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(1)Lectures on Young measure theory and its applications in economics Citation for published version (APA): Balder, E. J. (1998). Lectures on Young measure theory and its applications in economics. (Rijksuniversiteit Utrecht. Mathematisch Instituut : preprint; Vol. 1052). Utrecht University.. Document status and date: Published: 01/01/1998 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne. Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim.. Download date: 10. Sep. 2021.

(2) Lectures on Young Measure Theory and its Applications in Economics (nal version) Erik J. Balder Mathematical Institute, University of Utrecht, Netherlands September, 1998. 1 Introduction The rst four sections of these notes form a quick, incisive introduction to the subject of Young measure theory. The term Young measures refers to transition probabilities that are studied in connection with a certain weak topology (i.e., the narrow topology for Young measures). This name honors L.C. Young, whose seminal work on generalized solutions in the calculus of variations in 1937 98] formed the starting point of such considerations. Our presentation involves very little functional analysis, and is largely based on a transfer of the classical theory of narrow convergence from the domain of probabilities (section 2) to the more general domain of transition probabilities (section 4) by means of K -convergence and an associated key Prohorov-type extension of Komlos' theorem (Theorem 3.7). Such an extension of Komlos' theorem applies, much more generally than displayed here, to certain classes of abstract-valued scalarly integrable functions 18, 19, 38]. However, in the Young measure context it is particularly eective to transfer narrow convergence properties. This is because tightness, a crucial condition for Theorem 3.7, is, under mild restrictions, an automatic feature of narrow convergence of sequences of Young measures 25]. The useful portmanteau and product convergence theorems for classical narrow convergence, as well as Prohorov's theorem (an important device for relative narrow compactness) and certain limiting support properties are thus made available for Young measures. These results of section 4 form the point of departure for the second part of the notes, where lower closure (section 5), and variational inequalities and equilibria (section 6) are studied in connection with some existence questions in economics (viz., optimal growth, optimal consumption, CournotNash equilibrium distributions and Nash equilibria in continuum games and games with incomplete information). To keep these notes within a reasonable size, the choice has been made to discuss those applications at a great level of generality, and with little regard for the economical context. However, adequate references are suggested to ll this gap. Other surveys of Young measure theory include the account given in J. Warga's textbook 97] (largely control-theoretical and mostly limited to a compact image space, but going well beyond existence and lower closure issues), the study by H. Berliocchi and J.M. Lasry 42] (oering a locally compact image space and a Scorza-Dragoni-type connection with classical narrow convergence, but in many respects a very innovative study), M. Valadier's survey in 94] (presenting much of the material treated in sections 2 to 4 via a more functional-analytic approach and with rather dierent applications) and the present author's lecture notes 25], which cover more ground than the present paper, but do not address economical applications at the level of generality presented here. Let us also 67] for an apparently dierent approach to sequential narrow convergence on product spaces that can nevertheless be reduced to the present one 33], 61, p. 2]. For recent important applications in nonlinear analysis (that started with 91]) we refer to 83]. Nine lectures given at the School on Measure Theory and Real Analysis (GNAFA, CNR) in Grado, Italy (15-26 September, 1997).. 1.

(3) 2 Narrow convergence of probability measures This section recapitulates some results on narrow convergence of probability measures on a topological space cf. 4, 43, 44, 55, 75, 82]. We discuss these results in two settings: (i) a metrizable one, for which the material presented is rather standard and (ii) a nonmetrizable one, which includes the situation where the topological space is completely regular and Suslin. As a rule, we extend from (i) to (ii) via tightness. Let S be a completely regular topological space, whose topology we indicate by . Let B(S ) be the Borel -algebra on (S ) and let Cb (S ) be the set of all bounded -continuous functions on S. Throughout this paper we work with the following hypothesis:. Hypothesis 2.1 There exist a separable metric space P and a continuous mapping  : P ! S such that S = (P ).. Clearly, this hypothesis implies that the space (S ) is separable.. Proposition 2.2 There exists an (at most) countable collection (ci ) in Cb(S ) that separates the points of S (i.e., x = x if and only if ci (x) = ci (x ) for all i 2 N). Consequently, there exists a weak metric  on S whose topology  is such that    . 0. 0. Proof. Since P  P is second countable, it has the the Lindelof property. That is, every open subset of P  P has the countable subcover property. But then S  S, being the continuous surjective image of P  P, also has the Lindelof property. In particular, the complement C of the diagonal in S  S has the countable subcover property. Now C is covered by the collection of all open sets f(x x ) 2 S  S : c(x) 6= c(x )g, c 2 Cb(S). Hence, C is already covered by a countable subcollection, and this evidently corresponds to thePfact that there is a countable subset (ci) of Cb (S ) separating the points of S. Setting (x x ) := i=1 2 i(supS jcij) 1jci (x) ; ci (x )j then produces a metric on S, and the inclusion    is trivial. QED While we accept that the topologies  and  may be dierent, the associated Borel -algebras are required to be identical: 0. 0. 0. 1. ;. ;. 0. Hypothesis 2.3 The metric  in Proposition 2.2 is such that B(S  ) = B(S ) =: B(S): Two dierent sucient conditions for Hypothesis 2.3 to hold are as follows:. Remark 2.4 (i) If (S ) is a separable metric space, then it meets Hypotheses 2.1 and 2.3 trivially (let  be the postulated metric on S then  =  ). (ii) Let (S ) be completely regular and Suslin (i.e., a Hausdor space that is the surjective image of a complete, separable metric space under a continuous mapping

(4) 55, 89]). Then Hypothesis 2.1 evidently holds, and Hypothesis 2.3 holds by a well-known property of Suslin spaces

(5) 89, Corollary 2, p. 101].. Many useful spaces, e.g., Euclidean spaces, compact metric spaces, separable Banach spaces with their strong or weak topology are completely regular and Suslin (observe that innite-dimensional separable Banach spaces are not metrizable for their weak topology { this example explains why we are not just interested in the metrizable case). Let P (S) be the set of all probability measures on (S B(S)). Let Cb(S ) be the set of all  -continuous bounded functions from S into R.. Denition 2.5 A sequence (n) in PR(S) converges R narrowly with respect to the topology  to  0 2 P (S) (notation: n ) 0) if limn S cdn = S cd0 for every c in Cb (S ). 2.

(6)  ", is dened by replacing C (S ) The corresponding notion of -narrow convergence, denoted by \ ) b in the above denition by Cb(S ). Clearly, -narrow convergence implies  -narrow convergence by Proposition 2.2, but in some interesting cases the two convergence modes will actually coincide. A useful tool is the following so-called portmanteau theorem for  -narrow convergence. Here Cu (S ) stands for the set of all uniformly -continuous and bounded functions from S into R.. Theorem 2.6 (i) Let (n) and 0 be in P (S). The following are equivalent:  (a) n )R 0. R (b) limn S cd R n = S cd R 0 for every c 2 Cu(S ). (c) lim infn S qdn S qd0 for every -lower semicontinuous function q : S ! (;1 +1]. which is bounded from below. (ii) Moreover, if (n ) is  -tight, then the above are also equivalent to the following:  . (d) n ) 0R R (e) lim infn S qdn S qd0 for every sequentially  -lower semicontinuous function q : S ! (;1 +1] which is bounded from below.. Recall that -tightness in the above theorem can be dened in two equivalent forms:. Denition 2.7 A sequence (n) in P (S) is  -tight if either one of the following two equivalent statements is true: (a) There exists a sequentially -inf-compact function h : S ! 0 +1] (i.e., a function h for whichR all lower level sets fx 2 S : h(x)  g,  2 R, are sequentially -compact) such that supn S hdn < +1. (b) For every > 0 there exists a sequentially -compact set K  S such that supn n(S nK ) . Of course, the denition of  -tightness goes likewise, simply by replacing the topology  by  , and clearly -tightness of a sequence implies its  -tightness (notice in (a) that h is a fortiori -infcompact). Returning to -tightness itself, note further that h is also -lower semicontinuous, whence B(S)-measurable. Similarly, it follows that the K in (b) belong to B(S). The equivalence of (a) and (b) in the above denition is a simple exercise 46, Exercise 10, p. 109] (see also the proof following Denition 3.3). To identify sets in S that are sequentially -compact, it is useful to observe that any -compact set K  S is automatically sequentially -compact (use Proposition 2.2 and the fact that  coincides with the metrizable topology  on K). Proof of Theorem 2.6: Part (i) is classical and can be found in e.g. 4, 4.5.1], 43, Proposition 7.21] or 44, Theorem 2.1]. As for part (ii), we note the following: (d) ) (a): This is a fortiori. (e) ) (d): Evident by applying (e) to both c and ;c. (d) ) (e): Let h be as in Denition 2.7. For q as specied we notice that for any > 0 the function q := q + h is sequentially -inf-compact, whence  -inf-compact R and thus alsoR  -lower semicontinuous. We may therefore apply (c) to q , which gives lim inf  n S qdn + supn S hdn R qd0. Letting go to zero nishes the proof. QED Remark 2.8 The above proof also justies the existence of the quasi-integrals RS qdn in (e). This goes as follows: in the above notation, q~ := supk q1=k is B(S)-measurable. Clearly, q~ coincides with q on the set fh < +1g and it is equal to +1 on fh = +1g. It remains to notice that Denition 2.7 forces the set fh = +1g to have n-measure zero for each n. It turns out that tightness is a sucient { and in a number of cases also necessary { condition for relative compactness in the narrow topology. Just as in Denition 2.7 we only state the sequential version of this result, even though there is also a fully topological analogue.. Theorem 2.9 (Prohorov) (i) Let ( n) in P (S) be  -tight. Then there exist a subsequence (n ) of (n) and  2 P (S) such that n )  .   can be achieved in (i). (ii) Moreover, if (n ) is  -tight, then in fact n ) 0. . 0. . 0. . 3.

(7) Proof. Part (i) is Prohorov's classical result in sequential format 44, Theorem 6.1], 75, Theo-. rem 12.3.A]. Part (ii) follows by Theorem 2.6(ii)). QED As a necessity complement to the above result, we remark that tightness is known to be a necessary condition for relative sequential narrow compactness when S is complete separable metric or locally compact 44, Theorem 6.2], 89, Theorem 4, p. 381]. See also Theorem 2.19 below. As a rule, in what follows the parts (i) of the above results, and also of those that still follow in this section, are essential for the transfer process. What is done in the parts (ii), all of which exploit -tightness to reduce the situation to that of the corresponding part (i), could also have been added ad hoc. However, it is hoped that the systematic inclusion of such parts (ii) underlines the harmony of the present approach. Next, we study narrow convergence of product measures. The essence of the results that we require is already available if we just consider probability measures on the product S  N^ . Here N^ := N

(8) f1g is the usual Alexandrov-compactication of the natural numbers. This is a compact metrizable space, which obviously satises Hypotheses 2.1, 2.3. From now on, let ^ be a xed metric ^ and denote the topology   ^ by ~. on N^ , let ~ be any compatible product metric on S  N, ~ " respectively. For We denote ~-narrow and ~-narrow convergence in P (S  N^ ) by \ )~ " and \ ) n 2 N^ , let n 2 P (N^ ) stand for the Dirac measure concentrated at the point n.. Proposition (n) and 0 be in P (S). The following are equivalent: P 2.10 (i)Let (a) N1 Nn=1 n ) 0. P (b) N1 Nn=1 (n P n) )~ 0  . 1. (ii) Moreover, if ( N1 Nn=1 n) is  -tight, then the above are also equivalent to the following: P  . (c) N1 Nn=1 n ) 0 P (d) N1 Nn=1 (n  n) )~ 0  . ^  ~) and

(9) > 0 be arbitrary. There exists p 2 N such that Proof. (a) ) (b): Let c 2 Cu(S  N jc(x n) ; c(x 1)j <

(10) =2 for all n > p, uniformly in x 2 S. Hence, the triangle inequality gives Z 1X N (N ; p)

(11)  j N c(x n) ; c(x 1)jn(dx) 2p sup j c j + N S 2N S n=1 1. where the right hand side is less than

(12) for N suciently large. So now (b) follows easily by invoking Theorem 2.6(i). (b) ) (a): Trivial, since any function c in Cb (S ) can be identied with the function c~ in Cb (S  N^ ) given by c~(x n) := c(x). (a) , (c): By Theorem 2.6(ii). (b) , (d): Also by Theorem 2.6(ii), since the sequence (N ) is trivially ~-tight. Here N := 1 PN (n  n). Indeed, by hypothesis there exists h : S ! 0 +1], sequentially -inf-compact, N n=1 P R such that s := supN N1 Nn=1 S hdn < +1. Then ~h(x n) := h(x) denes a function ~h :RS  N^ ! 0 +1] that is sequentially ~-inf-compact (by compactness of the space N^ ), with supN ~hdN = s < +1. QED. Corollary 2.11 (i) Let (n) and 0 be in P (S). The following are equivalent: (a) n ) 0 ~ (b) n  n ) 0  . 1. (ii) Moreover, if (n ) is  -tight, then the above are also equivalent to the following: ~   . (c) n  n ) 0 1. Proof. (a) ) (b): Suppose (b) were notR true. Then there would R exist > 0, c" 2 Cb(S  N^  ~) and a subsequence (n ) of (n) such that R S c"d(0  ) + <R S c"d(n  n ) for all n . Set P N := N1 Nn =1 (n  n ). Then evidently S c"d(0  ) + < S c"dN for all N. But n ) 0 0. 0. 0. 1. 0. 1. 4. 0. 0. 0. 0.

(13) P  ~ implies N1 Nn =1 n ) 0, so N ) 0  by Proposition 2.10. In the limit this contradicts the above inequality for the N . The implication (b) ) (a) is evident (see the proof of the same implication in Proposition 2.10). (b) , (c): As in the proof of Proposition 2.10, it follows easily that under the additional hypothesis (n  n ) is ~-tight by compactness of the space N^ . So the result follows by Theorem 2.6. QED Let (S   ) be another completely regular topological space for which the analogue of Hypotheses 2.1, 2.3 holds the associated metric on S is denoted by  (cf. Proposition 2.2). It is easy to see that the Hypotheses 2.1, 2.3 hold for S  S , which can either be equipped with the product metric    or the product topology    . 0. 0. 0. 1. 0. 0. 0. 0. 0. 0.  Theorem 2.12 (i) Let n ) 0 in P (S) and let n ) 0 in P (S ). Then n  n ) 0  0 in P (S  S ). (ii) Moreover, if (n ) is  -tight and (n) is  -tight, then in fact n  n ) 0  0. ^ ! R as follows: Proof. Let c 2 Cu(S  S     ) be arbitrary. Dene c~ : S  N R c(x x )k (dx ) if k < 1 c~(x k) := RS c(x x )0(dx ) if k = 1 S 0. 0. 0. 0. . 0. 0. 0. 0. 0. 0. 0. 0. . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. Then c~ is ~-continuous, thanks toRuniform continuity ofR c. Hence, the proof of part (i) is nished by invoking Corollary 2.11(i), since S N^ c~d(n  n) = S S cd(n  n). Under the extra tightness conditions of part (ii), the sequence (n  n) is clearly tight with respect to the product topology    on S  S . So the desired result follows from part (i) by virtue of Theorem 2.6(ii). QED After this, we study the support of the limit of a narrowly convergent sequence. Denition 2.13 The support -supp  of a probability measure  2 P (S) is dened by -supp  := \fF : F  S F -closed (F) = 1g: The  -support of a measure  in P (S), denoted by  -supp , is dened by replacing the topology  by  in the above formula of course, -supp  is always contained in  -supp . Proposition 2.14 Every  2 P (S) is carried by its support, i.e., (-supp ) = 1. Proof. By Denition 2.13 the set C := S n-supp  is the union of all -open sets G with (G) = 0. By Hypothesis 2.1, C evidently has the countable subcover property (see the proof of Proposition 2.2). So C, being the union of a countable collection of -null sets, is a -null set itself. QED Denition 2.15 The sequential  -limes superior of a sequence of subsets (An) of S, denoted by -Lsn An, is the set of all x 2 S for which there exists a subsequence (An ) of (An ), and corresponding elements xn 2 An , such that x = -limn xn . The denition of the  -limes superior is of course completely analogous. However, the metrizable nature of  causes an equivalent alternative formulation to be valid. The proof of this is an easy exercise, left to the reader. Lemma 2.16 Let (An) be a sequence of subsets of S . Then  -LsnAn := \p=1  -cl

(14) n p An : Theorem 2.17 (i) Let (n) and 0 be in P (S) with N1 PNn=1 n ) 0 in P (S) (this holds in par ticular when n ) 0 ). Then  -supp 0   -Lsn  -supp n : 0. . 0. 00. 0. 0. 0. 0. 0. 0. 0. 0. 1. 5. .

(15) P (ii) Moreover, if ( N1 Nn=1 n) is  -tight then in fact 0(-seq-cl -Lsn -supp n) = 1 and, consequently,. -supp 0  -cl -Lsn -supp n: Recall that the  -sequential closure -seq-cl A of a set A in S is dened as the intersection of all those -sequentially closed sets C in S for which C  A. Clearly, -seq-cl A  -cl A. Given Hypothesis 2.1, it is easy to check that for any sequence (An ) of subsets of S one has -seq-cl -Lsn An   -Lsn An . P (  ) )~   . Dene q : S  N^ ! Proof. (i) By Proposition 2.10 we have N := N1 N 0 0 n=1 n n f0 +1g by 8 if x 2  -supp k and k < 1, <0 if x 2  -Lsn -supp n and k = 1 q0(x k) := : 0 +1 otherwise. Then q0 is ~-lower semicontinuous in every point (x k) of S  N^ . Indeed, let (xj  kj ) ! (x k) (note that sequential arguments suce to verify lower semicontinuity). We must show that  := lim infn q0(xj  kj ) q0(x k). If k < 1, then eventually kj  k, so  q0(x k) follows by the fact that  -supp k is  -closed (Lemma 2.16). If k = 1, we distinguish two cases: if eventually kj  1, then  q0(x 1) follows by closedness of  -Lsn  -supp n, which in turn is an immediate consequence of Lemma 2.16. On the other hand, if kj < 1 innitely often, then the same inequality follows directly from Denition 2.15. So we conclude that q0 is indeed ~-lower semicontinuous. Now R R R q d( . ) = q (x n) (dx) = 0 for every n (by Proposition 2.14). Hence, q d ^ n n n ^ N= 0 0 0 S S N S N R 0 for every N. Thus, Theorem 2.6(i) gives S q0(x 1)0(dx) = 0, and the desired support properties for 0 follow. P (ii) Under the additional -tightness condition it follows that N := N1 Nn=1(n  n) )~ 0 . by Proposition 2.10(ii). Let q0 : S  N^ ! f0 +1g be given by 8 if x 2 -supp k and k < 1, <0 if x 2 -seq-cl -Lsn -supp n and k = 1 q0(x k) := : 0 +1 otherwise. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. . 0. 0. . 0. 1. With a little careful work, this function is seen to be ~-sequentially lower semicontinuous on S  N^ (observe that, unlike the previous part, -Lsn -supp n is not sequentially closed { hence the additional sequential closure operation has been added in the denition of q0). By Proposition 2.14 R and Theorem 2.6(ii) we nd S q0(x 1)0(dx) = 0. The desired properties of 0 then follow with ease. QED Remark 2.18 If in Theorem 2.17 there exists a  -compact set K containing

(16) n supp n, then the set -Lsn-supp n is  -closed and the following simplication can be made: -seq-cl -Lsn -supp n = -Lsn -supp n: Indeed, on K the topologies  and  coincide, which gives -Lsn -supp n =  -Lsn  -supp n, and the latter set is  -closed, whence  -closed (cf. Lemma 2.16). In order to connect narrow and K-convergence of Young measures in section 4, the following sucient condition for  -tightness is quite useful. Recall that a probability measure  in P (S) is said to be  -Radon if the singleton f g is  -tight (cf. Denition 2.7). The set of all such Radon probability measures is denoted by PRadon (S  ).. Theorem 2.19 Let (n) and 0 be in PRadon(S  ). Then n ) 0 implies that (n) is -tight.. This is 44, Theorem 8, Appendix III] and 92, Theorem 9.3] the proof depends critically on both the metric nature of  and the fact that one considers only sequential narrow convergence. 6.

(17) 3 K -convergence of Young measures. This section develops K-convergence, an auxiliary, nontopological convergence mode for Young measures introduced in 19, 20, 25]. This will be of great use in the next section when we transfer narrow convergence results of the previous section from probability measures to Young measures. Thus, the present section can be regarded as an intermediate stage in the transfer process. As in section 2, results are developed both in a metrizable and in a nonmetrizable setting. Let (# A ) be a nite measure space. Let us remark that much of what is done here extends without further ado to a -nite measure space such a measure is equivalent to a nite one, and one can always premultiply the integrands below by the appropriate Radon Nikodym derivative and an appropriate extension of uniform integrability is also available]. Let (S ) be as in the previous section (i.e., a completely regular topological space satisfying Hypotheses 2.1, 2.3). Let R(# S) be the set of all transition probabilities from (# A) into (S B(S)) 81, III.2]. That is to say, R(# S) consists of all functions  : # ! P (S) such that ! 7! (!)(B) is A-measurable for every B 2 B(S). Note that this notion subsumes that of probability measure: P (S) can be identied with the constant functions in R(# S) in fact, R(# S) coincides with P (S) when A is trivial, i.e., A = f #g.] In association with the central topology of these lecture notes (Denition 4.1), transition probabilities are also called Young measures, and we shall adopt this terminology (other names used for Young measures in the literature are, depending on the context: Markov kernels, randomized decision functions, relaxed control functions, etc.). For some elementary measure-theoretical properties of Young measures the reader is referred to 81, III.2] or 4, 2.6] (see also Appendix A). In particular, the product measure that is induced on (#  S AB(S)) by  and any  2 R(# S) (cf. 81, III.2]) is denoted by    cf. Theorem A.1. Let L0(# S) be the set of all measurable functions from (# A) into (S B(S)). A Young measure  2 R(# S) is said to be Dirac if it is a degenerate transition probability, i.e., if there exists a function f 2 L0(# S) such that for every ! in # (!) = f (!) := Dirac measure at the point f(!): In this special case  is denoted by f and is called the Young measure relaxation of the function f. The set of all Dirac Young measures in R(# S) is denoted by RDirac (# S). The fundamental idea behind Young measure theory is that, in some sense, R(# S) forms a completion of L0(# S), when the latter is identied with RDirac (# S). In the context of the previous section, the much less fruitful analogue of this would be to view P (S) as an extension of S, because the latter can be identied with the set f x : x 2 S g of all Dirac measures, to which it is homeomorphic. Let us agree to the following terminology: an integrand on #  S is a function g : #  S ! (;1 +1] such that for every ! 2 # the function g(! ) on S is B(S)-measurable. Moreover, such an integrand g is said to be integrably bounded below if there exists  2 L1 (# R) such that g(! x) (!) for all ! 2 # and x 2 S. Further, a function g : #  S ! (;1 +1] is said to be a (sequentially)  -lower semicontinuous

(18)  -continuous]

(19)

(20)  -inf-compact]] integrand on #  S if for every ! 2 # the function g(! ) on S is (sequentially) -lower semicontinuous -continuous] -infcompact]] respectively. Let g be an integrand on #  S. The following expression is meaningful for any  2 R(# S): Z Z Ig () := g(! x)(!)(dx)](d!) .  S. provided that the two integral signs are interpreted as follows: (1) for every xed ! the integral over the set S of the function g(! ), which is B(S)-measurable by denition of the term integrand, is a quasi-integral in the sense of 81, p. 41] and Appendix B, (2) the integral over # is interpreted as an outer integral in the sense of Appendix B (note that outer integration comes down to quasi-integration when measurable functions are involved). The resulting functional Ig : R(# S) ! ;1 +1] is called the Young measure integral functional associated to g. Another integral functional associated to g, this time on the set L0 (# S) of all measurable functions. 7.

(21) from # into S, is given by the formula. Z. Jg (f) :=. . . g(! f(!))(d!) = Ig ( f ):. The following notion of convergence was introduced and studied in a more abstract context in 18, 19].. Denition 3.1 A sequence (n ) in R(# S) K -converges with respect to the topology  to 0 2 K R(# S) (notation: n ;! 0 ) if for every subsequence (n ) of (n ) 0. N 1 X  N n =1 n (!) ) 0 (!) as N ! 1 for a.e. ! in #. 0. 0. Note that the exceptional null set is allowed to vary with the subsequence (n ). Of course, the  " above refers to -narrow convergence in P (S) in the sense of Denition 2.5. short arrow \ ) Unlike narrow convergence, K-convergence is nontopological . If in the above denition ) , i.e.,  the mode of pointwise convergence mode, is replaced by ) , we obtain a corresponding notion of K-convergence with respect to  that is denoted by \ K ! ". Since \ ) " is implied by \ ) ", it K follows that \ K ! " is implied by \ ;! ". 0. Example 3.2 Let (# A ) be ( 0 1] L( 01]) 1) (i.e., the Lebesgue unit interval). Let (fn ) be the sequence of Rademacher functions, dened by fn (!) := sign sin(2n !) (here S := R, of course). K Then fn ;! 0 , where 0 2 R( 0 1] R) is the constant function 0 (!)  21 1 + 12 1. In fact, here one could argue that the strong law of large numbers applies to the sequence ( fn ) of P (R)-valued ;. random variables, but one can also give a proof of the above by means of the standard (scalar) strong law of large numbers and scalarization, analogous to the proof of Theorem 3.7 below.. A crucial instrument for the transfer process of these notes is the following generalization of Denition 2.7.. Denition 3.3 A sequence (n ) in R(# S) is  -tight if either one of the following two equivalent statements is true: (a) There exists a nonnegative, sequentially -inf-compact integrand h on #  S such that sup Ih (n ) < +1: n (b) For every > 0 there exists a multifunction ; : # ! 2S , with ; (!) sequentially compact for every ! 2 #, such that Z sup n (!)(S n; (!))(d!) : . n. . Recall from the previously given denition of integrands that a sequentially -inf-compact integrand h is simply a function on #  S with the following property: for every ! 2 # the function h(! ) is sequentially -inf-compact on S (i.e., all sets fx 2 S : h(! x)  g,  2 R, are sequentially -compact). As is by now usual, the alternative, weaker notion of  -tightness of a sequence of Young measures is obtained by replacing the topology  by  in the above denition. Proof of equivalence of (a) and (b) in Definition 3.3 68]. (a) ) (b): Let s := supn Ih (n ) then s 2 R+ . For every > 0, let ; (!) be the set of all x 2 S for which h(! x) s=  then ; (!) is sequentially -compact for every !. Also, for every n s Z  (!)(S n; (!))(d!) I ( ) s  h n.  n . 8.

(22) and this proves that the denition as given in part (b) holds. (b) ) (a): Let ;m be the given multifunction corresponding to = 3 m , m 2 N. With no loss of generality we may suppose that ;m (!)  ;m+1 (!) for every ! and m (otherwise, we could always take nite unions of the ;m ). Now set ;0   and dene  m 2 if x 2 ;m (!)n;m 1 (!), m 2 N h(! x) := + 1 if x 62

(23) m ;m (!) ;. ;. Then h(! ) is sequentially -inf-compact on S for every ! and supn Ih (n ) 6. QED. Example 3.4 (a) Let E be a separable re%exive Banach space with norm k  k. Let E be the dual space of E. Observe that (E ) is a completely regular R Suslin space for  := (E E ). Suppose that (fn )  L1(# E) is bounded in L1-seminorm: supn  kfn (!)k(d!) < +1. Then the corresponding sequence ( fn ) in R(# E) is -tight: just take h(! x) := kxk in Denition 3.3. (b) Let E be a separable Banach space with norm k  k. Then (E ) is a completely regular Suslin space for  := (E E ). Suppose that (fn )  L1(# E) is bounded in L1-seminorm and that there exists a multifunction R : # ! 2S such that for a.e. ! both ffn (!) : n 2 Ng  R(!) and R(!) is -ball-compact i.e., the intersection of R(!) with every closed ball in E is (E E )-compact]. Then ( fn ) is -tight, as is seen by considering hR (! x) := kxk if x 2 R(!), and hR (! x) := +1 otherwise. For notice that for every ! 2 # and  2 R+ the set of all x 2 E such that hR (! x)  0. 0. 0. 0. coincides with the intersection of R(!) and the closed ball with radius  around 0. This set is (E E )-compact, whence sequentially (E E )-compact by the Eberlein-S&mulian theorem. 0. 0. Part (b) in the above example generalizes part (a): simply observe that in part (a) E itself is (E E )-ball-compact (by re%exivity), so that there one can set R(!) := E for all ! 2 #. A very important property of K-convergence for Young measures is the following Fatou-Vitalitype result: 0. Proposition 3.5 (i) Let n K ! 0 in R(# S). Then lim infn Ig (n ) Ig (0 ) for every  -lower semicontinuous integrand g on #  S such that s() := sup n. Z Z. . . g (! x)n(!)(dx)](d!) ! 0 for  ! 1:. (3:1). ;. g. g!. f ;. (ii) Moreover, if (n ) is  -tight, then in fact lim infn Ig (n ) Ig (0 ) for every sequentially  -lower semicontinuous integrand g on #  S such that (3.1) holds. Here g := max(;g 0) and fg ;g! denotes the set fx 2 S : g(! x) ;g. Remark 3.6 (i) If g is integrably bounded from below, then (3.1) holds automatically. (ii) In case n = fn for all n 2 N (this specication does not include the limit 0 ) the condition ;. (3.1) runs as follows:. lim sup !1. n. Z. . . 1 g( fn ( ))  g (! fn (!))(d!) = 0: f. ;. g. ;. Clearly, for every ! 2 # we have g(! fn(!)) ; if and only if g (! fn (!)) . This means that (3.1) simply comes down to uniform (outer) integrability of the sequence (g ( fn())) in the case of a Dirac sequence. If g is T  B(S)-measurable in addition, this coincides with the usual classical formulations of uniform integrability a la Vitali of the sequence of negative parts (g ( fn())) cf.

(24) 65, 9]. ;. ;. ;. Proof of Proposition 3.5. The proof of part (i) will be given in two steps.. Step 1: the case g 0. Set  :=Plim infRn Ig (n ) then there is a subsequence R (n ) such that 0.  = limn Ig (n ). Dene N (!) := N1 Nn =1 S g(! x)n (!)(dx) and 0(!) := S g(!P x)0 (!)(dx). Then lim infN N (!) 0(!) for a.e. ! by Theorem 2.6(i), because by Denition 3.1 N1 Nn =1 n (!) 0. 0. 0. 0. 0. 9. 0.

(25) ) 0 (!) in P (S) for a.e. !. Thus, R Fatou's Rlemma for outer integration (Proposition B.4) can be applied. This gives lim infN  N d  0d. Here the right-hand side is equal to Ig (0 ), . . !1. and the left-hand side is at most , by subadditivity of outer integrals (Lemma B.5) and by the choice of (n ). Step 2: the general case. We essentially follow Ioe 65] by pointing out that the simple inequality g + 1 g  g g := max(g ;) on #  S leads to Z Z Z g(! x)n(!)(dx) + 1 g  (! x)g (! x)n (!)(dx) g (! x)n (!)(dx): 0. f ;. g. ;. S. S. f ;. ;. g. S. After one more (outer) integration this gives, in the notation of (3.1), Ig (n )+s() Ig (n ), where we use again the subadditivity of outer integration (Lemma B.5). Now observe that step 1 trivially extends to any g that is bounded from below, such as g. This gives limninf Ig (n ) + s() limninf Ig (n ) Ig (0 ) Ig (0 ) where the last inequality follows from g g. In view of (3.1), the proof of (i) is now nished by letting  go to innity. (ii) Let h be as in Denition 3.3 and denote s := supn Ih (n ). We augment g, similar to the proof of Theorem 2.6(ii): For > 0 dene g := g + h. Then g g and g (! ) is  -lower semicontinuous on S for every ! 2 # (see the proof of Theorem 2.6(ii)). Thus, part (i) gives lim infn Ig (n ) + s lim infn Ig (n ) Ig (0 ) Ig (0 ) for any > 0. Letting go to zero gives the desired inequality. QED The following important Prohorov-type \relative sequential compactness criterion for K-convergence" (apostrophes are in order because K-convergence is nontopological) is a crucial tool for these notes. It extends Prohorov's classical Theorem 2.9 to K-convergence of Young measures and was rst obtained in 19, Theorem 5.1] as a specialization of an abstract Komlos' theorem (i.e., an abstract version of Theorem 3.9 below) to Young measures.. Theorem 3.7 (i) Let (n ) be a -tight sequence in R(# S). Then there exist a subsequence (n ) !. of (n ) and  2 R(# S) such that n K K (ii) Moreover, if (n ) is  -tight, then in fact n ;!  can be achieved in (i). 0. 0. . . 0. . The following example, which extends Example 3.2, demonstrates the power of this result. Clearly, this brings K-convergence (for subsequences!) to settings where the law of large numbers stands no chance at all.. Example 3.8 Let (# A ) be ( 0 1] L( 01]) 1) (cf. Example 3.2). Let f1 2 L1 ( 0 1] R) be arbitrary it can be extended periodically from 0 1] to all of R. We dene fn+1 (!) := f1 (2n!). Clearly, the sequence ( fn ) is tight in the sense of Denition 3.3 e.g., use h(! x) := jxj to meet part (a) or K  ;1 1] to satisfy part (b)]. Therefore, by Theorem 3.7 there exist a subsequence K (fn ) of (fn ) and some  2 R( 0 1] R) such that fn ;!  . The precise nature of  could now 0. . 0. . . be determined by means of Proposition 3.5, but we shall defer this to Example 4.3 later on. To prove Theorem 3.7 we use an outstanding theorem, due to J. Komlos 71].1. Theorem 3.9 (Komlos) Let (n) be a sequence in L1(# R) such that Z. jnjd < +1: n  1 The original proof in 71] went by subtle truncation arguments and application of a martingale limit theorem. It is not hard to show that Komlos' theorem implies the strong law of large numbers. What is much more interesting is that, conversely, Theorem 3.9 also follows from the strong law of large numbers by invoking \subsequence principle theory" 1, 53]. sup. 10.

(26) Then there exist a subsequence (n ) of (n ) and a function  2 L1 (# R) such that for every further subsequence (n ) of (n ) 0. 00. . 0. N 1 X lim N N n =1 n (!) =  (!) for a.e. ! in #. 00. !1. . 00. Observe here that  is universal with respect to the possible choices of a subsequence (n ) from (n ), but that the associated exceptional -null set in the limit statement is allowed to vary with the subsequence. Lemma 3.10 There exists a countable set C0  Cu(S ) such that for every (n) and 0 in P (S) Z Z lim cdn = cd0 for all c 2 C0 n 00. . 0. S. S.  if and only if n ) 0. In particular, C0 separates the points of P (S).. Proof. As was observed following Hypothesis 2.1, (S ) is separable. Hence, (S  ) is a separable metric space (apply Proposition 2.2). Therefore, the result follows from 43, Proposition 7.19]. QED. Lemma 3.11 Let (n) in P (S) be -tight and let C0  Cu(S ) be as in Lemma 3.10. If Z. lim cdn exists for every c 2 C0 , n S.  then there exists  2 P (S) such that n ) . . . Proof. By RTheorem 2.9 there exist R a subsequence (n ) of (n) and  2 P (S) such that  n )  . Then S cd = c := limn S cdn for every c 2 C0. Now if (n) as a whole were not to 0. 0. . . . converge to  , there R would exist R c~ 2 Cb(S ) and > 0 such that for some subsequence (m ) of (n) one would have j S c~dm ; S c~d j for all m. Since (m ) is  -tight, there would then exist,  by another application of Theorem R 2.9, a subsequence (m ) and  2 P (S) such that m )  . Just as above, this would entailR S cd =R c for all c 2 C0, so  =  by the point-separating property of C0. But since also j S c~d ; S c~d j , a contradiction would follow. QED Proof of Theorem 3.7. (i) By Lemma 3.10 there exists R a countable subset C0 = fci : i 2 Ng of Cu (S ) that separates the points of P (S). Clearly, sup n  jinjd < +1 for every i 2 N, where we R set in(!) := S ci (x)n (!)(dx). Let h be as in Denition 3.3 (case of  -tightness). By Lemma B.3 1(# R) such that 0n(!) R h(! x)n(!)(dx) there exists for each n 2 N a function  2 L 0 n S R for all ! 2 # and  0nd = Ih (n ). Applying the Komlos Theorem 3.9 in a diagonal extraction procedure, wePobtain a subsequence (n ) of (n ) and functions i 2 L1 (# R), i 2 N

(27) f0g, such that limN N1 Nn =1 in = i a.e. for every further subsequence (n ) and for all i 2 N

(28) f0g. It follows therefore that for every such subsequence (n ) for a.e. ! in # Z N 1 X lim h(! x) (3:2) N S N n =1 n (!)(dx) = 0 (!) < +1 Z N 1 X lim c (x)  (!)(dx) = i (!) for all i 2 N. (3:3) N S i N n =1 n Let us begin by considering (n ) itself as the subsequence in question. Fix ! outside the exceptional null set M, associated with this particular choice of a subsequence in (3.2){(3.3). Then (3.2) implies P that for a.e. ! the sequence (!N ) in P (S), dened by !N :=R N1 Nn =1 n (!), is  -tight in P (S) in the sense of Denition 2.7. Also, (3.3) implies that limN S ci d!N exists for every i. By . . 0. . . . 00. . . 0. 00. 0. . . 00. . 00. 00. . 00. 00. . 00. 0. 0. 11. 0. .

(29) Lemma 3.11(i), there exists ! in P (S) such that !N ) ! . Dene  (!) := ! for ! 2 #nM. Also, on M we dene  to be equal to an arbitrary, but xed element from P (S). Then it is elementary, in view of Proposition A.2, that  belongs to R(# S). Finally, the argument following (3.3) can be repeated if one starts out with an arbitrary subsequence (n ) of (n ), instead of (n ) itself. Except for the change in the exceptional null set M, for which the denition of K-convergence allows, nothing changes. This nishes the proof of part (i). Part (ii) then follows immediately by Theorem 2.6(ii), in view of the fact that for every subsequence (n ) of the above (n ) (3.2) implies P that (!N ) is -tight for a.e. !, where !N := N1 Nn =1 n (!). QED . . . . . . 00. 0. 00. 0. 0. 00. 0. 0. 00. Remark 3.12 From (3.2) in the above proof it is seen thatPthe sequence (n ) in Theorem 3.7 is such that for every further subsequence (n ) the sequence ( N1 Nn =1 n (!)) in P (S) is either  -tight 0. 00. for a.e. ! (part (i)) or even  -tight for a.e. ! (part (ii)).. 00. 00. As the nal results in this intermediate section, we present direct consequences of Proposition 2.10 and Theorem 2.17 for K-convergence of Young measures. Such results rst gured in 20] they will be used in the next section.. Proposition 3.13 (i) Let (n) and 0 be in R(# S). The following are equivalent: (a) n K ! 0 . (b) n  n K !~ 0  . 1. (ii) Moreover, if (n ) is  -tight, then the following two equivalent statements are implied by the. above:. K (c) Every subsequence (n ) of (n ) contains a further subsequence (n ) such that n ;! 0 . (d) Every subsequence (n ) of (n ) contains a further subsequence (n ) such that n  K~. n ;! 0  . 0. 00. 0. 00. 00. 00. 00. 1. Proof. (a) , (b) follows by pointwise application of Proposition 2.10(i). In part (ii) (c) , (d) follows by pointwise application of Proposition 2.10(ii), by taking into consideration Remark 3.12. Finally, the implication (a) ) (c) of part (ii) follows by pointwise application of Theorem 2.6, again taking into consideration Remark 3.12. QED. Theorem 3.14 (i) Let (n ) and 0 be in R(# S). Then n K ! 0 implies  -supp 0 (!)   -Lsn  -supp n (!) for a.e. ! in #. (ii) Moreover, if (n ) is  -tight, then in fact 0 (!)(-seq-cl -Lsn-supp n (!)) = 1 for a.e. ! in #, so that in particular. -supp 0 (!)  -cl -Lsn -supp n (!) for a.e. ! in #. Proof. Part (i) of this result follows directly from a pointwise application of Theorem 2.17(i). Part. (ii) also follows by a pointwise application of Theorem 2.17(ii), in view of the tightness observation in Remark 3.12. QED. 4 Narrow convergence of Young measures In this section our program to transfer narrow convergence results for probability measures (section 2) to Young measures is completed. We use the same fundamental hypotheses as in the previous section: (# A ) is a nite measure space and (S ) is a topological space for which Hypotheses 2.1, 2.3 hold. We start out by giving the denition of narrow convergence for Young measures: 12.

(30) Denition 4.1 A sequence (n ) in R(# S) converges  -narrowly to 0 in R(# S) (this is denoted by n =) 0 ) if for every A 2 A and for every c in Cb(S ) Z Z Z Z lim c(x)n (!)(dx)](d!) = c(x)0 (!)(dx)](d!): n A S. A S. The obviously weaker notion  -narrow convergence is dened by replacing  by  . Similar to section 2, the latter notion is denoted by \ =) ". In analogy to section 2, we shall see that for tight sequences of Young measures -narrow and  -narrow convergence are actually the same. For further benet, note carefully the dierence in notation between the narrow convergences for probability measures (indicated by short arrows) and Young measures (indicated by long arrows). In its above form the denition of narrow convergence is classical in statistical decision theory 96, 74]. It merges two completely dierent classical modes of convergence:. Remark 4.2 Let (n ) and 0 be in R(# S). The following are obviously equivalent: (a) n =) 0 in R(# S). (b) For every A 2 A with (A) > 0   n ](A  )=(A) )   0 ](A  )=(A) in P (S). (c) For every c 2 Cb (S ) Z. S. c(x)n ()(dx)* . Z. c(x)0 ()(dx) in L (# R), 1. S. where \* " denotes convergence in the topology (L (# R) L1(# R)). . 1. The following example continues the previous Examples 3.2 and 3.8.. Example 4.3 Let (# A ) be ( 0 1] L( 01]) 1) (cf. Example 3.2). As in Example 3.8, let f1 2 L1( 0 1] R) be arbitrary and extended periodically from 0 1] to all of R. We dene fn+1 (!) := f1 (2n!). Then fn =) 0 , where 0 2 R( 0 1] R) is the constant function given by 0 (!)  f11 . Here f1 2 P (R) is the image of 1 under the mapping f1  i.e., f1 (B) := (f1 1 (B)). To prove the above convergence statement, let c 2 Cb(R) be arbitrary, and let A be of the form A = 0 ] with ;.  > 0. Then a simple change of variable gives Z Z Z 2n n n c(fn+1 (!))d! = c(f1 (2 !))d! = 2 c(f1 (! ))d!  ;. 0. A. 0. 0. 0. R R and by periodicity of f1 the latter expression equals  01 c(f1 (! ))d! = 1 (A) R c(x)f11 (dx) in the limit. So it has been shown that Z Z Z lim c(f (!))d! = c(x)0 (!)(dx)]d! (4:1) n n 0. !1. A. A. 0. R. for A = 0 ]. By subtraction, (4.1) continues to be valid for A's of the form A = ( ], and, by summation, also for A's that are a nite disjoint union of such intervals. Finally, by 4, 1.3.11] for any A 2 A and any > 0 there exists a nite union A of intervalsR ( ] suchR that the symmetric dierence of A and A has Lebesgue measure at most . But then j A c(fn ) ; A c(fn )j supS jcj, so, by letting go to zero, we conclude that (4.1) continues to hold in the general case. 0. 0. 0. The above example shows that  in Example 3.8 is equal to the above 0 , modulo a 1 -null set. In fact, the narrow limit of a sequence of Young measures in R(# S) can only be essentially unique (that is to say, unique modulo a -null set). This follows immediately from the following general result: . 13.

(31) Proposition 4.4 For every ,  in R(# S) the following are equivalent: (a) For every A 2 A and c 2 C0 0. Z Z Z Z c(x)(!)(dx)](d!) = c(x) (!)(dx)](d!): 0. A S. A S. (b) (!) =  (!) for a.e. ! in #. 0. The essentially sequential setup chosen for these lecture notes leads to frequent use of a semimetric d on R(# S), as dened in the next result. This allows us to use sequentially oriented R. approaches when we apply the narrow topology (the latter is of course dened by rereading Denition 4.1 with generalized sequences in mind).. Theorem 4.5 Suppose that the -algebra A on # is countably generated. Then there exists a semimetric d on R(# S) such that for every (n ) and 0 in R(# S) the following are equivalent: (a) n =) 0 . R. (b) limn d (n  0) = 0. R. Proof. Dene a semimetric on R(# S) by. d (  ) := R. 0. XX 1. 1. i=1 j =1. 2. i jj. ; ;. Z Z Z Z ci (x)(!)(dx)](d!) ; ci (x) (!)(dx)](d!)j=(Aj ): 0. Aj S. Aj S. Here (ci ) is an enumeration of the functions, conveniently normalized so as to give supS jcij = 1 for each i, in the narrow convergence determining set C0 used in Lemma 3.10. Also, (Aj ) is an at most countable algebra which generates A. (a) ) (b): By using the approximation result 4, 1.3.11] in the same way as in the above Example 4.3, it follows that Z Z Z Z lim. c (x) (!)(dx)](d!) = ci(x)0 (!)(dx)](d!): i n n A S. A S. for every A 2 A and every i. By the narrow convergence determining property of C0 in Lemma 3.10, this implies   n ](A  )=(A) )   0 ](A  )=(A) in P (S) for every A 2 A with (A) > 0. By Remark 4.2 this implies n =) 0 . The converse implication (a) ) (b) is very simple. QED From Proposition 3.5 and its proof we immediately obtain that K-convergence implies narrow convergence:. Remark 4.6 Let (n ) and 0 be in R(# S). The following hold: (a) If n K ! 0 , then n =) 0 . (b) If n K ! 0 and if (n ) is  -tight, then n =) 0 . K (c) If n ;! 0 , then n =) 0 . The implications in this remark cannot be reversed: the following example shows that a narrowly convergent sequence does not have to K-converge, even when S is the set of real numbers. Let us already mention that, nevertheless, in Theorem 4.13 below a partial converse will be achieved in terms of subsequences.. Example 4.7 Consider the sequence (fn) of Rademacher functions from Example 3.2. Dene the following sequence (fn ) in L1(# R): for each m 2 N dene fn := fm for 2m 1 n 2m ; 1. From Examples 3.2 and 4.3 it is clear that fn =) 0 , where 0  21 1 + 12 1 a.e. By Remark 4.6 we 0. 0. 0. ;. ;. 14.

(32) know that if ( fn ) were to K-converge to some Young measure, it would have to be 0 (modulo null sets). But it is easy to check the following: for N = 2m ; 1 0. m;2 ;. N 1X 1 2 X 1 (!) + 2m 2 (!) + 2m 1 (!):. (!) = f N n=1 n 2m ; 1 n=1 fn 2m ; 1 fm 1 2m ; 1 fm ;. 0. ;. 0. ;. K This shows that fn ;! 0 is not possible, since 2m i =(2m ; 1) ! 2 i for i = 1, 2, and 1(f! 2 # : fm (!) = fm 1 (!)g) > 0 for all m 2 N. Concatenation of Theorem 3.7 and Remark 4.6 gives immediately a Prohorov-type result for narrow convergence of Young measures: Theorem 4.8 (i) Let (n ) be a -tight sequence in R(# S). Then there exist a subsequence (n ) of (n ) and  2 R(# S) such that n =)  . (ii) Moreover, if (n ) is  -tight, then in fact n =)  can be achieved in (i). Example 4.9 We continue with Example 3.4(b). By (E E )-tightness of ( fn ) we get from Theorem 4.8 that there exist a subsequence (fn ) of (fn ) and  2 R(# E) such that fn =)  . (a) We now introduce a function f 2 L1E that is \barycentrically" associated to  , simply by inspecting the consequences of the tightness inequality s := supn IhR ( fn ) < +1 that was established there. For hR is a fortiori a (E R E )-lower semicontinuous integrand, so Theorem 4.10(e) gives IhR ( ) s < +1, which implies S hRR(! x) (!)(dx) < +1 for a.e. !. So by the denition of hR it follows that both  (!)(R(!)) = 1 and E kxk (!)(dx) < +1 for a.e. !. By Theorem A.10(i) it follows that the barycenter f (!) := bar  (!) of the probability measure  (!) is dened for a.e. !. Thus, if we set f := 0 on the exceptional null set, we obtain a function f 2 L0(# E). Finally we notice that, as announced, f is -integrable, i.e., f 2 L1(# E). This follows simply from IhR ( ) < +1 by use of Jensen's inequality and the inequality hR (! x) kxk. (b) Suppose that in part (a) one has in addition that (kfn k) is uniformly integrable in L1(# R). w f 2 L1(# E) (weak convergence in L1 (# E)). This follows directly from another Then fn ! application of Theorem 4.10(e), namely, to all integrands g of the type g(! x) =  < x b(!) >, b 2 L (# E ) E]. The latter symbol denotes the set of all scalarly measurable bounded E -valued 1 E). This yields limn Ig ( f ) = Ig ( ), with functions on # it forms n R the prequotient dual of L (# R Ig ( fn ) = Jg (fn ) =  < fn  b > d and Ig ( ) =  < f  b > d (cf. Theorem A.10(i)). ;. 0. ;. ;. 0. 0. . . 0. . 0. 0. . . 0. . . 0. . . . . . . . . . . . . 0. 0. 1. . 0. 0. 0. 0. 0. 0. . 0. . . w Part (b) in the above example implies R that fn ! f0 in Example 4.3, where f0 is the constant function given by f0 (!) := bar 0 (!) = R f1 d1 (apply 55, II.12]). Proposition 3.5 and Theorem 3.7 imply the following transfer of the earlier portmanteau Theorem 2.6 to Young measures (see 16, Theorem 2.2] for other equivalences of this sort). Theorem 4.10 Suppose that (S ) is Suslin. Let (n ) and 0 be in R(# S). The following are. equivalent:. (a) (b) (c). n =) R R0 . R R limn A S c(x)n (!)(dx)](d!) = A S c(x)0 (!)(dx)](d!) for every A 2 A, c 2 Cu (S ). lim infn Ig (n ) Ig (0 ) for every -lower semicontinuous integrand g on #  S such that Z Z lim sup. g (! x)n (!)(dx)](d!) = 0:  . !1. n. ;. . g. g!. f ;. (ii) Moreover, if (n ) is  -tight, then the above are also equivalent to the following: (d) n =) 0 . (e) lim infn Ig (n ) Ig (0 ) for every sequentially  -lower semicontinuous integrand g on #  S such that Z Z g (! x)n (!)(dx)](d!) = 0: lim sup.  . !1. n. ;. . g. g !. f ;. 15.

(33) Observe that (a) ) (c) and (d) ) (e), which are the most powerful implications of the above theorem, constitute a very general theorem of Fatou-Vitali type for narrow convergence of Young measures. Results of this kind are usually obtained by means of approximation procedures for the lower semicontinuous integrands 48, 42, 7, 67, 9, 16, 94, 95]. In contrast to the present result, such procedures depend on approximation arguments requiring the measurable projection theorem and related Suslin conditions for S. The following important lemma establishes that  -narrow convergence implies  -tightness when (S ) is a Suslin space (note: this is the case in particular when (S ) itself is a Suslin space).. Lemma 4.11 Suppose that (S ) is Suslin. Let (n ) and 0 be in R(# S) with n =) 0 . Then (n ) is  -tight.. 2 PRadon (S  ) for every n 2 N

(34) f0g, since PRadon (S  ) = P (S) by 55, III.69]. By Remark 4.2 it follows that n ) 0. Therefore, Theorem 2.19 implies that (n) is  -tight in P (S). By Denition R 2.7(a), this means that there exists a  -inf-compact function h : S ! 0 + 1 ] such that sup n S h dn < +1. Now by denition R of n we have S h dn = Ih (n )=(#) for every n, where h(! x) := h (x). Thus supn Ih (n ) < +1, Proof. Set n :=   n ](#  )=(#) then n 0. 0. 0. 0. which demonstrates that (n ) is  -tight. QED Proof of Theorem 4.10. We start with the proof of part (i). (a) , (b): The equivalence follows immediately from the equivalence of (a) and (b) in Theorem 2.6 and Remark 4.2. (c) ) (b): Obvious, for (b) follows by applying (c) to both g(! x) := 1A (!)c(x) and g (! x) := ;1A (!)c(x), with A 2 A and c 2 Cu(S ). (a) ) (c): For g as stated, let  := lim infn Ig (n ). Then  = limn Ig (n ) for a suitable subsequence (n ) of (n ). By Lemma 4.11 we have that (n ), whence (n ), is  -tight, so by Theorem 3.7(i) there exists a subsequence (n ) of (n ) such that n K !  for some  in R(# S). But in combination with (a) this implies  (!) = 0 (!) a.e. (apply Remark 4.6 and Proposition 4.4), so in fact n K ! 0 . The desired Fatou-Vitali inequality  Ig (0 ) then follows from Proposition 3.5. Next, we prove part (ii) of the theorem. (e) ) (d) ) (c) ) (b) ) (a): These all hold a fortiori (see also the proof of (i)). (a) ) (e): The proof is virtually the same as the proof of (a) ) (c) that was given above. This time, tightness is forced ab initio let h correspond to the condition of -tightness as in Denition 3.3. In the remainder of the proof of (a) ) (c) we now substitute g := g + h, which is certainly a  lower semicontinuous integrand (see the proof of Proposition 3.5). Letting go to zero then gives (e). QED 0. 0. 0. 0. 0. 00. 0. 00. . . . 00. Remark 4.12 Note that in the above proof the Suslin space hypothesis for S (in the shape of Lemmma 4.11) was only used one time, namely for the proof of the implication (a) ) (c). From Remark 4.6 we already know that K-convergence implies narrow convergence of Young measures. The above proof of Theorem 4.10 enables us now to characterize narrow convergence completely in terms of K-convergence:. Theorem 4.13 (i) Suppose that (S ) is Suslin. Let (n ) and 0 be in R(# S). The following are are equivalent:. (a) n =) 0 . (b) Every subsequence (n ) of (n ) contains a further subsequence (n ) such that n (ii) Moreover, if (n ) is  -tight, then the above are also equivalent to the following: (c) n =) 0 . (d) Every subsequence (n ) of (n ) contains a further subsequence (n ) such that n 0. 00. 00. 0. 00. 00. 16. K. ! 0 .. K ;! 0 ..

(35) In parts (b) and (d) the use of subsequences cannot be replaced by the use of the entire sequence (n ) itself, because of Example 4.7. Observe also that in part (ii) the Suslin space hypothesis is actually not needed by Remark 4.12. Theorem 4.14 (i) Suppose that (S ) is Suslin. Let (n ) and 0 be in R(# S). The following are are equivalent:. n =) 0 . ~   . n  n =) 0 (ii) Moreover, if (n ) is  -tight, then the above are also equivalent to the following: (c) n =) 0 . ~   . (d) n  n =) 0 This result, which is the Young measure analogue of Corollary 2.11, follows simply from Proposition 3.13 by Theorem 4.13. Observe once more that in part (ii) the Suslin space hypothesis is actually not needed by Remark 4.12. The transfer of the support Theorem 2.17 to Young measures is now immediate because of the intermediate support Theorem 3.14 and Theorem 4.13: (a) (b). 1. 1. Theorem 4.15 (i) Suppose that (S ) is Suslin. Let (n ) and 0 be in R(# S) with n =) 0. Then  -supp 0 (!)   -Lsn  -supp n (!) for a.e. ! in #. (ii) Let (n ), 0 be in R(# S), with n =) 0 and (n )  -tight. Then 0 (!)(-seq-cl -Lsn-supp n (!)) = 1 for a.e. ! in #. As before, in part (ii) the Suslin space hypothesis is actually not needed (Remark 4.12). Next, we examine narrow convergence when it is restricted to the set RDirac (# S). Recall rst that a sequence (fn ) in L0(# S) is dened to converge in measure to f0 2 L0(# S) (we denote this. as fn ! f0 ) if for every > 0 lim (f! 2 # : (fn (!) f0 (!)) > g) = 0: n Recall also that for any f 2 L0(# S) the image measure f of  under f is dened by f (B) := (f 1 (B)), B 2 B(S) by f (!)(B) = 1B (f(!)) this implies   f ](#  ) = f (). Proposition 4.16 Suppose that (S ) is Suslin. Let (fn ) and f0 be in L0(# S). Then the following ;. are equivalent: (a) fn =) f0 in RDirac (# S).. (b) fn ! f0 in L0(# S).. Proof. (a) ) (b): Let > 0 be arbitrary. Dene a lower semicontinuous integrand on #  S by. g(! x) :=. . ;1 if (x f0 (!)) ,. 0 otherwise. By Lemma 4.11 and Theorem 4.10(i) we have lim infn Jg (fn ) Jg (f0 ) = 0 i.e., lim supn (f! 2 # : (fn (!) f0 (!)) g) = 0. R (b) ) (a):R Let A 2 A, c 2 Cb (S ) be arbitrary. It is enough to prove that  = A c(f0 )d for  := lim infn A c(fn )d (for R the same argument applies to ;c). Clearly, there exists a subsequence (fn ) such that  = limn A c(fn )d. By (b), ((fn  f0 )) certainly converges in measure to zero in L0(# R). So by 4, Theorem 2.5.3] (fn ) has a subsequence (fn ) that -converges a.e. to f0 . The desired identity for  thus follows from the dominated convergence theorem. QED Next, Theorem 2.12 is transferred to tensor products of Young measures. Let (#  A   ) be another nite measure space and let (S   ) be another topological space for which the obvious analogues of Hypotheses 2.1, 2.3 hold we denote the associated metric on S by  (observe that the 0. 0. 0. 0. 0. 00. 0. 0. 0. 0. 17. 0. 0. 0.

(36) topological space S  S then also meets the analogue of Hypotheses 2.1, 2.3). The tensor product    of  2 R(# S) and  2 R(#  S ) is dened by 0. 0. 0. 0. 0. (   )(! ! ) := (!)   (! ) 0. 0. 0. 0. i.e., (   )(! ! ) is the product of the two probability measures (!) and  (! ). It is clear that    , thus dened, is a transition probability from (#  #  A  A ) into S  S  hence, it belongs to R(#  #  S  S ). We now present a continuity result for the tensor product with respect to narrow convergence. There is also a fully topological analogue see 16] where these results were rst introduced (see also 97, Ch. IX]). 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. Theorem 4.17 (i) Let n =) 0 in R(# S) and let n =) 0 in R(#  S ). Then 0. 0. 0. 0. 0. n  n =) 0  0 in R(#  #  S  S ): 0. . 0. 0. 0. 0. (ii) Moreover, if (n ) is  -tight and (n ) is  -tight, then 0. 0. n  n =) 0  0 in R(#  #  S  S ): Lemma 4.18 For every A~ 2 A  A and every there exist nitely many disjoint measurable rectangles Ai  Ai in AA , i = 1 : : : m, such that the symmetric di erence of A~ and

(37) m i=1Ai  Ai has    -measure at most . Proof. The algebra consisting of nite disjoint unions of measurable rectangles generates A  A  hence, the result follows by 4, 1.3.11]. QED Proof of Theorem 4.17. (i) Let A~ 2 A  A and c 2 Cb (S  S     ), and set g(! !  x x ) := 1A~ (! ! )c(x x ). Since uniform limits of nite sums of continuous functions are continuous, the result obtained in Lemma 4.18 enables us to just consider the case A~ = A  A , with A 2 A and A 2 A . We may also suppose (A) > 0,  (A ) > 0. Then Ig (n  n ) = (A) (A ) RS S cd(n  n),  where n :=   n ](A  )=(A) and n :=   n ](A  )= (A ) satisfy n ) 0 and n ) 0, in view of Remark 4.2. By Theorem 2.12(i) this gives Ig (n  n ) ! Ig (0  0 ). This nishes the proof of part (i). Part (ii) directly follows by Theorem 4.8(ii), since (n  n ) is evidently tight for    . Alternatively, it can be obtained as above by using Theorem 2.12(ii) this time. QED As shown by the following counterexample, Theorem 4.17 need not hold when the measure on (#  #  A  A ) is not a product measure, even when  and  are its marginals. Example 4.19 Take for (# A) and (#  A ) the space ( 0 1] B( 0 1])). Let (fn ) be the sequence of Rademacher functions on # and let (fn ) be the sequence of Rademacher functions on # (see Example 3.2). Equip #~ := 0 1]2 with A~ := B( 0 1]2) and ~, dened to be the uniform measure concentrated on the diagonal of 0 1]2. Equip (# A) and (#  A ) each with the Lebesgue measure. Then by Example 3.2, we have fn =) 0 in R(# R) and fn =) 0 in R(#  R), but ( fn  fn ) does not narrowly converge to 0  0 in R(#~  R2). To see the latter, apply Denition 4.1 with A := #~ and c(x x ) := xx  then in Denition 4.1 the limit on the left equals 1, but the expression on the right is equal to 0. 0. . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. . 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 5 Lower closure. Let (# A ) be as in section 4 and let (S ) be a completely regular Suslin space (cf. Remark 2.4(ii)). In this section we combine the main results from section 4 in the form of so-called lower closure results. As an abstract starting point for lower closure we have the following immediate consequence of Theorems 4.8, 4.14 and 4.15: 18.

(38) Theorem 5.1 Let (n) be a  -tight sequence in R(# S). Then there exist a subsequence (n ) of (n ) and  2 R(# S) such that ~   : n  n =) 0. . 0. . 1. Besides,  has the following pointwise support property: .  (!)(-seq-cl -Lsn-supp n (!)) = 1 for a.e. ! in #. Somewhat more concretely Theorem 5.1 can be stated as follows. Let (D dD ) be an arbitrary metric space. . Theorem 5.2 Let (n) in R(# S) be  -tight and let dn ! d0 in L0(# D) (convergence in measure). Then there exist a subsequence (n ) of (n ) and  in R(# S) such that 0. . Z Z Z Z limn inf `(! x dn (!))n (!)(dx)](d!) `(! x d0(!)) (!)(dx)](d!) . . 0.  S. 0. 0. .  S. for every sequentially   dD -sequentially lower semicontinuous integrand ` on #  (S  D) such that Z Z s () := sup ` (! x dn(!))n (!)(dx)](d!) ! 0 for  ! 1: (5:1) . 0. ;. n. . `. f ;. g!n. Besides,  has the following pointwise support property: .  (!)(-seq-cl -Lsn-supp n (!)) = 1 for a.e. ! in #. Here f` ;g!n stands for the set of all x 2 S for which `(! x dn(!)) ;. Proof. Theorem 4.8 and well-known facts about convergence in measure ( 4, Theorem 2.5.3]) imply the existence of a subsequence (n  dn ) of (n  dn) and existence of a  2 R(T  S) such that n =)  and dD (dn (!) d0(!)) ! 0 for a.e. !. By Theorem 4.15 this implies the stated pointwise ~ ~ in R(# S), ~ with S~ := S  N^ , support property for  . By Theorem 4.14 this gives ~n =) ~n := n  n and ~ :=   . Rather than to renumber, we suppose without loss of generality that (n ) enumerates all the numbers in N. Let ` be as stated. We dene g` : #  S~ ! (;1 +1] by  x dk(!)) if k < 1 g`(! x~) := `(! `(! x d0(!)) if k = 1 Then g` is a ~-lower semicontinuous integrand, modulo an insignicant null set (note that for k = 1 lower semicontinuity of g` (! ) at (x 1) follows from dn (!) ! d0(!) and the lower semicontinuity of `(!  ) at (x d0(!))). Since (5.1) coincides with (3.1) for g = g` , we may apply Theorem 4.10 to g` . This gives lim infn Ig` (~n ) Ig` (~ ). Since the following identities hold elementarily for each n and !: Z Z ~ g (! x~)n (!)(d~x) = `(! x dn (!))n (!)(dx) ~ ` ZS ZS g (! x~)~ (!)(d~x) = `(! x d0(!)) (!)(dx) ~ ` . 0. 0. 0. . 0. . 0. . . . . 1. 0. 0. 0. 0. . 0. 0. 0. . S. S. 0. . the main inequality of the theorem has also been proven. QED Remark 5.3 Let h be the nonnegative, sequentially  -inf-compact integrand h on #  S that corresponds as in Denition 3.3 to the  -tight sequence (n ) in Theorem 5.2 i.e., with s := supn Ih (n ) < +1. Then the uniform integrability condition (5.1) applies whenever the integrand ` has the following growth property with respect to h: for every > 0 there exists  2 L1(# R) such that for every n2N ` (! x dn(!)) h(! x) +  (!) on #  S: Indeed, we can observe that the set f` R;g!n in (5.1) is contained in the union of f < hg and f =2g, which gives s () 3 s +

(39)  =2  d, whence s () ! 0 for  ! 1, as claimed. ;. 0. 0. f. . g. 19.

(40) Let us show that the so-called fundamental theorem for Young measures in 40] follows from Theorem 5.2. To this end, let L be a locally compact space that is countable at innity its usual Alexandrov compactication is denoted by L^ := L

(41) f1g. Although it could be avoided by the additional introduction of transition subprobabilities (see the comments below), the Alexandrov compactication L^ of L gures explicitly in the result. The space L^ is metrizable, and its metric is denoted by d.^ On L we use the natural restriction of d,^ and denote it by d. Let C0(L) be the usual space of continuous functions on L that converge to zero at innity. Also, below  denotes a -nite measure on (# A).. Corollary 5.4 (i) Let (fn) in L0(# L) and the closed set C  L be such that limn (fn 1(LnG)) = 0 for every open G, C  G  L. Then there exist a subsequence (fn ) of (fn ) and  in R(# L^ ) such ;. 0. Z. Z Z lim (!)c(fn (!))(d!) = (!)c(x) (!)(dx)](d!) n. that. 0. . . .  L. for every  2 L1(# R) and every c 2 C0 (L). Besides, we have  (!)(LnC) = 0 for a.e. ! in #. (ii) Moreover, if for that subsequence (fn ) there exists a sequence (Kr ) of compact sets in L such that limr supn (f! 2 # : fn (!) 62 Kr g = 0 then  (!)(f1g) = 0 for a.e. ! in # and . 0. !1. 0. 0. Z. . Z Z lim (!)c(fn (!))(d!) = (!)c(x) (!)(dx)](d!) n 0. for every A 2 A,  L1(A R).. A 2 L1 (A R) and. . A L. c 2 C (L) for which (1A c(fn )) is relatively weakly compact in 0. In 40] both L and # are Euclidean, and the Kr 's are closed balls around the origin with radius r. As was done in 40], the result could be equivalently restated in terms of the transition subprobability  from (# A) into (L B(L)), dened by obvious restriction to L, i.e.,  (!)(B) :=  (!)(B

(42) f1g), B 2 B(L). In this connection the tightness condition in part (ii) guarantees that  is an authentic transition probability (Young measure). Rather than via (i), part (ii) can also be derived directly from Theorem 3.7 or 5.2. Proof. (i) By -niteness of , there exists a nite measure  that is equivalent to . Let ~ be a version of the Radon-Nikodym density d=d. Now (n ), dened by n := fn 2 R(#  L^ ), is trivially tight by compactness of L^ (set h  0). By Theorem 4.8 or 5.2 there exist a subsequence (fn ) of (fn ) and  2 R(# L^ ) for which fn =)  . Every c 2 C0 (L) has a canonical extension c^ 2 Cb(L^ ) by setting c^(1) = 0. Now ~ is -integrable for any  2 L1(# A  R), and Theorem 4.10 (or 5.2) can be applied to g : # RL^ ! R R ~(!)^c(x). This gives the desired R given by g(! x) :=R(!) ~ equality, because of the identity   L c^(x) ()(dx)d =   L c(x) ()(dx)d. Next, let C be as stated. For any i 2 N the set Fi , consisting of all x 2 L with d-dist(x C) i 1 , is closed in L. Note already that \i Fi = C, by the given d -closedness of C in L. Further, F^i := Fi

(43) f1g is closed in L^ . Set g^i (! x) := ~(!)1L^ F^i (x). This denes a nonnegative lower semicontinuous integrand g^i on #  L^ . Hence, Ig^i ( ) i := lim infn Ig^i ( fn ) by Theorem 4.10(c). By L^ nF^i = LnFi the denitions of g^i and fn give Ig^i ( fn ) = (fn 1(LnFi )). So i = lim infn (fn 1(LnFi )) (fn 1 (LnGi)), where Gi, Gi  Fi , is the d -open Rset of all x 2 L with d-dist(x C) < i 1 . Since Gi  C, the hypotheses imply 0 = i Ig^i ( ) =   ()(LnFi )d. Hence  (!)(LnC) = 0 -a.e. because of \iFi = C, which was demonstrated above. (ii) The additional condition is then a tightness condition for ( fn ), when viewed as a subset of R(# L) (take ;  Kr for large enough r in Denition 3.3(b)). Hence, there is a  -inf-compact integrand h on #  L with s := supn Ih (n ) < +1. Dene the inf-compact integrand ^h on #  L^ by h^ (! x) := h(! x) if x 2 L and ^h(! 1) := +1. Since ^h is in particular a lower semicontinuous integrand on #  L^ , we have Ih^ ( ) lim infn Ih^ ( fn ) by Theorem 4.10. Trivially, Ih^ ( fn ) = Ih ( fn ), so we get Ih^ ( ) s < +1. The latter shows that  (!)(f1g) = 0 for -a.e. ! in #, whence for -a.e. !. So  can also be viewed as an element of R(# L), for which we then get. fn =d)  in R(# L) by the above. To conclude, observe that for any A 2 A with (A) < +1 0 . 0 . . . 0. . . 0. . . ;. n. 0. . 0. ;. 0. 0. 0. 0. ;. 0. ;. 0. . . . 0. . 0. . 0. 0. . . 0. ;. . 20.

(44) Theorem 4.10 applies to g(! x) := 1A (!)(!)~(!)c(x), which is a continuous integrand on #  L that is -integrably bounded. In view of part (i), this gives the desired limit statement if A has nite measure. If (A) = +1 and A is as stated, there exists, by 's -niteness, a sequence (Aj ) of subsets of A with nite -measure, with Aj " A. The previous result applies to each R of the Aj and the weak relative compactness hypothesis implies uniform -additivity, i.e., supn A Aj jc(fn)jd # 0 47]. So also in this case the desired limit statement follows. QED If in the above lower closure Theorem 5.2 additional conditions are imposed upon the Young measures (n ), then extra \barycentric" information about  may become available in terms of its marginals. In this way, Theorem 5.2 will be turned into a very general lower closure result \with convexity". Let E and F be separable Banach spaces, each of which is equipped with a locally convex Hausdor topology, respectively denoted by E and F , that is not weaker than the weak topology and not stronger than the norm topology. As usual, L1(# E) denotes the space of all Bochner integrable E-valued functions (here this is precisely the space of all e 2 L0(# E) such that ke()kE is -integrable). Let (D dD ) be a metric space. Functions that are \barycentrically" associated to Young measures can play a special role in lower closure and existence results. This is demonstrated by our proof of the following result. n. . w 1 Theorem 5.5 Let dn ! d0 in L0 (# D) (convergence R in measure), en ! e0 in L (# E) (weak 1 convergence), and let (fn ) in L (# F ) satisfy supn  kfnkF d < +1. Suppose that there exist E and F -ball-compact multifunctions RE : # ! 2E and RF : # ! 2F (cf. Example 3.4) such that f(en(!) fn(!)) : n 2 Ng  RE (!)  RF (!) -a.e. Then there exist a subsequence (dn  en  fn ) of (dn en  fn) and f 2 L1(# F ) such that. limn inf 0. 0. Z. . 0. `(! en (!) fn (!) dn (!))(d!) 0. . 0. 0. 0. Z. . . . `(! e0 (!) f (!) d0(!))(d!) . for every sequentially E  F  D -lower semicontinuous integrand ` on #  (E  F  D) such that the following hold:. (` ( en() fn() dn())) is uniformly (outer) integrable ;. (see Remark 3.6(ii)) and. `(!   d0(!)) is convex on E  F for a.e. !. Besides, the functions e0 and f can be localized as follows: 2 (e0 (!) f (!)) 2 cl co-w-Lsnf(en (!) fn (!))g for a.e. ! in #: Observe, as was already done following Example 3.4, that the ball-compactness condition involving RE and RF is automatically satised in case the Banach spaces E and F are re%exive. Proof. To apply Theorem 5.2 we set S := E  F ,  := E  F and n := (en fn ) . Observe that S is a separable Banach space for the product norm k  kS , so (S ) is a Suslin space, and by the Hahn-Banach theorem (S ) is completely regular. Next, we note that (ken k) in L1(# R) is uniformly integrable this follows from the weak convergenceR hypothesis (apply 47, Theorem 1] and 81, Proposition II.5.2]). In particular, this implies supn  k(en  fn)kS d < +1. By -ballcompactness of R := RE  RF this proves that (n ) is -tight, in view of Example 3.4. We can now apply Theorem 5.2. Let the subsequence (n  dn ) of (n  dn) and  in R(# S) be as guaranteed by that theorem, i.e., with n =)  . Then it is elementary to establish from Denition 4.1 that,  \E-marginally", en =)  E and, \F -marginally", fn =)  F . Here  E (!) :=  (!)(E  ) and  F (!) :=  (!)(  F ). So E-marginally we then have the situation of Example 4.9(b), which gives that bar  E = e0 a.e. Also, F -marginally we have the more primitive situation of Example 4.9(a), . . 0. 0. 0. 0. . . 0. . . . . . 2. In case E and F are nite-dimensional one may replace here \cl co" by \co".. 21. . .

(45) which gives existence of f 2 L1(# F ) such that f = bar  F a.e. (note that E - and F -ballcompactness imply (E E )- and (F F )-ball-compactness respectively). Recombining the above two marginal cases, we nd bar  = (e0  f ) a.e. (note that barycenters decompose marginally). We now nish the proof. For an integrand ` of the stated variety Theorem 5.2 gives Z Z Z limn inf `(! en (!) fn (!) dn (!))(d!) `(! x y d0(!)) (!)(d(x y))](d!)  0. . . . . . 0. . 0. . 0. 0. 0. .  E F . (see also Remark 3.6(ii)). In the inner integral above, the convexity of `(!   d0(!)) gives Z `(! x y d0(!)) (!) g(! bar  (!) d0(!)) = g(! e0(!) f (!) d0(!)) . E F. . . for a.e. !, by Jensen's inequality and our previous identity bar  = (e0  f ) a.e. The desired inequality thus follows. QED The above lower closure result \with convexity" is quite general: it further extends the results in 9, 14], which in turn already generalize several lower closure results in the literature, including those for orientor elds (cf. 52]). See 22] for another development, not covered by the above result. Results of this kind are very useful in the existence theory for optimal control and optimal growth theory. Corollaries of Theorem 5.5 are so-called weak-strong lower semicontinuity results for integral functionals in the calculus of variations and optimal growth theory cf. 45, 52, 65]. Recently, similar-spirited versions that employ quasi-convexity in the sense of Morrey have been derived from Theorem 5.2 in 72, 90] (these have for en the gradient function of dn and depend on a characterization of so-called gradient Young measures 83]). Another result that is generalized by the above theorem is as follows. w f in L1(# Rd ) (weak convergence). Then Corollary 5.6 Let fn ! 0 f0 (!) 2 co-Lsnffn (!)g for a.e. ! in #: This result is due to Z. Artstein 3, Proposition C]. It is obtained from Theorem 5.5 by setting E := Rd and activating the footnote in its statement. We turn brie%y to an extension of the Dunford-Pettis theorem (suciency part) this comes from 25, 32] and generalizes 49] and 36, Lemma 4.3]. Again E denotes a separable Banach space. Theorem 5.7 Let (fn ) in L1(# E) be uniformly integrable and such that for every > 0 there is a multifunction ; : # ! 2E , having norm-compact values with  (f! 2 # : fn (!) 62 ; (!)Rg) for all R n. Then there exist a subsequence (fn ) of (fn ) and f 2 L1 (# E) such that limn k A fn d ; A f dk = 0 for every A 2 A. Above  stands for outer -measure. Obviously, when E is nite-dimensional, the tightness condition in the above result holds automatically and we get the Dunford-Pettis theorem (suciency part). Proof. We set S := E and  := norm-topology. By Denition 3.3(b), the sequence ( fn ) is tight. Also, by uniform integrability, (fn ) is of course bounded in L1-seminorm. Theorem 3.7 gives existence of a subsequence (fn ) and  2 R(# E) such that fn =)  . Because of (E E )  , w Example 4.9(b) implies and set R fn ! f := bar  . But more can be said. Let A 2 A be arbitrary R  := ; lim supn k A (fn ; f )kd. Without loss of generality we may suppose ;k A (fn ; f )dk !  0. By the Hahn-Banach theorem, there exists a sequence (xn ) in the unit sphere of the dual space E such that Z Z Z ;k (fn ; f )dk =< (fn ; f )d xn >= < fn ; f  xn > d . . . 0. 0. . 0. . . 0. 0. . 0. 0. . 0. . 0. . 0. . 0. . 0. 0. A. 0. . A. 0. . 0. 0. A. 0. . 0. 0. for every n . By the Alaoglu-Bourbaki theorem it then follows that a subsequence of (xn ) converges in the weak star topology to some x in the closed unit ball of E (note that this ball is metrizable) 0. 0. 0 1. 0. 22. 0.

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