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Characterization of L1-Closed Decomposable Sets in L∞
Journal of Mathematical Analysis and Applications 238, 491᎐515 Ž1999. Article ID jmaa.1999.6531, available online at http:rrwww.idealibrary.com on
Characterization of L1-Closed Decomposable Sets in L⬁* Zsolt Pales ´ Institute of Mathematics and Informatics, L. Kossuth Uni¨ ersity, H-4010 Debrecen, Hungary E-mail: [email protected]
and Vera Zeidan Department of Mathematics, Michigan State Uni¨ ersity, East Lansing, Michigan 48824 E-mail: [email protected] Submitted by Helene ´` Frankowska Received December 19, 1997
In this paper the set of essentially bounded measurable selections of a measurable set-valued map is considered. One of the main results states that only those subsets of L⬁ that are L1-closed in L⬁ and decomposable can be represented in this way. The connection between topological and convexity properties of the set-valued map and its L⬁-selection set is also studied in details. The adjacent and Clarke’s tangent cones to L1-closed and decomposable sets and their polar cones are described in terms of the corresponding set-valued map as well. 䊚 1999 Academic Press
Key Words: L1-closed in L⬁ sets; decomposable sets; measurable set-valued maps; L⬁-selections.
1. INTRODUCTION Throughout this paper let Ž ⍀, A, . denote a finite complete measure space, and for 1 F p F ⬁ let L p Ž ⍀ . s L p Ž ⍀, ⺢ m . be the space of equiva*Supported by Hungarian Soros Foundation grant 022-2998r95; by the Hungarian National Foundation for Scientific Research ŽOTKA., grant T-016846; and by the National Science Foundation, under grant NSF-DMS-9404591. 491 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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lence classes of L p-integrable functions equipped with the usual norm 5 ⭈ 5 p . For the sake of brevity, we write a.e. for ‘‘-almost everywhere.’’ The notion of decomposable sets in L p spaces appeared in the work of Hiaı¨ and Umegaki w8x: A set Q : L p Ž ⍀ . is called decomposable if, for x, y g Q and for A g A, we have 1 A x q 1 ⍀ _ A y g Q. It is easily seen that if Q: ⍀ ª 2 ⺢ is an arbitrary set-valued function, then the set of L p-measurable selections m
p Ž Q . s x g L p Ž ⍀ . N x Ž t . g Q Ž t . for a.e. t g ⍀ 4
Ž 1.
is always a decomposable subset of L ⍀ .. Therefore, it is natural to ask if the converse of this statement is also true. In the case 1 - p F ⬁, Hiaı¨ and Umegaki w8x proved that the closed decomposable sets can be derived from set-valued maps. More precisely they proved the following: pŽ
THEOREM A w8, Theorem 3.1x. Let 1 F p - ⬁ and Q ; L p Ž ⍀ . be a nonempty closed set. Then there exists a measurable nonempty closed set m ¨ alued map Q: ⍀ ª 2 ⺢ such that Q s p Ž Q ., where p Ž Q . is defined in Ž1., if and only if Q is decomposable. For the case p s ⬁, there is no similar result available in the literature.1 In fact, if we consider a nonempty, closed, and decomposable subset Q of L⬁Ž ⍀ ., it is not in general represented by a measurable set-valued map Q with nonempty and closed images. As we shall show in Lemma 1 below, the image ⬁Ž Q . of such a set-valued map Q is not only closed in L⬁Ž ⍀ ., but also L1-closed. Therefore, in L⬁Ž ⍀ ., the L1-closed decomposable sets are the proper candidates for the representability via set-valued functions. DEFINITION 1. A set Q : L⬁Ž ⍀ . is called L p-closed in L⬁ if, whenever a sequence x n4 in Q converges to x 0 g L⬁Ž ⍀ . in the L p-norm, then x 0 g Q. The L p-closure in L⬁ of a set Q g L⬁Ž ⍀ . will be denoted by cl pQ. In Theorem 2 we describe the class of set-valued maps whose members are in a bijective correspondence with L1-closed in L⬁ decomposable sets. The next results of the paper offer an analogous description for L1-closed decomposable sets Q with the following additional properties: 䢇 䢇 䢇
1
Q has a nonempty interior. Q is convex. Q is convex and has a nonempty interior.
The authors thank the anonymous referee who called their attention to an unpublished paper of Hiaı¨ w7x, distributed among the participants at an international conference held in Catania, 1983, in which the case p s ⬁ is considered.
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In the last two cases, the corresponding set-valued map has convex images; therefore it can also be described via its supporting functional. In the last section, we develop the notions of convex analysis for the sets that are L1-closed and decomposable. These results are essential for the investigations of extremum problems, where constraints of the form x Ž t . g QŽ t .
Ž x g L⬁ Ž ⍀ . .
for a.e. t g ⍀
are present. The need for investigating constraints of this form stems from control theory. Some of the results in this paper could be extended to the case where the underlying measure space is -finite. However, most of the applications we are interested in have finite measure spaces. For instance, this is the case when the results of this paper are applied to the optimal control problem. This application will be established in a subsequent paper w11x. A summary of the results derived in this paper has been published in w10x. 2. DECOMPOSABLE AND L1-CLOSED SUBSETS IN L⬁Ž ⍀ . The notion of equality between measurable set-valued maps over ⍀ is defined as Q1 s Q 2
means
Q1 Ž t . s Q 2 Ž t .
for a.e. t g ⍀ .
A characterization of the nonemptiness of the image ⬁Ž Q . is given by the following. LEMMA 1. Let Q: ⍀ ª 2 ⺢ be a measurable set-¨ alued map with nonempty closed images. Then, ⬁Ž Q . is L1-closed in L⬁ and decomposable. Furthermore, ⬁Ž Q . / ⭋ if and only if m
᭚ r ) 0: Q Ž t . l Br / ⭋
for a.e. t g ⍀ .
Ž 2.
Ž Here Br denotes the ball of radius r in ⺢ m centered at the origin.. Proof. The decomposability of ⬁Ž Q . is obvious. For the L1-closedness, let x n g ⬁Ž Q . be a sequence such that x n tends in L1-norm to x 0 g L⬁Ž ⍀ .. Then there exists a subsequence of x n , denoted by x n k, such that, as k ª ⬁, x n kŽ t . ª x 0 Ž t .
for a.e. t g ⍀ .
By the closedness of QŽ t ., x 0 Ž t . g QŽ t . for a.e. t g ⍀. Therefore x 0 g ⬁Ž Q ..
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If ⬁Ž Q . / ⭋, then there exists x 0 g ⬁Ž Q .. Clearly, < x 0 Ž t .< F 5 x 0 5 ⬁ \ r for a.e. t g ⍀. Thus QŽ t . l Br is nonempty for a.e. t g ⍀. On the other hand, if condition Ž2. holds on a set A of full measure, then the set-valued map Q r , defined by Qr Ž t . s
½
Q Ž t . l Br ⺢m
t g A, t g ⍀ _ A,
is measurable with nonempty and closed values. Therefore, by the measurable selection theorem Žsee w2x., there exists a measurable function x 0 : ⍀ ª ⺢ m such that, for all t in ⍀, x 0 Ž t . g Qr Ž t . . Clearly 5 x 0 5 ⬁ F r and x Ž t . g QŽ t ., a.e. Thus, x 0 g ⬁Ž Q .. Remark 1. Condition Ž2. is equivalent to the essential boundedness of the distance function from 0 to QŽ t .. For the case where p g w1, ⬁., this fact is given in w1x. Remark 2. The L1-closedness in L⬁ of ⬁Ž Q . also follows from the equality
⬁Ž Q . s L⬁ Ž ⍀ . l 1 Ž Q . , because 1Ž Q . is easily seen to be L1-closed. Consider a nonempty and decomposable subset Q of L⬁Ž ⍀ .. We furnish first a characterization of the L p-closure in L⬁ of Q. As a consequence, we will obtain that, for each p g w1, ⬁., the L1-closure in L⬁ of Q coincides with its L p-closure in L⬁. THEOREM 1. Let Q ; L⬁Ž ⍀ . be a nonempty decomposable set. Then, for p g w1, ⬁., x g cl pQ if and only if there exists a sequence of measurable sets A n g A and g Q such that
Ž An . ª Ž ⍀ .
as n ª ⬁
and 1 A nx q 1 ⍀ _ A n g cl⬁Q
for all n g ⺞.
Proof. Assume that x g cl pQ. Then there exists a sequence x k g Q such that 5 x k y x 5 p ª 0 as k ª ⬁. By taking subsequences if necessary, we can assume that x k tends to x a.e. in ⍀. By Egorov’s theorem, for all n g ⺞, there exists A n g A such that Ž ⍀ _ A n . - 1rn and x k converges uniformly to x on A n . Let g Q be arbitrary and let
k , n [ 1 A nx k q 1 ⍀ _ A n
Ž k, n g ⺞ . .
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Then k, n g Q by the decomposability. On the other hand, for all fixed n g ⺞, the sequence k, n4⬁ks1 converges uniformly on ⍀ to 1 A nx q 1 ⍀ _ A n . Therefore 1 A nx q 1 ⍀ _ A n g cl⬁Q for all n g ⺞. Conversely, assume that there exists a sequence of sets A n g A and g Q such that Ž ⍀ _ A n . ª 0 and x n [ 1 A nx q 1 ⍀ _ A n g cl⬁Q
for all n g ⺞.
Then there exists n g Q such that 5 n y x n 5 ⬁ - 1rn. We show that 5 n y x 5 p ª 0: 5 n y x 5 p F 5 n y x n 5 p q 5 x n y x 5 p F F
1 n 1 n
1rp
Ž Ž ⍀ . .
1rp
Ž Ž ⍀ . .
1rp
q
žH
⍀_A n
< y x < pd
/
q 5 y x 5 ⬁p Ž Ž ⍀ _ A n . .
1rp
ª 0.
Therefore, x g cl p ŽQ.. COROLLARY 1.
If Q : L⬁Ž ⍀ . is nonempty and decomposable, then
Ži. For all p g w1, ⬁., cl 1Q s cl pQ. Žii. For all p g w1, ⬁., ⬁
cl pQ s x g L⬁ Ž ⍀ . N ᭚ n 4 ns 1 ; Q:
½
lim 5 n y x 5 p s 0 and sup 5 n 5 ⬁ - ⬁ .
nª⬁
ng⺞
5
Proof. The characterization of the L p-closure in L⬁ of decomposable sets obtained in the above theorem does not depend on p. Therefore Ži. is obvious. For Žii., it suffices to show only the : inclusion. Let x g cl pQ. then the characterization presented by the theorem holds true. Thus the sequence n g Q constructed in the ‘‘only if’’ part of the proof of theorem tends to x in L p-norm. On the other hand, for all n g ⺞, 5 n 5 ⬁ F 5 n y x n 5 ⬁ q 5 x n 5 ⬁ F 1 q max Ž 5 x 5 ⬁ , 5 5 ⬁ . .
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Remark 3. It follows from statement Žii. of the above corollary that a subset Q of L⬁Ž ⍀ . is L p-closed in L⬁ Žfor some 1 F p - ⬁. if and only if Q is closed under the boundedly a.e. convergence. That is, n g Q for all n g ⺞, the sequence n converges to x almost everywhere, and sup ng ⺞ 5 n 5 ⬁ - ⬁, and then x also belongs to Q. This latter notion of convergence was used in Hiaı’s ¨ paper w7x. Indeed, if x g cl pQ s Q, then by using Žii. and extracting an a.e. convergent subsequence if necessary, we get that Q is boundedly a.e. closed. Conversely, if n g Q boundedly a.e. converges to x, then, due to the finiteness of the measure space, n tends to x in L p-norm as well. Let us introduce the following classes: ⌫ Ž ⍀ . [ Q: ⍀ ª 2 ⺢ N Q is measurable with closed nonempty m
images satisfying Ž 2 . 4 and ⌺ Ž ⍀ . [ Q ; L⬁ Ž ⍀ . N Q is L1-closed in L⬁ , nonempty and decomposable4 . From Lemma 1, we know that these two classes are related. The exact nature of their relationship is established in the next theorem. THEOREM 2. The map ⬁ is a bijection between ⌫ Ž ⍀ . and ⌺ Ž ⍀ .. Proof. By Lemma 1, ⬁ maps ⌫ Ž ⍀ . into ⌺ Ž ⍀ .. The injectivity of ⬁ is the consequence of a result by Hiaı¨ and Umegaki w8, Corollary 1.2x. Now we prove that ⬁ is surjective. Let Q g ⌺ Ž ⍀ . be arbitrary. We can consider Q as a subset of L1 Ž ⍀ .. Let Q denote the closure of Q in L1 Ž ⍀ . Žthis set may not coincide with Q because Q is only L1-closed in L⬁ .. Since Q is decomposable, Q is also decomposable. Therefore, by w8, Theorem m 3.1x, there exist a measurable nonempty closed set-valued map Q: ⍀ ª 2 ⺢ such that Q s 1Ž Q . . We show that Q s ⬁Ž Q .. If x g Q, then x g Q; therefore x g 1ŽQ.. However, x g L⬁Ž ⍀ ., and hence x g ⬁Ž Q .. On the other hand, if x g ⬁Ž Q ., then x g 1Ž Q . s Q. That is, x g Q l L⬁Ž ⍀ .. Since Q is L1-closed, Q l L⬁Ž ⍀ . s Q, which yields x g Q. Remark 4. In the unpublished paper w7, Theorem 1.5x, Hiaı¨ obtained an analogous result where he assumed the underlying measure space to be -finite and used the notion of the boundedly a.e. closedness instead of L1-closedness in L⬁.
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Our next aim is to consider L1-closed decomposable sets with a nonempty interior. LEMMA 2. Let Q s ⬁Ž Q ., where Q g ⌫ Ž ⍀ .. Then x g L⬁Ž ⍀ . belongs to int Q Ž the interior of Q. if and only if ᭚ ) 0: B Ž x Ž t . . : Q Ž t .
for a.e. t g ⍀ .
Ž 3.
Proof. Let x g int Q. Then there exists ) 0 such that B Ž x . : Q. Let yn4⬁ns1 be a dense subset of the ball B : ⺢ m . Then x n Ž t . [ x Ž t . q yn
Ž n g ⺞.
defines an essentially bounded measurable function such that 5 x n y x 5 ⬁ F . Therefore x n g B Ž x . : Q. Thus xnŽ t . g QŽ t .
for t g A n ,
where A n g A is of full measure. Then xnŽ t . g QŽ t .
᭙ n g ⺞ and ᭙ t g A [
⬁
F An ,
ns1
and A is of full measure, In other words, x Ž t . q yn g Q Ž t .
᭙ n g ⺞, ᭙ t g A.
The set QŽ t . is closed; therefore this inclusion implies B Ž x Ž t . . : Q Ž t .
᭙ t g A.
Conversely, if Ž3. holds and y g B Ž x ., then < y Ž t . y x Ž t .< F a.e. on ⍀. Therefore, y Ž t . g B Ž x Ž t . . : Q Ž t .
for a.e. t g ⍀ .
Thus y g ⬁Ž Q ., which yields B Ž x . : Q. As we shall prove in the result below, the nonemptiness of the interior of ⬁Ž Q . is characterized by the following property of Q: ᭚ r G ) 0 and, for a.e. t g ⍀ , ᭚ x t g ⺢ m such that B Ž x t . : Q Ž t . l Br .
Ž 4.
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498 Therefore, set
⌫0 Ž ⍀ . [ Q g ⌫ Ž ⍀ . N Q satisfies Ž 4 . 4 and ⌺ 0 Ž ⍀ . [ Q g ⌺ Ž ⍀ . N int Q / ⭋ 4 . THEOREM 3. The map ⬁ is a bijection between ⌫0 Ž ⍀ . and ⌺ 0 Ž ⍀ .. Proof. We show first that ⬁Ž ⌫0 Ž ⍀ .. : ⌺ 0 Ž ⍀ .. Let Q g ⌫0 Ž ⍀ . be arbitrary. Let r G ) 0 and A g A be a set of full measure such that Q r Ž t . [ QŽ t . l Br satisfies ᭙ t g A ᭚ x t : B Ž x t . : Q r Ž t . . Define, for t g A, R Ž t . s y N B Ž y . : Q r Ž t . 4 . By our assumption, RŽ t . / ⭋ for all t g A. We are going to show that R is a measurable closed set-valued mapping. The set-valued map with images ⭸ Q r Ž t . Žwhere ⭸ stands for the boundary. is clearly a measurable closed set-valued map. Therefore, by the characterization theorem of measurable set-valued maps Žsee w2x., there exists a sequence of measurable selections n4⬁ns1 of ⭸ Q r such that
⭸ Q r Ž t . s cl n Ž t . N n g ⺞ 4
for t g A.
Let ⌽n Ž t . s ⺢ m _ U Ž n Ž t . .
for t g A, n g ⺞,
where U Ž z . denotes the open ball in ⺢ m of radius centered at z. Then ⌽n is a measurable closed set-valued mapping. Therefore, the set-valued map t ª Qr Ž t . l
⬁
ž
F ⌽n Ž t .
ns1
/
\ SŽ t .
Ž t g A.
is also measurable. We show that, for all t g A, RŽ t . s SŽ t . .
Ž 5.
If x t g RŽ t ., then B Ž x t . ; Q r Ž t .. Hence x t g Q r Ž t . and the distance of x t from any boundary points of Q r Ž t . is at least ; that is if y g ⭸ Q r Ž t ., then
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x t f U Ž y . . Thus, x t f U Ž nŽ t .., i.e., x t g ⺢ m _ U Ž nŽ t .. for all n g ⺞. Therefore, x t g SŽ t ., and hence, we have RŽ t . : SŽ t .. On the other hand, if x t f RŽ t . but x t g Q r Ž t ., then B Ž x t . Q r Ž t .. Therefore, there exists y g ⭸ Q r Ž t . such that < y y x t < - . The sequence nŽ t .4n is dense in ⭸ Q r Ž t .; hence, for some n g ⺞, < n Ž t . y x t < - , that is, x t g U Ž nŽ t ... Therefore, x t f SŽ t .. We have proved that R is a measurable nonempty closed valued map. Therefore, there exists a measurable selection x of R. That is, x Ž t . g RŽ t .
for all t g A.
By the definition of R, B Ž x Ž t . . : Q r Ž t .
for all t g A.
Thus, x g L⬁Ž ⍀ . and condition Ž3. is satisfied. Therefore, x g int ⬁Ž Q .. Conversely, let Q g ⌺ 0 Ž ⍀ .. Then, there exists Q g ⌫ Ž ⍀ . such that Ž ⬁ Q . s Q. Let x be an interior point of Q. By Lemma 2, ᭚ ) 0 such that Ž3. is valid. Let r s 5 x 5 ⬁ q . Then B Ž x Ž t . . : Br
for a.e. t g ⍀ ;
hence Ž4. is satisfied with x t s x Ž t .. Now consider an element Q of ⌫ Ž ⍀ ., and Q s ⬁Ž Q .. We are interested in investigating the connection between the tangent cones associated with each of Q and Q. For this purpose we focus on two types of tangent cones to a subset M of a normed vector space X, namely, the adjacent cone T Ž x 0 N M . and the Clarke tangent cone C Ž x 0 N M . Žwhere x 0 belongs to the closure of M . which are defined by T Ž x 0 N M . [ ¨ g X N ᭙ n ª 0q, ᭚ ¨ n ª ¨ such that x 0 q n¨ n g M, ᭙ n g ⺞ 4 and C Ž x 0 N M . [ ¨ g X N ᭙ n ª 0q, ᭙ x n g M : x n ª x 0 , ᭚ ¨ n ª ¨ with x n q n¨ n g M, ᭙ n g ⺞ 4 .
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Both tangent cones are nonempty and closed; the cone C Ž x 0 N M . is a convex subset of T Ž x 0 N M .. Let Q in ⌫ and x 0 g ⬁Ž Q .. We can associate with Q two set-valued maps T Ž x 0 N Q . and C Ž x 0 N Q ., defined, for t in ⍀, by T Ž x 0 N Q . Ž t . [T Ž x 0 Ž t . N Q Ž t . .
and C Ž x 0 N Q . Ž t . [C Ž x 0 Ž t . N Q Ž t . . .
It is shown by Giner in w5x that these set-valued maps are measurable. Thus, they are elements of ⌫ Ž ⍀ .. On the other hand, we can define these tangent cones for the set ⬁Ž Q ., that is, T Ž x 0 N ⬁Ž Q .. and C Ž x 0 N ⬁Ž Q ... These cones are decomposable and closed subsets in L⬁Ž ⍀ .. However, they are not necessarily L1-closed in L⬁. Hence, by Theorem 2, they cannot be represented via the set-valued map T Ž x 0 N Q . and C Ž x 0 N Q ., respectively. As we shall show below, these set-valued maps correspond via ⬁ to the L1-closures in L⬁ of those tangent cones. The proof of the results in based on the work by Giner w5x. THEOREM 4.
Let Q g ⌫ Ž ⍀ .. Then, for all x 0 g ⬁Ž Q ., we ha¨ e
cl 1T Ž x 0 N ⬁Ž Q . . s⬁Ž T Ž x 0 N Q . .
and cl 1C Ž x 0 N ⬁Ž Q . . s⬁Ž C Ž x 0 NQ . . .
Proof. Let K denote either T or C. From w5, Proposition 1.6, Chap. VIIx it follows that K Ž x 0 N ⬁Ž Q . . ; ⬁Ž K Ž x 0 N Q . . . Since ⬁Ž K Ž x 0 N Q .. is L1-closed in L⬁ , we obtain the inclusion : in Ži. and Žii.. Conversely, let x g ⬁Ž K Ž x 0 N Q ... By w5, Proposition 1.6, Chap. VIIx, there exists a sequence A k 4⬁ks1 of measurable sets of ⍀ such that lim Ž A k . s Ž ⍀ .
kª⬁
and 1 A k x g K Ž x 0 N ⬁Ž Q . . .
This means that, for s 0, x satisfies the condition in Theorem 1 characterizing the L1-closure in L⬁ of K Ž x 0 N ⬁Ž Q ... Hence, x g cl 1 K Ž x 0 N ⬁Ž Q ... Remark 5. In w5x, it is shown that for p g w1, ⬁.,
p Ž T Ž x 0 N Q . . : T Ž x 0 N p Ž Q . .
and
C Ž x 0 N p Ž Q . . : p Ž C Ž x 0 N Q . . ,
and these inclusions are strict in general. Hence, Theorem 4 states that in L⬁Ž ⍀ ., certain equalities do occur, yielding no gap between the concerned sets.
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3. L1-CLOSED DECOMPOSABLE AND CONVEX SETS IN L⬁Ž ⍀ . This section is devoted to the study of convexity properties of decomposable sets. m Let Q: ⍀ ª 2 ⺢ be a set-valued map on ⍀ with convex images. Then ⬁Ž Q . is decomposable and convex. However, this subset of L⬁Ž ⍀ . enjoys a richer property than the two combined that we call the L-con¨ exity. A set Q ; L⬁Ž ⍀ . is called L-con¨ ex if for all x, y g Q, and for all : ⍀ ª w0, 1x measurable, we have
x q Ž 1 y . y g Q. This notion defined for sets in L⬁Ž ⍀ . is the analog of the C-convexity defined in w9x for sets of continuous functions over a compact Hausdorff space. The next result shows how this notion is related to convexity for closed sets in L⬁Ž ⍀ .. Let Q be a closed subset of L⬁Ž ⍀ .. Then, the following are
LEMMA 3. equi¨ alent:
Ži. Q is con¨ ex and decomposable. Žii. Q is L-con¨ ex. Žiii. For all k g ⺞, for all 1 , . . . , k g L⬁Ž ⍀, w0, 1x. such that Ý kis1 i s 1, and for all x 1 , . . . , x k g Q, the L-con¨ ex combination Ý kis1 i x i belongs to Q. Proof. Clearly Žiii. « Žii. and Žii. « Ži.. It remains to show that Ži. « Žiii.. Assume Ži.. We first show that Žiii. is true when 1 , . . . , k are step functions on ⍀. Let 1 , . . . , k be in L⬁Ž ⍀, w0, 1x. with a range consisting of finite elements. Without loss of generality, we can assume that there exists a measurable partition A1 , . . . , A l of ⍀ such that, for all i s 1, . . . , k, l
i s
i j1 A j ,
Ý js1
where k
Ý
i j s 1 and
i j G 0
᭙ i , j.
is1
Hence k
Ý is1
k
i x i s
l
ž
Ý Ý is1
js1
l
i j1 A j x i s
/
k
ž
Ý Ý js1
is1
i j x i 1 A j .
/
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Since x i g Q for all i, and Q is convex, it results that y j [ Ý kis1 i j x i g Q, for all j s 1, . . . , l. By using the decomposability of Q, it follows that l
y j 1 A j g Q.
Ý js1
Now, let 1 , . . . , k be arbitrary elements in L⬁Ž ⍀, w0, 1x. with Ý kis1 i s 1. Then, for each i, there exists a sequence of step functions ni 4⬁ns1 such that k
ni ª i as n ª ⬁,
ni G 0, and
Ý
ni s 1.
is1
Thus k
Ý
k
i x i s lim
Ý ni x i .
nª⬁ is1
is1
Since Q is closed, it follows that Ý kis1 i x i g Q. Now let Q be in ⌫ Ž ⍀ .. We can associate with Q another set-valued map defined by
Ž co Q . Ž t . s co Q Ž t . . In w8x, it is shown that for p g w1, ⬁.,
p Ž co Q . s cl p co Ž p Ž Q . . . The case where p s ⬁ has been an open question. Our goal now is to solve this open question. As we shall see below, the above identity is not true when p s ⬁. However, it is known Žcf. w1x. that co Q is measurable and hence is in ⌫ Ž ⍀ .. The associated set ⬁Žco Q . is not only L⬁ -closed but also L1-closed in L⬁ and L-convex. Thus, as in Theorem 4, we shall show that by replacing, in the sought identity, the L⬁-closure with the L1-closure in L⬁, we obtain a valid statement. THEOREM 5.
Let Q be in ⌫ Ž ⍀ .. Then
⬁Ž co Q . s cl 1co Ž ⬁Ž Q . . . Proof. It is clear that ⬁Žco Q . = cl 1coŽ ⬁Ž Q ... It remains to show the opposite inclusion. Let x g ⬁Žco Q .. Then x Ž t . g co QŽ t . for t g A, for some set A of full measure. Let t g A; then co Q Ž t . s
⬁
D co Ž Q Ž t . l Bn . .
ns1
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503
Hence, rn Ž t . [ dist Ž x Ž t . , co Ž Q Ž t . l Bn . . ª 0
as n ª ⬁.
The set coŽ QŽ t . l Bn . is convex and compact. Furthermore, the fact that Q is in ⌫ Ž ⍀ . implies that there exists n 0 g ⺞, such that coŽ QŽ t . l Bn . is nonempty for all n G n 0 and for all t in A. Fix t g A and n G n 0 . Then, there exists a unique element x nŽ t . g coŽ QŽ t . l Bn . such that rn Ž t . s < x n Ž t . y x Ž t . < . Clearly
x n Ž t . 4 s Br Ž t . Ž x Ž t . . l co Ž Q Ž t . l Bn . , n
and, hence, x n is a bounded measurable function. Since rn4 is a decreasing sequence of measurable function that tends to 0 a.e., then 5 xn y x 51 s
H⍀ r d ª 0 n
as n ª ⬁.
On the other hand, x n g ⬁ŽcoŽ Q l Bn .. and the Caratheodory representation w1, Theorem 8.2.15x imply the existence of Ž m q 1. measurable n selections y 1n, . . . , ymq1 of Q l Bn and nonnegative measurable functions n n nŽ . 1 , . . . , mq1 with Ý mq1 is1 i t s 1 for a.e. t g ⍀, such that mq1
xn s
Ý
ni yin .
is1 n But, y 1n, . . . , ymq1 g cl⬁co ⬁Ž Q ., which is closed, convex, and decomposable, and hence it is also L-convex, by Lemma 3. Therefore, x n g cl⬁co ⬁Ž Q . : cl 1co ⬁Ž Q ., whence, x g cl 1co ⬁Ž Q ..
Let us introduce the following subsets of ⌫ Ž ⍀ . and ÝŽ ⍀ .: ⌫ Ž ⍀ . [ Q g ⌫ Ž ⍀ . N Q Ž t . is convex for a.e. t g ⍀ 4 ⌫0 Ž ⍀ . [ Q g ⌫0 Ž ⍀ . N Q Ž t . is convex for a.e. t g ⍀ 4 ⌺ Ž ⍀ . [ Q g ⌺ Ž ⍀ . N Q is convex 4 ⌺ 0 Ž ⍀ . [ Q g ⌺ 0 Ž ⍀ . N Q is convex 4 . The connection between these sets is given by COROLLARY 2. The map ⬁ is a bijection between ⌫ Ž ⍀ . and ⌺Ž ⍀ ., and between ⌫0 Ž ⍀ . and ⌺ 0 Ž ⍀ ..
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504
Proof. It is clear that if Q g ⌫ Ž ⍀ . Žrespectively ⌫0 Ž ⍀ .. and has convex images, then ⬁Ž Q . is convex. Conversely, if Q s ⬁Ž Q . g ⌺ Ž ⍀ . Žrespectively ⌺ 0 Ž ⍀ .. is convex, then, by Theorem 5, we have
⬁Ž co Q . s cl 1co Ž ⬁Ž Q . . s ⬁Ž Q . . Using the injectivity of ⬁ , it results that co Q s Q. Hence Q is convex valued. Therefore, ⬁ is a bijection for both cases. 4. SUPPORT FUNCTIONALS FOR SET-VALUED MAPS Let X be a normed vector space and let X * be its dual. For a nonempty set Q : X we define the support functional on X * as
␦ * Ž x* N Q . s sup ² x*, x : N x g Q 4 . Then
␦ *Ž⭈N Q . is a w*-lower semicontinuous and sublinear function from X * to x y ⬁, q⬁x. ␦ *Ž x* N Q1 . F ␦ *Ž x* N Q2 . for all x* g X * iff co Q1 : co Q2 . co Q s x g X N ␦ *Ž x* N Q . G ² x*, x :᭙ x* g X *4 . int coQ s x g X N ␦ *Ž x* N Q . ) ² x*, x :᭙ x* / 0, x* g X *4 , when X is finite-dimensional. 䢇
䢇 䢇 䢇
Let Q be a measurable set-valued map on ⍀ with nonempty images in ⺢ m . Then, we can associate with Q the function q: ⍀ = ⺢ m ªx y ⬁, q⬁x defined by q Ž t , . s ␦ *Ž N QŽ t . . . Set ⌬*Ž Q . [ q. Hence, one can easily see that Ži. For all t g ⍀, q Ž t, ⭈ . is lower semicontinous and sublinear on ⺢ . Žii. q is Ž A = B .-measurable, where B denotes the collection of Borel subset of ⺢ m. m
On the other hand, for Q in ⌫ Ž ⍀ ., we have the support functional of ⬁Ž Q . defined on Ž L⬁Ž ⍀ ..*. As we shall see below, this support functional, restricted to elements from L1 Ž ⍀ ., can be calculated via ⌬*Ž Q .. LEMMA 4. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ .. Then, for g L1 Ž ⍀ ., we ha¨ e
␦ * Ž N cl 1Q . s ␦ * Ž N Q . .
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505
Proof. It suffices to show that ‘‘F ’’ is true. Let x g cl 1Q. Then, by Corollary 1, there exists a sequence x n g Q and a constant C such that 5 x n y x 5 1 ª 0 as n ª ⬁ and sup ng ⺞ 5 x n 5 ⬁ F C. Taking a subsequence of x n Žwhich we do not relabel . that converges almost everywhere and applying the Lebesgue dominated convergence theorem, we obtain lim H ² , x : d F ␦ * Ž N Q . , H⍀² , x : d F nª⬁ ⍀ n
from which the result follows. THEOREM 6. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ . such that cl 1 co Q s ⬁Ž Q . for some Q g ⌫ Ž ⍀ ., and let g L1 Ž ⍀ .. Then, H⍀ ␦ *Ž Ž t . N QŽ t .. d Ž t . is well defined and
␦ * Ž N Q. s
H⍀␦ * Ž Ž t . N Q Ž t . . d Ž t . .
Proof. By the second listed property of support functions and by Lemma 4, we have
␦ * Ž N Q . s ␦ * Ž N co Q . s ␦ * Ž N cl 1 co Q . s ␦ * Ž N ⬁Ž Q . . . The rest of the proof follows immediately from w8, Theorem 2.2x. The following result is an obvious consequence of the above theorem. COROLLARY 3.
Let Q g ⌫ Ž ⍀ .. Then, for g L1 Ž ⍀ .,
␦ * Ž N ⬁Ž Q . . s
H⍀␦ * Ž Ž t . N Q Ž t . . d Ž t . .
To completely characterize set-valued maps Q in ⌫ Ž ⍀ . in terms of their support functional ⌬*Ž Q ., it is only natural to consider the images of Q to be convex, that is, Q must be in ⌫ Ž ⍀ .. The following is a characterization, in terms of ⌬*Ž Q ., of the nonemptiness of ⬁Ž Q .. LEMMA 5.
Let Q be in ⌫ Ž ⍀ ., and q s ⌬*Ž Q .; then, ⬁Ž Q . / ⭋ iff
᭚ r ) 0: for a.e. t g ⍀ , q Ž t , . q r < < G 0 ᭙ g ⺢ m .
Ž 6.
Proof. From Lemma 1, it suffices to show that condition Ž2. is equivalent to condition Ž6.. Assume that Ž2. holds. Then, there exists r ) 0 such that QŽ t . l Br / ⭋ for t g A, where Ž A. s Ž ⍀ .. Fix t g A, and consider x g QŽ t . l Br and g ⺢ m . Then q Ž t , . s ␦ * Ž N Q Ž t . . G ² , x : G inf ² , u: s yr < < . ugB r
That is, Ž6. holds true.
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PALES AND ZEIDAN ´
Conversely, assume that Ž6. is satisfied for r ) 0. If condition Ž2. is false, then there exists a set A of positive measure such that Q Ž t . l Br s ⭋
for t g A.
Fix t g A; then by the strong form of the Hahn᎐Banach separation theorem there exists g ⺢ m such that
␦ * Ž N Q Ž t . . - inf ² , u: N u g Br 4 s yr < < . Therefore condition Ž6. is violated. The above lemma inspires the consideration of the following sets for support functionals to elements of ⌫ Ž ⍀ .. Set ⌳ Ž ⍀ . [ q : ⍀ = ⺢ m ª x y ⬁, ⬁ x N q Ž t , ⭈ . is sublinear and lsc, q is Ž A = B . -measurable and satisfies condition Ž 6 . 4 . The equality q1 s q2 in ⌳Ž ⍀ . means that, for a.e. t g ⍀, q1Ž t, . s q2 Ž t, . for all g ⺢ m . The intimate connection between ⌳Ž ⍀ . and ⌫ Ž ⍀ . is presented below. THEOREM 7. The map ⌬* is a bijection from ⌫ Ž ⍀ . onto ⌳Ž ⍀ .. Proof. From Lemma 5 and the previous properties of ⌬*Ž Q ., it results that ⌬*Ž ⌫ Ž ⍀ .. ; ⌳Ž ⍀ .. For the injectivity note that if Q1 , Q2 in ⌫ Ž ⍀ . satisfies Q1 / Q2 , then there exists a set A of positive measure over which Q1Ž t . / Q2 Ž t .. Hence, ␦ *Ž⭈N Q1Ž t .. / ␦ *Ž⭈N Q2 Ž t .. for t g A, and thus ⌬*Ž Q1 . / ⌬*Ž Q2 .. For the surjectivity, let q g ⌳Ž ⍀ .. Define the set-valued map Q by Q Ž t . s x g ⺢ m N q Ž t , . G ² x, : ᭙ g ⺢ m 4 . Clearly ⌬*Ž Q . s q, and Q is measurable with nonempty, closed convex values. Furthermore, by Lemma 5, Q satisfies condition Ž2.. Thus, Q g ⌫ Ž ⍀ .. Remark 6. By combining Theorem 7 with Corollary 2, we obtain the equivalence between the three sets ⌫ Ž ⍀ ., ⌺Ž ⍀ ., and ⌳Ž ⍀ .. The rest of this section is devoted to the characterization of ⌫0 Ž ⍀ . in terms of elements of ⌳Ž ⍀ .. First of all, let us translate condition Ž3. on the elements of ⌫ Ž ⍀ . in terms of the elements of ⌳Ž ⍀ ..
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L -CLOSED DECOMPOSABLE SETS IN L
LEMMA 6.
507
Q g ⌫ Ž ⍀ . and q s ⌬*Ž Q .. Then x g int ⬁Ž Q . if and only if
᭚ ) 0: for a.e. t g ⍀ ,
qŽ t, . G < < q ² , xŽ t.:
᭙ g ⺢ m . Ž 7 .
Proof. By Lemma 2, it is equivalent to show that condition Ž3. holds true iff condition Ž7. is satisfied. However, B Ž x Ž t .. ; QŽ t . is equivalent to
␦ * Ž N B Ž x Ž t . . . F ␦ * Ž N Q Ž t . . ,
for all g ⺢ m ,
that is,
< < q ² , xŽ t.: F qŽ t, . ,
for all g ⺢ m ,
whence the result follows. As we shall see in the result below, the nonemptiness of the interior of ⬁Ž Q . is characterized by this property of q g ⌳Ž ⍀ .: ᭚ r G ) 0: for a.e. t g ⍀ , q Ž t , . q q Ž t , . q r < q < G w < < q < < q < q < x
᭙ , g ⺢ m . Ž 8.
Set ⌳ 0 Ž ⍀ . [ q g ⌳ Ž ⍀ . N q satisfies Ž 8 . 4 . The following result shows that there is a one-to-one correspondence between ⌫0 Ž ⍀ . and ⌳ 0 Ž ⍀ .. THEOREM 8. The map ⌬* is a bijection from ⌫0 Ž ⍀ . onto ⌳ 0 Ž ⍀ .. Proof. Let Q g ⌫0 Ž ⍀ . and set q [ ⌬*Ž Q .. Then Q satisfies condition Ž4.. By Theorem 3, int ⬁Ž Q . / ⭋. Let x g int ⬁Ž Q .; then, Lemma 6 gives that Ž7. is satisfied for some ) 0. Set r [ 5 x 5 ⬁ q . Then, there exists a set A of full measure such that, for all t g A, Br = B Ž x Ž t . . and qŽ t, . G < < q ² , xŽ t.:
᭙ g ⺢ m .
Ž 9.
Thus, for , g ⺢ m , the inclusion above gives
␦ * Ž y y N Br . G ␦ * Ž y y N B Ž x Ž t . . . , that is, r < q < G ² y y , x Ž t . : q < q < .
Ž 10 .
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508
By substituting for in Ž9. and , respectively, and by adding the two inequalities so obtained to Ž10., we get that Ž8. holds true; therefore ⌬* maps ⌫0 Ž ⍀ . into ⌳ 0 Ž ⍀ .. ⌬* is also injective; hence it suffices to show only that ⌬*Ž ⌫0 Ž ⍀ .. = ⌳ 0 Ž ⍀ .. Let q g ⌳ 0 Ž ⍀ . and let Q g ⌫ Ž ⍀ . such that ⌬*Ž Q . s q. We are going to show that Q also belongs to ⌫0 Ž ⍀ .. Then, there exist r G ) 0 and a set A of full measure such that the inequality in Ž8. is valid for all , g ⺢ m. Putting s , we obtain qŽ t, . q Ž r y 2 . < < G 0
for g ⺢ m , t g A.
It follows from this inequality that qŽ t, . q Ž r y . < < G 0
᭙ g ⺢ m , ᭙ t g A.
Using the argument of the second part of the proof of Lemma 6, this yields that Q Ž t . l Bry / ⭋
᭙ t g A.
Therefore, for t g A, the relative interior of QŽ t . and Br are not disjoint. Using w13, Corollary 16.4.1x, we infer that the support functional of QŽ t . l Br s Q r Ž t . can be expressed as the infimal convolution of the supporting functionals of QŽ t . and Br , that is, qr Ž t , . [ ␦ * Ž N Q r Ž t . . s inf q Ž t , . q r < y < : g ⺢ m 4 . We shall now show that qr satisfies the following inequality: qr Ž t , . q qr Ž t , y . G 2 < <
᭙ t g A, ᭙ g ⺢ m .
It is enough to show that qŽ t, . q r < y < q qŽ t, . q r < q < G 2 < <
᭙t g A
᭙ , g ⺢ m . Using Ž8. and triangle inequalities, we get qŽ t, . q qŽ t, . q r < y < q r < q < G Ž < < q < < q < q < . y r < q < q r < y < q r < q < s Ž r y . Ž < y < q < q < y < q <. q Ž < < q < y <. q Ž < y < q < q < . G 2 < < . Thus Ž11. is proved.
Ž 11 .
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509
Our next aim is to prove that if Ž11. is valid, then, for each t g A, Q r Ž t . contains a ball of radius m2q 1 . In fact, we prove that the ball of radius m2q 1 centered at the center of mass of Q r Ž t . is contained in Q r Ž t .. If P : ⺢ m is a bounded Lebesgue measurable set, then cŽ P ., the center of mass of P, is defined by cŽ P . [
1 Vm Ž P .
HPx dx,
where VmŽ P . s HP 1 dx is the m-dimensional Lebesgue measure of P, which is assumed to be positive. We need the following lemma. LEMMA 7.
If P is a compact con¨ ex set in ⺢ m , then, for all g ⺢ m ,
HP² , x : dx F
m ␦ * Ž N P . y ␦ * Ž y N P . mq1
HP1 dx.
Ž 12 .
Proof. Observe first, that if Ž12. is valid for some compact convex P then it is also valid for the translates of P. For, let u g ⺢ m. Then
HPqu² , x : dx s HP² , x q u: dx s HP² , x : q ² , u:HP1 dx. On the other hand,
␦ * Ž N P q u . s ␦ * Ž N P . q ² , u:
for all g ⺢ m .
Hence m ␦ * Ž N P q u . y ␦ * Ž y N P q u . mq1 s
ž
␦ * Ž N P . y ␦ * Ž y N P . mq1
HPqu1 dx
q ² , u:
/H
1 dx.
P
Thus Ž12. implies the analogous inequality for P q u. Therefore, without loss of generality, we may assume that 0 g P and
␦ * Ž y N P . s sup ² y , x : N x g P 4 s 0. Both sides of inequality Ž12. are positively homogeneous in ; therefore, < < s 1 can also be assumed. There is always an orthogonal transformation T : ⺢ m ª ⺢ m such that T Ž . s Ž1, 0, . . . , 0. s e1. Then VmŽ P . s VmŽT Ž P .. and
HP² , x : s HP²T , Tx : s HP² e , Tx : dx s HT Ž P .² e , x : dx. 1
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510 Similarly,
␦ * Ž " N P . s sup ² " T , Tx : N x g P 4 s sup ² " e1 , u: N u g T Ž P . 4 s ␦ * Ž "e1 N T Ž P . . . Therefore, Ž12. is equivalent to saying that Ž12. Žreplacing P by T Ž P . if necessary . holds for s e1 and for compact convex sets P such that 0 g P and ␦ *Žye1 N P . s 0. For analogous reasons, replacing P by P Ž ) 0., we can assume that ␦ *Ž e1 N P . s 1. Summarizing all the reductions made above, it suffices to show that
HPx
1
m
dx F
HP1 dx
mq1
Ž 13 .
for all compact convex sets P such that 0 g P, ␦ *Žye1 N P . s 0, ␦ *Ž e1 N P . s 1. Define P0 [ x g P x 1 s
½
m mq1
5
R[
and
D
0F F Ž mq1 .rm
P0 .
We first show that Ž13. holds with equality for P s R. Observe that the elements of R can be written as
½
R s Ž x 1 , . . . , x m . 0 F x 1 F 1, Ž x 2 , . . . , x m . g x 1
mq1 m
5
1Ž P . ,
where 1 is the orthogonal projection defined via
1 Ž u1 , . . . , u m . [ Ž u 2 , . . . , u m . . Therefore, for any integrable function f : w0, 1x ª ⺢, we have 1
HR f Ž x . dx s H0 f Ž x . Hx ŽŽ mq1.rm . Ž P . 1d Ž x 1
1
s s
1
H0
1
H0
ž
1
ž
f Ž x 1 . Vmy1 x 1 f Ž x1 . x1 ⭈
ž
2,...,
1
mq1 m
mq1
s Vmy 1 Ž 1 Ž P . . ⭈
m
ž
1 Ž P . dx 1
/
my 1
⭈ Vmy1 Ž 1 Ž P . . dx 1
/
mq1 m
/
x m . dx 1
my 1
/
⭈
1
H0
f Ž t . t my 1 dt.
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511
Taking f Ž t . s t and f Ž t . s 1, we obtain that both side of Ž13. are equal for P s R. Now we prove that Ž13. is valid for general P. Define Py[ x g P x 1 F
½ ½
Pq[ x g P x 1 )
m mq1 m mq1
5 5
,
Ry[ x g R x 1 F
,
Rq[ x g R x 1 F
½ ½
m mq1 m mq1
5 5
, .
First, using the convexity of P, we show that Ry: Py
Pq: Rq .
and
Ž 14 .
Let x g Ry. Then there exist g w0, m mq 1 x and u g P0 such that x s 0 u. Since u1 s m mq 1 then, x 1 F m mq 1 and hence, g w0, 1x. The elements 0 and u belong to Py, and Py is also convex. Thus x s Ž 1 y . 0 q u g Py . To prove the inclusion Pq: Rq, let x g Pq. Then s mrwŽ m q 1. x 1 x - 1 and m x s x q Ž 1 y . 0 g P. Ž m q 1. x1 Hence m
Ž m q 1. x1
x g P0 .
Therefore xg
mq1 m
x 1 P0 ; Rq
because m mq 1 x 1 F m mq 1 . Thus Ž14. is proved. Now, using that Ž13. holds with equality for P s R, we are able to prove that Ž13. is valid for any P:
HPx
1
dx s
HP _R y
F
HP _R y
s
x 1 dx q y
m y
mq1
HR x
1
dx q
dx y
HR _P q
x 1 dx 1 q
m
HR m q 1 dx y HR _P
m
HP m q 1 dx.
Thus the lemma is completely proved.
q
m q
mq1
dx
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512
Now we can continue the proof of the theorem. Assume that Ž11. is valid. Then the dimension of Q r Ž t . cannot be smaller than m; otherwise there exists / 0 in ⺢ m such that qr Ž t , . q qr Ž t , y . s 0, and this contradicts Ž11.. Therefore, the measure of Q r Ž t . is positive. Applying Lemma 7 to the set P s Q r Ž t ., we obtain
HQ Ž t .² , x : dx F
mqr Ž t , . y qr Ž t , y . mq1
r
s qr Ž t , . y
ž ž
F qr Ž t , . y
⭈ Vm Ž Q r Ž t . .
qr Ž t , . q qr Ž t , y . mq1 2< < mq1
/
/
⭈ Vm Ž Q r Ž t . .
⭈ Vm Ž Q r Ž t . . .
Hence, dividing by VmŽ Q r Ž t .. and using the definition of the center of mass for Q r Ž t ., we get ² , c Ž Qr Ž t . . : q
2 mq1
< < F qr Ž t , .
for all g ⺢ m , t g A. Thus, for all g ⺢ m ,
␦ * Ž N B2 rŽ mq1. Ž c Ž Q r Ž t . . . . F ␦ * Ž Q Ž t . l Br . . Therefore B2 rŽ mq1. Ž c Ž Q r Ž t . . . : Q Ž t . l Br for a.e. t g ⍀, which means that property Ž4. is satisfied. Hence Q belongs to ⌫0 Ž ⍀ ., as was to be proved. 5. NORMAL AND POLAR CONES OF DECOMPOSABLE SETS IN L⬁Ž ⍀ . Let X be a normed vector space and let Q be a nonempty subset of X. For x 0 g Q we define the normal cone N Ž x 0 N Q . to Q at an element x 0 g cl Q by N Ž x 0 N Q . [ x* g X * N ² x*, x : F ² x*, x 0 : ᭙ x g Q 4 .
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513
Clearly, N Ž x 0 N Q . is a closed convex nonempty cone. When Q is a cone and x 0 s 0, N Ž0 N Q . is called the polar cone of Q, denoted by Q 0 . For the rest of this section, we take X s L⬁Ž ⍀ .. We first focus on the investigation of the normal cone of decomposable sets Q. We introduce the multiplication in Ž L⬁Ž ⍀ ..* by an essentially bounded real-valued function f as follows: for x* g Ž L⬁Ž ⍀ ..*, the linear functional fx* is defined by ² fx*, x : s ² x*, fx :
᭙ x g L⬁ Ž ⍀ . .
Hence, the notions of decomposability and L-convexity extend naturally to subsets ⌽ of Ž L⬁Ž ⍀ ..*. We say that ⌽ : Ž L⬁Ž ⍀ ..* is an L-cone if for now f g L⬁Ž ⍀, ⺢q. and x* g ⌽, fx* g ⌽. The characterization obtained in Lemma 3 for the L⬁Ž ⍀ . setting remains valid for the Ž L⬁Ž ⍀ ..* case. Furthermore, similar arguments show the following. LEMMA 8. equi¨ alent:
Let ⌽ be a closed cone of Ž L⬁Ž ⍀ ..*. The following are
Ži. ⌽ is decomposable and con¨ ex. Žii. ⌽ is a con¨ ex L-cone. Žiii. Ý kis1 f i xUi g ⌽ whene¨ er, for all i, xUi g ⌽ and f i g L⬁Ž ⍀, ⺢q. . LEMMA 9. Let Q ; L⬁Ž ⍀ . be a nonempty decomposable set. Then, for x 0 g Q, N Ž x 0 N Q. is a closed con¨ ex L-cone. Proof. It remains to show that N Ž x 0 N Q. is an L-cone. By Lemma 8, this is equivalent to showing that N Ž x 0 N Q. is decomposable. Let A g A and x g Q be arbitrary. Then 1 A x q 1 ⍀ _ A x 0 g Q, by the decomposability of Q. Thus, if x* g N Ž x 0 N Q., then ² x*, 1 A x q 1 ⍀ _ A x 0 : F ² x*, x 0 : , that is, ² x*, 1 A x : F ² x*, 1 A x 0 : for all x g Q. This yields that 1 A x* g N Ž x 0 N Q . . If y* g N Ž x 0 N Q., then, similarly, 1 ⍀ _ A y* g N Ž x 0 N Q . .
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514
Since N Ž x 0 N Q. is a cone, we obtain that 1 A x* q 1 ⍀ _ A y* g N Ž x 0 N Q . .
Let Q be a set-valued map in ⌫ Ž ⍀ ., let Q s ⬁Ž Q ., and let x 0 be in Q. Then, with Q we can associate a set-valued map N Ž x 0 N Q . defined via the pointwise normal cones to QŽ t . at x 0 Ž t ., that is, N Ž x 0 N Q .Ž t . s N Ž x 0 Ž t . N QŽ t ... This map is measurable, since ␦ *Ž N QŽ⭈.. is measurable and N Ž x 0 Ž t . N Q Ž t . . s g ⺢ m N ␦ * Ž N Q Ž t . . s ² , x 0 Ž t . :4 for a.e. t g ⍀ . The goal is to describe the L1-elements of N Ž x 0 N Q. in terms of the set-valued map N Ž x 0 N Q .. However, we shall show that such a characterization in terms of N Ž x 0 N Q . is true for subsets Q of L⬁Ž ⍀ ., which are not necessarily L1-closed, but rather their L1-closures are decomposable. This property is met by both the Clarke and the adjacent tangent cones used in Section 2. THEOREM 9. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ . such that cl 1Q s ⬁Ž Q . for some Q g ⌫ Ž ⍀ .. Then, for x 0 g Q, N Ž x 0 N Q . l L1 Ž ⍀ . s 1 Ž N Ž x 0 N Q . . . Proof. A function g L1 Ž ⍀ . belongs to N Ž x 0 N Q. if and only if ␦ *Ž N Q. s H⍀ ² N x 0 : d . By Theorem 6, this is equivalent to
H⍀ Ž ␦ * Ž N Q . y ² , x :. d s 0. 0
Being nonpositive, the integrand must be zero a.e. on ⍀. This means
Ž t . g N Ž x0 Ž t . N QŽ t . .
for a.e. t g ⍀ ,
which proves the theorem. COROLLARY 4.
If Q g ⌫ Ž ⍀ ., then
N Ž x 0 N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž N Ž x 0 N Q . . . The characterization of the polar cones of the Clarke and the adjacent tangent cones is presented in what follows.
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Let Q g ⌫ Ž ⍀ . and x 0 g ⬁Ž Q .; then
COROLLARY 5.
žT Ž x žCŽ x
515
0
0
N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž T 0 Ž x 0 N Q . .
0
N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž C 0 Ž x 0 N Q . . ,
and
0
where, for t g ⍀, T 0 Ž x0 N Q. Ž t . [ Ž T Ž x Ž t . N QŽ t . . .
0
and
0
C 0 Ž x0 N Q. Ž t . [ Ž C Ž x0 Ž t . N QŽ t . . . . Proof. Set Q s T Ž x 0 N ⬁Ž Q .. and Q s C Ž x 0 N ⬁Ž Q .., respectively. Then the results follow immediately from Theorems 4 and 9.
REFERENCES 1. J.-P. Aubin and H. Frankowska, ‘‘Set-Valued Analysis,’’ Birkhauser, BostonrBasel, 1990. ¨ 2. C. Castaing and M. Valadier, ‘‘Convex-Analysis and Measurable Multifunctions,’’ Springer-Verlag, Lecture Notes in Math., 580 Ž1977.. 3. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Wiley, New York, 1983. 4. I. Ekeland and R. Temam, ‘‘Analyse convexe et problemes variationelles,’’ Hermann, ` Paris, 1974.2 5. E. Giner, ‘‘Etudes des fonctionnelles integrables,’’ Thesis, Universite ´ de Pau, France, 1985. 6. E. Giner, Sous-differentiabilite´ des fonctionnelles integrables ŽII., Thesis, Sem. ´ Anal. Num. No. 6, 83᎐84. 7. F. Hiaı, ¨ Multivalued conditional expectations, multivalued Radon-Nikodym theorems, integral representation of additive operators, and multivalued strong laws of large numbers, unpublished manuscript. 8. F. Hiaı¨ and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multi¨ ariate Anal. 7 Ž1977., 149᎐182. 9. Zs. Pales, ´ and V. Zeidan, Characterization of closed and open C-convex sets in C ŽT, ⺢ r ., Acta Sci. Math. Szeged 65 Ž1999., 339᎐357. 10. Zs. Pales, ´ and V. Zeidan, On L1-closed decomposable sets in L⬁, Systems modelling and optimization ŽDetroit, MI, 1997., Chapman & HallrCRC Res. Notes Math. 396 Ž1999., 198᎐206. 11. Zs. Pales, ´ and V. Zeidan, Optimization problems with measurable set-valued constraints, SIAM Journal on Optim., to appear. 12. A. W. Roberts and D. E. Varberg, ‘‘Convex Functions,’’ Academic Press, New YorkrLondon, 1973. 13. R. T. Rockafellar, ‘‘Convex Analysis,’’ Princeton Univ. Press, Princeton, NJ, 1970. 14. R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 Ž1968., 525᎐539. 15. R. T. Rockafellar, Integrals which are convex functionals, II, Pacific J. Math. 39 Ž1971., 439᎐469.
2 English translation: ‘‘Convex Analysis and Variational Problems,’’ North-Holland, Amsterdam, 1977.
Characterization of L1-Closed Decomposable Sets in L⬁* Zsolt Pales ´ Institute of Mathematics and Informatics, L. Kossuth Uni¨ ersity, H-4010 Debrecen, Hungary E-mail: [email protected]
and Vera Zeidan Department of Mathematics, Michigan State Uni¨ ersity, East Lansing, Michigan 48824 E-mail: [email protected] Submitted by Helene ´` Frankowska Received December 19, 1997
In this paper the set of essentially bounded measurable selections of a measurable set-valued map is considered. One of the main results states that only those subsets of L⬁ that are L1-closed in L⬁ and decomposable can be represented in this way. The connection between topological and convexity properties of the set-valued map and its L⬁-selection set is also studied in details. The adjacent and Clarke’s tangent cones to L1-closed and decomposable sets and their polar cones are described in terms of the corresponding set-valued map as well. 䊚 1999 Academic Press
Key Words: L1-closed in L⬁ sets; decomposable sets; measurable set-valued maps; L⬁-selections.
1. INTRODUCTION Throughout this paper let Ž ⍀, A, . denote a finite complete measure space, and for 1 F p F ⬁ let L p Ž ⍀ . s L p Ž ⍀, ⺢ m . be the space of equiva*Supported by Hungarian Soros Foundation grant 022-2998r95; by the Hungarian National Foundation for Scientific Research ŽOTKA., grant T-016846; and by the National Science Foundation, under grant NSF-DMS-9404591. 491 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
PALES AND ZEIDAN ´
492
lence classes of L p-integrable functions equipped with the usual norm 5 ⭈ 5 p . For the sake of brevity, we write a.e. for ‘‘-almost everywhere.’’ The notion of decomposable sets in L p spaces appeared in the work of Hiaı¨ and Umegaki w8x: A set Q : L p Ž ⍀ . is called decomposable if, for x, y g Q and for A g A, we have 1 A x q 1 ⍀ _ A y g Q. It is easily seen that if Q: ⍀ ª 2 ⺢ is an arbitrary set-valued function, then the set of L p-measurable selections m
p Ž Q . s x g L p Ž ⍀ . N x Ž t . g Q Ž t . for a.e. t g ⍀ 4
Ž 1.
is always a decomposable subset of L ⍀ .. Therefore, it is natural to ask if the converse of this statement is also true. In the case 1 - p F ⬁, Hiaı¨ and Umegaki w8x proved that the closed decomposable sets can be derived from set-valued maps. More precisely they proved the following: pŽ
THEOREM A w8, Theorem 3.1x. Let 1 F p - ⬁ and Q ; L p Ž ⍀ . be a nonempty closed set. Then there exists a measurable nonempty closed set m ¨ alued map Q: ⍀ ª 2 ⺢ such that Q s p Ž Q ., where p Ž Q . is defined in Ž1., if and only if Q is decomposable. For the case p s ⬁, there is no similar result available in the literature.1 In fact, if we consider a nonempty, closed, and decomposable subset Q of L⬁Ž ⍀ ., it is not in general represented by a measurable set-valued map Q with nonempty and closed images. As we shall show in Lemma 1 below, the image ⬁Ž Q . of such a set-valued map Q is not only closed in L⬁Ž ⍀ ., but also L1-closed. Therefore, in L⬁Ž ⍀ ., the L1-closed decomposable sets are the proper candidates for the representability via set-valued functions. DEFINITION 1. A set Q : L⬁Ž ⍀ . is called L p-closed in L⬁ if, whenever a sequence x n4 in Q converges to x 0 g L⬁Ž ⍀ . in the L p-norm, then x 0 g Q. The L p-closure in L⬁ of a set Q g L⬁Ž ⍀ . will be denoted by cl pQ. In Theorem 2 we describe the class of set-valued maps whose members are in a bijective correspondence with L1-closed in L⬁ decomposable sets. The next results of the paper offer an analogous description for L1-closed decomposable sets Q with the following additional properties: 䢇 䢇 䢇
1
Q has a nonempty interior. Q is convex. Q is convex and has a nonempty interior.
The authors thank the anonymous referee who called their attention to an unpublished paper of Hiaı¨ w7x, distributed among the participants at an international conference held in Catania, 1983, in which the case p s ⬁ is considered.
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493
In the last two cases, the corresponding set-valued map has convex images; therefore it can also be described via its supporting functional. In the last section, we develop the notions of convex analysis for the sets that are L1-closed and decomposable. These results are essential for the investigations of extremum problems, where constraints of the form x Ž t . g QŽ t .
Ž x g L⬁ Ž ⍀ . .
for a.e. t g ⍀
are present. The need for investigating constraints of this form stems from control theory. Some of the results in this paper could be extended to the case where the underlying measure space is -finite. However, most of the applications we are interested in have finite measure spaces. For instance, this is the case when the results of this paper are applied to the optimal control problem. This application will be established in a subsequent paper w11x. A summary of the results derived in this paper has been published in w10x. 2. DECOMPOSABLE AND L1-CLOSED SUBSETS IN L⬁Ž ⍀ . The notion of equality between measurable set-valued maps over ⍀ is defined as Q1 s Q 2
means
Q1 Ž t . s Q 2 Ž t .
for a.e. t g ⍀ .
A characterization of the nonemptiness of the image ⬁Ž Q . is given by the following. LEMMA 1. Let Q: ⍀ ª 2 ⺢ be a measurable set-¨ alued map with nonempty closed images. Then, ⬁Ž Q . is L1-closed in L⬁ and decomposable. Furthermore, ⬁Ž Q . / ⭋ if and only if m
᭚ r ) 0: Q Ž t . l Br / ⭋
for a.e. t g ⍀ .
Ž 2.
Ž Here Br denotes the ball of radius r in ⺢ m centered at the origin.. Proof. The decomposability of ⬁Ž Q . is obvious. For the L1-closedness, let x n g ⬁Ž Q . be a sequence such that x n tends in L1-norm to x 0 g L⬁Ž ⍀ .. Then there exists a subsequence of x n , denoted by x n k, such that, as k ª ⬁, x n kŽ t . ª x 0 Ž t .
for a.e. t g ⍀ .
By the closedness of QŽ t ., x 0 Ž t . g QŽ t . for a.e. t g ⍀. Therefore x 0 g ⬁Ž Q ..
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If ⬁Ž Q . / ⭋, then there exists x 0 g ⬁Ž Q .. Clearly, < x 0 Ž t .< F 5 x 0 5 ⬁ \ r for a.e. t g ⍀. Thus QŽ t . l Br is nonempty for a.e. t g ⍀. On the other hand, if condition Ž2. holds on a set A of full measure, then the set-valued map Q r , defined by Qr Ž t . s
½
Q Ž t . l Br ⺢m
t g A, t g ⍀ _ A,
is measurable with nonempty and closed values. Therefore, by the measurable selection theorem Žsee w2x., there exists a measurable function x 0 : ⍀ ª ⺢ m such that, for all t in ⍀, x 0 Ž t . g Qr Ž t . . Clearly 5 x 0 5 ⬁ F r and x Ž t . g QŽ t ., a.e. Thus, x 0 g ⬁Ž Q .. Remark 1. Condition Ž2. is equivalent to the essential boundedness of the distance function from 0 to QŽ t .. For the case where p g w1, ⬁., this fact is given in w1x. Remark 2. The L1-closedness in L⬁ of ⬁Ž Q . also follows from the equality
⬁Ž Q . s L⬁ Ž ⍀ . l 1 Ž Q . , because 1Ž Q . is easily seen to be L1-closed. Consider a nonempty and decomposable subset Q of L⬁Ž ⍀ .. We furnish first a characterization of the L p-closure in L⬁ of Q. As a consequence, we will obtain that, for each p g w1, ⬁., the L1-closure in L⬁ of Q coincides with its L p-closure in L⬁. THEOREM 1. Let Q ; L⬁Ž ⍀ . be a nonempty decomposable set. Then, for p g w1, ⬁., x g cl pQ if and only if there exists a sequence of measurable sets A n g A and g Q such that
Ž An . ª Ž ⍀ .
as n ª ⬁
and 1 A nx q 1 ⍀ _ A n g cl⬁Q
for all n g ⺞.
Proof. Assume that x g cl pQ. Then there exists a sequence x k g Q such that 5 x k y x 5 p ª 0 as k ª ⬁. By taking subsequences if necessary, we can assume that x k tends to x a.e. in ⍀. By Egorov’s theorem, for all n g ⺞, there exists A n g A such that Ž ⍀ _ A n . - 1rn and x k converges uniformly to x on A n . Let g Q be arbitrary and let
k , n [ 1 A nx k q 1 ⍀ _ A n
Ž k, n g ⺞ . .
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495
Then k, n g Q by the decomposability. On the other hand, for all fixed n g ⺞, the sequence k, n4⬁ks1 converges uniformly on ⍀ to 1 A nx q 1 ⍀ _ A n . Therefore 1 A nx q 1 ⍀ _ A n g cl⬁Q for all n g ⺞. Conversely, assume that there exists a sequence of sets A n g A and g Q such that Ž ⍀ _ A n . ª 0 and x n [ 1 A nx q 1 ⍀ _ A n g cl⬁Q
for all n g ⺞.
Then there exists n g Q such that 5 n y x n 5 ⬁ - 1rn. We show that 5 n y x 5 p ª 0: 5 n y x 5 p F 5 n y x n 5 p q 5 x n y x 5 p F F
1 n 1 n
1rp
Ž Ž ⍀ . .
1rp
Ž Ž ⍀ . .
1rp
q
žH
⍀_A n
< y x < pd
/
q 5 y x 5 ⬁p Ž Ž ⍀ _ A n . .
1rp
ª 0.
Therefore, x g cl p ŽQ.. COROLLARY 1.
If Q : L⬁Ž ⍀ . is nonempty and decomposable, then
Ži. For all p g w1, ⬁., cl 1Q s cl pQ. Žii. For all p g w1, ⬁., ⬁
cl pQ s x g L⬁ Ž ⍀ . N ᭚ n 4 ns 1 ; Q:
½
lim 5 n y x 5 p s 0 and sup 5 n 5 ⬁ - ⬁ .
nª⬁
ng⺞
5
Proof. The characterization of the L p-closure in L⬁ of decomposable sets obtained in the above theorem does not depend on p. Therefore Ži. is obvious. For Žii., it suffices to show only the : inclusion. Let x g cl pQ. then the characterization presented by the theorem holds true. Thus the sequence n g Q constructed in the ‘‘only if’’ part of the proof of theorem tends to x in L p-norm. On the other hand, for all n g ⺞, 5 n 5 ⬁ F 5 n y x n 5 ⬁ q 5 x n 5 ⬁ F 1 q max Ž 5 x 5 ⬁ , 5 5 ⬁ . .
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Remark 3. It follows from statement Žii. of the above corollary that a subset Q of L⬁Ž ⍀ . is L p-closed in L⬁ Žfor some 1 F p - ⬁. if and only if Q is closed under the boundedly a.e. convergence. That is, n g Q for all n g ⺞, the sequence n converges to x almost everywhere, and sup ng ⺞ 5 n 5 ⬁ - ⬁, and then x also belongs to Q. This latter notion of convergence was used in Hiaı’s ¨ paper w7x. Indeed, if x g cl pQ s Q, then by using Žii. and extracting an a.e. convergent subsequence if necessary, we get that Q is boundedly a.e. closed. Conversely, if n g Q boundedly a.e. converges to x, then, due to the finiteness of the measure space, n tends to x in L p-norm as well. Let us introduce the following classes: ⌫ Ž ⍀ . [ Q: ⍀ ª 2 ⺢ N Q is measurable with closed nonempty m
images satisfying Ž 2 . 4 and ⌺ Ž ⍀ . [ Q ; L⬁ Ž ⍀ . N Q is L1-closed in L⬁ , nonempty and decomposable4 . From Lemma 1, we know that these two classes are related. The exact nature of their relationship is established in the next theorem. THEOREM 2. The map ⬁ is a bijection between ⌫ Ž ⍀ . and ⌺ Ž ⍀ .. Proof. By Lemma 1, ⬁ maps ⌫ Ž ⍀ . into ⌺ Ž ⍀ .. The injectivity of ⬁ is the consequence of a result by Hiaı¨ and Umegaki w8, Corollary 1.2x. Now we prove that ⬁ is surjective. Let Q g ⌺ Ž ⍀ . be arbitrary. We can consider Q as a subset of L1 Ž ⍀ .. Let Q denote the closure of Q in L1 Ž ⍀ . Žthis set may not coincide with Q because Q is only L1-closed in L⬁ .. Since Q is decomposable, Q is also decomposable. Therefore, by w8, Theorem m 3.1x, there exist a measurable nonempty closed set-valued map Q: ⍀ ª 2 ⺢ such that Q s 1Ž Q . . We show that Q s ⬁Ž Q .. If x g Q, then x g Q; therefore x g 1ŽQ.. However, x g L⬁Ž ⍀ ., and hence x g ⬁Ž Q .. On the other hand, if x g ⬁Ž Q ., then x g 1Ž Q . s Q. That is, x g Q l L⬁Ž ⍀ .. Since Q is L1-closed, Q l L⬁Ž ⍀ . s Q, which yields x g Q. Remark 4. In the unpublished paper w7, Theorem 1.5x, Hiaı¨ obtained an analogous result where he assumed the underlying measure space to be -finite and used the notion of the boundedly a.e. closedness instead of L1-closedness in L⬁.
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Our next aim is to consider L1-closed decomposable sets with a nonempty interior. LEMMA 2. Let Q s ⬁Ž Q ., where Q g ⌫ Ž ⍀ .. Then x g L⬁Ž ⍀ . belongs to int Q Ž the interior of Q. if and only if ᭚ ) 0: B Ž x Ž t . . : Q Ž t .
for a.e. t g ⍀ .
Ž 3.
Proof. Let x g int Q. Then there exists ) 0 such that B Ž x . : Q. Let yn4⬁ns1 be a dense subset of the ball B : ⺢ m . Then x n Ž t . [ x Ž t . q yn
Ž n g ⺞.
defines an essentially bounded measurable function such that 5 x n y x 5 ⬁ F . Therefore x n g B Ž x . : Q. Thus xnŽ t . g QŽ t .
for t g A n ,
where A n g A is of full measure. Then xnŽ t . g QŽ t .
᭙ n g ⺞ and ᭙ t g A [
⬁
F An ,
ns1
and A is of full measure, In other words, x Ž t . q yn g Q Ž t .
᭙ n g ⺞, ᭙ t g A.
The set QŽ t . is closed; therefore this inclusion implies B Ž x Ž t . . : Q Ž t .
᭙ t g A.
Conversely, if Ž3. holds and y g B Ž x ., then < y Ž t . y x Ž t .< F a.e. on ⍀. Therefore, y Ž t . g B Ž x Ž t . . : Q Ž t .
for a.e. t g ⍀ .
Thus y g ⬁Ž Q ., which yields B Ž x . : Q. As we shall prove in the result below, the nonemptiness of the interior of ⬁Ž Q . is characterized by the following property of Q: ᭚ r G ) 0 and, for a.e. t g ⍀ , ᭚ x t g ⺢ m such that B Ž x t . : Q Ž t . l Br .
Ž 4.
PALES AND ZEIDAN ´
498 Therefore, set
⌫0 Ž ⍀ . [ Q g ⌫ Ž ⍀ . N Q satisfies Ž 4 . 4 and ⌺ 0 Ž ⍀ . [ Q g ⌺ Ž ⍀ . N int Q / ⭋ 4 . THEOREM 3. The map ⬁ is a bijection between ⌫0 Ž ⍀ . and ⌺ 0 Ž ⍀ .. Proof. We show first that ⬁Ž ⌫0 Ž ⍀ .. : ⌺ 0 Ž ⍀ .. Let Q g ⌫0 Ž ⍀ . be arbitrary. Let r G ) 0 and A g A be a set of full measure such that Q r Ž t . [ QŽ t . l Br satisfies ᭙ t g A ᭚ x t : B Ž x t . : Q r Ž t . . Define, for t g A, R Ž t . s y N B Ž y . : Q r Ž t . 4 . By our assumption, RŽ t . / ⭋ for all t g A. We are going to show that R is a measurable closed set-valued mapping. The set-valued map with images ⭸ Q r Ž t . Žwhere ⭸ stands for the boundary. is clearly a measurable closed set-valued map. Therefore, by the characterization theorem of measurable set-valued maps Žsee w2x., there exists a sequence of measurable selections n4⬁ns1 of ⭸ Q r such that
⭸ Q r Ž t . s cl n Ž t . N n g ⺞ 4
for t g A.
Let ⌽n Ž t . s ⺢ m _ U Ž n Ž t . .
for t g A, n g ⺞,
where U Ž z . denotes the open ball in ⺢ m of radius centered at z. Then ⌽n is a measurable closed set-valued mapping. Therefore, the set-valued map t ª Qr Ž t . l
⬁
ž
F ⌽n Ž t .
ns1
/
\ SŽ t .
Ž t g A.
is also measurable. We show that, for all t g A, RŽ t . s SŽ t . .
Ž 5.
If x t g RŽ t ., then B Ž x t . ; Q r Ž t .. Hence x t g Q r Ž t . and the distance of x t from any boundary points of Q r Ž t . is at least ; that is if y g ⭸ Q r Ž t ., then
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L -CLOSED DECOMPOSABLE SETS IN L
499
x t f U Ž y . . Thus, x t f U Ž nŽ t .., i.e., x t g ⺢ m _ U Ž nŽ t .. for all n g ⺞. Therefore, x t g SŽ t ., and hence, we have RŽ t . : SŽ t .. On the other hand, if x t f RŽ t . but x t g Q r Ž t ., then B Ž x t . Q r Ž t .. Therefore, there exists y g ⭸ Q r Ž t . such that < y y x t < - . The sequence nŽ t .4n is dense in ⭸ Q r Ž t .; hence, for some n g ⺞, < n Ž t . y x t < - , that is, x t g U Ž nŽ t ... Therefore, x t f SŽ t .. We have proved that R is a measurable nonempty closed valued map. Therefore, there exists a measurable selection x of R. That is, x Ž t . g RŽ t .
for all t g A.
By the definition of R, B Ž x Ž t . . : Q r Ž t .
for all t g A.
Thus, x g L⬁Ž ⍀ . and condition Ž3. is satisfied. Therefore, x g int ⬁Ž Q .. Conversely, let Q g ⌺ 0 Ž ⍀ .. Then, there exists Q g ⌫ Ž ⍀ . such that Ž ⬁ Q . s Q. Let x be an interior point of Q. By Lemma 2, ᭚ ) 0 such that Ž3. is valid. Let r s 5 x 5 ⬁ q . Then B Ž x Ž t . . : Br
for a.e. t g ⍀ ;
hence Ž4. is satisfied with x t s x Ž t .. Now consider an element Q of ⌫ Ž ⍀ ., and Q s ⬁Ž Q .. We are interested in investigating the connection between the tangent cones associated with each of Q and Q. For this purpose we focus on two types of tangent cones to a subset M of a normed vector space X, namely, the adjacent cone T Ž x 0 N M . and the Clarke tangent cone C Ž x 0 N M . Žwhere x 0 belongs to the closure of M . which are defined by T Ž x 0 N M . [ ¨ g X N ᭙ n ª 0q, ᭚ ¨ n ª ¨ such that x 0 q n¨ n g M, ᭙ n g ⺞ 4 and C Ž x 0 N M . [ ¨ g X N ᭙ n ª 0q, ᭙ x n g M : x n ª x 0 , ᭚ ¨ n ª ¨ with x n q n¨ n g M, ᭙ n g ⺞ 4 .
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500
Both tangent cones are nonempty and closed; the cone C Ž x 0 N M . is a convex subset of T Ž x 0 N M .. Let Q in ⌫ and x 0 g ⬁Ž Q .. We can associate with Q two set-valued maps T Ž x 0 N Q . and C Ž x 0 N Q ., defined, for t in ⍀, by T Ž x 0 N Q . Ž t . [T Ž x 0 Ž t . N Q Ž t . .
and C Ž x 0 N Q . Ž t . [C Ž x 0 Ž t . N Q Ž t . . .
It is shown by Giner in w5x that these set-valued maps are measurable. Thus, they are elements of ⌫ Ž ⍀ .. On the other hand, we can define these tangent cones for the set ⬁Ž Q ., that is, T Ž x 0 N ⬁Ž Q .. and C Ž x 0 N ⬁Ž Q ... These cones are decomposable and closed subsets in L⬁Ž ⍀ .. However, they are not necessarily L1-closed in L⬁. Hence, by Theorem 2, they cannot be represented via the set-valued map T Ž x 0 N Q . and C Ž x 0 N Q ., respectively. As we shall show below, these set-valued maps correspond via ⬁ to the L1-closures in L⬁ of those tangent cones. The proof of the results in based on the work by Giner w5x. THEOREM 4.
Let Q g ⌫ Ž ⍀ .. Then, for all x 0 g ⬁Ž Q ., we ha¨ e
cl 1T Ž x 0 N ⬁Ž Q . . s⬁Ž T Ž x 0 N Q . .
and cl 1C Ž x 0 N ⬁Ž Q . . s⬁Ž C Ž x 0 NQ . . .
Proof. Let K denote either T or C. From w5, Proposition 1.6, Chap. VIIx it follows that K Ž x 0 N ⬁Ž Q . . ; ⬁Ž K Ž x 0 N Q . . . Since ⬁Ž K Ž x 0 N Q .. is L1-closed in L⬁ , we obtain the inclusion : in Ži. and Žii.. Conversely, let x g ⬁Ž K Ž x 0 N Q ... By w5, Proposition 1.6, Chap. VIIx, there exists a sequence A k 4⬁ks1 of measurable sets of ⍀ such that lim Ž A k . s Ž ⍀ .
kª⬁
and 1 A k x g K Ž x 0 N ⬁Ž Q . . .
This means that, for s 0, x satisfies the condition in Theorem 1 characterizing the L1-closure in L⬁ of K Ž x 0 N ⬁Ž Q ... Hence, x g cl 1 K Ž x 0 N ⬁Ž Q ... Remark 5. In w5x, it is shown that for p g w1, ⬁.,
p Ž T Ž x 0 N Q . . : T Ž x 0 N p Ž Q . .
and
C Ž x 0 N p Ž Q . . : p Ž C Ž x 0 N Q . . ,
and these inclusions are strict in general. Hence, Theorem 4 states that in L⬁Ž ⍀ ., certain equalities do occur, yielding no gap between the concerned sets.
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501
3. L1-CLOSED DECOMPOSABLE AND CONVEX SETS IN L⬁Ž ⍀ . This section is devoted to the study of convexity properties of decomposable sets. m Let Q: ⍀ ª 2 ⺢ be a set-valued map on ⍀ with convex images. Then ⬁Ž Q . is decomposable and convex. However, this subset of L⬁Ž ⍀ . enjoys a richer property than the two combined that we call the L-con¨ exity. A set Q ; L⬁Ž ⍀ . is called L-con¨ ex if for all x, y g Q, and for all : ⍀ ª w0, 1x measurable, we have
x q Ž 1 y . y g Q. This notion defined for sets in L⬁Ž ⍀ . is the analog of the C-convexity defined in w9x for sets of continuous functions over a compact Hausdorff space. The next result shows how this notion is related to convexity for closed sets in L⬁Ž ⍀ .. Let Q be a closed subset of L⬁Ž ⍀ .. Then, the following are
LEMMA 3. equi¨ alent:
Ži. Q is con¨ ex and decomposable. Žii. Q is L-con¨ ex. Žiii. For all k g ⺞, for all 1 , . . . , k g L⬁Ž ⍀, w0, 1x. such that Ý kis1 i s 1, and for all x 1 , . . . , x k g Q, the L-con¨ ex combination Ý kis1 i x i belongs to Q. Proof. Clearly Žiii. « Žii. and Žii. « Ži.. It remains to show that Ži. « Žiii.. Assume Ži.. We first show that Žiii. is true when 1 , . . . , k are step functions on ⍀. Let 1 , . . . , k be in L⬁Ž ⍀, w0, 1x. with a range consisting of finite elements. Without loss of generality, we can assume that there exists a measurable partition A1 , . . . , A l of ⍀ such that, for all i s 1, . . . , k, l
i s
i j1 A j ,
Ý js1
where k
Ý
i j s 1 and
i j G 0
᭙ i , j.
is1
Hence k
Ý is1
k
i x i s
l
ž
Ý Ý is1
js1
l
i j1 A j x i s
/
k
ž
Ý Ý js1
is1
i j x i 1 A j .
/
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502
Since x i g Q for all i, and Q is convex, it results that y j [ Ý kis1 i j x i g Q, for all j s 1, . . . , l. By using the decomposability of Q, it follows that l
y j 1 A j g Q.
Ý js1
Now, let 1 , . . . , k be arbitrary elements in L⬁Ž ⍀, w0, 1x. with Ý kis1 i s 1. Then, for each i, there exists a sequence of step functions ni 4⬁ns1 such that k
ni ª i as n ª ⬁,
ni G 0, and
Ý
ni s 1.
is1
Thus k
Ý
k
i x i s lim
Ý ni x i .
nª⬁ is1
is1
Since Q is closed, it follows that Ý kis1 i x i g Q. Now let Q be in ⌫ Ž ⍀ .. We can associate with Q another set-valued map defined by
Ž co Q . Ž t . s co Q Ž t . . In w8x, it is shown that for p g w1, ⬁.,
p Ž co Q . s cl p co Ž p Ž Q . . . The case where p s ⬁ has been an open question. Our goal now is to solve this open question. As we shall see below, the above identity is not true when p s ⬁. However, it is known Žcf. w1x. that co Q is measurable and hence is in ⌫ Ž ⍀ .. The associated set ⬁Žco Q . is not only L⬁ -closed but also L1-closed in L⬁ and L-convex. Thus, as in Theorem 4, we shall show that by replacing, in the sought identity, the L⬁-closure with the L1-closure in L⬁, we obtain a valid statement. THEOREM 5.
Let Q be in ⌫ Ž ⍀ .. Then
⬁Ž co Q . s cl 1co Ž ⬁Ž Q . . . Proof. It is clear that ⬁Žco Q . = cl 1coŽ ⬁Ž Q ... It remains to show the opposite inclusion. Let x g ⬁Žco Q .. Then x Ž t . g co QŽ t . for t g A, for some set A of full measure. Let t g A; then co Q Ž t . s
⬁
D co Ž Q Ž t . l Bn . .
ns1
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L -CLOSED DECOMPOSABLE SETS IN L
503
Hence, rn Ž t . [ dist Ž x Ž t . , co Ž Q Ž t . l Bn . . ª 0
as n ª ⬁.
The set coŽ QŽ t . l Bn . is convex and compact. Furthermore, the fact that Q is in ⌫ Ž ⍀ . implies that there exists n 0 g ⺞, such that coŽ QŽ t . l Bn . is nonempty for all n G n 0 and for all t in A. Fix t g A and n G n 0 . Then, there exists a unique element x nŽ t . g coŽ QŽ t . l Bn . such that rn Ž t . s < x n Ž t . y x Ž t . < . Clearly
x n Ž t . 4 s Br Ž t . Ž x Ž t . . l co Ž Q Ž t . l Bn . , n
and, hence, x n is a bounded measurable function. Since rn4 is a decreasing sequence of measurable function that tends to 0 a.e., then 5 xn y x 51 s
H⍀ r d ª 0 n
as n ª ⬁.
On the other hand, x n g ⬁ŽcoŽ Q l Bn .. and the Caratheodory representation w1, Theorem 8.2.15x imply the existence of Ž m q 1. measurable n selections y 1n, . . . , ymq1 of Q l Bn and nonnegative measurable functions n n nŽ . 1 , . . . , mq1 with Ý mq1 is1 i t s 1 for a.e. t g ⍀, such that mq1
xn s
Ý
ni yin .
is1 n But, y 1n, . . . , ymq1 g cl⬁co ⬁Ž Q ., which is closed, convex, and decomposable, and hence it is also L-convex, by Lemma 3. Therefore, x n g cl⬁co ⬁Ž Q . : cl 1co ⬁Ž Q ., whence, x g cl 1co ⬁Ž Q ..
Let us introduce the following subsets of ⌫ Ž ⍀ . and ÝŽ ⍀ .: ⌫ Ž ⍀ . [ Q g ⌫ Ž ⍀ . N Q Ž t . is convex for a.e. t g ⍀ 4 ⌫0 Ž ⍀ . [ Q g ⌫0 Ž ⍀ . N Q Ž t . is convex for a.e. t g ⍀ 4 ⌺ Ž ⍀ . [ Q g ⌺ Ž ⍀ . N Q is convex 4 ⌺ 0 Ž ⍀ . [ Q g ⌺ 0 Ž ⍀ . N Q is convex 4 . The connection between these sets is given by COROLLARY 2. The map ⬁ is a bijection between ⌫ Ž ⍀ . and ⌺Ž ⍀ ., and between ⌫0 Ž ⍀ . and ⌺ 0 Ž ⍀ ..
PALES AND ZEIDAN ´
504
Proof. It is clear that if Q g ⌫ Ž ⍀ . Žrespectively ⌫0 Ž ⍀ .. and has convex images, then ⬁Ž Q . is convex. Conversely, if Q s ⬁Ž Q . g ⌺ Ž ⍀ . Žrespectively ⌺ 0 Ž ⍀ .. is convex, then, by Theorem 5, we have
⬁Ž co Q . s cl 1co Ž ⬁Ž Q . . s ⬁Ž Q . . Using the injectivity of ⬁ , it results that co Q s Q. Hence Q is convex valued. Therefore, ⬁ is a bijection for both cases. 4. SUPPORT FUNCTIONALS FOR SET-VALUED MAPS Let X be a normed vector space and let X * be its dual. For a nonempty set Q : X we define the support functional on X * as
␦ * Ž x* N Q . s sup ² x*, x : N x g Q 4 . Then
␦ *Ž⭈N Q . is a w*-lower semicontinuous and sublinear function from X * to x y ⬁, q⬁x. ␦ *Ž x* N Q1 . F ␦ *Ž x* N Q2 . for all x* g X * iff co Q1 : co Q2 . co Q s x g X N ␦ *Ž x* N Q . G ² x*, x :᭙ x* g X *4 . int coQ s x g X N ␦ *Ž x* N Q . ) ² x*, x :᭙ x* / 0, x* g X *4 , when X is finite-dimensional. 䢇
䢇 䢇 䢇
Let Q be a measurable set-valued map on ⍀ with nonempty images in ⺢ m . Then, we can associate with Q the function q: ⍀ = ⺢ m ªx y ⬁, q⬁x defined by q Ž t , . s ␦ *Ž N QŽ t . . . Set ⌬*Ž Q . [ q. Hence, one can easily see that Ži. For all t g ⍀, q Ž t, ⭈ . is lower semicontinous and sublinear on ⺢ . Žii. q is Ž A = B .-measurable, where B denotes the collection of Borel subset of ⺢ m. m
On the other hand, for Q in ⌫ Ž ⍀ ., we have the support functional of ⬁Ž Q . defined on Ž L⬁Ž ⍀ ..*. As we shall see below, this support functional, restricted to elements from L1 Ž ⍀ ., can be calculated via ⌬*Ž Q .. LEMMA 4. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ .. Then, for g L1 Ž ⍀ ., we ha¨ e
␦ * Ž N cl 1Q . s ␦ * Ž N Q . .
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505
Proof. It suffices to show that ‘‘F ’’ is true. Let x g cl 1Q. Then, by Corollary 1, there exists a sequence x n g Q and a constant C such that 5 x n y x 5 1 ª 0 as n ª ⬁ and sup ng ⺞ 5 x n 5 ⬁ F C. Taking a subsequence of x n Žwhich we do not relabel . that converges almost everywhere and applying the Lebesgue dominated convergence theorem, we obtain lim H ² , x : d F ␦ * Ž N Q . , H⍀² , x : d F nª⬁ ⍀ n
from which the result follows. THEOREM 6. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ . such that cl 1 co Q s ⬁Ž Q . for some Q g ⌫ Ž ⍀ ., and let g L1 Ž ⍀ .. Then, H⍀ ␦ *Ž Ž t . N QŽ t .. d Ž t . is well defined and
␦ * Ž N Q. s
H⍀␦ * Ž Ž t . N Q Ž t . . d Ž t . .
Proof. By the second listed property of support functions and by Lemma 4, we have
␦ * Ž N Q . s ␦ * Ž N co Q . s ␦ * Ž N cl 1 co Q . s ␦ * Ž N ⬁Ž Q . . . The rest of the proof follows immediately from w8, Theorem 2.2x. The following result is an obvious consequence of the above theorem. COROLLARY 3.
Let Q g ⌫ Ž ⍀ .. Then, for g L1 Ž ⍀ .,
␦ * Ž N ⬁Ž Q . . s
H⍀␦ * Ž Ž t . N Q Ž t . . d Ž t . .
To completely characterize set-valued maps Q in ⌫ Ž ⍀ . in terms of their support functional ⌬*Ž Q ., it is only natural to consider the images of Q to be convex, that is, Q must be in ⌫ Ž ⍀ .. The following is a characterization, in terms of ⌬*Ž Q ., of the nonemptiness of ⬁Ž Q .. LEMMA 5.
Let Q be in ⌫ Ž ⍀ ., and q s ⌬*Ž Q .; then, ⬁Ž Q . / ⭋ iff
᭚ r ) 0: for a.e. t g ⍀ , q Ž t , . q r < < G 0 ᭙ g ⺢ m .
Ž 6.
Proof. From Lemma 1, it suffices to show that condition Ž2. is equivalent to condition Ž6.. Assume that Ž2. holds. Then, there exists r ) 0 such that QŽ t . l Br / ⭋ for t g A, where Ž A. s Ž ⍀ .. Fix t g A, and consider x g QŽ t . l Br and g ⺢ m . Then q Ž t , . s ␦ * Ž N Q Ž t . . G ² , x : G inf ² , u: s yr < < . ugB r
That is, Ž6. holds true.
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PALES AND ZEIDAN ´
Conversely, assume that Ž6. is satisfied for r ) 0. If condition Ž2. is false, then there exists a set A of positive measure such that Q Ž t . l Br s ⭋
for t g A.
Fix t g A; then by the strong form of the Hahn᎐Banach separation theorem there exists g ⺢ m such that
␦ * Ž N Q Ž t . . - inf ² , u: N u g Br 4 s yr < < . Therefore condition Ž6. is violated. The above lemma inspires the consideration of the following sets for support functionals to elements of ⌫ Ž ⍀ .. Set ⌳ Ž ⍀ . [ q : ⍀ = ⺢ m ª x y ⬁, ⬁ x N q Ž t , ⭈ . is sublinear and lsc, q is Ž A = B . -measurable and satisfies condition Ž 6 . 4 . The equality q1 s q2 in ⌳Ž ⍀ . means that, for a.e. t g ⍀, q1Ž t, . s q2 Ž t, . for all g ⺢ m . The intimate connection between ⌳Ž ⍀ . and ⌫ Ž ⍀ . is presented below. THEOREM 7. The map ⌬* is a bijection from ⌫ Ž ⍀ . onto ⌳Ž ⍀ .. Proof. From Lemma 5 and the previous properties of ⌬*Ž Q ., it results that ⌬*Ž ⌫ Ž ⍀ .. ; ⌳Ž ⍀ .. For the injectivity note that if Q1 , Q2 in ⌫ Ž ⍀ . satisfies Q1 / Q2 , then there exists a set A of positive measure over which Q1Ž t . / Q2 Ž t .. Hence, ␦ *Ž⭈N Q1Ž t .. / ␦ *Ž⭈N Q2 Ž t .. for t g A, and thus ⌬*Ž Q1 . / ⌬*Ž Q2 .. For the surjectivity, let q g ⌳Ž ⍀ .. Define the set-valued map Q by Q Ž t . s x g ⺢ m N q Ž t , . G ² x, : ᭙ g ⺢ m 4 . Clearly ⌬*Ž Q . s q, and Q is measurable with nonempty, closed convex values. Furthermore, by Lemma 5, Q satisfies condition Ž2.. Thus, Q g ⌫ Ž ⍀ .. Remark 6. By combining Theorem 7 with Corollary 2, we obtain the equivalence between the three sets ⌫ Ž ⍀ ., ⌺Ž ⍀ ., and ⌳Ž ⍀ .. The rest of this section is devoted to the characterization of ⌫0 Ž ⍀ . in terms of elements of ⌳Ž ⍀ .. First of all, let us translate condition Ž3. on the elements of ⌫ Ž ⍀ . in terms of the elements of ⌳Ž ⍀ ..
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L -CLOSED DECOMPOSABLE SETS IN L
LEMMA 6.
507
Q g ⌫ Ž ⍀ . and q s ⌬*Ž Q .. Then x g int ⬁Ž Q . if and only if
᭚ ) 0: for a.e. t g ⍀ ,
qŽ t, . G < < q ² , xŽ t.:
᭙ g ⺢ m . Ž 7 .
Proof. By Lemma 2, it is equivalent to show that condition Ž3. holds true iff condition Ž7. is satisfied. However, B Ž x Ž t .. ; QŽ t . is equivalent to
␦ * Ž N B Ž x Ž t . . . F ␦ * Ž N Q Ž t . . ,
for all g ⺢ m ,
that is,
< < q ² , xŽ t.: F qŽ t, . ,
for all g ⺢ m ,
whence the result follows. As we shall see in the result below, the nonemptiness of the interior of ⬁Ž Q . is characterized by this property of q g ⌳Ž ⍀ .: ᭚ r G ) 0: for a.e. t g ⍀ , q Ž t , . q q Ž t , . q r < q < G w < < q < < q < q < x
᭙ , g ⺢ m . Ž 8.
Set ⌳ 0 Ž ⍀ . [ q g ⌳ Ž ⍀ . N q satisfies Ž 8 . 4 . The following result shows that there is a one-to-one correspondence between ⌫0 Ž ⍀ . and ⌳ 0 Ž ⍀ .. THEOREM 8. The map ⌬* is a bijection from ⌫0 Ž ⍀ . onto ⌳ 0 Ž ⍀ .. Proof. Let Q g ⌫0 Ž ⍀ . and set q [ ⌬*Ž Q .. Then Q satisfies condition Ž4.. By Theorem 3, int ⬁Ž Q . / ⭋. Let x g int ⬁Ž Q .; then, Lemma 6 gives that Ž7. is satisfied for some ) 0. Set r [ 5 x 5 ⬁ q . Then, there exists a set A of full measure such that, for all t g A, Br = B Ž x Ž t . . and qŽ t, . G < < q ² , xŽ t.:
᭙ g ⺢ m .
Ž 9.
Thus, for , g ⺢ m , the inclusion above gives
␦ * Ž y y N Br . G ␦ * Ž y y N B Ž x Ž t . . . , that is, r < q < G ² y y , x Ž t . : q < q < .
Ž 10 .
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508
By substituting for in Ž9. and , respectively, and by adding the two inequalities so obtained to Ž10., we get that Ž8. holds true; therefore ⌬* maps ⌫0 Ž ⍀ . into ⌳ 0 Ž ⍀ .. ⌬* is also injective; hence it suffices to show only that ⌬*Ž ⌫0 Ž ⍀ .. = ⌳ 0 Ž ⍀ .. Let q g ⌳ 0 Ž ⍀ . and let Q g ⌫ Ž ⍀ . such that ⌬*Ž Q . s q. We are going to show that Q also belongs to ⌫0 Ž ⍀ .. Then, there exist r G ) 0 and a set A of full measure such that the inequality in Ž8. is valid for all , g ⺢ m. Putting s , we obtain qŽ t, . q Ž r y 2 . < < G 0
for g ⺢ m , t g A.
It follows from this inequality that qŽ t, . q Ž r y . < < G 0
᭙ g ⺢ m , ᭙ t g A.
Using the argument of the second part of the proof of Lemma 6, this yields that Q Ž t . l Bry / ⭋
᭙ t g A.
Therefore, for t g A, the relative interior of QŽ t . and Br are not disjoint. Using w13, Corollary 16.4.1x, we infer that the support functional of QŽ t . l Br s Q r Ž t . can be expressed as the infimal convolution of the supporting functionals of QŽ t . and Br , that is, qr Ž t , . [ ␦ * Ž N Q r Ž t . . s inf q Ž t , . q r < y < : g ⺢ m 4 . We shall now show that qr satisfies the following inequality: qr Ž t , . q qr Ž t , y . G 2 < <
᭙ t g A, ᭙ g ⺢ m .
It is enough to show that qŽ t, . q r < y < q qŽ t, . q r < q < G 2 < <
᭙t g A
᭙ , g ⺢ m . Using Ž8. and triangle inequalities, we get qŽ t, . q qŽ t, . q r < y < q r < q < G Ž < < q < < q < q < . y r < q < q r < y < q r < q < s Ž r y . Ž < y < q < q < y < q <. q Ž < < q < y <. q Ž < y < q < q < . G 2 < < . Thus Ž11. is proved.
Ž 11 .
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509
Our next aim is to prove that if Ž11. is valid, then, for each t g A, Q r Ž t . contains a ball of radius m2q 1 . In fact, we prove that the ball of radius m2q 1 centered at the center of mass of Q r Ž t . is contained in Q r Ž t .. If P : ⺢ m is a bounded Lebesgue measurable set, then cŽ P ., the center of mass of P, is defined by cŽ P . [
1 Vm Ž P .
HPx dx,
where VmŽ P . s HP 1 dx is the m-dimensional Lebesgue measure of P, which is assumed to be positive. We need the following lemma. LEMMA 7.
If P is a compact con¨ ex set in ⺢ m , then, for all g ⺢ m ,
HP² , x : dx F
m ␦ * Ž N P . y ␦ * Ž y N P . mq1
HP1 dx.
Ž 12 .
Proof. Observe first, that if Ž12. is valid for some compact convex P then it is also valid for the translates of P. For, let u g ⺢ m. Then
HPqu² , x : dx s HP² , x q u: dx s HP² , x : q ² , u:HP1 dx. On the other hand,
␦ * Ž N P q u . s ␦ * Ž N P . q ² , u:
for all g ⺢ m .
Hence m ␦ * Ž N P q u . y ␦ * Ž y N P q u . mq1 s
ž
␦ * Ž N P . y ␦ * Ž y N P . mq1
HPqu1 dx
q ² , u:
/H
1 dx.
P
Thus Ž12. implies the analogous inequality for P q u. Therefore, without loss of generality, we may assume that 0 g P and
␦ * Ž y N P . s sup ² y , x : N x g P 4 s 0. Both sides of inequality Ž12. are positively homogeneous in ; therefore, < < s 1 can also be assumed. There is always an orthogonal transformation T : ⺢ m ª ⺢ m such that T Ž . s Ž1, 0, . . . , 0. s e1. Then VmŽ P . s VmŽT Ž P .. and
HP² , x : s HP²T , Tx : s HP² e , Tx : dx s HT Ž P .² e , x : dx. 1
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PALES AND ZEIDAN ´
510 Similarly,
␦ * Ž " N P . s sup ² " T , Tx : N x g P 4 s sup ² " e1 , u: N u g T Ž P . 4 s ␦ * Ž "e1 N T Ž P . . . Therefore, Ž12. is equivalent to saying that Ž12. Žreplacing P by T Ž P . if necessary . holds for s e1 and for compact convex sets P such that 0 g P and ␦ *Žye1 N P . s 0. For analogous reasons, replacing P by P Ž ) 0., we can assume that ␦ *Ž e1 N P . s 1. Summarizing all the reductions made above, it suffices to show that
HPx
1
m
dx F
HP1 dx
mq1
Ž 13 .
for all compact convex sets P such that 0 g P, ␦ *Žye1 N P . s 0, ␦ *Ž e1 N P . s 1. Define P0 [ x g P x 1 s
½
m mq1
5
R[
and
D
0F F Ž mq1 .rm
P0 .
We first show that Ž13. holds with equality for P s R. Observe that the elements of R can be written as
½
R s Ž x 1 , . . . , x m . 0 F x 1 F 1, Ž x 2 , . . . , x m . g x 1
mq1 m
5
1Ž P . ,
where 1 is the orthogonal projection defined via
1 Ž u1 , . . . , u m . [ Ž u 2 , . . . , u m . . Therefore, for any integrable function f : w0, 1x ª ⺢, we have 1
HR f Ž x . dx s H0 f Ž x . Hx ŽŽ mq1.rm . Ž P . 1d Ž x 1
1
s s
1
H0
1
H0
ž
1
ž
f Ž x 1 . Vmy1 x 1 f Ž x1 . x1 ⭈
ž
2,...,
1
mq1 m
mq1
s Vmy 1 Ž 1 Ž P . . ⭈
m
ž
1 Ž P . dx 1
/
my 1
⭈ Vmy1 Ž 1 Ž P . . dx 1
/
mq1 m
/
x m . dx 1
my 1
/
⭈
1
H0
f Ž t . t my 1 dt.
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511
Taking f Ž t . s t and f Ž t . s 1, we obtain that both side of Ž13. are equal for P s R. Now we prove that Ž13. is valid for general P. Define Py[ x g P x 1 F
½ ½
Pq[ x g P x 1 )
m mq1 m mq1
5 5
,
Ry[ x g R x 1 F
,
Rq[ x g R x 1 F
½ ½
m mq1 m mq1
5 5
, .
First, using the convexity of P, we show that Ry: Py
Pq: Rq .
and
Ž 14 .
Let x g Ry. Then there exist g w0, m mq 1 x and u g P0 such that x s 0 u. Since u1 s m mq 1 then, x 1 F m mq 1 and hence, g w0, 1x. The elements 0 and u belong to Py, and Py is also convex. Thus x s Ž 1 y . 0 q u g Py . To prove the inclusion Pq: Rq, let x g Pq. Then s mrwŽ m q 1. x 1 x - 1 and m x s x q Ž 1 y . 0 g P. Ž m q 1. x1 Hence m
Ž m q 1. x1
x g P0 .
Therefore xg
mq1 m
x 1 P0 ; Rq
because m mq 1 x 1 F m mq 1 . Thus Ž14. is proved. Now, using that Ž13. holds with equality for P s R, we are able to prove that Ž13. is valid for any P:
HPx
1
dx s
HP _R y
F
HP _R y
s
x 1 dx q y
m y
mq1
HR x
1
dx q
dx y
HR _P q
x 1 dx 1 q
m
HR m q 1 dx y HR _P
m
HP m q 1 dx.
Thus the lemma is completely proved.
q
m q
mq1
dx
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Now we can continue the proof of the theorem. Assume that Ž11. is valid. Then the dimension of Q r Ž t . cannot be smaller than m; otherwise there exists / 0 in ⺢ m such that qr Ž t , . q qr Ž t , y . s 0, and this contradicts Ž11.. Therefore, the measure of Q r Ž t . is positive. Applying Lemma 7 to the set P s Q r Ž t ., we obtain
HQ Ž t .² , x : dx F
mqr Ž t , . y qr Ž t , y . mq1
r
s qr Ž t , . y
ž ž
F qr Ž t , . y
⭈ Vm Ž Q r Ž t . .
qr Ž t , . q qr Ž t , y . mq1 2< < mq1
/
/
⭈ Vm Ž Q r Ž t . .
⭈ Vm Ž Q r Ž t . . .
Hence, dividing by VmŽ Q r Ž t .. and using the definition of the center of mass for Q r Ž t ., we get ² , c Ž Qr Ž t . . : q
2 mq1
< < F qr Ž t , .
for all g ⺢ m , t g A. Thus, for all g ⺢ m ,
␦ * Ž N B2 rŽ mq1. Ž c Ž Q r Ž t . . . . F ␦ * Ž Q Ž t . l Br . . Therefore B2 rŽ mq1. Ž c Ž Q r Ž t . . . : Q Ž t . l Br for a.e. t g ⍀, which means that property Ž4. is satisfied. Hence Q belongs to ⌫0 Ž ⍀ ., as was to be proved. 5. NORMAL AND POLAR CONES OF DECOMPOSABLE SETS IN L⬁Ž ⍀ . Let X be a normed vector space and let Q be a nonempty subset of X. For x 0 g Q we define the normal cone N Ž x 0 N Q . to Q at an element x 0 g cl Q by N Ž x 0 N Q . [ x* g X * N ² x*, x : F ² x*, x 0 : ᭙ x g Q 4 .
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513
Clearly, N Ž x 0 N Q . is a closed convex nonempty cone. When Q is a cone and x 0 s 0, N Ž0 N Q . is called the polar cone of Q, denoted by Q 0 . For the rest of this section, we take X s L⬁Ž ⍀ .. We first focus on the investigation of the normal cone of decomposable sets Q. We introduce the multiplication in Ž L⬁Ž ⍀ ..* by an essentially bounded real-valued function f as follows: for x* g Ž L⬁Ž ⍀ ..*, the linear functional fx* is defined by ² fx*, x : s ² x*, fx :
᭙ x g L⬁ Ž ⍀ . .
Hence, the notions of decomposability and L-convexity extend naturally to subsets ⌽ of Ž L⬁Ž ⍀ ..*. We say that ⌽ : Ž L⬁Ž ⍀ ..* is an L-cone if for now f g L⬁Ž ⍀, ⺢q. and x* g ⌽, fx* g ⌽. The characterization obtained in Lemma 3 for the L⬁Ž ⍀ . setting remains valid for the Ž L⬁Ž ⍀ ..* case. Furthermore, similar arguments show the following. LEMMA 8. equi¨ alent:
Let ⌽ be a closed cone of Ž L⬁Ž ⍀ ..*. The following are
Ži. ⌽ is decomposable and con¨ ex. Žii. ⌽ is a con¨ ex L-cone. Žiii. Ý kis1 f i xUi g ⌽ whene¨ er, for all i, xUi g ⌽ and f i g L⬁Ž ⍀, ⺢q. . LEMMA 9. Let Q ; L⬁Ž ⍀ . be a nonempty decomposable set. Then, for x 0 g Q, N Ž x 0 N Q. is a closed con¨ ex L-cone. Proof. It remains to show that N Ž x 0 N Q. is an L-cone. By Lemma 8, this is equivalent to showing that N Ž x 0 N Q. is decomposable. Let A g A and x g Q be arbitrary. Then 1 A x q 1 ⍀ _ A x 0 g Q, by the decomposability of Q. Thus, if x* g N Ž x 0 N Q., then ² x*, 1 A x q 1 ⍀ _ A x 0 : F ² x*, x 0 : , that is, ² x*, 1 A x : F ² x*, 1 A x 0 : for all x g Q. This yields that 1 A x* g N Ž x 0 N Q . . If y* g N Ž x 0 N Q., then, similarly, 1 ⍀ _ A y* g N Ž x 0 N Q . .
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Since N Ž x 0 N Q. is a cone, we obtain that 1 A x* q 1 ⍀ _ A y* g N Ž x 0 N Q . .
Let Q be a set-valued map in ⌫ Ž ⍀ ., let Q s ⬁Ž Q ., and let x 0 be in Q. Then, with Q we can associate a set-valued map N Ž x 0 N Q . defined via the pointwise normal cones to QŽ t . at x 0 Ž t ., that is, N Ž x 0 N Q .Ž t . s N Ž x 0 Ž t . N QŽ t ... This map is measurable, since ␦ *Ž N QŽ⭈.. is measurable and N Ž x 0 Ž t . N Q Ž t . . s g ⺢ m N ␦ * Ž N Q Ž t . . s ² , x 0 Ž t . :4 for a.e. t g ⍀ . The goal is to describe the L1-elements of N Ž x 0 N Q. in terms of the set-valued map N Ž x 0 N Q .. However, we shall show that such a characterization in terms of N Ž x 0 N Q . is true for subsets Q of L⬁Ž ⍀ ., which are not necessarily L1-closed, but rather their L1-closures are decomposable. This property is met by both the Clarke and the adjacent tangent cones used in Section 2. THEOREM 9. Let Q be a nonempty decomposable subset of L⬁Ž ⍀ . such that cl 1Q s ⬁Ž Q . for some Q g ⌫ Ž ⍀ .. Then, for x 0 g Q, N Ž x 0 N Q . l L1 Ž ⍀ . s 1 Ž N Ž x 0 N Q . . . Proof. A function g L1 Ž ⍀ . belongs to N Ž x 0 N Q. if and only if ␦ *Ž N Q. s H⍀ ² N x 0 : d . By Theorem 6, this is equivalent to
H⍀ Ž ␦ * Ž N Q . y ² , x :. d s 0. 0
Being nonpositive, the integrand must be zero a.e. on ⍀. This means
Ž t . g N Ž x0 Ž t . N QŽ t . .
for a.e. t g ⍀ ,
which proves the theorem. COROLLARY 4.
If Q g ⌫ Ž ⍀ ., then
N Ž x 0 N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž N Ž x 0 N Q . . . The characterization of the polar cones of the Clarke and the adjacent tangent cones is presented in what follows.
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Let Q g ⌫ Ž ⍀ . and x 0 g ⬁Ž Q .; then
COROLLARY 5.
žT Ž x žCŽ x
515
0
0
N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž T 0 Ž x 0 N Q . .
0
N ⬁Ž Q . . l L1 Ž ⍀ . s 1 Ž C 0 Ž x 0 N Q . . ,
and
0
where, for t g ⍀, T 0 Ž x0 N Q. Ž t . [ Ž T Ž x Ž t . N QŽ t . . .
0
and
0
C 0 Ž x0 N Q. Ž t . [ Ž C Ž x0 Ž t . N QŽ t . . . . Proof. Set Q s T Ž x 0 N ⬁Ž Q .. and Q s C Ž x 0 N ⬁Ž Q .., respectively. Then the results follow immediately from Theorems 4 and 9.
REFERENCES 1. J.-P. Aubin and H. Frankowska, ‘‘Set-Valued Analysis,’’ Birkhauser, BostonrBasel, 1990. ¨ 2. C. Castaing and M. Valadier, ‘‘Convex-Analysis and Measurable Multifunctions,’’ Springer-Verlag, Lecture Notes in Math., 580 Ž1977.. 3. F. H. Clarke, ‘‘Optimization and Nonsmooth Analysis,’’ Wiley, New York, 1983. 4. I. Ekeland and R. Temam, ‘‘Analyse convexe et problemes variationelles,’’ Hermann, ` Paris, 1974.2 5. E. Giner, ‘‘Etudes des fonctionnelles integrables,’’ Thesis, Universite ´ de Pau, France, 1985. 6. E. Giner, Sous-differentiabilite´ des fonctionnelles integrables ŽII., Thesis, Sem. ´ Anal. Num. No. 6, 83᎐84. 7. F. Hiaı, ¨ Multivalued conditional expectations, multivalued Radon-Nikodym theorems, integral representation of additive operators, and multivalued strong laws of large numbers, unpublished manuscript. 8. F. Hiaı¨ and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multi¨ ariate Anal. 7 Ž1977., 149᎐182. 9. Zs. Pales, ´ and V. Zeidan, Characterization of closed and open C-convex sets in C ŽT, ⺢ r ., Acta Sci. Math. Szeged 65 Ž1999., 339᎐357. 10. Zs. Pales, ´ and V. Zeidan, On L1-closed decomposable sets in L⬁, Systems modelling and optimization ŽDetroit, MI, 1997., Chapman & HallrCRC Res. Notes Math. 396 Ž1999., 198᎐206. 11. Zs. Pales, ´ and V. Zeidan, Optimization problems with measurable set-valued constraints, SIAM Journal on Optim., to appear. 12. A. W. Roberts and D. E. Varberg, ‘‘Convex Functions,’’ Academic Press, New YorkrLondon, 1973. 13. R. T. Rockafellar, ‘‘Convex Analysis,’’ Princeton Univ. Press, Princeton, NJ, 1970. 14. R. T. Rockafellar, Integrals which are convex functionals, Pacific J. Math. 24 Ž1968., 525᎐539. 15. R. T. Rockafellar, Integrals which are convex functionals, II, Pacific J. Math. 39 Ž1971., 439᎐469.
2 English translation: ‘‘Convex Analysis and Variational Problems,’’ North-Holland, Amsterdam, 1977.