . R \ B 0 / D A\B 0 gd. Therefore R For any A 2 A with .A/ D 0, .A/ D .A 0 0 A\B 0 .1  g/d D 0 and, since g < 1 on B , .A \ B / D .A/ D 0. In order to demonstrate the uniqueness, let’s put  D C D 1 C 1 . Then  1 D 1  . The measures 1  and  are singular; therefore there exists D 2 A such that j 1  j is concentrated on the set D and .D/ D 0. On the other hand  1 . Consequently, since .D/ D 0, j  1 j.D/ D j 1  j.D/ D 0; because j 1  j is concentrated on the set D, it follows that 1 D and then 1 D . (b) Now we suppose that  and  are two -additive, -finite and positive measures.SThen, we will appeal to the previous case, considering an A-partition of X D 1 nD1 Xn with  and  be finite on every Xn . If n and n are the restrictions of  and  respectively to .Xn ; An /.An D A \ Xn /. Then n and n belong to caC .An / and, from (a), there exist n and n such that n D n C n where n and n satisfy the properties (ii) and (iii); i.e. n n and n ? n . Let ; W A ! RC ; defined by .A/ D

1 X 1

n .A \ Xn /

and

.A/ D

1 X 1

n .A \ Xn /:

Section 1.2 Radon–Nikodym Theorem. The Dual of L1

35

P1 P1 for all A 2PA, .A/ D 1 .A \ Xn / D 1 n .A \ Xn / D PThen, 1 1 .A \ X / C .A \ X / D .A/ C .A/ therefore n n n 1 1 n  D C :

(4)

It is easy to see that and are -additive. S1 Indeed, if .Bi /i 2N  A is a sequence of pairwise disjoint sets and if B D 1 Bi , then .B/ D D

1 X

n .B nD1 1 1 X X

\ Xn / D

1 1 X X nD1 i D1 1 X

n .Bi \ Xn / D

i D1 nD1

n .Bi \ Xn / .Bi /:

i D1

Similarly, we demonstrate that is -additive. We still need to demonstrate that  and ? . Let A 2 A with .A/ D 0; for every n 2 N  , .A \ Xn / D n .A \ Xn / D 0 and therefore, since P 1n n , n .A \ Xn / D 0. Hence .A/ D nD1 n .A \ Xn / D 0I which means that : For every n 2 N  , n and n are singular measures; so thereSexists Dn 2 An such that n is concentrated on Dn and n .Dn / D 0. If D D n2N Dn then it is clear that .D/ D 0 and is concentrated on D. Therefore we obtain ? . (c) Let now  be a -additive, -finite and positive measure and let  be a -additive and -finite signed measure on A. With the help of the relation  D C   , the proof ends in an obvious way. Definition 1.48. The decomposition presented in the previous theorem is named Lebesgue decomposition of  relatively to . We will end this section with a final important application of the Radon– Nikodym theorem: the identification of the dual of L1 ./. Theorem 1.49 (Theorem IV.8.5 in [62]). Let  W A ! RC be a -additive, -finite and positive measure on A. There is an isometric isomorphism between the topological dual ŒL1 ./ of 1 L ./ and L1 ./. Proof. (a) Firstly, we suppose that  2 caC .A/, which means that .X/ < C1. 1 1 R For every f 2 L ./, 1let us define a map Ff W L ./ ! R by R Ff .g/ D fgd, for every g 2 L ./. Then F is linear and jF .g/j D j f f X X fgdj  kf k1 kgk1 . Therefore Ff is a linear continuous map and kFf k  kf k1 . We will show that kFf k D kf k1 .

36

Chapter 1 Weak Compactness in Measure Spaces

8 < C1; f .x/  kf k1  "; 1; f .x/  kf k1 C "; For every " > 0, let g.x/ D : 0; elsewhere: Let A D ¹xR 2 X W jf .x/j  kf k1  "º; then jgj D A . Therefore g 2 L1 ./ and kgk1 D jgjd D .A/. R R Hence jFf .g/j D j X fgdj D A jf jd  .A/.kf k1  "/ D kgk1 .kf k1  "/, from where kFf k  kf k1  ". Because " > 0 is arbitrary, it follows that kFf k D kf k1 . Let now T W L1 ./ ! ŒL1 ./ be the mapping defined by T .f / D Ff , for every f 2 L1 ./. Obviously, T is a linear continuous mapping of norm 1 and therefore it is an injective mapping. We still have to demonstrate that T is surjective. For every x  2 ŒL1 ./ , let  W A ! R be defined by .A/ D x  . A /, for all A 2 A. It is obvious that  is additive and because j.A/j  kx  k  k A k1 D kx  k  .A/  kx  k  .X/;

for every A 2 A; (1)

 is bounded; therefore  2 ba.A/. It is clear that  ; according to Proposition 1.31,  is a -additive measure. Therefore, according to the Radon–Nikodym theorem (Theorem 1.40),  admits a Radon–Nikodym Rderivative f with respect to ; it means that f 2 L1 ./ and x  . A / D .A/ D A f d, for every A 2 A. Let E.A/ be the space of all A-simple functions; it is clear that we have Z  x .g/ D fgd; for every g 2 E.A/: (2) X

If we prove that f 2 L1 ./, then R T is surjective. Indeed, in this case Ff W ! R, defined by Ff .g/ D X fgd, is continuous on L1 ./; because it coincides with x  on E.A/ which is dense in L1 ./, x  D Ff D T .f /. 1  Let’s prove that f 2 R L ./. Forevery n 2 N , let An D ¹x 2 X W jf.x/j  nº. Then n  .An /  An f d D x . An /  kx k  .An /: Therefore kx k  n, for every n 2 N with .An / ¤ 0. Because kx  k is finite, there exists n0 2 N such that .An0 / D 0. Therefore jf j  n0 , -a.e. which means that f 2 L1 ./. We have demonstrated the surjectivity of T which ends the proof in the case where  is a finite measure on A. (b) Let now  be a -additive, -finite and positive measure on A. Just as in ! ŒL1 ./ is defined by T .f / D Ff , where Ff W the case (a), T W L1 ./ R L1 ./ ! R; Ff .g/ D X fgd, for every g 2 L1 ./. Because kFf k  kf k1 , T is a linear and continuous mapping. L1 ./

Section 1.2 Radon–Nikodym Theorem. The Dual of L1

37

1 R Let N.T / D ¹f 2 L ./ W 1T .f / D 0º; then f 2 N.T / , Ff D 0 , X fgd D 0, for every g 2 L ./ , f D 0; -a.e. Therefore N.T / D ¹0º and so T is injective. We still need to proof the surjectivity of T . Because  is -finite, there exists an increasing sequence .En /n2N  A whose union is X such that .En / < C1, for every n 2 N. For every n 2 N, let L1En ./ be the Banach space of all integrable functions on En and, for every g 2 L1En ./, let gN 2 L1 ./ be the extension of g to X that takes the value 0 on X n En . For every x  2 ŒL1 ./ let xn W L1En ./ ! R defined by xn .g/ D x  .g/. N 1  Then xn is linear and continuous on LEn ./ and so, by (a), there exists fn 2 L1 En ./ such that

xn .g/

Z D

for every g 2 L1En ./:

fn gd; En

(3)

Obviously, kfn k1 D kFfn k D kxn k  kx  k, for every n 2 N.  .g/; hence For every n 2 N and every g 2 L1En ./, xn .g/ D xnC1 Z

Z En

fn gd D

for every g 2 L1En ./:

fnC1 gd; En

Then fn D fnC1 , -a.e. on En and therefore there exists f W X ! R, f D limn fNn , -a.e. on X. For every n 2 N, f  En D fn and so .jf j > kx  k/ D lim .jfn j > kx  k/ D 0: n

Therefore jf j  kx  k, -a.e. and then kf k1  kx  k < C1 which means that f 2 L1 ./. According to (3), for every n 2 N and every g 2 L1 ./ Z  g  f  E d: x .g  E / D n

n

X

kk1

On the other hand g  En ! g; therefore 

x .g/ D lim x .g  En / D lim n

Z

Z



n

X

f  g  En d D

Hence x  D Ff D T .f /, where f 2 L1 ./.

fgd: X

38

Chapter 1 Weak Compactness in Measure Spaces

Remark 1.50. .i/ Using an analogous method, we can demonstrate the following statement: Let 1 < p < C1 and q such that p1 C q1 D 1. For every -additive, -finite and positive measure  W A ! RC and R for every f 2 Lq ./, the mapping Ff W Lp ! R, defined by: Ff .g/ D X fgd, is a linear continuous mapping on Lp . The mapping f 7! Ff is an isometric isomorphism between the Banach spaces Lq and the dual .Lp / of Lp . .ii/ If  W A ! RC is a -additive and positive measure, but not a -finite one then J. Schwartz ([151]) showed, using a totally different method, that this result is still true for 1  p < C1. Example 1.51. In Example 1.44, we have introduced the measure  on N and we showed that L1 ./ is the space `1 of all absolutely summable sequences. From the previous theorem, it follows that the dual space of `1 is L1 ./, the space of all bounded sequences; this space is noted with `1 .

1.3

Convergences in L1 ./ and ca.A/

In this section, we will characterize the strong convergence in L1 ./ through of convergence in measure and weak convergence. In the last part, we will present the weak convergence in L1 ./ and in ca.A/. We first recall the definition of the strong convergence and that of the convergence in measure, as well as the notion of -equicontinuity of a family of functions. Let  W A ! RC be a -additive and positive measure and let .fn /n2N  1 L ./ and f 2 L1 ./; we say that: R .i/ .fn / is strongly convergent to f ” kfn  f k1 D X jfn  f jd ! 0. n!1

The strong convergence (or norm convergence) of the sequence .fn / to f is kk1

noted: fn ! f .ii/ .fn / converges in measure to f , .¹x W jfn .x/  f .x/j  "º/ ! 0, n!1

for every " > 0. The convergence in measure of the sequence .fn / to f is 

! f: denoted by fn  Definition 1.52. A family F  L1 ./ of integrable functions is -equicontinuous if, for every " > 0, there R exists ı > 0 such that, for any E 2 A with .E/ < ı and for all f 2 F , j E f dj < ":

Section 1.3 Convergences in L1 ./ and ca.A/

39

Remark 1.53. .i/ This definition is coherent with Definition 1.35. Indeed, let us R denote, for every f 2 F , f D ‰.f /, where f W A ! R; f .E/ D E f d; then F is -equicontinuous if and only if ¹f W f 2 F º is -equicontinuous. .ii/ According to Remark 1.29 and to (i), F  RL1 ./ is -equicontinuous if, for every " > 0, there exists ı > 0 such that E jf jd < ", for every E 2 A verifying .E/ < ı and for all f 2 F . .iii/ It is obvious that all finite family of functions F  L1 ./ is -equicontinuous. The following theorem gives us the conditions that will be added to the convergence in measure in order to obtain the strong convergence in L1 ./ (see Theorem III.6.15 of [62]). Theorem 1.54 (Vitali). Let  W A ! RC be a -additive and positive measure, let .fn /n2N  L1 ./ and let f W X ! R be an A-measurable function. Then kk1

f 2 L1 ./ and fn ! f if and only if the following conditions are accomplished: 

.i/ fn  !f, .ii/ ¹fn W n 2 Nº is -equicontinuous, .iii/ for R each " > 0, there exists A 2 A, with .A/ < C1 such that, X nA jfn jd < ", for every n 2 N. kk1

Proof. Firstly, we suppose that f 2 L1 ./ and fn ! f . The convergence in measure results immediately from the following inequality: .jfn  f j  "/  1" kfn  f k1 , for all n and for all " > 0. kk1

Let’s show that ¹fn W n 2 Nº is -equicontinuous. As fn ! f , for every " > 0, there exists n0 2 N such that, for all n  n0 ; kfn  f k1 < ":

(1)

According to Remark 1.53(iii), there exists ı > 0 such that, for all E 2 A such that .E/ < ı, Z jf jd < " (2) E

40

Chapter 1 Weak Compactness in Measure Spaces

and

Z E

jfn jd < ";

for every n  n0  1:

(3)

From (1) and (2), for every n  n0 , Z Z Z Z jfn jd  jfn  f jd C jf jd  kfn  f k1 C jf jd < 2": E

E

E

E

(4) (3) and (4) show that ¹fn W n 2 Nº is -equicontinuous. Similarly, there is A 2 A with .A/ < C1 such that Z jf jd < " and X nA

(5)

Z X nA

jfn jd < ";

for every n  n0  1:

(6)

From (1) and (5), we obtain, for every n  n0 , Z Z Z jfn jd  jfn  f jd C jf jd < kfn  f k1 C " < 2": (7) X nA

X nA

X nA

(6) and (7) lead to Condition (iii). Reciprocally, we assume (i), (ii) and (iii). According to (iii), for every " > 0, there exists A0 2 A with 0 < .A0 / < C1 satisfying Z jfn jd < "; for every n 2 N: (8) X nA0



Since fn  ! f , there exists a subsequence .fkn /n of .fn /n such that fkn ! f -a.e. According to Fatou’s lemma, we obtain Z R jf jd  lim inf n X nA0 jfkn jd  " and therefore X nA0

Z

X nA0

jf jd  ":

(9)

The family ¹fn W n 2 Nº is -equicontinuous; then there exists ı 20; "Œ such that, for every E 2 A satisfying .E/ < ı, Z jfn jd < "; for every n 2 N: (10) E

Section 1.3 Convergences in L1 ./ and ca.A/

41

Since the sequence .fp / is convergent in measure to f ,   " lim  jfp  f j > D 0; p .A0 / " / < ı < ", Consequently, we can find p0 2 N such that .jfp  f j > .A 0/ " for every p  p0 ; let "1 D .A0 / . According to (10), we have Z jfn jd < "; for every p  p0 and every n 2 N: (11) .jfp f j>"1 /

Applying again the Fatou’s lemma to the sequence .fkn / on the sets .jfp f j > "1 / with p  p0 , we obtain Z Z jf jd  lim inf jfkn jd  " n

.jfp f j>"1 /

.jfp f j>"1 /

and therefore Z .jfp f j>"1 /

jf j  ";

for every p  p0 :

(12)

Then, by (8), (9), (11) and (12), for every p  p0 , Z

Z X

jfp  f jd D Z 

Z

X nA0

X nA0

jfp  f jd C jfp  f jd A0 Z Z jfp jdC jf jdC X nA0

Z

C

A0 \.jfp f j"1 /

Z

 2" C

.jfp f j>"1 /

< 2" C 2" C

A0 \.jfp f j>"1 /

jfp  f jd Z jfp jd C

.jfp f j>"1 /

jfp f jd

jf jd C "1 .A0 /

"  .A0 / D 5": .A0 /

Therefore fp  f 2 L1 ./ from where f 2 L1 ./ and kfp  f k1 < 5"; kk1

Hence fp ! f .

for every p  p0 :

(13)

42

Chapter 1 Weak Compactness in Measure Spaces

Remark 1.55. According to Corollary 1.14, if  2 caC .A/, then .X/ < C1; therefore the Condition (iii) of previous theorem is satisfied. We have, therefore, the following theorem: Theorem. Let  2 caC .A/; the sequence .fn /  L1 ./ is strongly convergent to f 2 L1 ./ if and only if it is convergent in measure to f and it is equicontinuous. As we show in the following examples, the convergence in measure or the equicontinuity, separately taken, do not lead to the strong convergence. Example 1.56. (a) Let  be the Lebesgue measure on Œ0; 1 and let us define the mapping fn W Œ0; 1 ! R, letting fn D n

1 Œ0; n 



. Then fn  ! 0 but kfn  0k1 D 1 ¹ 0

and therefore ¹fn W n 2 Nº it is not -equicontinuous. (b) Let .rn /n2N be the Rademacher’s sequence: for every n 2 N; rn W Œ0; 1 ! R is defined by 8 S2n1 1 h 2k 2kC1 h < C1; t 2 kD0 ; 2n ; 2n h h rn .t/ D S n1 : 1; t 2 2 1 2kC1 ; 2kC2 : kD0 2n 2n ForR any " > 0, there exists ı D " such that, for every A 2 A with .A/ < ı; A jrn jd D .A/ < ı D ". Therefore, the sequence .rn / is -equicontinuous but it does not converge strongly in L1 ./. In fact, if we assume that .rn / is strongly convergent in L1 ./, then it is a Cauchy sequence in L1 ./; but it is not, because, for every n 2 N, 8 S2n1 1 h 4k 4kC1 h h 4kC3 4kC4 h ˆ ˆ ; [ 2nC1 ; 2nC1 ; 0 ; t 2 kD0 ˆ 2nC1 2nC1 ˆ < h h S n1 2 ; t 2 2kD0 1 4kC1 ; 4kC2 ; rn .t/  rnC1 .t/ D 2nC1 2nC1 ˆ h ˆ ˆ S2n1 1 4kC2 4kC3 h ˆ : 2; t 2 kD0 ; 2nC1 2nC1 and then

Z Œ0;1

jrn  rnC1 jd D 4  2n1 

1 2nC1

D 1 ¹ 0:

According to the previous theorem, one can also deduce that the Rademacher’s sequence .rn / is not convergent in measure on Œ0; 1.

Section 1.3 Convergences in L1 ./ and ca.A/

43

In the following theorem, we will characterize the weak convergence in L1 ./ (as references, see Theorem IV.8.7 of [62] or corollary on page 91 of [57]). Theorem 1.57. Let  W A ! RC be a -additive and -finite measure; a sequence .fn /  L1 ./ is weakly convergent to f 2 L1 ./ if and only if Z Z fn d ! f d; for every A 2 A: (*) A

A

Proof. According to Theorem 1.49, .fn / is weakly convergent to f if and only if, for every g 2 L1 ./, Z Z fn gd ! fgd: X

X

Since, for all A 2 A, A 2 L1 ./, the condition . / is verified. To proof the sufficiency of . /, we will first show that .fn /n2N is bounded in 1 L ./. R For every h 2 L1 ./, let h W A ! R be defined by h .A/ D A h d, for every A 2 A and let ‰ W L1 ./ ! ca.A/, defined by ‰.h/ D h . According to Theorem 1.43, ‰ is an isometric isomorphism between L1 ./ and ca .A/  ca.A/. Therefore, for every h 2 L1 ./, k‰.h/k D kh k D jh j.X/ D khk1 . Let’s note n D ‰.fn / and  D ‰.f /. By . /, for every A 2 A, n .A/ ! .A/. Therefore, according to Nikodym’s Theorem 1.37, we obtain that .n / is bounded in ca.A/ and, because ‰ is an isometric isomorphism, .fn / is bounded in L1 ./. Let M > 0 such that kfn k1  M;

for every n 2 N:

(1)

From . /, it follows immediately that, for every h 2 E.A/, Z X

Z fn hd !

f hd:

(2)

X

Let g be arbitrary in L1 ./. We must show that Z Z fn gd ! fgd: X

X

Since E.A/ is dense in L1 ./, for every " > 0, there exists h 2 E.A/ such that kg  hk1 < ":

(3)

44

Chapter 1 Weak Compactness in Measure Spaces

By (2), there exists n0 2 N such that, for every n  n0 , ˇ ˇZ Z ˇ ˇ ˇ fn hd  f hdˇˇ < ": ˇ X

(4)

X

Then, by (4), (1) and (3), for every n  n0 , we have ˇ ˇZ ˇ ˇZ ˇ ˇZ Z Z Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ fn gd  fgdˇˇ  ˇˇ fn gd fn hdˇˇ C ˇˇ fn hd f hdˇˇ ˇ X X X X X ˇ X ˇZ Z ˇ ˇ fgdˇˇ C ˇˇ f hd  X

X

 kfn k1  kg  hk1 C " C kf k1  kg  hk1  M  " C " C kf k1  " D ".M C kf k1 C 1/ who shows that .fn / converges weakly to f . Corollary 1.58 (Theorem IV.9.5 of [62] or Theorem 11, p. 90, of [57]). Let .n /  ca.A/ and let  2 ca.A/. .n / is weakly convergent to  in ca.A/ if and only if n .A/ ! .A/, for every A 2 A. Proof. If we suppose that .n / is weakly convergent to , then, for any continuous linear functional x  on ca.A/, x  .n / converges to x  ./ for n tending to infinity. If xA W ca.A/ ! R is defined by: xA ./ D .A/, for every  2 ca.A/, then xA is a linear functional on ca.A/. For every  2 ca.A/, jxA ./j D j.A/j  jj.A/  kk. Therefore xA 2 Œca.A/ and then xA .n / ! xA ./ which means that n .A/ ! .A/, for every A 2 A. Conversely, let .n /  ca.A/ and let  2 ca.A/ with, n .A/ ! .A/, for every A 2 A. The Nikodym’s Theorem 1.37 says that .n / is bounded in .ca.A/; k:k/. Let now  W A ! RC ; defined by: 1 X 1 jn j.A/; .A/ D 2n

for every A 2 A:

nD1

Therefore, through a similar proof to that of Nikodym’s theorem, we show that  2 caC .A/. Because, for all n 2 N and for all A 2 A, jn .A/j  2n  .A/, n , for every n 2 N. Therefore, according to Radon–Nikodym’s theorem 1 (see R Theorem 1.40), for every n 2 N, there exists fn 2 L ./ such that, n .A/ D A fn d, for every A 2 A. As stated by Vitali–Hahn–Saks’ theorem (see Theorem 1.36 (iii)),   and 1 so, R from Radon–Nikodym’s theorem, there exists f 2 L ./ such that, .A/ D A f d, for every A 2 A.

Section 1.3 Convergences in L1 ./ and ca.A/

45

R R Then, for all A 2 A; A fn d D n .A/ ! .A/ D A f d and so, according to Theorem 1.57, the sequence .fn / is weakly convergent to f in the space L1 ./. Since L1 ./ is a normed space isometrically isomorphic with ca .A/ (see Theorem 1.43), .n / converges weakly to  in ca .A/ and so in ca.A/. Corollary 1.59 (Theorem IV.9.4 of [62] or Theorem 12, p. 91, [57]). ca.A/ is sequentially weakly complete. Proof. Let .n /  ca.A/ be a weakly Cauchy sequence; this means that, for every x  2 Œca.A/ , .x  .n //n2N is a Cauchy sequence in R. Let now, for every A 2 A, xA 2 Œca.A/ , defined by xA ./ D .A/, for every  2 ca.A/. Then xA .n / D n .A/ and .n .A//n2N is a Cauchy sequence in R, for every A 2 A. We note, for all A 2 A, .A/ D limn n .A/. According to Nikodym’s theorem (see Theorem 1.37),  2 ca.A/ and since n .A/ ! .A/, for all A 2 A, the previous corollary assures that .n / converges weakly to . Remark 1.60. Let  W A ! RC be a -additive and -finite measure. ca .A/ is a strongly closed subspace of ca.A/; therefore it is a weakly closed one. It follows that ca .A/ is weakly sequentially complete and therefore L1 ./ is weakly sequentially complete, too. It is known that the weak topology coincides with the strong topology on a normed space if and only if it is of finite dimension. However one may happen in an infinite dimensional space that every weakly convergent sequence to be strongly convergent. This phenomenon occurs in the space `1 . Theorem 1.61 (Schur; see Corollary IV.8.14 of [62]). Let  W P .N/ ! RC be the measure of Example 1.44 and let L1 ./ D `1 be the space P of all absolutely convergent sequences x D .xn /  R, with the norm kxk D n2N jxn j. A sequence .x p /p2N  .`1 ; k  k1 / is strongly convergent to x 2 `1 if and only if .x p / is weakly convergent to x. Proof. For all normed space, the strong convergence implies the weak convergence. Conversely, let .x p /p2N be a sequence weakly convergent to x D .xn /n2N p in `1 ; we note that, for every p 2 N, x p D .xn /n2N . We prove that the sequence .x p /p2N is strongly convergent to x D .xn /n2N by using the Vitali’s Theorem 1.54.

46

Chapter 1 Weak Compactness in Measure Spaces

For every n 2 N; ¹nº 2 P .N/ and so, according to Theorem 1.57, Z Z p x d ! xd; ¹nº

¹nº

p which means that .xn /p converges to xn , for every n p We show that xn ! xn uniformly in n 2 N. p!1

2 N.

R .N/ ! R defined by p .A/ D A x p d D p 2 N, letR p ;  W PP PFor every p n2A xn and .A/ D A xd D n2A xn , for every A  N. R Then .p /R ca.P .N// and, according to Theorem 1.57, p .A/ D A x p d converges to A xd D .A/, for every A 2 P .N/. According to Nikodym’s theorem (1.37), the family ¹p W p 2 Nº is uniformly -additive, and from Theorems 1.36 and 1.34, ¹p W p 2 Nº is -equicontinuous. Then the condition (ii) of Vitali’s theorem is satisfied. Now, since ¹p W p 2 Nº is uniformly -additive, for every " > 0, there exists n0 2 N such that, for all n  n0 , ˇ n ˇ ˇX ˇ ˇ ˇ p .¹kº/  p .N/ˇ < "; for all p 2 N (1) ˇ ˇ ˇ kD1

what is equivalent with ˇR ˇ ˇ ˇ ˇ p ˇp .¹n C 1; n C 2; : : : º/ˇ D ˇˇ Nn¹0;1;:::;nº x dˇ ˇP pˇ D ˇ k>n xk ˇ < "; 8p 2 N;

8n  n0 : (2)

Therefore ˇ ˇ ˇ X X p ˇˇ ˇ p p xk  xk ˇ < 2"; for every p 2 N; and every n  n0 C 1: jxn j D ˇ ˇ ˇ k>n1

k>n

(3) We recall that x 2 `1 ; so we can choose n0 such as jxn j < ";

for every n  n0 C 1:

(4)

By (3) and (4), we have, jxnp  xn j < 3";

for every p 2 N;

and every n  n0 C 1:

(5)

p

Since .xn / is punctually convergent to xn , there exists p0 2 N such that jxnp  xn j < ";

for every p  p0 ;

and every n  n0 :

(6)

Section 1.4 Weak Compactness in ca.A/ and L1 ./

47 p

By (5) and (6), we obtain that, for all p  p0 and all n 2 N, jxn  xn j < p 3" which means that .x p / D .xn /p converges uniformly in n to x D .xn / in .N; P .N/; /. Therefore .x p / is convergent in measure to x. Moreover, by (2), the condition (iii) of Vitali’s theorem is satisfied and so, the sequence .x p / is strongly convergent to x.

1.4

Weak Compactness in ca.A/ and L1 ./

Lp ./ is a reflexive space for all p 21; C1Œ. This is an easy consequence of the theorem given in Remark 1.50. In this situation, a set K  Lp ./ is relatively weakly compact if and only if K is a bounded set. For the space L1 ./, which is not reflexive, the issue of weak compactness is more complicated. We begin with the following lemma (see lemma at page 91 in [57]): Lemma 1.62. Let G be an algebra which generates the -algebra A and let .n /  ca.A/ be a sequence of measures. The limit limn n .A/ 2 R exists, for all A 2 A, if and only if the following two conditions are verified: (1) limn n .E/ 2 R, (2) ¹n W n 2 Nº

for every E 2 G ,

is uniformly -additive.

Proof. The necessity of (1) is obvious and that of condition (2) is a consequence of the Nikodym’s theorem (see Theorem 1.37). To show that the conditions (1) and (2) are also sufficient, we use the fact that -algebra generated by an algebra is just the monotone class generated by this algebra [93, Th.B, p. 27]. Let L D ¹A 2 A W there exists limn n .A/ 2 Rº. Then, (a) G  L  A and (b) L is a monotone class of sets. Indeed, let .Em /  L be a monotone sequence such that S E D limm Em . If we suppose that the sequence is increasing, then ED 1 0 Em and, since ¹n W n 2 Nº is uniformly   additive, for every " > 0, there exists m0 2 N such that ˇ ˇ m1 ˇ ˇ X ˇ ˇ n .Ei C1 n Ei /ˇ < "; for every m  m0 and every n 2 N ˇn .E/  ˇ ˇ i D0

which can be written: jn .E/  n .Em /j < ";

for every m  m0

and every n 2 N: (1)

48

Chapter 1 Weak Compactness in Measure Spaces

It means that .n .Em //m converges to n .E/, uniformly in n 2 N. We have the same result if we assume that the sequence .Em /m is decreasing to E. If Em0 2 L, then .n .Em0 //n2N is a Cauchy sequence; then there exists n0 2 N such that, jp .Em0 /  q .Em0 /j < ";

for every p; q  n0 :

(2)

Therefore, by (1) and (2), for every p; q  n0 ; jp .E/  q .E/j  jp .E/  p .Em0 /j C jp .Em0 /  q .Em0 /j C jq .Em0 /  q .E/j < 3": Then .p .E//p2N is convergent in R and therefore E 2 L; this proves (b). (c) L is a monotone class containing G . Then the monotone class generated by G , M.G /, is contained in L. But, the -algebra .G / D A is just the monotone class M.G /  L  A. Which proves the result. Remark 1.63. As we show in the following example, the condition of “uniformly -additivity” in the previous lemma is indispensable. Let G D ¹A  N W A finite or N n A finiteº; G is an algebra that generates the -algebra P .N/. For every n 2 N, let n D ın the unit mass concentrated at n. Then .n /  ca.P .N// and ² 0; E is finite; for all E 2 G : lim n .E/ D 1; X n E is finite; n!C1 If P is the set of all even numbers of N, then the limit limn!C1 n .P / doesn’t exist. It is clear that ¹n W n 2 Nº is not uniformly -additive on N. Using the previous lemma, we can give the following weak compactness result on ca.A/ (see Theorems IV.9. 1 and 2 of [62] and Theorem 13, p. 92, of [57]). Theorem 1.64. Let K  ca.A/; the following conditions are equivalent: .i/ K is relatively weakly compact in (ca.A/; k:k), .ii/ K is bounded and uniformly -additive, .iii/ K is bounded and there exists a measure  2 caC .A/ such that K is equicontinuous.

Section 1.4 Weak Compactness in ca.A/ and L1 ./

49

Proof. (i)H)(iii) We start by noticing that, in every normed space .X; k:k/, the relatively weakly compact sets are bounded. Let then M > 0 such that, for every  2 K, kk  M . Let us show that for every " > 0; there exist ¹1 ; : : : ; n º  K and ı > 0 such that (*) jk j.E/ < ı; for every k D 1; : : : ; n; H) j.E/j < "; for every  2 K: By reducing it to absurd, let us suppose that the condition . / is not satisfied: there is " > 0 such that, if we fix 1 2 K and ı1 D 12 , there exist E1 2 A and 2 2 K with j1 j.E1 / < 12 and j2 .E1 /j  ". Similarly, for ¹1 ; 2 º  K and ı2 D 212 , there exist E2 2 A and 3 2 K with j1 j.E2 / < 212 ; j2 j.E2 / < 212 and j3 .E2 /j  ": 1 , there exist En1 2 A Recursively, for ¹1 ; : : : ; n1 º  K and ın1 D 2n1 1 and n 2 K with ji j.En1 / < 2n1 ; 8i D 1; : : : ; n1, and jn .En1 /j  ": Therefore, we have found two sequences: .n /  K and .En /  A such that, for every n 2 N, ji j.En /
0 such that ˇ ˇ ˇk .A/ˇ < "; for every A 2 A with .A/ < ı; and for every n 2 N: (3) n For all p 2 N, p 1 1 X   X X  1 1 M j j E j j E  C  EkpC1 1 D kn kpC1 1 kn kpC1 1 n n 2 2 2n nD1

nD1

1 M p M 1  p   k 1 C p D k C p: 2 2 pC1 2 2 2 pC1

nDpC1

50

Chapter 1 Weak Compactness in Measure Spaces

Therefore .EkpC1 1 / ! 0 and so there exists p0 2 N such that, for all p  p0 , .EkpC1 1 / < ı. According to (3), ˇ  ˇ ˇk Ek 1 ˇ < "; for all n 2 N and for all p  p0 : n pC1 For n D p C 1, we obtain, ˇ  ˇ ˇ k EkpC1 1 ˇ < "; pC1

for every p  p0

(4)

which contradicts (2). Then, we have demonstrated . /. Now we use inductively the condition . /: ² " D 1; there exist ¹1 ; : : : ; n1 º  K such that, if A 2 A satisfies j j.A/ D : : : D jn1 j.A/ D 0; then j.A/j < 1; for any  2 K: ² 1 1 " D 2 ; there exist ¹n1 C1 ; : : : ; n2 º  K such that, if A 2 A satisfies jn1 C1 j.A/ D : : : D jn2 j.A/ D 0; then j.A/j < 12 ; for any  2 K: .......................... ´ " D p1 ; there exist ¹np1 C1 ; : : : ; np º  K s. t., if A 2 A satisfies jnp1 C1 j.E/ D : : : D jnp j.E/ D 0; then j.E/j < p1 ; for any  2 K: .......................... For every p 2 N, let p D np1 C1 . Then the sequence . p /p  K satisfies: j p j.A/ D 0; for any p 2 N H) .A/ D 0; for every  2 K: Let now  D

P1

1 pD1 2p j p j;

(5)

 2 caC .A/: By (5),

 ;

for every  2 K:

(6)

Let us show, by reducing it to absurd, that K is -equicontinuous. If K is not -equicontinuous, according to Theorem 1.34, it results that K is not -equicontinuous at ; and therefore there exists " > 0 such that, for every ı > 0, there exist Fı 2 A and ı 2 K satisfying: .Fı / < ı

and

j ı .Fı /j  ":

For ı D n1 , n 2 N, we obtain two sequences .Fn /n  A; . n /n  K such that .Fn / ! 0

and

j n .Fn /j  ";

for all n 2 N:

(7)

Because K is weakly sequentially compact (see Eberlein–Šmulian theo rem [62], V.6.1), perhaps extracting a subsequence, we may assume that . n /n is weakly convergent to a measure 2 ca.A/. By (6), n , for all n 2 N; then . n /n satisfies the hypotheses of the Vitali–Hahn–Saks theorem (see Theorem 1.36). Therefore . n / is -equicontinuous and this contradicts (7).

Section 1.4 Weak Compactness in ca.A/ and L1 ./

51

(iii)H)(ii) The implication is a consequence of Theorem 1.34. (ii)H)(i) Let K  ca.A/ be a bounded and uniformly -additive family of measures; let .n /n2N be any sequence in K and let D

1 X 1 jn jI 2n

 2 caC .A/:

nD1

For every n 2 N; n  and, according to Radon–Nikodym (Theorem 1.40), for any n 2 N, there exists a function fn 2 L1 ./ such that Z fn d; for every E 2 A: n .E/ D E

For all n 2 N, there exists a sequence of simple functions .fpn /p2N  E.A/ such that fpn ! fn , -a.e. If p!1

fnp D

m.n;p/ X kD1

n;p

ak

n;p

Ak

; n;p

then, for every n 2 N, let S1us denote with Fn D ¹Ak W p 2 N; k D 1; : : : ; m.n; p/º and let F D nD1 Fn  A. F is a countably family of sets and therefore the algebra B generated by F , is also countable. So, let B D ¹E1 ; : : : ; Ep ; : : : º. By using a diagonal method, we obtain a subsequence .kn /n2N such that .kn .Ep //n2N is convergent in R, for every p 2 N. .kn /n being still uniformly -additive, .kn .E//n2N is convergent, for every E 2 .B/ - the -algebra generated by B (see Lemma 1.62). We note by 1 .B/ the -completion of .B/ and by L1 .; 1 .B// the space R of all 1 .B/-measurable functions f , with X jf jd < C1. p Then fn is .B/-measurable, for all p; n 2 N; therefore, for every n, fn is 1 .B/-measurable. Hence we have: .fn /  L1 .; 1 .B//  L1 ./ and, 1 according to Theorem R 1.57, .fkn / is weakly convergent in L .; 1 .B// (we need to remark that E fkn d ! 0, for every E 2 A with .E/ D 0, and R therefore . E fkn d/n2N is convergent, for every E 2 1 .B/). Since L1 .; 1 .B//  L1 ./, .fkn / is weakly convergent in L1 ./; it means that .kn / is weakly convergent in ca.A/ (see Remark 1.43). Therefore, K is weakly sequentially compact and then we can apply again Eberlein–Šmulian theorem ([62], V.6.1) to obtain that it is relatively weakly compact.

52

Chapter 1 Weak Compactness in Measure Spaces

Theorem 1.65 (Dunford–Pettis; see Theorem 2.54 of [85] and [57], p. 93). Let  2 caC .A/ and K  L1 ./. The following conditions are equivalent: .i/ K is relatively weakly compact in (L1 ./; k  k1 ), .ii/ K is bounded and the integrals of members of K are uniformly -additive, .iii/ K is bounded and -equicontinuous. Proof. According to the notations introduced at the beginning of the Section 1.2, by ‰.f / D f , where, for every f 2 L1 ./, let ‰ W L1 ./ ! ca.A/ defined R f is defined by f .A/ D A f d, for every A 2 A. ‰ is an isometric isomorphism between the normed spaces L1 ./ and ca .A/  ca.A/ (Theorem 1.43). Therefore the subset K  L1 ./ is relatively weakly compact in .L1 ./; k  k1 / if and only if K D ‰.K/ is relatively weakly compact in ca .A/; since ca .A/ is a closed subset of ca.A/, this is equivalent with ‰.K/ is relatively weakly compact in ca.A/. Obviously, the condition (ii) of Theorem 1.64 on K is equivalent with the above condition (ii) on K. For the conditions (iii) of these two theorems, we need to demonstrate the equivalence between the two following conditions: (a) K is -equicontinuous; (b) 90 2 caC .A/ for which K is 0 -equicontinuous. We can rewrite these conditions: R (a) For every " > 0, there exists ı > 0 such that A jf jd < ", for every A 2 A with .A/ < ı and every f 2 K. 0 C (b) There R exists  2 ca .A/ such that, for 0every " > 0, there exists ı > 0 such that A jf jd < ", for every A 2 A with  .A/ < ı and every f 2 K. Obviously, (a)H)(b) (with 0 D ). (b)H)(a) Let 0 D 01 C 02 be the Lebesgue decomposition of 0 relatively to  (Theorem 1.47) where 01 ; 02 2 ca.A/, 01  and 02 ? ; therefore there exists a set A0 2 A such that 02 .A0 / D 0 D .X n A0 /. According to (b), for every " > R 0, there exists ı1 > 0 such that, for every A 2 A with 0 .A/ < ı1 , we have A jf jd < ", for any f 2 K. Since 01 , there exists ı > 0 such that .A/ < ı ) 01 .A/ < ı1 . Then, for any A 2 A with .A/ < ı, 0 .A \ A0 / D 01 .A \ A0 / C 02 .A \ A0 /  01 .A/ < ı1

Section 1.5 The Bidual of L1 ./

and therefore

Z A\A0

or yet Z Z jf jd D A

53

A\A0

jf jd < ";

for every f 2 K

Z jf jdC

AnA0

Z jf jd D

A\A0

jf jd < "; for every f 2 K:

Which was to be demonstrated. Remark 1.66. .i/ If  W A ! RC is a -finite measure, then K  L1 ./ is relatively weakly compact if and only if .a/ K is bounded, .b/ K is -equicontinuous, .c/

8" > 0; 9A" 2 A with .A" / < C1 and f 2 K.

R X nA"

jf jd < ", for any

.ii/ If .fn /n2N  L1 ./ is weakly convergent, then K D ¹fn W n 2 Nº is -equicontinuous. Indeed, K is relatively weakly compact in .L1 ./; k  k1 / and therefore, by Theorem 1.65, K is -equicontinuous. .iii/ We can now restate the Vitali’s theorem (see Theorem 1.54) in the following manner: Let  2 caC .A/; a sequence .fn /n  L1 ./ is strongly convergent to f if and only if .fn / converges in measure and .fn / is weakly convergent to f .

1.5

The Bidual of L1 ./

Let  be a -additive, -finite, positive measure on the -algebra A, let L1 ./ be the space of all -integrable mappings, let L1 ./ be the dual space of L1 ./ and, in accordance with Definition 1.28, let ba .A/ D ¹ 2 ba.A/ W  º: In this section, we will identify ba .A/ with the bidual of L1 ./. Then, we will give the characteristic properties of the measures of this bidual and we will identify a purely finite-additive measure of ba.A/ as being a measure orthogonal

54

Chapter 1 Weak Compactness in Measure Spaces

on every measure of ca.A/. We will conclude this part with the Hewitt–Yosida decomposition of every additive measure in a -additive part and a purely finiteadditive one. Theorem 1.67 (Theorem IV.8.16 of [62]). The dual space of L1 ./ is isometrically isomorphic as normed space with ba .A/. Proof. Let x  2 ŒL1 ./ ; then x  W L1 ./ ! R is a continuous linear mapping on L1 ./. Since, for all A 2 A, A 2 L1 ./ , we can define x  W A ! R by: x  .A/ D x  . A /, for every A 2 A. Obviously, x  is additive. jx  .A/j  kx  k  k A k1  kx  k < C1;

for every A 2 A:

x  is therefore bounded. For every A 2 A with .A/ D 0, A D 0, -a.e. We have, therefore, x  .A/ D x  . A / D 0; therefore x  2 ba .A/. Let now S W ŒL1 ./ ! ba .A/ defined by: S.x  / D x  ;

for every x  2 ŒL1 ./ :

We will show that S is an isometric isomorphism of normed spaces.  It is evident that S is linear and that kS.x  /k D kx  k D jx  j.X/ ´ n X jx  . A /j W ¹A1 ; : : : ; An º D A  D sup i D1

μ partition of

i

X :

For every A-partition ¹A1 ; : : : ; An º of X, we can rewrite the sets in such a manner as x  . A /; : : : ; x  . Ap /  0 and 1

x  . A

pC1

Then n X

jx  . A /j D

p X

i

1

x  . A / 

n X

i

1

x  . A

i

/; : : : ; x  . An / < 0:

0 p X @ /Dx

pC1



X

p n X





 kx k 

A 

A

i i



1 pC1

1

1

D kx  k;

Ai



n X

1

A A i

pC1

Section 1.5 The Bidual of L1 ./

55

therefore kS.x  /k  kx  k;

for every x  2 ŒL1 ./ :

(1)

 S is surjective.   for every  2 ba .A/, PIndeed, Pnwe define x W E.A/ ! R letting x .f / D n i D1 ai .Ai /, for every f D 1 ai Ai 2 E.A/, where ¹a1 ; : : : ; an º  R and ¹A1 ; : : : ; An º is a A-partition of X. Then x  is a linear mapping on E.A/ and, since  , for every f 2 E.A/, ˇ n ˇ n ˇX ˇ X ˇ ˇ  ai .Ai /ˇ  jai j  j.Ai /j jx .f /j D ˇ ˇ ˇ 1

1

 sup¹jai j W .Ai / ¤ 0º   sup¹jai j W .Ai / ¤ 0º 

n X 1 n X

j.Ai /j j.Ai /j  kf k1  kk:

1

We have shown that: jx  .f /j  kk  kf k1 ;

for every f 2 E.A/:

(2)

Hence x  is a continuous linear mapping on E.A/. We can extend by conti1 nuity this mapping to L1 ./ D E.A/ . The extension, written again x  , is a continuous linear real mapping on .L1 ./; k  k1 /, therefore x  2 ŒL1 ./ . For every g 2 L1 ./, let us denote Z gd: x  .g/ D X

S.x  /

Then D x  , where, for every A 2 A, x  .A/ D x  . A / D .A/. Therefore, we obtain S.x  / D .  Now we show that the map S preserves the norm. For every f 2 L1 ./ with kk1

kf k1  1, there exists a sequence .fn /  E.A/ such that fn ! f . Then, by (2), jx  .f /j D lim jx  .fn /j  lim kk  kfn k1 D kk  kf k1  kk D kS.x  /k n

n

hence kx  k D

sup kf k1 1

jx  .f /j  kS.x  /k:

(3)

56

Chapter 1 Weak Compactness in Measure Spaces

From (1) and (3), we obtain: kS.x  /k D kx  k;

for every x  2 ŒL1 ./ :

S is therefore an isometric isomorphism between the normed spaces ŒL1 ./ and ba .A/: According to Theorem 1.49, we deduce the following corollary. Corollary 1.68. The space .ba .A/; k  k/ is the bidual of .L1 ./; k  k1 /. Remark 1.69. ca .A/  ba .A/ and the mapping ‰ W L1 ./ ! ba .A/, defined by ‰.f / D f   for every f 2 L1 ./, is the natural embedding of L1 ./ in its bidual. As a consequence of Radon–Nikodym theorem (1.40), we have shown (Theorem 1.43) that L1 ./ is isometric isomorphic with ca .A/. According to Alaoglu theorem [62, V.4.2] and its immediate consequence, the Goldstine’s theorem ([62] V.4.5 and 6), ‰.L1 .// is .L1 ./; ba .A// -dense in ba .A/. This will allow us to characterize the measures of ba .A/ through nets (generalized sequences). Theorem 1.70. Let  W A ! RC be a -additive and - finite measure and let  2 ba.A/.  2 ba .A/ ” there exists a net .fi /i 2I  L1 ./ such that Z .A/ D lim fi d; for every A 2 A: i

A

If this net is -equicontinuous, then  2 ca .A/. Proof. ()) According to Theorem 1.43 and with the notations already used, the mapping ‰ W L1 ./ ! ba .A/, defined by ‰.f / D f D f  , is an isometric isomorphism between the normed spaces L1 ./ and ca .A/. Let again ‰ be the natural embedding of L1 ./ in its bidual ba .A/ and let w be the topology .ba .A/; L1 .// on ba .A/; according to the previous remark, ca .A/ D ‰.L1 .// is w - dense in ba .A/ . Therefore, for all  2 ba .A/, there exists a w net .fi /i 2I  L1 ./ such that ‰.fi / ! . R Since, for all A R2 A; A 2 L1 ./, ‰.fi /. A / D E fi d ! .A/. Therefore .A/ D limi A fi d, for every A 2 A. 1 (() R for every A 2 A, .A/ D R Let now .fi /i 2I  L ./ be a net such that, limi A fi d. For every A 2 A with .A/ D 0; A fi d D 0; 8i 2 I ; therefore .A/ D 0, from where   and  2 ba .A/.

Section 1.5 The Bidual of L1 ./

57

Now, if .fi /i 2I is -equicontinuous, then, by Dunford–Pettis’ theorem (Theorem 1.65), ¹fi W i 2 I º is relatively weakly compact. So there exists aR subnet of .fi /i 2I weakly convergent to a function f 2 L1 ./. Then .A/ D A f d, for every A 2 A; therefore  D f: and, according to Proposition 1.31,  2 ca .A/. Remark 1.71. .i/ In accordance with the notations of the previous theorem, let i W A ! R, i D fi  . Therefore i .E/ ! .E/, for all E 2 A and  does not necessarily belong to ca.A/. This shows that the Nikodym’s theorem (1.37) is no longer true if we replace the sequences by nets. .ii/ Let X be a countable set, P .X/ P be the family of all subsets of X and 0 W P .X/ ! Œ0; C1, 0 .A/ D x2X ıx .A/, where ıx is the unit mass concentrated in x (0 .A/ gives the number of elements of A). Then 0 is a positive, -additive and -finite measure on P .X/. Let B.X/ D ¹f W X ! R W f is bounded on Xº and k  k W B.X/ ! RC ; kf k D supx2X jf .x/j. Then .B.X/; k  k/ .L1 .0 /; k  k1 / is a Banach space. According to Theorem 1.67, the dual space of .B.X/; k  k/ is ba0 .P .X//. Because 0 .A/ D 0 , A D ;; ba0 .P .X// D ba.P .X// ( 0 ; 8 2 ba.P .X//). Therefore .ba.P .X/; k  k/ is the dual space of .B.X/; k  k/. Definition 1.72. (Definition III.7.7 of [62]) A measure  2 baC .A/ is said to be purely finitely additive if the inequality 0    , with  countably additive, imply  D 0. Let  2 ba.A/ and let C ;  2 baC .A/ be the positive and the negative variation of  (see Definition 1.19).  is called purely finitely additive if C and  are purely finitely additive. Definition 1.73. Two additive measures ; on A are called orthogonal measures if, for every " > 0, there exists A" 2 A such that jj.A" / < " and j j.X n A" / < "; this is denoted by  ? . Remark 1.74. .i/ This definition is in according to Definition 1.45; indeed if  and are countable additives (; 2 ca.A/), then, according with Proposition 1.46, the following assertions are equivalent:

58

Chapter 1 Weak Compactness in Measure Spaces

.a/ There exists a set D 2 A such that jj is concentrated on D and j j is concentrated on D c D X n D. .b/ For every " > 0, there exists a set A" 2 A such that jj.A" / < " and j j.X n A" / < ". .ii/ If  and are only finitely additive, the previous properties are no longer equivalent. We have the following implication: .a/ ) .b/. As shown in the following example, the implication .b/ ) .a/ is not true. Example 1.75. Let X be an arbitrary set and let  W P .X/ ! ¹0; 1º be an additive measure such that .X/ D 1. Let us denote U D ¹A  X W .A/ D 1º. It is easy to see that U is an ultrafilter on X. Reciprocally, for every ultrafilter U on X, , defined by .A/ D 1 if A 2 U and .A/ D 0 if A … U, is an additive measure on X associated to the ultrafilter U. Let now F D ¹A  N W N n A is finite º. F is a filter on N; let U be the ultrafilter which contains F and let  be the additive measure associated to U.  is purely finitely additive. Indeed,  2 baC .P .N// and, for P every countably additive measure  2 caC .P .N// with 0    , .N/ D n2N .¹nº/ D 0 (if there were a number n0 2 N with .¹n0 º/ > 0 then .¹n0 º/ D 1 and so ¹n0 º 2 U what is impossible because N n ¹n0 º 2 FP U) . 1 Let m W P .N/ ! RC defined by m.A/ D n2A .nC1/2 if A ¤ ; and m.;/ D 0. It is clear that m 2 caC P.P .N//. 1We show that m ? . For every " > 0, there exists n0 2 N such that nn0 .nC1/2 < " and therefore m.A0 / < ", where A0 D ¹n0 ; n0 C 1; : : : º. N n A0 is finite so that A0 2 U; then N n A0 … U and so .N n A0 / D 0 < ". Therefore m ? . On the other hand, if there were a set B0 2 P .N/ with m.B0 / D 0 D .N n B0 /, then N n B0 … U so that B0 2 U and then .B0 / D 1; because m.B0 / D 0, B0 D ; and then N n B0 2 U which is impossible. So .b/ » .a/. We have seen in the previous example that the purely finitely additive measure  is orthogonal to the -additive measure m. The following theorem shows that this is a general property for all purely finitely additive measure (see [175]). Theorem 1.76. Let  2 ba.A/; then  is purely finitely additive if and only if  is orthogonal on every -additive measure  2 ca.A/. Proof. We can notice that  ?  if and only if  ? jj and then we can suppose that  2 caC .A/.

Section 1.5 The Bidual of L1 ./

59

(H)) a) Let  2 baC .A/ be a purely finitely additive measure and let  2 caC .A/. We first define W A ! RC , letting .E/ D inf¹.A/ C .E n A/ W A 2 A; A  Eº, for every E 2 A. Then 0  .E/  .E/;

for every E 2 A:

(1)

Therefore is bounded. is an additive measure. Indeed, for every E; F 2 A, such that E \ F D ; we note G D E [ F . For every " > 0, there exist A; B 2 A; A  E; B  F such that .A/ C .E n A/ < .E/ C " and .B/ C .F n B/ < .F / C ". If C D A [ B, C 2 A, C  G and .G/  .C / C .G n C / D .A/ C .B/ C .E n A/ C .F n B/ < .E/ C .F / C 2" from where .G/  .E/ C .F /:

(2)

On the other hand, there exists D 2 A with D  G such that .D/ C .G n D/ < .G/ C ". Then .G/ C " > .D/ C .G n D/ D .D \ E/ C .D \ F / C .E n .D \ E// C .F n .D \ F //  .E/ C .F / from where .E/ C .F /  .G/:

(3)

(1), (2) and (3) lead to .E [ F / D .E/ C .F / and so 2 baC .A/. Let us show that is -additive. For every sequence .En /n2N  A of pairwise disjoint sets, let us denote E D n n [1 1 En . From (1), 0  .E n [1 Ek /  .E n [1 Ek /, for every n 2 N. Because n n C  2 ca 1 Ek / ! 0 when n ! 1. Then .E/ D limn .[1 Ek / D Pn.A/, .E nP[1 limn 1 .Ek / D 1 .En /, and so, 2 caC .A/. From the definition of , for every E 2 A, 0  .E/  .E/. Since  is purely finitely additive, D 0. Therefore .X/ D 0 and then, for every " > 0, there exists A" 2 A such that .A" / C .X n A" / < ". Then .A" / < " and .X n A" / < ". So  ? . b) If  2 ba.A/ is purely finitely additive then C ;  are purely finitely additive and positive. From the part (a), C and  are orthogonal to . Hence, for every  2 caC .A/, and for all " > 0, there exist A" and B" 2 A such that C .A" / < 2" , .X n A" / < 2" and  .B" / < 2" , .X n B" / < 2" . Let C" D A" \ B" 2 A; then jj.C" / D C .A" \ B" / C  .A" \ B" /  C .A" / C  .B" / < " and .X n C" /  .X n A" / C .X n B" / < ". Therefore  ? . ((H) Suppose that, for every  2 caC .A/,  ? . Then, for every " > 0, there exists A" 2 A with jj.A" / < "; and so C .A" / < ";  .A" / < " and .X n A" / < " from where, for every  2 caC .A/, C ?  and  ? .

60

Chapter 1 Weak Compactness in Measure Spaces

Let now  2 caC .A/ with 0    C ; since C ? , for every " > 0, there exists A" 2 A such that C .A" / < " and .X n A" / < ". Hence, for every " > 0, .X/ D .A" / C .X n A" /  C .A" / C .X n A" / < 2"; then .X/ D 0 and so  D 0. Therefore C is purely finitely additive. Similarly we show that  is purely finitely additive and so  is purely finitely additive. Corollary 1.77. The subset bap .A/  ba.A/ of all purely finitely additive measures is a subspace of the real vector space ba.A/. Proof. For every 1 ; 2 2 bap .A/,  D 1 C 2 2 ba.A/. From the previous theorem, for any  2 ca.A/; 1 ?  and 2 ? . Therefore, for every " > 0, there exist A1 ; B1 2 A such that j1 j.A1 / < 2" and jj.X n A1 / < 2" , j2 j.A2 / < " " 2 and jj.X n A2 / < 2 . Since jj D j1 C 2 j  j1 j C j2 j, jj.A1 \ A2 /  j1 j.A1 /Cj2 j.A2 / < " and jj.X n.A1 \A2 //  jj.X nA1 /Cjj.X nA2 / < ": Therefore, for every  2 ca.A/,  ?  and, by previous theorem,  2 bap .A/. For every  2 bap .A/, for all a 2 R, let D a  , 2 ba.A/. Then j j D jaj  jj is evidently orthogonal to every  2 caC .A/, hence 2 bap .A/. Now we can present an important result of decomposition for the finitely additive measures. It was proved by Hewitt and Yosida in [175] (see also Theorem 1.12 in [85] and Theorem III.7.8 in [62]). Theorem 1.78 (Hewitt–Yosida). Each finitely additive measure  2 ba.A/ is uniquely decomposed as the sum of a countably additive and a purely finitely additive measure:  D  C  ; where  2 ca.A/ and  2 bap .A/: Proof. (i) Firstly, we suppose that  2 baC .A/. Let  D ¹ 2 caC .A/ W 0    º and let ˛ D sup 2 .X/; then 0  ˛  .X/ < C1: Let .n /n2N   be a sequence such that limn!1 n .X/ D ˛. We associate an increasing sequence .n /n2N   so that n .X/ ! ˛. Let 1 D 1 and 2 W A ! RC defined by: 2 .E/ D sup¹1 .A/ C 2 .E n A/ W A 2 A; A  Eº;

for every A 2 A:

For every E; A 2 A with A  E, 1 .A/ C 2 .E n A/  .A/ C .E n A/ D .E/. Therefore 0  1  2   and also, for every E 2 A, 2 .E/ 2 RC .

Section 1.5 The Bidual of L1 ./

61

(a) Let us show that 2 is additive. For every E; F 2 A with E \ F D ;, let G D E [ F . For any " > 0, there exist A; B 2 A; A  E; B  F so that 1 .A/ C 2 .E n A/ > 2 .E/  "; 1 .B/ C 2 .F n B/ > 2 .F /  ": Then C D A [ B  G and 2 .G/  1 .C / C 2 .G n C / D 1 .A/ C 1 .B/ C 2 .E n A/ C 2 .F n B/ > 2 .E/ C 2 .F /  2". Therefore 2 .E [ F /  2 .E/ C 2 .F /:

(1)

On the other hand, according to the definition of 2 .G/ D 2 .E [F /, for every " > 0, there exists D  G; D 2 A such that 2 .G/  " < 1 .D/ C 2 .G n D/ D 1 .D \E/C1 .D \F /C2 .E n.D \E//C2 .F n.D \F //  2 .E/C2 .F / and so 2 .E [ F /  2 .E/ C 2 .F /:

(2)

(1) and (2) show that 2 is finitely additive and therefore that 2 2 baC .A/. (b) Let us show that 2 is -additive.S Let .En /n2N  A be a sequence 1 n of pairwise disjoint sets and let E D nD1 En . .2 .[kD1 Ek //n is an inexists creasing bounded Sn sequence in R. Therefore, because 2 is additive, Sthere n limn 2 .E n kD1 Ek / D l  0. If l > 0 then, for every n, 2 .E S n kD1 Ek / > l > 0. Then, by definition of 2 , there exist An 2 A; An  E n nkD1 Ek such that, " ! # n [ 1 .An / C 2 Ek n An > l; for every n 2 N: En (3) kD1

But 1 ; 2 2 caC .A/ and so, 1 .An /  1 E n

n [

! Ek

! 0 and n!1

kD1

" 2

En

n [

!

#

Ek n An  2 E n

kD1

n [

! Ek

! 0; n!1

kD1

which is incompatible with (3). Then l D 0 and so 2 is -additive or 2 2 caC .A/. It is obvious that 0  1  2  

and 2  2 :

62

Chapter 1 Weak Compactness in Measure Spaces

We construct then, by recurrence, the sequence .n /n , where, for every n 2 N, n W A ! RC is defined by: n .E/ D sup¹n1 .A/ C n .E n A/ W A 2 A; A  Eº;

for every E 2 A:

With an analogous demonstration, we show that, for every n, n 2 caC .A/ and 0  1  : : :  n  : : :  

and

n  n :

.n /n   and, since n .X/  n .X/  ˛, we have: lim n .X/ D ˛: n

For every E 2 A, .n .E//n2N is an increasing sequence, bounded by .E/. So, for every E 2 A; there exists  .E/ D limn n .E/ 2 R: According to Nikodym’s theorem (1.37), we have  2 caC .A/. It results also that    and therefore  D    2 baC .A/. (c) Let us show that  is purely finitely additive. Let  2 ca.A/ so that 0     . We have therefore:  C    and  C  2 caC .A/; hence  C  2  and . C  /.X/  ˛. On the other hand, . C  /.X/ D .X/ C  .X/ D .X/ C lim n .X/ D .X/ C ˛; n

and so .X/  0. Hence, there results that .X/ D 0 and therefore  D 0. (ii) If  2 ba.A/, then let C be the positive variation and  be the negative variation of . Then C ;  2 baC .A/ and, from (i), there exist C ;  2 ca.A/; C ;  2 bap .A/ such that C D C C C

and

 D  C  :

We have therefore  D C   D .C   / C . C   /. Let   D   C   

and

 D C   :

Then  D  C  where  2 ca.A/ and  2 bap .A/. (iii) In order to demonstrate the uniqueness of the decomposition, let us suppose that there are two measures  2 ca.A/ and 2 bap .A/ such that  D  C  D  C . Then    D   where    2 ca.A/ and   2 bap .A/. According to Theorem 1.76,   ?   , therefore, for every " > 0, there exists A" 2 A such that j   j.A" / < "

and

j  j.X n A" / < ":

We have therefore: j  j.X/ D j  j.X n A" / C j  j.A" / < " C j   j.A" / < 2", from where j  j.X/ D 0 and so  D  and D  .

Section 1.5 The Bidual of L1 ./

63

We will now use the two results of decomposition (the theorem of Lebesgue decomposition 1.47 and the theorem of Hewitt–Yosida Theorem 1.78) to obtain an extension of Lebesgue decomposition theorem. We will first introduce a new notion (see [73]). Definition 1.79. Let  2 caC .A/ and let c W ba.A/ ! RC defined by: c ./ D inf sup jj.A/: ">0 .A/ 0, jj.X/  sup.A/ 0, there exists A" 2 A such that jj.X n A" / < " and .A" / < ". Then sup.A/t º

#

jf jd :

Proof. Let " 1 D lim

#

Z sup

t !1 f 2H

.jf j>t /

"

jf jd D inf

#

Z sup

t >0 f 2H

.jf j>t /

jf jd

and let M D sup¹kf k1 W f 2 H º. Firstly, we show that .H /  1 . For every " > 0, there exists t0 > 0 such that M t0 < ". According to the definition of 1 , for every ı > 0, there exists f0 2 H such that Z 1  ı < jf0 jd: (1) .jf j>t / Z Z 0 0 jf0 jd  jf0 jd > t0  .jf0 j > t0 /I hence M  .jf0 j>t0 /

X

.jf0 j > t0 /
t0 /

E

jf0 jd > 1  ı;

for every ı > 0 and then .H /  1 :

(2)

On the other hand, let ı > 0, t > 0 and let " > 0 such that " < ıt . According to the definition of .H /, there exist A0 2 A and f0 2 H such that .A0 / < " and Z jf0 jd: (3) .H /  ı < A0

Then Z .jf0 j>t /

Z jf0 jd 

.jf0 j>t /\A0

Z jf0 jd 

A0

Z jf0 jd 

A0 \.jf0 jt /

jf0 jd

 .H /  ı  t  .A0 \ ¹jf0 j  t º/  .H /  ı  t.A0 / > .H /  ı  t " > .H /  2ı:

66

Chapter 1 Weak Compactness in Measure Spaces

R

So, for every t > 0 and every ı > 0, supf 2H Œ We have therefore

.jf j>t / jf

jd > .H /  2ı.

1  .H /:

(4)

By (2) and (4), .H / D 1 . Theorem 1.84 (See [58]). Let  2 caC .A/ and let H  L1 ./. The following conditions are equivalent: .i/ H is relatively weakly compact in (L1 ./; k  k1 ). .ii/ H is bounded and -equicontinuous. .iii/ For R every " > 0, there exists t" > 0 such that, for every f 2 H , .jf j>t" / jf jd < ". .iv/ There exists ˆ W RC ! RC an Rincreasing function as ˆ.0/ D 0, limx!1 x1  ˆ.x/ D 1 and supf 2H X ˆ.jf j/d < C1. Proof. (i) ” (ii) was demonstrated in the Dunford–Pettis theorem (1.65). (ii) ” (iii) is a consequence of Proposition 1.83. (iii) H) (iv) According to the hypothesis (iii), for every k 2 N  , there exists tk > 0 such that, Z 1 jf jd < k ; for every f 2 H: (1) 2 .jf j>tk / Let t0 D 0; we can choose the sequence .tk /k such that, for every k 2 N  , tk > 2tk1 . Let ˆ W Œ0; C1Œ! Œ0; C1Œ be continuous and linear on every interval Œtk ; tkC1  so that ˆ.tk / D k  tk , for every k 2 N. Then ˆ is an increasing mapping and ˆ.0/ D 0. On every interval Œtk ; tkC1 ,   tkC1 tk  tkC1 ˆ.x/ D k C x tkC1  tk tkC1  tk and then tkC1 tkC1 ˆ.x/ kC  D k: x tkC1  tk tkC1  tk Therefore limx!C1

ˆ.x/ x

D C1.

67

Section 1.6 Extensions of Dunford–Pettis’ Theorem

We also have, on every interval Œtk ; tkC1 , k C 1, from where, for every f 2 H , Z X

ˆ.jf j/d D

1 Z X kD0

.tk jf j 0, there exists ı 20; "Œ such that Z fn d <  C "; for every A 2 A with .A/ < ı and for every n 2 N: A

(*)

69

Section 1.6 Extensions of Dunford–Pettis’ Theorem

Let gn D supkn fk  f and An D ¹x 2 X W gn .x/  ıº; then An 2 A. .An /n2N is an increasing sequence; hence .[1 1 An / D limn .An /. Since gn # 0; -a.e., .[1 A / D .X/. Therefore there exists n0 2 N such that .X n n 1 An0 / < ı. Then, by . /, for every n  n0 ; # " Z Z Z Z X

C

fn d D Z An0

X nAn0

fn d C

An0

fn d <  C " C

f d   C " C ı.An0 / C

and so we obtain

Z An0

n

X

sup .fk  f / d kn0

f d   C ".1 C .X// C

Z f d X

Z

Z lim sup

An0

fn d 

X

f d C :

Corollary 1.87. If .fn /  L1C ./ is a -equicontinuous sequence, then Z Z fn d  lim sup fn d: lim sup n

X

n

X

Remark 1.88. The condition of -equicontinuity in the previous corollary is indispensable. Indeed, let X D Œ0; 1 and let  be the Lebesgue measure on X; for every n 2 N  , let fn D n 1 : Then lim supn fn D 0; -a.e. and Œ0; n  R lim supn X fn d D 1. We can easily demonstrate that ..fn // D 1. Corollary 1.89 (Proposition 1 in [76]). Let .fn /  L1 ./ and let f 2 L1 ./; then: R .i/ lim supn kfn  f k1  ..fn // C X lim supn jfn  f jd: 

! f , then lim supn kfn  f k1  ..fn //. .ii/ If fn  Proof. (i) It is sufficient to apply Proposition 1.86 to the sequence .jfn  f j/n2N and to notice that ..fn  f // D ..fn //. (ii) Let .fn /  L1 ./ be a sequence convergent in measure to f 2 L1 ./. Let .fn0 /n be a subsequence of .fn / such that lim sup kfn  f k1 D lim kfn0  f k1 : n

n

Then there is a subsequence .fn00 / of .fn0 / such that .fn00 / converges to f , -a.e.

70

Chapter 1 Weak Compactness in Measure Spaces

Part (i) applied to the sequence .fn00 / allows us to write: Z 00 00 lim sup jfn00  f jd lim sup kfn  f k1 D lim kfn  f k1  ..fn // C n

n

D

X

..fn00 //

n

 ..fn //:

Remark 1.90. In the part (ii) of the previous corollary, if we suppose that .fn / is kk1

-equicontinuous, then fn ! f . This gives another proof for the sufficiency of Vitali’s theorem 1.54. In [73], we introduced the a-convergence as a generalization of weak convergence. Definition 1.91. Let .fi /i 2I  L1 ./ be a net, f 2 L1 ./ and a  0. The sequence .fi /i 2I is called a- convergent to f if, for every A 2 A, ˇ ˇZ ˇ ˇ lim sup ˇˇ .fi  f /dˇˇ  a; i 2I

A

a

 f. which we write fi ! H  L1 ./ is said relatively a-compact if every net .fi /i 2I  H admits at least one subnet a-convergent to a function f 2 L1 ./. Remark 1.92. .i/ Obviously, a net is 0-convergent if and only if it is weakly convergent in L1 ./. Similarly, a set H  L1 ./ is relatively 0-compact if and only if H is relatively weakly compact. .ii/ According to (ii) of Corollary 1.89, if .fn /  L1 ./ is convergent in measure to f , then .fn / is ..fn //-convergent to f . The following theorem is a first extension of the Dunford–Pettis theorem; it was demonstrated in [73] (see Theorem 1). Theorem 1.93. Let H  L1 ./ be a bounded set and let .H / be his modulus of uniform integrability. Then H is relatively .H /-compact. Proof. Let ‰ be the natural embedding of L1 ./ in its bidual ba .A/ and let .fi /i 2I  H be a net. Since H is bounded in L1 ./, ‰.H / it is bounded in

71

Section 1.6 Extensions of Dunford–Pettis’ Theorem

ba .A/ also; therefore ‰.H / is relatively w*- compact, according to the notations already used in Remark 1.69 and in Theorem 1.70. So .‰.fi //i 2I admits a subnet .‰.fj //j 2J w*- convergent to a measure  2 ba .A/. Therefore we have, Z fj d ! .A/; for every A 2 A: (1) j 2J

A

Thanks to Theorem 1.81, there exist f 2 L1 ./ and 2 ba.A/, ?  such that, Z f d C .A/; for every A 2 A and (2) .A/ D A

c ./ D k k:

(3)

.H /

Let us show that fj ! f . According to the definition of .H /, for every " > 0, there exists ı 20; "Œ such that Z jf jd < .H / C "; for every A 2 A with .A/ < ı and for every f 2 H: A

(4) After the definition of c ./, there exists A0 2 A such that c ./  ı < jj.A0 /

and .A0 / < ı:

(5)

According to the definition of jj, there exists an A-partition ¹A1 ; : : : ; An º of A0 such that jj.A0 /  "
0 .A/ 0, R there exists ı > 0, such that, for any A 2 A with .A/ < ı and for any n 2 N, A jfn jd <  C ": Since .Bp / ! 0, there exists p0 2 N such that .Bp / < ı, for every p  p0 and therefore Z jfn jd <  C "; for every n 2 N and every p  p0 : Bp

R Then lim supn Bp jfn jd <  C ", for every p  p0 ; hence, for every " > 0, 1 <  C " and so 1  :

(1)

If  D 0, 1 D . If 0 <  < C1, according to the definition of 1 , for every " 20; 2 Œ, there exist p0 2 N and n0 2 N such that Z jfn jd < 1 C "; for every n  n0 : (2) Bp0

Since .fn  X nBp0 /n2N is weakly convergent on X n Bp0 , ¹fn  X nBp0 W n 2 Nº is -equicontinuous on X n Bp0 (see (ii) of Remark 1.66). The finite set ¹f0 ; : : : ; fn0 1 º is -equicontinuous on X. Then there exists ı 20; "Œ such that, for every n 2 N, Z jfn jd < "; for every A 2 A; A  X n Bp0 with .A/ < ı (3) A

and, for any i D 0; : : : ; n0  1, Z jfi jd < "; for every A 2 A with A

.A/ < ı:

(4)

78

Chapter 1 Weak Compactness in Measure Spaces

According to the definition of , there exist A1 2 A n1 2 N such that Z ı < jfn1 jdI

with .A1 / < ı and

(5)

A1

since ı < " < 2 ,  "< <ı < 2

Z A1

jfn1 jd:

(6)

From (6) and (4), n1  n0 and so, according to (5), (3) and (2), we have Z Z Z jfn1 jd D jfn1 jd C jfn1 jd "<ı < Z 0,  < 1 C 3" and so   1 :

(7)

By (1) and (7)  D 1 . Proposition 1.99. Let .fn /  L1 ./ be a sequence w 2 -convergent. .fn / is weakly convergent if and only if .fn / is -equicontinuous. Proof. We have already seen (see (ii) of Remark 1.66) that every weakly convergent sequence is -equicontinuous. Let now .fn /n2N be a -equicontinuous sequence w 2 -convergent to f . Then, for every " > 0, there exists ı > 0 such that, Z Z " " and jf jd < jfn jd < ; 8A 2 A; 3 3 A A with .A/ < ı; 8n 2 N: (1) Let .Bp /  A be a decreasing sequence which localizes the concentration of mass of .fn /. Then there exists p0 2 N such that, for any p  p0 , .Bp / < ı and so, Z Z " " jfn jd < and jf jd < ; for every n 2 N and every p  p0 : 3 3 Bp Bp (2)

79

Section 1.6 Extensions of Dunford–Pettis’ Theorem

2 for every p and every A 2 A; A  X n Bp , R R Because of w -convergence, f d converges to f d. Therefore, for p D p0 , there exists n0 2 N such A n A that Z " .fn  f /d < ; for every n  n0 and every A 2 A; A  X n Bp0 : 3 A (3)

According to (1), (2) and (3), for every n  n0 and every A 2 A, we have ˇ Z ˇZ ˇ ˇˇZ Z ˇ ˇ ˇ ˇ ˇ ˇ .fn  f / dˇ  ˇ .fn  f / dˇ C jfn jd C jf jd ˇ ˇ ˇ ˇ A AnBp0 A\Bp0 A\Bp0 ˇ Z ˇZ Z ˇ ˇ ˇ ˇ .fn  f / dˇ C jfn jd C jf jd  ": ˇ ˇ ˇ AnBp Bp Bp 0

Therefore, for every A 2 A,

0

R

A fn d

!

R

A f d

0

w

i.e. fn ! f . L1 ./

Theorem 1.100 (Theorem 5 of [76]). For every bounded sequence .fn /n , w 2 convergent in L1 ./, there exist a subsequence .gn / and a sequence of pairwise disjoint sets .An /  A such that: R (i) ..fn // D limn An jgn jd and   is weakly convergent in L1 ./. (ii) X nA  gn n

n2N

Proof. Let .Bp /  A be a decreasing sequence which localizes the concentration of mass of .fn /. According to Theorem 1.98, Z jfn jd:  D ..fn // D lim lim sup p

n

Bp

Then, for every n 2 N  , there exist pn 2 N and kn  n such that Z 1 1  < jfkn jd <  C : n n Bpn

(1)

As the sequence .Bp / is decreasing, we can select the sequence .pn / to be increasing and so that lim pn D C1. For n D i0 D 1, by (1), there exist p1 D pi0 and k1  1 such that Z jfk1 jd <  C 1: (2) 1< Bpi

0

80

Chapter 1 Weak Compactness in Measure Spaces

Since fk1   , for "1 D 1, there exists ı1 > 0 such that Z jfk1 jd < 1 D "1 ; for every A 2 A with .A/ < ı1 :

(3)

A

Let i1 > i0 such that .Bpi1 / < ı1 ; then, by (3), we have: Z Bpi

and by (2) and (4), Z 11
0, there exist p0 2 N and k0 2 N, such that Z jgnk jd < ..gnk // C "; for every k  k0 : Bp0

But, for k sufficiently large, through the definition of An D Bpin n BpinC1 , Z

Z An

jgnk jd 

Bpi

Z jgnk jd  n

Bp0

jgnk jd < ..gnk // C ":

83

Section 1.6 Extensions of Dunford–Pettis’ Theorem

If we pass to the limit for n ! 1, we obtain Z jgnk jd  ..gnk // C "; ..fn // D lim n

for every " > 0:

An

Then ..fn //  ..gnk //. Since .gnk / is a subsequence of .fn /, ..gnk //  ..fn //. We have therefore ..gnk // D ..fn //. Let us remark that the concentration of mass for .gn /n is the maximum between the concentration of mass for all subsequences of .fn /n . .ii/ As we show in the following example, the subsequence .gn /n of .fn /n does not generally coincide with the sequence .fn /n . Let  be the Lebesgue measure on 0; 1; for every n 2 N, let fn D n

1 0; n 

.

Then fn 2 L1 ./ and kfn k1 D 1, for every n 2 N. Therefore .fn / is a w2

bounded sequence in L1 ./, fn ! 0 and ..fn // D 1. If we suppose that the sequence .gn / coincides with the sequence .fn /, then there Rexists a sequence of pairwise disjoint subsets .An / of 0; 1 such that f d D 1 D ..fn //. Therefore, there exists n0 2 N such that lim R n An n 1 1 An fn d > 2 , for every n  n0 . We have therefore .An / > 2n , for every n  n0 . Then P the infinite series of1general term .An / is divergent. On the other hand, 1 nDn0 .An / D .[nDn0 An /  1 which is a contradiction. Therefore the sequence .gn / is different from the sequence .fn /. 1 1 For every n 2 N, we take gn D fnn and An D .nC1/ nC1 ; nn ; then R n n 1 gn d D 1  . nC1 /  nC1 ! 1 D ..fn // and, for all A 2 A, RAn 1 nn n A .0;1nA / gn d D n .A\0; .nC1/nC1 /  .nC1/nC1 ! 0: n

w

Therefore .0;1nA / gn ! 0. Here .gn / D .fnn / is the subsequence n

L1 ./

that suits to problem. w2

Proposition 1.102. Let .fn /  L1 ./ and f 2 L1 ./ such that fn ! f . Then, .i/ .fn /n is ..fn //-convergent to f ; .ii/ If .fn /n is convergent in measure, then .fn / converges in measure to f . Proof. Let .Bp /  A be a decreasing sequence which localizes the concentration of mass of .fn /.

84

Chapter 1 Weak Compactness in Measure Spaces

(i) According to Theorem 1.98, we have

Z

..fn // D lim lim sup p

n

jfn jd:

Bp

(1)

For every A 2 A, every p 2 N and every n 2 N, one can write ˇ Z ˇZ ˇ ˇˇZ Z ˇ ˇ ˇ ˇ ˇ ˇ .fn  f /dˇ  ˇ .fn  f /dˇ C jfn jd C jf jd: ˇ ˇ ˇ ˇ A .AnBp / Bp Bp

(2)

w

Since, for every p 2 N, fn .X nBp / ! f  .X nBp / , ˇ ˇZ ˇ ˇ ˇ ˇ lim ˇ .fn  f /dˇ D 0; n ˇ .AnB / ˇ p

X nBp

for every A 2 A;

and every p 2 N: (3)

By (2) and (3), we have ˇ ˇZ Z ˇ ˇ lim sup ˇˇ .fn  f /dˇˇ  lim sup n

n

A

Z

Bp

jfn jd C

Bp

jf jd;

for every p 2 N: Hence, using (1), ˇ ˇZ Z ˇ ˇ ˇ ˇ lim sup ˇ .fn  f /dˇ  ..fn // C lim n

p

A

Bp

jf jd D ..fn //:

..fn //

According to Definition 1.91, fn ! f . This proves the (i). 

(ii) Let g be a measurable function such that fn  ! g. Then, for all p 2 N, 

.X nB / fn  ! .X nB / g and since . .X nB / fn /n2N is -equicontinuous, for all p p p p 2 N, it results from the Vitali’s theorem (see Theorem 1.54) that kk1

.X nB / fn ! .X nB / g; p

p

for every p 2 N:

Then, for every p 2 N and every A 2 A with A  X n Bp , ˇ ˇZ ˇ ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ ˇ .f  g/dˇ  ˇ .f  fn /dˇ C ˇ .fn  g/dˇ ˇ ˇ ˇ ˇ ˇ ˇ A ˇ AZ ˇZA ˇ ˇ ˇ ˇ .f  fn /dˇ C ˇ jfn  gj d: ˇ ˇ AnBp X nBp

(4)

85

Section 1.6 Extensions of Dunford–Pettis’ Theorem

R R By (3), limn AnBp .f  fn /d D 0 and, by (4), limn X nBp jfn  gj d D 0, R for every p 2 N. Therefore, for every A 2 A with A  X nBp , A .f g/d D 0. 

!f. Since .Bp / ! 0, f D g, -a.e.; hence fn  Now we are going to give another extension to the Dunford–Pettis’ theorem of weak compactness. This extension, known under the name of “biting lemma”, was demonstrated in 1980 by J. K. Brooks and R. V. Chacon ([37]). We can also mention other demonstrations of this result: M. Słaby [154], J. M. Ball and F. Murat [23], D. Q. Luu [115] and L. C. Florescu [75]. The demonstration given here is that of M. Słaby and was reproduced by M. Saadoune and M. Valadier in [149] (see also Theorem 6 in [76]). Theorem 1.103 (Biting Lemma). Every bounded sequence .fn /n  L1 ./ admits a subsequence .gn /n w 2 -convergent. Besides, for all subsequence .gnk /k of .gn /n , ..gnk /k / D ..fn /n /. Proof. Let .fn /n be a bounded sequence in L1 ./ and let M D supn kfn k1 ; we have shown in Proposition 1.83 that  Z jfn jd :  D ..fn // D lim sup t !1

.jfn j>t /

n

For every i 2 N, let Fi W Œ0; C1Œ! Œ0; C1Œ; Z Fi .t/ D sup jfn jd: ni

.jfn j>t /

It is clear that, for all i, Fi is a decreasing function on Œ0; C1Œ and that lim t !1 Fi .t/ D .¹fn W n  iº/. As .¹fn W n < iº/ D 0, for every i, we have: lim Fi .t/ D ..fn // D ;

t !1

for all

i 2 N:

(1)

Thus, for i D 0, we can choose an increasing sequence .tq /  Œ0; C1Œ, tq " C1 such that 1 F0 .tq /   C ; q

for every q 2 N  :

Then, by (1) and (2), it results that, for every i D q Z  D inf Fq .t/  Fq .tq / D sup t >0

nq .jfn j>tq /

jfn jd:

(2)

86

Chapter 1 Weak Compactness in Measure Spaces

Then, for all q 2 N  1   < sup q nq

Z .jfn j>tq /

jfn jd:

Therefore, for all q 2 N  , there exists nq  q such that Z 1 jfnq jd; for every q 2 N  :  < q .jfnq j>tq / If Aq D ¹x 2 X W jfnq .x/j > tq º, then Z Z jfnq jd  jfnq jd  tq .Aq /; M D sup kfn k1  n

X

(3)

for all q 2 N  :

Aq

Hence .Aq / ! 0. We will now show that the sequence . .X nA / fnq /q is -equicontinuous. q Let us define F .t / by Z jfnq jd; for every t > 0: F .t / D sup q2N

.jfnq j>t /nAq

From Proposition 1.83, we have .. .X nA / fnq // D lim t !1 F .t /: Since F is q a decreasing function, lim t !1 F .t / D limj !1 F .tj ), for every increasing sequence .tj /j with tj " C1. Then, by definition of Aq , for every q 2 N and every j 2 N, .jfnq j > tj / n Aq D .jfnq j > tj / \ .jfnq j  tq / D .tj < jfnq j  tq /. Thus, for all j  q; .jfnq j > tj / n Aq D ; and from (2) and (3), we obtain, for every j 2 N, Z jfnq jd F .tj / D sup q>j

D sup q>j

.tj 0 and every A 2 BT , there exist F D F  A and D 2 , D A such that jj.D n F / < ". Proof. (i))(ii) For every A 2 BT and every a > jj.A/, jj.T n A/ > jj.T /  a. By hypothesis, there exists F D F  T n A such that jj.F / > jj.T /  a or a > jj.T n F /. If G D T n F , G 2 , then G A and a > jj.G/. Therefore jj.A/  inf D2 ;DA jj.D/ and so we have (ii). (ii))(iii) For every A 2 BT ; T n A 2 BT . By (ii), for every " > 0, there exist D; G 2  such that A  D; T n A  G; jj.D/ < jj.A/ C 2" and jj.G/ < jj.T n A/ C 2" . If F D T n G, then F D F  A and jj.D n F / D jj.D/  jj.F / < jj.A/ C 2"  jj.A/ C 2" D ". (iii))(i) For every A 2 BT and every a < jj.A/, let " D jj.A/  a > 0. Then, by (iii), there exist F D F  A and D 2 ; D A such that jj.D n F / < ". Therefore we have jj.F / > jj.D/  "  jj.A/  " D a and so jj.A/ D supF DF A jj.F /. Definition 2.2. A measure  W BT ! R is regular if it fulfils one of the equivalent conditions (i)–(iii) of the previous proposition. Let us note rca.BT / the set of all regular measures of ca.BT / and by rcaC .BT / the cone of all positive measures of rca.BT /. Obviously, rca.BT / is a vector subspace of ca.BT /. Remark 2.3. .i/ rca.BT / is a closed vector subspace of .ca.BT /; k  k/; therefore it is a Banach space.

92

Chapter 2 Bounded Measures on Topological Spaces

Indeed, let .n /n  rca.BT / be a sequence convergent in .ca.BT /; k  k/ to  2 rca.BT / ; by Theorem 1.23, jn  j.T / ! 0. Then, for every " > 0, there exists n0 2 N such that " jn0  j.T / < : 2

(1)

As n0 is a regular measure, for every A 2 BT , there exists F" D F "  A such that " jn0 j.A n F" / < : 2

(2)

By (1) and (2), we obtain jj.A n F" /  j  n0 j.A n F" / C jn0 j.A n F" /
0; 9F" D F "  A  D" 2  and .D" n F" / < "º. Firstly, let us show that A is a -algebra. Indeed, for every A 2 A and every " > 0, there exist F" D F "  A and D" 2 ; A  D" such that .D" n F" / < ". 0 Let D"0 D T n F" and F"0 D T n D" ; then D"0 2 ; D"0 T n A; F"0 D F "  T n A and .D"0 n F"0 / D .D" n F" / < "; therefore T n A 2 A.

93

Section 2.1 Regular Measures

S Let now .An /n2N  A and A D 1 1 An 2 BT ; for every n 2 N and ; Dn 2  such that Fn  S A  Dn and every " > 0, there exist Fn D F nS n " .Dn n Fn / < 2nC1 . The sequence . kD1 Fk /n2N converges to 1 1 Fn and so S " there exists n0 2 N such that . 1 F / < . kDn 2 S 0 S0 C1 k Fk and D" D 1 D ; then F" D F "  A; D" 2 , Let F" D nkD1 n nD1 A  D" and 2 3 ! ! n0 1 1 1 1 [ [ [ [ [ Dn n Fk D  4 Dn n Fk [ Fk 5 .D" n F" / D  " 

nD1 1 [

kD1

#

.Dn n Fn / C  4

1 X nD1

" 2nC1

nD1 1 [

3

kD1

Fk 5 

kDn0 C1

nD1


0, there exists n0 with .F n0 n F / < ". Because F n0 2 , we have F 2 A. Therefore A  BT is a -algebra that contains the closed sets, so that A D BT and then  is regular on BT . Remark 2.5. The proof of the previous theorem allows us to give the following result: For every Hausdorff topological space T for which the closed sets are Gı -set (a countable intersection of open sets), every -additive measure is regular, which means rca.BT / D ca.BT /. Definition 2.6 (see [61], p. 224). A measure  2 ca.BT / is called tight if, for every " > 0, there exists K" 2 KT such that jj.T n K" / < ". We will note by t ca.BT / the set of all tight measures in ca.BT /. By using an analogous demonstration to that of Remark 2.3 (i), it easy to see that t ca.BT / is a closed vector subspace of .ca.BT /; k  k/ and so, a Banach subspace.

94

Chapter 2 Bounded Measures on Topological Spaces

Definition 2.7 (see [152] and [32]). A measure  2 ca.BT / is called a Radon measure if for every A 2 BT , jj.A/ D

sup

jj.K/:

K2KT ;KA

We will note the set of all Radon measures on BT by Rca.BT /. Obviously, Rca.BT /  tca.BT /, Rca.BT /  rca.BT /; if T is compact, then rca.BT / D Rca.BT /. Proposition 2.8. rca.BT / \ tca.BT / D Rca.BT /: Proof. Let  2 rca.BT / \ tca.BT /. As  is regular, for every A 2 BT and every " > 0, there exists F" D F "  A such that jj.A n F" / < 2" . As  is tight, there exists K" 2 KT such that jj.T n K" / < 2" . Let K D F" \ K" 2 KT ; then K  A and jj.AnK/ D jjŒ.AnF" /[.AnK" /  jj.AnF" /Cjj.T nK" / < ". Therefore jj.A/ D supK2KT ;KA jj.K/ and  2 Rca.BT / . Definition 2.9 (Definition 8 in [152] and Definition 2, §3, [32]). A topological space T is said Radon space if every measure  2 ca.BT / is a Radon measure. For a Radon space T , ca.BT / D rca.BT / D t ca.BT / D Rca.BT /. In the following section, we will specify several types of Radon spaces. However, according to Proposition 2.1, we can now give the next proposition that early will be generalized (see Proposition 6 in [152]): Proposition 2.10. Every metrizable compact space is a Radon space. Remark 2.11. It is easy to see that .i/  2 Rca.BT / , C and  2 Rca.BT /. .ii/ For every t 2 T , ı t 2 Rca.BT /. .iii/ Rca.BT / is a closed vector subspace of .ca.BT /; k  k/ and so, a Banach subspace. We now recall the notion of support of a measure that be will used in the following sections.

95

Section 2.2 Polish Spaces. Suslin Spaces

Definition 2.12. Let  2 ca.BT / non identically null; we call the support of  the smallest non-void closed set of T on which is concentrated the measure jj (see Definition 1.11). It is noted supp . In accordance with Definition 1.11, the support of  is ® ¯ supp  D \ F W F D FN  T; jj.T n F / D 0 : Proposition 2.13. For every measure  2 Rca.BT /; jj.T n supp / D 0 (which means that every Radon measure is concentrated on its support). Proof. For every K 2 KT with K  T n supp , we have K  [ ¹D 2 ; jj.D/ D 0º : From this S cover of K, we can extract then a finite subcover D1 ; : : : ; Dn 2  such that K  niD1 Di and, for i  n, jj.Di / D 0; therefore jj.K/ D 0. As  is Radon, jj.T n supp / D

sup K2KT ;KT n supp 

jj.K/ D 0:

Remark 2.14. The previous remark is not generally true for the measures of ca.BT / that are not Radon measures. We can find in [59] an example of Borel measure on a compact space taking only the 0 and 1 values and without support. This example is mentioned also in [152] , p. 45.

2.2

Polish Spaces. Suslin Spaces

The aim of this section is to present several important classes of Radon spaces: Polish spaces and, more general, Suslin spaces. In this paragraph, we use the monographs [31, 68, 152]. We will first recall the definitions to be used in the following. Definition 2.15. A topological space T is said .i/ second-countable if T has a countable base, .ii/ separable if T contains a countable dense set, .iii/ Lindelöf space if every open cover of T has a countable subcover, .iv/ hereditarily Lindelöf if every subspace of T is a Lindelöf space.

96

Chapter 2 Bounded Measures on Topological Spaces

It is well known that a second-countable space is separable (see Proposition 12 of §2 in [31]); moreover: In a metrizable space T , the following properties are equivalent: .i/ T is second-countable, .ii/ T is separable, .iii/ T is homeomorphic to a subset of the Hilbert cube Œ0; 1N . In the following, we present the definitions of Polish and Suslin spaces; see Definitions 1 and 2, §6 in [31] and Definitions 1, p. 93 and 3, p. 96 in [152] . Definition 2.16. A topological space .P; / is said to be a Polish space if it is separable and if there exists a metric d on P compatible with  (the topology induced by d coincides with ) for which .P; d / is complete. We can always choose d  1. A Hausdorff topological space S is said to be a Suslin space if there exists a Polish space P and a continuous surjection from P to S . It is clear that, if .S; / is a Suslin space, then .S; / is Suslin also, for every Hausdorff topology   . Example 2.17. As examples of Polish spaces, we can mention the compact metrizable spaces as well as the separable Banach spaces. Every Polish space is a Suslin space, but there are important examples of Suslin spaces that are not Polish spaces, as the one to follow. Let .X; k  k/ be a separable Banach space of infinite dimension; then the weak topology .X; X  / is not metrizable and, as .X; .X; X  // is the continuous image of the Polish space .X; kk /, .X; .X; X  // is a Suslin space which is not a Polish one. In the following theorems, we will present some important families and properties of the Polish and the Suslin spaces (see Propositions 1 and 2, §6, in [31] and Proposition 3, p. 104, in [152] ). Theorem 2.18. .i/ Every space homeomorphic to a Polish space is Polish. .ii/ Every finite or countable product of Polish spaces is Polish. Every finite or countable sum of Polish spaces is a Polish space. .iii/ Every closed (open) subspace of a Polish space is Polish. .iv/ Every Polish space is hereditarily Lindelöf.

Section 2.2 Polish Spaces. Suslin Spaces

97

Proof. (i) Let P be a Polish space and let d be a complete metric compatible with the topology of P . If Q is a topological space and if f is a homeomorphism of P on Q, then g W Q  Q ! RC , defined by g.x; y/ D d.f 1 .x/; f 1 .y//; is a metric on Q compatible with the topology of Q and .Q; g/ is complete; then Q is a Polish space. (ii) It is easy to seeQthat, if ¹.P Qn ; n / W n 2 I º is a finite or countable family of Polish spaces, then . n2I Pn ; n2I n / is a Polish space . For the sum, we suppose that the sets Pn are pairwise disjoint (if it is not the case, then one can replace the Pn by Pn0 D Pn  ¹nº and n with n0 D n  ¹;; ¹nºº). S Let P D 1  D ¹D  P W D \ Pn 2 n ; for every n 2 Nº. nD0 Pn and L Let us show that .P; / D 1 nD0 .Pn ; n / is a Polish space. Indeed, if, for every n 2 N; An is a countable and dense set in .Pn ; n /, then S1 A D nD0 An is countable and dense in .P; /. Moreover, if dn  1 is a metric on Pn such that n D dn and that .Pn ; dn / is complete, then d W P  P ! RC , ² dn .x; y/; x; y 2 Pn ; d.x; y/ D ; 1; x 2 Pn ; y 2 Pm ; n ¤ m is a metric on P with  D d and .P; d / is complete (if .xk /k is a Cauchy sequence in P , then there exists a n such that, from a certain rank, the sequence .xk /k is contained in Pn ). (iii) Because every closed subspace of a complete space is complete, it is clear that every closed subspace of a Polish is Polish. Let us now show the property for the open subspace. Let P be a Polish space, d be a complete metric compatible with the topology of P and Q be an open subset of P . Then R  P is Polish and therefore G D ¹.x; p/ 2 R  P W x  d.p; P n Q/ D 1º is Polish, also (G is a closed subspace of R  P ). The projection  W G ! Q; ..x; p// D p is a homeomorphism of G on Q. By (i), Q is Polish. (iv) Every Polish space P is a second-countable and metrizable space; then the subspaces of P are metrizable and second-countable spaces and, according to Proposition 13 of §2, no 8 in [31], they are Lindelöf spaces. Theorem 2.19 (Corollary p. 94 in [152]). A topological space is Polish if and only if it is homeomorphic to a Gı subset of Hilbert cube Œ0; 1N (a countable intersection of open sets in Œ0; 1N ).

98

Chapter 2 Bounded Measures on Topological Spaces

Proof. By (iii) and (ii) of Theorem 2.18, H D Œ0; 1N is a Polish space. Firstly, let us show that a Gı subset of Œ0; 1N is a Polish space. T Let .Dn /n2N be a sequence of open subsets of Œ0; 1N and Q D Qn2N Dn . By (iii) of 2.18, for every n 2 N, Dn is PolishQand by (ii) of 2.18, n2N Dn is Polish, also. The diagonal  of the product n2N Dn is a closed subset of Q Q D .xn /n2N , where, for n2N Dn , therefore it is Polish. For every x 2 Q, let x every n, xn D x; then the mapping x ! xQ is a homeomorphism of Q on  so that Q is Polish. Conversely, let .P; / be a Polish space, let d be a complete metric compatible with  such that d  1 and let A D ¹pn W n 2 Nº be a countable dense subset of P . The mapping f W P ! H , defined by f .p/ D .d.p; pn //p2N is a homeomorphism of P on f .P / D Q. Indeed, because A is dense in P , it is clear that f is a continuous injective mapping. Let now B D Bd .p; r/ and let n0 such that d.p; pn0 / < 3r . If V D ¹x 2 P W jd.x; pn0 /  d.p; pn0 /j < 3r º, then V is a neighborhood of p whose image by f is a neighborhood of f .p/ in Q D f .P /. By (i) of Theorem 2.18, Q is Polish. Now let g be a complete metric compatible with the topology of Q. Then g D 0 Q , where 0 is the product topology on H D Œ0; 1N . For every n 2 N  , let ³ [² 1 ; Un D U 2 0 W ıg .U \ Q/ < n where ıg .E/ is the diameter of E in the metric space .Q; g/. We first notice that, for every n 2 N; Q  Un . Indeed, if y 2 Q, then 1 / is an open subset of Q; hence there exists U 2 0 such that the ball Bg .y; 2n T 1 Bg .y; 2n / D U \ Q and therefore y 2 Un . Then we have Q  1 nD1 .Un \ Q/. We will show that QD

1 \

.Un \ Q/:

(*)

nD1

which was to be demonstrated. T For every x 2 1 nD1 .Un \ Q/, there exists a sequence .xk /  Q such that

0

xk ! x. For every " > 0 let n0 2 N  with

1 n0 1 n0 .

< ". As x 2 Un0 , there exists

Therefore, there exists k0 2 N U 2 0 such that x 2 U and ıg .U \ Q/ < such that, for every k  k0 ; xk 2 U and, for every k; l  k0 , g.xk ; xl / < n10 < "; then .xn / is a Cauchy sequence in the complete metric space .Q; g/. So, there g

0 exists y 2 Q such that xk  ! y. We have therefore xk ! y and x D y 2 Q and so . / is demonstrated.

Section 2.2 Polish Spaces. Suslin Spaces

99

As Q is a closed subset in the metrizable space H DTŒ0; 1N , it is a countable intersection of open subsets of H from where, Q D 1 nD1 .Un \ Q/ is a Gı subset of H . The following proposition gives a family of Polish spaces which will be used later. Proposition 2.20. A locally compact space is Polish if and only if it is secondcountable. Proof. Every Polish space is second-countable. If T is a locally compact space which is second-countable, then its Alexanb is metrizable (see corollary of Proposition 16 of §2 droff’s compactification T b is Polish and, as T is open in T b, by in [31]) and second-countable. Therefore T (iii) of Theorem 2.18, T is Polish. Let us now give some properties of Suslin spaces (see Proposition 4, Section 6, in [31]). Theorem 2.21. .i/ Every Suslin space is separable. .ii/ Every Suslin space is hereditarily Lindelöf. .iii/ Every regular Suslin space is a normal space. Proof. Let .S; / be a Suslin space; there exist a Polish space P and a surjective continuous mapping f W P ! S . (i) If A is a countable dense subset of P , then f .A/ is countable and dense in S , hence S is separable. (ii) To show that S is hereditarly Lindelöf it is sufficient to show that every open subset D  S is Lindelöf. Let ¹D W  2 º be an open cover of D. Then ¹f 1 .D / W  2 º is an open cover of f 1 .D/ in P . As P is hereditarily Lindelöf (see (iv) of Theorem 2.18), there exists a countable family 0   such that ¹f 1 .D W  2 0 º is a cover of f 1 .D/. Then ¹D W  2 0 º is a countable cover of D. (iii) Let .S; / be a regular Suslin space and let A and B two closed and disjoint subsets of S . For every x 2 A and every y 2 B, there exist Ux ; Vy 2  such that x 2 Ux  U x  S n B and y 2 Vy  V y  S n A. The family ¹Ux W x 2 Aº is an open cover of A and the family ¹Vy W y 2 Bº an open cover of B. As S isS hereditarly Lindelöf,Sthere exist .xn /n2N  A and .yn /n2N  B such 1 that A  1 nD1 Uxn and B  nD1 Vyn .

100

Chapter 2 Bounded Measures on Topological Spaces

S S For every n 2 N, let Dn DSUxn n mn V ym and Gn D Vyn n mn U xm . S 1 Dn ; Gn 2  and if D Dn and G D 1 nD1 Gn 2 , then A  D, S1D SnD1 1 B  G and D \ G D nD1 mD1 .Dn \ Gm /. Let n; m 2 N; if we suppose that m  n, then Dn  S n V ym  S n Vym  S n Gm and so Dn \ Gm D ;. Similarly in the case where n  m. Therefore Dn \ Gm D ;, for every n; m 2 N and then D \ G D ;. Hence .S; / is a normal space. In the following sections, the property of a space to be completely regular will be indispensable in order to obtain significant results; then we will give the next corollary: Corollary 2.22. Every regular Suslin space is a completely regular space. Proof. Since any normal space is completely regular, the corollary results immediately from (iii) of previous theorem. The following theorem lists some important properties of Suslin spaces (see Propositions 5, 7 and 8, §6, in [31]). Theorem 2.23. .i/ Every space homeomorphic to a Suslin space is Suslin. .ii/ Every closed or open subspace of a Suslin space is Suslin. .iii/ A continuous image of a Suslin space in a Hausdorff space is Suslin. .iv/ Every finite or countable product of Suslin spaces is Suslin. The sum of a finite or countable family of Suslin spaces is Suslin. .v/ Every union and every intersection of a finite or countable family of Suslin subspaces of a Hausdorff space is Suslin. Proof. The proofs of (i), (ii) and (iii) are obvious. (iv) Let ¹.Sn ; n / W n 2 Nº be a countable family of Suslin spaces and let, for every n 2 N; .Pn ; n / be a Polish space and fn W Pn ! Sn be a Q surjective con1 tinuous mapping; then, according to Theorem Q12.18 (ii), .P; / D nD0 .Pn ; n / is a Polish space and, if f W P ! S D Q 0 Sn is defined by: f ..xn /n2N / D .fn .xn //n2N and if  the product topology n n , then f is a continuous surjection from .P; / on .S; /. Therefore .S; / is Suslin. L Let now suppose that Sn are pairwise disjoint and let .S; / D 1 nD0 .Sn ; n /. We can suppose Lalso that Pn are pairwise disjoint. According to Theorem 2.18 (ii), .P; / D 1 nD0 .Pn ; n / is Polish. Then the mapping f W P ! S , defined

Section 2.2 Polish Spaces. Suslin Spaces

101

by: f .x/ D fn .x/, if x 2 Pn , is a continuous surjection and therefore .S; / is Suslin. (v) Let .Sn ; n /n2N be a sequence of Suslin subspaces of the Hausdorff space XN and let S D [1 nD0 Sn . For every n 2 N, let An D Sn  ¹nº and n D n ¹;; ¹nºº. Then, for every L1n 2 N, n is a topology on An and .An ; n / is a Suslin space. By (iv), nD0 .An ; n / D .A; / is Suslin. The mapping ' W A ! S , defined by: '..x; n// D x, for every .x; n/ 2 An , is a continuous surjection and therefore S is Suslin. Let now .X; / be a Hausdorff space; for every nQ2 N; let .Sn ; n / be a Suslin subspace, where n D  Sn . Then, by (iv), S D 1 nD0 Sn is a Suslin subspace of X N . The diagonal  D ¹.xn /n2N W where xn D x 2 X; for every n 2 Nº, is a closed subset of the product space X N ; then T  \ S is a closed subset of S and so, by (ii), \S is Suslin. The mapping ' W 1 nD0 Sn ! 4\S , that, to every x associates .x / where x D x, for every n 2 N, is a homeomorphism; by n n2N n T1 (i), nD0 Sn is Suslin. Theorem 2.24 (Proposition 11, §6, in [31] and Corollary 1, p. 101, in [152]). .i/ Every Borel subspace of a Suslin space is Suslin. .ii/ If .S; / is a Suslin space then B 2 BS ./ , B and S n B are Suslin subspaces of S . Proof. (i) Let .S; / be a Suslin space and let A D ¹A  S W A and S n A are Suslinº. Thanks to (v) of Theorem 2.23 and from the definition of A, it is clear that A is a -algebra. By (ii) of Theorem 2.23,   A and so BS ./  A. (ii) The ) is a consequence of (i). The implication ( has a long demonstration and we omit it; one can found it in Corollary 1, p. 101 of [152] . Theorem 2.25. .i/ For every Suslin space .S; /, there exists a topology    such that .S; / is a second-countable Suslin space. .ii/ If .S; / is a regular Suslin space, then there exists a metric d on S such that: .a/ d   , .b/ d is second-countable and .c/ BS .d / D BS ./. Proof. (i) Let .S; / be a Suslin space; since S is a Hausdorff space, 4 D ¹.x; x/ W x 2 S º is a closed subset in S  S and so D D .S  S / n 4 is

102

Chapter 2 Bounded Measures on Topological Spaces

open in S  S . For every .x; y/ 2 D, x ¤ y; therefore there exist U.x;y/ and VS .x;y/ 2  such that x 2 U.x;y/ , y 2 V.x;y/ , U.x;y/ \ V.x;y/ D ; and D D .x;y/2D ŒU.x;y/  V.x;y/ . According to Theorems 2.21 (ii) and 2.23 (iii), S  S is hereditarily Lindelöf and then there exist .xn ; yn /n2N  D such that DD

1 [ 

 U.xn ;yn /  V.xn ;yn / :

(*)

nD1

Let now S D ¹U.xn ;yn / ; V.xn ;yn / W n 2 Nº  . By . /, for every .x; y/ 2 D, there exists n 2 N such that .x; y/ 2 U.xn ;yn /  V.xn ;yn / . Then, for every x 2 S , there is a n 2 N with x 2 U.xn ;yn / . Therefore S is a countable open cover of S . Let  be the topology on S generated by the subbase S. Then  is a Hausdorff topology,    and  has a countable base. .S; / is a Suslin space, also. (ii) If .S; / is a regular Suslin space, then, by (iii) of 2.21, .S; / is normal space and so a completely regular one. Therefore, for every .x; y/ 2 D D .S  S / n , there exists a continuous mapping f W .S; / ! Œ0; 1 such that f .x/ ¤ f .y/. Let ¹fi W i 2 I º be a family of continuous mappings from S in Œ0; 1 which separates the points of S and, for any i 2 I , let S Di D ¹.x; y/ 2 S  S W fi .x/ ¤ fi .y/º. The sets Di are open in S S and D D S i 2I Di . As S S is hereditarily Lindelöf, there exist .in /n2N  I such that DPD 1 nD1 Din . 1 1 Let now d W S  S ! RC ; d.x; y/ D nD1 2n jfin .x/  fin .y/j. d is a metric on S and, as the mappings ¹fin W n 2 Nº are –continuous, d  . Since a continuous image of a Lindelöf space is again a Lindelöf one, .S; d / is Lindelöf. If Bd .t; r/ designates the open d -ball of center t and ray r, then, for every p 2 N  ; ¹Bd .x; p1 / W x 2 S º is a d –open cover of S ; then there exists a countable set Ap  S such that   [ 1 : Bd x; SD p x2Ap

S1

The set A D pD1 Ap  S is countable d –dense in S and so d has a countable base. Now we show (c). Since d  , BS .d /  BS ./. Let B 2 BS ./; by (i) of Theorem 2.24, B and S n B are Suslin sets with respect to the topology . As d   and d is a Hausdorff topology, B and S n B are Suslin with respect to the topology d . According to (ii) of Theorem 2.24, B 2 BS .d /. Therefore BS .d / D BS ./. Corollary 2.26 (Corollary 2, p. 106 and Theorem 6, p. 111, in [152]). Every locally compact Suslin space is a Polish space. Particularly, every compact subspace of a Suslin space is metrizable and so it is sequentially compact.

103

Section 2.2 Polish Spaces. Suslin Spaces

Proof. 1. If .S; / is a compact Suslin space, then it is regular and, according to (ii) of Theorem 2.25, there exists a metric d on S such that d   and d has a countable base. Therefore, the identity map of .S; / on .S; d / is a continuous bijection of compact space .S; / on the Hausdorff space .S; d / and so it is a homeomorphism. Then  D d . As the metrizable space .S; d / is compact, .S; d / is complete and then .S; / is a compact Polish space. 2. If .S; / is a locally compact Suslin space, then, by (i) of Theorem 2.21, it is separable. Let ¹xn W n 2 Nº be a dense subset of S ; for every n 2 N, let Vn be an open and relatively compact neighborhood of xn . Vn is a compact Suslin space and, after the first part of the demonstration, a Polish space. Let now ¹Unm W m 2 Nº be a countable base of -open sets of the topology of Vn . Then ¹Unm W n; m 2 Nº is a countable base of S , so that S is second-countable locally compact space; according to Proposition 2.20, it is a Polish space. In the following we will also show that the Polish spaces and, more generally, the Suslin spaces are Radon spaces. The next preliminary result was established by A. D. Alexandrov (1940) and was rediscovered by Yu. Prohorov (1956). Theorem 2.27 (Theorem 1.4 in [28]). Let .P; / be a Polish space; then every measure  2 ca.BP .// is tight. Proof. Let d be a complete metric which generates the topology of P and let A D ¹tn W n 2 N  º be a dense subset of .P; d /. For every n 2 N  , P D S1 1   kD1 Bd .tk ; n / and so, for every " > 0 and every n 2 N , there exists kn 2 N such that: 1 0   kn [ " 1 A < n: Bd tk ; (*) jj @P n n 2 kD1

T Skn B .t ; 1 /  P ; as A" is totally bounded, K" D A" is Let A" D 1 nD1 kD1 d k n totally bounded also and, .P; d / being complete, K" is compact. Then, by . /, 2 13 0 kn 1 [ [ 1 @P n Bd .tk ; A5 jj.P n K" /  jj.P n A" / D jj 4 n nD1 kD1 1 0 kn 1 1 X [ X 1 "  jj @P n Bd .tk ; /A < D "; n 2n nD1

which finishes the proof.

kD1

nD1

104

Chapter 2 Bounded Measures on Topological Spaces

We recall that a Radon space is a topological space T for which ca.BT / D Rca.BT / (see Definition 2.9). The following corollary was first proved by J. C. Oxtoby and S. M. Ulam. Corollary 2.28 (Theorem 9, p. 122, in [152]). Any Polish space is a Radon space. Proof. Let .P; / be a Polish space; according to 2.27, ca.BP .// D t ca.BP .// and by Theorem 2.4, ca.BP .// D rca.BP .//. Then the result is a consequence of Proposition 2.8. Theorem 2.29 (Theorem 10, p. 122, in [152] and Proposition 3, §3, in [32]). Any Suslin space is Radon space. Proof. Let P be a Polish space and let f W P ! S be a continuous surjection on S . Let us consider the multivalued mappings F W S ! P0 .P / defined by: F .s/ D f 1 .s/, for every s 2 S , where P0 .P / designates the family of all not empty subsets of P ; then the graph of F , GF D ¹.s; p/ 2 S  P W p 2 F .s/º D ¹.s; p/ W s D f .p/º D ¹.f .p/; p/ W p 2 P º, is a closed subset of S  P and therefore GF 2 BS P D BS ˝ BP . According to Aumann–Sainte Beuve theorem ([42], T. III.22), there exists a mapping g W S ! P , .BO S  BP /– measurable such that, for every s 2 S , g.s/ 2 F .s/ (g is a measurable selection of F ). Then, for every s 2 S , g.s/ 2 f 1 .s/ or, equivalently, f .g.s// D s; for every s 2 S . Let now  2 ca.BS / and let O 2 ca.BO S / be the unique extension of  on BO S . O 1 .A//, it is clear For every A 2 BP , g 1 .A/ 2 BO S . If we note .A/ D .g that  W BP ! R belongs to ca.BP / and that, for any C 2 BS , f 1 .C / 2 O 1 .f 1 .C /// D .C O / D .C /: Therefore  D BP and .f 1 .C // D .g 1 ıf is the image of  under the mapping f . According to Corollary 2.28,  2 ca.BP / D Rca.BP /. We have therefore .A/ D

sup

.K/;

for every A 2 BP :

K2KP ;KA

Then, for each C 2 BS and each a < .C / D .f 1 .C //, there exists K 2 KP ; K  f 1 .C / such that a < .K/ D .g O 1 .K//  .f O .K// D .f .K//, because f .K/ 2 KS  BS . Thus, for any a < .C /, there exists a compact set f .K/ 2 KS ; f .K/  C with a < .f .K// from where .C / D

sup K2KS ;KC

Which means that  2 Rca.BS /.

.K/;

for every C 2 BS :

105

Section 2.3 Narrow Topology

Remark 2.30. Let S be a Suslin space, P be a Polish space and f be a continuous surjection of P on S ; let us define the mapping fN W Rca.BP / D ca.BP / ! Rca.BS / D ca.BS / fN./.A/ D .f 1 .A//;

by:

for every  2 ca.BP / and every A 2 BS :

We show that fN is a surjection of Rca.BP / on Rca.BS /. In fact, according to the proof of Theorem 2.29, for all  2 ca.BS /, there exists  2 ca.BP / such that .A/ D .g O 1 .A// (where O is the extension of  on BO S and g is a O .BS  BP /-measurable selection of multivalued mapping f 1 ). fN./ D . The restriction of fN to RcaC .BP / is again a surjection of RcaC .BP / on RcaC .BS /. This remark will be used later to show that, if S is a Suslin space, RcaC .BS / is also Suslin.

2.3

Narrow Topology

In the following, we intend to introduce on ca.BT / a topology weaker than that induced by the norm. Let us first recall that ca.BT / (as well as rca.BT /, Rca.BT / and t ca.BT /), provided with the norm k  k, given by kk D jj.T / - where jj is variation of , is a Banach space (see Theorem 1.23). The notation .ca.BT // refers to the topological dual of .ca.BT /; k  k/. Notations and recalls. In the following, T designates a Hausdorff topological space, BT the –algebra of all Borel sets of T and KT the family of all compacts of T . We will note: C.T / D ¹f W T ! R W f continuous on T º, Cb .T / D ¹f 2 C.T / W f bounded on T º, Cc .T / D ¹f 2 C.T / W suppf 2 KT º—the subspace of all functions with compact support, C0 .T / D ¹f 2 C.T / W 8" > 0; 9K 2 KT s.t. jf .x/j < "; 8x 2 T nKº—the subspace of all functions vanishing at infinity, C1 .T / D ¹f 2 C.T / W 9a 2 R such thatf  a 2 C0 .T /º—the subspace of all functions having a limit at infinity. Here, a designates the constant function equal to a on T . If f  a 2 C0 .T /, then a is said to be the limit of f at infinity. Obviously, Cc .T /  C0 .T /  C1 .T /  Cb .T /  C.T / and, if T is a compact space, then the previous inclusions become equalities.

106

Chapter 2 Bounded Measures on Topological Spaces

Let T be a locally compact space and let TO D T [ ¹!º be the Alexandroff compactification of T , obtained by addition of a point to the infinity; then f 2 C1 .T / if and only if there exists g 2 C.TO / such that f D gT . In this case, the limit of f at infinity is g.!/. Let now T be a Hausdorff space; Cb .T / (and its subspaces) can be endowed with the uniform convergence norm k  k1 : if f 2 Cb .T /, then kf k1 D supx2T jf .x/j. .Cb .T /; k  k1 / is a Banach space; C0 .T / and C1 .T / are closed subspaces of Cb .T / and therefore they are Banach spaces, also. Generally, Cc .T /  C0 .T /. If T is locally compact, then Cc .T / D C0 .T / and so C0 .T / is the completion of Cc .T /; in this case the topological dual of Cc .T / with respect to uniform convergence norm is identical with the dual space of C0 .T / and coincides with .Rca.BT /; k  k/. Definition 2.31 (Definition 1, §5, in [32]). If f 2 Cb .T / and  2 ca.BT /, then f 2 RL1 ./ (see Definition 1.1.22). In this part, .f / will designate f .T / D T f d (cf. the notations of Chapter 1). For every f 2 Cb .T /, let R If W ca.BT / ! R defined by If ./ D T f d D .f /; obviously, If is a linear map on ca.BT /. The weakest topology on ca.BT / making continuous the mappings ¹If W f 2 Cb .T /º is said narrow topology on ca.BT /; this topology is noted by T.T / or T if there is no ambiguity on the space T . We also say that T is generated by Cb .T /. A net .i /i 2I  ca.BT / is narrowly convergent (or T–convergent) to  2 ca.BT / if and only if i .f / ! .f /, for every f 2 Cb .T /; we note this by T

T

! . Particularly, since 1 2 Cb .T /, if i  ! , then i .T / ! .T /. i  We will note Tc .T /; T0 .T / and T1 .T / the topologies generated on ca.BT / by the families of mappings Cc .T /; C0 .T / and C1 .T /, respectively. The topology Tc .T / is called the vague topology on ca.BT /. Remark 2.32. .i/ In general case, the mapping  7! jj of ca.BT / on caC .BT / is not narrowly continuous. Indeed, let n W BR ! R; n D ı 1  ı 1 , for every n n n 2 N  ; then the sequence .n /n is narrowly convergent to the measure 0 T

but jn j  ! 2ı0 . .ii/ Because of the inclusions between the generating families, we have Tc .T /  T0 .T /  T1 .T /  T.T /: If T is compact, it is obvious that these five topologies coincide. Without any supplementary hypotheses on T , the above-mentioned inclusions are,

Section 2.3 Narrow Topology

107

generally, strict. For example, the sequence .ın /n2N is T0 .R/–convergent to the null measure and it is not T1 .R/–convergent. In this section, we will study cases in which these inclusions are equalities. .iii/ For every f 2 Cb .T / and every  2 ca.BT /, Z ˇ ˇ ˇIf ./ˇ  jf jd jj  kf k1  jj.T / D kf k1  kk: T

Then, for all f 2 Cb .T /, If is a linear continuous functional on the norm space .ca.BT /; k  k/; hence If 2 .ca.BT // . We have therefore T.T /  .ca.BT /; .ca.BT // /  kk We will see later (see Remark 2.39) that, in general, these inclusions are strict. R .iv/ The family of applications ¹k  kf W f 2 Cb .T /º, where kkf D j T f dj; is a family of semi-norms on ca.BT / that generates the narrow topology; therefore .ca.BT /; T.T // is a locally convex vector space. We can make the same statement about the spaces .ca.BT /; T1 .T //, .ca.BT /; T0 .T // and .ca.BT /, Tc .T //. .v/ If T is compact, then Tc .T / D T0 .T / D T1 .T / D T.T /; then we have rca.BT / D Rca.BT / D ŒC.T / and T.T / is the weak* topology on ŒC.T / . Therefore, we have the following property: For every ˛ 2 R, let rca˛ .BT / D ¹ 2 rca.BT / W jj.T /  ˛º; if T is compact, then rca˛ .BT / is a compact subset of .ca.BT /; T/. Indeed, according to the Banach–Alaoglu theorem, rca1 .BT / (the closed unit ball in the normed space .rca.BT /; k  k/ D .ŒC.T / ; k  k/) is weak*-compact and therefore it is T-compact. Since rca˛ .BT / D ˛  rca1 .BT /, it is T-compact, too. We also have the following property: If T is compact, then PT (the set of all probabilities on T ) is T- compact. To demonstrate it, it is sufficient to prove that PT D ¹ 2 rcaC .BT / W .T / D 1º is a closed subset in the compact space rca1 .BT /. Indeed, for every net .i /i 2I  T

! , i .T / ! .T /, so that  is a probability on T . PT with i  For every ˛ 2 RC ; let us note by rca˛C .BT / D rca˛ .BT / \ rcaC .BT / D ¹ 2 rcaC .BT / W .T /  ˛º: Proposition 2.33. T0 .T / D T.T / if and only if T is compact.

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Chapter 2 Bounded Measures on Topological Spaces

Proof. .(/ This is evident because, if T is compact, then the generating families of these two topologies coincide. R .)/ The mapping 1 2 Cb .T /; hence I1 W ca.BT / ! R; I1 ./ D T 1d D .T /, is T–continuous and so it is T0 –continuous. For " D 12 , there exist ı > 0 and R f1 ; : : : ; fn 2 C0 .T / such that, for every i  n and every  2 ca.BT / with j T fi dj < ı, we have jI1 ./j D j.T /j < 12 . For every i, the mapping fi vanish at infinity; there exists Ki 2 KT such that, jfi .t /j < ı, for every i  n and every t 2 T n Ki . Let now K D [niD1 Ki 2 KT ; we will demonstrate by the reduction to absurd that T D K. If we suppose that R there exists t0 2 T n K, then ı t0 2 ca.BT /. Besides, for all i D 1; : : : ; n, j T fi d ı t0 j D jfi .t0 /j < ı (t0 … Ki ); therefore ı t0 .T / < 12 < 1, what is impossible. Therefore T D K is compact. Corollary 2.34. Tc .T / D T.T / if and only if T is compact. Proof. If T is compact, then Tc .T / D T.T /. Conversely, from (ii) of Remark 2.32, immediately results that, if Tc .T / D T.T /, then T0 .T / D T.T / and the previous proposition allows us to finish the demonstration. In order to demonstrate the next theorem, we need the following lemma: Lemma 2.35. Let .T; d / be a metric space; for every lower semicontinuous mapping f W T ! R, bounded from below and for every n 2 N, let fn W T ! R be the Yosida’s transform of f : fn .t/ D inf¹f .s/ C n  d.t; s/ W s 2 T º; for any t 2 T. Then the sequence .fn /n2N has the following properties: .i/ inf¹f .s/ W s 2 T º  fn .t/  f .t/, for every t 2 T , and every n 2 N. .ii/ fn .t/  fnC1 .t/, for every t 2 T and every n 2 N. .iii/ jfn .t/  fn .s/j  n  d.t; s/, for every t; s 2 T and every n 2 N. .iv/ fn .t/ " f .t/, for every t 2 T . Proof. (i) Let m D inf¹f .s/ W s 2 T º 2 R; for every t; s 2 T and every n 2 N; m  f .s/  f .s/ C n  d.t; s/, from where m  fn .t /. By the definition of fn , fn .t/  f .t /, for every t 2 T . (ii) follows immediately. (iii) For every t; s 2 T and every " > 0, there exist t1 ; s1 2 T such that, f .t1 / C n  d.t1 ; t/ < fn .t / C "; f .s1 / C n  d.s1 ; s/ < fn .s/ C ":

109

Section 2.3 Narrow Topology

We have therefore fn .t/  fn .s/ < f .s1 / C n  d.t; s1 /  f .s1 /  n  d.s; s1 / C "  n  d.t; s/ C "; fn .s/  fn .t/ < f .t1 / C n  d.t1 ; s/  f .t1 /  n  d.t1 ; t / C "  n  d.t; s/ C ": Then, for every " > 0; jfn .t/  fn .s/j < n  d.t; s/ C " what demonstrates (iii). (iv) By the definition of Yosida’s transform results that, for every t 2 T and every n 2 N  , there exists sn 2 T such that f .sn / C n  d.t; sn / < fn .t/ C

1 1  f .t / C : n n

(1)

Let us suppose that sn ¹ t ; with perhaps extracting a subsequence noted again .sn /n , there exists "0 > 0 such that d.sn ; t/  "0 , for any n 2 N  . Therefore, we obtain from (1) m C n  "0  f .sn / C n  "0 < fn .t/ C

1 1  f .t / C ; n n

for all n 2 N  :

Then f .t / D C1; this contradicts the hypothesis that f .T /  R. Therefore sn ! t and, as f is lower semicontinuous, f .t /  lim inf n f .sn /. By using (1), we obtain lim sup n  d.t; sn /  lim sup .f .t /  f .sn //  0; n

(2)

n

from where limn n  d.t; sn / D 0. Therefore, by making in (1) n tend to 1, f .t/  lim inf f .sn /  lim fn .t /  f .t /: n

n

Therefore, for every t 2 T; limn fn .t/ D f .t /. Theorem 2.36. Let T be a locally compact space whose topology is secondcountable; then: .i/ T.T /caC .BT / D T1 .T /caC .BT / and T.T /rcaC .BT / D T1 .T /rcaC .BT / .ii/ T.T /PT

D Tc .T /PT .

110

Chapter 2 Bounded Measures on Topological Spaces

Proof. (i) If T is a second-countable locally compact space, by Proposition 2.20, T is a Polish space. Let dO be a compatible metric on the Alexandroff one-point compactification TO D T [ ¹!º of T ; as TO is compact, dO is a bounded metric and his restriction to T  T is a bounded metric on T compatible with its topology. Let f 2 Cb .T / and m D inf¹f .t/ W t 2 T º; the mapping fO W TO ! R; defined by ² f .t/; t 2 T O ; f .t / D m; t D ! is bounded from below and lower semicontinuous on TO . Let .fOn /n2N be the sequence of Yosida’s transforms of fO; by (iii) of the previous lemma, .fOn /n2N  C.TO /. Therefore, if we note fn D fOn T , then .fn /n  C1 .T / (the limit of fn at infinity is fOn .!/). So, we have proved that, for every f 2 Cb .T /, there exists a sequence .fn /n  C1 .T / such that fn " f . .fn  m/n2N is, therefore, an increasing sequence of continuousRand positive mappings converging to f  m; R C .B /; lim so, for every  2 ca .f  m/d D .f  m/d, from where n T n T T R R limn T fn d D T f d or If ./ D supn Ifn ./. Because, for every n 2 N, Ifn is T1 .T /–continuous on caC .BT /, If is T1 .T /-l.s.c. on caC .BT /, for every f 2 Cb .T /. We have then the same property for f and, as If D If , If is T1 .T /– continuous on caC .BT /. According to Theorem 2.4, the second relation is identical with the first. (ii) From (ii) of Remark 2.32, there results immediately R that Tc PT  TPT . For every f 2 Cb .T /, let If W P .T / ! R, If ./ D T f d; according to the definition of narrow topology T, If is T–continuous. In order to demonstrate that TPT  Tc PT , let us show that If is Tc –continuous. Let 0 2 P .T / and let Tc

.i /i 2I  P .T / be a net with i ! 0 . Since T is Polish, for every " > 0, there exists K" 2 KT such that 0 .T n K" / < " (see 2.28). Then there exists g 2 Cc .T / with g.T /  Œ0; 1 and g K" D 1. As i .g/ ! 0 .g/, there exists i1 2 I such that, for every i  i1 , ji .g/  0 .g/j < ". Let S" D supp g K" . As g  S" and K"  g, we have: i .T n S" / D 1  i .S" /  1  i .g/ D 1 C 0 .g/  i .g/  0 .g/  1  0 .g/ C "  1  0 .K" / C " D 0 .T n K" / C " < 2": Therefore, for i  i1 , i .T n S" / < 2": Since S" is compact, there exists h 2 Cc .T / such that h.T /  Œ0; 1; h

(3) S" D Tc

1.

Let now f0 D h  f ; because supp f0  supp h, f0 2 Cc .T /. As i ! 0 ,

111

Section 2.3 Narrow Topology

i .f0 / ! 0 .f0 /; so there exists i2 2 I such that, for every i  i2 , ji .f0 /  0 .f0 /j < ":

(4)

Let i0  i1 , i0  i2 and i  i0 . Then ˇ ˇ ˇ ˇ ˇ ˇ ˇIf .i /  If .0 /ˇ  ˇIf .i /  If .i /ˇ C ˇIf .i /  If .0 /ˇ 0 0 0 ƒ‚ … „ ƒ‚ … „ T

T2

1 ˇ ˇ ˇ C If0 .0 /  If .0 /ˇ ƒ‚ … „

T3

²

0; x 2 S" ; hence  kf k; x 2 T n S" ; ˇ ˇ T1 D ˇIf .i /  If0 .i /ˇ Z .3/ jf  f0 jdi  kf ki .T n S" /  2"kf k: 

jf  f0 j.x/ D

T

ˇ ˇ .4/ T2 D ˇIf0 .i /  If0 .0 /ˇ < ": Z ˇ ˇ ˇ ˇ T3 D If0 .0 /  If .0 /  jf  f0 jd0  kf k0 .T n S" / T

 kf k0 .T n K" / < "kf k: So, for all i  i0 ,

ˇ ˇ ˇIf .i /  If .0 /ˇ < ".3kf k C 1/

and then If .i / ! If .0 /. Therefore If PT is Tc –continuous, for every f 2 Cb .T /. The proof is finished if we remark that T PT is the weakest topology making continuous all mappings of family ¹If W f 2 Cb .T /º; therefore TcPT TPT . Remark 2.37. .i/ Let .i /i 2I  rcaC .BT / be a net and let 0 2 rcaC .BT /; if T is a secondcountable locally compact space, then (i) of Theorem 2.36 can be rewritten: T

 0 , i .f / ! 0 .f /; i !

for every f 2 C1 .T /:

We remark that f 2 C1 .T / if and only if there exists a 2 R such that f  a 2 C0 .T /. Therefore, if T is a second-countable locally compact space, then ² T i .T / ! 0 .T / and  0 ” i ! i .f / ! 0 .f /; for every f 2 C0 .T /:

112

Chapter 2 Bounded Measures on Topological Spaces

.ii/ We can rewrite the second part of the previous theorem in the following manner: If T is a second-countable locally compact space, then a net of probabilities is narrowly convergent if and only if it is vaguely convergent. Let us note that this result, which is true for the probabilities, cannot be applied for caC .BT / D rcaC .BT /; in fact, the sequence .ın /n converges vaguely to 0, but it is not narrowly convergent. .iii/ We will show later (see 2.52) that, in the case where T is only locally compact (that is, without being second-countable), then, for every net .i /i 2I  RcaC .BT / and every measure  2 RcaC .BT /; T

  ” i !  vaguely and i !

i .T / ! .T /

(even though, in this case, Rca.BT / ¤ ca.BT /). We will now mention some properties of the Radon measure spaces on a completely regular space. Let us remind that a Hausdorff space T is completely regular (or Tychonoff space) if, for every x 2 T and every closed set F with x … F , there exists a continuous mapping f W T ! Œ0; 1 such that f .x/ D 0 and f .y/ D 1, for any y 2 F . Proposition 2.38. Let .T; / be a completely regular space and let ‰ W T ! Rca.BT / be the application defined by: ‰.t / D ı t , for every t 2 T . Then ‰ is a homeomorphism of .T; / on .‰.T /; T‰.T / /. Proof. Since T is completely regular, for every pair .s; t / 2 T  T with s ¤ t , there exists f 2 Cb .T / such that f .s/ D 0 and f .t / D 1 what implies that ıs ¤ ı t . Then, ‰ is injective and therefore a bijection of T on ‰.T /. Let us show that ‰ is bicontinuous. Let .ti /i 2I be a net of .T; / which converges to t . For every " > 0 and every f 2 Cb .T /, Uf;" .ı t / D ¹ 2 Rca.BT / W j.f /  ı t .f /j < "º is a T-neighborhood of ‰.t / D ı t . Since f is continuous, f .ti / ! f .t/ and so there exists i0 2 I such that jı ti .f /  ı t .f /j D jf .ti /  f .t/j < ";

for every i  i0 : T

 ‰.t /; therefore ‰ is Then, for every i  i0 , ı ti 2 Uf;" .ı t / and so ‰.ti / ! continuous. T  ‰.t /; for every f 2 Cb .T /, ı ti .f / ! ı t .f / or, Now suppose that ‰.ti / ! equivalently, f .ti / ! f .t/. For every –neighborhood V of t , let f0 W T ! Œ0; 1 be a continuous mapping such that f0 .t/ D 0 and f0T nV D 1.

113

Section 2.3 Narrow Topology

As f0 .ti / ! f0 .t/, there exists i0 2 I such that, for every i  i0 , jf0 .ti /  f0 .t/j D jf0 .ti /j < 12 . Then, for every i  i0 , ti 2 V which means that the sequence .ti /i 2I is  -convergent to t. Therefore ‰ is a homeomorphism of T on ‰.T /. Remark 2.39. If T is completely regular and if tn  ! t , with tn ¤ t , for every T

T

n 2 N, then, by Proposition 2.38, ı tn !  ı t . But ı tn .¹t º/ D 0 ¹ ı t .¹t º/ D 1. So, according to Corollary 1.58, .ı tn /n2N is not weakly convergent to ı t in ca.BT /. Hence, we infer that when T is completely regular, the narrow topology on ca.BT / is strictly less fine than weak topology of ca.BT /. In the following, we will present some topological properties of Radon measures space .Rca.BT /; T/, where we will still note T instead of TRca.BT / , the trace of the narrow topology on Rca.BT /. Proposition 2.40. Let T be a completely regular space and let  be a positive Radon measure on T ; then: .i/ for every lower semi-continuous (l.s.c.) mapping f W T ! RC , .f / D sup¹.g/ W g 2 Cb .T /, 0  g  f º, .ii/ for every bounded from above upper semicontinuous mapping f W T ! R, .f / D inf¹.g/ W g 2 Cb .T /; g  f º. Proof. (i) Let G D ¹g 2 Cb .T / W 0  g  f º. Let us first show that sup g D f:

(*)

g2G

It is obvious that supg2G g  f . Suppose that there is t0 2 T such that a D supg2G g.t0 / < f .t0 /; if 0 < ı < f .t0 /  a, then, as f is l.s.c., there exists an open neighborhood V of t0 , such that a C ı < f .t /, for every t 2 V . As T is completely regular, there exists a continuous mapping h W T ! Œ0; 1 such that h.t0 / D 1 and hT nV D 0. Let now g D .a C ı/  h; then g 2 Cb .T / and g  f so that g 2 G. Therefore, we would have a  g.t0 / D a C ı which is impossible. Therefore the property . / is demonstrated. R R Now, let us show R that .fR/ D T f d D supg2G T gd D supg2G .g/. For every g 2 G, T gd  T f d and so sup .g/  .f /:

g2G

(1)

114

Chapter 2 Bounded Measures on Topological Spaces

Let ' be a positive BT -simple function such that '  f . ' can be written ' D P n i D1 ai Ai , where, for every i D 1; : : : ; n, ai ¤ 0 and ¹Ai W i D 1; : : : ; nº  BT are pairwise disjoint nonempty sets. As  2 RcaC .BT /, for every " > 0 and every i  n, there exists Ki 2 KT ; Ki  Ai such that .Ai n Ki /
0, there exists D 2 T such that K  D and jj.D n K/ < ". Let f W T ! Œ0; 1 function such that R be a continuous R R f K D 1 and f T nD D 0. Then .f / D T f d D D f d D .K/ C DnK f d. R Therefore .K/ D .f /  DnK f d  0  jj.D n K/ > ", for every " > 0. Then .K/  0, for every K 2 KT . As  is a Radon measure, it is inner regular with respect to the family of compact sets so that  2 RcaC .BT /. Corollary 2.49. If T is a completely regular space, then .RcaC .BT /; T/ is also a completely regular space. Proposition 2.50. Let T be a completely regular space, let  2 RcaC .BT / T

  if and only if .f /  and let .i /i  RcaC .BT / be a net; then i ! lim inf i i .f /, for every function f W T ! R, bounded from below and l.s.c. T

  and if f W T ! R is bounded from below and lower Proof. .H)/ If i ! semicontinuous, then there exists m 2 R such that m C f is a positive and l.s.c. function. According to (i) of Remark 2.42, .m C f /  lim inf i i .m C f /. As i .m/ D m  i .T / ! m  .T /, we have .f /  lim inf i i .f /, for every bounded from below and l.s.c. function f W T ! R. .(H/ For every f 2 Cb .T /, f is bounded and l.s.c. and so .f /  lim inf i .f /: i

(1)

On the other hand, as f is also bounded and l.s.c., .f /  lim inf i i .f /, from where lim sup i .f /  .f /:

(2)

i

By (1) and (2), we have limi i .f / D .f / and, therefore, .i /i is T-convergent to .

119

Section 2.3 Narrow Topology

Let .T; d / be a metric space; we note by BL.T; d / the set of all bounded, realvalued Lipschitz functions on .T; d /; the following proposition is a consequence of the previous proposition and of Lemma 2.35. Proposition 2.51 (Theorem 11.3.3 in [61]). Let .T; d / be a metric space, let .i /i  RcaC .BT / be a net and let  2 RcaC .BT /; then T

  , i .f / ! .f /; i !

for every f 2 BL.T; d /:

Proof. The implication ) is a consequence of the inclusion BL.T; d /  Cb .T /. .(/ Let us suppose that i .f / ! .f /, for every f 2 BL.T; d / and let f W T ! R be a bounded l.s.c. function. Thanks to Lemma 2.35, there exists a sequence .fn /n of Lipschitz functions such that fn .t / " f .t /, for every t 2 T . According to (i) of the same lemma, fn is bounded, for every n 2 N and therefore .fn /  BL.T; d /. As .fn / " .f /, for every " > 0, there exists n0 2 N such that .f /  " < .fn0 / D lim i .fn0 /  lim inf i .f /: i

i

T

 . Proposition 2.50 allows us to conclude that i ! Proposition 2.52. Let T be a locally compact space, let  2 RcaC .BT / and let .i /i 2I  RcaC .BT / be a net; then ² T 1: i !  vaguely and  ” i ! 2: i .T / ! .T /: T

 , then, according to (ii) of Remark 2.32, .i /i is vaguely Proof. .H)/ If i ! convergent to  and i .T / ! .T /. .(H/ Let us suppose that the net .i /i is vaguely convergent to  and i .T / ! .T /. For every bounded l.s.c. function f W T ! R, there exists m 2 R such that m C f  0. Therefore, through an analogous demonstration to that of Proposition 2.40, we have .m C f / D sup¹.g/ W g 2 Cc .T /; 0  g  m C f º: So that, for every " > 0, there exists g 2 Cc .T /; 0  g  m C f such that .m C f /  "  .g/ D lim i .g/  lim inf i .m C f /: i

i

As " is arbitrary and i .m/ D m  i .T / ! m  .T / D .m/, .f /  lim inf i i .f /. Proposition 2.50 allows us to conclude.

120

Chapter 2 Bounded Measures on Topological Spaces

The following theorem characterizes the narrow topology on RcaC .BT /; the theorem can be find in the literature as Portmanteau theorem (see Theorem 2.1 in [28]). Theorem 2.53. Let .T; / be a completely regular space, let  2 RcaC .BT / and let .i /i 2I  RcaC .BT / be a net; the following properties are equivalent: .i/ .i /i 2I is narrowly convergent to . .ii/ limi i .T / D .T / and .D/  lim inf i i .D/, for every D 2 T .

.iii/ limi i .T / D .T / and .F /  lim supi i .F /, for every F D F . Each of these conditions leads to: .iv/ For every A 2 BT such that .AN n Aı / D 0; limi i .A/ D .A/. If T is metrizable, then (iv) is equivalent to the previous conditions. Proof. (i) ) (ii) Since .i /i 2I is narrowly convergent to , i .T / D i .1/ ! .1/ D .T /. On the other hand, for every R and posiR D 2 ; D is l.s.c. tive; therefore, following (i) in Remark 2.42, T D d  lim inf i T D di or .D/  lim inf i i .D/. (ii) ) (iii) For every F D FN  T; D D T n F 2  and so, by (ii), .T n F /  lim inf i i .T n F / and, as i .T / ! .T /, we have (iii). (iii) ) (i) Let f 2 Cb .T / and let m; M 2 R such that m < f .x/ < M , for every x 2 T . If g W T !0; 1Œ is defined by g.x/ D M 1m Œf .x/  m, for every x 2 T , then g 2 Cb .T / and so, for every n 2 N  and every k D 0; : : : ; n, Fkn D ¹x 2 T W kn  g.x/º is a closed subset of T . As T D R Sn Sn n n 1 . 0; 1Œ / D 1 .Œ k1 ; k Œ / D g g .F n F /, we have kD1 kD1 T gd D n n k1 k Pn R C n kD1 Fk1 nFkn gd , for every 2 Rca .BT /. Since, for every k D 1; : : : ; n, Z k k1 n n n  .Fk1 n Fk /  gd   .Fk1 n Fkn /; n n n n Fk1 nFk we have Z n n X X k1 k n n n gd  n Fkn /:  .Fk1 n Fk /   .Fk1 n n T

kD1

kD1

(1)

121

Section 2.3 Narrow Topology

As F0n D T and Fnn D ;; n X  1 k n n  .Fk1 n Fkn / D .F0n n F1n /C2 .F1n n F2n /C  Cn .Fn1 n Fnn / n n

kD1

 1 n .T / C .F1n / C    C .Fn1 /  n .Fnn / n n 1 1X .Fkn / C .T /: D n n

D

kD1

Similarly, we obtain: n n X k1 1X n n Fkn / D .Fkn /:  .Fk1 n n

kD1

kD1

Therefore, (1) leads to: Z n n 1X 1 1X n .Fk /  gd  .T / C .Fkn /: n n n T kD1

(2)

kD1

If we use (2) for i and for , due to the hypothesis (iii), then we obtain, for every n 2 N : # " Z n 1X 1 gdi  lim sup i .Fkn / lim sup  i .T / C n n i i T kD1



1 1  .T / C n n

1 1  .T / C n n

n X

kD1 n X

lim sup i .Fkn / i

.Fkn /

kD1

1  .T / C n

Z gd: T

If this takes place for all n 2 N  , we have lim sup i .g/  .g/:

(3)

i

Returning to f D .M  m/g C m, we obtain lim sup i .f /  .f /; i

for every f 2 Cb .T /:

(4)

122

Chapter 2 Bounded Measures on Topological Spaces

By replacing f 2 Cb .T / with f in (4), we obtain lim inf i .f /  .f /:

(5)

i

Finally, by (4) and (5), .f / D limi i .f /, for every f 2 Cb .T /; therefore .i /i is narrowly convergent to . (iii)) (iv) In the hypothesis (iii), the properties (i) and (ii) are satisfied; then, N D .Aı / D .A/. Since i for every A 2 BT such that .AN n Aı / D 0; .A/ and  are positive measures, due to properties (iii) and (ii), N  lim sup i .A/  lim inf i .A/ N  lim sup i .A/ .A/ D .A/ i

i

i

ı

ı

 lim inf i .A /  .A / D .A/: i

Therefore .A/ D limi i .A/. Let us suppose that .T; T/ is a metrizable space and let d be a metric on T such that  D d . In this case, we show that (iv)) (iii). For every F D FN  T; 8a > 0, let F a D ¹x 2 T W d.x; F / < aº; then Ha D F a n .F a /ı  ¹x 2 T W d.x; F /  aº n ¹x 2 T W d.x; F / < aº D ¹x 2 T W d.x; F / D aº: It is obvious that Ha \ Hb D ;, if a ¤ b. By reducing to the absurd, we show that there exists a decreasing sequence .an /n  R; an # 0 such that

.Han / D 0;

for every n 2 N:

(6)

In fact, if there is a > 0 such that, .Hb / > 0, for every b 20; aŒ, then 0; aŒD S1 1 nD1 An , where An D ¹b 20; aŒW .Hb /  n º; there exists n0 2 N such that An0 is infinite. We can, therefore, find a sequence ¹bp W p 2 Nº  An0 such that S1bp ¤ bq , for p ¤ q and, since  is a measure positive and finite, if H D pD0 Hbp 2 BT , then C1 > .T /  .H / D

1 X

.Hbp /  C1 

pD1

1 D C1 n0

which is impossible. Therefore the property (6) is proved. We have, therefore, for all n 2 N; .F an n .F an /ı / D 0 and, due to hypothesis (iv), lim i .F an / D .F an /; i

for every n 2 N:

(7)

123

Section 2.3 Narrow Topology

From (7), there results that lim supi i .F /  limi i .F an / D .F an /, for every n 2 N. As limn .F an / D .F /, we have shown that lim supi i .F /  .F /, for every F D FN  T . Finally, since .TN n T ı / D 0, from hypothesis (iv), we immediately obtain that limi i .T / D .T /. (iii) is then proved. Remark 2.54. The properties of the previous theorem do not remain equivalent on Rca.BT /. Indeed, let u be the usual topology on R and let Bu be the Borel -algebra on .R; u /. For every n 2 N, let us note n D ı 1  ı 1 ; then .n /n  n

T

n

  D 0 2 Rca.Bu /. Therefore the condition (i) is satisfied. Rca.Bu /, and n ! On the other hand, D D .1; 0/ 2 u , .D/ D 0 and lim inf n n .D/ D 1, therefore the condition (ii) is not satisfied. Proposition 2.55. Let .T; / be a completely regular space, let  2 RcaC .BT / T

and let .n /n  RcaC .BT /. If n !  , then supp   Lsn .supp n / (where 1 Lsn .An / D \pD1 [1 nDp An is the Kuratowski upper limit of .An /n2N ).

Proof. According to Definition 2.12, supp  D \¹F D FN W .T n F / D 0º (the support of ). Suppose that there is t 2 supp  and t … Lsn .supp n /. S1 S .T n 1 We have, therefore, t 2 T n Lsn .supp n / DS pD1 nDp supp n /. Let p0 2 N and D 2 T such that t 2 D and D \ 1 .supp  n / D ;. nDp0 By (ii) of Theorem 2.53, it results that: .D/  lim inf n .D/  lim inf n .T n supp n / D 0: n

n

According to Proposition 2.13, n are concentrated on their supports; so that n .T n suppn / D 0, for every n 2 N. Then F D T n D is a closed set and .T n F / D 0; since t 2 supp ; t 2 F D T n D which is a contradiction. T

Remark 2.56. From the previous proposition, it results that, if n !  ; then  is concentrated on the Kuratowski upper limit Lsn .supp n /. Proposition 2.57 (Proposition 8, §5, in [32]). Let .X; X / be a completely regular space, let T be a set and let i W T ! X be an injection. Let us note by: T BT BX MT

D i 1 .X /—the preimage topology of X by i, —the Borel sets of (T; T ), —the Borel sets of (X; X ) and D ¹ 2 Rca.BX / W j j .X n i.T // D 0º.

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Chapter 2 Bounded Measures on Topological Spaces

Then: .i/ BT D i 1 .BX / D ¹i 1 .A/ W A 2 BX º. .ii/ For every measure  2 Rca.BT /, let I./ W BX ! R defined by I./.A/ D .i 1 .A//, for every A 2 BX ; then I./ is a Radon measure on BX ; jI./j D I.jj/ and I./ 2 MT . Let I W Rca.BT / ! MT  Rca.BX /,  7! I./; then: .iii/ I is a bijection, .iv/ kI./k D kk, for every  2 Rca.BT / and .v/ I is a homeomorphism between the subspaces RcaC .BT / and MTC provided with their narrow topology. Before the demonstration, let us remind that  j j is the total variation of measure (see Definition 1.19).  j j W P .X/ ! RC indicates the outer measure generated by j j, defined by j j .A/ D inf¹j j.B/ W B 2 BX ; A  Bº.  MT  Rca.BX / is the subset of all measures on X concentrated on T (according to the terminology of [32], Chap. 9). Proof. (i) Let BT0 D i 1 .BX /. Since T D i 1 .X / D ¹i 1 .A/ W A 2 X º, BT D B.i 1 .X //. On the other hand, BT0 is a -algebra on T generated by i 1 .X / since X generates BX . Therefore, we have BT0 D BT . (ii) It is obvious that I./ is a positive measure on X and that I./.X/ D .T /. Therefore I./ 2 ca.BX /. According to Theorem 1.17, for every A 2 BX , jI./j.A/ D 8 9 n = I.jj/.A/  " D jI./j.A/  ". Then, for every A 2 BX , jI./j.A/ D sup¹jI./j.K 0 / W K 0 2 KX ; K 0  Aº and so I./ 2 Rca.BX /. We have  2 Rca.BT /; therefore, for every " > 0, there exists K1 2 KT such that jj.K1 / > jj.T /  ". If K10 D i.K1 / 2 KX , then jI./j.K10 / D I.jj/.K10 / D jj.K1 / > jj.i 1 .X//  " D I.jj/.X/  " D jI./j.X/  ". So jI./j.X n K10 / < " and, as X n K10 X n i.T /, jI./j .X n i.T // < ". Hence jI./j .X n i.T // D 0 and therefore I./ 2 MT . (iii) The mapping I W Rca.BT / ! MT , which associates to every , I./, is well defined and is injective. Now, let us show that I is a surjection of Rca.BT / on MT . If 2 MT , then 2 Rca.BX / and j j .X n i.T // D 0. Let  W BT ! R defined by .B/ D .A/, where A 2 BX is so that B D i 1 .A/.  is correctly defined since if A; C 2 BX and i 1 .A/ D i 1 .C /, then AC  X ni.T /. Hence j j.A 4 C / D 0; as j .A/  .C /j  j j.A 4 C /, .A/ D .C /. Firstly, we will show that  is a measure on BT and that I./ D . It is obvious that .;/ D .;/SD 0. Let .Bn /n2N  BT be a pairwise disjoint sequence of such that, sets and let B D 1 1 Bn 2 BT ; then, by (i), there exists .An /n  BXS 0 1 0 Bn D i .An /, for every n 2 N. We note A1 D A1 and An D An n n1 kD1 Ak , for every n  2; we obtain recursively that, for every n 2 N, Bn D i 1 .A0n /, whereS .A0n /n  BX is a pairwise disjoint sequence of S Ssets. Therefore, P1 we 0have 1 1 0 / D i 1 . 1 A0 / and .B/ D . 1 A0 / D i .A B D n n n 1 1 1 .An / D P1 1 1 .Bn /. So  is a measure on BT and .T / D .X/. For every B 2 BT , there exists A 2 BX such that B D i 1 .A/; because A n i.B/  X n i.T /, we have j j .A n i.B//  j j .X n i.T // D 0. Therefore i.B/ is –measurable and there exist C 2 BX and N  X such that j j .N / D 0, C \ N D ; and i.B/ D C [ N  i.T /. We have then j j .A n C /  j j .A n i.B// C j j .N / D 0 and so .A/ D .C /: Since 2 Rca.BX /, for every " > 0, there exists K 2 KX ; K  C such that j j.K/ > j j.C /  ". Then jj.B/ D j j.A/ D j .C /j < j j.K/ C ". As i is a homeomorphism of T on i.T /  X and as K  i.T /, H D i 1 .K/ 2 KT . Then H D i 1 .K/  B and j j.K/ D jj.H /. For every B 2 BT and every " > 0, we find H 2 KT , H  B such that jj.B/ < jj.H / C ". Therefore  2 Rca.BT /. For every A 2 BX ; I./.A/ D .i 1 .A// D .A/ which means that I./ D . Therefore, I is a surjection of Rca.BT / on MT . (iv) kI./k D jI./j.X/ D I.jj/.X/ D jj.T / D kk. (v) Let us show that I is continuous on RcaC .BT /. Let .j /j 2J  RcaC .BT / be a net narrowly convergent to  2 RcaC .BT /; for every D 2 X ; i 1 .D/ 2 T

126

Chapter 2 Bounded Measures on Topological Spaces

and so, by (ii) of Theorem 2.53, .i 1 .D//  lim inf j .i 1 .D// from where j

I./.D/  lim inf I.j /.D/: j

Moreover, I.j /.X/ D j .T / ! .T / D I./.X/ and, still according to Theorem 2.53, .I.j //j 2J is narrowly convergent to I./ in MTC . Therefore, I is a continuous mapping. We have to show yet that I 1 is continuous. Let . j /i 2J  MTC be a net narrowly convergent to 2 MTC ; for every D 2 T , there exists G 2 X such that D D i 1 .G/. Therefore I 1 . /.D/ D .G/  lim infj j .G/ D lim infj I 1 . j /.D/. According to (ii) of Theorem 2.53, .I 1 . j //j is narrowly convergent to I 1 . / in RcaC .BT / and, therefore, I is a homeomorphism of RcaC .BT / on MTC . In the rest of this section, we present several properties that can be transferred from .T; / towards .RcaC .BT /; T/. Theorem 2.58 (Proposition 10 and corollaries 1 and 2, §5, in [32]). a) If .T; / is a Polish space, then .RcaC .BT /; T/ is also Polish. b) If .T; / is a metrizable second-countable space, then .RcaC .BT /; T/ is also metrizable and second-countable. c) If .T; / is a Suslin space, then .RcaC .BT /; T/ is also Suslin space. Proof. (a) (i) We first assume that .T; / is a metrizable compact space. By (v) of Remark 2.32 and according to Theorem 2.4, we obtain: ca.BT / D rca.BT / D Rca.BT / D ŒC.T / and then T D .Rca.BT /; C.T // is the weak* topology on Rca.BT /. Still according to (v) of Remark 2.32, the closed unit ball rca1 .BT / of Rca.BT / is, therefore, T–compact. As T is metrizable compact, it is separable and so the closed unit ball rca1 .BT / provided with the weak* topology T is metrizable; hence .rca1 .BT /; T/ is a metrizable compact space. According to Proposition 2.48, RcaC .BT / D rcaC .BT / is a T–closed subset of rca.BT /; then rca1C .BT / D ¹ 2 RcaC .BT / W .T /  1º is also a metrizable compact space and then a Polish one. Let 1 be the constant function equal to 1 on T ; the mapping I1 W rca1C .BT / ! R, defined by: I1 ./ D .1/ D .T /, is T–continuous and, therefore, M D ¹ 2 rca1C .BT / W .T / < 1º D I11 .   1; 1Œ / is an open subset of .rca1C .BT /; T/.

127

Section 2.3 Narrow Topology

From (iii) of Theorem 2.18 it results that M is a T–Polish space. 1  ; is a homeFinally, the application ' W RcaC .BT / ! M; './ D 1C.T / C omorphism and so, following (i) of Theorem 2.18, Rca .BT / is Polish. (ii) Let now .T; / be an arbitrary Polish space; according to Theorem 2.19, T is homeomorphic to a Gı set P  H D Œ0; 1N . T Therefore there exist the H –open sets ¹Gn W n 2 Nº; such that P D 1 nD0 Gn . According to (i), rcaC .BH / D RcaC .BH / is a Polish space. For every n 2 N and every p 2 N  , let us note Dnp D ¹ 2 rcaC .BH / W .H n Gn / < 1 p º. As Gn is open in H , H nG is a bounded l.s.c. function; then, by (ii) of n

Corollary 2.41, for every n 2 N, I As Dnp D I1

H nGn

H nGn

is T–u.s.c. on rcaC .BH /.

.   1; p1 Œ /, Dnp is T–open in rcaC .BH /, for every

n 2 N and every p 2T N . T1 1 Then A D nD0 pD1 Dnp is a Gı subset of Polish space rcaC .BH /. By (ii) and (iii) of Theorem 2.18, A is Polish space. We remark that A D ¹ 2 rcaC .BH / W .H nP / D 0º (the set of all measures concentrated on P ). According to Proposition 2.57, RcaC .BP / is homeomorphic to A, therefore RcaC .BP / is Polish. As T is homeomorphic to P , RcaC .BT / is homeomorphic to RcaC .BP / and RcaC .BP / is also a Polish space. (b) Let .T; / be a metrizable second-countable space and let d be a metric on T compatible with  (i.e. d D ). Let .TO ; dO / be a completion of the metric space .T; d /; then .TO ; dO / is a Polish space and, by a), RcaC .BTO / is also Polish. According to Proposition 2.57, RcaC .BT / is homeomorphic to MTC D ¹ 2 RcaC .BTO / W  .TO n T / D 0º: As MTC is a subspace of RcaC .BTO /, MTC is a metrizable separable space and then RcaC .BT / is also separable and metrizable. (c) Let .T; / be a Suslin space, let P be a Polish space and let f W P ! T be a continuous surjection on T . Just as in Remark 2.30, let fN W RcaC .BP / ! RcaC .BT / be the mapping defined by fN./ W BT E ! RC ; fN./.A/ D .f 1 .A//. Then, for every net .i /i 2I  RcaC .BP / with i !  2 RcaC .BP / and every D 2 T ; f 1 .D/ 2 P and so .f 1 .D//  lim inf i i .f 1 .D//, from where fN.i / ! fN./. Therefore fN is continuous on RcaC .BP /. RcaC .BP / is Polish and, according to Remark 2.30, fN is a surjection; then RcaC .BT / is a Suslin space. Corollary 2.59. If T is a Suslin space and if H  RcaC .BT / is T– compact, then H is sequentially T–compact. Proof. According to (c) of Theorem 2.58, .RcaC .BT /; T/ is a Suslin space; the result is an immediately consequence of Corollary 2.26.

128

Chapter 2 Bounded Measures on Topological Spaces

Remark 2.60. According to Proposition 2.38, the converse of b) of Theorem 2.58 is also true. Therefore we can present the following result: .T; / is a metrizable second-countable space if and only if .RcaC .B.T //; T/ is metrizable and second-countable.

2.4

Compactness Results

Definition 2.61 (Definition 2, §5, in [32]). Let T be a completely regular space and let H  ca.BT /; H is tight if satisfies: (a) sup2H jj.T / < C1. (b) For every " > 0, there exists K" 2 KT such that jj.T n K" /  ", for every,  2 H. A sequence .n /n  ca.BT / is said be tight if ¹n W n 2 Nº is tight. Remark 2.62. .i/ This definition is coherently with Definition 2.1.6. .ii/ Let A  T and H D ‰.A/, where ‰ is the application defined in Proposition 2.38; then H D ¹ıx W x 2 Aº  RcaC .BT /. H is tight if and only if A is relatively compact in T . Proposition 2.63 (Exercise 10, §5, in [32]). Let H be a bounded subset of ca.BT /. H is tight if and only if there exists an inf-compact application ' W T ! Œ0; C1 such that Z 'd jj < C1: sup 2H

T

Proof. We recall that an application ' W T ! Œ0; C1 is inf-compact if, for every a 2 RC , the a-level set ' 1 .Œ0; a/ is a compact of T . Let H  ca.BT / be a bounded set. If H is tight, then, according to condition b) of the previous definition, there exists an increasing sequence .Kn /n of compact P subsets of T such that, for every n 2 N, sup2H jj.T nKn /  21n . Let ' D 1 nD0 T nK . If Œa note the integer n

C1Œ. Therefore ' is part of a, ' 1 .Œ0; a/ D KŒaC1 2RKT , for every a 2 Œ0;P inf-compact. Obviously, sup2H T 'd jj D sup2H 1 nD0 jj.T n Kn /  P1 P1 1 sup jj.T n K /  < C1. n 2H nD0 nD0 2n

129

Section 2.4 Compactness Results

Conversely, if we suppose application R that ' W T ! Œ0; C1 is an inf-compact M 1 such that M D sup2H T 'd jj < C1, then K" D ' .Œ0; " / 2 KT , for every " > 0; therefore, for every  2 H ,   Z " M jj.T n K" / D jj ' >  'd jj  ": " M T Hence H is tight. Proposition 2.64 (Proposition 11, §5, in [32]). If H  ca.BT / is tight, then the T-closure of H , HN , is also tight. Proof. Let H be a tight set and let M D sup2H jj.T / < C1. For every T

 . As 1 is l.s.c. and positive,  2 HN , there exists .i /i 2I  H such that i ! according to Corollary 2.45, jj.T /  lim inf ji j.T /  M: i

For every " > 0, there exists K 2 KT such that jj.T nK/ < ", for every  2 H . The mapping T nK is l.s.c. and positive; therefore, according to the same Corollary 2.45, jj.T n K/  lim inf ji j.T n K/  ": i

Hence, for every  2 HN , jj.T n K/ < "; therefore HN is tight. We introduced the following notions in [78], Definition 3.7. Definition 2.65. For every bounded set H  ca.BT /, we note t.H / D inf

sup jj.T n K/:

K2KT 2H

t .H / 2 RC is called the modulus of narrow compactness of H . It is clear that H is tight if and only if t.H / D 0. In what will follow, we present the compactness results related to weaker convergences than the narrow convergence. Definition 2.66. Let a  0; a net .i /i 2I  ca.BT / is said a–convergent to  2 ca.BT / if lim sup ji .f /  .f /j  a  kf k; i

a

 . which will be noted i !

for every f 2 Cb .T /;

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Chapter 2 Bounded Measures on Topological Spaces

A set H  ca.BT / is said relatively a-compact if every net .i /i 2I  H has at least one subnet a–convergent to a measure  2 ca.BT /. Remark 2.67. In 1.91, we have introduced a notion of a-convergence for the sequences of integrable functions:  2 caC .BT / and a  0 being fixed, a net .fi /i 2I  L1 ./ is said a–convergent to f 2 L1 ./ if ˇ ˇZ ˇ ˇ ˇ lim sup ˇ .fi  f /dˇˇ  a; for every A 2 BT : i 2I

A

If i ;  W BT ! R are defined by i .A/ D

R

a

A fi d; .A/

D

R

A f d,

then

 f if and only if .i /  ca.BT /;  2 ca.BT / and fi ! ˇ ˇ lim sup ˇi . A /  . A /ˇ  a  k A k; i 2I

for every A 2 BT :

There is, therefore, no inconvenience in using the same notion for the two convergences. Remark 2.68. .i/ .i /i 2I is 0–convergent to  if and only if .i /i 2I is narrowly convergent to . Hence, a set H  ca.BT / is relatively 0-compact if and only if H is relatively narrowly compact. .ii/ We can interpret inf¹a  0 W .i /i 2I is a  convergentº as indicating the deficit of narrow convergence of the net .i /i . The following theorem generalizes the Prohorov’s theorem (see Theorem 3.8 of [78]). Theorem 2.69. Let T be a completely regular space, let H be a bounded subset of Rca.BT / and let t.H / be its modulus of narrow compactness; then H is relatively t .H /-compact. ˘ Proof. Let X be the Cech–Stone compactification of T , let i be a homeomorphic embedding of T in X (i.T / D X) and let I be the injection of Rca.BT / in rca.BX / D Rca.BX / associated to injection i (see Proposition 2.57). Then I is a isometric injection; moreover, for every R F 2 C.X/, fR D F ı i 2 Cb .T / and, for every  2 Rca.BT /, I./.F / D X F d.I.// D T f d D .f /. Since H  Rca.BT / is bounded, I.H / is also bounded in rca.BX /. Therefore, the set I.H / is relatively weak* compact, which means that it is relatively T.X/-compact (see (v) of Remark 2.32).

131

Section 2.4 Compactness Results

Let .j /j 2J be a net in H and, for every j 2 J , let j D I.j /. Then . j /j 2J is a net in I.H / so that there exists a subnet . jl /l2L , T.X/–convergent to a measure 2 Rca.BX /. By the definition of t .H /, for every  2 H and every n 2 N  , there exists Kn  KT such that 1 : n

(1)

i.Kn / D i.T0 /:

(2)

jj.T n Kn / < t.H / C Let T0 D

S1

nD1 Kn

and X0 D

1 [ nD1

For every n 2 N; i.Kn / 2 KX hence X0 2 BX and X ni.Kn / is an open subset of X. Therefore, according to Corollary 2.45, the mapping  7! jj.X n i.Kn // of rca.BX / in RC is T.X/–l.s.c. on rca.BX / and so, j j.X n i.Kn //  lim inf j jl j.X n i.Kn //; l

for every n 2 N  :

(3)

By (1), for every l 2 L and every n 2 N  , j jl j.X n i.Kn // D jI.jl /j.X n i.Kn // D jjl j.T n Kn / < t .H / C n1 . Then, by (3), we obtain: j j.X n i.Kn //  t.H / C

1 ; n

for every n 2 N  :

(4)

Therefore, according to (2), j j.X n X0 /  t .H /:

(5)

˘ As X is the Cech–Stone compactification of T , every continuous mapping f 2 Cb .T /, is extendable to a mapping F 2 C.X/ (F ı i D f ); moreover, kF k D kf k (see Theorem 3.6.1. of [68]). Let now J W Cb .T / ! R defined by Z F d : J.f / D .F  X / D 0

X0

Obviously, J is a continuous linear mapping on Cb .T /. Since is a Radon measure on X, i is a homeomorphism of T on i.T /  X and X0 D i.T0 / 2 BX , then, for every " > 0, there exists K 2 KT ; K  T0 (and so i.K/  X0 ) such that j j.X0 n i.K// < ".

132

Chapter 2 Bounded Measures on Topological Spaces

For every g 2 Cb .T / such that jgj  1 and gK D 0, let G 2 C.X/ such that G ı i D g and kGk D kgk  1. Then Z Z Z Z J.g/ D .G  X / D Gd D Gd C Gd D Gd and 0 X0 X0 ni.K/ i.K/ X0 ni.K/ ˇ ˇZ ˇ ˇ ˇ Gd ˇˇ  j j.X0 n i.K// < ": jJ.g/j  ˇ X0 ni.K/

Then, according to Proposition 5 of Chap. 9, §5 of [32] , there exists a measure  2 Rca.BT / such that .f / D J.f /, for every f 2 Cb .T /. For every f 2 Cb .T /, let us note as previously F 2 C.X/, the extension of f ; F ıi D f and kF k D kf k. Then jjl .f /.f /j D jjl .F ıi/ .F  X /j  0 j jl .F /  .F /j C j .F  X nX /j. 0 As . jl /l2L is T.X/–convergent to , by passing to limit, we obtain: lim sup jjl .f /  .f /j  kF k  j j.X n X0 / l

and then, using the inequality (5), lim sup jjl .f /  .f /j  t.H /  kf k;

for every f 2 Cb .T /:

l

Therefore, .jl /l is t.H /–convergent to  and so H is t .H /–compact. Remark 2.70. From previous theorem, it results that: For every bounded net .i /i 2I  Rca.BT /, there exist a subnet .ij /j 2J and a measure  2 Rca.BT / such that lim sup jij .f /  .f /j  t.¹i W i 2 I º/  kf k; j

for every f 2 Cb .T /:

Now, we obtain as corollary an important result of compactness—Prohorov’s theorem (see Theorem 1, §5, in [32] and Theorem 11.5.4 of [61]). Corollary 2.71 (Prohorov’s Theorem). If T is a completely regular space, then every tight of Rca.BT / is relatively T-compact. Proof. If H is tight, then H is bounded and t .H / D 0. From Corollaries 2.59 and 2.71, we obtain: Corollary 2.72. Let T be a Suslin space; every tight .n /n2N  RcaC .BT / admits one subsequence .kn /n2N T–convergent to a measure  2 RcaC .BT /.

133

Section 2.4 Compactness Results

We can formulate a reciprocal of the Prohorov’s theorem in the case of locally compact spaces or of Polish spaces but, only for the positive measures (see Theorem 2, §5, in [32] ). Theorem 2.73. Let T be a locally compact space and let H  RcaC .BT / be a relatively T.T /-compact set; then H is tight. Proof. The mapping  7! .T / is continuous on .RcaC .BT /; T.T //. Hence, since HN is T.T /-compact, sup2H .T / < C1. For every " > 0 and every  2 HN , there exists K 2 KT such that .T n K / < ". Now let U be a T.T /-open relatively compact neighborhood of K . Since T nU is u.s.c. and bounded, in the light of (ii) of Corollary 2.41, the  mapping  7! .T n U / is T.T /-u.s.c. Therefore V D ¹ 2 HN W .T n U / < "º is T.T / HN -open and, as  2 V , V is a neighborhood T.T /-open of . ¹V W  2 HN º is a T.T /-open cover of the compact set HN ; then there exist N such that HN D [n Vi . 1 ; : : : ; n 2 HS i D1 Then K D niD1 UN i 2 KT and, for every  2 H , there exists i so that  2 Vi and so .T n Ui / < ". Then, we have .T n K/ < " which leads to the fact that H is tight. Theorem 2.74. Let T be a Polish space and let H  RcaC .BT / be a relatively T.T /-compact set; then H is tight. Proof. Just as in the proof of the previous theorem, sup2H .T / < C1. As T is Polish, according to Theorem 2.19, T is homeomorphic to a Gı subset of Œ0; 1N . For every p 2 N, let Tp be an open subset of Œ0; 1N and let i W 1 T ! \pD0 Tp be a homeomorphism. For every p 2 N, let ip W T ! Tp be the application defined by ip .t/ D i.t /, for any t 2 T . Then, we define Ip W RcaC .BT / ! RcaC .BTp / by Ip ./.A/ D .ip1 .A//, for every  2 RcaC .BT / and every A 2 BTp . According to (v) of Proposition 2.57, Ip is continuous and Hp D Ip .HN / is a compact set in .RcaC .BTp /; T.Tp //. Since Tp is an open set in the compact space Œ0; 1N , Tp is locally compact. From Theorem 2.73 it results that Hp is tight. Therefore, for every p 2 N and every " > 0, there exists Cp 2 KTp  KŒ0;1N such that .Tp n Cp /
0 such that kfn k1 D sup jfn .t/j  N; t 2T

for every n 2 N:

(1)

As T is Polish, .RcaC .BT /; T/ is also Polish (see a) of Theorem 2.58). Then ¹n W n 2 Nº is narrowly compact and hence it is tight (see Theorem 2.74). Therefore M D sup n .T / < C1:

(2)

n2N

For every " > 0, there exists K 2 KT such that n .T n K/ < "1 D

" ; 6N

for every n 2 N:

(3)

u

As fn ! f0 and n .f0 / ! 0 .f0 /, there exists n0 2 N such that K

jfn .t/  f0 .t/j < "2 D

" ; 3M

for every n  n0

and every t 2 K

" jn .f0 /  0 .f0 /j < "3 D ; for every n  n0 : 3

and

(4)

(5)

138

Chapter 2 Bounded Measures on Topological Spaces

Then, for every n  n0 , jn .fn /  0 .f0 /j



jn .fn /  n .f0 /j C jn .f0 /  0 .f0 /j ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ ˇ .fn  f0 /dn ˇ C ˇ .fn  f0 /dn ˇˇ C "3 ˇ ˇ ˇ

.5/


.D/  "; 2

(3)

for every n  n1 ;

from where lim inf n n .D/  .D/. Remark 2.82. .RcaC .BT /; T/ is a metrizable space and D is a metric compatible with the narrow topology T; so T is the trace of strong topology of BL.T / . T

Therefore, for every net .i /i  RcaC .BT / and every  2 RcaC .BT /; i !   ” D.i ; / ! 0.

142

2.5.2

Chapter 2 Bounded Measures on Topological Spaces

Lévy–Prohorov’s Metric

According to Theorem 2.58, if .T; / is a Polish space, then .RcaC .BT /; T/ is also Polish. In the following theorem, we present a complete metric compatible with the narrow topology on RcaC .BT / (see Exercise 8, §5, in [32]). Theorem 2.83. Let .T; / be a Polish space and let d be a metric on T compatible with . For every " > 0 and every F  T let F " D ¹x 2 T W d.x; F / < "º be the open ball of radius " centred at F . Let L W RcaC .BT /  RcaC .BT / ! RC be a mapping defined by: ² ³ .F /  .F " / C " L.; / D inf " > 0 W ; for every F D F  T : .F /  .F " / C " Then L is a complete metric on RcaC .BT / that generates the narrow topology on RcaC .BT /. The metric L is called the Lévy–Prohorov metric. Proof.  Firstly, we will show that L is a metric on RcaC .BT /. For every ; 2 RcaC .BT /, L.; /  maxŒ.T /; .T / < C1 ( and are bounded measures); therefore L is real valued. If  D , then, for every " > 0 and every F D FN , .F /  .F " / C " and .F /  .F " / C ", from where L.; / D 0. Conversely, if L.; / D 0, then there exists a decreasing sequence "n # 0 such that, for every n 2 N and every F D FN , .F /  .F "n / C "n and .F /  .F "n / C "n . Then the sequence "n "n .F "n /n is decreasing and F D \1 nD1 F . Therefore, .F / D limn .F / and " n .F / D limn .F /. We have then, for every F D FN , .F / D .F / what assures that  D since  and are regular measures. The equality L.; / D L. ; / is obvious. Let ; ; 2 RcaC .BT /; for every " > 0, there exist ı1 > 0 and ı2 > 0 such that ı1 < L.; / C ", ı2 < L.; / C " and, for every F D FN ; ² .F /  .F ı1 / C ı1 ; (1) .F /  .F ı1 / C ı1 : ²

.F /  .F ı2 / C ı2 ; .F /  .F ı2 / C ı2 :

(2)

An elementary calculation leads to F ı1

ı2

 F ı1 Cı2 :

(3)

Section 2.5 Metrics on the Space .RcaC .BT /; T/

143

Then, by (1) - (3), we obtain, for every F D FN , .F /  .F ı1 / C ı1  .F ı1 / C ı1 ı2

 .F ı1 / C ı1 C ı2  .F ı1 Cı2 / C ı1 C ı2 and similarly .F /  .F ı1 Cı2 / C ı1 C ı2 : These last relations leads us to: L.; /  ı1 C ı2 < L.; / C L.; / C 2" from where, L.; /  L.; / C L.; /. Therefore, L is a metric on RcaC .BT /. Let us show that TL D T. Since T is a metrizable topology, it is sufficient to show that, for every sequence .n /n  RcaC .BT / and every  2 RcaC .BT /; T

  ” L.n ; / ! 0: n ! T

  and let r > 0: The open balls of T having the  Let us suppose that n ! same center t and different radii have their boundaries disjoint. Then, since  is bounded, the -measure of the boundaries of these balls is null, except perhaps for at most countable family of balls.  Since T is separable, there exists a family ¹B n W n 2 N º of open balls Bn of S radius rn < 6r such that .@Bn / D 0 and T D 1 nD1 SBn . Let K 2 KTSsuch that .T n K/ < 3r . As K  1 nD1 Bn , there exists p 2 N p such that K  nD1 Bn D A0 and r .T n A0 / < : 3

(4)

Let A be the family of all finite union of balls of the set ¹B1 ; : : : ; Bp º. A is j j finite and, if A 2 A, then A D [kD1 Bik , @A D AN n Aı  [kD1 .BN ik n Biık / D j

[kD1 @Bik , from T where .@A/ D 0. Let U D A2A ¹ 2 RcaC .BT / W j .A/  .A/j < 6r º \ ¹ 2 RcaC .BT / W j .T /  .T /j < 6r º. According to (iv) of Theorem 2.53, for every A 2 A, n .A/ ! .A/ and n .T / ! .T /; then, there exists n0 2 N such that, n 2 U;

for every n  n0 :

(5)

Let us show that U  T .; r/ D ¹ W L.; /  rº:

(6)

144

Chapter 2 Bounded Measures on Topological Spaces

For every F D FN , let A1 D [¹Bn W n D 1; : : : ; p; Bn \ F ¤ ;º. Then A1 2 A, r F  A1 [ .T n A0 / and A1  F 3 . Therefore, for every 2 U , we have .F /  .A1 / C .T n A0 / D .A1 / C .T /  .A0 / r r r r < .A1 / C C .T / C  .A0 / C D .A1 / C .T n A0 / C 6 6 6 2 r r r r < .A1 / C C < .A1 / C r < .F 3 / C r  .F / C r: 3 2 Therefore, we showed that, for every 2 U , .F / < .F r / C r. By applying an analogous method, we obtain .F / < .F r / C r from where L.; /  r and so 2 T .; r/. Hence we have U  T .; r/. Therefore, by (5), for every r > 0, there exists n0 such that, for any n  n0 , L.n ; /  r, which means that L.n ; / ! 0.  Now, let us suppose that L.n ; / ! 0. We show that, for every F D FN , lim supn n .F /  .F / and that n .T / ! .T / which will ensure, according T

 . to (iii) of Theorem 2.53, that n ! Let F D FN  T ; since limr#0 .F r / D .F /, there exist " > 0 and r > 0 such that " " r< and .F r / < .F / C : (7) 2 2 Let n0 2 N such that, L.n ; / < r;

for every n  n0 :

By (8), for every n  n0 , there exists ın < r such that ² n .F / < .F ın / C ın ; .F / < n .F ın / C ın :

(8)

(9)

By (7) and (9), we obtain: n .F / < .F ın /Cın  .F r /Cr < .F /C 2" C 2" D .F / C "; for every n  n0 , from where lim sup n .F /  .F /:

(10)

n

For every " > 0, there exists n0 2 N such that, for every n  n0 ; L.n ; / < ". Hence, for every n  n0 , there exists ın < " such that n .T / < .T ın / C ın D .T / C ın and .T / < n .T ın / C ın D n .T / C ın , from where jn .T /  .T /j < ın < "; T

 . (10) and (11) lead to n !

for every n  n0 :

(11)

Section 2.5 Metrics on the Space .RcaC .BT /; T/

145

Therefore, we have shown the equality of topologies: TL D T. In the rest of the proof, we will show directly that .RcaC .BT /; L/ is a Polish space. Since TL D T, the space .RcaC .BT /; L/ is separable.  We only have to show that .RcaC .BT /; L/ is complete. Let .n /n2N  .RcaC .BT /; L/ be a Cauchy sequence; then .n /n is tight. Indeed, since .n /n is Cauchy, for every " > 0 and every a > 0, there exists n0 2 N such that L.n ; n0 / < 2" , for every n  n0 . Therefore, for every n  n0 , there exists ın < 2" ; ın < a2 such that n .F / < n0 .F ın / C ın ;

for every F D FN  T:

(12)

Let K 2 KT such that " n .T n K/ < ; 2

for every n D 1; : : : ; n0 :

(13)

According to (12), for every n > n0 ; n .T / < n0 .T ın / C ın D n0 .T / C ın and n0 .T / < n .T / C ın , from where sup jn .T /  n0 .T /j 

nn0

" 2

(14)

and then supn2N jn j.T / D supn n .T / < C1. By (12) and (13), we obtain:   a a n .T n K 2 / < n0 .T n K 2 /ın C ın  n0 .T n K/ C ın < "; 8n  n0 : (15) a

As K is compact, there exists a finite set F  T such that K 2  F a and so, by (15), n .T n F a / < ", for every n  n0 . For any n < n0 , n .T n F a /  n .T n K/ < " and then, by (13) and (15), for every " > 0 and every a > 0, there exists a finite set F  T such that n .T n F a / < ";

for every n 2 N:

(16)

Let " > 0; by (16), to every p 2 N, we associate an a D 21p and a finite set Fp  T such that   1 " 2n (17) < p ; for every n 2 N: n T n Fp 2 1 T1 p Fp2 is a closed and totally bounded set in the Therefore, the set K" D pD1 complete metric space .T; d /, hence a compact one: K" 2 KT . By (17) it follows

146

Chapter 2 Bounded Measures on Topological Spaces

that, for every n 2 N, 0 n .T n K" / D n @

1 [

1 2p

T n Fp

pD1

!1   1 X 1 2p A n T n Fp < ": pD1

Therefore, the sequence .n /n is tight; according to Corollary 2.71, there exists kn " C1 such that .kn /n to be narrowly convergent to a measure  2 RcaC .BT /. T

Since .n /n is L–Cauchy, n !  . Remark 2.84. .i/ Let ‰ W T ! RcaC .BT /, ‰.t / D ı t (see Proposition 2.38). It is obvious that, for all s; t 2 T , L.‰.t/; ‰.s// D d.t; s/. Therefore, ‰ is an isometry of T on its image ‰.T /. .ii/ The completness of .T; d / was used only to show that .RcaC .BT /; L/ is complete. Therefore, if .T; d / is a second-countable metric space, L remains a compatible metric for .RcaC .BT /; T/. We have noticed in Example 2.76, that a narrowly convergent sequence of measures on a Suslin space is not necessarily tight. As we will show in the following theorem, in the case of a metrizable space, every narrowly convergent sequence is tight. Theorem 2.85 (Le Cam, see [61], 11.5.3). Let .T; d / be a metric space, let T

 ; then .n /n .n /n2N  RcaC .BT / and let  2 RcaC .BT / such that n ! is tight. Proof. Every n is tight; so it is concentrated on a  - compact set Sn which is separable (Sn is a countable union of separable sets). Then S D [1 nD0 Sn is also separable. Since every bounded continuous function on S can be extended to a bounded continuous function on T , .n S /n2N remains narrowly convergent to S . Therefore, we can suppose that T is second-countable itself. According to (ii) of the previous remark, L.n ; / ! 0 and n .T / ! .T /. Since  2 RcaC .BT /, for each " > 0, there exists K 2 KT such that .T n K/ < 4" or " .K/ > .T /  : 4

(1)

Section 2.5 Metrics on the Space .RcaC .BT /; T/

147

For every n 2 N, ¯ ® ı > 0 W .F /  n .F ı / C ı; n .F /  .F ı / C ı; for every F D F  T ¯ ®  ı > 0 W .F /  n .F ı / C ı; for every F D F  T ¯ ®  ı > 0 W .K/  n .K ı / C ı : Then ln D inf¹ı > 0 W .K/  n .K ı / C ıº  L.n ; /;

for all

n 2 N:

(2)

Since L.n ; / ! 0, there exist n0 2 N and a sequence .ın /n2N 0; C1Œ such that .K/  n .K ın / C ın ; ın
n .K ın /  : (4) 4 Let n1 2 N such that, for every n  n1 , " .T / > n .T /  : 4

(5)

According to (4), (3), (1) and (5) n .T n Kn / < ";

for every n  max¹n0 ; n1 º D n2 :

(6)

Since ¹0 ; : : : ; S n2 1 º is tight, we can suppose that (6) is satisfied for all n 2 N. Let C D K [ 1 nD0 Kn ; then n .T n C / < ", for every n 2 N. We only have to show that C is compact. Let .xp /p2N  C . If .xp /p2N has a subsequence with all the terms in the same compact K or Kn , then it also has a subsequence convergent in C . If no, let .pk /k2N and .nk /k2N two sequence in N such that xpk 2 Knk , for every k 2 N. Since Knk  K ınk , for every k 2 N, there exists yk 2 K with d.xpk ; yk / < ınk . Then .yk /k has a subsequence, still noted .yk /k , convergent to y 2 K. Since ınk ! 0 (see (3)), we have xpk ! y 2 C . Remark 2.86. Generally speaking, a metrizable space is not a Prohorov space (see Remark 2.75) but, according to the previous theorem, it is sequentially Prohorov.

148

2.6

Chapter 2 Bounded Measures on Topological Spaces

Wiener Measure

In this section, we will present an important application of compactness theorem of Prohorov in the construction of the Wiener measure on C Œ0; 1. The presentation of Wiener measure represents a processing of Paragraphs 8 and 9 of Chapter 2 of Billingsley’s monograph [28]. Let C D C Œ0; 1 be the space of all real-valued continuous functions on Œ0; 1 equipped with norm of uniform convergence, noted k:k. If kk is the norm topology, then .C; kk / is a Polish space. Before establishing the existence of Wiener measure, we begin this section by studying the narrow convergence of measures on C in connection to the narrow convergence of the image of this measures on the finite dimensional subspaces. In the following, B is the -algebra of all Borel sets of C and, for every n 2 N  , Bn indicates the -algebra of Borel sets of Rn . Definition 2.87. For every t1 ; t2 ; : : : ; tn 2 Œ0; 1 with t1 < t2    < tn , let  t1 ;:::;tn W C ! Rn be the natural projection from C to Rn defined by:  t1 ;:::;tn .x/ D .x.t1 /; : : : ; x.tn // ;

for every x 2 C:

A set C  C is said to be a cylinder set if there are t1 ; t2 ; : : : ; tn 2 Œ0; 1 with .H /. By C we denote t1 < t2 <    < tn and H 2 Bn , such that C D  t1 1 ;:::;tn the family of all cylinder sets: .H / W n 2 N  ; 0  t1 <    < tn  1; H 2 Bn º: C D ¹ t1 1 ;:::;tn Proposition 2.88. C is an algebra that generates B. C is said the algebra of cylinder sets. Proof. Since every projection  t1 ;:::;tn is continuous,  t1 .H / 2 B, for every 1 ;:::;tn H 2 B therefore C  B  For every I; J 2 C, there exist n; m 2 N, 0  t1 <    < tn  1, 0  s1 < .B/ and J D    < sm  1 and B 2 Bn , C 2 Bm such that I D  t1 1 ;:::;tn .C /. Let ¹r ; : : : ; r º D ¹t ; : : : ; t º [ ¹s ; : : : ; s º where max¹n; mº  s1 1 p 1 n 1 m 1 ;:::;sm p  n C m and 0  r1 < r2 <    < rp  1. We suppose that , for every k D 1; : : : ; n, rik D tk and let 'n;p W Rn ! Rp be the embedding defined by: ² uk ; j D ik ; k D 1; : : : ; n; 'n;p .u1 ; : : : ; un / D .v1 ; : : : ; vp /; where vj D 0; j … ¹i1 ; : : : ; in º: 1 .B /, from where ' According to (i) of Proposition 2.57, Bn D 'n;p p n;p .B/ 2 Bp 1 and I D r1 ;:::;rp .'n;p .B//.

149

Section 2.6 Wiener Measure

Similarly, we define the embedding 'm;p W Rm ! Rp so that 'm;p .C / 2 Bp and J D r1 .'m;p .C //. 1 ;:::;rp 1 I [ J D r1 ;:::;rp .'n;p .B/ [ 'm;p .C // and I n J D r1 .'n;p .B/ n 1 ;:::;rp 'm;p .C //. Therefore I [ J; I n J 2 C. Since C D  t1 .R/, for every t 2 Œ0; 1, C 2 C, and therefore C is an algebra on C.  Let A be the -algebra generated by C; then A  B. For every x 2 C and every " > 0, let T .x; "/ D ¹y 2 C W kx  yk  "º be the closed ball of radius " centered at x and let   n    Y i i  "; x C" : x Hn D n n i D1

Hn is a closed subset of Rn ; therefore, for every n 2 N, Hn 2 Bn . Let us show that T .x; "/ D

1 \ nD1

If y 2

T1

1 nD1  1 ; 2 ;:::; n .Hn /, n n

n

 1 .Hn /: 1 2 ; ;:::; n n n

(*)

n

then, for every n 2 N  , .y. n1 /; : : : ; y. nn // 2 Hn ,

form where, for every n 2 N  and every i D 1; : : : ; n, jy. ni /x. ni /j  ". Since x and y are uniformly continuous on Œ0; 1, for every  > 0, there exists ı > 0 such that jt  sj < ı implies jx.t/  x.s/j <  and jy.t /  y.s/j < . Let now n 2 N  i with n1 < ı; for every t 2 Œ0; 1, there exists i 2 ¹1; : : : ; nº such that i 1 n  t < n. i i i i So we have jx.t /y.t /j  jx.t/x. n /jCjx. n /y. n /jCjy. n /y.t /j < 2C" from where kx yk D sup t 2Œ0;1 jx.t/y.t/j  2C", for any  > 0. Therefore kx  yk  " and y 2 T .x; "/. Conversely, if y 2 T .x; "/, then, for every n 2 N  and every i D 1; : : : ; n,  jx. ni /  y. ni /j  kx  yk  " and so, y 2  1 1 2 n .Hn /, for every n 2 N . n ; n ;:::; n

By . /, for every x 2 C and every " > 0, T .x; "/ 2 A. Since C is separable, every open set of C is a countable union of closed balls and, therefore, every open set belongs to A and, consequently, A D B. Remark 2.89. According to the previous proposition, if the measures  and 2 RcaC .B/ coincide on C, then  D : We will now characterize the narrow convergence on RcaC .B/ with the help of the narrow convergence of the image measures on the finite dimensional subspaces.

150

Chapter 2 Bounded Measures on Topological Spaces

Let us remember that, according to Theorem 2.58, .RcaC .B/; T/ is a Polish space. For every  2 RcaC .B/ and every t1 ; : : : ; tp 2 Œ0; 1, let  t1 ;:::;tp ./ D  ı 1  t1 ;:::;tp W Bp ! RC , defined by: . ı  t1 /.A/ D . t1 .A//, for every 1 ;:::;tp 1 ;:::;tp C A 2 Bp . It is obvious that  t1 ;:::;tp ./ 2 Rca .Bp /. Let us denote by Tp the narrow topology on RcaC .Bp /. Theorem 2.90. For every .n /n2N  RcaC .B/ and every  2 RcaC .B/; 8 Tp ˆ < 1:  t1 ;:::;tp .n / !  t1 ;:::;tp ./; T  ” n ! for every p 2 N  and every 0  t1 <    < tp  1; ˆ : 2: .n /n2N is tight : T

 ; for every p 2 N  , every 0  t1 <    < tp  1 and Proof. .H)/ Let n ! every n 2 N, let n D  t1 ;:::;tp .n / and let D  t1 ;:::;tp ./: According to (iv) of Theorem 2.53, T

p n ! , lim n .A/ D .A/; for every A 2 Bp such that .AN n Aı / D 0:

n

.A/ 2 B. Let A 2 Bp with .AN n Aı / D 0 and let B D  t1 1 ;:::;tp 1 1 ı ı 1 Since  t1 ;:::;tp .A/ n . t1 ;:::;tp .A// D BN n B   t1 ;:::;tp .AN n Aı /, we obtain .BN n B ı / D 0. T

 ; limn n .B/ D .B/, from where, for every A 2 Bp such that As n ! ı N .AnA / D 0, limn n .A/ D .A/ and so . n /n2N is Tp -convergent to . Which demonstrates 1. T  , .n /n is relatively T– Since .RcaC .B/; T/ is Polish space and n ! compact; according to Theorem 2.73, .n /n is tight. .(H/ According to Corollary 2.72, as .n /n is tight and .RcaC .B/; T/ is a Polish space, .n /n is relatively T-compact. Therefore, every subsequence .0n /n of .n /n has a subsequence .00n /n T–convergent to  2 RcaC .B/. For every C 2 C  B, there exist p 2 N  , 0  t1 <    < tp  1 and A 2 Bp such that C D  t1 .A/. 1 ;:::;tp Tp

Tp

Since  t1 ;:::;tp .00n / !  t1 ;:::;tp ./ and  t1 ;:::;tp .00n / !  t1 ;:::;tp ./,  t1 ;:::;tp ./ D  t1 ;:::;tp ./ (.RcaC .Bp /; Tp / is a Hausdorff space). Therefore, for every C 2 C, .C / D .C /. Then, according to Remark 2.89,  D . Therefore, every subsequence of .n /n admits a subsequence T-convergent to ; T

what assures that n !  .

151

Section 2.6 Wiener Measure

Remark 2.91. In Theorem 2.90, the quality of being tight is essential and cannot be removed, as it is demonstrated in the following example: 8 0  t  n1 ; < n  t; Let xn W Œ0; 1 ! R; defined by xn .t/ D 2  n  t; n1 < t  n2 ; : 2 0; n 0º, then  t1 ;:::;tp .ıxn /.A/ D 1 D ı0  t1 ;:::;tp .ı0 /.A/, for every p  p0 . N then  t1 ;:::;tp .ıxn /.A/ D 0 D ı0  t1 ;:::;tp .ı0 /.A/: If 0p … A,

p

Therefore, we have  t1 ;:::;tp .ıxn / !  t1 ;:::;tp .ı0 /, for every p 2 N  and every 0  t1 <    < tp  1. It is obvious that the sequence .ıxn /n2N is not tight. Now, we will give some characterizations of tightness for a sequence of positive Radon measures on C. Definition 2.92. For every x 2 C; let wx W0; 1 ! RC , defined by: wx .ı/ D supjst j 0, there exists K 2 KC with .C n K/ < ", such that limı!0 wx .ı/ D 0, uniformly on K .limı!0 wx .ı/ D 0, -almost uniformly on C). .ii/ there exists N  C such that  .N / D 0 and limı!0 wx .ı/ D 0, for every x 2 C n N (limı!0 wx .ı/ D 0, -a.e. on C). .iii/ for every " > 0 and every  > 0, there exists ı 20; 1 such that (¹x 2 C W wx .ı/  "º/ <  .limı!0 wx .ı/ D 0 in -measure). Proof. (i) Since  2 RcaC .B/, for every " > 0, there exists K 2 KC such that .C n K/ < ": By the Arzelà–Ascoli theorem, as K is compact in C, K is uniformly equicontinuous; therefore there exists ı0 20; 1 such that, for every s; t 2 Œ0; 1 with js  tj < ı0 and for every x 2 K; jx.s/  x.t /j < ", which means that limı!0 wx .ı/ D 0, uniformly on K. (ii) and (iii) are immediate consequences of (i). According to Remark 2.93 (ii), we can see that w: .ı/ is uniformly continuous, for every ı > 0, hence ¹x 2 C W wx .ı/  "º is a closed subset of C, and, therefore, a Borel subset of C. Theorem 2.95. The sequence .n /  RcaC .B/ is tight if and only if .i/ .0 .n //n2N is tight on R and .ii/ 8";  > 0; 9ı 20; 1 such that n .¹x 2 C W wx .ı/  "º/ < ; 8n 2 N. Proof. (i) Firstly, let us show that the sequence .0 .n //n2N is tight on R if and only if: (1) supn n .C/ < C1 and (2) 8" > 0; 9a > 0 such that n .¹x 2 C W jx.0/j > aº/ < "; 8n 2 N. According to Definition 2.61, .0 .n //n2N is tight on R if and only if: (a) supn .0 .n //.R/ < C1 and, (b) 8" > 0; 9K 2 KR such that .0 .n //.R n K/ < "; 8n 2 N. (1) ” (a) since .0 .n //.R/ D n .¹x 2 C W x.0/ 2 Rº/ D n .C/ and, therefore, .0 .n //n2N is bounded if and only if .n /n2N is bounded. (2) ” (b) Indeed, it is clear that (b) ” for every " > 0, there exists a > 0 such that, for every n 2 N; .0 .n //.R n Œa; a/ D n .¹x 2 C W jx.0/j > aº/ < ". (ii) According to (ii) of Remark 2.93, for every ı 2 .0; 1, w: .ı/ is an uniformly continuous function, hence ¹x 2 C W wx .ı/  "º is a closed subset of C and so a Borel subset. Therefore, we can consider the measure n .¹x 2 C W wx .ı/  "º/.

153

Section 2.6 Wiener Measure

(H)) Let us suppose that the sequence .n /n is tight. Then supn n .C/ < C1 and (1) is proved. By (b), for every  > 0, there exists K 2 KC such that, for all n 2 N, n .C n K/ < . By Arzelà–Ascoli theorem, K is bounded and uniformly equicontinuous. Since K is bounded, there exists a > 0 such that, for every x 2 K, kxk0  a from where ¹x 2 C W jx.0/j > aº  C n K and so, for every n 2 N, n .¹x 2 C W jx.0/j > aº/ < . (1) and (2) being satisfied .0 .n //n2N is tight on R. Since K is uniformly equicontinuous, there exists ı 20; 1 such that, for every s; t 2 Œ0; 1 with js  tj < ı and for every x 2 K; jx.t /  x.s/j < ", i.e., for every x 2 K, wx .ı/ < ". Therefore, we have ¹x 2 C W wx .ı/  "º  C n K and, for every n 2 N, n .¹x 2 C W wx .ı/  "º/ < . (ii) is then satisfied. ((H) Let us suppose that .n /n2N satisfies the conditions (i) and (ii). Since .0 .n //n2N is tight, (1) is satisfied: supn n .C/ < C1. Also we have: For every  > 0, there exists a > 0 such that n .¹x 2 C W jx.0/j > aº/
"º/ <  and, for every n  n0  1, n .¹x 2 C W wx .ı/ > "º/  n .¹x 2 C W wx .ı/  2" º/  n .¹x 2 C W wx .ın /  2" º/ <  what gives (ii). Proposition 2.97. Let .n /n  RcaC .B/ be a sequence satisfying the following two conditions: .a/ .0 .n //n2N is tight on R, .b/ for every " > 0 and every  > 0, there exist ı 20; 1 and n0 2 N such that, for every n  n0 , and every t 2 Œ0; 1, 1  n .¹x 2 C W sup t st Cı jx.s/  x.t /j  "º/  . ı Then .n /n is tight. Proof. According to Theorem 2.95 and to (ii) of Remark 2.96, it is sufficient to show that (b)H)(ii’). By (b), for every " > 0, there exist  > 0, ı0 20; 1 and n0 2 N such that, for every n  n0 and every t 2 Œ0; 1, ´ μ! " :ı0 x 2 C W sup jx.s/  x.t /j   : (*) n 3 2 t st Cı0

155

Section 2.6 Wiener Measure

For every t 2 Œ0; 1/, let A t D ¹x 2 C W sup t st Cı0 jx.s/  x.t /j  3" º. Then [

¹x 2 C W wx .ı0 / > "º 

Ai ı0 :

(**)

i2N

i ı0 ", there exist s; t 2 Œ0; 1 with js  t j < ı0 and jx.s/  x.t /j > ". We can suppose that t < s  1 and t < s < t C ı0 . Let p 2 N such that p  S ı0 < 1  .p C 1/  ı0 . pC1 Since t 2 Œ0; 1Œ i D1 Œ .i  1/ı0 ; iı0 Œ, there exists i0 2 ¹1; : : : ; p C 1º such that t 2 Œ.i0  1/ı0 ; i0 ı0 Œ. Let us show that: .1/ If i0  p; then x 2 A.i0 1/ı0 [ Ai0 ı0 : .2/ If i0 D p C 1; then x 2 Apı0 : Indeed, in the case (1), if we suppose that x … A.i0 1/ı0 [ Ai0 ı0 , then sup .i0 1/ı0 ui0 ı0

jx ..i0  1/ı0 /  x.u/j
0 and every  > 0, there exist ı 20; 1 and n0 2 N such that, for all n  n0 and all t 2 Œ0; 1; μ! ´ 1  ; (T1 ) P ! 2  W sup jXn .!/.t /  Xn .!/.s/j  " ı t st Cı then .Pn /n2N is tight. Now, we transform the condition .T1 / to one easier to handle. Theorem 2.108. Let .Xn /n2N be the sequence of measurable applications and let .Pn /n2N be the sequence of probabilities defined in 2.105 and specified in Remark 2.106. If .Pn /n satisfies the condition: For every " > 0, there exist  > 1 and n0 2 N such that, for every n  n0 and every k 2 N, ² ³ p " P ! 2  W max jkC1 .!/ C    C kCi .!/j   n  2; (T2 ) i n  then the sequence .Pn /n is tight. Proof. For every i D 1; : : : ; n, let us note Si D 1 C 2 C    C i : For every t 2 Œ0; 1 and every n 2 N, there exists a single i 2 ¹0; : : : ; nº such that ni  t < i C1 n . Let ı > 0 and s 2 Œt; t C ı; then there exists j  i such that jn  t C ı < j C1 n Sj and so s 2 Œt; t C ı  kDi Œ kn ; kC1 Œ . Since X .!/ is affine on every interval n n , Œ ni ; i C1 n ˇ ˇ  ˇ ˇ   ˇ ˇ ˇ ˇ i i ˇ C ˇXn .!/ Xn .!/.s/ˇˇ jXn .!/.t/Xn .!/.s/j  ˇˇXn .!/.t/Xn .!/ ˇ ˇ n n ˇ    ˇ ˇ i C1 i ˇˇ  Xn .!/  ˇˇXn .!/ n n ˇ ˇ    ˇ ˇ i k ˇˇ ˇXn .!/  Xn .!/ C max ˇ n n ˇ i C1kj C1 ˇ    ˇ ˇ i l C i C 1 ˇˇ ˇ  2 max ˇXn .!/  Xn .!/ ˇ n n 0lj 1 D 2 max

0lj i

1 p jSi ClC1 .!/  Si .!/j: n

164

Chapter 2 Bounded Measures on Topological Spaces

As j  i < nı C 1 < j  i C 1, we have, for every s 2 Œt; t C ı, jXn .!/.t/  Xn .!/.s/j  2

1 p jSi ClC1 .!/  Si .!/j : n 0l m0 and every i 2 N, μ! ´ r m2 "  ı: (3) P ! 2  W max jSi Cl .!/  Si .!/j  ı 1lm q Let now  D p"  m2 m ; since ı be chosen as small,  > 1. ı So, we can reduce (3) to: for every " > 0 and every  > 0, there exist m0 2 N and  > 1 such that, for every m  m0 and every i 2 N, ³ ² p 1 "2   2; (4) P ! 2  W max jSi Cl .!/  Si .!/j  m   2  1lm which is equivalent to .T2 /. Theorem 2.109. The sequence .Pn /n2N defined in 2.105 is tight. Proof. Let us mention again that .n /n2N is a independent sequence normally distributed N .0; 1/ on the space .; A; P /, that .Xn /n2N is the sequence of mappings Xn W  ! C, .A  B/-measurable such that: 1 nt  Œnt   Œnt C1 .!/; Xn .!/.t/ D p SŒnt  .!/ C p n n

165

Section 2.6 Wiener Measure

where Sk .!/ D 1 .!/ C    C k .!/ and S0 .!/ D 0 and that, for all n 2 N, Pn W B ! R is image of P under Xn . According to Theorem 2.108, to show that the sequence .Pn /n2N is tight, it is sufficient to show that, for every " > 0, there exist  > 1 and n0 2 N such that, for every n  n0 and every k 2 N, ² ³ p " P ! 2  W max jSkCi .!/  Sk .!/j   n  2: (T2 ) i n  Let k 2 N;  > 1 and n 2 N be fixed. For every i  n  1, we note ² ³ ˇ ˇ p ˇ ˇ Ei D ! 2  W max SkCj .!/  Sk !/ <  n  jSkCi .!/  Sk .!/j : j i 1

If i ¤ j , then Ei \ Ej D ; and ² ³ p ± p n ! 2  W maxSkCi .!/Sk .!/j  n  ! W jSkCn .!/Sk .!/j  i n 2

°

[

n1 [ i D1

² ³ p Ei \ ! 2  W jSkCn .!/  SkCi .!/j  n : 2

(1)

Indeed, let ! 2  so that p max jSkCi .!/  Sk .!/j   n

p n: i n 2 p If we note i0 the smallest index i  n for which jSkCi .!/  Sk .!/j   n, then i0  n  1 and ! 2 Ei0 . Also, we have ˇ ˇ ˇ ˇ ˇSkCn .!/  SkCi .!/ˇ  ˇSkCi .!/  Sk .!/ˇ  jSkCn .!/  Sk .!/j 0 0 p p p  n nD n 2 2 and

jSkCn .!/  Sk .!/j
1 such that ³ ²  1 jSj j < 3; P ! W p .!/  2  j

1  2

e

x2 2

dx D 0;

for every   0

(5)

(6)

and every j 2 N  : (7)

Then, by (4) and (7), we obtain ! ² ³ n1 X p 1 P ! W max jSkCi .!/  Sk .!/j   n  3 1C P .Ei / i n  i D1 !! n1 [ 2 1 Ei for every   0 :  3; D 3 1CP   i D1

(8)

167

Section 2.6 Wiener Measure

For every " > 0 and  > max¹0 ; 2" º,  > 1 and so   p " P max jSkCi .!/  Sk .!/j   n  2 ; for every k; n 2 N: i n  The sequence .Pn /n2N satisfies (T2 ) and, therefore, it is tight. Now we can introduce the Wiener measure on C. Definition 2.110. A measure W W B ! Œ0; 1 is called a Wiener measure on C if it satisfies the two following conditions: R ˛  u2 2t du, .i/ W .¹x 2 C W x.t/  ˛º/ D p 1 8t 20; 1, 8˛ 2 R. 1 e 2 t

.ii/ 8n 2 N, 8 0  t0 < t1 <    < tn  1, 8i D 0; : : : ; n  1, the random variables i W C ! R, defined by i .x/ D x.ti C1 /  x.ti /, 8x 2 C, are independent with respect to the measure W . Remark 2.111. .i/ W .C/ D limn!1 W .11 .1; n/ D limn!1

p1 2

Rn

 1 e

u2 2

du D 1.

Therefore, W is a probability on C. .ii/ The first condition of Definition 2.105 implies that, for every t 20; 1, the random variable p  t W .C; B; W / ! R (see Definition 2.87) is normally distributed N .0; t/. Indeed, for every t 20; 1 and every ˛ 2 R, Z  1 u2 1 e  2t du; W  t .1; ˛ D p 2 t .1;˛ so that, for every t 20; 1 and every B 2 BR , Z  1 u2 1 e  2t du: W  t .B/ D p 2 t B .iii/ W .¹x 2 C W x.0/ D 0º/ D 1. Indeed, let tn # 0 and, for every ˛ 2 R, let An .˛/ D ¹x 2 C W x.tn /  ˛º. Then \ [ Am .˛/  ¹x 2 C W x.0/  ˛º lim sup An .˛/ D n

n mn

168

Chapter 2 Bounded Measures on Topological Spaces

so that W .¹x 2 C W x.0/  ˛/º/  W .lim sup An .˛//  lim sup W .An .˛// n n Z ˛ u2 1 e  2tn du D lim sup p 2 tn 1 n Z p˛ 2tn 1 2 e v dv D 1: D lim p n  1 It results that, for every ˛ > 0, W .¹x 2 C W x.0/  ˛/º/  1 and so, W .¹x 2 C W x.0/  ˛/º/ D 1: Particularly, for ˛ D 0, we obtain (*) W .¹x 2 C W x.0/  0/º/ D 1: S1 On the other hand, .¹x 2 C W x.0/ < 0º/ D pD1 .¹x 2 C W x.0/ <  p1 º/ from where ³ ² 1 : x 2 C W x.0/ <  W .¹x 2 C W x.0/ < 0º/ D lim W p p Since .¹x 2 C W x.0/ <  p1 º/  lim inf n An . p1 /, for every p 2 N  , ³    ² 1 1  W lim inf An  W x 2 C W x.0/ <  n p p Z  p1 p 2tn 1 2 e v dv D 0:  lim inf p n  1 Therefore, for every p 2 N  , W .¹x 2 C W x.0/ <  p1 º/ D 0 from where we obtain: W .¹x 2 C W x.0/ < 0º/ D 0 and, by . /, W .¹x 2 C W x.0/ D 0º/ D 1:

(**)

According to the relation .

/, the theory of Wiener measure is unchanged if the space C is replaced by its subspace C0 D ¹x 2 C W x.0/ D 0º. .iv/ From condition (ii) of Definition 2.110 and according to the notations of Definition 2.87, for every t; s 2 Œ0; 1 with 0  s < t  1, the random variables s 0 and  t s are independent. From the above property .

/, it results that s  0 D s , W -a.e. hence s and  t  s are independent. According to Proposition 2.103, we have E. t / D E.s / C E. t  s / and V . t / D V .s / C V . t  s /. Therefore, by (ii), E. t  ps / D 0 and V . t  s / D t  s, i.e.  t  s is normally distributed N .0; t  s/.

169

Section 2.6 Wiener Measure

.v/ For the same reason, for all t0 D 0 < t1 <    < tn  1, the set of random variables ¹ tiC1   ti W i D 0; : : : ; n  1º is independent. We can therefore write, for every ˛1 ; : : : ; ˛n 2 R, ¯ ® W x 2 C W  tiC1 .x/   ti .x/  ˛i C1 W i D 0; : : : ; n  1 D D

n1 Y i D0 n1 Y

 W  tiC1   ti  ˛i C1 p

i D0

Z

1 2.ti C1  ti /

˛iC1

e

u2 iC1 ti /

 2.t

du

1

1 Dp n .2/ t1 .t2  t1 / : : : .tn  tn1 /   2 Z u u2 u2 n 2  2t1 C 2.t t CC 2.tn tn1 / 1 2 1/ e du1 : : : dun :  Q n iD1 .1;˛i 

Proposition 2.112. .i/ Let W W B ! Œ0; 1 be a Wiener measure on C and let C be the algebra of all cylinder sets on C; for every 0 D t0 < t1 <    < tn  1 and every .B/ 2 C, then B 2 Bn , let C D  t1 1 ;:::;tn W .C / D q

1 .2/n …nkD1 .tk  tk1 /

Z



e

Pn

u2 k kD1 2.tk tk1 /

!

du1 : : : dun :

B

(*) .ii/ Conversely, if W W B ! Œ0; 1 is a measure satisfying . /, for every C 2 C, then W is a Wiener measure on C. Proof. Let n 2 N  and let Tn W Rn ! Rn be defined by Tn .x/ D .x1 ; x2  x1 ; : : : ; xn  xn1 /, for every x D .x1 ; : : : ; xn / 2 Rn . It is clear that Tn is a homeomorphism of Rn on Rn . The -algebra Bn is generated by the family Qn S D ¹ i D1   1; ˛i  W ˛1 ; : : : ; ˛n 2 Rº; since Tn is a homeomorphism, Tn1 .S/ generates Bn . (i) Let 0 D t0 < t1 <    < tn  1 and B 2 Tn1 .S/. Then, there exists .˛1 ; : : : ; ˛n / 2 Rn such that B D Tn1 .

n Y

  1; ˛i /

i D1

D ¹.x1 ; : : : ; xn / W x1  ˛1 ; x2  x1  ˛2 ; : : : ; xn  xn1  ˛n º:

170

Chapter 2 Bounded Measures on Topological Spaces

Let C D  t1 .B/ 2 C. According to (v) of Remark 2.111, we have 1 ;:::;tn  .B/ W .C / D W  t1 ;:::;t n 1 1 Dq .2/n …nkD1 .tk  tk1 / 

Z 

e

Qn



2 2 2 v1 v2 vn 2t1 C 2.t2 t1 / CC 2.tn tn1 /



dv1 ; : : : ; dvn :

iD1 1;˛i 

By using the change of the variable u D .u1 ; : : : ; un / D Tn .v1 ; : : : ; vn / D v we obtain . /: 1 W .C / D q .2/n …nkD1 .tk  tk1 /

Z



e

Pn

u2 k kD1 2.tk tk1 /

!

du1 ; : : : ; dun :

B

Because Tn1 .S/ generates Bn and . / is satisfied for all sets C 2 ¹ t1 .B/ W B 2 Tn1 .S/º, W satisfies . /, for all C 2 ¹ t1 .B/ W B 2 1 ;:::;tn 1 ;:::;tn Bn º. (ii) Let  W B ! Œ0; 1 be a measure that satisfies . / on C; then, for every t 20; 1, every ˛ 2 R and C D  t1 .   1; ˛/ 2 C: Z ˛ u2 1 .C / D p e  2t du; 2 t 1 which is condition (i) of Definition 2.110. Let n 2 N, t0 D 0 < t1 <    < tn  1 and i W C ! R; i .x/ D x.ti C1 /  x.ti /. According to (iii) of Remark 2.111, we can suppose that x.0/ D 0. Then, 0 .x/ D x.t1 /. In order to show that 0 ; : : : ; n1 are independent it is sufficient, according to (iii) of Proposition 2.100, to show that, for every ˛0 ; : : : ; ˛n1 2 R; ! n1 n1 \ Y  1  i .   1; ˛i / D  i1 .   1; ˛i / : (1) i D0

i D0

On the one hand, we have:   n1 \ 1 i .   1; ˛i /  i D0

´ D

x 2 C W Tn .x.t1 /; : : : ; x.tn // 2 ´

D

x 2 C W .x.t1 /; : : : ; x.tn // 2

n1 Y

μ!   1; ˛i 

i D0

Tn1

n1 Y i D0

!μ!   1; ˛i 

:

171

Section 2.6 Wiener Measure

Q If we note B D Tn1 . n1 i D0   1; ˛i /, then we can rewrite the previous equality: ! n1 \  1 i .   1; ˛i / D   t1 .B/ :  ;:::;t n 1 i D0

Therefore, we have  T n1 1  i D0 i .   1; ˛i  / Z

1

Dq

(2)  

Pn

.uk uk1 /2 kD1 2.tk tk1 /



e du1 ; : : : ; dun .2/n …nkD1 .tk  tk1 / B 1 Dq .2/n …nkD1 .tk  tk1 /   2 2 2 Z v v2 vn  2t1 C 2.t t CC 2.tn tn1 / 1 2 1/ e dv1 ; : : : ; dvn  Q n1 iD0 1;˛i 

D

n1 Y i D0

p

Z

1

˛i

e

2.ti C1  ti /

 D  01 .   1; ˛0 / 

1 n1 Y i D1

v2 iC1 iC1 ti /

 2.t

p

dvi C1 Z

1 2.ti C1  ti /

˛i

e

 2.t

v2 iC1 iC1 ti /

1

dvi C1 :

On the other hand, for every i D 1; : : : ; n  1;   i1 .   1; ˛i / D  .¹x 2 C W x.ti C1 /  x.ti /  ˛i º/ D  .¹x 2 C W T2 .x.ti /; x.ti C1 // 2 R  1; ˛i º/  D  ¹x 2 C W .x.ti /; x.ti C1 // 2 T21 .R  1; ˛i /º : If we note Bi D T21 .R  1; ˛i /, then    .B /  i1 .   1; ˛i / D   t1 i ;t i iC1 Dp

1 .2/2 ti .ti C1  ti /

Z

u2

e

.u u1 /2 iC1 ti /

 2t1  2.t2 i

du1 du2 :

Bi

Using the change of variable v D .v1 ; v2 / D T2 .u1 ; u2 / D T2 .u/, we obtain  2 Z ˛i Z 2 v2 v1  1  1  2.tiC1 ti / e 2ti dv1  e dv2  i .1; ˛i  D p 2 .2/ ti .ti C1  ti / R 1 Z ˛i v2 1  2.t iC1 ti / iC1 Dp e dvi C1 ; 2.ti C1  ti / 1 which transferred back in (2), for i D 1; : : : ; n  1, leads to relation (1).

172

Chapter 2 Bounded Measures on Topological Spaces

Therefore, the random variables 0 ; : : : ; n1 are independent and so  is a Wiener measure. Remark 2.113. .i/ It results that, in order to study if one measure is a Wiener measure, we can replace conditions (i) and (ii) of Definition 2.110 with condition . / of Proposition 2.112. .ii/ If the measures ; W B ! Œ0; 1 on C verify . /, they coincide on C and, by Remark 2.89,  D . Therefore, if there is a measure which satisfies conditions (i) and (ii) of Definition 2.110, this measure is unique. In other words, there is at least a Wiener measure on C. Now, we demonstrate the existence of such a measure. Theorem 2.114. There exists only one Wiener measure on C. Proof. Let .n /n2N be a independent sequence of random variables on .; A; P /, every one normally distributed N .0; 1/; let .Xn /n2N be the sequence of .A  B/measurable mappings defined in 2.105. We recall that 1 nt  Œnt  Xn .!/.t/ D p SŒnt  .!/ C p  Œnt C1 .!/; n n where Sk .!/ D 1 .!/ C    C k .!/ and S0 .!/ D 0. For each n 2 N, let Pn W B ! R be the image of P under the mapping Xn . We will identify the Wiener measure as the T-limit of .Pn /n2N . We have shown in Theorem 2.109 that .Pn /n2N is tight. Therefore, since C is Polish, and so completely regular, we can apply Prohorov’s theorem (see 2.71). Then the family ¹Pn W n 2 Nº is relatively T-compact. So every subsequence .Pn0 /n2N of .Pn /n2N admits at least a subsequence .Pn00 /n2N narrowly convergent to a measure  2 RcaC .B/. We only have to demonstrate that  is the Wiener measure under research. According to Theorem 2.90, for each p 2 N  and each 0  t1  t2      tp  1, Tp

 t1 ;:::;tp .Pn00 / !  t1 ;:::;tp ./: Particularly, for every t 2 .0; 1, T1

 t .Pn00 / !  t ./:

Section 2.6 Wiener Measure

173

We have therefore, according to (ii) and (iv) of Theorem 2.53, on the one hand, . t .Pn00 //.R/ D Pn00 .C/ D 1 ! 1 D . t .//.R/ D .C/, from where it results that  is a probability on C. On the other hand, for every ˛ 2 R such that . t .//.@.   1; ˛// D .¹x 2 C W x.t / D ˛º/ D 0,   t .Pn00 / .   1; ˛/ D Pn00 .¹x 2 C W x.t/  ˛º/  D P ¹! 2  W Xn00 .!/.t /  ˛º !  .¹x 2 C W x.t /  ˛º/ : According to (ii) of Remark 2.106, for every n 2 N, Xn .:/.t / is normally  .nt Œnt /2 distributed N .0; n .t// where n2 .t/ D Œnt . Then we have n2 .t / ! n C n t and Z ˛ 2   u 1 002 .t/ 00 2 n e du: P ¹! 2  W Xn .!/.t/  ˛º D q 2n00 2 .t / 1 So, for every t 20; 1 and every ˛ 2 R with .¹x 2 C W x.t / D ˛º/ D 0, Z ˛  u2 1 00 lim  t .Pn / .   1; ˛/ D p e  2t du: (1) n 2 t 1 Now, we note that there is a countable set E  R such that, for all ˛ 2 R n E, .¹x 2 C W x.t / D ˛º/ D 0 and so, by (1), Z ˛ u2 1  .¹x 2 C W x.t/  ˛º/ D p e  2t du; for every ˛ 2 R n E: (2) 2 t 1 Since R n E is dense in R, (2) is satisfied, for every ˛ 2 R and every t 20; 1. The condition (i) of Definition 2.110 is therefore fulfiled by . We still have to show that  satisfies (ii) of Definition 2.110; this means that, for every 0  t0 < t1 <    < tn  1 and every i D 0; : : : ; n  1, the random variables x 7! x.ti C1 /  x.ti /, are independent. According to (iii) of Remark 2.111, we can suppose that, for every x 2 C, x.0/ D 0. Let us start by n D 2. Let 0 < t < s  1 and T2 W R2 ! R2 ; T2 .x1 ; x2 / D .x1 ; x2  x1 /: For every ˛; ˇ 2 R, we note A D  1; ˛  1; ˇ. Using an analogous demonstration to the previous one, we obtain:  .¹x 2 C W x.t/  ˛; x.s/  x.t/  ˇº/ D  .¹x 2 C W T2 .x.t /; x.s// 2 Aº/  D  ¹x 2 C W .x.t/; x.s// 2 T21 .A/º     D  t;s ./ T21 .A/ D lim  t;s .Pn00 / T21 .A/ n ®  00 ¯ D lim P ! 2  W Xn .!/.t /; Xn00 .!/.s/ 2 T21 .A/ n ® ¯ D lim P ! 2  W Xn00 .!/.t /  ˛; Xn00 .!/.s/Xn00 .!/.t /  ˇ : n

174

Chapter 2 Bounded Measures on Topological Spaces

Since the random variables .n /n2N are independent, ¯ ® P ! 2  W Xn00 .!/.t/  ˛; Xn00 .!/.s/  Xn00 .!/.t /  ˇ ¯ ® D P ! 2  W Xn00 .!/.t /  ˛ ¯ ®  P ! 2  W Xn00 .!/.s/  Xn00 .!/.t /  ˇ Therefore, we have:  .¹x 2 C W x.t/  ˛; x.s/  x.t/  ˇº/ ® ¯ D lim P ! 2  W Xn00 .!/.t /  ˛ n ¯ ®  P ! 2  W Xn00 .!/.s/  Xn00 .!/.t /  ˇ D  .¹x 2 C W x.t/  ˛º/   .¹x 2 C W x.s/  x.t /  ˇº/ : Therefore, we have shown that the random variables x 7! x.t / and x 7! x.s/  x.t / are independent. Similarly, we show that, for every n 2 N and every 0  t0 < t1 <    < tn  1, the set of random variables ¹x 7! x.ti C1 /  x.ti / W i D 0; : : : ; n  1º is independent with respect to . Hence  satisfies the condition (ii) of Definition 2.110. Therefore,  D W is the unique Wiener measure on C. Since every subsequence .Pn0 /n2N of .Pn /n2N has a subsequence .Pn00 /n2N T

narrowly convergent to W , then Pn !  W.

Chapter 3

Young Measures

In this last chapter, we will use the compactness results obtained in the two previous chapters in order to study the Young measures. On the one hand, the Young measures generalize the measurable functions. Even if they were introduced in order to obtain relaxed solutions for the variational problems, the Young measures represent an important tool for other branches of mathematical analysis. In the first paragraph, we start with the presentation of the preliminaries connected to the disintegration of measures as well as of the various families of integrands. In the second paragraph, we present the definitions and several characteristics of Young measures and we give some examples. We will study, in the third and fourth paragraphs, the stable topology on the Young measure spaces and the properties of its trace on the subspace of measurable functions. The two main results of the fourth paragraph point out that the trace of the stable topology on the subspace of measurable functions is the topology of convergence in measure and that this subspace is dense in the space of Young measures. In the fifth paragraph, we will present Prohorov’s theorem of compactness for the Young measures. The main element in this theorem is the condition of tightness; the simplicity of this condition represents the main attraction of the results of compactness in the Young measure spaces. In the first chapter, we have introduced the scalar version of the biting lemma; the sixth paragraph is dedicated to the study of the vector version of this lemma. By adding to the tightness condition that of boundedness, we can state the general result of compactness of Saadoune–Valadier. We will extend these two results of compactness (biting lemma and theorem of Saadoune–Valadier) to an unbounded set of measurable functions—the finite-tight sets. In the seventh paragraph, we will study the two types of products for the Young measures and will give the fiber product lemma. In the eighth paragraph, we will introduce, in the Euclidean case, the Jordan finite-tight sets. The main result of the paragraph states that a tight subset H  W 1;1 .; Rm /, for which the set of its gradients rH is a Jordan finite-tight set,

176

Chapter 3 Young Measures

is relatively compact in measure. This result will offer excellent conditions of application of fiber product lemma in order to obtain of lower semicontinuity results for integral functionals. In the ninth paragraph, we will give several applications of Young measure theory in the study of strong compactness in Lp . In the last two sections of the book, we consider some aspects of relaxed variational calculus. Are studied the Young measures generated by sequences and particularly the gradient Young measures. We pay special attention to quasiconvexity and its various equivalent definitions. The quasiconvexity is essentially used in the Kinderlehrer–Pedregal’s characterization of gradient Young measures, but also in the study of lower semicontinuity of energy functional that appears in variational calculus. Finally, we present some results of existence of solutions in a relaxed variant of variational calculus. The theory of the Young measures is presented in the general frame of the regular Suslin spaces which allows us to included the special case of a separable Banach spaces equipped with the weak topology (see Example 2.17). To simplify the demonstrations, we will present some results in the particular case of Polish spaces or even of the Euclidean spaces, by mentioning, at the same time, the bibliography for the more general case.

3.1

Preliminaries

In this chapter, we will use the following notations:   is an arbitrary set,  A is a -algebra of subsets of ,   2 caC .A/ is a complete, -additive measure,  .S; S / is a regular Suslin space. We recall that such a space S is a normal space and, according to Theorem 2.29, S is a Radon space (ca.BS / D Rca.BS /).  BS is the family of Borel sets of S,  PS is the set of all probabilities on S endowed with the narrow topology,  C is the -algebra of Borel sets of PS .  BR is the family of Borel sets of R endowed with the usual topology,  BŒ0;1 is the -algebra of Borel sets of subspace Œ0; 1  R.

Section 3.1 Preliminaries

177

If X is any topological space and BX is the -algebra of Borel sets, then a mapping f W  ! X is .A  BX /-measurable if f 1 .B/ 2 A, for every B 2 BX .  M.X/ notes the set of all .A  BX /-measurable mappings.  Particularly, if X D R, then we note M.R/ with M.  For every f 2 M.X/; fN denotes the set of all mappings of M.X/ equal with f -almost everywhere (-a.e.). If .E; k  k/ is a separable Banach space and if p  1, then  Lp .; ; E/ D Lp .; E/R D Lp .; E/ D ¹f 2 M.E/ W kf kp D kf kp d < C1º and  Lp .; ; E/ D Lp .; E/ D Lp .; E/ D ¹fN W f 2 Lp .; E/º. More particular, Lp ./ D Lp .; R/ and Lp ./ D Lp .; R/.  L1 .; ; E/ D L1 .; E/ D ¹f 2 M.E/ W kf k1 D inf¹c > 0 W kf k  c;   p.p.º < C1º,  L1 .; ; E/ D L1 .; E/ D ¹fN W f 2 L1 .; E/º.

3.1.1

Disintegration

Theorem 3.1. Let  W  ! PS ,  .t/ D  t , for every t 2 ; the following properties are equivalent: .i/  is .A  C/-measurable. .ii/ For every B 2 BS , the mapping gB W  ! Œ0; 1, defined by gB .t / D  t .B/, is .A  BŒ0;1 /-measurable. .iii/ For every C 2 A ˝ BS , the mapping fC W  ! Œ0; 1, defined by fC .t / D  t .C t /, is .A  BŒ0;1 /-measurable, where, for every t 2 ; C t D ¹x 2 S W .t; x/ 2 C º denotes the section of C determined by t . Proof. (i) ) (ii) Let E D ¹B 2 BS W gB is .A  BŒ0;1 /-measurable}. We show that E D BS . (a) For every open set D  S , D is l.s.c. from where the mapping F W .PS ; T/ ! RC , defined by F . / D .D/, is l.s.c. (see Corollary 2.41, (i)). Then, for any ˛ 2 R; G D ¹ 2 PS W .D/ > ˛º D F 1 . ˛; C1Œ / is narrowly open and, since  is .A  C/-measurable,  1 .G/ D ¹t 2  W gD .t / > ˛º 2 A from where it results that gD is .A  BŒ0;1 /-measurable and, therefore, D 2 E.

178

Chapter 3 Young Measures

(b) For every open set D and every closed set F in S, gD\F .t / D  t .D\F / D  t .D/   t .D n F / D gD .t/  gDnF .t/. By (a), D; D n F 2 E and so gD\F is .A  BŒ0;1 /-measurable, hence D \ F 2 E. (c) We can easily show that E is a monotone class of sets, closed under countable pairwise disjoint union of sets. (d) By (b) and (c), it results that E contains the -algebra generated by the open sets of S , therefore it contains BS , which proved (ii). (ii) ) (iii) Let F D ¹C 2 A ˝ BS W fC is .A  BŒ0;1 /-measurable}. (a) For every A 2 A and every B 2 BS , fA B .t / D  t ..A  B/ t / ²  t .B/; t 2 A D D A .t /   t .B/ D A .t /  gB .t /; 0; t …A from where it results that fA B is .A  BŒ0;1 /-measurable and A  B 2 F . (b) For every sequence .Cn /n2N of pairwise S1 S1 disjoint subsets of F , let C D t 2 , C t D nD1 .C nD1 S Cn ; then, for Pevery Pn1/ t , from where fC .t / D 1 t . 1 .C / / D  ..C / /. Then f D n t t n t C nD1 nD1 nD1 fCn is measurable and so C 2 F . S (c) Let R D ¹ nkD1 .Ak  Bk / W n 2 N; .Ak  Bk / \ .Al  Bl / D ; if k ¤ l; Ak 2 A and Bk 2 BS º; by (a) and (b), R is a sub-algebra of F . (d) Let .Cn /n  F be an increasing sequence and let C D [1 nD1 Cn ; then fC .t/ D  t .[1 .C / / D lim  ..C / / D lim f .t /. Therefore fC is n t n t n t n C n nD1 .A  BŒ0;1 /-measurable and C 2 F . In the same way, we will show that F is closed under the limit of decreasing sequences. Then F is a monotone class and, according to (c), the -algebra generated by R, A ˝ BS  F . It follows that, for every C 2 A ˝ BS , fC is .A  BŒ0;1 /measurable. R (iii) ) (i) For every B 2 BS , let C D   B; fC .t / D  t .B/ D S B .x/d  t .x/; by hypothesis, the mapping t 7! fC .t / is .A  BŒ0;1 /-measurable. It follows R that, for every simple function f (with regards to BS ), the mapping t 7! S f .x/d  t .x/ is .A  BŒ0;1 /-measurable. For every function f 2 Cb .S /, there exists a sequence of simple functions, .fn /n2N , uniformly convergent to f ; then the sequence .fn /n2N is uniformly bounded and so, according to Lebesgue theorem, Z Z fn .x/d  t .x/ ! f .x/d  t .x/; 8t 2 : S

S

179

Section 3.1 Preliminaries

R Therefore, the mapping t 7! S f .x/d  t .x/ is .A  BŒ0;1 /-measurable. As .PS ; Cb .S // is the narrow topology on PS , we obtain that the mapping  W  ! PS is measurable. The following theorem presents the disintegration result. Its proof follows the one of [167]. Theorem 3.2. Let p W   S !  be the canonical projection on . For every measure  2 caC .A ˝ BS / such that  D p . /, there exists a mapping  W  ! PS , .A  C/-measurable such that Z  .A  B/ D  t .B/d.t/; for every A 2 A; B 2 BS : A

Proof. (1) Firstly, let S be a compact Suslin space and let C.S / be the space of all real-valued continuous mapping on S endowed with the norm of uniform convergence k  k0 . For every f 2 C.S /, we can introduce the mapping Z 1 Ff W L ./ ! R; by Ff .g/ N D g.t /  f .x/d  .t; x/; for every g 2 L1 ./: S

N  kf k0  kgk N 1 , Ff belongs Then Ff is a linear map on L1 ./ and, since jFf .g/j 1 1 to the dual of L ./ which is L ./ (see Theorem 1.49). Let hf 2 L1 ./ such that Z Ff .g/ N D g.t/  hf .t/d.t/; for every g 2 L1 ./:

Obviously, khf k1 D kFf k  kf k0 . Let ' W C.S / ! L1 ./, defined by '.f / D hf . We can see that ' is a linear map on C.S / and, since k'.f /k1  kf k0 , for every f 2 C.S /, ' is continuous; moreover '.1/ D 1. According to Theorem IV.2.3 of [97], there exists a linear lifting W L1 ./ ! 1 L ./, i.e. an application having the following properties: .i/ .h/ D h; -a.e., .ii/ .h1 / D .h2 /, if h1 D h2 , -a.e., .iii/ .˛h1 C ˇh2 / D ˛ .h1 / C ˇ .h2 /, 8 ˛; ˇ 2 R; h1 ; h2 2 L1 ./,

180

Chapter 3 Young Measures

.iv/ .h/  0, if h  0, .v/ .1/ D 1, .vi/ .h1  h2 / D .h1 /  .h2 /. N D .h/, Therefore, we can define the map ‰ W L1 ./ ! L1 ./ letting ‰.h/ 1 for every h 2 L ./. ‰ is correctly defined because the definition does not depend on our choice of representatives in the class hN containing h. ‰ is a linear map and, since khk1  h  khk1 ; -a.e., khk1  .h/  khk1 : N 1  khk1 D khk N 1 and therefore Then k .h/k1  khk1 , from where k‰.h/k ‰ is continuous. For every t 2  and every f 2 C.S /, let now    t .f / D ‰ .'.f // .t/ D ‰ hf .t / D .hf /.t /: We can easily remark that  t is linear and that, for every f 2 C.S / ˇ ˇ j t .f /j D ˇ .hf /.t/ˇ  khf k1  kf k0 ; -a.e. t 2 : Therefore  t is a Radon measure on S ; since  t .1/ D ‰.'.1//.t / D .1/.t / D 1,  t is a probability on S . Let now the mapping  W  ! PS , defined by  .t / D  t , for every t 2 . Since S is a compact Suslin space, according to Corollary 2.26 and to Theorem 2.58, RcaC .BS / is a Polish space; as PS is closed in RcaC .BS /, it is Polish, also. Hence the narrow topology on PS has a countable base. C is the family of all Borel sets on PS ; so,  is .A  C/-measurable if and only if, ¹t 2  W  t .f / > ˛º 2 A, for every ˛ 2 R and every f 2 C.S /. But  t .f / D ‰.'.f //.t/ and, since ‰.'.f // 2 L1 ./, the mapping ‰.'.f // is measurable and the condition stated above is proved. Therefore  is a mapping .A  C/-measurable of  in PS . For every A 2 A and every f 2 C.S /, Z Z Z Z  t .f /d.t/ D ‰ .'.f // .t/d.t/ D .hf /.t /d.t / D hf .t /d.t / A A A ZA

A .t/  hf .t/d.t/ D Ff . A / D Z Z

A .t/  f .x/d  .t; x/ D f .x/d  .t; x/: D S

A S

181

Section 3.1 Preliminaries

Since f is arbitrary in C.S /, for every open set D  S , Z Z  t .D/d.t/ D

D .x/d  .t; x/ D  .A  D/ A

A S

(we must remark that, according to Theorem 3.1 (ii), the mapping t 7!  t .D/ is positive and .A  BŒ0;1 /-measurable). Therefore, the previous relation is also proved for all D 2 BS . (2) Let now S be an arbitrary regular Suslin space and let D pS . /, where pS W   S ! S is the canonical projection on S . According to Theorem 2.29, 2 caC .BS / D RcaC .BS /. Hence there exists a sequence of pairwise disjoint compact subsets of S , .Kn /n2N , such that .S n

1 [

Kn / D 0:

nD1

For every n 2 N, let  n W A ˝ BKn ! RC ;  n .M / D  ..  Kn / \ M / and let n D p . n / be the projection of  n on  (n .A/ D  .A  Kn /); according to Radon–Nikodym’s theorem (Theorem 1.40), since n is -continuous, there exists an application gn W  ! RC ; gn 2 L1 ./ such that Z n gn d; for every A 2 A:  .A/ D A

For every A 2 A,  .A  S / D .A/ D

1 X

n .A/ D

P1

nD1

Z X 1

nD1

 .A  Kn / and so we obtain gn .t/d.t /;

A nD1

for every A 2 A;

from where 1 X

gn D 1;   a.e.

nD1

By (1), for every n 2 N, there exists an application  n W  ! PKn ; .A  Cn /measurable such that Z n  .A  B/ D  tn .B/dn .t/; for every A 2 A; B 2 BKn : A

According to 2.57 (i), for every B 2 BS and every n 2 N, B \ Kn 2 BKn and so we can define, for every t 2 ,  t .B/ D

1 X nD1

gn .t/   tn .B \ Kn /:

(*)

182

Chapter 3 Young Measures

P Then, for every t 2 ,  t 2 caC .BS / D RcaC .BS / and, as  t .S / D 1 nD1 gn .t /  P  tn .Kn / D 1 g .t/ D 1,  2 P . t S nD1 n According to Proposition 2.57 (v), for every n 2 N, PKn ,! PS and then  is measurable. Finally, for every A 2 A and every B 2 BS , 1 1 Z X X n  .A  B/ D  .A  .B \ Kn // D  tn .B \ Kn /dn .t / D

nD1 1 Z X nD1 A

nD1 A

gn .t/   tn .B \ Kn /d.t / D

Z  t .B/d.t /: A

Remark 3.3. (i) Let  2 caC .A ˝ BS / and let  W  ! PS be obtained by using the previous theorem. Let  0 W  ! PS an other application .A  C/-measurable such that, for every A 2 A and B 2 BS , Z  .A  B/ D  t0 .B/d.t /: A

Then, for every B 2 BS ,  t .B/ D  t0 .B/; -almost everywhere on . The relation    0 defined by: : .B/ D :0 .B/;   a.e. and for every B 2 BS is an equivalence relation. Then, if we accept to identify two equivalent applications, the mapping  of the previous theorem is the unique application .A  C/-measurable for which Z  .A  B/ D  t .B/d.t/; for every A 2 A and every B 2 BS : A

(ii) Let ;  0 W  ! PS two equivalent .A  C/-measurable mappings and, for every B 2 BS , let [ AB D ¹t 2  W  t ¤  t0 º: AB D ¹t 2  W  t .B/ ¤  t0 .B/º and A D B2BS

If BS is countably generated (BS is generated by a countable algebra), then A 2 A and .A/ D 0. Indeed, if B0 D ¹Bn W n 2 Nº  BS is a countable algebra which generates BS , then A D [n2N ABn .

183

Section 3.1 Preliminaries

In this case : D :0 , -almost everywhere. Therefore, if BS is countably generated (particularly, if .S; S / is secondcountable), then    0 if and only if  D  0 ; -a.e. Definition 3.4. The mapping  W  ! PS of Theorem 3.2 is called the disintegration of measure  2 caC .A ˝ BS / with respect to . According to (i) of Remark 3.3,  has an unique disintegration : . Proposition 3.5. Let  2 caC .A ˝ BS / such that p . / D  and let  W  ! PS the disintegration of  ; then: R .i/  .C / D  t .C t /d.t/, for every C 2 A ˝ BS ; C t D ¹x 2 S W .t; x/ 2 C º is the section of C determined by t . R R R .ii/ S f .t; x/d  .t; x/ D Œ S f .t; x/d  t .x/d.t /, for every .A ˝ BS /measurable and positive mapping f , or, for every mapping f 2 L1 . /. Proof. (i) Let E D ¹A  B W A 2 A; B 2 BS º, F D ¹C 2 A ˝ BS W  .C / D R  t .C t /d.t/º: We need to remark that, according to Theorem 3.1 (iii), the mapping t 7!  t .C t / is positive and .A  BŒ0;1 /-measurable. According to Theorem 3.2, E  F . E is a semiring and then the algebra that it generates is ´ n μ [ G D Ci W n 2 N  ; C1 ; : : : ; Cn 2 E; Ci \ Cj D ;; 8i ¤ j : iD1

Let Ci D Ai  Bi 2 E; i D 1; 2, with C1 \ C2 D ; and let C D C1 [ C2 ; then A1 \ A2 D ; or B1 \ B2 D ;. If A1 \ A2 D ;, then 8 < B1 ; t 2 A1 ; B2 ; t 2 A2 ; Ct D and therefore : ;; t 2  n .A1 [ A2 / Z Z Z  t .C t /d.t/ D  t .B1 /d.t/C  t .B2 /d.t / D  .C1 /C  .C2 / D  .C /;

A1

from where C 2 F .

A2

184

Chapter 3 Young Measures

If B1 \ B2 D ; and A1 \ A2 ¤ ;, then 8 B1 ; t 2 A1 n A2 ; ˆ ˆ < B2 ; t 2 A2 n A1 ; and then Ct D B [ B2 ; t 2 A1 \ A2 ; ˆ ˆ : 1 ;; t 2  n .A1 [ A2 / Z Z Z  t .C t /d.t/ D  t .B1 /d.t / C  t .B2 /d.t / A1 nA2 A2 nA1 Z  t .B1 [ B2 /d.t / C A1 \A2 Z Z  t .B1 /d.t/ C  t .B2 /d.t / D A1

A2

D  .C1 / C  .C2 / D  .C / and again C 2 F . A similar reasoning applied to any element of the family G leads to the inclusion G F. Since F is a monotone class, A ˝ BS —the -algebra generated by G — coincides with F . (ii) According to (i), the condition (ii) is proved for the simple functions; therefore, according to the theorem of monotonous convergence, it is proved for measurable and positive functions and, according to Lebesgue’s theorem of dominated convergence, for the functions of L1 . /. Remark 3.6. Let  W  ! PS be an .A  C/-measurable function; the mapping  W A ˝ BS ! RC defined by Z  .C / D  t .C t /d.t/; for every C 2 A ˝ BS ;

is a positive and finite measure on A ˝ BS for which p . / D . The disintegration of  is equivalent with  (according to Remark 3.3 (i)). By identifying the equivalent disintegrations, we can observe that the mapping  7!  is a bijection between the measures of caC .A ˝ BS /, for which p . / D , and their disintegrations.

185

Section 3.1 Preliminaries

3.1.2

Integrands

Definition 3.7. An integrand on S is an .A˝BS /-measurable map, ‰ W S ! R. An integrand ‰ is called L1 -bounded is there is an application  2 L1 ./ such that j‰.t; x/j  .t /, for every t 2  and every x 2 S . ‰ is said lower semicontinuous-l.s.c. (upper semicontinuous-u.s.c.) if, for every t 2 ; the application ‰.t; / is l.s.c. (u.s.c.) on S . The integrand ‰ is a Carathéodory integrand if, for every t 2 ; ‰.t; / is continuous on S . The set of all bounded Carathéodory integrands on S is noted by C t hb . S /. The following proposition shows that we can see the Carathéodory integrands as functions measurable in the first variable and continuous in the second one. Proposition 3.8 ([42], Lemma III.14 and [45], Lemma 1.2.3). A mapping ‰ W   S ! R is a Carathéodory integrand if and only if the following two conditions are satisfied: .i/ ‰.; x/ is A-measurable, for every x 2 S . .ii/ ‰.t; / is continuous, for every t 2 . Proof. It is sufficient to show that the two conditions mentioned above are sufficient so that ‰ to be .A ˝ BS /-measurable. Firstly, let us suppose that S is a metrizable Suslin space; let d be a metric which generates the topology of S (S D d ) and let A D ¹xn W n 2 Nº be a countable dense subset of .S; d / (every Suslin space is separable). For every p 2 N  and every x 2 S , let ²  ³ 1 n.p; x/ D min n 2 N W x 2 B xn ; ; p where B.xn ; p1 / is the open ball of radius p1 centered in xn in the metric space .S; d /. For any p 2 N  , let ‰p W   S ! R, ‰p .t; x/ D ‰.t; xn.p;x/ /. Since, for every x 2 S , d.x; xn.p;x/ / < p1 ; limp!1 xn.p;x/ D x; then, according to (ii), for every .t; x/ 2   S , lim ‰p .t; x/ D ‰.t; x/:

p!1

(*)

186

Chapter 3 Young Measures

For every p 2 N  and every Borel set A  R, μ ´ "   [  # 1 [ 1 1 ‰p1 .A/ D B xk ; .t; x/ 2  B xn ; n W ‰p .t; x/ 2 A p p nD0 k 0; there exists K 2 KS such that

 .  .S n K// < ":

Section 3.2 Definitions and Examples

189

.vi/ A Young measure is also a family of probabilities ¹ t W t 2 º on S such that, for every C 2 A ˝ BS , the mapping t 7!  t .C t / is measurable. Thus, we can explain the various denominations, such as parametric measures, transition probabilities, sometimes used to indicate Young measures. If : W  ! PS is a Young measure, then almost all values of : are in a tight subset of PS . More exactly, we have the following result: Proposition 3.12. Let S be a regular Suslin space, let  2 Y.S / and let : W  ! PS be the disintegration of  ; then, for every " > 0, there exists a tight set H"  PS such that :1 .PS n H" / 2 A

and  .¹t 2  W  t 2 PS n H" º/ < ":

Proof. Since D  .  / 2 RcaC .BS /; ¹ º is tight; according to Proposition 2.63, R there exists an inf-compact application ' W S ! Œ0; C1 such that M D S 'd < C1. For every " > 0, let H" D ¹ 2 PS W .'/  M " º. The same Proposition 2.63 says that H" is tight. Since the mapping of PS in RC , defined by

2 PS 7! I' . / D .'/, is narrowly l.s.c. (see (i) of Corollary 2.45), H" is a narrowly closed subset of PS and then, as the mapping : W  ! PS is .A ˝ C/-measurable, ² ³ M 1 : .PS n H" / D t 2  W  t .'/ > 2 A; and " Z Z Z 'd D '.x/d  .t; x/ D  t .'/d.t / M D S S Z  M  t .'/d.t /     :1 .PS n H" / " . t .'/> M" / and so  .¹t 2  W  t … H" º/ < ": Now we will present two particularly important cases of the Young measures.

3.2.1

Young Measure Associated to a Probability

Definition 3.13. For every probability 2 PS , the product measure   D  ˝ is a Young measure whose disintegration is the constant mapping  t D , for all t 2 . We say that   is the Young measure associated to the probability . The mapping 7!   is an injection of PS in Y.S /; identifying with   , we allow to write: PS  Y.S /.

190

Chapter 3 Young Measures

More generally, let n 2 N  , let ¹A1 ; : : : ; An º be an A-partition Pn of  and let 1 ; : : : ; n 2 PS ; the mapping  W  ! PS , defined by  t D i D1 i  A .t /, is i a Young measure. Indeed,  is an P A-simple function with values in PS ; therefore, for every B 2 BS ; t 7!  t .B/ D niD1 i .B/ A .t /, is .ABŒ0;1 /-measurable. i According to Theorem 3.1,  is a Young measure.  is the disintegration of measure  with respect to  , where

 .A  B/ D

n X

i .B/  .A \ Ai /;

for every A 2 A and every B 2 BS :

i D1

In this case, we say that  (as well as  ), is a simple Young measure and we note by E.A; PS / the subset of all simple Young measures. We show in first part of the following theorem that, in the case where S is a metrizable Suslin space, we can approximate any Young measure by simple Young measures; the second part is a result of Lusin type. Theorem 3.14 ([77]). Let S be a metrizable Suslin space and let  W  ! PS . .i/  is a Young measure if and only if there exists a sequence of simple Young measures, . n /n2N , such that, for all t 2 , . tn /n2N is narrowly convergent to  t . .ii/ Moreover, let  be a topological space, let  be a Radon measure on  and let A be the -completed of -algebra B of Borel sets on ; then  is a Young measure if and only if, for every compact K of  and every " > 0, there exists a compact set L  K such that .K n L/ < " and  L is narrowly continuous. Proof. (i) According to b) of 2.58, .RcaC .BS /; T/ is a second-countable metrizable space and then .PS ; T/ is a second-countable metrizable subspace. Therefore  W  ! PS is a Young measure (i.e. a measurable mapping) if and only if there exists a sequence of simple mappings . n /n2N with values in PS such that . tn /n2N converges narrowly to  t , for every t 2 . (ii) ()) Let  W  ! PS be a Young measure. PS is separable and metrizable space so that there exists an homeomorphic embedding  W PS ! I N , where I D Œ0; 1. For every n 2 N, let n be the nth -projection of I N in I . Then, for every n 2 N, the mapping n ı  ı  W  ! I is .A  BŒ0;1 /-measurable. Since  is a regular measure, for every compact set K  , for every " > 0 and every n 2 N, there exists a compact subset Ln  K such that .K n Ln / < 2"n and n ı  ı  is continuous on Ln .

191

Section 3.2 Definitions and Examples

T P1 The set L D 1 nD1 Ln  K is compact and .KnL/  nD1 .KnLn / < ". For every n 2 N, n ı  ı L is continuous and therefore,  ı L is continuous; then L is continuous. (ii) (()  is tight so that, for every n 2 N, there exists a compact set Kn   such that .nKn / < n1 . We can choose the sequence .Kn /n2N to be increasing. Let Ln  Kn be a compact set such that .Kn n Ln / < n1 and  is narrowly continuous on Ln ; we can choose the sequence .Ln /n2N to be increasing, also. Let ²  .t /; t 2 Ln ; n n ;  W  ! PS ;  .t/ D ı0 ; t …Ln S where ı0 is the Dirac measure concentrated at 0 and let L D 1 nD1 Ln . n n For every n 2 N,  is .AC/-measurable and  .t / !  .t /, for every t 2 L; since . n L/  . n Ln / < n2 , for every n 2 N  ,  n ! , -a.e. so that  is measurable. Therefore,  is a Young measure.

3.2.2

Young Measure Associated to a Measurable Mapping

The second important case of the Young measures is that of the measures which are associated to the measurable applications of  in S . Let M.S / be the set of all .A  BS /-measurable mappings; for every u 2 M.S / let  u W  ! PS ;  u .t/  tu D ıu.t / , for every t 2  (ı: is again the u Dirac measure). For every B 2 BS , the mapping gB W  ! Œ0; 1, defined by u u is .A  BŒ0;1 /-measurable and so, according to (ii) of gB D : .B/ D 1 u .B/ Theorem 3.1,  u is .A  C/-measurable and then a Young measure. Definition 3.15.  u is called the Young measure associated to the measurable application u. We will also say that a Young measure  comes from an application if there is a measurable application u 2 M.S / such that  D  u . Remark 3.16. .i/ According to Remark 3.6,  u is the disintegration of the Young measure  u , where:

 u .A  B/ D .A \ u1 .B//;

for every A 2 A

and every B 2 BS :

According to (iv) of Remark 3.11, which specifies the integration with respect to a Young measure, for every positive integrand ‰ (or, for every

192

Chapter 3 Young Measures

‰ 2 L1 . u /), Z

Z u

S

‰.t; x/d  .t; x/ D

‰.t; u.t //d.t /:

.ii/ The mapping u 7!  u is an injection of the space M.S / in Y.S / (we identify the applications that coincide -a.e.); therefore, by identifying u with  u , we can consider that M.S /  Y.S /. .iii/ The identification of PS and M.S / as subsets of Y.S / leads to the identification of S as subset of Y.S / (x 2 S being identifies with ıx ). Indeed, for every x 2 S; ıx 2 PS and also ıx is the Young measure associated to constant mapping x 2 M.S /. Conversely, if 2 PS , then   2 M.S /  Y.S / if we can find a mapping u 2 M.S / such that   D  u , from where ıu.t / D , for every t 2 . It means that u is a constant function x and so D ıx . So, according to this identification, we can consider that the Suslin space S is a subset of Y.S /. .iv/ If u W  ! S is a measurable function and if  u is the Young measure associated, then  u ./ D ¹ıu.t / W t 2 º is a tight subset of PS if and only if u./ is relatively compact in S . Indeed, if  u ./ is tight, then there exists a compact subset K 2 KS such that ıu.t / .S n K/ < 1, for every t 2 , which means that u./ D ¹u.t / W t 2 º  K. Conversely, if K D u./ is compact in S then ıu.t / .S n K/ D 0 < ", for every " > 0 and every t 2  which means that  u ./ is tight. We now establish two conditions assuring that a Young measure comes from a measurable application. Proposition 3.17. Let  be a Young measure and let u 2 M.S /. If Gu is the graph of u, then Gu0 will denote its complement with respect to   S . A necessary and sufficient condition so that  D  u is that  .Gu0 / D 0 or, equivalent,  .Gu / D ./. Proof. If fu W  !   S is defined by fu .t / D .t; u.t //, then  u .C / D .fu1 .C //, for every C 2 A ˝ BS . .)/ Let  D  u ; for every C 2 A ˝ BS ,  .C / D  u .C / D .fu1 .C //. Since u 2 M.S /, Gu0 D ¹.t; x/ 2   S W u.t / ¤ xº 2 A ˝ BS and fu1 .Gu0 / D ¹t 2  W .t; u.t// 2 Gu0 º D ;; therefore  .Gu0 / D 0. .(/ By hypothesis,  .Gu0 / D 0. We need to show that, for every C 2 A ˝ BS ,  .C / D  u .C / D .fu1 .C //.

193

Section 3.3 The Stable Topology

Let A D fu1 .C /; then C n Gu0  A  S and so  .C / D  .C n Gu0 /   .A  S / D .A/ and, because .A  S / n Gu0  C , .A/ D  .A  S / D  ..AS /nGu0 /   .C /. Therefore  .C / D .A/ D .fu1 .C // D  u .C /. Theorem 3.18 ([77]). Let  be a Young measure;  comes from a measurable application if and only if inf¹ .Gu0 / W u 2 M.S /º D 0 or, equivalently, sup¹ .Gu / W u 2 M.S /º D ./. Proof. .(/ If  comes from a measurable application v, according to the previous proposition, it is evident that inf¹ .Gu0 / W u 2 M.S /º D 0. .)/ Let . t / t 2 the disintegration of  . We will show that, for every t 2 , there exists x t 2 S such that  t .¹x t º/ D supx2S  t .¹xº/. Obviously, this is true if supx2S  t .¹xº/ D 0. Let supx2S  t .¹xº/ > 0: If the sup is not reached, then, for every y 2 S , there exists z 2 S such that  t .¹yº/ <  t .¹zº/. Then, we can define a sequence .xn /n2N  S Psuch that the sequence . t .¹xn º//n2N to be positive, strictly increasing and 1 nD1  t .¹xn º/ D  t .¹x1 ; : : : ; xn ; : : : º/  1 what is impossible. By hypothesis, inf¹ .Gu0 / W u 2 M.S /º D 0. Then, there exists a sequence .un /n  M.S / such that  .Gu0 n / ! 0, when n ! 1. Z Z  .Gu0 n / D  t ..Gu0 n / t /d.t/ D  t .S n ¹un .t /º/d.t /;



from where, t 7!  t .S n ¹un .t/º/ defines a sequence strongly convergent to 0 in L1 ; let .nk / be such that t 7!  t .S n unk .t// to define a subsequence convergent to 0, -a.e. Therefore, t 7!  t .unk .t// converges -a.e. to 1. Since  t .¹x t º/ D supx2S  t .¹xº/   t .¹unk .t /º/, for every k 2 N we have,  t .¹x t º/ D 1, for -almost every t 2 , which means that  t D ıx t , -a.e. Then let u W  ! S, defined by u.t / D x t .  t .¹unk .t /º/ ! 1, -a.e. Hence we have  t .¹x t º/ D  t .¹u.t /º/ D 1 D lim  t .unk .t //: k

Therefore, ıu.t / .¹unk .t/º/ ! 1 when k ! 1, so that there exists k0 such that unk D u a.e., for every k  k0 . .unk / converges therefore -a.e. to u and then u is measurable. Obviously,  u D  .

3.3

The Stable Topology

Let S be a regular Suslin space, let A be a -algebra on the set  and let  2 caC .A/ be a positive measure, complete with respect to A. The family of

194

Chapter 3 Young Measures

applications

® ¯ C thb0 .  S / D A ˝ f W A 2 A; f 2 Cb .S /

is a subset of the family C thb .  S / of all bounded Carathéodory integrands on   S. Definition 3.19. Let Y.S / be the space of Young measures on   S; for every ‰ D A ˝ f 2 C thb0 .  S /, let I‰ W Y.S / ! R, defined by Z Z I‰ . / D ‰.t; x/d  .t; x/ D  t .f /d.t /; S

A

where : is the disintegration of  . The stable topology on Y.S / is the projective limit topology on Y.S / generated by the family of mappings FS D ¹I‰ W ‰ 2 C t hb0 .  S /º, i.e. the weakest topology on Y.S / under which every mapping of the family FS is continuous; this topology is noted by S.Y.S //, or simply S if there is not ambiguity on the Suslin space S. S

If the net . i /i 2I  Y.S / is S-convergent to  2 Y.S /, then we write  i !  .

Remark 3.20. .i/

S

i !   if and only if  i .‰/ !  .‰/;

for every ‰ 2 C t hb0 .  S /

either:

 i . A ˝ f / !  . A ˝ f /;

for every A 2 A

and every f 2 Cb .S /:

If we use the disintegrations, then, according to Theorem 1.57, this is equivalent with the weakly convergence of the net .:i .f //i 2I  L1 ./ to : .f / 2 L1 ./, for every f 2 Cb .S /. .ii/ Let . i /i 2I  Y.S / be a net and let  2 Y.S /. For every A 2 A; . i .A  //i 2I  RcaC .BS / and  .A  / 2 RcaC .BS /. Then S

i !  

T

”  i .A  / !   .A  /;

for every A 2 A:

It means that the net . i .A  //i 2I is narrowly convergent to  .A  /.

195

Section 3.3 The Stable Topology

.iii/ The stable topology on Y.S / is noted in [45] by YW1 . .iv/ In [9] this topology is called the narrow topology. We prefer to use the name of stable topology to not confuse with the narrow topology on ca.B S / in the particular case where  is a Hausdorff topological space (see also the remark of the Section 2.1 in [45]). Proposition 3.21. Let A0 be an algebra which generated A; for every net . i /i 2I  Y.S / and every  2 Y.S /, S

i !  

T

”  i .A  / !   .A  /;

for every

A 2 A0 :

Proof. The implication “H)” results immediately from (ii) of the previous remark. T   .A  /, for every A 2 A0 and let us (H: Let us suppose that  i .A  / ! T

  .E  /º. denote F D ¹E 2 A W  i .E  / ! For every increasing sequence .En /n2N  F , let E D [1 nD1 En . For every open set D 2 S and every " > 0, there exists n0 2 N such that "  .En0  D/ >  .E  D/  : 2 Since En0 2 F , according to (ii) of Theorem 2.53, there exists i0 2 I such that "  i .E  D/   i .En0  D/ >  .En0  D/  >  .E  D/  "; 2 for every i  i0 and then lim inf i  i .E  D/   .E  D/: Since  i .E S / D .E/ D  .E S /, according again to (ii) of Theorem 2.53, T

 i .E  / !   .E  /: Therefore E 2 F . The same reasoning, using this time (iii) of 2.53, shows us that the family F also contains the limit of every decreasing sequence of sets of F . Therefore F is a monotone class and then A D F . T Therefore  i .A  / !   .A  /, for every A 2 A and so . i /i 2I is Sconvergent to  . Because of (ii) of Remark 3.20, we can provide several characterizations of the stably convergence.

196

Chapter 3 Young Measures

Proposition 3.22. Let S be a regular Suslin space, let . i /i 2I  Y.S / be a net and let  2 Y.S /; the following conditions are equivalent: S

(1)

i !  .

(2)

 . A ˝ f /  lim inf i  i . A ˝ f /, for every A 2 A and every bounded

from below l.s.c. function f W S ! R. (3)

 .AD/  lim inf i  i .AD/, for every A 2 A and every open set D  S .

(4)

 .A  F / F  S.

 lim supi  i .A  F /, for every A 2 A and every closed set

If, in addition, S is metrizable and d is a metric which generated the topology of S , then the conditions (1)–(4) from above are also equivalent to each of the following conditions: (5)

 . A ˝ f / D limi  i . A ˝ f /, for every A 2 A and every f

(6)

 . A ˝ f /

2 BL.S; d /.

D limi  i . A ˝ f /, for every A 2 A, uniformly w.r.t f 2 BL.S; d / with kf kBL  1.

If S is a locally compact Suslin space, then the conditions (1)–(6) from above are equivalent with the following one: (7)

 . A ˝ f / D limi  i . A ˝ f /, for every A 2 A and f

2 Cc .S /.

Proof. (1)”(2) is a consequence of Proposition 2.50. (1)”(3) is a consequence of (ii) of Theorem 2.53. (1)”(4) is a consequence of (iii) of Theorem 2.53. (1)”(5) is a consequence of Proposition 2.51. (1)”(6) is a consequence of Theorem 2.81 (the definition of BL.S; d / is given in the paragraph 2.5). (1)”(7): according to Theorem 6 of Chapter 2 of [152], in this case S is a Polish space; the result comes from (ii) of Theorem 2.36, while observing that, 1   . A  / 2 PS , for every  2 Y.S / and every A 2 A with .A/ ¤ 0. .A/ Remark 3.23. .i/ The condition (2) of the previous proposition characterizes the convergence in the topology YN1 of [45]; according to D) of Theorem 2.1.3 of [45], the conditions (1) and (2) listed above are equivalent in more general spaces than the regular Suslin spaces.

197

Section 3.3 The Stable Topology

.ii/ If A0 is an algebra that generates A, then, according to Proposition 3.21, we can replace, in all the conditions (2)–(7) of the previous proposition, A 2 A by A 2 A0 . Proposition 3.24. For every regular Suslin space S , .Y.S /; S/ is a completely regular space. Proof. Let FS D ¹I‰ W ‰ 2 C thb0 .  S /º be the family who generates S and let W Y.S / ! RFS be defined by

. / D .'. //'2FS D . .‰//‰2C t hb . S / : 0

Then is an injection of Y.S / in RFS ; indeed, if we suppose that . / D . 0 /, then  .‰/ D  0 .‰/, for every ‰ 2 C thb0 .  S / and therefore

 . A ˝ f / D  0 . A ˝ f /;

for every A 2 A

and every f 2 Cb .S /: (*)

C We can R notice that  .A  / 2 Rca .BS /, for every A 2 A (see 3.11, (v)) and that S f .x/d  .A  /.x/ D  . A ˝ f /. Then . / becomes Z Z f .x/d  .A  /.x/ D f .x/d  0 .A  /.x/; for every f 2 Cb .S /; S

S

or If . .A  / D If . 0 .A  /;

for every f 2 Cb .S /:

(**)

The family ¹If W f 2 Cb .S /º defines the topology on RcaC .BS / which is separated. According to Proposition 2.46, we obtain from .

/

 .A  B/ D  0 .A  B/;

for every A 2 A

and every B 2 BS ;

from where  D  0 . A net . i /i 2I  Y.S / is S-convergent to  2 Y.S / if and only if . . i //i 2I is convergent to . / in the product space RFS and so is a homeomorphism of .Y.S /; S/ on a subspace of the product space RFS . Therefore .Y.S /; S/ is completely regular. In the stronger conditions on S , we can obtain metrizability of Y.S /. Proposition 3.25. Let S be a metrizable Suslin space; if A is countably generated, then .Y.S /; S/ is metrizable.

198

Chapter 3 Young Measures

Proof. Let A0 D ¹A1 ; A2 ; : : : ; An ; : : : º be a countable algebra which generated A and let d be a compatible metric on S . For every  0 ;  00 2 Y.S / and every n 2 N  ,  0 .An  :/;  00 .An  :/ 2 RcaC .BS /. According to Definition 2.80, the Dudley’s distance between these two measures is: D. 0 .An  :/;  00 .An  :// D

sup kf kBL 1

j 0 . A ˝ f /   00 . A ˝ f /j n

n

 2.An /  2./: So, we can define  W Y.S /  Y.S / ! RC by: . 0 ;  00 / D D

1 X 1  D. 0 .An  :/;  00 .An  :// 2n

nD1 1 X nD1

1  sup j 0 . An ˝ f /   00 . An ˝ f /j; 2n kf kBL 1

for every  0 ;  00 2 Y.S /. . 0 ;  00 / D 0 if and only if D. 0 .An :/;  00 .An :// D 0, for every n 2 N  , and so if and only if  0 .An  :/ D  00 .An  :/, for every n 2 N  ; therefore, according to (iii) of Remark 3.11, . 0 ;  00 / D 0 if and only if  0 D  00 . The other axioms of distance being clearly proved,  is a metric on Y.S /. Let . i /i 2I  Y.S / be a net and let  2 Y.S /. According to Proposition 3.21, S

i !  

T

”  i .An  / !   .An  /;

for every n 2 N 

and, since .S; d / is a second-countable metric space, according to Theorem 2.81 and to Remark 2.82, S

i !  

” D. i .An  :/;  .An  :// ! 0;

for every n 2 N 

which is equivalent to . i ;  / ! 0. Therefore,  is a metric which generates the topology of the space .Y.S /; S/. According to Definition 3.13, PS  Y.S /; the following result presents the trace of stable topology S on PS . Proposition 3.26. (1) The trace of stable topology S on PS is the narrow topology T. (2) PS is a closed subspace of .Y.S /; S/.

199

Section 3.3 The Stable Topology

Proof. (1) If . i /i 2I  PS is a net, then T

 2 PS ” i .f / ! .f /; i !

for every f 2 Cb .S /

which means that  i . A ˝ f / D .A/  i .f / ! .A/  .f / D   . A ˝ f /, T

 ” . i /i is for every A 2 A and every f 2 Cb .S /. Therefore i ! S-convergent to   what demonstrates (1).

S

(2) Let . i /i 2I  PS be such that  i !   2 Y.S /. W BS ! Œ0; 1, defined by Z 1  t .B/d.t/; for every B 2 BS ; .B/ D ./ is a probability on S so that 2 PS . Let   D  ˝ ; for every A 2 A and every f 2 Cb .S /,  Z Z .A/   . A ˝ f / D .A/  .f / D f .x/d  t .x/ d.t /  ./ S .A/ .A/ D   . ˝ f / D  lim  i . ˝ f / ./ ./ i D .A/  lim i .f / D lim  i . A ˝ f /: i

i

Then . i /i is S-convergent to   and, since the narrow topology is separated,  D   2 PS . Therefore PS is closed. Remark 3.27. Since SPS is the narrow topology on PS , the stable topology S is also called the narrow topology on Y.S /. Theorem 3.28. Let T and S be two regular Suslin spaces such that T be homeomorphic to a subspace of S and let ST and SS be the stable topologies of Y.T / and Y.S /, respectively; then .Y.T /; ST / is homeomorphic to a subspace of .Y.S /; SS /. Proof. Let i W T ! S be a homeomorphism between T and i.T /  S . According to Proposition 2.57, BT D i 1 .BS / and I W RcaC .BT / ! RcaC .Bs /, defined by I. /.B/ D .i 1 .B//, for every 2 RcaC .BT / and every B 2 BS , is a homeomorphism between RcaC .BT / and I.RcaC .BT //  RcaC .BS / provided with their narrow topologies. The restriction of I to the subset of all probabilities, noted again with I , is a homeomorphism between PT and the subspace I.PT /  PS provided with its

200

Chapter 3 Young Measures

narrow topology. Let CT and CS be the Borel subsets of PT and PS , respectively; then CT D I 1 .CS /. Since the Young measures on   T are .A  CT /-measurable, for every : 2 Y.T /; : D I ı : is .A  CS /-measurable and so a Young measure on   S . For every t 2  and every B 2 BS ;  t .B/ D  t .i 1 .B//. Let J W Y.T / ! Y.S /, defined by J.: / D I ı : , for every : 2 Y.T /. If : is the disintegration of  and J.: / is the disintegration of  , for every C 2 A˝BS , Z Z  .C / D .I ı : / t .C t /d.t/ D  t .i 1 .C t //d.t / Z  t .¹x 2 T W .t; i.x// 2 C º/d.t / D Z D  t .j 1 .C / t /d.t / D  .j 1 .C //;

where j W T ! S; j.t; x/ D .t; i.x//, for every .t; x/ 2 S . Therefore J. / D  ı j 1 . Obviously, J is an injection. Let . d /d 2D  Y.T / be a net and let  2 Y.T /; according to (3) of Proposition 3.22, . d /d 2D is stably convergent to  in Y.T / if and only if

 .A  G/  lim inf  d .A  G/; d

for every A 2 A and every open set G 2 T ;

which is equivalent with I. /.A  U / D  .A  i 1 .U //  lim inf  d .A  i 1 .U // d

d

D lim inf I. /.A  U /; d

for every A 2 A and every U 2 S . Therefore . d /d 2D is stable convergent to  in Y.T / if and only if .J. d //d 2D is stably convergent to J. / in Y.S /. Then J is a homeomorphism between Y.T / and J.Y.T //  Y.S /. Remark 3.29. .i/ A Young measure  2 Y.S / belongs to J.Y.T //  Y.S / if and only if

  .  .S n i.T /// D inf¹ .  B/ W B 2 BT ; B S n i.T /º D 0: .ii/ If S is a regular Suslin space and if T is Suslin subspace of S , then the stable topology on Y.T / is the trace of stable topology on Y.S /.

201

Section 3.3 The Stable Topology

In Proposition 3.22, we have mentioned several equivalent properties with the stably convergence. Another important characterization of the stably convergence can be expressed with the help of the Carathéodory integrands. Theorem 3.30. Let S be metrizable Suslin space; for every net . i /i 2I  Y.S / and every  2 Y.S /, the following two conditions are equivalent. S

(1)

i !  .

(2)

 i .‰/ !  .‰/, for every ‰ 2 C thb .  S /.

Proof. Obviously, (2) H) (1).

S

(1) H) (2): Let . i /i 2I  Y.S /;  2 Y.S / such that  i !  . (a) Firstly, we suppose that S is a compact Suslin space; then S is Polish (see 2.26). The space C.S / D Cb .S / of all real valued continuous functions on S is equipped with norm of uniform convergence k  k0 and .C.S /; k  k0 / is a separable Banach space (see Theorem I.5.1 of [173]). For every ‰ 2 C thb .  S /, let '‰ W  ! C.S / be the mapping defined by '‰ .t/ D ‰.t; :/, for every t 2 . Let A be a countable set dense in S . For every f 2 C.S / and every r 2 QC , let T .f; r/  C.S / be the closed ball with center f and radius r; then \ 1 .T .f; r// D ¹t 2  W j‰.t; x/  f .x/j  rº 2 A '‰ x2A

and, since C.S / is separable, '‰ is .A  BC.S / /-measurable. On the other hand, since ‰ is bounded, Z Z k'‰ k1 D k'‰ .t/k0 d.t / D sup j‰.t; x/jd.t /  k‰k0  ./ < C1:

x2S

Then '‰ 2 L1 .; C.S //. According to Theorem I.4.30 of [173], the space E.A; C.S // of all C.S /valued simple functions is dense in the space L1 .; C.S //. Then, for every " > 0, there exists ' 2 E.A; C.S // such that Z " k'  '‰ k1 D (1) k'.t/  ‰.t; :/k0 d.t / < : 4 P Let ' D nkD1 A ˝ fk , where fk 2 C.S / and ¹Ak W k D 1; : : : ; nº  A. k Since . i /i 2I is S-convergent to  , . i .'//i 2I is convergent to  .'/; let so i0 2 I such that " (2) j i .'/   .'/j < ; for every i  i0 : 2

202

Chapter 3 Young Measures

After (1), for every  2 Y.S /, ˇZ Z ˇ ˇ ˇ .‰.t; x/  '.t /.x//d t .x/ d.t /ˇˇ j .‰/   .'/j D ˇˇ S R  k‰.t; :/  '.t /k0 d.t / < 4"

(3)

and then, according to (3) and (2), for every i  i0 , j i .‰/   .‰/j  j i .‰/   i .'/j C j i .'/   .'/j C j .'/   .‰/j " " " < C C D ": 4 2 4 Therefore we have  i .‰/ !  .‰/: (b) Let now S be a metrizable Suslin space; then S is second-countable and so S is homeomorphic to a subset of Hilbert cube I N D Œ0; 1N . Let SN be a metrizable compactification of S and let d be a metric generating the topology of SN . The embedding i W S ! SN is a homeomorphism between S and i.S /  SN ; let j W   S !   SN , defined by j.t; x/ D .t; i.x//, for every .t; x/ 2   S . According to Theorem 3.28, the mapping J W Y.S / ! Y.SN /, defined by J. / D  ı j 1 , is a homeomorphism of .Y.S /; SS / on the subspace J.Y.S //  .Y.SN /; SSN /. SS

SSN

Since  i !  , N D J. i / ! J. / D N : i

N n W   SN ! R, defined For every ‰ 2 C thb .  S / and every n 2 N, let ‰ by N n .t; x/ N D inf ¹‰.t; x/ C n  d.x; x/º; N ‰ x2S

for every .t; x/ N 2   SN :

Nn 2 Following the same reasoning as in Proposition 3.9, we obtain that ‰ b N and that ‰ N n .t; x/ " ‰.t; x/, for every .t; x/ 2   S: C t h .  S/ N Then  .‰n S / !  .‰/. For every " > 0, there exists n0 2 N such that  " "  .‰/ <  ‰N n S C D N .‰N n0 / C : (4) 2 2 i .N /i 2I is SSN -convergent to N in the space .Y.SN /; SSN / and, according to the part i N N n0 /; therefore there exists i0 2 I such that N .‰ (a) of this proof, N .‰ n0 / ! 

N .‰N n0 / < N i .‰N n0 / C

 " N n0  S C "   i .‰/ C " ; D i ‰ 2 2 2

8i  i0 : (5)

203

Section 3.3 The Stable Topology

By (4) and (5),

 .‰/  lim inf  i .‰/: i

(6)

If we repeat the same reasoning for ‰, we obtain lim sup  i   .‰/:

(7)

i

By (6) and (7),  i .‰/ !  .‰/, for every ‰ 2 C t hb .  S /. Remark 3.31. Condition (2) of the previous theorem characterizes the convergence in the topology YM1 of [45]; the point G of Theorem 2.1.3. of [45] contains the previous result. In the following theorem, we will present other characterizations of the stably convergence in the frame of the metrizable Suslin spaces. Theorem 3.32. Let S be a metrizable Suslin space; for every  2 Y.S / and every net . i /i 2I  Y.S /, the following conditions are equivalent: S

(1)

i !  .

(2)

 i .‰/ !  .‰/, for every ‰ 2 C thb .  S /.

(3)

 .‰/  lim inf i  i .‰/, for every bounded l.s.c. integrand ‰.

(4)

 .‰/  lim inf i  i .‰/, for every L1 -bounded l.s.c. integrand ‰.

(5)

 i .‰/ !  .‰/, for every L1 -bounded Carathéodory integrand ‰.

Proof. According to Theorem 3.30, (1)”(2). Since the implications (4)H)(5)H)(2) are evident, we only have to demonstrate that .2/ H) .3/ H) .4/. N be a bounded l.s.c. integrand. According (2)H)(3): Let ‰ W   S ! R to Proposition 3.9, there exists a sequence .‰n /n2N  C t hb .  S / such that ‰n " ‰. Then  .‰n / "  .‰/ and so, for every " > 0, there exists n0 2 N such that "  .‰/ <  .‰n0 / C : (i) 2 By (2),  i .‰n0 / !  .‰n0 / and then there exists i0 2 I such that

 .‰n0 / <  i .‰n0 / C

" ; 2

for every i  i0 :

(ii)

204

Chapter 3 Young Measures

By (i) and (ii), for every i  i0 ,  .‰/ <  i .‰n0 / C " <  i .‰/ C ", from where  .‰/  lim inf i  i .‰/. N be a L1 -bounded l.s.c. integrand and let (3)H)(4): Let ‰ W   S ! R 1  2 L ./ be such that j‰.t; x/j  .t /, for every t 2  and every x 2 S . For every " > 0, there exists M > 0 such that Z .t /d.t / < ": ..t />M /

Let A D ¹t 2  W .t /  M º 2 A. Then ‰1 D A  ‰ is a bounded l.s.c. integrand (k‰1 k0  M ) and, by (3),

 .‰1 /  lim inf  i .‰1 /:

(iii)

i

For every  2 Y.S /,

Z

j .‰/   .‰1 /j D j . nA  ‰/j  nA Z .t /d.t / < ": 

Z S

j‰.t; x/jd t .x/d.t /

(iv)

nA

According to (iii) and (iv), for every " > 0;

 .‰/ <  .‰1 / C "  lim inf  i .‰1 / C "  lim inf  i .‰/ C 2": i

For other particular spaces, we can give supplementary characterizations of the stable convergence. Theorem 3.33. Let S be a locally compact Suslin space; for every net . i /i 2I  Y.S / and every  2 Y.S /, the following conditions are equivalent: S

(1)

i !  .

(2)

 i .‰/ !  .‰/, for every bounded integrand ‰ such that ‰.t; :/ 2 Cc .S /,

for every t 2 .

Proof. Since S is a locally compact Suslin space, it is a Polish space (see Corollary 2.26). According to Theorem 3.30,

i

S

! 



”

 i .‰/

!

C t hb .  S /; then (1)H)(2). (2)H)(1): Let ‰ 2 C thb .  S /; then M D

sup .t;x/2 S

j‰.t; x/j < C1:

 .‰/, for every ‰

2

205

Section 3.3 The Stable Topology

According to (v) of Remark 3.11, for every " > 0, there exists a compact set K 2 KS such that  ..S nK// < ". Let g 2 Cc .S / such that g.S /  Œ0; 1 and gK D 1 and let C D suppg 2 KS . The mapping ‰1 W   S ! R; ‰1 .t; x/ D g.x/, is a bounded integrand, continuous and with compact support with respect to the second variable. By (2), there exists i1 2 I such that j i .‰1 /  .‰1 /j < ", for every i  i1 . Since g  C ; S nC  1  g and then

.S nC /  1  ‰1  .S nK/ : Therefore, for every i  i1 ,

 i .  .S n C //   i .1  ‰1 / D ./   i .‰1 /  ./   .‰1 / C " D  .1  ‰1 / C "   .  .S n K// C " < 2":

(a)

Since C 2 KS , there exists h 2 Cc .S / such that h.S /  Œ0; 1 and hC D 1. Let ‰2 D ‰  h; then ‰2 is a bounded integrand and ‰2 .t; :/ 2 Cc .S /, for every t 2 . According to (2),  i .‰2 / !  .‰2 / and then there exists i2 2 I; i2  i1 such that j i .‰2 /   .‰2 /j < ";

for every i  i2 :

(b)

On the other hand, for every .t; x/ 2   S , j‰.t; x/  ‰2 .t; x/j D j‰.t; x/j  j1  h.x/j  M  .S nC / .t /:

(c)

By (a), (b) and (c), for every i  i2 , j i .‰/   .‰/j  j i .‰/   i .‰2 /j C j i .‰2 /   .‰2 /j C j .‰2 /   .‰/j   i .j‰  ‰2 j/ C " C  .j‰  ‰2 j/  M   i .  .S n C // C " C M   .  .S n C //  2"M C " C M   .  .S n K// < ".3M C 1/: S

 . Then  i .‰/ !  .‰/ and therefore  i ! We complete this paragraph by giving several results of semi-continuity for the unbounded integrands. Proposition 3.34. Let S be a metrizable Suslin space; for every net . i /i 2I  Y.S / and every  2 Y.S /, S

i !  

if and only if

 .‰/  lim inf  i .‰/; i

for every l.s.c. integrand ‰ W   S ! R bounded from below.

206

Chapter 3 Young Measures S

Proof. Let us suppose that  i !   and let ‰ W   S ! R be a l.s.c. integrand bounded from below; for every n 2 N, let ‰n W   S ! R defined by ‰n D min¹‰; nº. Then ‰n is a l.s.c. bounded integrand. Since ‰n " ‰ and ‰ is bounded from below,  .‰n / "  .‰/. According to (3) of Theorem 3.32,

 .‰n /  lim inf  i .‰n /  lim inf  i .‰/; i

i

for every n 2 N;

from where

 .‰/  lim inf  i .‰/: i

The reciprocal is obvious. We can further relax the condition of being bounded from below. The following results are presented according to [169]. Theorem 3.35. Let S be a metrizable Suslin space and let .ui /i 2I  M.S /  i Y.S / be a net such that . u /i 2I is stably convergent to  2 Y.S /. Let ‰ W   S ! R be a l.s.c. integrand such that ¹‰  .; ui .// W i 2 I º is an uniformly integrable subset of L1 ./, where ‰  D sup¹‰; 0º is the negative part of ‰; if there exists  .‰/, then Z Z i  .‰/ D ‰.t; x/d  .t; x/  lim inf ‰.t; ui .t //d.t / D lim inf  u .‰/: i 2I

S



i 2I

Proof. According to Theorem 1.84 (see also Definition 1.85), for every " > 0, there exists a" > 0 such that Z sup ‰  .t; ui .t //d.t / < ": (1) i 2I

.‰  .;ui .//>a" /

For every i 2 I , let Ai D ¹t 2  W ‰.t; ui .t // < a" º; then t 2 Ai if and only if ‰  .t; ui .t// D ‰.t; ui .t// > a" and by (1) Z ‰.t; ui .t//d.t / > ": (2) inf i 2I Ai

207

Section 3.3 The Stable Topology

According to (2), for every i 2 I , Z Z Z i i ‰.t; u .t //d.t/ D ‰.t; u .t//d.t / C ‰.t; ui .t //d.t / Ai nAi Z sup¹‰.t; ui .t //; a" ºd.t / > " C nAi Z sup¹‰.t; ui .t //; a" ºd.t / D " C Z sup¹‰.t; ui .t //; a" ºd.t /  Ai Z sup¹‰.t; ui .t //; a" ºd.t / C a"  .Ai / D " C Z  " C sup¹‰; a" º.t; ui .t //d.t /:

The mapping ‰" W   S ! R, defined by: ‰" .t; x/ D sup¹‰.t; x/; a" º, for every .t; x/ 2   S , is a l.s.c. integrand bounded from below and Z Z i ‰.t; u .t//d.t/ > " C ‰" .t; ui .t //d.t /; for every i 2 I: (3)



Then, according to the previous proposition, i

 .‰" /  lim inf  u .‰" /:

(4)

i 2I

Since ‰  ‰" , by (3) and (4), for every " > 0, Z Z i  .‰/  lim inf ‰" .t; u .t//d.t/  lim inf ‰.t; ui .t //d.t / C ": i 2I

i 2I





Corollary 3.36 (see also Corollary 2.3.2 of [44]). Let  2 Y.S / and let .ui /i 2I  M.S /  Y.S / be a net such that

 ui

S

! 

.

Then, for every Carathéodory

integrand ‰ W   S ! R for which ¹‰.; ui .// W i 2 I º is an uniformly integrable subset of L1 ./ and such that there exists  .‰/, we have: Z Z i  .‰/ D ‰.t; x/d  .t; x/ D lim ‰.t; ui .t //d.t / D lim  u .‰/: S

i 2I



i 2I

Proof. The result follows by application of the previous theorem for ‰ and ‰.

208

Chapter 3 Young Measures

Proposition 3.37. Let S D Rd and let .ui /i 2I  L1 .; Rd / be an uniformly integrable net such that

 ui

S

! 

2 Y.Rd /; then .ui /i 2I is weakly conver-



d Rgent to the integrable function u W  ! R , where, for every t 2 , u.t / D Rd xd  t .x/ D bar t (the barycenter of  t ).

Proof. Indeed, for each A 2 A and each j 2 ¹1; : : : ; d º, the map ‰j W Rd ! R; ‰j .t; x/ D A .t/  pj .x/ (where pj W Rd ! R is the canonical j -projection) is a Carathéodory integrand for which ¹‰j .; ui .// W i 2 I º D ¹ A  pj .ui / W i 2 I º is uniformly integrable. ‰1 W   Rd ! RC ; ‰1 .t; x/ D kxk, is a l.s.c. integrand bounded from below; then, according to Proposition 3.34, Z ui  .‰1 /  lim inf  .‰1 / D lim inf kui .t /kd.t / < C1: i

i



R R R Then Œ Rd kxkd  t .x/d.t/ < C1 and so there exists u.t / D Rd xd  t .x/, almost for every t 2 ; obviously, u 2 L1 .; Rd /. For every j 2 ¹1; : : : ; d º, Z Z Z Z  .‰j / D pj .x/d  t .x/ d.t /  kxkd  t .x/ d.t /; Rd

A



Rd

so that  .‰j / there exists. According to the previous corollary, Z Z i lim pj .u .t//d.t/ D  .‰j / D pj .u.t //d.t /; for every j 2 ¹1; : : : ; d º: i 2I

A

A

R

R

Therefore limi 2I A ui .t/d.t/ D A u.t/d.t /, for every A 2 A; according to Theorem 1.57, .ui /i 2I is weakly convergent to u.

3.4

The Subspace M.S /  Y .S /

Let S be a regular Suslin space. In Definition 3.15, we have associated to each application u 2 M.S / a Young measure  u and, according to (ii) of Remark 3.16, M.S / was identified as a subspace of Y.S /. This subspace has interesting specific properties that we will present in this paragraph. We recall that the disintegrated of  u is an application which associates, for each t 2 , the Dirac probability ıu.t / 2 PS and that, for all positive integrand ‰ (or for all ‰ 2 L1 . u /), Z Z u u  .‰/ D ‰.t; x/d  .t; x/ D ‰.t; u.t //d.t /: S



Section 3.4 The Subspace M.S /  Y.S /

For every net .ui /i 2I  M.S /  Y.S /, S

209

 ui

S

! 

2 Y.S / will be noted by



S

ui !   or by ui !  u if  D  u . The main results of this paragraph are: (1) The trace of stable topology on M.S / is the topology of convergence in measure. (2) Under the natural condition that  is nonatomic, M.S / is dense in .Y.S /; S/. First of all, we introduce on Y.S / a finer topology than the stable topology. Definition 3.38 (see [45], p. 56). Let S be a regular Suslin space; for every f 2 Cb .S /, let f W Y.S /  Y.S / ! RC , defined by Z j t .f /   t .f /jd.t /; for all  ;  2 Y.S /: f . ;  / D

Then ¹f W f 2 Cb .S /º is a family of metrics on Y.S /; the topology generated by this family is said the topology of convergence in measure on Y.S /. It is noted P

!  notes that the net . i /i 2I  Y.S / is P-convergent to  2 Y.S /. If by P.  i  P

P

! u instead of  ui  !  u. .ui /i 2I  M.S / and u 2 M.S /, then we will note ui  Remark 3.39. .i/ A net . i /i 2I  Y.S / is P-convergent to  2 Y.S / if and only if Z i j ti .f /   t .f /jd.t / ! 0; for every f 2 Cb .S /I f . ;  / D

P

! in other words,  i 



if and only if .:i .f //i 2I is strongly convergent to

: .f / in L1 ./, for every f 2 Cb .S /. .ii/ For every f 2 Cb .S / and every  ;  2 Y.S /, ˇZ ˇ ˇ ˇ 1 ˇ f . ;  /  sup ˇ . t .f /   t .f //d.t /ˇˇ  f . ;  /; 2 A2A A which can be also written ˇ ˇ 1 f . ;  /  sup ˇ . A ˝ f /   . A ˝ f /ˇ  f . ;  /: 2 A2A

210

Chapter 3 Young Measures

Then

ˇ ˇ P ˇ ˇ  !  ” sup ˇ i . A ˝ f /   . A ˝ f /ˇ ! 0;

i

A2A

for every f 2 Cb .S /: .iii/ According to (i) and to Remark 3.20 (i),

i

P

S

 !  H)  i !  

and therefore S  P. .iv/ For every net .ui /i 2I  M.S / and every u 2 M.S /, kk1

P

! u ” f .ui / ! f .u/; ui 

for every f 2 Cb .S /:

L1 ./

We find so the definition given by Hoffmann–Jørgensen in [95] for the convergence in measure which explains the appellation of this convergence on Y.S /. In the case in where S is metrizable and d is a metric that generated the topology of S , the PM.S / -convergence is the classic convergence in measure. Proposition 3.40. Let S be a metrizable Suslin space and let d be a metric generating the topology of S ; for every sequence .un /n2N  M.S / and every u 2 M.S /, P

S

! u ” un !  u ” lim .¹t 2  W d.un .t /; u.t //  "º/ D 0; un  n

8" > 0:

Therefore, according to the notation already used in paragraph 1.3 in the real case, S



un !  u ” un  ! u: P

S

! u,  un !   u ; according Proof. (H)): By (iii) of previous remarks, since un  to Theorem 3.30,  un .‰/ !  u .‰/, for every ‰ 2 C t hb .  S /. Let ‰ W   S ! R, defined by ‰.t; x/ D min¹d.u.t /; x/; 1º, for every t 2  and every x 2 S . Then ‰ 2 C thb .  S / and therefore Z Z ‰.t; un .t//d.t/ ! ‰.t; u.t //d.t / D 0:



Section 3.4 The Subspace M.S /  Y.S /

On the other hand, for every " > 0, Z Z ‰.t; un .t//d.t/ 

¹t Wd.un .t /;u.t //"º



211

min¹1; d.u.t /; un .t //ºd.t /

 min¹1; "º  .¹t W d.un .t /; u.t //  "º/ from where .¹t W d.un .t/; u.t//  "º/ ! 0. 

! u; since .S; d / is separable, every subsequence .u0n /n2N ((H): Let un  of .un /n2N has a subsequence .u00n /n2N convergent -almost everywhere to u (Theorem 9.2.1 of [61]). Then, for every f 2 Cb .S /; .jf .u00n /  f .u/j/n2N is a sequence uniformly bounded of measurable mappings convergent -a.e. to 0 and so Z jf .u00n .t//  f .u.t//jd.t / ! 0:

Therefore every subsequence of .f .un //n2N has a subsequence convergent to kk1

f .u/ in .L1 ./; k  k1 /; so f .un / ! f .u/; for every f 2 Cb .S / and then L1 ./

P

! u. un  Proposition 3.41. Let S be a metrizable Suslin space; then SM.S / D PM.S / , which means that the trace of the stable topology on M.S / coincides with the topology of convergence in measure. Proof. According to (ii) of Remark 3.39, SM.S /  PM.S / :

S

 u; we must Let .ui /i 2I  M.S / be a net and let u 2 M.S / such that ui ! P

! u. show that ui  For every f 2 Cb .S / let ‰ W   S ! R; ‰.t; x/ D jf .x/  f .u.t //j; then ‰ 2 C t hb .  S /. According to Theorem 3.30,  ui .‰/ !  u .‰/ and therefore Z Z ‰.t; ui .t//d.t/ D  ui .‰/ !  u .‰/ D ‰.t; u.t //d.t / D 0:



R

jf .ui .t//  f .u.t//jd.t / P Remark 3.39, ui  ! u.

Then

! 0, for every f 2 Cb .S / and so, by (iv) of

Remark 3.42. Let S be a metrizable Suslin space and let .un /n2N  M.S / and u 2 M.S /; then, according to Propositions 3.40 and 3.41, S



 u ” un  ! u: un !

212

Chapter 3 Young Measures

In this case, we may say that, almost for all t 2 ; u.t/ 2 Ls.¹un .t/ W n 2 Nº/ D

1 \

¹uk .t / W k  pº D Lsn ¹un .t /º

pD1

(see [9]).

S

 More generally, if un !



2 Y.S /, then, almost for all t 2 , the support of

 t is contained in Ls.¹up .t/ W p 2 Nº/. Proposition 3.43. Let S be a metrizable Suslin space; if the sequence .un /n  M.S /  Y.S / is stably convergent to  2 Y.S / then, almost for all t 2 , supp  t  Ls.¹up .t / W p 2 Nº/: Proof. Let d be a metric generating the topology of S such that d  1 and, for every p 2 N and every t 2 , let Ap .t/ D ¹uk .t / W k  pº. The mapping ‰p W   S ! R, defined by ‰p .t; x/ D d.x; Ap .t //, for every t 2  and every x 2 S , is continuous with respect to the second variable and measurable with respect S to the first one (for every x 2 S and every ˛ 2 R; ¹t 2  W ‰p .t; x/ < ˛º D kp ¹t 2  W d.x; uk .t// < ˛º 2 A). According to Proposition 3.8, ‰ is a bounded Carathéodory integrand. Therefore, in accordance with Theorem 3.30, for all p 2 N, Z Z un  .‰p / D ‰p .t; un .t//d.t/ D d.un .t /; Ap .t //d.t / !  .‰p /:

n!1



Since, for every t 2  and every p 2 N, limn!1 d.un .t /; Ap .t // D 0, Z Z  .‰p / D d.x; Ap .t //d  t .x/d.t / D 0

S

and so, for all p 2 N, Z d.x; Ap .t//d  t .x/ D 0; S

almost for every t 2 :

Therefore, almost for every t 2 , Z d.x; Ap .t//d  t .x/ D 0; S

for every p 2 N:

Since every Ap .t/ is a closed set,  t .S n Ap .t// D 0;

for every p 2 N

and almost for every t 2 :

Then, almost for all t 2 , supp  t  Ap .t/;

for every p 2 N

or

supp  t  Ls.¹un .t / W n 2 Nº/:

Section 3.4 The Subspace M.S /  Y.S /

213

As shown in the following example, the stable topology is strictly weaker than the topology of the convergence in measure on Y.S /. Example 3.44. Let the Rademacher’s sequence .rn /n2N where rn W Œ0; 1 ! R is defined by 8 h h < C1; t 2 S2n1 1 2kn ; 2kC1 ; n kD0 2 2 rn .t/ D S2n1 1 h 2kC1 2kC2 h : 1; t 2 : kD0 2n ; 2n We have shown, in 1.56, b), that .rn /n2N is not convergent in measure on Œ0; 1. S

Let  D  ˝ . 12 .ı1 C ı1 // 2 Y.R/. We show that rn ! 



in .Y.R/; S/ by

establishing that, for every A 2 A and every f 2 Cb .R/, ˇZ ˇ  Z Z ˇ ˇ rn ˇ j . A ˝f /  . A ˝f /j D ˇ f .rn .t//d.t / f .x/d  t .x/ d.t /ˇˇ A R ˇZA ˇ ˇ ˇ 1 ˇ D ˇ f .rn .t//d.t / .A/.f .1/Cf .1//ˇˇ ! 0: 2 A Let f 2 Cb .R/ and A 2 A (A is the -algebra of all Lebesgue-measurable sets on 0; 1Œ). (a) For every " > 0, there exists D" D D an open subset of 0; 1Œ, such that A  D and .D n A/ < ": Then, ˇZ ˇ ˇ ˇ .A/ ˇ f .rn .t//d.t/  Œf .1/ C f .1/ˇˇ ˇ 2 A ˇ ˇ ˇZ ˇZ ˇ ˇ ˇ ˇ .D/ ˇ ˇ ˇ f .rn .t //d.t /ˇˇ Œf .1/ C f .1/ˇ C ˇ  ˇ f .rn .t//d.t/  2 D DnA ˇ ˇ ˇ ˇ .D n A/ Œf .1/ C f .1/ˇˇ C ˇˇ 2 ˇ ˇZ ˇ 3" ˇ .D/ ˇ Œf .1/ C f .1/ˇˇ C max.jf .1/j; jf .1/j/:  ˇ f .rn .t//d.t/  2 2 D Therefore, it is sufficient to demonstrate the property for the open sets of 0; 1Œ. (b) Any open set D of 0; 1Œ is a countable union of pairwise disjoint open intervals Ik and, therefore Z .D/ f .rn .t//d.t/   .f .1/ C f .1// 2 D  1 Z X .Ik / .f .1/ C f .1// : f .rn .t//d.t/  D 2 Ik kD1

214

Chapter 3 Young Measures

It is sufficient to demonstrate the property for the open intervals of Œ0; 1: Sh0 1 l lC1 0  2n ; 2n Œ where (c) Let I Da; bŒD 2kn0 ; 2kn0 Œ; for every n  n0 , I D lDh nn 0 0 nn 0 0 hDk2 and h D k  2 . Then 0 0 1 Z hX 1 Z hX 1 l f .rn .t//d.t/ D f ..1/ /d.t / D f ..1/l / n l lC1 2 I  2n ; 2n Œ lDh

lDh

h0  h k0  k D nC1 .f .1/ C f .1// D n C1 .f .1/ C f .1// 2 2 0 ba .f .1/ C f .1// D 2 if k and k 0 have the same parity. The case in which k and k 0 have different parity we make the demonstration in a similar manner. We obtain: ˇ ˇZ ˇ ˇ b  a ˇ f .rn .t//d.t/   .f .1/ C f .1//ˇˇ ˇ 2 a;bŒ 1  n  max.jf .1/j; jf .1/j/: 2 Therefore

Z a;bŒ

f .rn .t//d.t/ !

ba  .f .1/ C f .1//: 2

(d) If I Da; bŒ Œ0; 1, then, for every n 2 N, Z Z f .rn .t//d.t/ D i n h f .rn .t //d.t / a; Œ2

a;bŒ

Z

C

i

Œ2n b1 ;b 2n

h f .rn .t//d.t/

aC1 2n

Z

C

i

Œ2n aC1 Œ2n b1 ; 2n 2n

h f .rn .t //d.t /

from where it results that Z ba  .f .1/ C f .1// f .rn .t//d.t/ ! 2 a;bŒ which was to be demonstrated. The disintegration of  is the constant mapping  t D 12 .ı1C ı1 / and supp t D ¹1; 1º D Lsn ¹rn .t/ W n 2 Nº.

Section 3.4 The Subspace M.S /  Y.S /

215

Remark 3.45. According to the previous example, S ¨ P. Moreover, it results that M.S / ) is not a closed subset of Y.S /. The previous example is a special case of a more general construction, which is presented in the following proposition. Proposition 3.46. Let S be a regular Suslin space and let u W R ! S be a periodic measurable application of period T . Let Œ0; T Œ provided with the Lebesgue measure  and let un W Œ0; T Œ! S , defined by un .t / D u.nt /, for every t 2 Œ0; T Œ and every n 2 N. Then, the sequence .un /n2N  M.S /  Y.S / is stably convergent to  D ˝, where  2 PS is defined by .B/ D T1 .u1 .B/\Œ0; T Œ /, for every B 2 BS . p Proof. Let A0 D ¹Œ 2kp ; kC1 2p ŒW p 2 N; k D 0; 1; : : : ; Œ2 T   1º where Œa is the integer part of a 2 R. Then A0 is an algebra generating the -algebra A of all Lebesgue measurable sets on Œ0; T Œ. S

In accordance to Proposition 3.21, in order to prove that un !  is sufficient to establish that

 un . A ˝ f / !  . A ˝ f /;

for every A 2 A0



D  ˝ , it

and every f 2 Cb .S /:

Let A D Œ 2kp ; kC1 2p Œ2 A0 and let f 2 Cb .S /; then Z Z  un . A ˝ f / D f .un .t//d.t / D f .u.nt //d.t / A A Z 1 f .u.t//d.t /; D n nA nkCn where nA D Œ nk 2p ; 2p Œ. n Let qn D Œ 2p T  2 N; then qn T 

n 2p

and, since u is periodic,

qn 1 Z 1 X  . A ˝ f / D  f .u.t //d.t / nk nk n i D0 Œ 2p CiT; 2p C.iC1/T Œ Z 1 C  f .u.t //d.t / n Œ 2nkp Cqn T; 2nkp C 2np Œ Z 1 f .u.t //d.t / C In ; where D  qn  n Œ0;T Œ n  1 1 jIn j   kf k0  p  qn T <  kf k0  T; so that In ! 0: n 2 n un

216

Chapter 3 Young Measures

Then

Z

lim

n!1



un

qn f .u.t //d.t /  lim n!1 n Œ0;T Œ Z 1 f .u.t //d.t / D p 2 T Œ0;T Œ Z 1 f .u.t //d.t / D .A/   T Œ0;T Œ Z f .x/d.x/ D .A/  S Z . A ˝ f /.t; x/d  .t; x/: D

. A ˝ f / D

Œ0;T Œ S

Remark 3.47. .i/ Let u W R ! R, defined by u D [ is periodic of period 1.

1 k2Z Œ2k;2kC 2 Œ

 [

1 k2Z Œ2kC 2 ;2kC1Œ

; then u

Let  D 12 .ı1 C ı1 / and let un W Œ0; 1Œ! R defined by 8 h h S < C1; t 2 n1 2k ; 2kC1 ; kD0 2n 2n h h un .t/ D : 1; t 2 Sn1 2kC1 ; 2kC2 : kD0 2n 2n Obviously, the Rademacher’s sequence .rn /n2N is the subsequence .u2n1 /n2N of .un /n2N and therefore it is S-convergent to  ˝ . .ii/ If we take u W R ! Œ1; 1, u.t / D sin t , then  W BŒ1;1 ! Œ0; 1 is defined by Z 1 1 .B/ D d.x/; for every B 2 BŒ1;1 : p  B 1  x2 The sequence un W Œ0; 2 ! Œ1; 1, defined by un .t / D sin nt , is stably convergent to  ˝  on Œ0; 2ŒŒ1; 1. In what follows, we will first state a lemma that will be used to show two density results on the space Y.S /. Lemma 3.48. Let S be a regular Suslin space; for every  2 Y.S / and every " > 0, there exist a compact set K 2 KS and a measure  0 2 Y.K/ such that j .‰/   0 .‰ K /j < "  k‰k0 ;

for every ‰ 2 C t hb .  S /:

Section 3.4 The Subspace M.S /  Y.S /

217

Proof. According to (v) of Remark 3.11, for every  2 Y.S / and every " > 0, there exists a compact set K 2 KS such that Z " (1)  .  .S n K// D  t .S n K/d.t / < : 4 Let A D ¹t 2  W  t .K/ D 0º; since K 2 BS , the mapping t 7!  t .K/ is (A  BŒ0;1 )-measurable (see 3.1, (ii)) and therefore A 2 A. Moreover, Z Z  t .S n K/d.t/ C  t .S n K/d.t / .A/  A nA Z "  t .S n K/d.t/ < : (2) D 4 Let 2 PK be a fixed probability; then we can define  0 W  ! PK by ´ 1   t .B/; t 2  n A 0

t .K/ ; for every t 2  and every B 2 BK :  t .B/ D .B/; t 2 A The mapping t 7!  t0 .B/ is (A  BŒ0;1 )-measurable and so :0 is (A  CK )measurable. Then, it is the disintegration of a measure  0 2 Y.K/. For every integrand ‰ 2 C thb .  S /; ‰ 0 D ‰ K 2 C t hb .  K/. Then, according to (1) and (2), we have ˇ ˇZ Z Z ˇ ˇ 0 0 0 ˇ j .‰/   .‰ /j  ‰.t; x/d  t .x/ˇˇ d.t / ˇ ‰.t; x/d  t .x/  nA S K Z Z Z j‰.t; x/jd  t .x/ C j‰.t; x/jd .x/ d.t / C A S K ˇ ˇ Z Z Z ˇ ˇ ‰.t; x/ ˇ ‰.t; x/d  t .x/  d  t .x/ˇˇ d.t /  ˇ nA K K  t .K/  Z Z j‰.t; x/jd  t .x/ d.t / C 2k‰k0  .A/ C nA S nK  Z Z  t .S n K/ j‰.t; x/jd  t .x/ d.t /   t .K/ nA K " C k‰k0   .  .S n K// C 2  k‰k0 4 Z " "  k‰k0   t .S n K/d.t / C  k‰k0 C  k‰k0 4 2 nA " 3" <  k‰k0 C  k‰k0 D "  k‰k0 : 4 4

218

Chapter 3 Young Measures

Theorem 3.49. Let S be a regular Suslin space; the space E.A; PS / of all simple Young measures is dense in .Y.S /; S/. P Proof. According to Definition 3.13,  2 E.A; PS / if  D niD1 i  A , where i ¹A1 ; : : : ; An º is an A-partition of  and 1 ; : : : ; n 2 PS . (I) Let S be metrizable. According to (i) of Theorem 3.14, for every  2 Y.S /, T

there exists a sequence . n /n2N  E.A; PS / such that  tn !   t , for every t 2 . Then, for every A 2 A and every f 2 Cb .S /; Z Z n n  . A ˝ f / D  t .f /d.t/ !  t .f /d.t / D  . A ˝ f / A

A

S

S

therefore  n !   and  2 E.A; PS / . (II) Let now S be a regular Suslin space and let  2 Y.S /; since the stable topology on Y.S / is generated by the family FS D ¹I‰ W ‰ 2 C t hb0 .  S /º, for every S-neighborhood V of  , there exist " > 0; n 2 N  and ‰1 ; : : : ; ‰n 2 C t hb0 .  S / such that U D

n \

I‰1 .  .‰i /  ";  .‰i / C "Œ /  V; i

i D1

where U D ¹ 2 Y.S / W j .‰i /   .‰i /j < "; i D 1; : : : ; nº: For every i D 1; : : : ; n, let ‰i D A ˝ fi , where Ai 2 A and fi 2 Cb .S / and i let M D max¹kfi k0 W i D 1; : : : ; nº: According to Lemma 3.48, there exist a compact set K 2 KS and a measure  0 2 Y.K/ such that j .‰/   0 .‰ K /j
0 such that U D

p \ kD1

I. I Ak ; fk ; "/  V:

(1)

220

Chapter 3 Young Measures

Let "1 D

" 2. /

> 0; then the family

´ Dy D

p \

μ fk1 .

fk .y/  "1 ; fk .y/ C "1 Π/ W y 2 S

kD1

is an open cover of compact space S . Let now ¹Dyi W i D 1; : : : ; nº a finite subcover of S ; if Ci D Dyi n [j 0, there exists K 2 KS such that  .  .S n K// < ", for every  2 H . .T2 / There exists an inf-compact application ' W S ! Œ0; C1 such that sup 2H  . ˝ '/ < C1. .T3 / For every " > 0, there is a tight set H  PS such that, for : 2 H ; :1 .PSn H / 2 A and .¹t 2  W  t 2 PS n H º/  ". .T4 / For every " > 0 there exists C 2 A ˝ BS such that, for every t , C t D ¹x 2 S W .t; x/ 2 C º 2 KS and sup 2H  ..  S / n C / < ". .T5 / There exists an inf-compact integrand ‰ W   S ! Œ0; C1 such that sup 2H  .‰/ < C1. .T6 / For every " > 0 there R exists a measurable set-valued function  W  ! KS such that sup 2H  t .S n .t//d.t / < ". Then: .T1 / ” .T2 / ” .T3 / + .T4 / ” .T5 / ” .T6 / Proof. Condition .T1 / states that the family ¹ .  / W  2 H º  RcaC .BS / is tight; indeed the condition a) of Definition 2.61 says that sup 2H  .  S / D ./ < C1: Therefore, the equivalence .T1 / ” .T2 / is a consequence of Proposition 2.63. .T2 / H) .T3 / Let ' W S ! Œ0; C1 be an inf-compact mapping and let M D sup 2H  . ˝ '/ < C1. For every " > 0, let H D ¹ 2 PS W .'/  M " º. According to Proposition 2.63, H is tight and it is a narrowly closed subset of PS (see the proof of Proposition 3.12). Therefore, for every  2 H ;  1 .PS n H / 2 A and . 1 .PS n H // < ". .T3 / H) .T1 / According to .T3 /, for every " > 0 there exists a tight H  PS such that, for every  2 H ;  1 .PS n H / 2 A and . 1 .PS n H // < 2" . Let

224

Chapter 3 Young Measures

" K 2 KS such that .S n K/ < 2. / , for every 2 H . Then, for every  2 H ; Z  .  .S n K// D  t .S n K/d.t/ Z Z D  t .S n K/d.t / C  t .S n K/d.t /

1 .PS nH /

1 .H /

"  . 1 .H // < ":  . 1 .PS n H // C 2./ .T1 / H) .T4 / For every " > 0, let K 2 KS such that  .  .S n K// < ", for every  2 H . Then C D   K satisfies .T4 /. .T4 / ” .T6 / For every C 2 A ˝ BS with compact sections,  W  ! KS , defined by .t/ D C t , for every t 2 , is a measurable set-valued function (i.e. G D ¹.t; x/ 2   S W x 2 .t/º 2 A ˝ BS ). Conversely, for every measurable set-valued function  W  ! KS , if C D G , C 2 A ˝ BS and C t 2 KS ; for every t 2 . The equivalence of .T4 / and .T6 / results if we notice that, for every  2 H ; Z Z  ..  S / n C / D  t .S n C t /d.t / D  t .S n .t //d.t /:



.T5 / H) .T6 / Let ‰ W   S ! Œ0; C1 an inf-compact integrand such that M D sup 2H  .‰/ < C1. For every " > 0, we define the set-valued function  W  ! KS letting .t/ D ¹x 2 S W ‰.t; x/  M " º; for every t 2  (since ‰ is inf-compact, .t/ 2 KS , for every t 2 ). ‰ is .A ˝ BS /-measurable; then the graph of , G D ¹.t; x/ W x 2 .t /º D ¹.t; x/ W ‰.t; x/  M " º 2 A ˝ BS and therefore  is measurable. Moreover, we have, for every  2 H ; ³ ² Z Z M d.t /  t .S n .t//d.t/ D t x 2 S W ‰.t; x/ > " Z " "   t .‰/d.t / D    .‰/  ": M M .T6 / H) .T5 / For every n 2 N  , let n W  ! KS be a measurable setvalued function such that Z 1  t .S n n .t//d.t / < n sup 3

2H and let 0 .t/ D ;, for every t 2 . It is clear that we can choose the setvalued functions .n /n2N in a such manner that, for every t 2  and every n 2 N; n .t/  nC1 .t/. Let now ‰ W   S ! Œ0; C1 defined by ² 2n ; x 2 nC1 .t/ n n .t / ; for every .t; x/ 2   S: ‰.t; x/ D C1; x 2 S n [1 nD0 n .t/

225

Section 3.5 Compactness

For every a > 0, ¹.t; x/ W ‰.t; x/ < aº D

[

¹.t; x/ W x 2 nC1 .t / n n .t /º

2n 0, there exists K 2 KS such that 2 H . Then, for every  2 H , Z 1  t .S n K/d.t /  .¹t 2  W  t .S n K/  "º/  " . t .S nK/"/ Z 1 1  t .S n K/d.t / D   .  .S n K// < ":  " "

 .  .S n K// < "2 , for every 

((H) Let us now suppose that, for every " > 0, there exists K 2 KS such that  .¹t 2  W  t .S n K/  "º/ < ";

for every

Then, for every  2 H , Z  .  .S n K// D  t .S n K/d.t/ Z Z  t .S n K/d.t /C D . t .S nK/"/



2 H:

. t .S nK/ 0 and every u 2 H; M  .kukk/ kukd  k  .kuk  k/. Then, for every " > 0; there exists k > M " such that .kuk  k/ < ", for every u 2 H . Therefore, according to Definition 3.55, H is tight. (ii) E endowed with the weak topology is a regular Suslin space (see the Example 2.17); the proof comes from that of (i) by remarking that the set K D ¹x 2 E W kxk  kº is weakly compact and that  u .  .E n K// D .kuk > k/. Remark 3.57. If E is a separable Banach space, then, generally, a bounded set of L1 .; E/ is not a tight with respect to strong topology of E. Indeed, let E D CŒ0;1 be the space of all continuous real-valued mappings on Œ0; 1 provided with the norm of uniform convergence and, for every n 2 N, let fn W Œ0; 1 ! R; fn .x/ D x n . Then t 2 Œ0; 1 7! un .t/ D fn defines a E-valued function on Œ0; 1. The sequence .un /n  L1 .; E/ is bounded. If we suppose that .un /n is 1 tight, then there exists a compact set K  CŒ0;1 such that .u1 n .E n K// < 2 , for every n 2 N and so .fn /n  K which is impossible since .fn /n is not equicontinuous.

228

Chapter 3 Young Measures

Theorem 3.58. Let S be a compact Suslin space; then .Y.S /; S/ is a compact space. Proof. Let C.S / D Cb .S / be the space of all real-valued continuous functions 1 on S provided with the norm R of uniform convergence, k  k0 ; let L .; C.S // D ¹f 2 M.C.S // W kf k1 D kf k0 d < C1º. If L1 .; C.S // D ¹fN W f 2 L1 .; C.S //º, then L1 .; C.S // is a Banach space; let ŒL1 .; C.S // its dual space. In the proof of Theorem 3.30, we have associated to each ‰ 2 C t hb .  S / the mapping '‰ W  ! C.S /, defined by '‰ .t / D ‰.t; /, for every t 2  and we showed that '‰ 2 L1 .; C.S //. Since sup t 2 k'‰ .t/k0 D sup t 2 supx2S j‰.t; x/j < C1; '‰ is a bounded mapping. The mapping ‰ 7! '‰ is a bijection of C t hb .  S / on the subset E of all bounded functions of L1 .; C.S //. Indeed, for every bounded function f 2 L1 .; C.S //, the mapping ‰ W   S ! R; ‰.t; x/ D f .t /.x/, is measurable in t and continuous in x (if, for every x 2 S , x W C.S / ! R, x .g/ D g.x/, for every g 2 C.S /, then ‰.; x/ D x ı f is a composition of the continuous function x with the measurable function f ). Then, according to Proposition 3.8, ‰ is a Carathéodory integrand. Moreover, for every .t; x/ 2   S; j‰.t; x/j  kf .t/k0  sup kf .t /k0 < C1: t 2

Then ‰ 2 C t hb .  S /. Obviously, '‰ D f . For every  2 Y.S /, let f  W L1 .; C.S // ! R defined by  Z Z  f .t/.x/d  t .x/ d.t/; for every f 2 L1 .; C.S //: f .f / D

S

We remark that, for every f 2 L1 .; C.S //, Z Z sup jf .t/.x/j d.t / D kf .t/k0 d.t / < C1 x2S Z  kf .t/k0 d.t / D kf k1 : jf .f /j 

and therefore



Then the application  7! f  is an application of Y.S / in ŒL1 .; C.S // . Moreover, kf  k D supkf k1 1 jf  .f /j  1; which means that the range of this application, F D ¹f  W  2 Y.S /º; is a subset of the unit closed ball ofŒL1 .; C.S // . According to Theorem 3.30, a net . i /i 2I  Y.S / is stably convergent to  if and only if  i .‰/ !  .‰/, for every ‰ 2 C t hb .  S /, which is equivalent to say that f i .'‰ / ! f  .'‰ /, or that f i .f / ! f  .f /; for every f 2 E.

229

Section 3.5 Compactness

Since the set E is dense in L1 .; C.S //, it results that

i

S

! 



if and only

if .f i /i 2I is weak*-convergent to f  in ŒL1 .; C.S // . Then .Y.S /; S/ is homeomorphic to the subset F of the unit closed ball of ŒL1 .; C.S // endowed with the weak* topology. In order to complete the proof, it is sufficient to show that F is a weak*-closed set. Let f  be in the weak* closure of F ; there exists a net . i /i 2I  Y.S / such that .f i /i 2I to be weak*-convergent to f  . Then, for every ‰ 2 C t hb .  S /, f i .'‰ / ! f  .'‰ / or

 i .‰/ ! f  .'‰ /:

For every A 2 A; the mapping  .A  / W C.S / ! R, defined by:  .A  f / D limi  i . A ˝f / is a positive linear functional; therefore, according to Theorem 41 of Ch. 3.no 2 from [32], a Radon measure on S (i.e.  .A  / 2 RcaC .BS /). Moreover  .A  S / D limi  i .A  S / D .A/, for every A 2 A and then  2 Y.S /. Since, for each ‰ 2 C thb .  S /;  .‰/ D limi  i .‰/ D f  .'‰ /, f  D f  2 F . Theorem 3.59 (Prohorov’s theorem). (a) Let S be a regular Suslin space and let H  Y.S / be a tight; then H is relatively S-compact. (b) Let S be a regular Suslin space and a Prohorov space; if H is a relatively stably compact subset of Y.S /, then H is tight. Proof. (a) Let us recall that, according to (i) of Remark 3.53, H is tight if and only if there exists an inf-compact application ' W S ! Œ0; C1 such that M D sup 2H  . ˝ '/ < C1: (a1) Firstly, let us consider that S is metrizable; in this case, it can be embedded N According to Theorem 3.58, .Y.SN /; S N / is a in a compact metrizable space S. S compact space and according to Theorem 3.28, .Y.S /; S/ is homeomorphic to a subspace of .Y.SN /; SSN /. For every net . i /i 2I  H  Y.S /  Y.SN /, there exist a subnet . ij /j 2J S

SN and a Young measure  2 Y.SN / such that  ij !  . Let ˆ W SN ! Œ0; C1 be the mapping defined by ² '.x/ if x 2 S; ˆ.x/ D C1 if x 2 SN n S:

230

Chapter 3 Young Measures

Then, for every a 2 RC ; ˆ1 .Œ0; a/ D ' 1 .Œ0; a/ 2 KS  KSN ; therefore ˆ is an inf-compact application on SN and then ˆ is l.s.c. and bounded from below. According to (2) of Proposition 3.22,

 . ˝ ˆ/  lim inf  ij . ˝ ˆ/ D lim inf  ij . ˝ '/  M j 2J

j

and then

 ˆ.x/d  t .x/ d.t / C1 > M  SN  Z Z Z Z D '.x/d  t .x/ d.t / C Z Z



S



N SnS

 .C1/d  t .x/ d.t /:

We can see that, almost for every t 2 ;  t .SN n S / D 0 and therefore  2 Y.S /. S

Then  ij !   ; hence H is relatively S-compact. (a2) Let now .S; / be any regular Suslin space and let H be a subset -tight of Y.S / (i.e. tight with respect to compacts of .S; /). According to (ii) of Theorem 2.25, there exists a metric d on S such that the generated topology d is coarser than , second-countable and so that BS .d / D BS ./. Let Sd be the stable topology on Y.S / with respect to .S; d /. Since K.S; /  K.S; d / ; H is d -tight. According to (a1), since .S; d / is a metrizable Suslin space, H is relatively Sd -compact. A net . i /i 2I  Y.S / is Sd -convergent to  2 Y.S / if and only if, for every A 2 A and every f 2 Cb .S; d /;  i . A ˝f / !  . A ˝f /. Since Cb .S; d /  Cb .S; /, Sd  S. Since H is -tight, there exists a -inf-compact application ' W .S; / ! Œ0; C1 such that M D sup 2H  . ˝ '/ < C1. Let . i /i 2I  H be a net; since H is relatively Sd -compact there exist a Sd

subnet . ij /j 2J and a Young measure  2 Y.S / such that  ij !  . S

In order to show that  ij !   , we will go back to an argument used in the proof of Theorem 2.6 (ii) of [18]. Let A 2 A and let f W .S; / ! RC be a -l.s.c. application; for every " > 0, let '" D f C "  ' W .S; / ! RC . Then '" is -inf-compact; indeed, for every a 2 RC ; '"1 .Œ0; a/ D ¹x 2 S W f .x/ C "  '.x/  aº  ¹x 2 S W '.x/  a 1 .Œ0; a /. Since ' is -l.s.c., ' 1 .Œ0; a/ is a -closed subset of the " " "º D ' " -compact set ' 1 .Œ0; a" /. Then '"1 .Œ0; a/ 2 K.S; /  K.S; d / , from where '"1 .Œ0; a/ is a closed subset of .S; d / and so '" is an application d -l.s.c. and bounded from below. Therefore

 . A ˝ '" /  lim inf  ij . A ˝ '" / j

231

Section 3.5 Compactness

and then lim inf  ij . A ˝ f / C "  M  lim inf  ij . A ˝ '" /   . A ˝ '" / j

j

  . A ˝ f /;

for every " > 0I

therefore, for every positive -l.s.c. application, lim inf  ij . A ˝ f /   . A ˝ f /: j

It is evident that the inequality is also true for any application bounded from below S

and -l.s.c. and then, according to (2) of Proposition 3.22,  ij !  : (b) Let S be a regular Suslin space and a Prohorov space (according to the definition given in 2.75(i), this means that every relatively narrowly compact subset of RcaC .BS / is tight). The mapping ˆ W Y.S / ! RcaC .BS /, defined by ˆ. / D  .  /, for every  2 Y.S /, is .S  TS /-continuous (where TS is the narrow topology on RcaC .BS /). If H  Y.S / is relatively S-compact, then ˆ.H / is relatively TS -compact in RcaC .BS / and so ˆ.H / is tight. Therefore, for every " > 0, there exists K 2 KS such that, for every  2 H ; ˆ. /.S nK/ D  .  .S n K// < ": In the special case in which S is a Polish space, we can show, by using a similar method to that used in (a) of Theorem 3.59, that condition .T5 / of Theorem 3.51 is a sufficient condition of stable compactness. Theorem 3.60. Let S be a Polish space and let H  Y.S / be a set satisfying the condition .T5 / of Theorem 3.51; then H is relatively stably compact. Proof. According to condition .T5 / of Theorem 3.51, there exists an integrand inf-compact ‰ W   S ! Œ0; C1 such that M D sup  .‰/ < C1:

2H

Since S is Polish, it is homeomorphic to a Gı subset of compact space SN D Œ0; 1N (see Theorem 2.19) and therefore BS  BSN . According to Theorem 3.58, .Y.SN /; SSN / is a compact space and by Theorem 3.28, .Y.S /; S/ is homeomorphic to a subspace of .Y.SN /; SSN /. For every net . i /i 2I  H  Y.S /  Y.SN /, there exist a subnet . ij /j 2J S

SN and a Young measure  2 Y.SN / such that  ij !  .

232

Chapter 3 Young Measures

N W   SN ! Œ0; C1 be the mapping defined by Let ‰ ² ‰.t; x/; if t 2 ; x 2 S; N ‰.t; x/ D C1; if t 2 ; x 2 SN n S: N is an integrand on SN . Then ‰ N //1 .Œ0; a/ D ‰.t; /1 .Œ0; a/ 2 For every t 2  and every a 2 RC ; .‰.t; N KS  KSN , therefore ‰ is an integrand inf-compact on SN and so l.s.c. and bounded from below. According to Proposition 3.34, N  lim inf  ij .‰/  M  .‰/ j

and then

M   .‰/ ‰.t; SN   Z Z Z Z ‰.t; x/d  t .x/ d.t / C .C1/d  t .x/ d.t /: D Z Z



S



N SnS

Then  t .SN n S / D 0, almost for every t 2 , and so



S ij ! 



2 Y.S /. Therefore

 and then H is relatively stably compact.

Corollary 3.61. Let S be a Polish space; for every H  Y.S /, the conditions .T1 / to .T6 / of Theorem 3.51 are equivalent. Proof. In accordance to Theorem 3.51, we have to prove that .T5 / H) .T1 /. Let H  Y.S / be a set satisfying .T5 /; according to the previous theorem, H is relatively S-compact. Since every Polish space is a Prohorov one (see (i) of Remark 2.75), H is tight (see (b) of Theorem 3.59). In the following corollary, we will rewrite the equivalent conditions .T1 / to .T6 / of Theorem 3.51 for the particular case in which H  M.S /  Y.S /. Corollary 3.62. Let S be a Polish space and let H  M.S /  Y.S /; The following conditions are equivalent: .T1 / For every " > 0, there exists K 2 KS such that supu2H .u1 .S n K// < ". .T2 / There exists R an inf-compact mapping ' W S ! Œ0; C1 such that supu2H '.u.t//d.t/ < C1.

233

Section 3.5 Compactness

.T3 / For every " > 0, there exists a tight set L  PS such that supu2H .¹t 2  W ıu.t / … Lº/  ". .T4 / For every " > 0, there exists C 2 A ˝ BS such that, 8t 2 , C t D ¹x 2 S W .t; x/ 2 C º 2 KS and supu2H . n uN 1 .C // < " (where uN W  !   S; u.t/ N D .t; u.t ///. .T5 / There exists R an integrand inf-compact ‰ W   S ! Œ0; C1 such that supu2H ‰.t; u.t//d.t/ < C1. .T6 / For every " > 0, there exists a measurable set-valued mapping  W  ! KS such that supu2H .¹t 2  W u.t / … .t /º/ < ". Remark 3.63. .i/ Just as in the special case of the Prohorov theorem, we have: If S is a Polish space, then H  Y.S / is relatively S-compact if and only if H satisfies to one of equivalent conditions .T1 /–.T6 /. .ii/ The fact that any uniformly integrable subset of L1 .; Rd / is relatively weakly compact in .L1 .; Rd /; k  k1 / (see Dunford–Pettis’ Theorem 1.65 and Theorem 1.84) is now a consequence of the Prohorov theorem. Indeed, if H is an uniformly integrable subset of L1 .; Rd /, then H is bounded and, according to Proposition 3.56, H is tight. The Prohorov’s theorem 3.59 assures us that H is relatively stably compact. Then every net .ui /i 2I  H admits a subnet .uij /j 2J  S-convergent to a Young measure  2 Y.Rd /. Since .uij /j 2J is uniformly integrable and

 uij R

S

! 

 , .uij /j 2J

is weakly

convergent to u 2 L1 .; Rd /, where u.t / D Rd xd  t .x/, for every t 2  (see 3.37); therefore H is relatively weakly compact in .L1 .; Rd /; k  k1 /. We will now give two sequentially compactness results on .Y.S /; S/. Their demonstration is inspired by those of Lemma 4.4.1 and of Proposition 4.5.2 of [45]. Theorem 3.64. Let S be a regular Suslin space. If H  Y.S / is tight, then it is sequentially S-compact. Proof. (1) Firstly, let us assume that S is a metrizable Suslin space. According to Theorem 2.58, RcaC .BS / and PS are metrizable and second-countable. Then the Borel -algebra C of PS is countably generated. Let . n /n2N be a sequence of Young measures in H ; for every n 2 N; :n W .; A/ ! .PS ; C/ is then a measurable mapping. The smallest -algebra on

234

Chapter 3 Young Measures

 for which all applications :n are measurable, A0 , is then countably generated, also; obviously, A0  A. Let 0 be the restriction of  to A0 and let Y0 .S / be the space of Young measures on .; A0 ; 0 /, therefore the space of all .A0  C/measurable applications : W  ! PS or the space of measures  W A0 ˝ BS ! RC for which  .A  S / D 0 .A/; for every A 2 A0 . We can notice that 0 is not, generally, a complete measure on A0 . Let S0 be the stable topology on Y0 .S /; Y0 .S /  Y.S / and S0  SY0 .S / . Let ˆ W Y.S / ! Y0 .S / be the mapping defined by ˆ. / D  A0 ˝BS , for every  2 Y.S /; then ˆ is a surjection .S  S0 /-continuous. Since H  Y.S / is tight, it is relatively S-compact (Theorem 3.59) and then ˆ.H / is relatively S0 -compact. According to Proposition 3.25, .Y0 .S /; S0 / is metrizable, therefore ˆ.H / is sequentially compact. Let now . kn /n2N be a subsequence of . n /n2N and let  2 Y0 .S /  Y.S / S0

S

such that ˆ. kn / !  . We will show that  kn !  . For every A 2 A let W A0 ! RC defined by .B/ D .A \ B/, for every B 2 A0 ; is a finite, -additive measure, absolutely continuous with respect to 0 W A0 ! RC . According to Radon–Nikodym theorem (Theorem 1.40), there exists a mapping g 2 L1 .; A0 ; 0 / such that .B/ R D .A \ B/ D R g.t /d .t/, for every B 2 A ; particularly, .A/ D 0 0 B g.t /d0 .t /. Then, for every f 2 Cb .S /, Z Z  kn . A ˝ f / D A .t/   tkn .f /d.t / D g.t /  ˆ. kn / t .f /d0 .t / Z Z ! g.t/   t .f /d0 .t/ D

A .t /   t .f /d.t /



D  . A ˝ f /: (2) Let us now suppose that .S; / is a regular Suslin space. According to (ii) of Theorem 2.25, there exists a metric d on S such that the generated topology, d , is second-countable, less fine than  and such that BS .d / D BS ./. Then .S; d / is a metrizable Suslin space and, as we proved in Theorem 3.59, the stable topology Sd on Y.S; d / is less fine than the stable topology S on Y.S; /. Let H  Y.S / be -tight. The closure H of H in the space .Y.S /; S/ is Scompact and then SH D Sd H . But, according to the first part of proof, H is sequentially Sd -compact and then it is sequentially S-compact. The following proposition gives a reciprocal of this property. Proposition 3.65. Let S be a Polish space. If H  Y.S / is a sequentially Scompact set, then it is tight.

235

Section 3.5 Compactness

Proof. The mapping ˆ W Y.S / ! RcaC .BS /, defined by ˆ. / D  .  /, for every  2 Y.S /, is .S  TS /-continuous. Then ˆ.H / is sequentially TS compact in RcaC .BS /. Since .RcaC .BS /; TS / is metrizable (Theorem 2.58), ˆ.H / is relatively compact and, according to Theorem 2.74, ˆ.H / is tight; then H is tight. We will conclude this paragraph by giving two convergence results on M.S /. Theorem 3.66. Let S be locally compact Suslin space and let .un /n  M.S /; then .un /n is S-convergent in Y.S / if and only if ² .i/ .un /n is tight and .ii/ .f .un //n is weakly convergent in L1 ./; 8f 2 Cc .S /: Proof. ()) Let



S

2 Y.S / such that un ! 

;

the set H D ¹un W n 2 Nº

is sequentially S-compact. According to Corollary 2.26, every locally compact Suslin space is a Polish one; by the previous proposition H is tight. According to (7) of Proposition 3.22, for every f 2 Cc .S /, .f .un //n2N isRweakly convergent in L1 ./ to a function uf 2 L1 ./, defined by uf .t / D S f .x/d  t .x/, for every t 2 . (() Let .un /n  M.S / be a tight such that, for every f 2 Cc .S /, .f .un // is weakly convergent in L1 ./. Then, for every f 2 Cc .S /, there exists uf 2 R w L1 ./ such that f .un / ! uf . If f  0, then, for every A 2 A, A uf d  L1 ./

0; therefore uf  0, -almost everywhere. By replacing uf with max.uf ; 0/, we obtain: f  0 ) uf  0. For every t 2 , we define  t W Cc .S / ! R, by  t .f / D uf .t/; then  t is positive and linear and therefore a Radon measure on S (Theorem 4-1, Ch. 3, no. 2 of [32]). Moreover, for every A 2 A and every f 2 Cc .S /, Z Z n f .u .t//d.t/ !  t .f /d.t /: A

A

Since .un /n is tight, according to Theorem 3.64, it has a subsequence .ukn /n convergent to a Young measure  0 2 Y.S /; then, for every f 2 Cc .S /;  t .f / D  t0 .f /, for -almost every t 2 . Let .fp /p2N  Cc .S / be a sequence pointwise convergent to 1; then, almost for every t 2 ,  t .1/ D limp  t .fp / D limp  t0 .fp / D  t0 .1/ D 1. Therefore



S

2 Y.S / and, obviously, un !  .

236

Chapter 3 Young Measures

Theorem 3.67. Let S D Rd and let .un /n2N  M.Rd /; .un /n is convergent in measure in M.Rd / if and only if ² .i/ .un /n is tight and ; .ii/ .f .un //n is strongly convergent in L1 ./; 8f 2 Cb .S /: Proof. ()) The implication is a consequence of (iii) and (iv) of Remark 3.39 and of Proposition 3.65. (() Let .un /n  M.Rd / be a tight such that, for every f 2 Cb .Rd /, .f .un //n converges strongly in L1 ./. According to Definition 3.55, for every " > 0, there exists k > 0 such that, for every n 2 N, .kun k  k/ < 4" . For every i  m, let fi 2 Cb .Rd / such that, fi ¹kykkº .y/ D yi (yi is the canonical i-projection of y). For every i  m, .fi .un //n is strongly convergent so that there exists n0 such that, for every i  m, every n and every p  n0 ; kfi .un /  fi .up /k1 < 4" : Therefore, for every  > 0, every i  m, and every n; p  n0 , Z " > jfi .un /  fi .up /jd 4 Z jfi .un /  fi .up /jd  n p n p .jfi .u /fi .u /j> /\.jju jjk/\.jju jjk/   >    .jfi .un /  fi .up /j > / \ .jjun jj  k/ \ .jjup jj  k/ and so, for every i  m,   "  .jfi .un /  fi .up /j > / \ .jkun k  k/ \ .kup k  k/ < : 4 By considering jjyjj D supi m jyi j, we obtain: .jjun  up jj > ; jjun jj  k; jjup jj  k/
0; .jjun  up jj > /
0 A  Z ku.t /kd.t / D lim sup a!1 u2H

.kuk>a/

(see Definition 1.82 and Proposition 1.83). H is uniformly integrable if and only if .H / D 0: Theorem 3.68 (Biting Lemma). Let .E; k  k/ be a separable Banach space, let .un /n be a bounded sequence of L1 .; E/ and let  D ..un /n / be its modulus of uniform integrability. There exist a subsequence .ukn /n of .un /n and a decreasing sequence of “bits” .Bp /p2N  A with .Bp / ! 0 such that:

238

Chapter 3 Young Measures

(1) . nB  ukn /n2N is uniformly integrable. n

(2) ..ulkn /n / D ..un /n /, for every subsequence .ulkn /n of .ukn /n . R (3) limn Bn kukn .t/kd.t / D ..un /n /. If E is reflexive, then we can choose the sequences .ukn /n and .Bp /p such that . nB  ukn /n2N to be weakly convergent. For every p 2 N; .ukn  nBp /n2N is n

then weakly convergent in L1 . n Bp ; E/. Proof. For every n 2 N, let fn W  ! R; fn D kun k. Then .fn /n  L1 ./ is a bounded sequence; according to Corollary 1.106 (“Subsequence Splitting Lemma”), there exist a subsequence .fknR /n of .fn /n and a sequence of pairwise disjoint sets .An /n  A such that limn An jfkn .t /jd.t / D ..fn /n / D  and so that the sequence . nA  fkn /n to be weakly convergent. Then n Z kukn .t/kd.t / D  (1) lim n

An

and, according to Theorem 1.84 and Proposition 1.83,      . nA  fkn /n D  . nA  ukn /n D 0: n

(2)

n

1 A 2 A; it is clear that .B / is a For every p 2 N, we note Bp D [P p p nDp n decreasing sequence and that .Bp / D 1 .A / ! 0. n nDp Since nB  nA , by (2), .. nB  ukn /n / D 0 and then the sequence n n n . nB  ukn /n is uniformly integrable. n For every subsequence .ulkn /n of .ukn /n ,

..ulkn /n /  ..un /n / D : ²Z ..ulkn /n / D inf sup ">0

A

(3) ³

kulkn .t/kd.t / W n 2 N; .A/ < " :

Therefore, for every > 0, there exists " > 0 such that, for every n 2 N and every A 2 A with .A/ < ", Z kulkn .t/kd.t/ < ..ulkn /n / C : (4) R

A

By (1), An kukn .t/kd.t / !  and .An / ! 0; then there exists n 2 N such that .An / < " and Z kulkn .t/kd.t / >   : (5) An

239

Section 3.6 Biting Lemma

According to (4) and (5), ..ulkn // C >   , for every > 0 and, using (3), we obtain ..ulkn // D : By (2), for every " > 0, there exists ı > 0 such that, for every A 2 A with R .A/ < ı and every n 2 N; AnAn kukn .t/kd.t / < ". Since .Bn / ! 0, there exists n0 2 N such that .Bn / < ı, for every n  n0 and therefore Z kukn .t/kd.t / < "; for every n  n0 : (6) Bn nAn

According to (6), for every n  n0 , Z Z Z kukn .t/kd.t /  kukn .t/kd.t / D kukn .t /kd.t / An Bn An Z Z kukn .t/kd.t / < kukn .t /kd.t / C ": C Bn nAn

An

Therefore, from (1) we obtain 3 of the theorem. We recall that, according to Dunford’s theorem (Theorem 1, p. 101 of [56]), if E is reflexive, then the uniformly integrable sets are relatively weakly compact. Remark 3.69. Generally, the bounded sets of L1 .; E/ are not tight (see 3.57). If we add to the condition of “boundedness” that of “tightness”, then we can give a general result of convergence (for the Euclidean case, see Theorem 9 of [169]). Theorem 3.70. Let E be a separable and reflexive normed space and let .un /n be a bounded tight sequence in L1 .; E/; then we can find a subsequence .ukn /n , a Young measure  2 Y.E/ and a decreasing sequence of sets .Bp /p  A with .Bp / # 0 such that: SE

(1) ukn !  . (2) Almost for every t 2 ;  t has a barycenter, bar. t / D mapping t 7! u.t / D bar. t / is integrable.

R E

xd  t .x/ and the

w

(3) nB  ukn ! u. n

L1 .;E /

R (4) If M D ¹t 2  W E kx  u.t /kd  t .x/ D 0º D ¹t 2  W  t D ıu.t / º, then .ukn M /n is convergent in measure to uM . R (5) limn!1 kukn  uk1 D ..un // C E kx  u.t /kd  .t; x/.

240

Chapter 3 Young Measures

Proof. According to Theorem 3.68, there exist a subsequence .u0n /n of .un /n and a decreasing sequence .Bp /p  A with .Bp / # 0, such that the sequence . nB  u0n /n is weakly convergent to a mapping v 2 L1 .; E/, ..ukn // D n ..unR//, for every subsequence .ukn /n of .u0n /n and ..un // D 0 limn Bn kun .t/kd.t /. Since .u0n /n is tight, there exists a subsequence .ukn /n of .u0n /n stably convergent to a Young measure  2 Y.E/ (see Theorem 3.64). Let f W E ! R, defined by f .x/ D kxk, for every x 2 E. Then f is continuous and bounded from below; according to (2) of Proposition 3.22,

 . ˝ f /  limninf  ukn . ˝ f /; Z Z

E

or, since .ukn /n

is bounded,

 Z kxkd  t .x/ d.t /  lim inf kukn .t /kd.t /  m < C1: n



R

Therefore, for almost every t 2 ; E kxkd  t .x/ < C1. Subsequently, for almost all t 2 , Z xd  t .x/ D bar. t / 2 E u.t / D E

and u 2 L1 .; E/. Let us show that u D v. Since E is reflexive, the dual space of L1 .; E/ is 1 L .; E  / D ¹g W  ! E  W g is measurable and kgk1 D inf¹c > 0 W kg.t /kE   c;   a.e. º < C1º (see Theorem 1, p. 98 of [56]). w Let g be arbitrary in L1 .; E  /; since nB  ukn ! v, n

Z

L1 .;E /

Z   g.t / nB .t/  ukn .t/ d.t / ! g.t / .v.t // d.t /: n

(1)



Then ‰ W   E ! R; ‰.t; x/ D g.t/.x/, for every x 2 E and every t 2 , is measurable in t and continuous in x; according to Proposition 3.8, ‰ is a Carathéodory integrand. Moreover, for all t 2  and all n 2 N, ˇ ˇ ˇ  ˇ ˇ ˇ ˇ ˇ ‰.t;

.t/  u .t// D g.t /

.t /  u .t / ˇ ˇ ˇ ˇ kn kn nBn nBn  kg.t/kE   k nB .t /  ukn .t /kE n

 kgk1  k nB .t /  ukn .t /kE n

and, since . nB  ukn /n is uniformly integrable, .‰.:; nB .:/  ukn .:///n is n n uniformly integrable too.

241

Section 3.6 Biting Lemma

According to Corollary 3.36,  Z Z Z ‰.t; nB .t/  ukn .t//d.t/ ! ‰.t; x/d  t .x/ d.t / n



or

Z







E

Z Z



g.t / nB .t/  ukn .t/ d.t / ! n



 g.t /.x/d  t .x/ d.t /:

(2)

E

According to (1) and (2),  Z Z Z g.t/.v.t//d.t/ D g.t/.x/d  t .x/ d.t /  Z E Z Z D g.t/ xd  t .x/ d.t / D g.t /.u.t //d.t /

E



and so, as g is arbitrary in the dual space, u D v. For every t 2 M;  t .¹u.t /º/ D 1, which means that  t D ıu.t / . According to Proposition 3.41, .ukn M /n is convergent in measure to uM . In order to show (5), let us first remark that Z kukn .t /kd.t /:  D ..un // D ..ukn // D inf sup sup ı>0 n2N .A/   "1 ;

for every n  p:

(4)

R Since .Bp / # 0 Rand E kxkd  .t; x/  m, one can choose p 2 N so that .Bp / < ı and that Bp E kxkd  .t; x/ < "2 . Then, by (3) and (4), for every n  p, Z Z   "1 < kukn .t/kd.t /  kukn .t /kd.t / <  C "1 : Bkn

Bp

Therefore, for every " > 0, there exists p 2 N such that ˇZ ˇ ˇ ˇ ˇ ˇ kukn .t/kd.t /  ˇ < "1 ; for all n  p: ˇ ˇ Bp ˇ

(5)

242

Chapter 3 Young Measures

Then, for every n  p, Z kukn  uk1 D kukn .t/  u.t /kd.t / Z Z kukn .t/  u.t /kd.t / C D Bp

nBp

kukn .t /  u.t /kd.t /:

By (5), for every n  p, Z Z Z   " 1  "2 < kukn .t/kd.t / ku.t /kd.t /  kukn .t /u.t /kd.t / Bp Bp Bp Z Z kukn .t/kd.t / C ku.t /kd.t / <  C "1 C "2  Bp

Bp

and therefore ˇ ˇZ ˇ ˇ ˇ ˇ kukn .t/  u.t /kd.t /  ˇ < "1 C "2 ; ˇ ˇ Bp ˇ

for all n  p:

(6)

Let ‰p W   E ! R; ‰p .t; x/ D nB .t /  kx  u.t /k; then ‰p is a Carap théodory integrand and, for every n  p and every t 2 , ˇ ˇ ˇ‰p .t; uk .t//ˇ 

.t /  kuk .t /  u.t /k: nBp

n

n

Since . nB  ukn /n is uniformly integrable, . nB  .ukn  u//n is uniformly p

p

integrable too and then .‰p .:; ukn .:///n is uniformly integrable in L1 ./. According to Corollary 3.36, Z lim kukn .t/  u.t /kd.t / D  .‰p / n nBp  Z Z kx  u.t /kd  t .x/ d.t / D nBp E Z Z kx  u.t /kd  .t; x/ C kx  u.t /kd  .t; x/: D Bp E

E

Then there exists n0  p such that, for every n  n0 , ˇZ ˇ Z ˇ ˇ ˇ kukn .t/  u.t /kd.t /  kx  u.t /kd  .t; x/ˇˇ ˇ ˇ nBp E Z kx  u.t /kd  .t; x/ < Bp E Z Z kxkd  .t; x/ C ku.t /kd.t /  Bp E Bp  Z Z < "2 C kxkd  t .x/ d.t / < 2"2 : Bp

E

243

Section 3.6 Biting Lemma

Therefore, for every n  n0 , ˇZ ˇ Z ˇ ˇ ˇ ˇ kukn .t/  u.t /kd.t /  kx  u.t /kd  .t; x/ˇ < 2"2 : ˇ ˇ nBp ˇ E

(7)

Then, by (6) and (7), for all n  n0  p, ˇ Z ˇ ˇ ˇkuk  uk1   kx  u.t /kd  .t; x/ˇˇ n ˇZ E ˇ ˇ ˇ ˇ ˇ ˇ ku .t/  u.t /kd.t /  ˇ ˇ Bp kn ˇ ˇZ ˇ Z ˇ ˇ ˇ ˇ Cˇ kukn .t/  u.t /kd.t /  kx  u.t /kd  .t; x/ˇ ˇ nBp ˇ E < "1 C "2 C 2"2 D "; from where

Z lim kukn  uk1 D  C n

E

kx  u.t /kd  .t; x/:

Remark 3.71. .i/ In the conditions of the previous theorem: (1) If .un /n is uniformly integrable, then Z kukn  uk1 ! kx  u.t /kd  .t; x/: E

(2) If .un /n is convergent in measure, then kukn  uk1 ! ..un //: 

! u and then Indeed, in this case, according to Proposition 3.40, un  . n M / D 0. (3) If .un /n is uniformly integrable and convergent in measure, then kk1

ukn ! u L1 .;E /

and we obtain again the Vitali’s Theorem 1.54, this time in the vector case.

244

Chapter 3 Young Measures

.ii/ Even if the sequence .un /n is SE -convergent to a Young measure  2 Y.E/, it is necessary to extract a subsequence. Indeed, if  D Œ0; 1,  is the Lebesgue measure on  and if un W  ! R is defined by u2n D 2n 1 and u2nC1 D 0;

Œ0; 2n 

SR



8n 2 N, then un  ! 0 and un !  ˝ ı0 2 Y.R/,

M D  and ..un // D 1 but limn kun k1 does not exist. R .iii/ limn kukn  uk1 D ..un // C . E /nGu kx  u.t /kd  .t; x/; where Gu is the graph of u (see Proposition 3.17 and Theorem 3.18). We can show that .Gu0 / D supa>0 lim supn .kun  uk  a/: Therefore, the previous formula highlights a decomposition of limn kukn  uk1 in a part that can be called “weak” and one that can be called “in measure”. R  ! u ” E kx  u.t /kd  .t; x/ D 0: .iv/ u1n  Example 3.72. Let un W Œ2; 2 ! R be defined by un .t/ D n

1 Πn ;0

.t/ C sin nt  0;2 .t /:

In this case A is the family of all Lebesgue measurable subsets and  is the Lebesgue measure on  D Œ2; 2. SR

According to Proposition 3.40 and to (ii) of Remark 3.47, un !



where,

8A 2 A; B 2 BR ,

 .A  B/ D  .A \ Œ2; 0/  ı0 .B/ 1 C   .A\0; 2/ 

Z

B\Œ1;1

p

1 1  x2

d.x/:

The disintegration of  , . t / t 2 , is defined by ´ ı0 .B/; t 2 Œ2; 0; R  t .B/ D 1 p 1 d.x/; t 20; 2Œ:  B\Œ1;1 2 1x

u.t/ D bar. t / D 0; for all t 2 Œ2; 2. Bp D Œ p1 ; 0;

8p 2 N.

w

nB  un ! 0 and ..un // D 1. M D Œ2; 0 and n

L1 .;R/

Z lim kun k1 D 1 C n

R

Z jxjd  .t; x/ D 1 C 2

0

2

Z

1 0

x d  t .x/ d.t / D 5:

In the Euclidean case, we obtain Theorem 9 of [169] as a corollary of Theorem 3.70.

245

Section 3.6 Biting Lemma

Corollary 3.73. Let .un /n  L1 .; Rd / be a sequence weakly convergent to u 2 L1 .; Rd /; then, we can find a subsequence .ukn /n and a Young measure  2 Y.Rd / such that: SRd

(a) ukn !  ; (b) almost for every t 2 ;  t has a barycenter, bar. t / D bar. t / D u.t /; R (c) limn!1 kukn  uk1 D Rd kx  u.t /kd  .t; x/.

R Rd

xd  t .x/ and

The following result is a consequence of (5) of Theorem 3.70. Theorem 3.74 ([105]-T.2.7.). Let .un /  L1 .; I R/ be a bounded sequence and let u 2 L1 .; ; R/ such that lim supn kun  uk1  ..un //. Then there 

exists a subsequence .ukn /n such that ukn  ! u and ..un // D limn kukn  uk1 . Now we extend the compactness results presented for some non-bounded sets of measurable functions—the finite-tight sets. A finite-tight set localizes the great growths of a set of measurable functions on a finite family of sets of small measure. More precisely, we will give the following definition: Definition 3.75 ([81]). A subset H  M.E/ is said finite-tight set if, for every " > 0, there exist a compact K 2 KE and a finite subfamily Af  A with .A/ < ", for every A 2 Af , such that, for every u 2 H , there exists Au 2 Af with u1 .E n K/  Au . A sequence .un /n2N  M.E/ is a finite-tight sequence if the set H D ¹un W n 2 Nº is a finite-tight set. Proposition 3.76. Let  W 2 ! RC be the outer measure generated by  (for every A  ;  .A/ D inf¹.C / W C 2 A; A  C º). H  M.E/ is a finite-tight set if and only if, for every " > 0, there exist a compact K 2 KE S and a finite cover of H , ¹H1 ; : : : ; Hp º, such that, for every i 2 ¹1; : : : ; pº,  . u2Hi u1 .E n K// < ". In the case where H is countable, we can use the measure  instead of  . Proof. Let H  M.E/ be a finite-tight set; for every " > 0 there exist K 2 KE and a finite subfamily Af D ¹A1 ; : : : ; Ap º  A with .Ai / < ", for every i D 1; : : : ; p, such that, for any u 2 H , there exists Au 2 Af with u1 .E nK/  Au . For every i D 1; : : : ; p, let Hi D ¹u 2 H W u1 .E n K/  Ai º; then ¹H1 ; : : : ; Hp º is the required cover of H . Conversely, let K 2 KE and let ¹H1 ; : : : ; Hp º be a finite cover of H such that, for every i D 1; : : : ; p;  .[u2Hi u1 .E n K// < ". Then, for every i, there

246

Chapter 3 Young Measures

exists Ai 2 A such that [u2Hi u1 .E n K/  Ai and .Ai / < "; therefore Af D ¹A1 ; : : : ; Ap º is the finite subfamily of A required by Definition 3.75. Remark 3.77. .i/ If E is a Euclidean space then H  M.E/ is a finite-tight set if and only if, for every " > 0, there exist k > 0 and a finite subfamily Af  A with .A/ < ", for every A 2 Af , such that, for every u 2 H , there exists Au 2 Af with ¹t 2  W ku.t /k > kº  Au . Obviously, a bounded sequence in .L1 .; E/; k  k1 / is a finite-tight sequence. .ii/ According to Definition 3.75, for every u 2 H; .u1 .E n K// < "; thus any finite-tight set is a tight set. The following example shows that the reciprocal is not valid. P1 an;p  2p1 be the binary form of the natural Example 3.78. Let n D pD1 number n; thus, for every p 2 N  ; an;p 2 ¹0; 1º and there exists ln 2 N with an;ln D 1 and an;p D 0, for all p > ln . For all n; p 2 N  , we note kn;p D an;p C an;pC1  2 C    C an;2p1  2p1 I and, for every p > ln ; kn;p D 0. then 0  kn;p  2p  1; kn;ln D an;ln D 1P 1 For any n 2 N, let un W0; 1Œ! R; un D pD1 p  kn;p kn;p C1 : 

Therefore, for every t 20; 1Œ, un .t/ D

ln X pD1

p

i

kn;p kn;p C1 ; 2p 2p

h .t/

C

1 X pDln C1

2p

;

2p

Œ

p  1 .t / 2 RI 0; 2p Œ

thus un is well defined, for every n 2 N. Proposition 3.79. .un /n  M.R/ is a tight sequence but is not a finite-tight one. R1 P1 p 21p D 2; so .un /n is a bounded Proof. For every n 2 N; 0 jun .t/jdt D pD1 sequence in L1 . 0; 1Œ; R/ and then it is tight (see Proposition 3.56). Let " D 12 ; to show that .un /n is not finite-tight it is enough to notice that, for every k > 0, for every q 2 N and all A1 ; : : : ; Aq 2 A with .Ai / < "; 8i D 1; : : : ; q, there exists n 2 N such that .jun j > k/ ª Ai ; 8i D 1; : : : ; q. Let k > 0; q 2 N and A1 ; : : : ; Aq 2 A arbitrarily such that, for any i D k kn;p C1 1; : : : ; q, .Ai / < 12 ; fix p 2 N; p > k.  2n;p p ; 2p Œ .jun j > k/; for all n 2 N.

247

Section 3.6 Biting Lemma

Since .A1 / < 12 , there exists i1 2 ¹0; : : : ; 2p  1º such that  2i1p i12C1 p Œª A1 . Let n1 D 2p1  i1 ; then kn1 ;p D i1 and, according to   kn1 ;p kn1 ;p C 1 i1 i1 C 1 .jun1 j > k/ D p; ; ; 2p 2p 2 2p we can deduce that .jun1 j > k/ ª A1 : 2 i2 C1 Since .A2 / < 12 , there exists i2 2 ¹0; : : : ; 22p  1º such that  2i2p Œª A2 . 22p Let n2 D n1 C 22p1  i2 ; then kn2 ;p D i1 and kn2 ;2p D i2 . According to   i2 i2 C 1 i1 i1 C 1 .jun2 j > k/ p ; [ 2p ; 2p ; 2 2p 2 2 we deduce that .jun2 j > k/ ª A1 and .jun2 j > k/ ª A2 . Let us continue this reasoning; since .Aq / < 12 , there exists iq 2 q1 iq i C1 ; 2qq1 p Œª Aq . Let nq D n1 C : : : C ¹0; : : : ; 22 p  1º such that  2q1 p nq1 C 22

q1 p1

2

2

 iq ; then knq ;p D i1 , knq ;2p D i2 ; : : : ; knq ;2q1 p D iq and     i1 i1 C 1 iq C 1 i2 i2 C 1 iq junq j > k p ; ; [ 2p ; 2p [    [ : 2 2p 2 2 22q1 p 22q1 p

Therefore, for every i D 1; : : : ; q, .junq j > k/ ª Ai . In the following, we will present some sufficient conditions for a set to be finitetight. Proposition 3.80. Let E be a separable Banach space. .i/ Every finite set H  M.E/ is a finite-tight set. .ii/ Let H D ¹un W n 2 Nº  M.E/ be a tight set; if H satisfies one of the following two conditions: 9K0 2 KE

such that

1 X

.u1 n .E n K0 // < C1;

(S)

nD0 1 8K 2 KE ; 9nK 2 N such that u1 nC1 .E n K/  un .E n K/; 8n  nK ; (M)

then H is finite-tight.

248

Chapter 3 Young Measures

Proof. (i) For every u 2 M.E/, ıu1 W BE ! RC , defined by .ıu1 /.C / D .u1 .C //, for every C 2 BE , is a -additive measure; since E is a Polish space,  ı u1 is a Radon measure and then, for every " > 0, there exists K 2 KE such that .u1 .E n K// < ". Therefore, every finite set H  M.E/ is finite-tight. (ii) Let H be a tight set satisfying condition .S /; for every " > 0, there exists K 2 KE such that .u1 n .E n K// < ", for all n 2 N; obviously, we can suppose that K K0 . Let us note, for every n 2 N, An D u1 n .E Pn1K/. According to condition .S /, there is p 2 N such that nDp .An / < ". Therefore, the family Af D ¹A0 ; A1 ; : : : ; Ap1 ; [1 nDp An º satisfies the conditions of Definition 3.75. If H satisfies the condition .M /, then the family Af D ¹A0 ; A1 ; : : : ; AnK º satisfies also the conditions of 3.75. In the following examples, we will show that conditions .S / and .M / of the previous proposition are not necessary. Example 3.81. (1) Let  D Œ0; 1 and let A1 D 13 ; 23 Œ, A2 D 312 ; 322 Œ[ 372 ; 382 Œ, A3 D 313 ; 2 Œ[ 373 ; 383 Œ[ 19 ; 20 Œ[ 25 ; 26 Œ   be the sets extracted from the interval 33 33 33 33 33 Œ0; 1 to obtain the Cantor set; we recall that An is an union of 2n1 intervals of length 31n every one. Since .An / ! 0, the set H D ¹un D 3n  A W n 2 n N  º  M.R/ is a tight set. We note that H is not bounded in L1 .Œ0; 1; R/. P P1 1 n 2 KR ; then 1 Let K0 D Œ1; nD1 .un .R n K0 // D nD1 .j3  P1 1

An j > 1/ D nD1 .An / D 1 < C1 and, according to condition .S / of previous proposition, H is finite-tight. However, H does not satisfy condition .M / of (ii). In fact, for K D Œ1; 1 and for all n 2 N  ; u1 n .R n K/ D An , even though the sets An are pairwise disjoint. (2) For every n 2 N  , let un W Œ0; 1 ! R; un D n2 

1 Œ0; n 

; then .un /n 

M.R/. The set H D ¹un W n 2 N  º is tight but it is not bounded in L1 .Œ0; 1; R/. For every K 2 KR , there exists nK 2 N such that n2 … K, for all n  nK . Then, for every n  nK , ² Œ0; n1 ; 0 2 K .R n K/ D I u1 n Œ0; 1; 0 … K therefore .u1 n .R n K//nnK is a decreasing sequence of sets, so that H satisfies the condition .M / and then H is finite-tight.

249

Section 3.6 Biting Lemma

However, condition .S / is not satisfied; in fact 1 X nD1

1  1 X 1  un .R n ¹0º/ D D C1: n nD1

Definition 3.82. Let E be a separable Banach space; a sequence .un /n2N  M.E/ is w2 -convergent to a measurable function u W  ! E if the following two conditions are satisfied: (a) There exists a decreasing sequence .Bp /p2N  A with .Bp / ! 0 and, for all p 2 N; .un /n  L1 . n Bp ; E/; u 2 L1 . n Bp ; E/. (b) For every p 2 N, . nB  un /n2N is weakly convergent to nB  u. p

p

w2

In this case we note un ! u. If .un /n2N is w 2 -convergent to u, then every subsequence of .un /n2N is w 2 convergent to u. Remark 3.83. This definition slightly extends Definition 1.95 in which u 2 L1 .; E/ and un 2 L1 .; E/, for every n 2 N. The following result is an alternative of the biting lemma (Theorem 3.68) for the finite-tight sets. Theorem 3.84 (Theorem 2.10 of [81]). Let E be a separable Banach space such that E and its dual space E  have the Radon–Nikodym property (particularly, let E be a reflexive normed space) and let .un /n2N  M.E/ be a finite-tight sequence; then there exist a subsequence .ukn /n2N and a measurable mapping w2

u 2 M.E/ such that ukn ! u. Proof. (I) Since .un /n is finite-tight, for every " > 0, there exist K 2 KE and a finite subfamily Af of A with .A/ < ", for every A 2 Af , such that, for every n 2 N, there exists An 2 Af with u1 n .E n K/  An . Then, there exist a set A 2 Af and an infinite subset N  N such that u1 n .E nK/  A; for any n 2 N . Therefore, for every " > 0, there exist K 2 KE ; A 2 A with .A/ < " and an infinite subset N  N, such that u1 n .E n K/  A, for every n 2 N . For " D 1 let K1 2 KE ; B1 2 A with .B1 / < 1 and let N1 be an infinite subset of N such that u1 n .E n K1 /  B1 , for every n 2 N1 . For " D 12 let K2 2 KE ; B2 2 A with .B2 /  12 and let N2 be an infinite subset of N1 such that u1 n .E n K2 /  B2 , for every n 2 N2 ; obviously, we can choose K2 K1 and B2  B1 .

250

Chapter 3 Young Measures

Generally, for p 2 N  and " D p1 , let Kp 2 KE with Kp Kp1 , let Bp 2 A with Bp  Bp1 and .Bp / < p1 and let Np be an infinite subset of Np1 such that u1 n .E n Kp /  Bp , for every n 2 Np . Let now np 2 Np with np > np1 , for every p 2 N  ; .uni /i 2N is a subsequence of .un /n2N and, for every p 2 N and i  p; let ni 2 Np such that  u1 ni .E n Kp /  Bp . Then, for every p 2 N and every i  p, uni . n Bp /  Kp  E: Thus .uni jnBp /i p is uniformly bounded and then .uni jnBp /i 2N is bounded in L1 . n Bp ; E/ and absolutely continuous. For every A 2 A, with A   n Bp and .A/ > 0 and for every i  p, Z uni d 2 .A/  co.uni .A//  .A/  co.Kp /: A

R

Therefore ¹ A uni d W i 2 Nº is relatively weakly compact. According to Dunford’s theorem (see Theorem 1 of [56], pp. 101), .uni jnBp /i 2N is relatively weakly compact in L1 . n Bp ; E/, for every p 2 N. Using Eberlein– Šmulyan’s theorem, .uni jnBp /i 2N is relatively sequentially weakly compact in L1 . n Bp ; E/, for every p 2 N. (II) Let M1 be an infinite subset of N and let v1 2 L1 . n B1 ; E/ such that .uni /i 2M1 is weakly convergent to v1 . Let M2 be an infinite subset of M1 and let v2 2 L1 . n B2 ; E/ such that .uni /i 2M2 is weakly convergent to v2 in L1 . n B2 ; E/. Since  n B1   n B2 ; v2 D v1 almost everywhere on  n B1 . Generally, let Mp be an infinite subset of Mp1 and let vp 2 L1 . n Bp /, such that .uni /i 2Mp is weakly convergent to vp in L1 . n Bp ; E/; then vp D vp1 almost everywhere on  n Bp1 . For every p 2 N  , we can choose ip 2 Mp with ip > ip1 ; then .uniq /q2N is still a subsequence of .un /n and, for every p 2 N; .uniq /q2N is weakly convergent to vp in L1 . n Bp ; E/. 1 We define u W  ! E letting u D vp on  n Bp and u D 0 on \pD1 Bp . Then w2

u 2 M.E/ and uniq ! u. Now, let us give an alternative for the Saadoune–Valadier theorem (Theorem 3.70). Theorem 3.85 (Theorem 2.11 of [81]). Let E be a separable Banach space such that E and E  have the Radon–Nikodym property (particularly, this happens if E is a separable reflexive normed space) and let .un /n2N  M.E/ be a finite-tight

251

Section 3.6 Biting Lemma

sequence; there exist a subsequence .ukn /n2N and a Young measure  2 Y.E/ S

 . such that ukn ! If  W  ! PE is the disintegration of  , then, for almost every t 2 ,  t has a barycenter Z u.t / D bar. t / D xd  t .x/ 2 E: E

w2

The mapping u W  ! E is measurable .u 2 M.E//, ukn ! u and u.t / 2 co.Lsn .ukn .t///, for almost every t 2  (here, co.A/ is the closed convex hull of T1 ¹uki .t/ W i  pº). A and Lsn .ukn .t// D pD1 Proof. According to Theorem 3.84, there exist a subsequence .u0n /n2N w2

of .un /n2N and a mapping v 2 M.E/ such that u0n ! v. Let .Bp /p2N  A be a decreasing sequence with .Bp / # 0 such that, for every p 2 N, .u0n /n  L1 . n Bp ; E/; v 2 L1 . n Bp ; E/ and . nB  u0n /n2N p is weakly convergent to nB  v. From Prohorov’s theorem (Theorem 3.59), p there exist a subsequence .ukn /n2N of .u0n /n and a Young measure  such that S

w2

uk n !   ; obviously, ukn ! v. For every p 2 N, the mapping ‰p W   E ! R, defined by ‰p .t; x/ D

nB .t/  kxk, for every .t; x/ 2   E, is a positive l.s.c. integrand. Thus, p according to Proposition 3.34, Z Z ‰.t; x/d  .t; x/  lim inf kukn .t /kd.t / < C1 n

E

nBp

(.ukn /n is bounded in L1 . n Bp /; E/). Therefore  Z Z kxkd  t .x/ d.t / < C1: nBp

E

R So, for almost every t 2  n Bp , there exists up .t / D bar. t / D E xd  t .x/. up W  n Bp ! E is a measurable mapping. Since  n Bp   n BpC1 ; upC1 jnBp D up almost everywhere. Thus, we can define u W  ! E letting ² 1 bar. t /; if t 2 [pD1 . n Bp / and if there exists bar . t /; u.t/ D 0E ; otherwise. Then u 2 M.E/ and u 2 L1 . n Bp ; E/, for every p 2 N.

252

Chapter 3 Young Measures

Let us show that u D v. Since E  has the Radon–Nikodym property, is dual space of L1 .; E/ (Theorem 1, pp. 98, of [56]). Let g be an arbitrary function of L1 .; E  / and let p 2 N. w Since nB  ukn ! nB  v,

L1 .; E  /

p

L1 .;E /

Z

˝

p

˛ g.t/; ukn .t/ d.t / ! n!1

nBp

Z nBp

hg.t /; v.t /i d.t /:

(1)

Let ˆp W   E ! R; ˆ.t; x/ D nB  < g.t /; x >, for every t 2  and every p x 2 E; then ˆp is measurable in t and continuous in x. By Proposition 3.8, it is a Carathéodory integrand. Furthermore, for every t 2  and every n 2 N, ˇ  ˇ ˇ ˇ ˇˆp t; nB .t/  ukn .t/ ˇ  kgk1  nB .t /  kukn .t /kE p

p

and since . nB ukn /n is uniformly integrable, .ˆp .; nB ./ukn .///n is also p p uniformly integrable. According to Corollary 3.36, Z Z   ˆp t; nB .t/  ukn .t/ d.t / ! ˆp .t; x/d  .t; x/ or p



Z

˝

nBp

˛

g.t /; ukn .t/ d.t / !

By (1) and (2),

R

Z

Z nBp

E

< g.t/; v.t/ > d.t/ D 

Z Z xd  t .x/ d.t / D g.t/;

 hg.t /; xi d  t .x/ d.t /:

(2)

nBp

Z D

E

nBp

E

nBp

hg.t /; u.t /i d.t /:

Since g is arbitrary in the dual space, nB .t /  v.t / D nB .t /  u.t /, for p p almost every t 2  and for every p 2 N; hence v D u almost everywhere and w2

consequently ukn ! u. S

 Ls.ukn .t// D \1 ¹uk .t / W p  nº (see PropoRnD1 p R sition 3.43). Therefore u.t / D E xd  t .x/ D supp t xd  t .x/ 2 co.supp t /  coLs.ukn .t//, for almost every t 2 .  Since ukn !

 ; supp t

253

Section 3.7 Product of Young Measures

3.7

Product of Young Measures

In this section, we will introduce two products for Young measures.

3.7.1

Fiber Product

Let S and T be two regular Suslin spaces. According to (iv) of Theorem 2.23, S  T is a regular Suslin space; since S and T are hereditarily Lindelöf ((ii) of Theorem 2.21), S  T is hereditarily Lindelöf and then BS T D BS ˝ BT . For every  2 Y.S / and every  2 Y.T / we define  ˝ W  ! PS T letting . ˝ / t D  t ˝  t , for every t 2 , where : and : are the disintegrations of  and  . For every A 2 BS , every B 2 BT and every t 2 , . ˝ / t .A  B/ D  t .A/   t .B/: Therefore the mapping t 7! . ˝ / t .A  B/ is measurable and so the mapping t 7! . ˝ / t .C / is measurable, for every C 2 BS T . Therefore  ˝ is a Young measure on S T ; this mapping is the disintegration of a Young measure noted  ˝  2 Y.S  T /;  ˝  W A ˝ BS T ! RC and, N C, for every positive integrand ‰ W   .S  T / ! R  Z Z ‰.t; x; y/d. t ˝  t /.x; y/ d.t /: . ˝  /.‰/ D S T



If we suppose that  D  and that  D  v are the Young measures associated to the measurable applications u 2 M.S / and respectively v 2 M.T /, then, for every t 2 , . ˝ / t D ı.u.t /;v.t // . u

Definition 3.86. The measure  ˝  is called the fiber product of  and  . Let us note by SS ; ST and SS T the stable topologies on Y.S /; Y.T / and respectively on Y.S  T /. The following theorem has a special importance due to its multiple applications. The proof is that of [169]. Theorem 3.87 (fiber product lemma). Let H  Y.S / and L  Y.T / be two tight sets; then: (1) H ˝ L D ¹ ˝  W  2 H ;  2 Lº  Y.S  T / is tight. (2) Let . i /i 2I  H and . i /i 2I  L be two nets such that

i

SS

!  2 Y.S / and

i SST

ST

!  2 Y.T /:

If  D  u , where u 2 M.S /, then  i ˝  i !  ˝  :

254

Chapter 3 Young Measures

Proof. (1) Since H and L are tight, for every " > 0, there exists K 2 KS ; C 2 KT such that " ; 2 "  .  .T n C // < ; 2

 .  .S n K//
0, there exist K 2 KE and a finite subfamily Ef  E with .A/ < ", for every A 2 Ef , such that, for every u 2 H , there exists Au 2 Ef with u1 .A n K/  Au . A sequence .un /n  M.E/ is Jordan finite-tight if the set H D ¹un W n 2 Nº is Jordan finite-tight. Remark 3.92. .i/ Since E  A, every Jordan finite-tight set is finite-tight and therefore a tight set. .ii/ If E is a Euclidean space, then H  M.E/ is Jordan finite-tight if and only if, for every " > 0, there exist k > 0 and a finite subfamily Ef  E with .A/ < ", for every A 2 Ef , such that, for any u 2 H , there exists Au 2 Ef with ¹t 2  W ku.t /k > kº  Au .

262

Chapter 3 Young Measures

The following proposition gives a justification for the denomination Jordan finite-tight set. For every B   let J .B/ D inf¹.A/ W A 2 E; B  Aº be the Jordan outer measure of B; obviously, J .B/ D 0 if and only if B is a Jordan-negligible set. Proposition 3.93. H  M.E/ is a Jordan finite-tight set if and only if, for every " > 0, there exist a compact set K 2 KE and a finite cover of H , ¹H1 ; : : : ; Hp º, such that  J [u2Hi u1 .E n K/ < "; for every i D 1; : : : ; p: The demonstration is the same as that of Proposition 3.76. Theorem 3.94. Let E be a Euclidean space; for every H  M.E/ let ²  ³  N IH .1/ D t 2  W lim sup sup ku.s/k D C1 : s!t

u2H

A set H  M.E/ is Jordan finite-tight if and only if, for every " > 0, there exists a finite cover of H , ¹H1 ; : : : ; Hp º, such that, for any i D 1; : : : ; p,  J IHi .1/ < ": Proof. .H)/: Firstly, let us note that lim sups!t .supu2H ku.s/k/ D C1 is equivalent to the existence of two sequences .un /n2N  H and .sn /n2N   with sn ! t and kun .sn /k ! C1. Let us suppose that H  M.E/ is a Jordan finite-tight set; according to (ii) of Remark 3.92, for every " > 0, there exist k > 0 and a finite subfamily Ef  E with .A/ < ", for every A 2 Ef , such that, for all u 2 H , there exists Au 2 Ef with .kuk > k/  Au . For every A 2 Ef , let HA D ¹u 2 H W .kuk > k/  Aº; then ¹HA W A 2 Ef º is a finite cover of H . Let us show that, for every A 2 Ef ; IHA .1/  A. Indeed, for every t 2 IHA .1/; lim sups!t .supu2HA ku.s/k/ D C1 and thus there exist two sequences .sn /n2N   with sn ! t and .un /n2N  HA such that kun .sn /k ! C1; we can suppose that, for any n 2 N; kun .sn /k > k and therefore .sn /n2N  .kun k > k/  A. Since A is closed, t 2 A. Hence J .IHA .1// < ", for every A 2 Ef . .(H/: For every " > 0, let ¹H1 ; : : : ; Hp º be a finite cover of H such that, for any i D 1; : : : ; p; J .IHi .1// < " and let Ai 2 E such that IHi .1/  Ai and .Ai / < ". Obviously, we can suppose that every IHi .1/ is contained in the interior AVi of Ai . Then, for any i D 1; : : : ; p, there exists ki > 0 such that ku.t/k  ki , for every t 2  n AVi and every u 2 Hi (if, on the contrary, we

Section 3.8 Jordan Finite Tight Sets

263

suppose that there is a sequence .tn /n2N  n AVi and a sequence .un /n2N  Hi with kun .tn /k > n, for all n 2 N, then .tn /n2N has a subsequence .tkn /n2N N n AVi from where t 2 IH .1/; but IH .1/  AVi ). convergent to t 2  i i Let k D max¹k1 ; : : : ; kp º. For every u 2 H , there exists i 2 ¹1; : : : ; pº such that u 2 Hi ; thus .kuk > k/  .kuk > ki /  Ai . Remark 3.95. .i/ Let E be a Euclidean space and let u 2 M.E/; then, according to the previN W ous theorem, ¹uº is a Jordan finite-tight set if and only if Iu .1/ D ¹t 2  lim sups!t ku.s/k D C1º is a Jordan-negligible set. We can notice that Iu .1/ is the set of points in the vicinity of which u is not bounded. : : º be the set of all rational numbers of .ii/ Let Q\0; 1ŒD ¹q0 ; q1 ; : : : ; qn ; :P 0; 1Œ and let u W0; 1Œ! R; u D 1 nD0 n  ¹qn º . Then, according to (i) of Proposition 3.80, H D ¹uº is a finite-tight set. We remark that Iu .1/ D Œ0; 1. Indeed, for each t 2 Œ0; 1 and each n 2 N  , there exists kn  n such that jqkn  t j < n1 ; then, qkn ! t and u.qkn / D kn ! C1. Therefore, according to the previous remark, H is not a Jordan finite-tight set. We can notice that u D 0, almost everywhere and that H1 D ¹0º is a Jordan finite-tight set. This example is a special case of the following remark. .iii/ Let E be a Euclidean space; if the sequence .un /n  L1 .; E/ is bounded, then there exists a Jordan finite-tight sequence .vn /n such that, for every n 2 N, un D vn almost everywhere. Indeed, let k > 0 such that, for every n 2 N; kun k1  k. If we define ² un .t/; kun kE  k ; vn .t/ D 0E ; kun kE > k then .vn /n is an uniformly bounded sequence and therefore .vn /n is a Jordan finite-tight sequence. Obviously, for every n 2 N, vn D un almost everywhere. Particularly, any application of L1 .; E/ coincides almost everywhere with a Jordan finite-tight application.

264

Chapter 3 Young Measures

In conclusion, the property of being Jordan finite-tight can be seen as a property of the sets of measurable functions and not as a property of the sets of classes of functions equal almost everywhere. We will now present a sufficient condition that a set may be Jordan finite-tight. Proposition 3.96. Let   Rd be a measurable bounded set; for every A  , let ı.A/ D sup¹kt  skRd W t; s 2 Aº be the diameter of A. Every H  M.E/ satisfying the condition  8" > 0; 9K 2 KE tel que ı u1 .E n K/ < "; 8u 2 H; (ı) is a Jordan finite-tight set. Proof. Let a; b 2 Rd ; a D .a1 ; : : : ; ad /; b D .b1 ; : : : ; bd /; a < b such that Q   Œa; b D diD1 Œai ; bi I .Œa; b/ D .b1  a1 /    .bd  ad /. For every " > 0, let n 2 N such that 3d  .Œa; b/ < ": nd

(1)

Let  D ¹0; 1; : : : ; n  1ºd . For every i 2 ¹1; : : : ; d º and every j 2 ¹0; 1; : : : ; nº j i and, if  D .j1 ; : : : ; jd / 2 , let I D let us denote ai D ai C j  bi a n Qd ji ji C1 . i D1 Œai ; ai ¹I W  2 º is a partition of Œa; b; thus   [ 2 I . Furthermore .I 0 \ 1 d    bd a D .Œa;b/ . I 00 / D 0 if  0 ¤  00 and .I / D b1 a n n nd According to condition .ı/, there exists K 2 KE such that ² ³  1 bi  ai ı u .E n K/ < "1 D min W i D 1; : : : ; d : (2) n Q j 1 j C2 Let Ef D ¹ diD1 Œai i ; ai i  W .j1 ; : : : ; jd / 2 º; then Ef is a finite subfamily Q d j 1 j C2 < ", for every of E and, according to (1), . diD1 Œai i ; ai i / D 3 .Œa;b/ nd Qd ji 1 ji C2 ; ai  2 Ef . interval i D1 Œai For every function u 2 H for which u1 .E n K/ ¤ ;, there exists  D .j1 ; : : : ; jd / 2  such that u1 .E n K/ \ I ¤ ;. Q j 1 j C2 Then, according to (2), u1 .E n K/  diD1 Œai i ; ai i  2 Ef . Indeed, if t 0 D .t10 ; : : : ; td0 / 2 u1 .E n K/ \ I , then, for every t D .t1 ; : : : ; td / 2 1 u .E n K/ and every i D 1; : : : ; d , bi  ai j C2 D ai i and n bi  ai j j j 1 D ai i : ti > ti0  "1  ai i  "1  ai i  n j C1

ti < ti0 C "1  ai i

j C1

C "1  a i i

C

265

Section 3.8 Jordan Finite Tight Sets

Example 3.97. (1) Let Q\0; 1ŒD ¹q0 ; q1 ; : : : ; qn ; : : : º and, for every n 2 N  , let un D n2  

1 . Obviously, H D ¹un W n 2 N º is a tight set but it is not qn ;qn C n Œ

bounded in L1 . 0; 1Œ; R/. For every k > 0 and every n 2 N  , ² 2 ;;  n2  k; ¹t 20; 1ŒW jun .t/j > kº D  1 qn ; qn C n ; n > k: Then ı.jun j > k/  p1 , for every n 2 N  ; thus H D ¹un W n 2 N  º k satisfies condition .ı/ of the previous proposition and then H is a Jordan finite-tight set, so that a finite-tight set. P p 1 P D C1; thus H does not satisfy However, 1 nD1 .jun j > k/ D n k n condition .S / of Proposition 3.80; evidently, H does not satisfy condition .M / of the same proposition. (2) For every n 2 N  , let un W Œ0; 1 ! R; un D n 

2 n0

1 1 Œ0; n [Œ1 n ;1

; then, for

< ". Let K D Œ0; n0  2 KR every " > 0, there exists n0 2 N such that 1 1 and A D Œ0; n0  [ Œ1  n0 ; 1 2 E; then .A/ < ". For every n 2 N  , u1 n .R

n K/ D

u1 n .

² n0 ; C1ŒD

Œ0;

1 n

[ Œ1 

;; n  n0 ;  A: n > n0

1 n ; 1;

Thus H D ¹un W n 2 N  º is a Jordan finite-tight set. However, for every K 2 KR and n1 2 N with K  Œn1 ; n1 ,  for every n > n1 : ı u1 n .R n K/ D 1; The main result of this section states that a tight set H  W 1;1 .; Rm /, for which there exists a Jordan finite-tight set of gradients rH , is necessarily relatively compact in measure. At first, let us recall and clarify some notions and results. For every k 2 N, let us note by C k ./ the set of all functions f , k-times T k ./ and differentiable and with f .k/ continuous on ; let C 1 ./ D 1 C kD0 let Cc1 ./ be the subset of all functions  2 C 1 ./ with compact support. Definition 3.98. Let   Rd be a bounded open set. v D .vji / 1im 2 1j d

L1 .; Rmd / is the gradient of u D .ui /1i m 2 L1 .; Rm / if, for every

266

Chapter 3 Young Measures

i 2 ¹1; : : : ; mº, j 2 ¹1; : : : ; d º and every application  2 Cc1 ./, Z Z @ ui .t/  .t/d.t/ D  vji .t /  .t /d.t /: @t j If v 0 and v 00 are two gradients of u, then v 0 D v 00 almost everywhere. We will note the gradient of u by ru and then ru D .rui /1i m D .rj ui / 1im D .vji / 1im : 1j d

1j d

For every p  1, the Sobolev space W 1;p .; Rm / consists of all mappings u 2 Lp .; Rm / with ru 2 Lp .; Rmd /. For every H  W 1;p .; Rm / we write rH D ¹ru W u 2 H º and we say that rH is a gradient of H . Definition 3.99. For every mapping u W  ! Rm , let us denote ² ³ ku.t /  u.s/k L.u; / D sup W t; s 2 ; t ¤ s I kt  sk u is a Lipschitz mapping on  if and only if L.u; / < C1. We recall that, if  0  Rd is a bounded open convex set and if u 2 L1 .0 ; Rm / is a continuous mapping with ru D .rj ui / 1im 2 L1 . 0 ; Rmd /, then u is a Lipschitz application on  0 and

1j d

0 m X d X krj ui k2 1 L.u;  0 /  @ i D1 j D1

L

1 12 . 0 ;R/

A < C1:

Proposition 3.100 (Proposition 3.10 of [81]). Let   Rd be a bounded open convex set and let H  W 1;1 .; Rm /\C.; Rm / be a tight set. If, for every u 2 H , there exists a gradient ru such that rH D ¹ru W u 2 H º  L1 .; Rmd / is a Jordan finite-tight set, then H is also a Jordan finite-tight set. Proof. Let us suppose that, for every u 2 H , there exists a gradient ru such that rH D ¹ru W u 2 H º is a Jordan finite-tight set; we notice that, for any other gradient vu of u, vu D ru almost everywhere, but, according to (ii) of Remark 3.95, we cannot assert that the set ¹vu W u 2 H º is also a Jordan finitetight set.

267

Section 3.8 Jordan Finite Tight Sets

For every " > 0, let k > 0 and let Ef D ¹A1 ; : : : ; Ap º  E be a finite family of elementary sets, with .Ai / < ", for every i 2 ¹1; : : : ; pº, such that, for all u 2 H , there exists i 2 ¹1; : : : ; pº with ¹t 2  W kru.t /k > kº  Ai . For any i 2 ¹1; : : : ; pº, we can find a finite family of open convex sets ¹ji W Spi ji ; let ı D min¹.ji / W 1  i  j D 1; : : : ; pi º such that  n Ai D j D1 p; 1  j  pi º > 0. Since H is tight, there exists k1 > 0 such that .kuk > k1 / < ı;

for every u 2 H:

(1)

Fix an arbitrary u 2 H ; then u 2 L1 .; Rm / \ C.; Rm /. Let i 2 ¹1; : : : ; pº such that .kruk > k/  Ai ; then, for any j 2 ¹1; : : : ; pi º and any t 2 ji ; kru.t /k  k from where ru 2 L1 .ji ; Rmd /. Therefore u is a Lipschitz function on ji and 0 L.u; ji /  @

m X d X

aD1 bD1

1 12 krb ua k2L1 . i ;R/ A 

p

j

md  k:

(2)

Since .ji /  ı, by (1), ji ª .kuk > k1 /; then there exists t0 2 ji with ku.t0 /k  k1 :

(3)

According to (2) and (3), for every t 2 ji ,

p ku.t/k  ku.t /  u.t0 /k C ku.t0 /k  md  k  kt  t0 k C k1 p  md  k  diam./ C k1 k2 :

Since j is arbitrary in ¹1; : : : ; pi º, ku.t /k  k2 ;

for every t 2

pi [

ji :

(4)

j D1

By (4), .kuk > k2 /  Ai and therefore H is a Jordan finite-tight set. Remark 3.101. .i/ According to the demonstration of previous proposition, if H is a tight subset of W 1;1 .; Rm /, for which rH Jordan finite-tight, then, for every " > 0, there exist k > 0 and a finite family of elementary sets Ef D ¹A1 ; : : : ; Ap º with .Ai / < " for any i 2 ¹1; : : : ; pº, such that, for every u 2 H , there exists i 2 ¹1; : : : ; pº with ¹t 2  W ku.t /k > kº  Ai

and ¹t 2  W kru.t /k > kº  Ai :

268

Chapter 3 Young Measures

.ii/ We cannot remove the condition of continuity for the applications of H in 1;1 the previous P1 proposition. Indeed, if H D ¹uº  W . 0; 1Œ; R/, where u D nD0 n  ¹q º is the function of Remark 3.95(ii), then H is tight and n rH D ¹0º is Jordan finite-tight but H is not a Jordan finite-tight set. .iii/ In the previous proposition, the condition of being tight for H is a necessary condition. Indeed, if, for every n 2 N, un D n is the constant function n on 0; 1Œ, then H D ¹un W n 2 Nº  W 1;1 . 0; 1Œ; R/ \ C. 0; 1Œ; R/ and rH D ¹0º is a Jordan finite-tight set but H is not even a tight set. Theorem 3.102 (Theorem 3.12 of [81]). Let   Rd be a bounded open convex set and let H  W 1;1 .; Rm / \ C.; Rm / be a tight set such that there exists a Jordan finite-tight gradient rH ; then H is relatively compact in the topolgy of convergence in measure on M.Rm /. Proof. According to the previous proposition and to (i) of Remark 3.101, for every " > 0, there exist k > 0 and a finite family of d -dimensional intervals I D ¹I1 ; : : : ; Ip º such that, for every u 2 H , we can find a subfamily Iu  I with .[Iu / < " and .kuk > k/  [Iu ;

(1)

.kruk > k/  [Iu ;

(2)

where [Iu is the union of the intervals of the subfamily Iu  I. Q For every i 2 ¹1; : : : ; pº, let Ii D jdD1 Œaji ; bji  and let q 2 N be such that p (3) d  k  m < q  ": Let us show that H satisfies the conditions of Fréchet theorem of compactness in measure (see Theorem IV.11.1 of [62]). Q a b Let a1 ; : : : ; ad ; b1 ; : : : ; bd 2 Z be such that   jdD1 Œ qj ; qj . For all j 2 r

¹1; : : : ; d º, we note tj1 ; : : : ; tj j the elements of the set ³ ² bj  1 bj 1 aj aj C 1 d 1 d ; ;:::; ; ; aj ; : : : ; aj ; bj ; : : : ; bj q q q q r

so that tj1 < tj2 <    < tj j : Let N D ¹1; : : : ; r1  1º      ¹1; : : : ; rd  1º  N d and, for every n D .n1 ; : : : ; nd / 2 N , let An D Œt1n1 ; t1n1 C1 Œ     Œtdnd ; tdnd C1 Œ; then ¹ \ An W n 2 N º

is a partition of :

(4)

269

Section 3.8 Jordan Finite Tight Sets

For every u 2 H , let Fu D .[Iu / [ .[n2N @.An //, where @.An / is the boundary of An ; then .Fu / D  .[Iu / < ";

for every u 2 H

(5)

and, according to (1) sup ku.t /k  k;

t 2 nFu

for every u 2 H:

(6)

We can notice that, for every n 2 N and every u 2 H , . \ An / n Fu is either empty or equal to the open convex set  \ AVn . Therefore, according to (2), ru 2 L1 .. \ An / n Fu ; Rmd /, u is a Lipschitz function on . \ An / n Fu and p L.u; . \ An / n Fu /  md  k; for every n 2 N and every u 2 H: (7) Thus, for every n 2 N , every u 2 H and every t; s 2 . \ An / n Fu , p p ku.t /  u.s/k  md  k  kt  sk  md  k  diam.An / p p p d m  dk D  md  k  q q and, according to (3), sup

ku.t /  u.s/k < ":

(8)

t;s2. \An /nFu

The result comes from (4), (5), (6) and (8) by applying Theorem IV.11.1 of [62]. Under certain conditions, we can avoid the continuity hypothesis in the previous theorem. Corollary 3.103. Let   Rd be a bounded open convex set, let p  1 and let H be a tight subset of W 1;p .; Rm / such that rH is a Jordan finite-tight set; if either p > d or d D 1 is fulfilled, then H is relatively compact in measure. Proof. In both cases, p > d or d D 1, for every u 2 H , there exists uN 2 W 1;p .; Rm / \ C.; Rm / such that u D uN almost everywhere. Then HN D ¹uN W u 2 H º  W 1;1 .; Rm / \ C.; Rm / is a tight set for which there exists a Jordan finite-tight gradient r HN D rH . Consequently, HN , and then H , is relatively compact in measure.

270

Chapter 3 Young Measures

Remark 3.104. Let   Rd be a bounded open convex set and let H  W 1;1 .; Rm / \ C.; Rm / be tight such that rH  .L1 .; Rm /; k  k1 / is bounded. According to (iii) of Remark 3.95, for every sequence .un /n  H , there exists .vn /n  L1 .; Rm / such that, for every n 2 N, run D vn almost everywhere and ¹vn W n 2 Nº is a Jordan finite-tight set. Then, according to previous theorem, H is relatively compact in measure. But, in this case, we can obtain a stronger result. Following the Arzelà–Ascoli theorem, H is relatively compact in the topology of uniform convergence on C.; Rm /. In fact, H is uniformly Lipschitz and therefore it is uniformly bounded and uniformly equicontinuous. Under some relaxed conditions, we can obtain an alternative of the Rellich– Kondrachov theorem (see Theorem 12.12 in [50]). Corollary 3.105. Let   Rd be a bounded open convex set and let .un /n2N  W 1;1 .; Rm / \ C.; Rm / be a uniformly integrable sequence with the sequence of gradients, .run /n2N , Jordan finite-tight; then, up to a subsequence, .un /n2N converges in L1 .; Rm /. Proof. .un /n being uniformly integrable it is bounded in L1 .; Rm / and so .un /n2N is a tight sequence for which .run /n2N is a Jordan finite-tight sequence. Therefore .un /n2N has a subsequence .ukn /n2N convergent in measure to an application u. Since .un /n2N is uniformly integrable, u 2 L1 .; Rm / and L1

ukn ! u. The condition of being Jordan finite-tight offers a very good background for the application of the fiber product lemma; we can demonstrate it in the following corollary. Corollary 3.106. Let   Rd be a bounded open convex set and let H  W 1;1 .; Rm / \ C.; Rm / be a tight set such that rH is a Jordan finite-tight set; then, for every sequence .un /n  M.Rm /, there exist a subsequence .ukn /n of .un /n , a mapping u 2 M.Rm / and a Young measure  : 2 Y.Rmd / such that: .i/ .ukn /n is convergent in measure to u, S

 ; .ii/ rukn ! w2

.iii/ rukn ! bar : . .iv/ For every bounded l.s.c. integrand, ‰ W   Rm  Rmd ! R,

Section 3.9 Strong Compactness in Lp .; E/

271

Z

Z

Rmd

‰.t; u.t /; y/d  .t; y/  lim inf n



 ‰ t; ukn .t /; rukn .t / d.t /:

Proof. According to Theorem 3.102, H is relatively compact in measure in M.Rm / and, by 3.85, rH is sequentially S-compact in Y.Rmd /. Therefore, for every .un /n  M.Rm /, there exist a subsequence .ukn /n of .un /n , a mapping u 2 M.Rm / and a Young measure  : 2 Y.Rmd / such that (i), (ii) and (iii) are accomplished. According to fiber product lemma (Theorem 3.87),  S  ıu./ ˝ : ukn ; rukn ! which proves (iv). Such a result can be used in order to find relaxed solution for non-convex problems in the calculus of variations for which .un /n is a minimizing sequence.

3.9

Strong Compactness in Lp .; E /

In this paragraph, we will use the tool which is offered by the Young measures in order to study the strongly compact sets in L1 . Two main results are the Visintin–Balder theorem and the Rossi–Savaré theorem which may have, as corollaries, the Lions–Aubin and Gutman theorems.

3.9.1

Visintin–Balder’s Theorem

This theorem presents the sufficient conditions in which weak convergence leads to strong convergence. A first condition was given by Visintin in [171] without using Young measures. In [11], E. Balder demonstrated the same result in a more general hypothesis. In [169], we can find a detailed presentation of these results. Firstly, let us recall some definitions. Let A  Rd ; the closed convex hull of A—noted co A—is the intersection of all closed convex sets containing A. Let C  Rd be a convex set. We say that a point x 2 C is an extremal point of C if .x D .1  /y C z with  20; 1Œ and y; z 2 C / H) .y D z D x/: Finally, let usTrecall that, for every sequence .un /n  M.Rd / and every t 2 , 1 Lsn ¹un .t/º D pD1 ¹uk .t/ W k  pº (the Kuratowski upper limit). Proposition 3.107 (Proposition C of [5]). If the sequence .un /n is weakly convergent in L1 .; Rd / to u 2 L1 .; Rd /, then u.t / 2 co .Lsn ¹un .t/º/ ;

for almost every t 2 :

272

Chapter 3 Young Measures

Proof. Since .un /n is weakly convergent, according to Dunford–Pettis theorem (Theorem 1.65), the set ¹un W n 2 Nº is bounded in L1 .; Rd / and, according to Proposition 3.56, it is tight. By Theorem 3.64, there exists a subsequence .ukn /n of .un /n S-convergent to a Young measure  2 Y.Rd /; let ¹ t W t 2 º be Rthe disintegration of  . According to Proposition 3.37, for every t 2 ; u.t / D Rd xd  t .x/. Let now the mapping idRd W Rd ! Rd ; idRd .x/ D x, for every x 2 Rd ; then, for every t 2 , u.t / 2 co .idRd .supp t // D co .supp t /. The result follows from Proposition 3.43: supp t  Lsn ¹ukn .t/º  Lsn ¹un .t/º;

almost for every t 2 :

Remark 3.108. Let .un /n  L1 .; Rd / and u 2 L1 .; Rd /. According to Proposition 3.43, if .un /n is stably convergent to  2 Y.Rd /, then, almost for every t 2 , supp t  Ls¹un .t/º and, according to previous theorem, if .un /n is weakly convergent to u, u.t / 2 co Lsn ¹un .t /º, almost for every t 2 . Generally, u.t / … Lsn ¹un .t/º. Indeed, in Example 3.44, we have shown that the Rademacher’s sequence .rn /n  L1 .Œ0; 1; R/ is stably convergent to  D  ˝ . 12 .ı1 C ı1 // and weakly convergent to 0. For every t 2 Œ0; 1; 0 … Lsn ¹rn .t/º D ¹1; 1º, but 0 2 co.Lsn ¹rn .t/º/ D Œ1; 1. Lemma 3.109 (Lemma 6.20 of [15]). Let E be a separable Banach space and let R 2 PE ( is a probability on E) such that there exists x0 D bar D E xd .x/ 2 co .supp /  E. If x0 is an extremal point of co .supp /, then D ıx0 ( is the Dirac measure— the unit mass concentrated in x0 ). Proof. Let B be a closed convex subset of E such that x0 … B; then supp ª B and therefore .B/ < 1. If we suppose that .B/ > 0, we can define 1 ; 2 2 PE letting 1 .A/ D

1 1  .A \ B/ and 2 .A/ D  .A n B/; for every A 2 BE : .B/ 1  .B/

Then D .B/  1 C Œ1  .B/  2 and therefore Z xd D .B/  bar 1 C Œ1  .B/  bar 2 : x0 D E

Since bar 1 2 co .supp 1 /  co .supp /, bar 2 2 co .supp 2 /  co .supp / and x0 is an extremal point of co .supp / it results that x0 D bar 1 D bar 2 . Then x0 2 co.supp 1 /  co.B/ D B, who contradicts the hypothesis.

Section 3.9 Strong Compactness in Lp .; E/

273

Therefore, for every closed convex set B  E for which x0 … B, .B/ S D 0. Let S  E be an open ball of radius r centered at x; then S D n> 1 Bn , r

where, for all n 2 N  , Bn is the closed ball of radius r  n1 centered at x. If x0 … S; x0 … Bn , for every n > 1r , therefore .Bn / D 0 and then .S / D 0. Since every open subset D of E is a countable union of open balls, we have .D/ D 0, for every open set D for which x0 … D. It follows that .En¹x0 º/ D 0 and then D ıx0 . We can now give the Visintin–Balder theorem. Theorem 3.110 (Theorem 10 of [169]). Let .un /n  L1 .; Rd / be a sequence weakly convergent to u 2 L1 .; Rd / and let Ecc be the family of all closed convex subsets of Rd . Let us consider the following conditions: (V) There exists a set-valued mapping  W  ! Ecc such that, for every n 2 N and almost for every t 2 , un .t/ 2 .t / and u.t / 2 .t / is an extremal point of .t/. (B) u.t/ is an extremal point of co.Lsn ¹un .t /º/, almost for every t 2 . (C) .un /n is strongly convergent to u in L1 .; Rd /. Then (V) H) (B) H) (C).

T1 Proof. (V) H) (B): Since Lsn ¹un .t/º D pD1 ¹un .t / W n  pº  .t /, for almost every t 2  and since .t/ is a closed convex set, co .Lsn ¹un .t/º  .t/;

almost for every t 2 :

(1)

w

Since un ! u, according to Proposition 3.107, L1

u.t / 2 co .Lsn ¹un .t/º/;

almost for every t 2 :

(2)

By (1) and (2), because u.t / is an extremal point of .t / it is an extremal point of co .Lsn ¹un .t/º/, all the more. (B) H) (C): Since .un /n is weakly convergent it is tight and, according to Theorem 3.64, every subsequence .u0n /n of .un /n has a subsequence .u00n /n Sconvergent to a Young measure  2 Y.Rd /; letR : be the disintegration of  . According to Proposition 3.37, u.t / D bar  t D Rd xd  t .x/, for every t 2 . According to Proposition 3.43, supp  t  Lsn ¹u00n .t /º. In accordance with hypothesis (B), almost for every t 2 , u.t / D bar  t is an extremal point of co .Lsn ¹un .t/º and, since bar  t 2 co .supp  t /  co .Lsn ¹u00n .t /º/  co .Lsn ¹un .t /º/;

274

Chapter 3 Young Measures

bar  t is an extremal point of co .supp  t /, almost for every t 2 . From Lemma 3.109 it follows that  t D ıu.t / , almost for every t 2 ; therefore  D  u is the Young measure associated to the mapping u 2 L1 .; Rd / and, according to Proposition 3.40, .u00n /n is convergent in measure to u. According to Vitali theorem (Theorem 1.54), .u00n /n is strongly convergent to u. Therefore, every subsequence .u0n /n of .un /n has a subsequence .u00n /n strongly kk1

convergent to u; then un ! u. Remark 3.111. .i/ Condition (V) is introduced by Visintin in [171] where it is presented in the equivalent form: (V0 ) u.t / is an extremal point of co ¹¹u.t /º [ ¹un .t / W n 2 Nºº, almost for every t 2 . .ii/ The following example (Example 1 of Section 3 of [169]) shows that the Balder’s condition (B) (and also Visintin’s condition (V) much less) is not necessary for the strong convergence. For every m 2 N  and every k D 1; : : : ; m, let vn W Œ0; 1 ! R, vn D

Œ

k1 k m ;m



;

if

nD

m.m  1/ C k: 2

Let un W Œ0; 1 ! R; u2n D vn , and u2nC1 D vn . Then .un /n is strongly convergent to 0 but 0 is not an extremal point of co .Lsn ¹un .t /º/ D Œ1; 1. .iii/ In [171], one shows that the previous theorem cannot be generalized by replacing Rd by a infinite-dimensional normed space.

3.9.2

Rossi–Savaré’s Theorem

In this subsection,  D 0; T Œ  R is provided with the -algebra A of Lebesgue measurable sets and with the Lebesgue measure  D dt . Let .E; k  k/ be a separable Banach space and let 1  p < C1. We recall that RT p p L .; E/ D ¹u 2 M.E/ W 0 kukE dt < C1º is a Banach space with respect to the norm ! p1 Z k  kp W Lp .; E/ ! RC ; kukp D

T

0

p

kukE dt

:

A sequence .un /n  Lp .; E/ is strongly convergent to u 2 Lp .; E/ if kkp

kun  ukp ! 0; the strong convergence is noted by un ! u.

Section 3.9 Strong Compactness in Lp .; E/

275

A sequence .un /n  M.E/ is convergent in measure to u 2 M.E/ if .¹t 2 0; T Œ W kun .t/  u.t /kE  "º/ ! 0, for every " > 0; the convergence in measure 

of the sequence .un /n to u with respect to  is noted by un  ! u. We introduce an important condition in the study of the strong convergence in p L .; E/—the strong concentration condition. Definition 3.112. A set U  Lp .; E/ satisfies to the strong concentration condition if Z T h p ku.t C h/  u.t /kE dt D 0: (CF) lim sup h#0 u2U

0

In the Euclidean case, this condition is used in order to characterize the strong compactness in Lp .; E/ (see Theorem IV.26 of [36]): Theorem 3.113 (Riesz–Fréchet–Kolmogorov theorem). Let 1  p < C1 and let U  Lp .; Rm / be a bounded set; U is relatively strongly compact in Lp .; Rm / if and only if U satisfies to the strong concentration condition. If U  W 1;p .; Rm /, then a sufficient condition for (CF) is that rU to be bounded in L1 .; Rm /; we obtain the following corollary: Corollary 3.114. Let U  W 1;p .; Rm / be a bounded set in W 1;p ; if rU D ¹ru W u 2 U º is bounded in L1 .; Rm /, then U is relatively strongly compact in Lp .; Rm /. Remark 3.115. In the infinite dimensional case, the condition (CF) is not sufficient for the strong compactness. Indeed, let E D CŒ0;1 be the space of all real-valued continuous functions on Œ0; 1 provided with the norm of uniform convergence and let, for every n 2 N, fn W 0; 1Œ ! R; fn .x/ D x n . For every n 2 N, let un W 0; T Œ ! CŒ0;1 be the constant mapping un .t / D fn , for every t 2 0; T Œ and let U D ¹un W n 2 Nº. For every n 2 N, kun k1 D inf¹c > 0 W kun .t/kE  c;

for almost every t 2 0; T Œ º D 1

and then U  L1 .; CŒ0;1 /  L1 .; CŒ0;1 /. Obviously, U satisfies to condition (CF) but U is not relatively strongly comRT pact in L1 .; CŒ0;1 / because kun  um k1 D 0 kun .t /  um .t /kCŒ0;1 dt D RT m m nm  nm ! 1 and then .un /n does not 0 kfn  fm kCŒ0;1 dt D T  . n / n n!1 have strongly convergent subsequences.

276

Chapter 3 Young Measures

In the infinite dimensional case, in order to obtain the strong compacity, we need to add to the strong concentration condition the condition of being tight. We have already noticed in 3.57 that the set U defined above is not tight. Definition 3.116. A set U  Lp .; E/ is called p-uniformly integrable if Z p kukE dt D 0: (p-UI) lim sup a!C1 u2U

.kukE a/

U is called p-equicontinuous if lim

Z sup

.A/!0 u2U

A

p

kukE dt D 0:

(p-E)

A sequence .un /n  Lp .; E/ is p-uniformly integrable (p-equicontinuous) if the set ¹un W n 2 Nº is p-uniformly integrable (p-equicontinuous). Remark 3.117. p

.i/ U is p-uniformly integrable if and only if Up D ¹kukE W u 2 U º  L1 ./ is uniformly integrable (see Definition 1.85 and (iii) of Theorem 1.84). U is p-equicontinuous if and only if Up is equicontinuous in L1 ./ (see (ii) of Remark 1.53 and Definition 1.52). .ii/ Since  is atomless, then, according to Saks’ theorem (Th. I.4.10 of [173]), every p-equicontinuous set is bounded in Lp .; E/ and then Up is bounded in L1 ./. According to Theorem 1.84, U  Lp .; E/ is p-uniformly integrable if and only if it is p-equicontinuous. .iii/ Every p-uniformly integrable set U  Lp .; E/ is bounded in Lp . Indeed, if U is p-uniformly integrable, there exists a > 0 such that Z 1 p kukE dt < 1I then sup kukp < .1 C ap  T / p : sup u2U

.kukE a/

u2U

.iv/ If 1  p < q < C1 and if U is bounded in Lq .; E/  Lp .; E/, then U is p-uniformly integrable. Indeed, if supu2U kukq  M , then, for every A 2 A, Z  pq Z qp qp p q q kukE dt  ..A//  kukE dt  ..A// q  M p : sup u2U

A

A

Therefore U is p-equicontinuous and, according to (ii), it is p-uniformly integrable.

Section 3.9 Strong Compactness in Lp .; E/

277

We now give the vector version of the Vitali theorem (Theorem 1.54). Theorem 3.118 (Theorem III.3.6 of [62]). Let .un /n  Lp .; E/ and u 2 kkp

M.E/. Then u 2 Lp .; E/ and un ! u if and only if: 

.i/ un  ! u and .ii/ .un /n is p-uniformly integrable. kkp

Proof. Firstly, let us suppose that u 2 Lp .; E/ and un ! u. p

Since, for every " > 0; .kun ukE  "/  "1p kun ukp ; .un /n is convergent in measure to u. Let us show that the set ¹un W n 2 Nº is a p-equicontinuous set. For every " > 0, there exist n0 2 N and ı > 0 such that, for all A 2 A such that .A/ < ı, (1) kun  ukp < ", for every n  n0 , R p (2) A kukE dt < "p and R p (3) A kun kE dt < ", for every n < n0 . According to (1) and (2), for every n  n0 and A 2 A with .A/ < ı, R R R 1 1 p p p (4) A kun kE dt  Œ. A kun  ukE dt/ p C . A kukE dt / p p < .2"/p and then, by (3) and (4), .un /n is p-equicontinuous. According to (ii) of previous remark, .un /n is p-uniformly integrable. Reciprocally, let us suppose that conditions (i) and (ii) are satisfied. p p For every n 2 N, let vn D kun kE 2 L1 ./ and let v D kukE 2 M.R/. According to (i) of Remark 3.117, .vn /n is uniformly integrable. Every subsequence .ukn /n of .un /n has a subsequence .ulkn /n convergent to u, p p almost everywhere. Then .kulkn kE /n is convergent to kukE , almost everywhere p



p

! kukE . and, since ./ D T < C1; kulkn kE  

! v. Therefore vn  kk1

According to Vitali’s theorem (Theorem 1.54), v 2 L1 ./ and vn ! v. Then u 2 Lp .; E/ and Z T 0

Z

p

kun kE dt ! p

T 0

p

kukE dt: p

(*) p

For every n 2 N, let wn D 2p1 .kukE C kun kE /  ku  un kE  0; then .wn /n  L1 ./. Every subsequence .ukn /n of .un /n has a subsequence .ulkn /n convergent to p u, almost everywhere and therefore lim inf n!C1 wlkn D 2p  kukE .

278

Chapter 3 Young Measures

By applying the Fatou’s lemma to the sequence .wlkn /n and by using ( ), we obtain Z T Z T p 2p  kukE dt  lim inf wlkn dt n!C1 0 0 " Z T Z T p p kukE dt C 2p1  kulkn kE dt D lim inf 2p1 n!C1

0

Z  Z D 2p 

0

T

T 0

#

0

p

ku  ulkn kE dt Z

p

kukE dt  lim sup n!C1

T 0

p

ku  ulkn kE dt;

kkp

from where ulkn ! u. Therefore, every subsequence .ukn /n of .un /n has a subsequence .ulkn /n kkp

strongly convergent to u in Lp .; E/ and then un ! u. Corollary 3.119. A set U  Lp .; E/ is relatively strongly compact if and only if it is p-uniformly integrable and relatively compact in measure. We have noted in (iii) of Remark 3.117 that each p-uniformly integrable set is bounded in Lp .; E/. Reciprocally, a bounded set that satisfies to the strong concentration condition is p-uniformly integrable. Proposition 3.120. Let U be a bounded subset of Lp .; E/ which satisfies to the strong concentration condition; then U is p-uniformly integrable. Proof. Since, for every h > 0 and every t 2 T  h, j ku.t C h/kE  ku.t /kE j  ku.t C h/  u.t /kE , Z T h p j ku.t C h/kE  ku.t /kE j dt D 0; lim sup h#0 u2U

0

then V D ¹kukE W u 2 U º satisfies to the strong concentration condition in Lp ./. Since V is bounded in Lp ./, according to Theorem 3.113, V is relatively strongly compact in Lp ./. According to Corollary 3.119, V and then U are p-uniformly integrable. Since the strong concentration condition assures the condition .p  UI /, in order to obtain the strongly compactness of a set, it is sufficient, according to Corollary 3.119, to characterize the compactness in measure.

Section 3.9 Strong Compactness in Lp .; E/

279

For the following results, we will follow the line of reasoning of [144]. Firstly, we need two preliminary lemmas. Lemma 3.121. Let 0 < S < T; 1 D 0; S Œ  0; T ŒD  and A1 D A1 . For every Polish space P and every Young measure  . t / t 2 on   P , let  1 . t / t 2 1 2 Y.1  P /. For every h 2 0; T S Œ, the mapping  h W 1 ! PP ;  th D  t Ch , is .A1 C/measurable; it is the disintegration of a measure  h 2 Y.1  P /. S1

In addition,  h !  1 , where S1 is the stable topology on Y.1  P /. h!0

Proof. Firstly, let us remark that, for every C 2 BP , the mapping t 7!  th .C / is .A1  BP /-measurable. Indeed, for all B 2 BR , Œ:h .C /1 .B/ D h C Œ: .C /1 .B/ 2 A1 and then, according to Theorem 3.1, :h is .A1  C/-measurable. Therefore, for every h 2 0; T  S Œ, :h is the disintegration of a Young measure  h 2 Y.1  P /. The family of all finite disjoint unions of open intervals is an algebra which generated the -algebra A1 and then, according to Proposition 3.21, S

 h 1!  ,  h . A ˝ f / !  . A ˝ f /; h!0

h!0

8A Da; bŒ  0; S Œ;

8f 2 Cb .P /:

R Let now A D a; bΠ 11 and let f 2 Cb .P /; the function t 7! '.t / D P f .x/d  t .x/ belongs to L .; P / and then Z

 h . A ˝ f / D

D

A .t/  '.t C h/dt D

0

Z D

Z

S

S Ch

h Z b a

S Ch

h

Z

aCh;bChΠ.t/  '.t /dt D Z

'.t/dt C

Z

bCh

b

Because ˇ ˇZ ˇ ˇ bCh ˇ ˇ '.t/dt ˇ  kf k1  h and ˇ ˇ ˇ b it results immediately that  h . A ˝ f / !

'.t /dt 

hCA .t /  '.t /dt bCh

'.t /dt aCh aCh

'.t /dt: a

ˇZ ˇ ˇ aCh ˇ ˇ ˇ '.t /dt ˇ  kf k1  h; ˇ ˇ a ˇ Rb a

'.t /dt D  . A ˝ f /.

280

Chapter 3 Young Measures

Lemma 3.122. Let P be a Polish space, let  2 Y.  P / and let e W P  P ! Œ0; C1 be a l.s.c. application which vanishes only on the diagonal of P , P D ¹.x; x/ W x 2 P º. The mapping ‰ W 0; T Œ P ! Œ0; C1 defined by Z ‰.t; x/ D e.x; y/d  t .y/; for every .t; x/ 2 0; T Œ P; P

is a positive l.s.c. integrand. Proof. Firstly, let us suppose that e is continuous and bounded on P  P ; then, since : is a Young measure, ‰ is measurable with respect to t and continuous x (xn ! x implies e.xn ; y/ ! e.x; y/ and then R R in the second variable e.x ; y/d  .y/ ! e.x; y/d  t .y/). n t P P According to Proposition 3.8, ‰ is a Carathéodory integrand. In the general case, since P  P is metrizable, let, for every n 2 N, en W P P ! Œ0; C1Œ be the Yosida’s transform of e; .en /n is an increasing sequence of Lipschitz mappings such that en " e (see Lemma 2.35). According to the Lebesgue monotone convergence theorem “ ‰n .t; x/ D en .x; y/d  t .y/ " ‰.t; x/: P P

.‰n /n is a sequence of Carathéodory integrands and so ‰ D supn ‰n is measurable and l.s.c. in the second variable. The following result characterizes the compactness in measure by adding a weak concentration condition to that of being tight. Theorem 3.123 (Theorem 2 of [144]). (I) Let P be a Polish space and let e W P  P ! Œ0; C1 be a l.s.c. application with the property: e.x; y/ D 0 if and only if x D y. A tight set U  M.P / which satisfies to the weak concentration condition with respect to e: Z T h lim sup e.u.t C h/; u.t //dt D 0 (CF’) h#0 u2U

0

is relatively compact in measure. (II) Conversely, if U  M.P / is relatively compact in measure, then U is tight and satisfies to the weak concentration condition .CF 0 / with respect to any bounded continuous metric e on P .

Section 3.9 Strong Compactness in Lp .; E/

281

Proof. (I) Let U  M.P / be a tight set which satisfies to the weak concentration condition with respect to a l.s.c. application e W P P ! Œ0; C1 which vanishes only on the diagonal P of P . Since the topology of the convergence in measure is metrizable, it is sufficient to show that U is sequentially compact in measure. Let so .un /n  U ; since U is tight, it results from Theorem 3.64 that .un /n has a subsequence (still noted .un /n ) S-convergent to a Young measure  2 Y.P /. If  is associated to a measurable mapping u, then, according to Proposition 3.40, 

! u. un  We still need to proof that there exists u 2 M.P / such that  D  u . Firstly, let us show that, for almost every t 2 , supp  t contains only one point u.t/; the demonstration has three stages: “ Z Z 1 h T s e.u; v/d. t Cs ˝  t /.u; v/ dt ds D 0: (A) lim h#0 h 0 0 P P Z T “ e.u; v/d  t .u/d  t .v/ dt D 0: (B) P P

0

supp. t /

contains only one point, almost for every t 2 .0; T /:

(C)

R T s

e.un .t C s/; un .t //dt: According to conR T h 8" > 0; 9ı > 0 such that, 8h 2 0; ıŒ; 8n 2 N; 0 e.un .t C dition h/; un .t//dt < ": 8h 2 0; ıŒ, 8s 2 0; h; 0 < s < ı; therefore Z T s e.un .t C s/; un .t //dt  ": sup (A) Let r.h/ D sups20;h supn

0

.CF 0 /,

n

0

Hence

Z

T s

sup sup s2 0;h n

0

e.un .t C s/; un .t //dt  "

and then lim r.h/ D 0:

(1)

h#0

R T s 0

e.un .t C s/; un .t//dt  r.h/;

8h > 0;

8s 20; h;

8n 2 N.

282

Chapter 3 Young Measures

By integrating the last inequality with respect to s 2 0; h, we obtain # Z h "Z T s e.un .t C s/; un .t//dt ds  h  r.h/; or 0

0

“

e.un .t C s/; un .t//dtds  h  r.h/;

8h > 0;

8n 2 N;

(2)

A.h/

in which we noted A.h/ D ¹.s; t/ 2 0; T Œ  0; T Œ W s 2 0; h; 0 < t  T  sº. After the change of variables .s; t/ 7! .s  t; t /, relation (2) becomes “ e.un .s/; un .t//dsdt  h  r.h/; 8h > 0; 8n 2 N; (3) B.h/

where B.h/ D ¹.s; t/ 2 0; T Œ  0; T Œ W 0 < t < T; t < s  t C hº. S0;T Œ0;T Œ

According to Theorem 3.90,  un ˝ un !  ˝ . Since ƒ.s; t; x; y/ D

B.h/ .s; t/  e.x; y/ defines a positive l.s.c. integrand, according to Proposition 3.34, P  /.ƒ/  lim inf . un ˝ P  un /.ƒ/: . ˝ n

Then, using (3), we obtain “ “ P P

B.h/

e.x; y/d. t ˝ s /.x; y/ dt ds

“

 lim inf

e.un .t/; un .s//dt ds  h  r.h/:

n

(4)

B.h/

After a reversed change of variables .s; t/ 7! .t C s; t /, we obtain “ “ e.x; y/d. t ˝ s /.x; y/ dt ds P P

B.h/

“

“

D A.h/

P P

e.x; y/d. t Cs

and, according to (4), Z h Z T s “ 0

0

P P

e.x; y/d. t Cs

˝  t /.x; y/ dt ds

˝  t /.x; y/ dt ds  h  r.h/:

The last inequality and (1) lead us to (A). (B) According to RLemma 3.122, the mapping ‰ W 0; T Œ P ! Œ0; C1 defined by ‰.t; x/ D P e.x; y/d  t .y/, for every .t; x/ 2 0; T Œ P , is a positive

Section 3.9 Strong Compactness in Lp .; E/

283

l.s.c. integrand. We can therefore write the condition of stage (A) in the following form: Z h Z T s Z 1 lim  ‰.t; x/d  t Cs .x/dt ds D 0: h#0 h 0 0 P By making the change of variables .s; t/ 7! .hs; t / and by applying the Fatou lemma, we obtain, for every " > 0, Z h Z T s Z 1 ‰.t; x/d  t Cs .x/dt ds (5) 0 D lim inf  h#0 h P 0 0 Z D lim inf h#0

1 Z T hs 0

Z  lim inf h#0

h#0

Z 

1

0

0

P

‰.t; x/d  t Chs .x/dt ds

1 Z T " Z 0

Z D lim inf

Z

0 1

0

P

‰.t; x/d  t Chs .x/dt ds

 hs . 0;T "Œ  ‰/ds

lim inf  hs . 0;T "Œ  ‰/ds: h#0

S

Lemma 3.121 assures us that  hs ! h!0



and, since ‰ is positive and l.s.c., we

can apply again Proposition 3.34 to obtain lim inf  hs . 0;T "Œ  ‰/   . 0;T "Œ  ‰/: h#0

By going back to condition (5), we can write Z Z 1 0  . 0;T "Œ  ‰/ds D  . 0;T "Œ  ‰/ D 0

Z

D

T " Z

Z

T " Z

‰.t; x/ t .x/dt 0

P

e.x; y/d  t .x/d  t .y/dt: 0

P

P

Since e  0 is arbitrary, we obtain condition (B) Z T “ e.x; y/d  t .x/d  t .y/ dt D 0: 0

P P

284

Chapter 3 Young Measures

(C) Condition (B) leads us to “ e.x; y/d. t ˝  t /.x; y/ D 0; P P

for a.e. t 2 0; T Œ

or . t ˝  t / .¹.x; y/ 2 P  P W e.x; y/ > 0º/ D 0;

for a.e.

t 2 0; T Œ;

which, according to the properties of e, comes back to supp. t ˝  t /  4P ;

for a.e.

t 20; T Œ:

If we assume that supp  t contains two distinct points, x; y 2 P , then, there are two open sets U; V  P such that x 2 U; y 2 V and U \ V D ;. Since x; y 2 supp  t ,  t .U / > 0 and  t .V / > 0; therefore . t ˝  t /.U  V / D  t .U /   t .V / > 0. On the other hand . t ˝  t /.U  V / D . t ˝  t /..U  V / \ 4P / D . t ˝  t /.;/ D 0 which is a contradiction. So, supp  t D ¹u.t /º, almost for every t 2 . The application u W 0; T Œ! R is therefore well defined. For every A 2 BP , the mapping gA W 0; T Œ ! R, defined by gA .t / D  t .A/, is measurable; so u1 .A/ D ¹t 2 0; T ŒW u.t / 2 Aº D ¹t 2 0; T ŒW gA .t / D 1º D gA1 .1/ 2 A. Then u 2 M.P / and  D  u . (II) Let U  M.P / be a set relatively compact in measure; therefore its closure UN is compact in measure in M.P / and, according to Proposition 3.41, UN is Scompact. So UN is tight (Proposition 3.65). Let now e W P  P ! RC be a bounded continuous metric and let M D supx;y2P e.x; y/ < C1. Let us show that U satisfies to the weak concentration condition with respect to e. RT For every u; v 2 M.P /, let ıe .u; v/ D 0 e.u.t /; v.t //dt . Then ıe is a pseudo-metric on M.P /; let us show that ıe is continuous with respect to the topology of convergence in measure on M.P /. Let .un /n  M.P / be a sequence convergent in measure to a function u 2 M.P /; we can suppose that .un .t//n is convergent to u.t / and then e.un .t/; u.t// ! 0, almost for every t 2 0; T Œ. Since e is bounded, according to RT the bounded convergence theorem, ıe .un ; u/ D 0 e.un .t /; u.t //dt ! 0. If .vn /n is an other sequence convergent in measure to v, then jıe .un ; vn /  ıe .u; v/j  ıe .un ; u/ C ıe .vn ; v/ ! 0 therefore ıe is continuous. Then .UN ; ıe / is a compact space. R T h For every h 2 0; T Œ, let fh W UN ! R; fh .u/ D 0 e.u.t C h/; u.t //dt:

Section 3.9 Strong Compactness in Lp .; E/

285

For every u; v 2 UN and every h 2 0; T Œ, Z T h je.u.t C h/; u.t//  e.v.t C h/; v.t //jdt jfh .u/  fh .v/j  0 Z T h Œe.u.t Ch/; v.t C h//Ce.u.t /; v.t // dt  2ıe .u; v/:  0

Therefore, the mappings fh are uniformly Lipschitz on UN and then A D ¹fh W h 2 0; T Œ º  C.UN ; R/ is an uniformly equicontinuous set. Moreover, for every h 2 0; T Œ, jfh .u/j  M.T  h/  M  T;

8u 2 UN

and then A is uniformly bounded. According to Arzelà–Ascoli theorem, A is relatively compact in C.UN ; R/ with respect to the topology of uniform convergence. Let us show that, for every u 2 UN ; limh!0 fh .u/ D 0. (a) If u W 0; T Œ! P is continuous, then limh!0 u.t C h/ D u.t /, for every t 2 0; T  hŒ and, since e is bounded, Z T h fh .u/ D e.u.t C h/; u.t //dt ! 0: h!0

0

(b) Let now u 2 UN be a measurable mapping; there exists a sequence of continuous maps .up /p such that up ! u, almost everywhere and then ıe .up ; u/ ! 0. For every " > 0, let p0 2 N such that ıe .up0 ; u/ < 4" . By (a), fh .up0 / ! 0 and then there exists ı > 0 such that fh .up0 / < for every h with 0 < h < ı,

" 2,

h!0

for every h < ı. Therefore,

fh .u/ D fh .u/  fh .up0 / C fh .up0 / Z T h " je.u.t C h/; u.t//  e.up0 .t C h/; up0 .t //jdt C  2 Z T h Z0T h " e.u.t C h/; up0 .t C h//dt C e.u.t /; up0 .t //dt C  2 0 Z T h Z0 T " e.u.t /; up0 .t//dt C e.u.t /; up0 .t //dt C D 2 h 0 " < 2ıe .up0 ; u/ C < ": 2 Since A D ¹fh W h 2 0; T Œ º is relatively compact with respect to the topology of uniform convergence and limh!0 fh .u/ D 0, for every u 2 UN , the limit is 0

286

Chapter 3 Young Measures

uniformly with respect to u 2 UN and so Z T h e.u.t C h/; u.t //dt D 0 lim sup h!0 u2U

0

which is exactly the condition of weak concentration with respect to e. This last theorem allows us to demonstrate the result of strong convergence due to Rossi and Savaré. Theorem 3.124 (Theorem 1 of [144]). Let E be a separable Banach space, let 1  p < C1 and let U  Lp .; E/ be a Lp -bounded set. If U is tight and if it satisfies to strong concentration condition Z T h p lim sup ku.t C h/  u.t /kE dt D 0; (CF) h#0 u2U

0

then U is relatively compact in Lp .; E/. p

Proof. Let e W E  E ! RC defined by e.x; y/ D kx  ykE , for every x; y 2 E. Then e is l.s.c. on E  E and e.x; y/ D 0 if and only if x D y. Since U is tight and, according to (CF), it satisfies to weak concentration condition with respect to e, the previous theorem assures us that U is relatively compact in measure. According to Proposition 3.120, U is p-uniformly integrable and therefore, according to Corollary 3.119, U is relatively strongly compact in Lp .; E/. We obtain some known compactness results as corollaries of Rossi–Savaré theorem. The following theorem is presented in §4 of Ch. IV of [114] for the Hilbert spaces and in [7] for the Banach spaces. Theorem 3.125 (Lions–Aubin theorem). Let E0 be a Banach space such that E0 b E (every bounded subset of E0 is relatively compact in E) and let U be a bounded subset of Lp .;E0 / which satisfies to the strong concentration condition in Lp .;E/: Z T h p ku.t C h/  u.t /kE dt D 0: (CF) lim sup h#0 u2U

0

Then U is relatively strongly compact in Lp .; E/.

Section 3.9 Strong Compactness in Lp .; E/

287

Proof. According to Theorem 3.124, we only need to demonstrate that U is tight. Let ' W E ! Œ0; C1, defined by ´ p kxkE ; x 2 E0 ; 0 '.x/ D C1; x 2 E n E0 : 1

Then, for every a  0; ' 1 .Œ0; a/ D ¹x 2 E0 W kxkE0  a p º is bounded in E0 and therefore it is relatively compact in E (' 1 .Œ0; a/ is a closed set). Then ' is an inf-compact application. Since U is bounded in Lp .; E0 /, Z T Z T p '.u.t//dt D sup ku.t /kE dt < C1 sup u2U

0

u2U

0

0

and then U is tight (U satisfies to the condition .T2 / of Corollary 3.62). Theorem 3.126. Let E0 be a Banach space such that E0 b E and let U  Lp .; E0 /  Lp .; E/ be a bounded set in Lp .; E/ satisfying to the strong concentration condition .CF / in Lp .; E/. If   8" > 0; 9k > 0 such that sup  ¹t 2 0; T Œ W ku.t /kE0 > kº < "; u2U

then U is relatively strongly compact in Lp .; E/. Proof. If we remark that the set K D ¹x 2 E0 W kxkE0  kº is compact in E, then sup .u1 .E n K// D sup .¹t 2 0; T Œ W ku.t /kE0 > kº/ < "

u2U

u2U

and therefore, according to condition .T1 / of Corollary 3.62, U is tight. Theorem 3.127 (Gutman’s Theorem [92]). Let U  Lp .; E/ be a set satisfying to strong concentration condition .CF / such that 8" > 0; 9K 2 KE

with

sup .u1 .E n K// < ":

u2U

Then U is relatively strongly compact in Lp .; E/. Proof. According to condition .T1 / of Corollary 3.62, U is tight.

288

3.10

Chapter 3 Young Measures

Gradient Young Measures

Young measures were introduced in two different but equivalent ways. Initially, they were presented as being generated by sequences of measurable functions. This direction is based on the fundamental theorem of Young; it is followed in the works of L. Tartar [163, 164, 165], J. M. Ball [22], L. C. Evans [69], D. Kinderlehrer and P. Pedregal [102, 103], T. Roubi˘cek [145], I. Fonseca and S. Müller [83]. Among more recent works using this approach, we quote [158, 181, 161, 85, 8, 72, 109]. Here, we have used a recent definition, used by E. Balder [10, 12] and M. Valadier [168] (see also [45]). In this section, we compare the two definitions. In the first part of section, we present the fundamental theorem of Young. We define and characterize p-Young measures as limits of sequences of Lp . In the case where the Young measure  is generated by a sequence of gradients, we introduce the p-gradient Young measures. The p-gradient Young measures and lower semicontinuity are the main ingredients in the study of solutions existence in relaxed variational calculus. Both of these ingredients have characterizations with quasiconvex functions. The study of this class of functions was started by Ch. B. Morrey, Jr. (see [121, 122] and was continued and developed by many others of that we quote: I. Fonseca and S. Müller [83], J. M. Ball, B. Kirchheim and J. Kristensen [24], A. Braides, I. Fonseca and G. Leoni [35], P. Pedregal [131]; see also [84, 71, 117, 160, 66, 34]. In the second part of this section, we define and study quasiconvexity. We find in literature quasiconvexity defined in different ways; we present a comparative study of these different definitions and, finally, the theorem of Kinderlehrer– Pedregal for characterization of p-gradient Young measures (see [102]). The third part of section is devoted to the study of sequentially weakly lower semicontinuous (wlsc) functionals in W 1;p .; Rm / and their characterization by quasiconvexity (Morrey–Acerbi–Fusco theorem). Studies on wlsc can also be found in [9, 118, 100, 83, 108, 159, 53, 54, 170, 113] and [125]. In this section, we will use the following framework:   is a bounded open subset of the Euclidean space Rd (d  1).  A is the -algebra of Lebesgue measurable subsets of .   D dt is Lebesgue measure on .  B.Rm / is the -algebra of Borel subsets of the Euclidean space Rm (m  1).  M.Rm / notes the set of all .A  B.Rm //-measurable mappings.

289

Section 3.10 Gradient Young Measures

 P .Rm / is the set of probabilities on Rm endowed with the narrow topology.  C0 .Rm / is the set of all continuous functions f W Rm ! R such that, for every " > 0, there is k > 0, with jf .x/j < ", for every x 2 Rm , kxk > kº— the set of all continuous functions vanishing at infinity.  Lip.; Rm / is the set of all Lipschitz functions f W  ! Rm .  Cc1 .; Rm / is the set of all Rm -valued functions with compact support in , differentiable of all orders.

3.10.1

Young Measures Generated by Sequences

The density of the M.Rm / in Y.Rm / (see Theorem 3.50 for S D Rm ) is one of the most important properties of the stable topology; historically, the Young measures have been constructed as limits of the measurable functions. Thus, some mathematicians use the notion of Young measure associated to a sequence instead of that of Young measure. The existence of these measures is furnished by the fundamental theorem of Young; in [22], J. M. Ball stated the following version of this theorem (see also Theorem 8.2 from [85]): Theorem 3.128. Let   Rd be a measurable bounded set and let .un /n  M.Rm / a sequence satisfying the following condition: sup .kun k  k/ ! 0: k!C1

n2N

(T)

Then there exist a subsequence .ukn /n of .un /n and a measurable mapping t 7!  t 2 P .Rm / such that: (1) For every closed set K  Rm such that ukn .t / 2 K, for every n 2 N and for almost every t 2 , supp t  K;

for almost every t 2 :

(2) For every Carathéodory integrand ‰ W   Rm ! R bounded from below 1 m for which .‰.; ukn //n is R uniformly integrable in L .; R /, the mapping u‰ W  ! R, u‰ .t/ D Rm ‰.t; x/d  t .x/, is integrable and w

‰.; ukn / ! u‰ : L1 . /

m (3) For R every f 2 C0 .R /,1the mapping uf W  ! R, defined by uf .t / D Rm f .x/d  t .x/, is in L ./ and w

f .ukn / ! uf : L1 . /

290

Chapter 3 Young Measures

Proof. We observe that, in the hypotheses of the theorem, .un /n is tight in Y.Rm / and then, according to Prohorov’s theorem (see Theorem 3.64), it is sequentially S-compact. Let now .ukn /n be a subsequence of .un /n S-convergent to  2 Y.Rm /. (1) By Proposition 3.43: 1 \  ¹ukn .t / W k  pº  K; for a.e. t 2 : supp t  Ls ¹ukn .t/ W n 2 Nº D pD1

(2) .‰.; ukn //n is uniformly integrable in L1 .; Rm / so that it is bounded (see Definition 1.85 and Theorem 1.84; let M > 0 such that Z j‰.t; ukn .t//jdt  M < C1: sup n



According to Proposition 3.34, 1 <  .‰/  lim inf  ukn .‰/  M: n

Therefore  .‰/ D S

Since ukn ! Rm

R

u‰ .t/dt



2 R and then u‰ 2 L1 ./.

and .‰.; ukn //n is uniformly integrable, Corollary 3.36

assure us that, for every A 2 A; Z Z Z ‰.t; ukn .t//d.t/ ! A

Rm

A

 Z ‰.t; x/d  t .x/ d.t / D u‰ .t /d.t /; A

from where w

‰.; ukn / ! u‰ : L1 . /

(3) Let f 2 C0 .Rm /; then f is bounded and so .f .ukn //n  L1 ./. Since, for every t 2 , juf .t/j  kf k1 , uf 2 L1 ./. Let g 2 L1 ./ and let ‰ W   Rm ! R; ‰.t; x/ D g.t /  f .x/. Then ‰ is a Carathéodory integrand L1 -bounded (for every .t; x/ 2   Rm , j‰.t; x/j  kf k1  jg.t /j). According to (5) of Theorem 3.32,  ukn .‰/ !  .‰/ or Z Z Z Z g.t /  f .ukn .t//dt ! g.t/  f .x/d  t .x/ D g.t /  uf .t /dt:



Rm



Therefore w

f .ukn / ! uf : L1 . /

291

Section 3.10 Gradient Young Measures

Remark 3.129. From the previous theorem, every sequence .un /n  M.Rm / which satisfies the tightness condition .T /, generates a Young measure  on Rm with the properties (1)–(3). Conversely, if we suppose that  is atomless on , then, according to Theorem 3.50, M.Rm / is dense in .Y.Rm /; S/. Rm is a metrizable Suslin space and A is countable generated; then, according to Proposition 3.25, .Y.Rm /; S/ is metrizable. Therefore, every Young measure on Rm is the limit of a sequence of measurable functions .un /n . Because of Proposition 3.65, .un /n satisfies the condition .T / and, therefore, the properties (1)–(3) are satisfied. Now we give the definition of a p-Young measure generated by a sequence. Definition 3.130. Let 1  p < C1; a Young measure  2 Y.Rm / is a p-Young measure if there exists a sequence .un /n  Lp .; Rm / such that: ´ S   and (a) un ! (b) .kun kp /n2N

is uniformly integrable:

In this definition, k  k notes the usual norm on Rm ; according to Definition 3.116, (b) says that .un /n is p-uniformly integrable. With the above notations, we say that  is generated by the sequence .un /n  p L .; Rm /. Remark 3.131. Let .un /n  Lp .; Rm / be a sequence which generates the pYoung measure  . .i/ .un /n is uniformly integrable in L1 .; Rm /. Indeed, if p > 1, then, according to (iv) of Remark 3.117, .kun k/n is uniformly integrable in L1 .; Rm / and then .un /n it is. w

.ii/ By previous remark and by Proposition 3.37, un ! u 2 L1 .; Rm /, L1 . ;Rm / R m where u W  ! R is defined by u.t / D Rm xd  t .x/ D bar t (the barycenter of  t ). We use Theorem 3.128 to give in the following theorem two characterizations for p-Young measures (see [102] and [103]). Theorem 3.132. Let 1  p < C1 and let equivalent: (1)

 is a p-Young measure.



2 Y.Rm /; the following are

292

Chapter 3 Young Measures

(2) There is a sequence .un /n  Lp .; Rm / and a g 2 L1 ./ such that: 8 w p ˆ < (a) kun k ! g; L1 . /

w ˆ : (b) '.un / ! u' ;

for every ' 2 E p ;

L1 . /

where

²

³

'.x/ E D ' W R ! Rj' continuous on R ; lim 2R kxk!1 1 C kxkp Z and u' W  ! Rm ; u' .t/ D '.x/d  t .x/: p

m

m

Rm

(3) There exists a sequence .un /n  Lp .; Rm / such that: 8 < (a) .kun kp /n2N is uniformly integrable and w

! uf ; : (b) f .un / 1 L

. /

where uf W  ! R; uf .t/ D

R Rm

for every f 2 C0 .Rm /;

f .x/d  t .x/:

Proof. (1)H) (2): Let .un /n  Lp .; Rm /, a sequence which generates S

;

therefore .kun kp /n is uniformly integrable and un !  . (a) According to Definition 1.85 and Theorem 1.84, ¹kun kp W n 2 Nº is relatively weakly compact. So, with eventually extracting a subsequence, still noted w p .un /n , we can suppose that un ! g 2 L1 ./. L1 . /

(b) Let now ' 2 that

'.x/ j 1Ckxk p

Ep

and let L D limkxk!1

 Lj < 1, for every x 2

Rm

'.x/ 1Ckxkp

2 R. There is k > 0 such

with kxk > k. Since ' is continuous,

'.x/ supkxkk j 1Ckxk p j D K 2 RC and therefore ˇ ˇ ˇ '.x/ ˇ ˇ ˇ ˇ 1 C kxkp ˇ  max¹K; jLj C 1º D M;

for every x 2 Rm :

Then, for every n 2 N, ˇ ˇ ˇ '.un / ˇ ˇ ˇ  .1 C kun kp /  M  .1 C kun kp / j'.un /j D ˇ 1 C kun kp ˇ and, as .kun kp /n is uniformly integrable, .'.un //n is too uniformly integrable. Let ‰ W   Rm ! R, defined by ‰.t; x/ D '.x/. Then ‰ is a Carathéodory integrand bounded from below for which .‰.; un //n D .'.un //n is uniformly

293

Section 3.10 Gradient Young Measures

integrable. According to (2) of Theorem 3.128, u‰ D and

R Rm

'.x/d  .x/ 2 L1 ./

w

‰.; un / D '.un / ! u‰ D u' : L1 . /

(2)H)(3): Let .un /n  Lp .; Rm / be a sequence which satisfies (a) and (b) of (2). According to 1.84, .kun kp /n is uniformly integrable. w For every f 2 C0 .Rm /; f 2 E p and so, from (b), f .un / ! uf . ThereL1 . / R R fore, according to Theorem 1.57, for every A 2 A; A  f .un /dt ! A  uf dt: We can extend this to E.A/—the set of all simple functions on .; A; /. Therefore, Z Z h  f .un /dt ! h  uf dt; for every h 2 E.A/:



arbitrary; for every " > 0 there is h1 2 E.A/ such that Let now h 2 kh  h1 k1 < ". We remark that f is a bounded function on Rm ; then .f .un /n  L1 ./ and uf 2 L1 ./. Moreover, kf .un /  uf k1  2kf k1 , for every n 2 N. Then ˇ ˇZ ˇ ˇZ Z ˇ ˇ ˇ ˇ ˇ ˇ ˇ ˇ hf .un /dt  hu dt  .h  h /.f .u /  u /dt 1 n f f ˇ ˇ ˇ ˇ ˇ ˇZ ˇ ˇ C ˇˇ h1 .f .un /  uf /dt ˇˇ  kh  h1 k1 kf .un /  uf k1 ˇZ ˇ ˇ ˇ ˇ C ˇ h1 .f .un /  uf /dt ˇˇ : L1 ./,



Since the last term converges to 0, we obtain that Z Z h  f .un /dt ! h  uf dt; for every h 2 L1 ./: n!C1





w

Therefore, f .un / ! uf : L1 . /

(3)H)(1): Let .un /n  Lp .; Rm / be a sequence p-uniformly integrable and such that w

f .un / ! uf ; L1 . /

for every f 2 Cc .Rm /  C0 .Rm /:

Since, for every A 2 A, A 2 L1 ./; Z Z Z Z f .un /dt ! uf dt D A

A



Rm

f .x/d  t .x/: S

 . According to (7) of Proposition 3.22, this means that un !

294

Chapter 3 Young Measures

Remark 3.133. .i/ The equivalent form (2) for p-Young measures belongs to D. Kinderlehrer and P. Pedregal and has been given in [102]. .ii/ Many papers use as definition the equivalent form (3); see for example the Definition 8.3 in [85] and definition 2.1 in [100]. We remark that, in these definitions, the condition (a) are missing; but without it, (3) does not imply (1). Indeed, the sequence .un /n  L1 .Œ0; 1/; un D n  1 , is not uniformly integrable but it satisfies the condition (b) from (3).

Œ0; n 

In the case where a Young measure  is generated by the gradients of a sequence .un /n  W 1;p .; Rm /, we say that  is a gradient Young measure. This notion has been introduced by D. Kinderlehrer and P. Pedregal in [102]. This one is presented here in the equivalent form (1) (see the previous theorem). Definition 3.134. Let 1  p < C1,   Rd be a bounded open set and let W 1;p .; Rm / be the Sobolev space (the definition of which we already presented in 3.98). A Young measure  2 Y.Rmd / is a p-gradient Young measure if there exists a sequence .un /n  W 1;p .; Rm / such that: 8 < (a) .krun kp /n2N is uniformly integrable in L1 .; Rmd / and S

!  : : (b) run md Y.R

/

In this definition, k  k notes the usual norm in Rmd . Obviously,  is generated by .run /n . Remark 3.135. .i/ Adapting Remark 3.131 to the case of p-gradient Young measures, we obw tain: run ! bar 2 L1 .; Rmd /. L1 . ;Rmd /

.ii/ Theorem 3.132 allows us to obtain easily characterizations of type (2) and (3) for p-gradient Young measures, replacing the sequence .un /n with the sequence .run /n . Young measure and let Proposition 3.136. Let  2 Y.Rmd / be a p-gradient R bar: W  ! Rmd be its barycenter: bar t D yd  t .y/. Then Z kykp d  .t; y/ < C1 and so bar: 2 Lp .; Rmd /: Rmd

295

Section 3.10 Gradient Young Measures

Proof. According to (i) of previous remark, bar: 2 L1 .; Rmd /. Let .un /n  W 1;p .; Rm / such that .krun kp /n2N is uniformly integrable in S

L1 .; Rmd / and run ! Y.Rmd /

.

The mapping ‰ W   Rmd ! R, defined

by ‰.t; y/ D kykp , is a Carathéodory integrand and ¹‰.; run .// W n 2 Nº D ¹krun kp W n 2 Nº is uniformly integrable. According to Corollary 3.36, Z Z Z p kyk d  .t; y/ D kykp d  t .y/dt md md R R Z D lim krun .t /kp dt < C1 n



(every uniformly integrable subset of L1 ./ is bounded). R Then the mapping t 7! Rmd kykp d  t .y/ is finite almost every where and integrable. Now we can apply the Jensen’s inequality to obtain:

Z

p Z



p yd  t .y/

kykp d  t .y/: kbar t kRmd D



 Rmd

Rmd

Therefore, bar: 2 Lp .; Rmd /. Proposition 3.137. Let   Rd be a bounded open set with a Lipschitz boundary (see Definition 12.10 from [50]) and let 1  p < C1. Let  2 Y.Rmd / be a p-gradient Young measure generated by .run /n  1;p 1;p W0 .; Rm / (or .un  u0 /n  W0 .; Rm /, where u0 2 W 1;p .; Rm /). 1;p m There exist u 2 W .; R / and a subsequence of .un /n , still noted .un /n , such that kkp

(1) un ! u. Lp . ;Rm / w

(2) run ! ru D bar: . Lp . ;Rmd / w

(3) un ! u. W 1;p . ;Rm /

Proof. (1) Since .krun kp /n is uniformly integrable in L1 .; Rmd /, .run /n is bounded in Lp .; Rmd /. Appealing to Poincaré inequality (see, for example, Corollary IX.19 in [36] or Theorem 1.47 in [49]), .un /n is bounded in W 1;p .; Rm / and therefore, from Rellich–Kondrachov theorem (Theorem 12.12 in [50]), up to a subsequence, .un /n converges strongly to u 2 Lp .; Rm /.

296

Chapter 3 Young Measures

1 m (2) Let now un D .u1n ; : : : ; um n / and u D .u ; : : : ; u /. For every ' 2 every i D 1; : : : ; m and every j D 1; : : : ; d , Z Z @' @uin  'dt D  uin  dt: (1) @tj @tj

Cc1 .; R/,

w

2 From (i) of Remark 3.135, run ! bar D .vji /ji D1;:::;m D1;:::;d L1 . ;Rmd /

L1 .; Rmd / and, since ' 2 Cc1 .; R/  L1 .; R/, Z Z @uin  'dt ! vji  'dt: @t j

(2)

kkp

u, Since un ! p L

Z

uin

@'  dt ! @tj

Z

ui 

@' dt: @tj

(3)

Using (2) and (3) in (1), we obtain, for every ' 2 Cc1 .; R/ : Z Z @' ui  dt D  vji  'dt; for every i D 1; : : : ; m; j D 1; : : : ; d: @t j Therefore ru D bar: 2 Lp .; Rmd / (see the previous proposition). Then u 2 W 1;p .; Rm /. Moreover, Z Z @uin @ui  'dt !  'dt; for every ' 2 Cc1 .; R/: @t @t j j Since Cc1 .; R/ is dense in Lq .; R/ ( p1 C

1 q

w

D 1), run ! ru. Lp . ;Rmd /

(3) is a consequence of (1) and (2). Once we will introduce quasiconvexity, we will can present the Kinderlehrer– Pedregal’s characterization of p-gradient Young measures.

3.10.2

Quasiconvex Functions

Quasiconvex functions were introduced in 1952 by Ch. B. Morrey, Jr. (see [121] and [122]) in order to characterize lower semicontinuity of functional Z f .t; u.t /; ru.t //dt; I.u/ D

297

Section 3.10 Gradient Young Measures

commonly occurring in problems of variational calculus. Morrey’s results were later extended by Acerbi and Fusco ([1], 1984) and Marcellini ([118], 1985). In Chapter 5 of [50], we can find an excellent comparative study of various extensions of convexity. Definition 3.138. Let f W Rmd ! R be a continuous function; f is said to be quasiconvex if, for every y 2 Rmd , for every bounded open set   Rd and for every ' 2 Cc1 .; Rm /, Z 1  f .y C r'.t //dt: f .y/  ./ In the previous definition, 0 B r' D @

@'1 @t1 @'m @t1

1 1    @' @td C  A m    @' @t d

is the gradient of function ' D .'1 ; : : : ; 'm / W  ! Rm . In the next proposition, we show that the definition of quasiconvexity does not depend on the choice of . Let p 1 ; : : : ; p d 2 Rd such that < p i ; p j >D ıij , for every i; j 2 ¹1; : : : ; nº and let Q D Q.p 1 ; : : : ; p d / D ¹t 2 Rd W 0 < 1, for every i D 1; : : : ; nº be the open hypercube of side 1 which is based on versors p 1 ; : : : ; p d . Proposition 3.139. The continuous function f W Rmd ! R is quasiconvex if and only if, for every y 2 Rmd and every ' 2 Cc1 .Q; Rm /, Z f .y C r'.t //dt: f .y/  Q

Proof. The necessity is obvious. Let us prove the sufficiency; let y 2 Rmd , let  be a bounded open subset of Rd and let ' 2 Cc1 .; Rm /. Let t 0 2 Rd and P r > 0 such that S.t 0 ; r/  and let s 0 D r  jdD1 p j 2 Rd ; then   t 0 C s 0 C 2r  Q Q1 :

298

Chapter 3 Young Measures

Indeed, if t 2   S.t 0 ; r/; kt  t 0 k < r and so, for every i 2 ¹1; : : : ; nº;

 E 1 D E 1 D E 1 1 1 D  .t t 0 s 0 /; p i D  t t 0 ; p i   s0; pi D  t t 0 ; p i C : 2r 2r 2r 2r 2 ˇD Eˇ E D ˇ ˇ ˇ t  t 0 ; p i ˇ  kt  t 0 k < r ) r < t  t 0 ; p i < r E 1 1 1 1 D 1 1 ) 0 D r  C <  x  t 0; pi C < r  C D 1: 2r 2 2r 2 2r 2 1  .t  t 0  s 0 / 2 Q, from where t 2 t 0 C s 0 C 2r  Q Q1 : It follows that 2r We extend ' to Q1 , giving value 0 on Q1 n ; obviously, ' 2 Cc1 .Q1 ; Rm /. Z Z Z f .y C r'.t//dt D f .y C r'.t //dt  f .y/dt Q Q1 n Z 1 f .y C r'.t //dt  f .y/  .Q1 n /: D Q1

We make the change of variable t D t 0 C s 0 C 2r  s; then t 2 Q1 ” s D and so

1  .t  t 0  s 0 / 2 Q 2r Z

Z

f .y C r'.t//dt D .2r/d 

Q

f .y C .r'/.t 0 C s 0 C 2rs//ds

 f .y/  .Q1 n /: 1 1 0 1  '.t 0 C s 0 C 2rs/; supp D  2r .t C s 0 / C 2r  supp'  Q Let .t/ D 2r 0 0 r .s/ D .r'/.t C s C 2rs/ and, therefore, Z Z d f .y C r'.t//dt D .2r/  f .y C r .s//ds  f .y/  .Q1 n /

Q

 .Q1 /  f .y/  .Q1 n /  f .y/ D ./  f .y/: Remark 3.140. .i/ The cube Q in the preceding proposition, may be replaced by a cube translated at any point of Rd . So, if Q is a cube of side 1 in Rd , then the continuous function f W Rmd ! R is quasiconvex if and only if, for every y 2 Rmd and every ' 2 Cc1 .Q; Rm /, Z f .y/  f .y C r'.t //dt: Q

299

Section 3.10 Gradient Young Measures

.ii/ If, in the previous proposition, p 1 ; : : : ; p d are the versors e 1 ; : : : ; e d of the coordinate system of Rd , then Q D .0; 1/d . .iii/ Let p 1 D ¹p11 ; : : : ; pd1 º; : : : ; p d D ¹p1d ; : : : ; pdd º be d versors pairwise orthogonal in Rd and let Q.p 1 ; : : : ; p d / be the hypercube which is based on them; we denote by 1 0 1 p1    pd1 A  P D@ d d p1    pd and by T W Rd ! Rd the mapping defined by T .t / D P  t , for every t 2 Rd . T is a linear change of basis in Rd I 8i D 1; : : : ; d; T .p i / D e i s¸i T .Q.p 1 ; : : : ; p d // D .0; 1/d D Q. For every y 2 Rmd and every ' 2 Cc1 .; Rm /; 'ıT 2 Cc1 .Q.p 1; : : : ; p d /; Rd / (T is a homeomorphism and so supp.' ı T / D T 1 .supp'/ is a compact set in Q.p 1 ; : : : ; p d /); then y  P 2 Rmd and therefore Z f .y  P C r.' ı T /.t //dt f .y  P /  Q.p 1 ;:::;p d / Z Z f .y  P C r'.s/  P /ds D f Œ.y C r'.s//  P ds: D Q

Q

If f is quasiconvex, then f .T / is quasiconvex, also. We can now prove that any convex function is quasiconvex. Proposition 3.141. Every convex continuous function f W Rmd ! R is quasiconvex. Proof. Let Q D .0; 1/d , let y 2 Rmd and let ' 2 Cc1 .Q; Rm /; applying Jensen’s inequality, we get Z  Z f .y C r'.t//dt  f .y C r'.t //dt: Q

Q

On the other hand Z Z .y C r'.t//dt D y C Q

DyC

Z .0;1/d 1

Z 0

1

 @'j .t/dt 1id Q @ti 1j m   @'j .t/dti dt1 : : : dti 1 dti C1 : : : dtd D y: @ti 1id

According to Proposition 3.139, f is quasiconvex.

1j m

300

Chapter 3 Young Measures

The converse of Proposition 3.141 is not true. 2 2 ! R be the function defined by f .x/ D det.x/; Example 3.142. f WR  x1 x2Let i.e., if x D x3 x4 , then f .x/ D x1 x4  x2 x3 .

jf .y/  f .z/j  .kyk C kzk/  ky  zk;

for every y; z 2 R2 2 :

Therefore, f is continuous. Let Q D .0; 1/2 D .0; 1/  .0; 1/ and let ' D .'1 ; '2 / 2 Cc1 .Q; R2 /; ! @'1 @'1 .t/ .t / @t1 @t2 r'.t/ D @' I @'2 2 .t/ .t / @t1 @t2  then, for every y D yy13 yy24 2 R2 2 ,    @'1  @'2 .t/  y4 C .t / f .y C r'.t// D y1 C @t1 @t2     @'1 @'2 .t/  y3 C .t /  y2 C @t2 @t1 @'1 @'2 @'1 @'2 .t/ C y1 .t /  y3 .t /  y2 .t / D f .y/ C y4 @t1 @t2 @t2 @t1 @'2 @'1 @'2 @'1 .t/  .t/  .t /  .t /: C @t @t2 @t2 @t1 Z 1 Z Z Z @'1 @'2 @'1 @'2 .t/dt D .t/dt D .t /dt D .t /dt D 0 Q @t1 Q @t2 Q @t2 Q @t1 and

 @'2 @'1 @'2 @'1 .t/  .t/  .t/  .t / dt @t1 @t2 @t2 @t1 Q Z Z @2 '2 @2 '2 '1 .t/  .t/dt C '1 .t /  .t /dt D 0: D @t1 @t2 @t1 @t2 Q Q

Z 

It follows that

Z Q

f .y C r'.t//dt D f .y/

and, using Proposition 3.139, f is quasiconvex.  On the other hand, the function f is not convex; indeed, if y D 10 00 ; z D 0 0 2 2 , then f . 1 y C 1 z/ D 1 > 0 D 1 f .y/ C 1 f .z/: 01 2R 2 2 4 2 2 In the following, we broad the class of admissible functions ' in the definition of quasiconvexity.

301

Section 3.10 Gradient Young Measures

Theorem 3.143. A continuous function f W Rmd ! R is quasiconvex if and only if, for every y 2 Rmd , every bounded open subset   Rd and every ' 2 W01;1 .; Rm /, Z 1 f .y/   f .y C r'.t //dt: ./ Proof. The sufficiency is a consequence of the inclusion Cc1 .; Rm /  W01;1 .; Rm /. Let us prove the necessity. We recall the standard procedure of approximation. R Let 2 Cc1 .Rd /;  0; .t/ D .t /; supp  B1 ; Rd .t /dt D 1 and, for every " > 0; " .t/ D "1d  . "t /. We extend every u 2 L1 ./ to Rd giving the value 0 on the set Rd n  and weR note the extension againRwith u. Let u" D u " , i.e. u" .t/ D Rd u.s/  " .t  s/ds D Rd u.t  "s/  .s/ds. R L1 Then u" 2 C 1 ./, u" ! u and ri u" D 1"  Rd u.t  "s/ri .s/ds. For every u 2 W 1;1 .; Rm /; ri u" D u ri " D ri u " . L1

For every ' 2 W01;1 .; Rm /; r' 2 L1 .; Rmd / and so .r'/" ! r'; we recall that r' D .ri ' j / 1id . 1j m

It follows that '" 2 C 1 .Rd ; Rm / and supp'"  supp' C B" is compact. supp' D K  ; there exists "0 > 0 such that K " D supp' C B"  , for every "  "0 and then '" 2 Cc1 .; Rm /. From hypothesis, for every y 2 Rmd and every "  "0 , Z 1 f .y/  f .y C r'" .t //dt: (*)  ./ r'" D .r'/" ; indeed, for every 1  i  d; 1  j  m, Z  Z    j j ri ' .t/ D ' j .s/  ri . " .t  s//ds ri ' .s/  " .t  s/ds D  " Rd Rd Z D ' j .s/  .ri " /.t  s/ds D ri .'"i /.t /: Rd

L1

Since r'" D .r'/" ! r', there exists a subsequence of .r'" /">0 almost "!0

everywhere convergent to r'. On the other hand, r'" is continuous with compact support in ; it follows that there is M > 0 such that kr'" kRmd  M . Since f is continuous on Rmd , f is bounded on B.y; M / and then we can apply the Lebesgue’s theorem Z Z f .y C r'" .t//dt ! f .y C r'.t //dt:



302

Chapter 3 Young Measures

According to . /, it follows that f .y/ 

1  ./

Z

f .y C r'.t //dt:

Remark 3.144. We cannot relax condition ' 2 W01;1 .; Rm / of the previous theorem to condition ' 2 W 1;1 .; Rm /. Indeed, as we remarked in Example 3.142, the function f W R2 2 ! R; f .x/ D det.x/ is quasiconvex. Let Q D .0; 1/  .0; 1/ and ' W Q ! R2 defined by '.t1 ; t2 / D .t1  t2 ; 0/, for every .t1 ; t2 / 2 Q. Then ' 2 L1 .Q; R2 / and r'.t1 ; t2 / D t02 t01 . Therefore so ' 2 W 1;1 .Q; R2 /. r' 2 L1 .Q; R2 2/, and However, if y D 02 00 2 R2 2 , then Z Z 1 0 D f .y/ > f .y C r'.t//dt D  2t1 dt1 D 1: Q

0

But we can extend the characterization theorem to the class of periodic functions of W 1;1 .Rd ; Rm /. Definition 3.145. Let Q D .0; 1/d ; the function ' W Rd ! Rm is Q - periodic if, for every t D .t1 ; : : : ; td / 2 Rd and every i D 1; : : : ; d , the function gi W R ! R, defined by gi .s/ D '.t1 ; : : : ; ti 1 ; s; ti C1 ; : : : ; td /, is periodic with period 1. Theorem 3.146. Let f W Rmd ! R be a continuous function; f is quasiconvex if and only if, for every y 2 Rmd and every Q - periodic function ' 2 W 1;1 .Rd ; Rm /, Z f .y C r'.t //dt: f .y/  Q

Proof. Sufficiency follows from the fact that any function ' 2 Cc1 .Q; Rm / can be extended to a Q - periodic function of W 1;1 .Rd ; Rm /. Let us prove the necessity. We suppose that f W Rmd ! R is quasiconvex, y 2 Rmd and ' 2 W 1;1 .Rd ; Rm / is a Q-periodic function; for every k 2 N  , we define 'k W Rd ! Rm by 'k .t/ D k1  '.kt /; 8t 2 Rd . Then 'k 2 W 11 .Rd ; Rm /, k'k kL1 D k1  k'kL1 and kr'k kL1 D kr'kL1 . For every p 2 N  , let Qp D ¹t 2 Q W d.t; @Q/ > p1 ºI .Q n Qp / ! 0. Let p 2 Cc1 .Q; R/ such that 0  p  1, p jQp D 1 and let Mp D kr p kL1 < C1. There exists a subsequence .'kp /p of .'k /k such that k'kp kL1  Mp1C1 , for every p 2 N. We choose this subsequence such that Z Z f .y C r'k .t//dt D lim f .y C r'kp .t //dt: lim inf k

Q

p

Q

303

Section 3.10 Gradient Young Measures

We define p D p  'kp 2 W01;1 .Q; Rm /. Since f is quasiconvex, Z Z Z f .y Cr p .t //dt C f .y Cr p .t //dt f .y/  f .y Cr p .t//dt D Q Qp QnQp Z f .y C r'kp .t//dt D Qp Z f .y C p .t/  r'kp .t/ C 'kp .t / ˝ r p .t //dt C QnQp Z f .y C r'kp .t//dt C "p ; D Q

where Z "p D

 f .y C

QnQp

p .t/

 r'kp .t/C'kp .t/ ˝ r

p .t //f .y

 C r'kp .t // dt:

P p .t/  r'kp .t/kRmd  kr'kp .t/kRmd kr' kp kL1  kr'kL1 < C1: P kp kL1  kr p kL1 k'kp .t/ ˝ r p .t/kRmd  k'kp .t/kRm  kr p .t /kRd k'

k



Mp < 1: Mp C 1

From the above inequalities, it follows that, when t belongs to Q n Qp , y C p .t/r'kp .t/C'kp .t/˝r p .t/ belongs to the bounded set yCB.0;kr'kL1 /C B.0; 1/  Rmd ; also, y C r'kp .t/ 2 y C B.0; kr'kL1 /. Since f is continuous on Rmd , f preserves the bounded sets. Then let M D

sup

jf .y C

p .t/

x2QnQp

 r'kp .t / C 'kp .t / ˝ r

p .t /j

Therefore, j"p j  2M  .Q n Qp / ! 0 and so Z f .y/  limΠf .y C r'kp .t //dt C "p  p Q Z f .y C r'kp .t //dt C lim "p D lim p p Q Z D lim inf f .y C r'k .t //dt: k

2 RC :

(1)

Q

Now Z Z Z .t D 1 s/ 1 f .y Cr'k .t//dt D f .y C.r'/.kt//dt Dk  f .y Cr'.s//ds: k d kQ Q Q

304

Chapter 3 Young Measures d

But kQ D P [jkD1 .t j C Q/, where t j is the bottom left corner of each sub-cube of side one obtained by dividing the cube kQ (we use the almost everywhere equality, D, P because we neglect the faces of these sub-cubes). Since r' is Q-periodic, Z Z 1 f .y C r'k .t//dt D d  k d  f .y C r'.t //dt k Q Q and so

Z

Z lim inf k

Q

f .y C r'k .t//dt D

By (1) and (2), it follows

Q

f .y C r'.t //dt:

(2)

Z f .y/ 

Q

f .y C r'.t //dt:

Remark 3.147. Using (iii) of Remark 3.140, we can replace, in the previous theorem, the cube Q D .0; 1/d by a cube Q.p 1 ; : : : ; p d / based on the pairwise orthogonal versors p 1 ; : : : ; p d 2 Rd . The following two lemmas are useful for characterizing the quasiconvex functions through the Lipschitz functions, zero on the boundary. N and every Lemma 3.148. Let   Rd be a bounded open set; for every t 2  d N s 2 R n ; d.t; s/  d.t; @/. N s 2 Rd n  N such that Proof. We assume, by absurd, that there exist t 2 , N d.t; s/ < d.t; @/; then t 2  because, if we suppose that t 2  n  D @, then d.t; s/ < d.t; @/ D 0. Let ˛0 D sup¹˛ 2 Œ0; 1 W .1  ˛/t C ˛s 2 º and t 0 D .1  ˛0 /t C ˛0 s; there N If we assume exists ˛n " ˛0 such that .1  ˛n /t C ˛n s 2 , form where t 0 2 . 0 0 that t 2 , then there exists r > 0 such that the closed ball B.t ; r/  . In this r r situation, let ˛1 D ˛0 C kt sk > ˛0 and t 1 D .1˛1 /t C˛1 s D t 0 C kt sk .st /; r 1 0 1 0 then kt  t k D kt sk  ks  tk D r and so t 2 B.t ; r/  , which is absurd because ˛1 > ˛0 . N n  D @ and then d.t; s/ D kt  sk < d.t; @/  It follows that t 0 2  0 d.t; t / D ˛0  kt  sk, from where ˛0 > 1 ! Thus the previous assumption is false. N Rm / such that 'j@ D 0 and let 1  Rd be a Lemma 3.149. Let ' 2 Lip.; N N bounded open set with   1 ; we extend ' to 1 , giving value 0 on 1 n .

305

Section 3.10 Gradient Young Measures

Then ' 2 Lip.1 ; Rd / and N D sup L.'; 1 / D L.'; /

²

³ k'.t/  '.s/k N t ¤s : W t; s 2 ; kt  sk

Proof. Let t; s 2 1 arbitrary; N then k'.t/  '.s/k  L.'; / N  kt  sk. (1) If t; s 2 , N N (2) If t 2 ; s 2 1 n , then for every r 2 @, N  kt  rk; k'.t/  '.s/k D k'.t/k D k'.t/  '.r/k  L.'; / from where, using Lemma 3.148, N  d.t; @/  L.'; / N  kt  sk: k'.t/  '.s/k  L.'; / N k'.t/  '.s/k D 0  L.'; / N  kt  sk. (3) Finally, if t; s 2 1 n ; N  kt  sk; hence ' is a Lipschitz function In any case k'.t/  '.s/k  L.'; / N and L.'; 1 / D L.'; /. The following characterization of quasiconvexity is used as definition in [121] (see Definition 2.2). Theorem 3.150. A continuous function f W Rmd ! R is quasiconvex if and only if, for every y 2 Rmd , every bounded open subset   Rd and every N Rm / with 'j@ D 0, ' 2 Lip.; Z 1 f .y C r'.t //dt: f .y/   N N ./ Proof. Necessity. We suppose that f is quasiconvex and let y 2 Rmd , let a N Rm / with 'j@ D 0. bounded open subset   Rd and let ' 2 Lip.; N  1 and we We consider an arbitrary bounded open set 1  Rd with  N extend ' on 1 , by giving value 0 on 1 n . Lemma 3.149 assures us that ' 2 Lip.1 ; Rm /; then ' is uniformly continuous on 1 and then it is bounded N !). It follows that ' 2 W 1;1 .; Rm / and as supp' is a (' is bounded on  N ' 2 W 1;1 .; Rm /. subset of compact set , 0 According to Theorem 3.143, Z 1 f .y C r'.t//dt  f .y/  .1 / 1 Z 1 1 N f .y C r'.t //dt  f .y/  .1 n / C  D .1 / .1 / N from where

N 1 ./  f .y/   .1 / .1 /

Z N

f .y C r'.t //dt:

306

Chapter 3 Young Measures

Sufficiency. Let y 2 Rmd ;   Rd be a bounded open set and let ' D .'1 ; : : : ; 'm / 2 W01;1 .; Rm /; we consider a bounded open convex set 1  Rd such that   1 and we extend ' and r', giving to the both functions value 0 on 1 n ; then ' 2 W01;1 .1 ; Rm /. Indeed, because supp' D K   is compact, there exists an open set G such that K  G  GN  , GN is compact and there exists 0 2 Cc1 .1 ; R/; 0  0  1 such that 0 jGN D 1 and N 0 jRd n D 0. Then supp 0  ; r 0 jK D 0 and r'j nG D 0. 1 Therefore, for every 2 Cc .1 ; R/ and i 2 ¹1; : : : ; d º; j 2 ¹1; : : : ; mº, 8R R ˆ < 1R 'j .t/  ri .t/dt D K 'j .t/ Rri .t /dt (1) D K 'j .t/  0 .t/  ri .t/dt D K 'j .t /  ri .  0 /.t /dt ˆ R : R  K 'j .t/  .t/  ri 0 .t/dt D 'j .t /  ri .  0 /.t /dt: Since  0 2 Cc1 .; R/, 8R R ˆ j .t /  .t /  0 .t /dt < 'jR.t/  ri .  0 /.t/dt D  ri ' R D  G ri 'j .t/  .t/  0 .t/dt D  G ri 'j .t /  .t /dt ˆ R : D  1 ri 'j .t/  .t/dt:

(2)

By (1) and (2), it follows that, for every i 2 ¹1; : : : ; d º, every j 2 ¹1; : : : ; mº and every 2 Cc1 .1 ; R/, Z Z 'j .t/  ri .t/dt D  ri 'j .t /  .t /dt 1

1

(we have extended r' by ri 'j .t/ D 0, for every t 2 1 n ). It follows that ri 'j is the i-weak derivative of 'j on 1 ; hence r' is the gradient of '. It is obvious that '; r' 2 L1 .1 ; Rm / and so ' 2 W 1;1 .1 ; Rm /. N Rm / and 'j@ D 0. Since 1 is convex, ' 2 Lip.1 ; Rm /, hence ' 2 Lip.; 1;1 In fact, we proved that ' 2 W0 .; Rm / H) ' 2 Lip.1 ; Rm /: From the hypothesis, it follows that Z 1  f .y C r'.t//dt f .y/  N N ./ Z 1 1 N n /;   f .y/  . f .y C r'.t//dt C D N N ./ ./ from where 1 f .y/   ./

Z

f .y C r'.t //dt:

According to Theorem 3.143, f is quasiconvex.

307

Section 3.10 Gradient Young Measures

In the following, we introduce a notion more general than quasiconvexity, notion which has the advantage, however, to speak in a language closer to that of convexity. Definition 3.151. The function f W Rmd ! R is said rank one convex if the function g W R ! R, defined by g.t / D f .y C t .a ˝ b//, is convex, for every y 2 Rmd , every a 2 Rm and every b 2 Rd , where a ˝ b is the matrix 0 1 0 1 a1 a1 b1    a1 bd B C A ::: D aT  b: a ˝ b D @ ::: A  .b1    bd / D @ am b1    am bd m d am We remark that rank.a ˝ b/  1. If f 2 C 2 .Rmd /, then f is rank one convex if and only if, for every y 2 Rmd , every a 2 Rm and every b 2 Rd , .a ˝ b/T  d 2 f .y/.a ˝ b/  0: Proposition 3.152. The function f W Rmd ! R is rank one convex if and only if, for every y; z 2 Rmd with rank.y  z/  1, f ..1  /y C z/  .1  /f .y/ C f .z/;

for every  2 Œ0; 1:

Proof. We remark that if y; z 2 Rmd , then rank.y  z/  1 if and only if there exist a 2 Rm and b 2 Rd such that y  z D a ˝ b. Necessity. We suppose that the function f is rank one convex and let y; z 2 Rmd with rank.y  z/  1; let a 2 Rm ; b 2 Rd such that y  z D a ˝ b. Then the function g W R ! R; g.t/ D f .z C t.a ˝ b//, is convex and so, for every  2 Œ0; 1; g..1  /  1 C   0/  .1  /  g.1/ C   g.0/ which rewrites f .z C .1  /  .a ˝ b//  .1  /  f .z C a ˝ b/ C   f .z/ or f ..1  /y C z/  .1  /  f .y/ C   f .z/: Sufficiency. For every y 2 Rmd , every a 2 Rm and every b 2 Rd , we define g W R ! R by g.t / D f .y C t.a ˝ b//. Let us show that g is convex.

308

Chapter 3 Young Measures

For every t; s 2 R and every  2 Œ0; 1, let z D y Ct .a ˝b/; w D y Cs.a ˝b/; then z  w D .t  s/  .a ˝ b/ D Œ.t  s/a ˝ b and so f ..1  /z C w/  .1  /  f .z/ C   f .w/; from where g..1  /z C w/ D f .y C .1  /t.a ˝ b/ C s.a ˝ b// D f ..1  /z C w/  .1  /  f .y C t .a ˝ b// C f .y C s.a ˝ b// D .1  /  g.t/ C   g.s/: Therefore the function g is convex and so f is rank one convex. Now, we analyze the relations between the three types of convexity: the classical convexity, quasiconvexity and rank one convexity(see also theorem of [50]). Theorem 3.153. Let f W Rmd ! R be a continuous function and let the following assertions: .i/ f is convex; .ii/ f is quasiconvex; .iii/ f is rank one convex. Then (i) H) (ii) H) (iii). If m D 1 or n D 1 then the three statements are equivalent. Proof. The implication (i)H)(ii) was demonstrated in Proposition 3.141. (ii)H)(iii). We suppose that f W Rmd ! R is quasiconvex. Let y; z 2 Rmd , with rank.y  z/  1; according to Proposition 3.152, it must to show that f ..1  /y C z/  .1  /f .y/ C f .z/;

for every  2 Œ0; 1:

Let a 2 Rm and b 2 Rd such that y  z D a ˝ b. 1  b/, we can suppose that kbk D 1. Since a ˝ b D .kbk  a/ ˝ . jjbjj

Choose an orthonormal basis ¹p 1 ; : : : ; p d º on Rd with p 1 D b and let Q be the hypercube based on this basis. If W R ! R is a periodic function with period 1 such that Œ0;1=2/ D 1 and

Œ1=2;1 D 1, then we define ' W Rd ! Rm by ! Z ht;bi 1 '.t/ D 

.r/dr  a: 2 0

309

Section 3.10 Gradient Young Measures

For every t; s 2 Rd , k'.t/  '.s/kRm

ˇ ˇZ Z hs;bi ˇ ˇ ht;bi ˇ ˇ

.r/dr 

.r/dr ˇ  kak ˇ ˇ ˇ 0 0 ˇ ˇZ ˇ 1 ˇˇ ht;bi ˇ

.r/dr ˇ  kak D ˇ ˇ ˇ 2 hs;bi 1 D 2



1 1  jht  s; bij  kak   kak  kt  skRd : 2 2

It follows that ' is a bounded Lipschitz function on Rd . Then ' 2 W 1;1 .Rd ; Rm / and r'.t/ D 12  .< t; b >/  .a ˝ b/: Let us show that ' is Q-periodic. R˛ Let t 2 Rd ; t D ˛1 p 1 C : : : C ˛d p d ; then '.t / D 12  . 0 1 .r/dr/  a. R ˛ C1 If s D .˛1 C1/p 1 C˛2 p 2 C: : : C˛d p d , then '.s/ D 12 . 0 1 .r/dr/a D '.t/; if s D t C p i with 1 < i  d , then '.s/ D '.t / because < p i ; b >D 0. Therefore ' is Q-periodic. According to Theorem 3.146 and to Remark 3.147,    Z  yCz yCz f  C r'.t / dt: f 2 2 Q Let Q1 D ¹t W 0 < 12 º and Q2 D ¹t W 12 < t; b >< 1º; then Q D Q1 [ Q2 and r'jQ1 D 12 .a ˝ b/; r'jQ2 D  12 .a ˝ b/. Therefore   Z     Z yCz yCz yCz a˝b a˝b f  C dt C  dt f f 2 2 2 2 2 Q1 Q2 f .y/ C f .z/ : D f .y/.Q1 / C f .z/.Q2 / D 2 As f is continuous, it follows that f ..1  /y C z/  .1  /f .y/ C f .z/;

8 2 Œ0; 1:

Therefore, f rank one convex. Let now m D 1; then a ˝ b D .ab1    abd /1 d D a  b: In this case, for every y; z 2 R1 d D Rd ; y  z 2 Rd and we can find a 2 R and b 2 Rd such that y  z D a ˝ b so that rank.y  z/  1. According to Proposition 3.152, rank one convexity is equivalent in this case with convexity. In the case d D 1, 0 1 a1 b B C a ˝ b D @ ::: Dba A am b m 1

310

Chapter 3 Young Measures

and again, for every y; z 2 Rm 1 D Rm , we can find a 2 Rm and b 2 R such that y  z D a ˝ b. Remark 3.154. In Example 3.142, we present a quasiconvex function which is not convex. In 1952, Ch. Morrey formulated the conjecture under which rank one convexity does not imply quasiconvexity if m; d  2. Šverák demonstrated in 1992 (see [156]) that Morrey’s statement is true if m  3 and d  2. The case where m D 2; d  2 is still open. A detailed discussion on this issue can be found in Chapter 5 of [50]. At the end of this section, we present the Kinderlehrer–Pedregal characterization of p-gradient Young measures (see [102]). Theorem 3.155. Let  2 Y.Rmd / and let 1  p < C1;  is a p-gradient Young measure if and only if the following conditions are satisfied: (1) There is u 2 W 1;p .; Rm / such that Z yd  t .y/; ru.t / D bar t Rmd

for almost every

t 2 :

(2) Let X p be the set of all bounded from below, continuous and quasiconvex functions f W Rmd ! R, which satisfy jf .y/j  c.1 C kykp /, for every y 2 Rmd . Then Z f .y/d  t .y/; for every f 2 X p and almost every t 2 : f .ru.t//  (3)

Rmd

R R Rmd

3.10.3

kykp d  t .y/dt < C1;

almost for every t 2 :

Lower Semicontinuity

We recall some notions and results about Sobolev spaces (for more complete information, see [36, 49, 50]). Let   Rd be a bounded open set. (1) The case 1  p < C1. W 1;p .; Rm / D ¹u W  ! Rm ju 2 Lp .; Rm /; ru 2 Lp .; Rmd /º. p p 1 The mapping u 7! kukW 1;p D .kukp C krukp / p is a norm on W 1;p .; Rm / and .W 1;p .; Rm /; k  kW 1;p / is a separable Banach space. 8 w < a) un ! u kkp w w p L I un ! u H) un ! u: un ! u ” w : b) run ! ru Lp W 1;p W 1;p p L

311

Section 3.10 Gradient Young Measures

If 1 < p < C1, then this space is reflexive. 1;p

W0

W 1;p

.; Rm / D Cc1 .; Rm /

:

(2) The case p D C1. W 1;1 .; Rm / D ¹u W  ! Rm ju 2 L1 .; Rm /; ru 2 L1 .; Rmd /º. The mapping u 7! kukW 1;1 D max¹kukL1 ; krukL1 º is a norm on W 1;1 and .W 1;1 .; Rm /; k  kW 1;1 / is a Banach space. 8 w < a) un ! u 1

w

un ! u ”

L

w

w

: b) run ! ru 1

W 1;1

kk1

I un ! u H) un  ! u: 1 W 1;1

L

L

W01;1 .; Rm / D W 1;1 .; Rm / \ W01;1 .; Rm /:

Remark 3.156. .i/ If 1 < p  C1 and .un /n  W 1;p .; Rm / is a bounded sequence, then there exist u 2 W 1;p .; Rm / and a subsequence .ukn /n of .un /n such that w

w

W 1;p

W 1;1

ukn ! u .ukn ! u; if p D C1/: .ii/ W 1;1 .; Rm /  W 1;p .; Rm /, for every 1  p < C1. For every .un /n  W 1;1 .; Rm / and every u 2 W 1;1 .; Rm /, w

w

W 1;1

W 1;p

un ! u H) un ! u;

for every 1  p < C1:

Definition 3.157. Let f W   Rm  Rmd ! R; f is a Carathéodory function if (1) t 7! f .t; x; y/ is a measurable mapping, for every .x; y/ 2 Rm  Rmd : (2) .x; y/ 7! f .t; x; y/ is continuous on Rm  Rmd , for almost every t 2 . Definition 3.158. Let f W   Rm  Rmd ! R be R a Carathéodory function N defined by I.u/ D and let I W W 1;p .; Rm / ! R f .t; u.t /; ru.t //dt (we suppose that the integral exists).

312

Chapter 3 Young Measures

Let 1  p < C1; I is sequentially weakly lower semicontinuous (wlsc) on W 1;p .; Rm / if, for every sequence .un /n  W 1;p .; Rm / and every u 2 w W 1;p .; Rm / with un ! u, I.u/  lim inf n I.un /: W 1;p

Let p D C1; I is sequentially weak* lower semicontinuous (w*lsc) on W 1;1 .; Rm / if, for every sequence .un /n  W 1;1 .; Rm / and every u 2 w W 1;1 .; Rm / with un ! u, I.u/  lim inf n I.un /: W 1;1

In the following, we usually omit the word “sequentially” in the above definitions. N is Remark 3.159. According to (ii) of Remark 3.156, if I W W 1;p .; Rm / ! R wlsc, then I jW 1;1 . ;Rm / is w*lsc. In the following, we present the relationships between the wlsc of I and the quasiconvexity of f ; briefly, I is wlsc if and only if f .t; x; / is quasiconvex. We begin with a particular case (see Theorem 3.13 of [50]). Theorem 3.160. Let 1  p  C1, let f W Rmd ! RR be a continuous function, N be defined by I.u/ D and let I W W 1;p .; Rm / ! R f .ru.t //dt: 1;p m If I is wlsc in W .; R / (w*lsc in W 1;1 .; Rm /), then f is a quasiconvex function. Proof. (1) Firstly, we suppose that p D C1. Let Q   be a hypercube with the edge length a, let 0 1 y11 : : : y1d A ::: yD@ 2 Rmd ym1 : : : ymd m d and let ' 2 W 1;1 .; Rm / be a Q-periodic function. We define 'n W Rd ! Rm letting 'n .t/ D n1  '.nt/. Then .'n /n2R is a bounded sequence in W 1;1 .; Rm /. According to Remark 3.156 (i), there exist '0 2 W 1;1 .; Rm / w and a subsequence .'kn /n of .'n /n such that 'kn ! '0 . Since 'n ! 0, W 1;1

kk1

! '0 , it follows that '0 D 0. Therefore, almost everywhere onRd and 'kn  1 L

w

'kn ! 0.

1 t1 C B ; We remark that, every t D @ ::: A td d 1 W 1;1

0

y  t 2 Rm .

313

Section 3.10 Gradient Young Measures

Let now un W  ! Rm , defined by ² y  t; t 2  n Q; un .t/ D 'n .t/ C y  t; t 2 Q: Then .un /n  W 1;1 .; Rm / and ² y; t 2  n Q; run .t/ D r'.nt/ C y; t 2 Q: w

Obviously, ukn ! u, where u.t / D y  t . W 1;1

Since I is w*lsc in W 1;1 .; Rm /, I.u/  lim inf I.ukn /: R

n

(*)

D f .y/  ./ and, for every n 2 N  , Z Z Z I.ukn / D f .rukn .t//dt D f .rukn .t //dt C f .rukn .t //dt nQ Q Z f .y C r'.kn t //dt D f .y/  . n Q/ C Q Z 1 f .y C r'.s//ds D f .y/  . n Q/ C d kn kn Q Z f .y C r'.s//ds: D f .y/  . n Q/ C

Now, I.u/ D

f .y/dt

Q

Therefore, by . /,

Z

f .y/  ./  f .y/  . n Q/ C from where

Q

f .y C r'.s//ds;

Z f .y/  .Q/ 

Q

f .y C r'.s//ds:

Now, if we suppose that the edge length a is so that the hypercube Q1 D .0; 1/d is an union of l d hypercubes of edge length a, then we obtain Z f .y C r'.t //dt: f .y/  Q1

By Theorem 3.146, f is quasiconvex. (2) Let now 1  p < C1; if I is wlsc in W 1;p .; Rm /, then, according to Remark 3.159, I jW 1;1 . ;Rm / is w*lsc and so f is quasiconvex.

314

Chapter 3 Young Measures

For the general case, we need to define the following growth condition. Definition 3.161 (see Definition 8.10 of [50]). Let 1 < p < C1 and let f W   Rm  Rmd ! R be a Carathéodory function. We say that f satisfies the condition .Cp / if, for almost every t 2  and every .x; y/ 2 Rm  Rmd , ˛.kxk C kykq /  ˇ.t/  f .t; x; y/  g.t; x/.1 C kykp /;

(Cp )

where ˛; ˇ; g  0; ˇ 2 L1 ./, 1  q < p and g is a Carathéodory function. In the case p D 1, jf .t; x; y/j  ˛.1 C kyk/;

(C1 )

where ˛ > 0. Theorem 3.162 (Morrey–Acerbi–Fusco). Let  be a bounded open subset of Rd with a Lipschitz boundary, let 1  p < C1 and let f W   Rm  Rmd ! R be a Carathéodory function which satisfies the condition .Cp /. R N defined by I.u/ D The functional I W W 1;p .; Rm / ! R f .t; u.t /; ru.t//dt , is sequentially weakly lower semicontinuous in W 1;p .; Rm / if and only if f .t; x; / is quasiconvex, for almost every t 2  and every x 2 Rm . We omit the laborious demonstration of this theorem; it can be found in [50] (Lemma 3.18 and Theorem 8.11). Remark 3.163. .i/ This theorem is due to Morrey (see [121] and [122]). The variant for Carathéodory functions is due to Acerbi–Fusco (see [1] where it is proved for f  0). .ii/ In [100], A. Kałamajska has proved the sufficiency of this theorem in the case f  0 using a Jensen-type inequality for p-gradient Young measures.

3.11

Relaxed Solutions in Variational Calculus

Young measures were introduced as a need to provide generalized solutions to problems which do not admit classical solutions. With an illustrative purpose and without any intention of exhaustivity, we seek a classification of the bibliography, represented by works that use Young measures as a tool, in a few areas. Please note that these fields have ambiguous boundaries, some of the cited works could being able to belong to many of them.

Section 3.11 Relaxed Solutions in Variational Calculus

315

(1) Nonlinear differential equations and differential inclusions: Ch. Castaing, P. Raynaud de Fitte and A. Salvadori [47], S. Demoulini [55], H.-P. Gittel [90], P. Pedregal [135], G.-Q. Song and J. Xiao [155], S. Taheri, Q. Tang and K. Zhang [162], J. Yin and C. Wang [174], P. Zhang and Y. Zheng [180] and K. Zhang [182]. (2) Nonlinear homogenization problems and -convergence: L. Ambrosio, H. Frid and J. Silva [3], L. Ambrosio and H. Frid [4], M. Barchiesi [25], A. V. Demyanov [53], G. Michaille and M. Valadier [120], D. Piau [136], P. Pedregal [133], P. Pedregal and H. Serrano [134], D. Piau [136], H. Serrano [153], L. Tartar [165] and A. Visintin [172]. (3) Continuum mechanics, nonlinear elasticity, microstructures theory, magnetic phenomena in fero-magnetic materials and hysteresis: A. L. Bessoud [27], C. Carstensen and A. Prohl [40], C. Carstensen and M. O. Rieger [41], G. Dal Maso, A. DeSimone, M. G. Mora and M. Morini [51], L. Freddi and R. Paroni [87, 88], M. Hillairet [94], P. Kami´nski [101], Ph. G. LeFloch and M. Westdickenberg [111], C. De Lellis and L. Székelyhidi Jr. [112], C. Melcher [119], A. Münch, P. Pedregal and F. Periago [123, 124], S. Owczarek [127], P. Pedregal [132], M. O. Rieger and J. Zimmer [140, 142], M. O. Rieger [141], T. Roubi˘cek and M. Kru˘zik [146, 147] and F. Theil [166]. (4) Economic equilibrium theory and games theory: E. J. Balder [17, 20, 21], A. M. Ramos and T. Roubi˘cek [138] and A. Rustichini and P. Siconolfi [148]. However, we should note that the original reason for the emergence of Young measures has been to give relaxed solutions to variational problems. Many problems, formulated in the language of optimization theory, lack classical optimal solutions; the main reason is the rapid oscillations of its minimizing gradient sequences. This phenomenon requires the extension of the notion of solution of such problems. L. C. Young introduced in 1937 so called generalized curves (see [177]) and, afterwards, the generalized surfaces ([178]). The technique of Young measures was used then to obtain relaxed solutions by J. Warga ([173]), E. J. Balder ([9]), M. Valadier ([169]) and P. Pedregal ([128, 130]). We can also cite the works of C. Carstensen and T. Roubi˘cek ([39]), C. Castaing, A. Jofre and A. Salvadori ([46]), M. Chipot ([48]), J. F. Edmond ([64]), J. J. Egozcue, R. Meziat and P. Pedregal ([67]) and J. Mach ([116]). In this last section, we present applications of Young measures in this direction: the conditions for existence of relaxed solutions to problems of variational calculus.

316

Chapter 3 Young Measures

The standard problem of the calculus of variations is to find minimizers of functional Z I.u/ D f .t; u.t /; ru.t //dt;

where  is a bounded open subset of Rd , the admissible mappings u W  ! Rm belong to a set of competitors H  W 1;p .; Rm /; 1  p  C1 and f W   Rm  Rmd ! R is a Carathéodory integrand, i.e.: a) .x; y/ 2 Rm  Rmd 7! f .t; x; y/ is continuous, for almost every t 2 . b) t 2  7! f .t; x; y/ is measurable, for every .x; y/ 2 Rm  Rmd . In short, the direct method in variational calculus is the following: Let a minimizing sequence .un /n  H for the problem: inf¹I.u/ W u 2 H º D m:

(P)

This means that I.un / ! m. Using some compactness results (to be specified), we can obtain a subsequence (still noted .un /n ) weakly convergent to u0 in W 1;p .; Rm / (weak* convergent in the case p D C1). If I is sequentially weakly lower semicontinuous (weak* l.s.c. in the case p D C1), then I.u0 /  lim inf I.un / D m  I.u0 /: n!C1

Therefore, u0 is a minimizer for the problem .P /. This method is fruitful when a convexity condition in the gradient variable is assumed on the integrand f (I is weakly lower semicontinuous if and only if f .t; x; / is quasiconvex). For more detailed information on this approach, we indicate the books [49, 50, 85]. Unfortunately, many problems formulated in the language of optimization theory lack classical optimal solutions. So, non-convex optimization problems may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of classical solutions for such problems; often we can obtain a such relaxation by means of Young measures (see [45, 85, 128, 145, 163, 173, 176]). We will extend the class of competitors to HN and then we will extend the functional I to IN W HN ! R; therefore we arrive at the relaxed problem: inf¹IN.u/ N W uN 2 HN º D m: N

(PN )

317

Section 3.11 Relaxed Solutions in Variational Calculus

We will give some conditions that the problem .PN / to have a minimizer. If, moreover, F is quasiconvex in the gradient variable, then we obtain the known solutions for the problem .P /. Let 1  p < C1;   Rd be a bounded open set with a Lipschitz boundary and let f W   Rm  Rmd ! R be a Carathéodory integrand for which there exist a; b > 0 such that, for every x 2 Rm , every y 2 Rmd and almost every t 2 : a  kykp  1  f .t; x; y/  b  .1 C kykp /:

(C)

We remark that, in this case, the condition .Cp / from the above section is satisfied, for every 1  p < C1. The classical problem of variational calculus is: ² ³ Z 1;p m inf I.u/ D f .t; u.t/; ru.t//dt W u 2 W0 .; R / D m: (P)

1;p

We remark that, for every u 2 W0

.; Rm /,

jf .t; u.t /; ru.t //j  c  .1 C ku.t /kp /;

for almost every t 2 ;

where c D max¹a; b; 1º. Therefore f .; u./; ru.// 2 L1 ./. 1;p Moreover, for every u 2 W0 .; Rm / and for almost every t 2 , 1  f .t; u.t/; ru.t // hence

 ./  I.u/

f .t; 0; r0/  b

I.0/  b  ./:

and so

and

Then 1 < m < C1. The following example shows that, generally, the problem .P / has no minimizer. Example 3.164 (Bolza example). Let d D m D 1 and let f W .0; 1/RR ! R, defined by f .t; x; y/ D .y 2 1/2 Cmin¹x 2 ; 1º, for every .t; x; y/ 2 .0; 1/RR. Then f satisfies the condition .C /: 1 4  y  1  f .t; x; y/  3.1 C y 4 /; 2

8t 2 .0; 1/;

8x; y 2 R:

If we consider u0 D 0 2 W 1;4 ..0; 1/; R/, then the problem becomes: ² Z inf I.u/ D

1 0

³  ..u0 .t//2  1/2 C min¹u2 .t /; 1º dt W u 2 W 1;4 D m:

318

Chapter 3 Young Measures

Let .un /n  W01;4 ..0; 1/; R/, defined by un .t/ D

2n1 X1  kD0

2k t n 2



 

2kC1 Π22k n ; 2n /

.t/ C

 2k C 2  t  2kC1 2kC2 .t / : Π2n ; 2n / 2n

Obviously, 0  un  21n , for every n 2 N  . Then .un /n is uniformly convergent to 0 on .0; 1/ and u0n .t/ D signŒsin .2n  t /, for a.e. t 2 .0; 1/ (.u0n /n is the Rademacher sequence). Therefore u0n D ˙1, for a.e. t 2 .0; 1/ and so I.un / ! 0; then m D 0. The sequence .un /n is a minimizing one, but there is no function u 2 W 1;4 ..0; 1/; R/ for which I.u/ D 0; hence the problem .P / has no minimizer. We note that the integrand f is not convex in the gradient variable and, therefore, it is not quasiconvex (see Theorem 3.153). According to Theorem 3.162, I w is not wlsc on W 1;4 ..0; 1/; R/; we can remark that un ! 0 but I.0/ D 1 > W 1;4

0 D lim inf n I.un /.

To obtain a satisfactory minimum, we will enlarge the set of competitors 1;p W0 .; Rm / to a set HN and we will extend the functional I on HN . 1;p The following results can be easily extended if we replace W0 .; Rm / with 1;p the broader class of competitors u0 C W0 .; Rm /, where u0 2 W 1;p .; Rm / is a fixed function. 1;p Definition 3.165. Let HN  W0 .; Rm /  Y.Rmd / be the subset of all .u;  / 1;p for which there exists .un /n  W0 .; Rm / such that S

(a) .un ; run / !  u ˝  ; Rm Rmd

(b) .krun kp /n

is uniformly integrable in

L1 ./:

We say that .un /n generates .u;  /. S

Proposition 3.166. .un ; run / !  u ˝  if and only if Rm Rmd



un ! u M.Rm /

S

and run !  : Rmd

Proof. First, we recall that  u is the Young measure associated to u (see Definition 3.15) and  u ˝  is the fiber product of  u and  (see Definition 3.86).

319

Section 3.11 Relaxed Solutions in Variational Calculus S

Therefore, .un ; run / !  u ˝  if and only if, for every A 2 A and every Rm Rmd

f 2 Cb .Rm  Rmd /, Z Z Z f .un .t/; run .t//dt ! f .x; y/d. tu ˝  t /.x; y/dt m md A Z A Z R R D f .u.t /; y/d  t .y/dt: Rmd

A

S

Let us suppose that .un ; run / ! Rm Rmd

 u ˝  ; let A 2 A and f

2 Cb .Rm /.

We define f1 W Rm  Rmd ! R letting f1 .x; y/ D f .x/, for every .x; y/ 2 Rm  Rmd ; then f1 2 Cb .Rm  Rmd / and so Z Z f .un .t//dt ! f .u.t //dt: A

A

S



Therefore,  un !  u and, according to Proposition 3.40, un ! u. Rm

On the other hand, if A 2 A, f 2 Cb and if we note f2 R letting f2 .x; y/ D f .y/, then f2 2 Cb .Rm  Rmd / and so Z Z Z f .run /dt ! f .y/d  t .y/dt: .Rmd /

A

A

M.Rm / W Rm  Rmd

!

Rmd

S

Therefore,  run !  . Rmd

The sufficiency is a direct consequence of fiber product lemma (see Theorem 3.87). Remark 3.167. According to the previous proposition, .u;  / 2 HN if and only if 1;p there exists .un /n  W0 .; Rm / such that: 

(1) un ! u. M.Rm / S

(2) run !  . Rmd

(3) .krun kp /n is uniformly integrable in L1 ./. We remark that  is a p-gradient Young measure (see Definition 3.134) generated by .run /n and ru D bar: . According to (i) of Remark 3.135, w run ! bar: . L1 . ;Rmd /

320

Chapter 3 Young Measures

Using Proposition 3.137, we can obtain: 1;p Proposition 3.168. Let .u;  / 2 HN ; there exists .un /n  W0 .; Rm / such that: kkp

u. (1) un ! p L

w

ru D bar: . (2) run ! p L

w

(3) un ! u. W 1;p

Remark 3.169. If .u;  / 2 HN , then  u ˝  is a p-Young measure on Rm Rmd . 1;p

Indeed, let .un /n  W0

S

.; Rm / such that .un ; run / ! Rm Rmd

u ˝ 

and .krun kp /n is uniformly integrable in L1 ./. According to the previous kkp

proposition, un ! u; then .kun kp /n is uniformly integrable in L1 ./. Since p L

p

p

p

k.un ; run /kRm Rmd  2p  .kun kRm C krun kRmd /;

for every n 2 N;

p

.k.un ; run /kRm Rmd /n is uniformly integrable in L1 ./ and then  u ˝  is a p-Young measure. Proposition 3.170. 1;p (1) For every u 2 W0 .; Rm /; .u;  ru / 2 HN ; the mapping u 7! .u;  ru / is 1;p 1;p an injection of W0 .; Rm / in HN . Therefore, W0 .; Rm / ,! HN .

(2) Let v 2 M.Rmd /; then .u;  v / 2 HN ” v 2 Lp .; Rmd / 1;p

Proof. (1) For every u 2 W0

and v D ru: 1;p

.; Rm /, let .un /n  W0 

.; Rm / be defined 

by un D u, for every n 2 N. Then un ! u and run ! ru, so that M.Rm /

 run

S

! 

 ru .

M.Rmd /

Obviously, .kun kp /n is uniformly integrable and so .u;  ru / 2

HN . 1;p (2) Suppose that .u;  v / 2 HN ; there exists .un /n  W0 .; Rm / such that kkp

S

u, run !  v and .krun kp /n is uniformly integrable. Using Propoun ! p L

Rmd

321

Section 3.11 Relaxed Solutions in Variational Calculus 

sition 3.40, run ! v. Now we can apply the Vitali’s theorem (see TheoM.Rmd /

kkp

v. rem 3.118) to obtain that v 2 Lp .; Rmd / and run ! p L

For every ' 2 Cc1 .; R/ and every n 2 N, Z Z T un  r'dt D  run  'dt

(*)



(for the definition of uTn  ', see 3.151). R R kkp Since un ! u, uTn  r'dt ! uT  r'dt . On the other hand, p L

ˇZ ˇ   ˇ ˇ 1 1 ˇ .v  run /  'dt ˇ  kv  run kp  k'kq C D1 ˇ ˇ p q R R and, therefore, run  'dt ! v  'dt: Passing to limit in . /, we get Z Z T u  r'dt D  v  'dt; for every ' 2 Cc1 .; R/:



Therefore v D ru. Conversely, if v 2 Lp .; Rmd / and v D ru, then, from 1), we obtain .u;  v / 2 HN . Remark 3.171. .u;  /; .u;  / 2 HN »  D  . Indeed, let .un /n  W01;4 ..0; 1/; R/  W 1;1 ..0; 1/; R/ be the sequence defined in Example 3.164. .un /n is uniformly convergent to 0 on .0; 1/ and .run /n S

is the Rademacher’ sequence. In the Example 3.44 we have shown that run !  D  ˝ 12  .ı1 C ı1 /. Obviously, .jrun j/n is uniformly integrable. Therefore .0;  / 2 HN and bar: D 0 D r0. By (1) of the previous proposition, .0;  r0 / D .0;  0 / 2 HN . But the disintegration of  0 is ı0 ¤ 12  .ı1 C ı1 / which is the disintegration of  .



We consider now the following relaxation of problem .P /: ³ ² Z N N f .t; u.t/; y/d  .t; y/ W .u;  / 2 H D m; N inf I .u;  / D

(PN )

Rmd

where f is a Carathéodory integrand satisfying, for every x 2 Rm , every y 2 Rmd and almost every t 2 , the condition: a  kykp  1  f .t; x; y/  b  .1 C kykp /; with a; b > 0.

(C)

322

Chapter 3 Young Measures

1;p For every u 2 W0 .; Rm /, .u;  ru / 2 HN and I.u/ D IN.u;  ru /; therefore 1;p IN extends I from W0 .; Rm / to HN and then m N  m < C1. Next we present some existence results for the relaxed problem .PN /; in the particular case where f is quasiconvex in the gradient variable, we obtain the existence results for the classical problem .P /. For similar results, see Pedregal’s monograph [128] (Theorem 4.4 and Corollary 4.5).

Proposition 3.172. m D m. N Proof. We have remarked that m N  m. 1;p For every .u;  / 2 HN , there exists a sequence .un /n  W0 .; Rm / such that S

.un ; run / !  u ˝  and .krun kp /n is uniformly integrable in L1 ./. Rm Rmd

According to condition .C /, .f .:; un .:/; run .:///n is uniformly integrable and then, by Corollary 3.36, Z Z N f .t; u.t/; y/d  t .y/dt I .u;  / D md Z ZR D f .t; x; y/d. tu ˝  t /.x; y/dt m md R R Z f .t; un .t/; run .t //dt  m: D lim n



Therefore, m N  m and then m D m. N Theorem 3.173. Let f W   Rm  Rmd ! R be a Carathéodory integrand satisfying the condition .C /; there exists .u;  / 2 HN such that IN.u;  / D m. N m If, moreover, almost for every t 2  and for every x 2 R , f .t; x; / is quasiconvex, then .P / admits a minimizer. 1;p

Proof. Let .un /n  W0 .; Rm / be a minimizing sequence for problem .P /; i.e. I.un / ! m D m: N According to Lemma 4.3 from [128] (p. 66), we can always suppose that .krun kp /n is uniformly integrable. By condition .C /, a  krun kp  f .t; un .t/; run .t// C 1; from where a

for every n 2 N;

Z

krun kp dt  I.un / C ./;

for every n 2 N:

Therefore .run /n is a bounded sequence in Lp .; Rmd /. According to the 1;p Poincaré inequality (see Theorem 1.47 in [49], .un /n is bounded in W0 and,

323

Section 3.11 Relaxed Solutions in Variational Calculus

by Rellich–Kondrachov theorem (Theorem 12.12 of [50]), up to a subsequence, .un /n is strongly convergent to a function u 2 Lp .; Rm /. 1;p Let us show that u 2 W0 .; Rm /. (a) Let first suppose that p > 1; then W 1;p .; Rm / is reflexive and, since .un /n is bounded in W 1;p .; Rm /, up to a subsequence, .un /n is weakly conver1;p 1;p gent to a function u0 2 W0 .; Rm /; then u D u0 2 W0 .; Rm /. (b] Let p D 1. For every ' 2 Cc1 .; R/, every i 2 ¹1; : : : ; mº and every j 2 ¹1; : : : ; d º, Z Z @' @uin i un  dt D   'dt; for every n 2 N; (*) @tj @tj 1 m where un D .u1n ; : : : ; um n / and u D .u ; : : : ; u /. .run /n is uniformly integrable in L1 .; Rmd / and then, up to a subsequence, it is weakly convergent to a function v D .vji /1i m;1j d 2 L1 .; Rmd /. w

w

L1

L1

Since un ! u and run ! v, for every i 2 ¹1; : : : ; mº and every j 2 ¹1; : : : ; d º, Z Z @' @' i un  dt D ui  dt lim n @tj @tj

Z

Passing to limit in . /, we obtain Z Z i @' u  dt D  vji  'dt; @t j Then

@ui @tj

lim

and

n



@uin  'dt D @tj

Z

vji  'dt:

for every ' 2 Cc1 .; R/:

D vji 2 L1 ./, for every i 2 ¹1; : : : ; mº and every j 2 ¹1; : : : ; d º. kk1

Therefore, u 2 W 1;1 .; Rm /. Since un ! u; uj@ D 0 and then u 2 L1

W01;1 .; Rm /. 1;p

Therefore, for every 1  p < C1; u 2 W0

kkp

.; Rm / and un ! u. p L

.run /n being bounded in Lp .; Rmd / it is bounded in L1 .; Rmd /. According to Proposition 3.56, it is tight and then we can apply Prohorov’s theorem (see Theorem 3.64). Therefore, .run /n is S-convergent, up to a subsequence, to a Young measure  2 Y.Rmd /. S Then .un ; run /n !  .u;  / and so .u;  / 2 HN . On the other hand, using again the condition .C /, we obtain, for every n 2 N and for almost every t 2 , jf .t; un .t/; run .t//j  c  .1 C krun .t/kp /;

where

c D max¹a; b; 1º:

324

Chapter 3 Young Measures

Therefore, .f .:; un .:/; run .://n is uniformly integrable in L1 ./ and then, Corollary 3.36 allow us to write: Z Z N I .u;  / D f .t; u.t/; y/d  t .y/dt Rmd Z f .t; un .t/; run .t //dt D lim I.un / D m: N D lim n

n



Now, let us suppose that, almost for every t 2  and for every x 2 Rm , f .t; x; / is quasiconvex. At begin of this subsection, we remarked already that .C / implies .Cp / and, therefore the conditions of Morrey–Acerbi–Fusco theorem are accomplished (see Theorem 3.162). Then I is sequentially weakly lower semicontinuous in W 1;p .; Rm /. w With above notations, we can decide that un ! u (see also ProposiW 1;p

tion 3.168). Therefore, I.u/  lim inf I.un / D lim I.un / D m n

n

and so I.u/ D m. Remark 3.174. In the case of Bolza problem, the relaxed problem is: ² ³ Z   2 2 2 inf IN.u;  / D .y  1/ C min¹u .t /; 1º d  .t; y/ W .u;  / 2 HN : .0;1/ R

According to Theorem 3.173, .PN / admits as minimizer. The sequence .un /n from the Example 3.164 is uniformly convergent to the constant function u0 D 0 and the sequence of gradients (the Rademacher sequence) is stably convergent to Young measure  0 whose disintegration is constant  t0 D 12  .ı1 C ı1 /, for every t 2 .0; 1/. Therefore m D IN.u0 ;  0 /. We note that we have no solution for problem .P /.

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Index

W 1;1 .; Rm / 311 W 1;p .; Rm / 266, 310 W01;1 .; Rm / 311 W01;p .; Rm / 311 BL.T; d / 136 C.T /, Cb .T /, C1 .T / 105 C0 .T /, Cc .T / 105 C0 .Rm / 289 Cc1 ./ 265 Cc1 .; Rm / 289 C t hb .  S / 185 L1 -bounded integrand 185 Lp .; ; E/ 177 Q-periodic function 302 ı t 90 P ˝-product 258 BO T . / 91 BO T .; / 90 -equicontinuity of functions 38 of measures 20 -negative set 3 -null set 3 -positive set 3 BR , BŒ0;1 176 A-partition 1 B.Rm / 288 BS 176 BT 90 C 176 KT 90 Lp .; ; E/; Lp .; ; E/ 177 L1 .; ; E/; L1 .; ; E/ 177 M.X / 177 M.R/, M 177 M.Rm / 288 P .Rm / 289

PS 176 Y.  S /; Y.S /; Y 188 T.T /; T 106 Tc .T /; T0 .T /, T1 .T / 106 a-compactness of a set of functions 70 of a set of measures 130 a-convergence of a sequence of functions 70 of a sequence of measures 129 p-Young measure 291 p-equicontinuous 276 p-gradient Young measure 294 p-uniformly integrable 276 rca.BT /, rcaC .BT / 91 t ca.BT / 93 w 2 -convergence 72, 249 Lip.; Rm / 289 ba.A/, baC .A/, ca.A/, caC .A/ S.Y.S //; S 194 Rca.BT / 94 absolutely continuous measure 17 algebra of cylinder sets 148 barycenter of a measure 208, 272 bidual of L1 56 biting lemma vector case 237 biting lemma finite-tight case 249 scalar case 85 Bolza example 317 Carathéodory integrand 185 Carathéodory function 311

2

337

Index ˇ Cech-complete space 134 condition of strong concentration 275 of weak concentration 280 convergence w 2 72, 249 in measure 38 narrow 106 stable 194 strong 38 cylinder set 148 decomposition of a measure 5 of a set 4 derivative of a measure 32 Dirac measure 90 disintegration of a measure 183 disintegration theorem 179 dual of L1 35 Dudley’s metric 138 Dunford–Pettis theorem 52 extremal point 271 family of functions -equicontinuous 38 p-equicontinuous 276 p-uniformly integrable 276 uniformly integrable 68 of measures -equicontinuous 20 fiber product 253 fiber product lemma 253 finite-tight set 245 function .A  BX /-measurable 177 Q-periodic 302 -integrable 13 Carathéodory 311 quasiconvex 297 rank one convex 307 functional w*lsc 312 wlsc 312

Gutman theorem 287 Hahn decomposition of a set 4 hereditarily Lindelöf space 95 Hewitt–Yosida theorem 60 inf-compact integrand 222 integrand 185 L1 -bounded 185 Carathéodory 185 inf-compact 222 l.s.c. 185 u.s.c. 185 Jordan finite-tight set 261 Jordan decomposition of a measure 5 Kinderlehrer–Pedregal theorem 310 Kuratowski upper limit 123 l.s.c. integrand 185 Lévy–Prohorov metric 142 Le Cam’s theorem 146 Lebesgue decomposition 35 Lebesgue’s decomposition theorem 33 Lindelöf space 95 Lions–Aubin theorem 286 measure -continuous 19  -additive 1  -finite 2 absolutely continuous 17 additive 1 concentrated on a set 5 Dirac 90 disintegration 183 finite 2 negative variation 9 positive variation 9 pseudo-metric associated 15 purely finitely additive 57 Radon 94 regular 91

338 tight 93 total variation 9 Young 188 measures orthogonal 33, 57 singular 33 metric Dudley 138 Lévy–Prohorov 142 modulus of -continuity 63 of narrow compactness 129 of uniform continuity 151 of uniform integrability 64 Morrey–Acerbi–Fusco theorem 314 narrow convergence 106 narrow topology 106 Nikodym boundedness theorem 25 Nikodym theorem 22 orthogonal  -additive measures 33 additive measures 57 Polish space 96 product of Young measures fiber 253 tensor 258 Prohorov space 134 Prohorov’s theorem 132 Prohorov’s theorem 229 purely finitely additive measure 57 quasiconvex function 297 Rademacher’s sequence 42, 213 Radon measure 94 Radon space 94 Radon–Nikodym theorem 29 random variable 156 k-th central moment 156 k-th moment 156 cumulative distribution 156 cumulative frequency 156

Index distribution function 156 expectation 156 independence 157 mean 156 normal distributed 157 normal distribution 157 standard deviation 156 variance 156 rank one convex function 307 regular measure 91 Riesz–Fréchet–Kolmogorov theorem 275 Rossi–Savaré theorem 286 Saadoune–Valadier theorem 250 Schur theorem 45 second-countable space 95 separable space 95 singular measures 33 Sobolev space 266 space ˘ Cech-complete 134 hereditarily Lindelöf 95 Lindelöf 95 Polish 96 Prohorov 134 Radon 94 second-countable 95 separable 95 Sobolev 266 Suslin 96 stable convergence 194 stable topology 194 strong concentration condition 275 strong convergence in L1 38 subsequence splitting lemma 88 support of a measure 95 Suslin space 96 tensor product 258 theorem bidual of L1 56 biting lemma 85 boundedness Nikodym 25 disintegration 179

339

Index dual of L1 35 Dunford–Pettis 52 fiber product lemma 253 Gutman 287 Hewitt–Yosida 60 Kinderlehrer–Pedregal 310 Le Cam 146 Lebesgue’s decomposition 33 Lions–Aubin 286 Morrey–Acerbi–Fusco 314 Nikodym 22 of density 218, 219 Prohorov 132, 229 Radon–Nikodym 29 Riesz–Fréchet–Kolmogorov 275 Rossi–Savaré 286 Saadoune–Valadier 250 Schur 45 subsequence splitting lemma 88 Visintin–Balder 273 Vitali 39, 277 Vitali–Hahn–Saks 20 weak compactness in ca.A/ 48 weak convergence in ca.A/ 44 weak convergence in L1 43 tight finite-tight 245 Jordan finite-tight 261 sequence 128 set of measurable functions 227 set of measures 128 set of Young measures 225 tight measure 93 topology narrow 106 of convergence in measure 209 stable 194

vague 106 total variation 9 u.s.c. integrand 185 uniformly integrable 68 universally measurable sets 91 vague topology 106 Visintin–Balder theorem 273 Vitali theorem 39, 277 Vitali–Hahn–Saks theorem 20 w*lsc functional 312 weak compactness in ca.A/ 15, 48 in L1 ./ 52, 66 weak concentration condition 280 weak convergence in W 1;p .; Rm / 311 in `1 45 in ca.A/ 44 in L1 273 in L1 ./ 43, 78, 79, 88 weak* convergence in W 1;1 .; Rm / 311 Wiener measure 167 wlsc functional 312 Yosida’s transform 108 Young measure associated to a probability 189 Young measure 188 p-Young measure 291 p-gradient Young measure 294 associated to an application 191 generated by a sequence 291 simple 190

About the Authors

Prof. Dr. Liviu C. Florescu teaches measure theory, harmonic analysis on topological groups, compactness in measure spaces and Young measures, general topology, and functional analysis at the “Alexandru Ioan Cuza” University of Ia¸si (Romania). In published works, he studied problems related to probabilistic topological structures, compactness in measure spaces, Young measures and applications to relaxed variational calculus.

Prof. Dr. Christiane Godet-Thobie is member of the Math Research Laboratory UMR 6205 (CNRS) of the Université de Bretagne Occidentale, Brest (France). She was member of the Scientific Committee of the international conference “Mesures de Young et Contrôle Stochastique” and organizer of the part “Measures de Young” (Brest, 2002). Her research interests are Young measures, set-valued analysis, fixed point theory, and differential inclusions.