Set-Valued Integration and Set-Valued Probability Theory: An Overview

CHAPTER 14

Set-Valued Integration and Set-Valued Probability Theory: An Overview Christian Hess Universite Paris Dauphine, Viabilite, Jeux, Contrdle, 75775 Paris Cedex, France E-mail: hess@ viab.ufrmd.dauphine.fr

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction Notations and preliminaries Integration of strongly measurable multifunctions Weakly measurable multifunctions and graph-measurable multifunctions The Aumann integral The set-valued conditional expectation of closed valued multifunctions Set-valued measures The probability distribution of a measurable multifunction Set-valued strong laws of large numbers 9.1. Convergence in the Hausdorff metric topology 9.2. Convergence in the sense of Painleve-Kuratowski 10. Set-valued martingales 11. Gaussian multifunctions. The set-valued Central Limit Theorem 12. Set-valued versions of the Fatou Lemma 13. Epigraphical convergence 14. Concluding remarks References

HANDBOOK OF MEASURE THEORY

Edited by Endre Pap © 2002 Elsevier Science B.V. All rights reserved 617

619 620 622 629 634 639 643 647 652 652 654 658 660 662 663 665 666

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1. Introduction The mathematical modehzation based on multifunctions (aHas set-valued maps, correspondences, etc.) has shown its great adaptability and relevance for a long time. It allows one to take into account the multiplicity of possible choices, the lack of information and/or the uncertainty in a lot of situations ranging from Optimal Control to Economic Theory. For example, this can be seen in the monographs by Hildenbrand [118], and Aubin and Frankowska [16]. In particular, measurable multifunctions (alias set-valued random variables, random sets, etc.) are of interest in Probability Theory and in Statistics, not only from the theoretical point of view, but also for applications. Here, our aim is to present the main results from the theory of measurable multifunctions. We will look more closely at integration, conditional expectation and convergence theorems such as strong laws of large numbers and martingales convergence theorems. The notion of random set is intuitive and can be considered from different points of view. It is possible to go back as far as the classical Buffon's needle problem (see, e.g., [125]) that implicitly involves this notion. In the twentieth century, the first papers dealing with random sets seem to be those of Robbins [172,173] in the middle forties. Robbins proved a celebrated formula related to the measure of a random set in an Euclidean space. Later, Kudo [135] and Richter [171], motivated by Mathematical Statistics, examined measurability and integration problems. The literature of the sixties and the early seventies was more abundant. At this time, a great deal of works were inspired by problems arising from Control Theory and Mathematical Economics. We can cite the papers by Aumann [17], Debreu [65], Castaing [33], Castaing and Valadier [40], Van Cutsem [203205] and Valadier [199,200]. During the seventies and the next decades, the number of works on the field increased rapidly. Let us mention, for example, the monograph by Castaing and Valadier [41], and the paper by Hiai and Umegaki [117]; both were published in the late seventies and were highly influential. A lot of further references will be given in the sequel, but one can also cite general monographs on multifunctions (not especially devoted to integration theory), for example that by Klein and Thompson [130] in the eighties, and that by Aubin and Frankowska [ 16] in the early nineties. This paper is organized as follows. In Section 2, we collect some elementary facts and introduce our notations. Section 3 deals with the integration of strongly measurable multifunctions and, more particularly, with those that can be approximated by simple measurable multifunctions in the sense of the Hausdorff distance. Furthermore, these multifunctions are assumed to be integrably bounded, whence almost surely bounded valued. In Section 4, we present weakly measurable multifunctions in connection with graph-measurability, and we recall two versions of the Measurable Selection Theorem. Section 5 is devoted to the Aumann integral, whose construction is based on integrable selections and which has become the most popular set-valued integral. The Hiai-Umegaki set-valued conditional expectation is examined in Section 6. Set-valued measures (alias multimeasures) are considered in Section 7; as we shall see, three distinct definitions can be given. A few basic results on the probability distribution of measurable multifunctions are presented in Section 8. Section 9 is devoted to set-valued strong laws of large numbers; there, two convergence concepts are considered on the space of subsets, namely the Hausdorff metric convergence and the Painleve-Kuratowski convergence. In

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Section 10, we introduce the definition and the main properties of set-valued martingales. In Section 11, we recall some basic facts about Gaussian multifunctions and the setvalued Central Limit Theorem. Section 12 is devoted to set-valued versions of Fatou's Lemma. Last but not least, we give a short introduction to epigraphical convergence, which derives from set convergence and has appeared to be an interesting functional convergence. There, we shall give applications in a stochastic context. Finally, a few concluding remarks indicate other topics of interest in the field. They are followed by a substantial reference list which shows how wealthy this domain is. 2. Notations and preliminaries We consider a separable Banach space X, whose norm is denoted by || • || and Borel a-field by B{X). We define the following families of subsets of X • 2^: the family of all subsets of X, • ^: the family of open subsets of X, • B{X): the Borel a-field (or Borel a-algebra) of X, generated by Q, • C(X): the set of all closed subsets of X, • Cc(X): the set of all closed convex subsets of X, • Cb(X): the set of all closed bounded subsets of X, • /C(X): the family of all nonempty compact subsets of X • /Cc(X): the family of nonempty compact convex subsets of X, • /Cw(X): the family of all nonempty weakly compact subsets of X. The meaning of Cbc(X) and /Cwc(^) is clear. On 2^ we consider the Minkowski addition, denoted by " + " and the scalar multiplication, respectively defined by C^C'

= [x+x'\

J C G C , X' eC'\,

aC = [ax:

xeC),

where C, C e 2^ and a G R. These operations satisfy the following properties aC + aC' = a{C + C ) ,

{a -h p)C
fiC.

(2.1)

Easy examples show that the inclusion in (2.1) is strict when C is not convex. When C is convex, inclusion (2.1) becomes an equality, namely {a -h ^)C = a C -f pC,

c^, ^ E M.

Given C € 2^, the distance function J(-, C) and the support function s{-, C) of C are respectively defined by d{x, C) = M[\\x - y\\: y e C}, ^(j,C) = sup{(},jc): A-€C},

.v G X, y GX*,

where (y,x) stands for the duality pairing and X* for the dual space of X. We denote by clC (respectively, coC, coC) the closure (respectively, the convex hull, the closed

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convex hull) of C. Recall that the distance function (respectively, the support function) characterizes a closed set (respectively, a closed convex set). Indeed, for every closed set C (respectively, every closed convex set C), we have C={xeX:

d(x,C)=0},

and C= P | { j c € X : {y\x)^s(yX)},

(2.2)

yeB*

where B* denote the closed unit ball of X*. Equality (2.2) is a consequence of the HahnBanach Theorem. The support function satisfies the following simple properties that are stated for easy references. s(yX)=s(y,coC), s{y, C-\-C)=

CG2^, s(y, C) +s{y, C ) ,

y e X, C, C G 2^ \ {0}.

It is positively homogeneous and subadditive. s(ay, C) = as{y, C),

a ^ 0, y € X, Ce2^.

(2.3)

s{y^y\C)^s(yX)-i-s{y\C),

y.yeX,

(2.4)

Ce2^.

In particular, taking y' = —y in (2.4) we obtain s(y, C) + si-y, C) ^ 5(0, C) = 0,

C G 2^ \ {0}.

(2.5)

For every pair (C, C ) e Cb, we also define the Hausdorjf distance between C and C by h{C, C') = inf{a > 0: C c C' -f aB and C' c C -h a B } , where ^ denotes the closed unit ball of X. Equivalently, if the excess of C over C is defined by e{C, C) = sup{j(jc, C):

xeC},

we have h{C,C)=max{e{CX'),e{C\C)}. These formulas also makes sense when C and C are closed sets, possibly unbounded, but in this case, h is only a pseudo-distance in that it can take on the value +oo. The

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topology generated by h is called the Hausdorff metric topology and is denoted by T\\. For every C G 2^ \ {0} we set ||C|| = sup{||x||:xGC}, which is the Hausdorff distance between {0} and C. Some other properties of the Hausdorff distance are listed below for easy references. The proof can be found in [65, p. 362]. PROPOSITION 2.1. For every C, C\ K, K' e C^iX) and a >0 the following properties hold (a) h{c\{C + O , c\{K + K')) ^ hiC K) 4- h(C\ K')\ (b) hiaC,aK)=ah(C,K); (c) h(cdC,cdK)^h(C,K).

In particular, the maps (C,K) -^ cl(C + A:), (a, C ) -^ otC and C -^ coC are continuous in the Hausdorff metric topology on Cb(X). It can be shown that the space C endowed with the pseudo-distance h is complete, and that Cb, Cc, /C are closed subspaces of C (see, e.g., [41, Chapter II]). Using a lemma due to Grothendieck (see, e.g.. Lemma V.6 in [41]), the same property can be proved for /Cw Finally, we recall the Hormander formula (see, e.g.. Theorem 11.18 of [41]) which expresses the Hausdorff distance of bounded convex sets in terms of support functions. For every C, C G Cbc one has h{CX')=

sup|5(y,C)-^(j,C')|.

(2.6)

The above equality admits several variants or special cases that will be useful. Because of (2.3), the supremum can also be restricted to 5 = { J G X * : ||>'|| = 1}, the unit sphere of X*. In the special case where C = {0}, it is readily seen by using (2.5) and (2.6) that ||C|| = sup^(y,C). vefi

3. Integration of strongly measurable multifunctions Let ( ^ , v4, IX) be a probability' space. A multifunction is a map, defined on Q, whose values are subsets of some given set. In this section, we shall restrict our attention to the space Cb(X) (or Cb) of closed bounded subsets of X, endowed with the topology TH We have choose this setting for sake of simphcity. In fact, a lot of results of this Handbook remain valid when /i is a finite or a-finite positive measure.

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generated by the Hausdorff distance /i. Further, we consider the Borel a-field jB(Cb(X), TH) generated by the tH-open subsets of Cb(X). A multifunction F \Q ^ Ch{X) is said to be strongly A-measurable (or simply, strongly measurable) if, for every member W of ^(Cb, TH), one has F~\W) G ^ . A more precise criterion is given in Proposition 3.3 below for the case where F takes on its values in a tH-separable subspace of C\^{X). Multifunctions that enjoy some measurability property are also called '^random sets". The proofs of the results of this section can be found in [110]. We begin by defining the set-valued integral or expectation of a simple multifunction, i.e., of a multifunction F assuming only a finite number of values. More precisely, let {A 1,..., A/:} be an ^-measurable partition of Q and let F : a> ^ F((JO) be the measurable multifunction taking on the value AT, G Ch for any a> G A, (/ = 1,..., /:). This can be written

^E^^' ^'/• = ! F is called a simple multifunction. Clearly, F is strongly measurable. In this equality, 1^/ denotes the indicator function of A,, namely . . .

f 1, [0,

licDe A/, iicofAi.

In this case, the expectation (or integral) of F is the member of Cb defined by k

1

E(F)=cl Y^tiiA^KA.

(3.1)

l/=i

where the sum refer to the Minkowski addition. When X is finite dimensional, the closure operation is not necessary. The following lemma which is valid for a general metric space allows for identifying the class of strongly measurable multifunctions that can be approximated by a sequence of strongly measurable simple multifunctions. LEMMA ?>A. If F \Q -^ C is a closed valued multifunction, then the following two statements (a) and (b) are equivalent. (a) F is the pointwise limit on Q of a sequence of strongly measurable simple multifunctions. (b) F{Q) is a Tw-separable subspace of C and, for every K G X{Q), the map co -> h{K, F(co)) is measurable.

Recall that /C and /Cc are TH-separable subspaces of Cb- However, when X is infinite dimensional, neither Cb nor Cbc is TH-separable. For example, consider the space X — i^ of sequences (a«)„^ i such that 5Z/?> i 1^/' I converges and, for any subset / of N, the set of positive integers, the subset Cj =co{^^: n G / } .

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Recall that e^ denotes the sequence whose all terms are zero excepted the n\\\ equal to 1. It is readily seen that if / and J are two distinct subsets of N, one has h(Ci,Cj)^\. Since the family {C/: / C N } is uncountable, the nonseparability of Cbc follows. In particular, there exists strongly measurable multifunctions that cannot be approximated by a sequence of strongly measurable simple multifunctions. Given a subspace C of Cb, it is useful to introduce the following three classes (A), (B) and (C) of multifunctions: (A) 0{C\ A) = the class of strongly ^-measurable multifunctions with values in C such that E||F||<+oo, where *'E" stands for the expectation. Such multifunctions are said to be integrably bounded', (B) S{C\ A) = the subclass of C^ (C\ A) whose members are strongly ^-measurable simple multifunctions; (C) CliCA) = the subclass of C^CA) of those F that can be approximated by simple multifunctions, i.e., such that one can find a sequence (Fn)n^\ in S(C', A) verifying lim h(F(a)),F„(oj))=0

a.s.

Further, given F,G e C^ {C\ A) we set A(F,G)=Eh{F,G). Similarly to the case of random vectors, it is easy to show that A is 3. metric on C\Cb,A), modulo the /x-almost sure equality. In other words, A satisfies the following three properties, for every F, G, H in C^iCh^A) (i) A(F, G) = 0 if and only if F(a)) = G(CD) a.s. (ii) A{F,G) = A{G.F)\ (iii) Z\(F, G) ^ Z\(F, H) + A{H, G). In addition, if we denote by F' and G' two other multifunctions in C^{C\y,A), the following three properties can be easily deduced from Proposition 2.1. (iv) A{c\{F + F'), cl(G + G')) ^ A(F, G) -h A{F\ G')\ (v) z \ ( a F , a G ) = a z ^ ( F , G ) , a ^ O ; (vi) Z\(coF,coG)^Z\(F,G). Further, it can be proven that £UCb, A) endowed with A is complete. More precisely, if (f^n)n^i is a Cauchy sequence in (£' (Cb, ^4), A), i.e., if it satisfies lim Z\(F,„,F„) = 0,

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then there exists a strongly measurable multifunction F such that lim A(F,Fn) = 0. /?->CXD

Moreover, F is unique up to a modification on a null set, and there exists a subsequence (Fn(k))k^\ of (Fn) such that lim h{F, F,j(^i^)) = 0

a.s.

The following lemma, inspired from Lemma V.2.4 of [157], shows that for every TH-separable subspace C of Cb, every member of £ ' (C, A) can be approximated by a sequence of strongly ^-measurable simple multifunctions. The limit exists almost surely (in fact for every a; G ^ ) in the sense of the quasi-metric A. LEMMA 3.2. Let C be a rw-separable subset ofC\,{X). Then, for every F e C^ (C\ A), there exists at least a sequence (F„) of simple multifunctions in C) (C\ A), satisfying the following three properties (a) limn-^ooh(F{o)), F,,{co)) = 0,Wa) e ^\ (b) \\Fn(co)\\^\\F{w)iycoeQ; (c) \imn^^A(F,Fn)=0; (d) Moreover, when the values of F are convex (respectively, compact, weakly compact), the F,j can be chosen so as to be convex-valued (respectively, compactvalued, weakly compact-valued). It is interesting to mention the following characterization of Cl(Cb,A) in the space

C\C^,,A).

3.3. For every F e C^(Cb,A), the following two statements are equivalent: (i) F is a member of C^ (Cb, A); (ii) There exists a xw-separable subspace C ofC^ such that F(o)) e C a.s.

PROPOSITION

Now, let us explain how to construct the integral of a strongly measurable multifunction F:Q -^ C\ where C is a TH-separable subspace of Cb. For the sake of simplicity, we limit ourselves to the case of a convex valued multifunction, i.e., when C C Ccb- We also assume that C is TH-closed, and stable under the Minkowski addition and multiplication by positive scalar. This is not restrictive, because the closed convex cone generated by a TH-separable subset of Cbc is still Tn-separable. The difficulties encountered in the case of a multifunction whose values are not convex are briefly discussed in Remark 3.8. So, given a multifunction F : ^ ^- C, we define the map 0 : S(C\ A) -> C by 0 ( F ) = E(F),

(3.2)

where E(F) is defined by (3.1). The following result displays the main properties of the map 0 on S(C\ A).

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PROPOSITION 3.4. Let F and G be two strongly measurable simple multifunctions with values in C, i.e., members of S{C\ A), and let a ^ 0. Then, the map 0 defined by (3.2) satisfies the following properties (a) 0(F + G) = c\{0(F) -\- 0(G)}, (b) 0{aF)=a0{F), (c) h(0(F), 0(G)) ^ Eh(F, G), and, in particular,

\\E(F)\\^E{\\F\\), (d) for every y e X, one has s{yMF))

= Es(y^F),

(e) if F C.G a.s., then E(F) c E(G) (monotonicity property). The next result shows that the integral (or expectation) can be defined for every F e C\{C\ A), and that the properties listed in Proposition 3.4 still hold in this more general framework. THEOREM 3.5. The map 0 : S(C\ A) -> C defined by (3.2) can be extended to a map $ from C^ (C\ A) into C. It will be also denoted by 0(F) = E(F), for F e C^ (C, A). This map still enjoys properties (a) to (e) of Proposition 3.4. More precisely we have (a) 0(F -^ G) = c\{0(F) -h 0(G)], (b) 0(aF)=a0(F), (c) h(0(F), 0(G)) ^ Eh(F, G), and, in particular,

||E(F)||^£||F||. Moreover, the map 0 still enjoys properties (d) and (e) of Proposition 3.4. REMARK 3.6. The above approach for defining the set-valued integral is explicit, in that it starts with simple multifunctions and uses only elementary operations such as the Minkowski addition and the scalar multiplication. It is also intrinsic in that it does not involve any auxiliary space.

In spite of these nice features, it is interesting to sketch briefly an alternative approach for constructing the set-valued conditional expectation for a compact convex multifunction. This approach was adopted by Debreu [65]. Although of a more implicit and abstract nature, it is shorter. Indeed, it is based on the possibility of embedding isometrically the metric space (Cbc, h), endowed with the Minkowski addition and the scalar multiplication, in a Banach space. More precisely, let C(B*) denote the set of real-valued continuous functions defined on B*. Further, it is assumed that C(B*) is endowed with the uniform norm || • ||u, i.e., ||u;||u = sup{|u;(>0|: v e ^ * } ,

we C{B*).

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As already mentioned, we assume that Cbc is endowed with the Hausdorff distance /i. Then, the map ^ : Cbc ^ C(B*) defined by ^(C)=^(.,C),

CeCbc,

is an isometry from (Cbc, h) onto (^(Cbc), II • ID- This is an immediate consequence of the Hormander formula. Moreover, ^(Cbc) is a convex cone, and ^ is an isomorphism with respect to the convex cone structure. In other words, it satisfies

^ is called the Radstrom embedding. In addition, we have the implication C c C' ^ ^ ( C ) ^ ^{C),

C, C e Cbc.

If ^ is restricted to a TH-separable subspace C of Cbc then ^(C), as well as c\(^{C) — ^(C)), the Banach subspace of C(B*) generated by ^(C), are separable. Thus, it is possible to define the integral of a random vector f :Q -> C{B*). This is similar to the construction of the Bochner integral for random vectors. Let us recall that this integral is appropriate when one has to deal with functions taking on their values in an infinite dimensions Banach space (see, e.g., [68, p. 441). Then, given an integrably bounded multifunction F : ^ ^ C, the integral of F, can be defined by simply setting E(F) = i/^-»(E(^(F))).

(3.3)

In Debreu's approach, the properties of the set-valued integral are inherited from those of the integral of C(5*)-valued random vectors and from the properties of the map ^. Thus, our approach is a little more general than that of Debreu, because Equation (3.3) is not required to define E(F), but is only a consequence of the definition. REMARK 3.7. When the dimension of X is greater than one, the convex cone ^(Cbc) is infinite dimensional. Indeed, it contains an infinite sequence of linearly independent vectors. For example, assume that X = R- and consider the sequence (C„)„^i of compact convex subsets defined by

Cn = the polygon whose vertices are the points of coordinates {cos2kn/n, sin2kn/n), where /: = 0, 1 , . . . , ^2. It is easily checked that, for every « ^ 1, the support functions 5(-, C i ) , . . . , ^(•, C„), regarded as vectors in C(JB*), are linearly independent. 3.8. Let us say some words about a possible definition of the integral of a multifunction F \Q ^^ Cb(X), whose values are not necessarily convex. For this purpose. REMARK

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we recall that a member A of ^ is called an atom of (^, A, /JL) if P (A) > 0 and if for every B eA such that 5 c A, one has P(B) ==0 or P{A\B) = 0.If A\ and Ai are two distinct atoms, then P(A\ 0 Aj) = 0. It follows that the collection of atoms of a probability space is finite or countable. Consequently, given a probability space (Q, A, /JL), the set i? can be split into the purely atomic part ^pa and the nonatomic part ^na (see, e.g., [118, p. 45]). The purely atomic part ^pa expresses as a finite or countable union of atoms Ak, namely

If the multifunction F: ^pa -^ ChiX) is defined by

k^\

where Ck eCh (k^ 1), and if F is integrably bounded, i.e., f

||F||J/X = ^ / X ( A A - ) | | Q | | < + O O ,

then it is natural to define the integral of F over .^pa by E(lr2pa/^) = ^ M ( A ^ ) Q ,

(3.4)

k^\

where the series converges in the Hausdorff metric. It was shown in [110] that, for any F e Cl (Cb, v4), the integral can be defined by E(F) = cl{E(l^p,F) + E ( l ^ „ , c o F J / i ) } , where Eil^^.^F) is defined by (3.4) and E(l^^^coF) is defined as in Theorem 3.5. When X is finite dimensional, the closure operation is no longer necessary. REMARK 3.9. (i) Using the approach displayed in this section, several other properties of the set-valued integral could be proved. For example, it would not be difficult to prove set-valued analogs of the Dominated Convergence Theorem. On the other hand, the same construction also allows for defining the conditional expectation of strongly measurable multifunctions (see [110]). However, a other more general approach will be given in Section 6. (ii) Other approaches based on the Riemann integral were adopted by Hukuhara [122, 123], by McShane [149], and by Artstein and Bumes [8]. In Section 5, we shall present the Aumann integral, whose definition involves selections and which even makes sense for non integrably bounded multifunctions.

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4. Weakly measurable multifunctions and graph-measurable multifunctions This measurability concept is the most largely used, because it is not too strong, whence easy to check. It allows for a lot of stability properties and for general results of existence of measurable selections, especially in connection with the graph measurability that we shall also present. On 2^, we consider the Effros a-field S generated by the subsets U~ defined by

where U ranges over Q, the set of open subsets of X. Clearly, the restriction of S to X, considered as a subset of 2^, coincides with B{X). The restriction of the Effros cr-field to some subspace S of 2^ will be denoted by £{S). We continue to denote by {Q, A) an abstract measurable space. As already mentioned, a map F from Q into 2^ is also called a multifunction. The domain and the graph of F are, respectively, defined by (Xom{F) = [cDeQ\ F{(jo)^9i]

and

GV{F) = [((JO.X)

e Q x X:

xeF{a))].

F is said to be Ejfros measurable or simply measurable, if for every W in £, F ~ ' (W) is a member of ^ ("weakly measurable" in the terminology of Himmelberg [120]). From the definition of the Effros a-field, it follows that F is measurable if and only if, for any open subset UofX, the subset F-^{U')

= [coeQ\ F ( a ; ) n ^ 7 ^ 0 } ,

is a member of A. The sub-a-field F~^{£) generated by F is denoted by AFClearly, a multifunction F : ^ ^- 2^ is measurable if and only if the multifunction cl(F) is measurable. Further, if F is measurable then its domain is a member of A. This is a consequence of the equality dom(F) = F - ' ( X - ) . A strongly measurable multifunction is measurable, which justifies the terminology. This follows from the relationship {F-\u-)Y

= F-'[{CeC:

C
valid for all open sets U and from the iH-closedness of the set {C G C: C c L^^}, which shows that F~^{U~) is a member of B(Cb, TH) (in the above equahty, the superscript "c" stands for the complement operation). Conversely, if F is a weakly measurable multifunction taking on its values in a iH-separable subspace C of Cb(X), then it is also B{C\ TH)-measurable (see Remark 4.9(ii)). On the other hand, there are measurable multifunctions which are not strongly measurable (see, e.g.. Example 3.4 in [117] or [23]).

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The set of all measurable multifunctions with values in X is denoted by M(A, 2^) or simply Mi2^). The set of closed-valued measurable multifunctions is denoted by M{C{X)). A multifunction F :Q ^^ 2^ is said to be graph-measurable if Gr(F) is a member of the product a-field A 0 B{X). The connections between Effros measurability and graph measurability are given by Theorem 4.1 below. The proofs can be found in [41]. First, the following notion from measure theory is needed. The a-field of universally measurable subsets, associated with this measurable space, is denoted by A and is defined as the intersection of all a-fields A^i (the //-completion of A), where // ranges over the set of all positive bounded (or probability) measures on(Q,A). Consequently, one has the inclusion Ac. A and, when the a-field A is //-complete with respect to some fixed positive bounded measure /x, one has the equality

For example, analytic subsets and coanalytic subsets of a complete separable metric space are universally measurable (see, e.g., [67]). THEOREM 4.1.

(a) If a closed valued multifunction F is Effros measurable, then it is graph-measurable. (b) Conversely, every graph-measurable multifunction F:Q ^^ 2^ is Effros measurable with respect to A, the a-field of universally measurable subsets of{Q,A). (c) Consequently, if F is defined on a complete probability space {Q,A,ii), a closed valued multifunction F is Effros measurable if and only if it is graph-measurable. PROOF. Statement (a) is easily proved by relying on the equality Gx{F) = [{a),x)eQ

X X: d{x, F{co)) =0],

and by noting that the function (a;, A) -^ d{x, F{a))) is ^ 0 5(X)-measurable. The proof of statement (b) is a consequence of the Projection Theorem (see the proof of Theorem 4.6). D From Theorem 4.1(c), one can deduce a lot of stability properties for measurable multifunctions. For example, we mention the following result concerning countable unions and intersections. 4.2. Assume that {Q,A, P) is complete. If {Fn)n^\ is a sequence of measurable multifunction, then the multifunctions F and G defined by

THEOREM

F=|JF„ are measurable.

and G=p|F,;,

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A crucial notion for working with measurable multifunctions is that of selection and of Castaing representation. DEFINITION 4.3. Let F : ^ ^ 2^ be a multifunction.

(i) A selection of F is a map / from Q into X such that f{oj) €

F((JL>)

for every

CO G d o m ( F ) .

(ii) A Castaing representation is a sequence (/,/) of measurable selections, such that for every o) e dom(F), F{co) is equal to the closure of the countable subset {fnico): n^ 1} (see, e.g., [41, Chapter III] or [174]). The following theorem provides two existence results of a measurable selection. It does not involve the linear structure of X. The proof of part (a) can be found in, e.g., [41,124, 136,174,208,209]; that of part (b) is contained in [41, Chapter III]. THEOREM 4.4.

(a) IfF e M (A, C(X)) and if domiF) is nonempty, then F admits at least one A-measurable selection. (b) If F is a graph-measurable multifunction with values in 2^ and //Gr(F) is nonempty, then F admits at least one A-measurable selection.

In the above result, it can be noted that, when F is closed valued, it has at least one ^-measurable selection. When no particular hypothesis is made on the values of F, one can only assert the existence of an ^-measurable selection. As already mentioned, when {Q,A) is endowed with a positive finite measure //, this entails the existence of one ^^-measurable selection. As the next result shows, the measurability of a closed-valued multifunction can be expressed in terms of distance functions and is connected with the notion of Castaing representation. 4.5. If F is a closed valued multifunction, the following three statements are equivalent (a) F is Effros measurable, i.e., F~^{U~) e A for all open subsets U ofX. (b) For every x e X, the function d(x, F(-)) is A-measurable. (c) F admits a Castaing representation.

THEOREM

Equivalence (a) ^ (b) is valid in a more general setting. Indeed, a multifunction F whose values lie in a separable metric space is weakly measurable if and only if, for every x e X, the positive function d{x, F ( ) ) is measurable. This is a consequence of the following equality F-\Bix,oi))

= {(oeQ: d{x,Fi(jo)) < « } ,

valid for all jc e X and a > 0 (where B{x, a) denotes the open ball of radius a centered atjc).

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THEOREM 4.6. If F is a closed valued multifunction defined on a complete probability space {Q,A,ii), then each of the following statements is equivalent to anyone of Theorem 4.5. (d) F is graph measurable. (e) F'BeAforall BeB(X). (f) F-CeAforallCeC(X). PROOF. We only sketch the main implications. Implication (d) => (e) is an immediate consequence of the Projection Theorem (Theorem III.23 in [41]). Indeed, for every B e B(X), the subset

F-B=pro}^{Gr{F)n(Q

x B)),

is a member of ^ = ^ . As to the implication (b) =^ (d), simply observe that, for any countable dense subset D of X, the following equality holds Gr(F) =f]{coe^:

d{x, F(aj)) = O}.

XGD

Implication (f) =^ (a) follows from the fact that, in a metric space, every open subset U is the countable union of closed sets, namely

U=\JCn, which yields

REMARK 4.7. Consider the property (g) below. (g) F-K eAforaWK elCiX). When X is a metric space the implications (f) => (a) =^ (g) hold. The three properties are equivalent when X is a-compact (i.e., a countable union of compact sets), especially when X is a finite dimensional Banach space (see, e.g., [120] or [41]). Further results on the measurability of multifunctions and counter-examples can be found in [121]. One of the advantages of considering the Effros a-field in order to define the measurability of a multifunction is that this a-field is equal to the Borel a-field of a separable metrizable topology, namely the Wijsman topology. This topology was introduced by Wijsman [213,214] on the space of closed convex subsets in an Euclidean space, but it also makes sense in a general metric space (X, d). In this setting, the Wijsman topology is defined on C(X) as the weakest topology rw(j) determined by the family {d(x, •): X € X). Equivalently, rw(j) is the topology of pointwise convergence of distance functions on X.

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For every C e C(X), the distance function d(,C) is Lipschitz continuous (with Lipschitz constant one). It follows that, when the metric space iX,d) is separable, the Wijsman topology is metrizable and separable (see [101]). Further, Beer [26] proved that when (X, d) is Polish^ (C(X), TW(^)) is Polish too. The following result displays the Borel structure of the Effros cr-field. The proof can be found in [101, Proposition 3.1.1] or [27, Theorem 6.5.14]). THEOREM

4.8. If{X, d) is a separable metric space, then, one has

f(C(X))=S(C(X),rwu/)). REMARK 4.9.

(i) The equality of Theorem 4.8 is no longer true when the Wijsman topology is replaced with the Hausdorff metric topology TH. In general, the Effros a-field is strictly included in S(C(X), TH) (see [23]). (ii) However, when we restrict to the space /C(X) of compact subsets of X, we have the equality i:(/C(X)) = s(/C(X),TH).

The proof can be found, e.g., in [27,41,177]. More generally, according to a remark in the beginning of the present section, if C is a iH-separable subspace of C(X), then one has f(C')=S(C\rw(./)) = S(C\TH). This follows from the second countability of the restriction of TH to C and from the fact that the /i-closed balls are members of the Effros a-field. In turn, the latter is a consequence of the equality /z(C, C) = sup\d(x, C) - d{jc, C')\, XGD

valid for every pair (C, C) e C(X)^, where D denotes a countable dense subset of X. There are a lot of specific situations where one wants to find a measurable selection. For example, a result which can be useful in many situations, for example in Control Theory, is the so-called Measurable Implicit Function Theorem (Theorem III.38 in [41]). THEOREM 4.10. We consider another measurable space (T,T), a Suslin topological space Y (i.e., the continuous image of a Polish space), a multifunction F: Q -^ 2^ such that Gr(F) eA BiY), a multifunction G\Q ->2^ such that Gr(G) eA^T and a map g:Q xY -^ T which is measurable with respect to A B(Y) and T. If it is assumed that

g{(jo, F(a))) n G((o) 7^ 0,

V(w G ^ ,

^ A topological space is said to be Polish if it is separable, metrizable and complete for a suitable metric.

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then there exists a map f \Q -> T, whose graph is A 0 B(Y)-measurable, which is a selection of F and which satisfies g{aj,f(a)))eGioj),

"icoeQ.

REMARK 4.11. (i) Another useful notion for closed convex valued multifunctions is that of scalar measurability. A multifunction F : ^ ^^ Cc is said to be scalarly measurable if for every >' G X*, the function s(}\ F ( ) ) is measurable. It is not hard to show that a measurable multifunction is scalarly measurable. Indeed, for every v G X* and every real of, one has

[CeCc'. s{yX)>ot]

= Vy",

where W — {x e X\ {y,x) > a). The converse implication does not hold in general, but it holds for the class of countably supported multifunctions (see Definition 5.5 below) satisfying a suitable analyticity property. The reader may consult [24] for further details and [23] for the connections with strong measurability. (ii) Let us also mention the works of Leese [141,142] who presented another approach of measurable multifunction, involving the Suslin operation and Suslin spaces. The case of multifunctions whose values lie in a nonseparable Banach space was considered by Barcenas and Urbina [25]. 5. The Aumann integral Consider a probability space {Q.A. fi) and a sub-a-field B of A. By L^\Q, B, II\ X) we denote the space of all (classes of) measurable functions from {Q. B) into (X, B(X)). For every F € M{C{X)), we also define S{F, B) = {f e L^(Q, B, /x: X): f((o) e F{co). for/x-almost every (i; e dom(F)}. It is known that L^(Q, A, //; X) (= L^(X)) endowed with the topology of convergence in probability is a metrizable topological vector space, provided one identify two functions that coincide /x-almost surely. Since a sequence converging in probability admits an almost surely converging subsequence, it is clear that, for any sub-a-field B of A, the set S(F, B) is closed in L^(Q,B, ^l; X). We denote by L ' ( ^ , ^ , M ; ^ ) (or ^^(^) for short) the subspace of L^(X), whose members are Bochner integrable. Given a sub-a-field B of A and a multifunction F, we define the following L^ (X)-closed subset of L^ {Q,B, p; X) 5' (F, B) = {f e L\Q.

B. p\ X): f(aj) e F(co), for /x-almost every co e dom(F)}.

We begin by recalling a result on the measurability of infimums (respectively, supremums).

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PROPOSITION 5.1. Let F e M(C{X)) andcp.Q x X-^ R be an A® B(X)-measurable function. If, for every co e Q, 0(a;, •) is upper semicontinuous, then the function nifp defined by

co-> nifpico) — inf{0(a;,x): x G F(a;)}, is measurable. On the other hand, if(^{cx). •) is lower semicontinuous, m^ is A-measurable. A similar result can be stated for the supremum by replacing 0 with —0. I f 0 : ^ x X - > M i s ^ ( 8 ) S(X)-measurable then we can define the integral functional y0onL'(X)by Mf)=

f (t>{co,fico))dfi.

feC\X).

The following result allows for interchanging the infimum and the integration operations, and plays a crucial role in set-valued integration. Its proof can be found in [117, Theorem 2.2]. PROPOSITION

5.2. Let(l):QxX^Rbeasin

inf{y^(/): feS\F)]=

[

Proposition 5.1. Then one has m^ico)dfi.

JQ

Applying Theorem 5.2 to the function {co.x) -^ 0(->/2(F((w)) = sup{||jc||: x e

F((D)]

(also denoted by | | F M | | )

is integrable. In this case, every measurable selection of F is integrable, so that one has 5^F,^) = 5(F,^). Further, the values of F are almost surely bounded.

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The following notion of integral for multifunctions was introduced by Aumann [17]. For any measurable multifunction F and any sub-a-field B of A, the set-valued {Aumann) integral of F over Q, with respect to B, is denoted by / ( F , B) and is defined by

\L

I{F^B) = \ / fd^l:

feS'(F.B)

I{F,A) is simply denoted by / ( F ) ; it is nonempty if and only if F is integrable. The integral of F over AeA'is denoted by /^ (F). On the other hand, F is said to be Aumann-Pettis integrable if it admits at least a Pettis integrable selection. For the definition and elementary properties of the Pettis integral, the reader may consult [68, p. 52] and [155,156]. Let us only recall that in finite dimensional spaces the Pettis integral coincide with the usual integral, whereas in infinite dimensions there exist Pettis integrable functions that are not Bochner integrable. EXAMPLE 5.3. An integrably bounded multifunction is integrable, whence AumannPettis integrable, but the converse implication is false. Indeed, consider the measurable multifunction F defined by

F{a)) = B(0, r{a))) = the closed ball of radius r{a)) centered at the origin, where r: Q -^ (0, -hoc) is a given nonintegrable measurable function. In this case, the Aumann integral (see below) of F over ^ is equal to X. Here are some properties of the Aumann integral, whose proofs can be found in [17]. 5.4. Let F,G e M{C{X)). If we assume in addition that F and G are integrable, then the following properties hold. (a) /(cl(F + G)) = cl(/(F) + /(G)). (b) c l / ( c o F ) = c o / ( F ) . (c) If(Q,Ay fi) is nonatomic, then c\I{F) is convex and one has

THEOREM

cl/(coF)=cl/(F). (d) If(!^,A,fji)

is nonatomic and if X is finite dimensional then I{F) is convex and I(F) = I{coF).

Moreover, if F is integrably bounded, / ( F ) is compact. (e) For all y e X* one has

,{y,I(F))=

f

JQ

s(y^F)dp.

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(f) If F and G are integrably bounded then the following inequality holds

/i(cl/(F),cl/(G))^ I

JQ

h(F,G)d^.

(g) If F is integrably bounded and takes on its values in a xw-separable subspace of Cbc(^), then one has E(F) = c\I{F) where E(F) denotes the integral as defined in Section 3 (Theorem 3.5). The first part of property (c) is a consequence of the Lyapunov Convexity Theorem (see [117, Theorem 4.2]). The first part of statement (d) was proved by Richter [171]. The proof of statement (g) can be found in [ 117]. The others properties are due to Aumann [17]. Other properties related to statement (b) will be given in Theorem 5.9. Theorem 5.6 below concerns the class of countably supported multifunctions, whose definition is recalled hereafter. 5.5. A multifunction F:Q -^ CdX) is said to be countably supported if one can find a countable subset D of B* such that DEHNITION

Fico)=:f]{xeX:

{y,x) ^s{y,

F(a)))},

foralla;Gf2.

xeD

If we want to emphasize the role of D, we say that F is D-countably supported. For example, if T is a topology on X, compatible with the duality pairing, every multifunction whose values are weakly compact and convex is D-countably supported for every Tdense subset of B*. However, the class of countably supported multifunctions is much wider (see [24]). In particular, a countably supported multifunction may have unbounded values. On the other hand, every member F of £)^(Cc,^) is countably supported with respect to a countable subset of X*, generally depending on F (Theorem 3.6 in [23] and Proposition 2.4(g) in [24]). The first part of the following theorem characterizes integrable selections of a countably supported multifunction, the second part provides a criterion for two such multifunctions to be almost surely equal. Theorem 5.6 remains valid for Aumann-Pettis integrable multifunctions (see [78]). THEOREM 5.6. Let F be a measurable, integrable, closed convex valued multifunction and f eL\X). (a) If F is countably supported then f e S^(F) if and only if

/ fdfiecllAiF). JA

for all A e A.

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(b) Let G be another integrable, closed convex valued multifunction. If G is countably supported then F(co) = Gico) for almost every' co e Q if and only if c\lA(F) = c\lAiG),

for all A e A.

The set-valued Aumann integral is not always closed. Thus, it is interesting to indicate situations where it is. This problem has been considered by several authors, such as Castaing [33,37], Hiai and Umegaki [117], Byrne [31] and Klei [128] among others. 5.7. If F e M(C{X)) is integrably bounded and convex weakly compact valued then I{F) is weakly compact.

THEOREM

PROOF. Clearly, the set S^ (F) of integrable selections of F is L' (X)-closed and convex. By Theorem 3.6(ii) of [129], S^ (F) is also weakly relatively compact, whence weakly compact. Further, since the map xj/ : f -^ f^ f d/j. from L ' ( X ) into X is linear and strongly continuous, it is also continuous with respect to the weak topologies. The desired conclusion is obtained by noting that I(F) = il/iS^F)). D

Now, we turn to an interesting question: given a subset 5 of L' (X), is it possible to find a necessary and sufficient condition for the existence of an integrable multifunction F such that 5 = 5'(F)?The answer is provided by Theorem 5.8 below, which is Theorem 3.1 of [117]. Recall that a subset 5 c L' (X) is said to be decomposable (with respect to A)y if for every A eA and every fgeS. 1A f -h IA^S is a member of S. THEOREM 5.8. Let S be a nonempty' closed subset of L^iX). Then, there exists an integrable multifunction F e MiCiX)) such that S = S^(F) if and only if S is decomposable.

The following result addresses the problem of interchanging the set-valued integration and the convex hull in finite dimensions. 5.9. Assume that X = W^ and consider a multifunction F, whose values lies in W^, the positive cone ofX. IfGr(F) eA
THEOREM

J I Fdp= CO F dp= f I coF dfi. REMARK 5.10. (i) The equality of Theorem 5.9 was proved by Aumann [17] and was extended later by Wagner [210]. Another proof based on the theory of Young measures was given by Balder [20]. On the other hand, the case of multifunctions with a countable range was examined by Khan and Sun [126]. (ii) It is interesting to mention another area of research concerning the integration of both, single and set-valued functions, namely integration of random variables taking on their values in a metric space with no linear structure. This can be traced back to Frechet [84] and was taken further by S. Doss [72,73], Benes [28], Herer [90-93],

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H. Doss [71], Raynaud de Fitte [170], Es-Sahib and H. Heinich [79], and Hess [112]. In some of these works, one can find extensions of the strong law of large numbers and of martingales convergence theorems. (iii) Let us mention a result that has applications in Control Theory and Optimization. Given a measurable multifunction G defined on a product space ^ x T, the integral over ^ is a multifunction depending on the second variable, namely

Q

G((L>, t)

/i(da)).

In this situation, one may look for a measurable selection x// of the multifunction ^, which has the following form \lf:t-^

\ g{(jo,t)fx{da)),

JQ

where g is a suitable measurable selection of G. This question was raised and solved by Artstein [6] when X is finite dimensional. An infinite dimensional extension was established recently by Saint-Pierre and Sajid [178]. (iv) A notion of variance for random sets was introduced by Kruse [134] by means of selections. 6. The set-valued conditional expectation of closed valued multifunctions The construction of the set-valued expectation is a natural extension of the multivalued integral, but it is more delicate. It was examined by several authors. Let us mention Van Cutsem [203-205], Neveu [158], Bismut [30], Dynkin and Evstigneev [77], Castaing and Valadier [41], Hiai' and Umegaki [117], Hiai [114] and Papageorgiou [163]. Given a subcr-field B of A, and a integrable >l-measurable multifunction F, Hiai and Umegaki [117] showed the existence of a S-measurable integrable-^ multifunction G such that 5'(G,i3) = cl{E(/|B): f

eS\F,A)\,

the closure being taken in L^ (^; X). The multifunction G is the (set-valued) conditional expectation of F relatively to B and is denoted by E{F\B). Clearly, it is defined up to a S-null set. The existence of G is an immediate consequence of Theorem 5.8. Indeed, it is easy to check that the set

[E{f\By.

feS\F.A)],

is decomposable with respect to the sub-a-field B and that the L'(X)-closure of a decomposable set is still decomposable. '^In fact, Hiai" and Umegaki assumed that F is integrably bounded, but their proofs easily extend to the integrable case (see [114]).

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One may observe that the above definition is global and only defines E{F\B) by mean of the set of its ^-measurable integrable selections. Thus, it is desirable to provide examples where a 'pointwise' characterization is available, i.e., a formula allowing one to calculate E{F\B){(D) for almost every CD e ^. Several examples of that type are presented hereafter. The more general ones are examples (e) and (f). EXAMPLE 6.1.

(a) If the multifunction F reduces to a single-valued function / : X2 ^- X, the conditional expectation of F clearly coincides with the ordinary conditional expectation for vector-valued random variables. (b) If iB = {^, 0}, then for any integrable multifunction F we have E(F|S)((w) = c l / ( F )

a.s.

(c) Let r be a real positive integrable function and / : ^ ^ X be a vector-valued integrable function. If the multifunction F is defined by

then its conditional expectation with respect to B is given by E ( F | S ) M = B{E(f\B){(D). E{r\B){cD)). This can be seen by using Theorem 6.2(c) below. (d) Let i; be a real-valued integrable function and y eX. If we define the multifunction Fby F(a>) = {jceX: {y,x) = v{a))\, then E{F\B){co) =[xeX'.

(>',x) = E{v\B)(a))}.

(e) Assume that X* is strongly separable and that F is an >4-measurable integrable multifunction whose values lie in Cbc(X). Then, the following equahty holds (see [117, Theorem 5.5]) E(F\B)((v)=

f]{xeX:

{y,x) ^E{s{y,

F)\B)(a))].

xeD

A similar equahty holds when F takes on its values in /Cwc(X) and when D is a countable Mackey dense"^ subset D of X*. It is proved by replacing the strong topology on X* by the Mackey topology. "^Such a countable subset exists in X* as soon as X is separable. For the definition and the basic properties of the Mackey topology, see, e.g., [184, p. 131].

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(f) Assume now that there exists a regular conditional probability distribution of P given S, i.e., a map Q:B x ^ -^ [0,\] satisfying (i) for every 5 G S, the map co^^ Q {B\a>) is ^-measurable, (ii) for every ct; G ^ , the map B ^ Q {B\(o) isdi probability measure on ( ^ , S), (iii) VA G A V5 G e, Q(A n B) = f^ QiB\aj) ixidco). In this case, it was shown by Valadier [201, Theoreme 1] that the set-valued conditional expectation is given by

-"II

E(F\B){co) = c\\\

f{^)Q{dH\co): f^S\F.A)\

a.s.

We end this section by recalling some basic properties of the set-valued conditional expectation, which are natural extensions of those of real-valued or vector-valued random variables. The proofs can be found in [117] for the integrably bounded case and in [114] for the integrable case. As the reader will see, the classical properties of the conditional expectation are recovered only for convex valued multifunctions. THEOREM 6.2. IfF and G are two integrable multifunctions with closed values in X, and B is a sub-a-field of A, then the following properties holds. (a) E(cl(F + G)\B) = c\(E(F\B) -f E{G\B)) a.s.

(b) Ifr is a real B-measurable function such that rF is integrable, then E(rF\B) = rE{F\B)

a.s.

(c) Ifg is a bounded scalarly B-measurable^ function from Q into X* then s{gMF\B))

= E{s{g^F)\B)

a.s.

where s{',C) denotes the support function of the subset C {see Section 2). (d) E{c6F\B)=c6E{F\B)a.s. (e) If F and G are integrably bounded, then A{^{F\B)MG\B))^A(F.G). In other words, the map F -> E(F\B) is nonexpansive from C^(Ch,A) onto C'iC^^B). (f) If F^G a.s., then E{F\B) ^E{G\B)

a.s. {monotonicityproperty).

In the following theorem, the values of the multifunction F are required to be convex. ^That is, for every x eX, the function o) —>• {g(co),.x) is /3-measurable.

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6.3. Let F be an A-measurable integrable multifunction, with values in CQ. (a) IfBi c 6 c A then E(E(F\B)\Bi) = E(F|Bi). (b) If F is B-measurable and if r is an A-measurable positive bounded function such that r F is integrable, then

THEOREM

E{rF\B) = E{r\B)F

a.s.

In particular, E{F\B) = F. (c) If F is B-measurable and if Bi C. B ^ A, then the conditional expectation of F relative to B\ taken on the base space iQ,A,/J.) is equal to the conditional expectation of F relative to B\ taken on (Q.B, p). REMARK 6.4. (i) The assumption of r being positive cannot be removed in Theorem 6.3(b) as the following example shows. Consider the measure space ([0, l),A,p), where p is the Lebesgue measure, B the trivial a-field {.^,0} and F the constant multifunction defined by Fioj) = [—1, 1] (here X = R). Also consider the function r such that

r(a)) = -l

if0^co<\/2

and

r(aj) = \,

if 1 / 2 ^ 0 x 1 .

Then E(rF\B) = F, but E(r\B)F = {0}. (ii) If F is not assumed to be convex valued, properties (b) and (c) of Theorem 6.3 are no longer true. Indeed, assume that X = R and that ( ^ , A. ix) is nonatomic. Let B ~ {12, 0} and F be the constant multifunction such that F(co) = {0, 1}, 66) € Q. Theorem 5.4(c) shows that E(F|i3) = c l / ( F ) = [0. 1], so that the equality E(F\B) = F does not hold. In view of the next result, we recall that for every G e Mi2^) of A, we have set (see Section 5)

-\L

lBiG,J^) = { I fdti: feS'{G.T)\.

and every sub-a-field JF

BeT.

6.5. Let F be an integrable multifunction with closed values. (a) For every B e B one has

THEOREM

cllB{EiF\B).B)=c\lB(F.A). (b) If F is convex valued, then for every- B e B one has C\IB{E{F\B),A)=CIIB(F,A).

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(c) If F is a member of C\{Ci:,A) {see Proposition 3.3), then E ( F | ^ ) is uniquely determined as a member of Cj (C, B) satisfying the equality in {b)for every B eB. Here is an extension of Theorem 5.5 of [117] and of our Example 6.1(e). The proof is similar, but is based on our Theorem 5.4. THEOREM 6.6. Let D be a countable subset of B* and F .Q -^ Cc(X) be a D-countably supported, integrably bounded multifunction. Then one has, for almost all co e Q,

E{F\B){aj) = P I {x € X: (y, x) ^ E[s{y. F{co))\B)\. VGD

REMARK 6.7. The multivalued conditional expectation is closely connected with the conditional expectation of random variables depending on a parameter. This problem is of great importance in many applications. See, e.g., [41,43,81,201,202,204].

7. Set-valued measures The theory of set-valued measures (we shall rather use the synonym "multimeasures"), has been developed by Vind [206], Schmeidler [186], Debreu and Schmeidler [66], Artstein [2], Godet-Thobie [86-88], Coste [48-50], Pallu de la Barriere [162], Drewnowski [74], Coste and Pallu de la Barriere [52-55], Hiai [113,114], Thiam [193-196], Papageorgiou [164] and Kle'i [127] among others. One of the motivations was the applications to Mathematical Economics or to Statistics. Basically, multimeasures are measures whose values are subsets of some Banach space (or, more generally, topological vector space). As we shall see, there are at least three possible definitions depending on the series summability concept considered in the space of closed sets. We first recall that a sequence (x,;)„^i in X is said to be unconditionally convergent if for every one to one map 0 from N onto itself the series J]/z^i -^0(") ^^ convergent. An analogous definition can be given for a sequence (C,;) in Cbc(^) endowed with the Hausdorff distance. Given a sequence (C„)„^i in 2^, the infinite sum X]/;>i ^n is defined by ^C„ =

X € X: V . = ^ . v „ , .v„ e C„. n^\\.

(7.1)

where the convergence of the series X^;;>i -^/? is unconditional. Further, recall that a map Af: ^ -> 2^ \ {0} is said to be additive if for every pair (A,B) of disjoint members of A one has M{A UB) = c\{M(A) + M(B)),

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where the " + " sign stands for the Minkowski addition. When the values of M are weakly compact, the closure operation is no longer necessary. This definition can be extended to a pairwise disjoint family of members of A. The first definition of a multimeasure uses the Hausdorff distance on the space of closed sets. 7.1. An additive map M.A-^ C{X) is called an h-multimeasure if M(0) = {0}, and if for every sequence (A„)„^| of pairwise disjoint sets in A, we have

DEFINITION

;|m^AU (J^,, .cl^M(A,)Uo. \

\/2^1

/

/=1

/

The second definition refers to the summability notion introduced by equality (7.1). DEFINITION 7.2. An additive map M\A-^ 2^ \ {0} is called a strong multimeasure if M(0) = {0}, and for every sequence (A,,),,^ i in A of pairwise disjoint sets, we have

MJ | J A J = C 1 ^ M ( A , ) . The third definition involves support functions. DEFINITION 7.3. An additive map M: >4 -^ 2^ \ {0} is called a weak multimeasure if for every y e Z*, the map A -^ s{y, M{A)) is an extended real-valued measure.

Clearly, if M : ^ -^ 2^ is a weak multimeasure, then so are the maps M' — clM and M" = coM. The same property is valid for /z-multimeasures (for M'\ use Proposition 2.1(c)). The main connections between the above three definitions are provided in the next proposition. PROPOSITION 7.4.

(a) If M :A^^ Cb(X) is an h-multimeasure, then it is also a strong multimeasure. (b) IfM'.A-^ C\y(X) is a strong multimeasure, then it is also a weak multimeasure. (c) IfM: A -^ /Cwc(^) is a weak multimeasure, then it is also an h-multimeasure. Thus, in this case the three notions of multimeasure coincide. Proposition 7.4(a) follows easily from the following result which provides another formulation of the countable summability for /7-multimeasures. It is due to Drewnowski [74]. PROPOSITION 7.5. If(Cn)n^\ is a sequence in CbiX), then the following two statements are equivalent. (i) The series (€„) is unconditionally convergent in Ch(X) endowed with the Hausdorff distance.

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645

(ii) For every sequence {Xn)n^\ in X such that jc„ € d for all n ^ \, the series (Xn) is unconditionally convergent. Moreover, the following equality^ holds c l ^ C = c l | ^ x „ : Xn eCn\. Let M:A-^

2^ \ {0} be a strong multimeasure. For every A G ^ we define VM(A) by

VM(A) = s u p n ] | | M ( A / ) | | L where the supremum is taken over all finite ^-measurable partitions {A\,..., A,j} of A. From Proposition 1.1 of [113], it is known that VM is a positive measure. If VA/(i2) < +oo, then M is said to be of bounded variation. 7.6. Let us give an example of strong multimeasure. For this purpose, consider Q e Chc(X) and the collection S of vector measures m-.A^-X such that for all m € 5 and A G A onehasm(A) e iji(A)Q. Now, define the map M :>t ^ 2^ \ {0} by

EXAMPLE

f "

1

M(A) = I ^ m / ( A / ) : m, e 5 , A, € ^ (pairwise disjoint), « ^ 1 [. It is not difficult to check that M is additive and that it is a strong multimeasure. EXAMPLE 1.1. A simple example of /i-multimeasure can be given in the following way. Let F e M(Ch(X)). Assume that F is integrably bounded and define M:A-> Ch(X) by M ( A ) = C1/A(F),

AeA,

where IA(F) denotes the Aumann integral of F over A (see Section 5). EXAMPLE 7.8. If in Example 7.7 the multifunction F is no longer assumed to be integrably bounded, but only Aumann-Pettis integrable (i.e., the set of its Pettis integrable selections is nonempty), then the map M is a weak multimeasure, but not necessarily an /z-multimeasure. Indeed, as Example 5.3 shows, there exist Aumann-Pettis integrable multifunctions that are not integrably bounded.

Further, M is said to be fi-continuous if /i(A) = 0 implies M{A) — {0}. Given a weak multimeasure M -.A-^ 2^ \ {0}, a selection of M is a vector measure m:A-^X such that m{A) e M{A) for every A e A. The set of all selections of M is denoted by S^. The multimeasure M is said to be rich (of selections) if M(A) = c\{m{A): me^A/},

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for all A e A.\n this case, the multimeasure can be recovered from its selections. The following result is due to Coste [48]. 7.9. Let M \ A-^ Ch{X) be a strong multimeasure. (a) If A is countably generated and if M is ji-continuous, then M is rich. (b) IfX is separable and if M takes on its values in Cbc(^). fhen M is rich. (c) IfM takes on its values in /C^ (X), then for every A e A, one has

THEOREM

M(A)={m{A):

m e 5A/}.

A member A of ^ is called an atom of the multimeasure M :A-> 2^ \ [^} if M(A) 7^ {0} and if either M(B) = {0} or M{A \B) = {0} holds for every B'ZA^BeA.A measure space having no atom is said to be nonatomic. The Banach space X is said to have the Radon-Nikodym property (RNP) if for each finite measure space ( ^ , .4, /x) and each vector measure m.A-^X which is of bounded variation and /x-continuous, there exists f e 0(Q .A. ti'.X) such that m{A) = j f dix, JA for all A e A. For example, it is known that reflexive Banach spaces and separable dual spaces have the RNP (see [68] or [69]). The following result concerns the convexity of the values of a multimeasure. The first part is due to Hiai [113], while the second part can be found in [2] or [49]. THEOREM 7.10. Assume that X has the RNP. Let M :A-> 2^ \{0} be a nonatomic strong multimeasure of bounded variation. Then for every A € A c\M(A) is convex. Moreover, the set

cl

IJM(A) iAeA

\

IS convex. EXAMPLE

7.11. Let w : ^ ^ X be a vector measure. Define M: ^ ^ 2^ \ {0} by

M(A) = [miB): BeA,

B^A}.

A e A.

Then M is a strong multimeasure, and M is nonatomic if and only if m is so. Further, VM(A) is equal to the variation of m and MiQ) is equal to the range of m. Now, we turn to versions of the Radon-Nikodym theorem for multimeasures.

Set-valued integration and set-valued probability theory: An oveniew

7.12. Let M : ^ ^ 2^ \{0} be a multimeasure. F eM{2^) set-valued Radon-Nikodym derivative of M with respect to fi if

DEHNITION

M ( A ) = / FJ/x, JA

641

is said to be a

A€ A

and a generalized Radon-Nikodym derivative of M with respect to // if

clM(A) = cl / Fd^l,

A^A.

JA

Let us give a version of the Radon-Nikodym Theorem concerning multimeasures, whose values are weakly compact. It is due to Klei [127,129]. It provides a necessary and sufficient condition. Earher versions of this result had been proved by several authors (see, e.g., [2,50,54,66,86-88,164,206]). THEOREM 7.13. Let X be a Banach space. Then, the following two statements are equivalent. (a) X has the Radon-Nikodym property. (b) For every probability space {Q, A, p.), for every multimeasure M :A-^ /Cw(^) of bounded variation, there exists an integrably bounded Radon-Nikodym derivative F of M with weakly compact values. REMARK 7.14. Let us also mention the works of Thiam [193-196] who studied the Daniell integral for functions with values in an ordered semi-group and gave applications to the integration of real-valued functions with respect to a multimeasure. On the other hand, Puri and Ralescu [166] proved a version of the strong law of large numbers for vector-valued random variables with respect to a multimeasure.

8. The probability distribution of a measurable multifunction This section is devoted to the presentation of fundamental properties of the distribution of measurable multifunctions. This domain was explored by several authors for various purposes (applications to mathematical economics, stochastic optimization, etc.). Let us mention, for example, the works of Hart and Kohlberg [89], Hildenbrand [118], Artstein [3], Gine et al. [85], Artstein and Hart [10], Hess [96,97,103,111], Salinetti and Wets [182], Lavie [137,138,140] and Raynaud de Fitte [169]. Here, we denote by M{X) the space of all probability measures on (X, B(X)). M(X) is endowed with the weak topology, also called the topology of narrow convergence (see, e.g., [29]). For each measurable function / : ^ -> X, the distribution of / is denoted by Pf and defined on B(X) by -1 Pf(B) = p[f-^(B)},

BeBiX).

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C. Hess

Let 7^ be a a-field on the space 2^ and F : ^ -> 2^ an ^/7^-measurable multifunction. Like for real or vector valued random variables, the distribution of F is the measure /IF on (2^, 7^) defined by /iF{R) = f^{F-^{R)).

Ren.

(8.1)

Two measurable multifunctions F, G : ^ ^- 2-^ are said to be independent if for every R,S en onehsis / x ( F - ' ( / ? ) n G - ' ( 5 ) ) = /x(F-'(/?))/x(G-'(5)). This amounts to the equality

on the product space (C(X) x C(X), 7^ 0 7^)). In the sequel, it will be sufficient to consider the case where n is equal to the Effros cr-field £ (this will be explained in Section 9). It is natural to consider measurable selections that are measurable with respect to the afield AF = F~\£) (i.e., the a-field generated by the measurable multifunction F). This raises the question of the relations between the sets of distributions of ^/^^-measurable selections and those of ^-measurable selections. The answer is given in Theorem 8.3, and in Corollary 8.5(a) for integrable selections. In view of appUcation to the SLLN, the multivalued integrals are considered in Corollary 8.5(b). On the other hand, we shall provide results allowing one to extract equidistributed and independent selections from sequences of measurable multifunctions having similar properties. For this purpose, appropriate characterizations of equidistribution are proved at the end of this section (Proposition 8.6), as well as a connection with the multivalued integral. First, we recall a simple and useful criterion for two measurable multifunctions to have the same distribution. PROPOSITION 8.1. If F and G are members of MiC(X)), then the following three statements are equivalent: (a) F and G have the same distribution on (C(X),£). (b) For any open subset U of X, P{F-(U)} = P{G-{U)}. (c) For any finite subset Y = {x\,... ,.Xk] of X (or of some countable dense subset), the R^-valued random vectors {dix, F))_xeY ^"^ (d(x,G))xeY have the same distribution. PROOF. It follows from the definitions that statement (a) implies (b) and (c). To prove the implication (a) =^ (b), we define the subset W^ of C(X) by

W^ = {C eC(X): C c w } ,

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649

for every subset W of X, and we observe that for each open subset U of X one has {U-y = {U')^ = [Ce C(X): C c U'}, where U^ denotes the complement of U. This class generates £ and is stable under finite intersections, which by a classical result yields the desired implication. For proving implication (a) =^ (c), remember that the Effros a-field is also generated by the family of distance functions C ^ J(jc,C), where x ranges over X (or over a countable dense subset). Like in the first part of the proof, it is enough to observe that the class, whose members are defined by {C eC(X): dixiX)
V/= 1

k]

(where k eN.Xj e X and a/ e R), generates S and is stable under finite intersection.

D

For every sub-a-field T of A, consider the space L^\Q,T, P; X) of all (classes of) measurable functions from {Q,T) into (X,S(X)). Further, define the following subset associated with the multifunction F M(F,T)

= [fx = fifeM(Xy.

feS{F,T)},

where S(F,T) is the set of all ^-measurable selections of F (see the beginning of Section 5). So, M(F,T) the set of all probability measures fi on (X,S(X)) such that each /x € M(F, ^ ) is the distribution of some JT-measurable selection of F. Before stating the results on the set of selections and its associated set of probability measures, an example is useful. EXAMPLE 8.2. Consider the case where Q = [0, 1], A = S ( ^ ) , P = Lebesgue's measure, X = R and F = {0, 1} (i.e., F is a constant multifunction). Clearly we have AF = {^,^],

5(F,^/r) = {/ = 0, / = ! }

and

M{F.AF) = {SiuS\}-

On the other hand, one has 5(F, A) = {f =

1A: A G A},

M(F. A) = {pSo -h (1 - p)S\: /? G [0, 1]}.

This shows that the inclusions S(F,AF)^S(F,A)

may be strict.

and

M(F,AF)^M{F,A)

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C Hess

The following theorem is the main result of the present section. It was already stated in [96,97], but the proof contained a gap; the correct proof was given in [103,111]. It provides a fundamental equality, which is the starting point for the study of the distribution of multifunctions in connection with their measurable selections. Its consequences, especially Corollary 8.5(b) concerning the multivalued integral, play an important role in the proof of the multivalued SLLN for measurable multifunctions whose values may be unbounded. 8.3. If F is a measurable multifunction with closed values in X, then, in M(X) endowed with the topology of weak convergence of measures, the following equality holds true THEOREM

COM{F, A) = COM{F,AF)^

(8.2)

where " c o " denotes the closed convex hull operation. We denote by M* (X) the subset of M(X) whose members /i satisfy

/,

\\x\\dp < +00.

Given a sub-or-field ^ of ^ and a measurable multifunction F, we define the following subset of M(X) M\F,J^)

= {pf:

feS\F,T)},

where S^ {F,T) was defined in Section 5. Now, consider an integrable multifunction F. Obviously, the inclusion M\F,T) C M{F,T) holds for any sub-a-field T of A. The following simple lemma shows that the closure of both sides are equal in M(X). LEMMA 8.4. For any sub-a -field T of A and any integrable multifunction F whose values are members ofC(X) the following equality holds true C\M\F,T)=C\M{F,J^).

the closure being taken in M(X) in the weak (narrow) topology. From Theorem 8.3 and Lemma 8.4, we easily deduce the following corollary, whose first part makes equality (8.2) more precise when F is integrable. Part (b) is an application of (a) to the multivalued integral. It was given by Artstein and Hart [10, Theorem 2.2] when F is an integrable measurable multifunction with closed values in a finite dimensional space. Let us note that Artstein and Hart's proof relies on a previous result of Hart and Kohlberg [89] valid for integrably bounded multifunctions. A self-contained proof was given in [111].

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651

COROLLARY 8.5.

(a) For every integrable multifunction F whose values are in C(X) the following equality holds true CO M\F,

A) = CO M\F,AF).

(8.3)

the closure being taken in M{X) in the weak topology. (b) For any integrable multifunction F whose values lie inC(X), one has CO I{F, A) = CO I(F^AF)^

(8.4)

The following result states that two measurable multifunctions F and G have the same distribution if and only if the sets of distributions of selections that are measurable relatively to the a-fields AF and AG^ respectively, are equal. As we shall see, this property plays a crucial role in the proof of the multivalued SLLN in an infinite dimensional Banach space. Considering the a-fields generated by the multifunctions (instead of the a-field A) allows for more precise results. Indeed, unlike in [10], the equalities of statements (b) and (c) below do not involve any closure operation. This is important in view of the extension of the SLLN in infinite dimensional spaces. 8.6. Let F and G be two measurable multifunctions with closed values in X. Then, the nvo following statements (a) and (b) are equivalent: (a) F and G have the same distribution on the measurable space (C{X), £). (b) In M{X), the following equality holds true

PROPOSITION

M{F,AF)=-M{G.AG)-

Moreover, if F and G are integrable then each of the above statements is equivalent to (c) In M ' (X), the following equalit}' holds true MUF,AF)

=

M\G.AG).

Consequently, if F and G have the same distribution one has

I(F,AF) = nG.AG)REMARK 8.7. (i) An alternate proof of the last statement of Proposition 2.6(c) was given by Hiai [116, Lemma 3.1(2)] using the properties of the multivalued conditional expectation, as defined in Section 6. (ii) It is worthwhile to remark that Theorem 8.3 and Proposition 8.6 ((a) and (b)) remain valid when X is only assumed to be a complete separable metric space. (iii) Let us mention a result of Artstein [3], who considered the following problem: given a probability distribution a on (/C(X), 5), is it possible to find p e M(X) which is

652

C Hess

the distribution of some measurable selection / of a random set F \Q -^ JCiX) whose distribution is equal to a ? (iv) The study of the distribution of measurable multifunctions was initiated by Choquet [45]. Given F e MiC(X)), Choquet introduced the capacity functional Tr defined on IC(X) by

Tf(K) = iii{F-^{K-)) = fi{aje^:

Fiaj)nK^0},

KelC(X).

Choquet showed that, when X is finite dimensional, Tf characterizes the distribution of F. He also showed that if T is a capacity satisfying suitable additional requirements then there exists a unique probability distribution v on (C(X), £) such that T{K) = v{K-) = v{[C e C{X): CHK^

0}),

K e K{X).

Later, Matheron [148] gave a probabilistic proof of Choquet's result. See also Norberg [160,161] and Salinetti and Wets [182] for related results. A more abstract approach was considered by Ross [176] without topological assumptions. It was shown recently by H.T. Nguyen and N.T. Nguyen [159] that the above result on the existence of V may fail when X is not locally compact. Finally, let us note that the convergence in distribution of multifunctions was studied by Norberg [160,161], and by Salinetti and Wets [182]. The convergence in probability was examined by Salinetti and Wets [181], and Salinetti et al. [179]. 9. Set-valued strong laws of large numbers It is remarkable that the classical Kolmogorov strong law of large numbers (SLLN) makes sense in the set-valued case. Firstly, we shall present set-valued SLLN involving the convergence in the Hausdorff metric topology TH on Cb(A^). Secondly, we shall be interested in SLLN involving the Painleve-Kuratowski convergence, which allows for multifunctions taking on unbounded values. In both cases, we shall observe a convexification phenomenon. More precisely, in multivalued SLLN the limit is always convex, even if the given multifunctions are not convex-valued. On the average, the Cesaro sums tend to be convex. 9.1. Convergence in the Hausdorff metric topology As to the measurability issues, we have already noted in the beginning of Section 8 that the distribution and independence of multifunctions are relative to a given or-field, say 7^, on C(X). In this subsection, it is natural to take the a-field 11 equal to S(Cb, TH). However, since we shall restrict to a TH-separable subspace, we can take IZ equal to the Effros a-field as well (see Remark 4.9(ii)). 9.1.1. Consider a strongly measurable multifunction F with values in a Tw-separable subspace C of ChiX) and a sequence iFn)n^\ of pairwise independent multifunctions having the same distribution as F. Then, the following two statements are equivalent THEOREM

Set-valued integration and set-valued probability theory: An oven'iew

653

(a) F is integrably bounded. (b) There exists C G Cb such that lim /21 C, - V Fi (a))\=0

n-^oc

\

n ^

I

Moreover, when C exists it is equal to col(F) metric, i.e., lim AlC-Y^

n-^oc

\

n ^

a.s. and the convergence also holds in the A-

Fi I = 0. I

The proof of Theorem 9.1.1 can be done in two steps. When F is assumed to be convex valued, it suffices to use the embedding ^ from Cbc(^) into C(5*) (see Remark 3.6) and to invoke the SLLN for vector-valued random variables. This is the easier step, but it requires the TH-separability of C. The second step consists of a so-called convexification procedure, namely of a property of the following type, which is of a purely deterministic nature. CONVEXinCATlON PROPERTY (#). Consider a pair (XX'). where X is a Banach space and C a subspace of Cb(X). For every sequence (C„)„^ i in C and C eC satisfying (i) C is convex, (ii) \imn^ooh{C.]-^Y!U^Ci)=(). one has

THE

lim / z | C , - V C , 1 = 0 . 9.1.2. Let us mention three situations where property (#) holds. (a) X is finite dimensional and C = IC(X) (Artstein and Vitale [11]). Using the Shapley-Folkmann inequality, it can be shown that for every sequence C i , . . . , C„, in /C(X) one has

EXAMPLE

hi X ] c o C / , ^ C / I ^x/jmax{r(C,): / = 1,...,^?}, \i=\ i=\ I where d is the dimension of X and r{C) denotes the radius of the subset C, i.e., r(C)=inf sup||x-y||. An alternative proof in special cases was given by Cressie [59]. A variant of this theorem has been rediscovered recently by Uemura [198], and Taylor and Inoue [192].

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(b) X is a separable Hilbert space and C — K^{X). This was proved first by Cassels [32] using probabilistic arguments and later by Hess [95,99] by a geometric approach in the framework of (countable) Hilbert spaces. This was extended later by Puri and Ralescu [166,167] when X is a Banach space of type /7 G (1,2]. In all these papers, the proof is based on an inequality of the following form

( X]coC„^C/UO]''(^')i n

"

\

f "

1 ''^^

'

/=1 /^l / I /=1 J where k = l when X is Hilbert and A: = p e (1, 2] when X is of type p. (c) X is a separable Banach space and C — /C(X), the set of strongly compact subsets ofX. In this case, property (#) was proved independently by Artstein and Hansen [9] and by Hiai* [115]. Further, it was shown by Hess [98,99] that it is not possible to replace /C(X) by /Cw(X). It is interesting to mention a set-valued version of the Birkhoff ergodic theorem. Assume that a measurable map 7 : .T? -> f2 is given. T is said to be measure preserving if for every A^ A one has /x(r~' (A)) = /x(A). The cr-field J of invariant subsets of Q is the set of those A^A such that A and T~^{A) are equal up to a /x-nuU set. The following result is due to Hess [94-96]. THEOREM 9.1.3. Let F be a strongly measurable and integrably bounded multifunction with values in a inseparable subspace C ofCh(X). Then, for almost all co e ^ one has

lim h I coE(F|X), - Y " Flro)) ] = 0.

/2^>oo

\

ri ^—^

I

As it is known for single-valued random variables, this result can be stated equivalently in terms of stationary sequences (see [94-96]). Extensions and variants of this result have been proved by Schurger [189] and, more recently, by Krupa [131]. On the other hand, assuming conditions on the moments, set-valued SLLN have been proved for nonidentically distributed, independent sequences of integrably bounded multifunctions. See, e.g., Lyashenko [146] and Hiai* [115]. 9.2. Convergence in the sense of Painleve-Kuratowski Now, we consider the extension of the SLLN to the case of measurable multifunctions whose values are unbounded. In this case, the Hausdorff distance is no longer appropriate to formulate the convergence, because it is too strong. Instead, we consider the PainleveKuratowski convergence, whose definition is recalled hereafter. Let r be any topology on X. If (C„)„^i is a sequence in 2^ we set -LiC,7 = \x eX: x = T- hm x„, Xn eCn, n ^

\\,

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655

and r-LsC„ = \x eX\ X = T- lim .v;, Xk € C„(j^), /: ^ 11, where (C,2()^));(:^i is a subsequence of (C„). The set T-LiC„ (respectively, T - L S C „ ) is called the lower limit (respectively, the upper limit) of (C,;) relatively to the topology z. These subsets are also called the inferior limit and the superior limit, respectively, which explains the notation. When r is metrizable, these subsets are both r-closed. Further, r-Li Cn and r-LsC„ are unchanged if each C,, is replaced with its r-closure. The sequence (C„) is said to converge to C in the sense of Painleve-Kuratowski (PK), relatively to r, if one has C = r-Li Cn = r-Ls €„ (see, for example, [14,27,45,180,190]). This is denoted by C = PK(r)- lim Q , or simply C = PK- lim C„, when X is finite dimensional and r is the usual topology on X. A convergence criterion in the general case was given by Couvreux and Hess (Corollary 1, p. 944, in [56]). The convergence in the Hausdorff distance entails the PK-convergence, but the converse implication is false, as the following simple example shows. EXAMPLE

9.2.1. Let X = R^ be endowed with any norm and consider the subsets

C = a straight line passing through the origin, Cn = a straight line such that the angle between C and Cn is \/n. It is readily seen that the sequence (C,, )„^ i PK-converges to C, but does not TH-converges. In the present section, the cr-field 1Z considered in Section 8 is taken equal to £, the Effros cr-field on C(X). Thus, the definition of the distribution and independence of multifunctions are relative to £ and we consider an Effros-measurable multifunction F:Q -^ C{X). As to the integrability condition, integrable boundedness is no longer appropriate because an integrably bounded multifunction is almost surely bounded valued. Instead, we assume that F is integrable, namely that S^ (F) is nonempty. The proof of the SLLN for multifunctions whose values may be unbounded are mainly based on the results presented in Section 8, especially on Corollary 8.5(b) and Proposition 8.6(c).

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THEOREM 9.2.2. Let X be a separable Banach space, F e M{C(X)) be an integrable multifunction and (Fn)n^\ be a sequence of pairwise independent multifunctions having the same distribution as F. Then, there exists a null set N such that

1 " co/(F,^)=PK-lim - V F / M ,

cx)eQ\N.

i= \

A converse to the above theorem can be given when the values of the multifunctions do not contain any line. They may be unbounded, but may contain half-lines only. For sake of simplicity, we only present the finite dimensional version of a result due to Hess (Theorem 9 in [98,99]). THEOREM 9.2.3. Consider a finite dimensional normed linear space X and a closed subset L of X which contains no whole line. Let (Fn)n^\ be a pairwise independent, equidistributed sequence in M(C(X)). Moreover, assume that for almost all co e Q the following inclusion holds

F\{co)
(9.1)

Under the above assumptions, if F\ is not integrable, i.e., if S{F\) = 0, then 1 " PK- lim - y F/(cL>) = 0. i= \

For example, if X = M^, the subset L can be taken equal to the positive cone R^{.. When the F„ are single-valued, condition (9.1) simply means that the X-valued random variables Fn have positive values. REMARK 9.2.4. (i) Theorem 9.2.2 has been proved by a lot of authors. It was proved first by Artstein and Hart [10] when X is finite dimensional, for mutually independent, identically distributed sequences. In the early eighties, infinite dimensional versions were proved independently by Hiai [116] and Hess [98,99], when the PK-convergence is replaced by the Mosco convergence, an infinite dimensional extension of the PKconvergence (see the next remark). Moreover, using a result of Etemadi [80], Hess [98,99] relaxed the hypothesis of mutual independence, assuming only the pairwise independence. Afterwards, the set-valued SLLN was proved with respect to other extensions of the PKconvergence such as the Wijsman convergence [111] and the slice convergence [107,111], that are of interest in infinite dimensional spaces. (ii) The above mentioned Mosco convergence is one of the infinite dimensional extensions of the PK-convergence. It was introduced by Mosco [153,154] and has interesting variational properties that was exploited in the areas of calculus of variations (see, e.g., [62,64]) and optimization [175]. If X is infinite dimensional and if the sequence {Cn) PK-converges to C relatively to both the strong topology and the weak topology of X (denoted by s and w, respectively) then (C„) is said to be Mosco convergent to C. This

Set-valued integration and set-valued probability- theory: An oveniew

657

is denoted by C = M- lim„ C„. However, Mosco convergence has nice properties only in reflexive spaces so that several other topologies have been introduced. For an extensive study of convergences and topologies on spaces of subsets of a metric or a normed linear space, the reader is referred to the monograph by Beer [27] and to the paper by Sonntag and Zalinescu [190], who presented a classification of set-convergences. (iii) In some extent, it is possible to deduce the set-valued SLLN for unbounded valued multifunction from the integrably bounded case. This approach relies on a truncation argument. More precisely, let F e M(C(X)) be an integrable multifunction. For every integer k^ 1, we define the multifunction F^ by F^(oj) = c\{F(oj) n B^'io))),

(joeQ,

where B^{co) denotes the open ball of radius rk(co) = k{d{0, F(a>)) -h 1), centered at 0. Now, if (Fn)fj^\ is a sequence of pairwise independent measurable multifunctions having the same distribution as F, we set F,fM = c\{Fn{(v) n Bj^^ico)),

coeQ.

where B^^ {co) is defined as above except that F{co) is replaced by F„ (CD). For every k,n ^ \, FJ^ is ^yr^-measurable and integrably bounded (because J(0, F„()) is integrable). The Af,j -measurability follows from the Effros measurability of the map C->cl(Cn5(0,r)), for every r > 0. Thus, when X is finite dimensional, F,f takes on compact values, which allows us to apply Theorem 9.1.1 to the sequence (F,f )„^i, for every k^ \. This yields COE(FM=

1 lim -Y^Fl'ia))

a.s.

Then, using the simple equalities

A'^l

/==1

i = \ k^\

i= \

and Lemma 5.11 of [106], it is readily seen that this yields the PK-convergence as in Theorem 9.2.2. This technique can be used for proving other strong limit theorem for multifunctions. (iv) There are other kinds of set-valued SLLN where the multifunctions are no longer assumed to be identically distributed, but only independent (see, e.g., [115,116] and [169]). Let us also mention set-valued versions of the Komlos theorem (see [22,39,133,216]). Like in the SLLN, Komlos theorem involves Cesaro means, but is valid for L'-bounded sequences. An interesting application to fuzzy random sets was obtained by Colubi et al. [47].

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10. Set-valued martingales In this section, X generally stands for a finite dimensional Banach space. Infinite dimensional extensions will be mentioned in Remark 10.4. DEHNITION 10.1. Let (S„)„^i be a nondecreasing sequence of sub-a-fields of A. A sequence (F„)„^i of measurable multifunctions with values in CdX) is said to be adapted to (S„) if, for any A? ^ 1, F,; is S„-measurable, i.e., F'^dJ") e Bn for every U e G' An adapted sequence (F„) is said to be a set-valued martingale if the following two conditions (a) and (b) hold, for every integer n ^ \: (a) S^ (F,j,Bn) is nonempty (i.e., F„ is integrable), (b) F„=E(F„^i\B„). Further, if, instead of equality (b), one has the inclusion

Fn c E(F,+i \Bn)

(respectively, F„ D E(F„+, 1^,)),

we shall say that (F„) is a submartingale (respectively, supermartingale). EXAMPLE 10.2. In the following examples (G,,) denotes a sequence of integrable CdX)valued multifunctions and, for every integer /? ^ 1, iB„ = a (G i , G„) is the sub-a -field generated by the multifunctions G\ G„. (i) In this example, we assume that the G/j are centered in the sense that

OGE(G„+i|fi//)

a.s. for all/2 ^ 1.

(lO.l)

If we define the sequence F,, by (10.2)

F,=cl|^G/ ./=i

then (F„) is a submartingale. Indeed, from (10.1) and (10.2), and from Theorems 6.2 and 6.3, we obtain, for every « ^ 1, E(F,+,|B,) =

E(CI(^G,+G,H

Bn

.i = \

= cl{F„+E(G,+,|^,;)}2F,. (ii) Now, we consider the sequence (G„) and (Bn) as in (i). In addition, the G„ are assumed to be integrably bounded, but condition (10.1) is no longer assumed to hold. If the sequence (F„) defined by (10.2) is a supermartingale, then for all A? > 2 the multifunctions Gn are single-valued. Thus we have Fn =G] -^g2-\

-\-gn.

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where gi'.Q^^X are integrable random vectors with zero expectation. Only G \ can be set-valued. Indeed, for every n ^ 1, one has E(F,+ , \Bn) = F, + E{Gn-,\\Bn) C F,. which implies E(G,+i|B„) = {0},

foralln^l.

In turn, this yields the existence of a iB,;+i-measurable random vector g„+i such that G,7+i = {g/j+i}. This result is due to Ezzaki ([83, p. 17]). (iii) If we consider (G„) and {B,^) as at the beginning of the present example, then the sequence (F„) defined by

Fn^coiyjoX i^n

for all « > 1,

is a submartingale. Indeed, using Theorems 6.2 and 6.3 it is readily seen that for every n ^ 1 and almost every a)e Q, one has E(F,+ ,|i3„) = E ( F , U G , + , | S , ) 5 E(F,|e,) UE(G„+i|S„) = F„ UE(G„+i|B„) D F,. (iv) Let (G„) be a sequence of multifunctions as in example (iii). If we assume that OeGn

a.s. f o r a l l n ^ 1,

(10.3)

then it is readily seen that the sequence (F„) defined by

F, = P^Gi. i^n

is a supermartingale. Indeed, for every n ^ 1 one has almost surely E(F„^i\B„) =

E(F„nGn^i\Bn)

c E(Fn\Bn)nE(Gn\Bn)

= F, n E ( G J S „ ) c F,.

Condition (10.3) prevents F„ from taking empty values. (vi) Examples (iv) and (v) are special cases of more general situations. Indeed, consider an adapted sequence (F„) of multifunctions. If (F„) is a nondecreasing (respectively, nonincreasing) sequence of multifunctions, then it is a submartingale (respectively, a supermartingale). (vii) Finally, it is worthwhile to observe that the following property is an immediate consequence of our definitions: if a set-valued submartingale or supermartingale is singlevalued (i.e., its values are points of the space X), then it is a martingale.

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The following result presents a property of almost sure convergence of set-valued martingales whose values may be unbounded. It can be seen as an extension of the classical Doob's Convergence Theorem. 10.3. Let (Bn)n^\ be a nondecreasing sequence of sub-a-fields of A and (F,j),i^\ an adapted sequence of multifunctions with values in Cc(X). If (Fn) is a setvalued supermartingale satisfying the condition THEOREM

supEJ(0, F,,) < -hoo, then there exists an integrable multifunction F e M{C{X)) such that, for almost every coeQ, F((w)=PK- lim

F„{(D).

REMARK 10.4. (i) Theorem 10.3 is a special case of Theorem 5.12 in [106], where X is assumed to be an arbitrary separable Banach space (possibly infinite dimensional). There, the following additional condition was required. (WB) there exists a multifunction //, whose values are weak ball-compact, such that

Fn(a))C,H(a)),

n^\,

co e Q.

Recall that a subset C of Cc{X) is said to be weak ball-compact if its intersection with every closed ball is weakly compact. Condition (WB) is automatically satisfied when X is reflexive or, more particularly, finite dimensional. The proof is based on a truncation argument and involves Lemma 5.11 in [106], already described in Remark 9.2.4(iii). On the other hand, existence results for martingale selections can be found in [105,106]. Here, for the sake of simplicity, we have only presented convergence results when X is finite dimensional. (ii) There are other references where other set-valued versions of the Doob convergence theorem are proved. For example, let us mention the works of Neveu [158], Daures [63], Hiai and Umegaki [117], Coste [51], Luu [144,145], Bagchi [18], Choukairi [46], Wang and Xue [211], Dong and Wang [70], Hess [109], Krupa [131,132] and Lavie [139]. A version of the optional sampling theorem can be found in [1]. An appHcation to the convergence of fuzzy sets was given by Li and Ogura [143]. A lot of further references can be found in [131]. 11. Gaussian multifunctions. The set-valued Central Limit Theorem First recall that an X-valued random variable / : ^ ^- X is said to be Gaussian if for every finite sequence ( j i , . . . , yj) G (X*)^, where y ^ 1, the R^-valued random vector ((yi,/>,..., (yy,/)),

Set-valued integration and set-valued probability theory: An overx'iew

661

is Gaussian. More generally, an integrably bounded multifunction F \Q ^^ fC(X) is said to be Gaussian if, for every finite sequence {y\,..., yj) e (X*)A the E^-valued random vector (^(j,,F),...,^(j,,F)),

is Gaussian. The following result characterizes Gaussian multifunctions with compact values. THEOREM 11.1. Let F\Q -^ JCdX) be an integrably bounded multifunction. The following two statements are equivalent. (a) F is Gaussian. (b) F has the following form

F = K + f. where K G lCc{X) is nonrandom and f: ^ -^ X is a Gaussian vector with zero expectation. Theorem 11.1 was proved by Lyashenko [147] when X is finite dimensional and by Vitale [207]. It was extended later to the case where X is infinite dimensional by Puri and Ralescu [168], and by Meaya [150]. On the other hand, there exists a version of the central limit theorem for multifunctions whose values are compact. First, we need some notations. Let S^~ ^ be the unit sphere of X = R^ and C(5^~') be the space of continuous functions defined on S^~\ endowed with the uniform norm || • ||u- The dual space C(5^-')* is the space of Radon measures on S^~^. The duality pairing is defined by {k,u)=

[

udk,

ueClS^-^),

A6C(5^~')*.

Given a C(5^~')-valued random variable / , the expectation E ( / ) is defined as a Bochner integral. This is possible because C{S^~^) is separable. Further the covariance function Ff is defined on the product space C(5^"•)* x C(5^"')* by r/(/x,, /X2) = E{(/x,, / - E(/))(/X2, / - E(/))} = I {^l\. f((o) - E(/))(/X2, f(co) - E(/))Pidco). JQ

When F e M{1C{X)) is an integrably bounded multifunction, the covariance function of F is denoted by Ff and is defined as the covariance function of the C(5^~^ )-valued random variable y->

s{y,F).

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11.2. We assume that X = E^. Let F e M{1C(X)) be an integrably bounded multifunction and a sequence (F,j),j^\ of independent multifunctions having the same distribution as F. Then the sequence THEOREM

y^/z(-^F/,EcoFj, converges in distribution to \\g\\u, where g is a centered Gaussian C(S^ ^)-random variable with covariance function Fy. The proof of Theorem 11.2 appeals to the central limit theorem for C(5'^~')-random variables, which allows for proving the result in the case where F e M{lCc(X)), and to the Shapley-Folkmann inequality in the general case. Theorem 11.2 is due to W. Weil [212]. Previous versions had been settled by Cressie [60,61] in the case of multifunctions taking on a finite number of values. 12. Set-valued versions of the Fatou Lemma The classical Fatou Lemma of Integration Theory asserts that, given a sequence (fn)n^\ of positive measurable functions, the following inequality holds / lim'mffndP

^ liminf /

fndP.

Extensions of this inequality to the case where the /„ take on their values in X = W' (d ^ 2) or in infinite dimensional Banach spaces, were established in order to get the existence of equilibria in Economic Theory. In these extensions, the inequality must be changed into an inclusion. It is worthwhile to note that, because of the partial character of the order relation in W^, this sort of result cannot be deduced directly from the classical Fatou Lemma. For the same modelization purpose, the case where the /„ are replaced with measurable multifunctions Fn was examined by several authors (see, e.g., [118, Theorem 6, p. 68]). In the set-valued case, there are in fact two versions of the Fatou Lemma for multifunctions: one for the PK-lower limit and another for the PK-upper limit. Indeed, it is no longer possible to deduce one version from the other by merely changing Fn into — F„. On the other hand, it was shown by Balder and Hess [21] that there is a strong connection between the set-valued version and the vector version of Fatou's Lemma. In fact, one can be deduced from the other. As mentioned above, several authors have studied this subject. Let us mention the works by Aumann [17], Schmeidler [185], Hildenbrand and Mertens [119], Artstein [5], Balder [19], YanneHs [215], Castaing and Clauzure [38], and Balder and Hess [21] among others. We refer the reader to [21 ] for a more complete list of references and for a discussion of the various existing results. Basically, the first version of the following result is due to Aumann [17].

Set-valued integration and set-valued probability theory: An overview

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12.1. Let X be a separable Banach space and (F,,) be a sequence in M{C{X)). (a) If the sequence (^(0, F„)),,^i is uniformly integrable, then the following inclusion holds

THEOREM

cl/(s-LiF„)Cs-Li/(F,;). (b) If the sequence (|| F„ \\)n^\ is uniformly integrable and if there exists a multifunction G'.Q^^ /Cw(^) such that Fn(a))<^G(a)),

coe^,

n^\.

then one has w-Ls/(F;,) c cl/(cow-LsF,,). REMARK 12.2. (i) In statement (a) the multifunctions are only assumed to be integrable, whence can have unbounded values. In statement (b), the F„ are assumed to be integrably bounded, thus they are almost surely bounded valued. Versions of statement (b) for integrable multifunctions have been given, but they are of a more delicate nature. For example, they involves conditions on the asymptotic cones of the multifunctions (see [21,104]). On the other hand, results involving the convergence in distribution in connection with the approximation of the multivalued integral were given by Artstein and Wets [12]. (ii) Similar properties have been proved for the set-valued conditional expectation (see [102,104,116]).

13. Epigraphical convergence The epigraphical convergence, "epiconvergence" for short, is closely related to PKconvergence and its infinite dimensional extensions. Let w: X -> E be an extended realvalued function. Its epigraph (or upper graph) is the subset of X x M denoted by epi(M) and defined by epi(M) = {(jc,Qf)G X xR: u(x)

^a}.

The conjugate (or the polar) function of u is denoted by w* and defined on X* by u*(y) = sup{(y, x) - uix): x € X},

v G X*.

The map w ^- M* is often called the Young-Fenchel transform. Further, let (w„)„^i be a sequence of extended real-valued functions. Consider a topology r on X. The sequence (Un) is said to converge epigraphically (or ''epiconverge'' for short) to M, relatively to r, if epi(M) = PK(r)- lim epi(w„). n^-oc

(13.1)

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C Hess

When X is finite dimensional and r is the usual topology, this is denoted by U = l i m e W/z,

where the subscript "e" stands for "epigraphical". Of course, a symmetric notion involving hypographs (alias lower graph) can be introduced. This type of convergence was considered by a lot of authors in order to study the convergence and the approximation of optimization problems, especially in the calculus of variations. Let us mention the works of De Giorgi [64], Mosco [153,154], Attouch [14], Dal Maso [62], Rockafellar and Wets [175] and Artstein and Wets [13], where further references can be found. Also, applications to the convergence of estimators, in Statistics, were given by Hess [108] and Choirat et al. [43]. Without entering into details, let us mention the following facts (see, e.g., [14] and [62]). If (P,j) denotes the minimization problem min[un(x): x e X],

n eNU {oo},

and if (M„) epiconverges to M^C* then under additional compactness assumptions, it can be shown that inf xeX

^OO(-^)

= lim inf u,j(x). n-^ocxeX

Moreover, the solutions of (Voc) can be approximated by solutions of (P„) for n large enough. The connection of epiconvergence with that of measurable multifunctions is clear. Indeed, let 0 : ^ X X -> R, be an extended real-valued function. The multifunction F defined by Fico) = tpi{(t){aj,')) = [(x,a) e X xR: 0 ( a ; , j c ) ^ a } .

(13.2)

It is called the epigraphical multifunction associated with 0. When 0 is ^ 0 B{X)measurable, it is called an integrand. It is not difficult to show the following equivalence (see Lemma VII. 1 in [41]) Gr(F) G ^ (g) B{X)

^

0 is ^ 0 e(X)-measurable.

Let us give an example involving epiconvergence in connection with the SLLN. First, we need the following simple auxiliary result, which is of independent interest. The proof can be found in [107].

Set-valued integration and set-valued pwhabilit}' theory: An overx'iew

665

PROPOSITION 13.1.

(a) The multifunction F defined by (13.2) is integrable if and only if there exists f e 0{X) such that the function (\){(JL>, f{co))^ is integrable.^ (b) In X xR, the distance function from the origin to the epigraphical multifunction is given by (o -^ J((0, 0), F{oj)) = inf{ ||jcII -f 0(a>, x)^:

THEOREM

xeX}.

13.2.

(a) The stochastic discrete infimal convolution \l/„ (co, x) defined on Q x X by

I

I "

1 "

]

^ /=i " /=i J epiconverges to the convex lower semicontinuous regularization of the continuous infimal convolution \j/ which is the deterministic function defined by V^(x) = infj / (t>{co,fico))d^l: feL\X).

j

f(aj)dfi=xV

(b) For almost all coe Q, the so-called mean functional defined by

L

(I){co, x) dIJL,

is the epi-limit of the sequence of the sample mean given by {o),x) -^ - 7 0(CL>,X). n /=1

Statement (a) can be easily proved by reformulating Theorem 9.2.2 in terms of epigraphs. The proof of statement (b) is based on the continuity of the Young-Fenchel transform with respect to epiconvergence (see [14,62]). REMARK 13.3. There is another approach allowing one to prove more general versions of Theorem 13.2(b). It is based on the formulation of the epigraphical limit in terms of Lipschitz approximation. It has applications to Statistics [76,108], Optimization Theory [13,57,58], Ergodic Theory and in Econometrics [43].

14. Concluding remarks As illustrated above, the theory of measurable multifunctions raises a great variety of problems involving probability theory and mathematical analysis arguments. It allows ^For every real a, we set or"'" = max(0, or).

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C Hess

for new points of view and is often at origin of new approaches for classical questions. Of course, we could have presented other topics of interest. Let us mention, for example, the measurability issues concerning the extreme points of multifunctions (see, e.g., [41, Chapter IV]). Further, measurable multifunctions depending measurably or continuously on a second parameter play a important role in several applications. We refer the reader to [34,35] and [165] for continuous dependence (Caratheodory multifunctions), and to Godet-Thobie [87] and Artstein [7] for measurable dependence. Concerning convergence problems, one can mention the weak convergence of multifunctions studied by Artstein [4], using two different approaches: one based on support functions, the other on the embedding technique. A result on large deviations for sums of random sets was proved recently by Cerf [42]. The theory of random sets has also beautiful and rich geometric aspects, especially when X is finite dimensional. Let us mention, for example, the books of Kendall and Moran [125], Matheron [148], Santalo [183], and that of Stoyan et al. [191]. In this context, a lot of interesting probabilistic and statistical applications have been given, for example, to Sterology and to Image Processing. In fact, the Statistics of random sets, which has connections with spatial stochastic processes, is a very lively domain. One can find works dealing with theoretical aspects, and others with engineering and numerical applications. Let us mention [ 148,191 ], Molchanov [151,152], Schmitt and Mattioli [ 187], Aubin [15], and Choirat and Seri [44]. The interested reader may consult the proceedings of the international symposium on "Advances in Theory and Applications of Random Sets", edited by D. Jeulin (World Scientific, 1997), where a large variety of recent papers are available along this line. References [1] R. Alo, A. de Korvin and R. Roberts, The optional sampling theorem for convex set-valued martingales, J. Reine Angew. Math. 310 (1979), 1-6. [2] Z. Artstein, Set-valued measures. Trans. Amer. Math. Soc. 165 (1972). [3] Z. Artstein, Distribution of random sets and random selections, Israel J. Math. 46 (4) (1983). [4] Z. Artstein, Weak convergence of set valued fimctions and control, SIAM J. Control Optim. 13 (4) (1975), 865-878. [5] Z. Artstein, A note on Fatou's lemma in several dimensions, J. Math. Econom. 6 (1979), 277-282. [6] Z. Artstein, Parametrized integration of multifunctions with applications to control and optimization, SIAM J. Control Optim. 27 (6) (1989), 1369-1380. [7] Z. Artstein, Relaxed multifunctions and Young multimeasures. Set-Valued Anal. 6 (3) (1998), 237-255. [8] Z. Artstein and J. A. Bumes, Integration of compact set-valued functions. Pacific J. Math. 58 (2) (1975), 297-307. [9] Z. Artstein and J.C. Hansen, Convexification in limit laws of random sets in Banach spaces, Ann. Probab. 13(1985), 307-309. [10] Z. Artstein and S. Hart, LMW of large numbers for random sets and allocation processes. Math. Oper. Res. 6 (1981), 482-492. [11] Z. Artstein and R.A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab. 3 (1975), 879-882. [12] Z. Artstein and R. Wets, Approximating the integral of a multifunction, J. Multivariate Anal. 24 (1988), 285-308. [13] Z. Artstein and R. Wets, Consistency of minimizers and the SLLNfor stochastic programs, J. Convex Anal. 2 (1-2) (1995), 1-17.

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[14] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman Advanced Publishing Program, New York (1984). [15] J.P Aubin, Mutational and Morphological Analysis, Birkhauser, Boston (1999). [16] J.P Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston (1990). [17] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. [18] S. Bagchi, On a.s. convergence of classes of multivalued asymptotic martingales, Ann. Inst. H. Poincare, Probab. Statist. 21 (4) (1985), 313-321. [19] E. Balder, A unifying note on Fatou's Lemma in several dimensions. Math. Oper. Res. 9 (1984), 267-275. [20] E. Balder, A unified approach to several results involving integrals of multifunctions. Set-Valued Anal. 2 (1994), 63-75. [21] E. Balder and C. Hess, On the unbounded multivalued version of Fatou 's Lemma, Math. Oper. Res. 20 (1) (1995). [22] E. Balder and C. Hess, Two generalizations of Komlos' theorem with lower closure-type applications, J. Convex Anal. 3 (1996), 1-20. [23] A. Barbati,G. Beer and C. Hess, J. Convex Anal. 1(0(1994), 107-119. [24] A. Barbati and C. Hess, On the largest class of closed convex valued multifimctions for which Effros measurability and scalar measurability coincide. Set-Valued Anal. 6 (1998), 309-326. [25] D. Barcenas and W. Urbina, Measurable multifimctions in nonseparable Banach spaces, SIAM. J. Math. Anal. 28 (5) (1997), 1212-1226. [26] G. Beer, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991), 1123-1133. [27] G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications, Kluwer Acad. Publ., Dordrecht (1993). [28] V.E. Benes, Martingales on metric spaces. Theory Probab. Appl. 7 (1962), 81-82. [29] P. Billingsley, Convergence of Probability Measures, Wiley (1968). [30] J.M. Bismut, Integrales convexes et probabiites, J. Math. Anal. Appl. 42 (1973), 639-673. [31] C.L. Byrne, Remarks on the set-valued integrals of Debreu and Aumann, J. Math. Anal. Appl. 62 (1978), 243-246. [32] J.W.S. Cassels, Measure of nonconvexity of sets and the Shapley-Folkman-Starr theorem. Math. Proc. Cambridge Phil. Soc. 78 (1975), 433-436. [33] C. Castaing, Sur les multi-applications mesurables, R.I.R.O. 1 (1967), 91-126. [34] C. Castaing, Sur Vexistence des sections separement mesurables et separement continues d'une multifonction, Sem. Anal. Convexe de TUniversite de Montpellier, No. 14 (1975). [35] C. Castaing, A propos de Vexistence des sections separement mesurables et separement continuez d'une multifonction separement mesurable et separement semi-continue inferieurement, Sem. Anal. Convexe de rUniversite de MontpeUier, No. 6 (1976). [36] C. Castaing, Methode de compacite et de decomposition, applications: minimisation, convergence des martingales, lemme de Fatou multivoque, Ann. Mat. Pura Appl. (4) 164 (1993), 51-75. [37] C. Castaing, Weak compactness and convergences in Bochner and Pettis integration, Vietnam J. Math. 24 (3) (1996). [38] C. Castaing and P. Clauzure, Lemme de Fatou multivoque, Atti. Sem. Mat. Pis. Univ. Modena 39 (1991), 303-320. [39] C. Castaing and F. Ezzaki, Convergence of convex weakly compact random sets in super-reflexive Banach spaces. Preprint, Universite de Montpellier II (1996). [40] C. Castaing and M. Valadier, Equations differentielles multivoques dans les espaces vectoriels localement convexes, R.I.R.O. 16 (1969), 3-16. [41] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer, Beriin (1977). [42] R. Cerf, Large deviations for sums ofi.i.d. random compact sets, Proc. Amer. Math. Soc. (1998). [43] C. Choirat, C. Hess and R. Seri, A functional version of the Birkhoff ergodic theorem for a normal integrand: A variational approach. Working paper No. 0101, Centre de Recherche Viabilite, Jeux, Controle, Universite Paris Dauphine (2000), to appear in Ann. Probab. [44] C. Choirat and R. Seri, Confidence sets for the integral of random closed sets, in: Proceedings of the 8th International Conference IPMU, Madrid, July 3-7 (2000), 509-514.

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