Journal of Theoretical Probability, Vol. 12, No. 4. 1999

A Levy Type Martingale Convergence Theorem forRandom Sets with Unbounded Values

Jerome Couvreux1 and Christian Hess1

Received October 3. 1997; revised April 20, 1999

1 Centre de Recherche Viabilite, Jeux, Controle, ERS 644 CNRS. Universite Paris Dauphine,75775 Paris Cedex 16, France. E-mail: hess@ viab.ufrmd.dauphine.fr.

Given a nondecreasing sequence (Jln) of sub-<r-fields and a real or vector valuedrandom variable /, the Levy Martingale convergence Theorem (LMCT) assertsthat E(f/Jin) converges to E ( f / J i x ) almost surely and in L1, where Jlx standsfor the rr-field generated by the 41n. In the present paper, we study the validityof the multivalued analog this theorem for a random set F whose values aremembers of ,F(X), the space of nonempty closed sets of a Banach space X,when .&(X) is endowed either with the Painleve-Kuratowski convergence orits infinite dimensional extensions. We deduce epi-convergence results forintegrands via the epigraphical multifunctions. As it is known, these results areuseful for approximating optimization problems. The method relies on count-ability supportness hypotheses which are shown to hold when the values of therandom set E(F/^a .) do not contain any line. On the other hand, since thevalues of F are not assumed to be bounded, conditions involving barrier andasymptotic cones are shown to be necessary. Moreover, we discuss the relationswith other multivalued martingale convergence theorems and provide examplesshowing the role of the hypotheses. Even in the finite dimensional setting, ourresults are new or subsume already existing ones.

KEY WORDS: Measurable multifunctions; random sets; multivalued condi-tional expectations; multivalued martingales; set convergences; epi-convergence;integrands; asymptotic cone.

1. INTRODUCTION

The Levy Martingale Convergence Theorem (LMCT) is certainly one ofthe simplest and the most useful convergence results existing in ProbabilityTheory. The classical statement is: if ( ,#n) denotes a nondecreasing

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934 Couvreux and Hess

sequence of sub-<r-fields of some probability space (T, -&,fi) and f a real orvector valued integrable random variable then one has

almost surely, where 3SX stands for the sub-a-field generated by the 38n.This result was first proved by Levy(37,38) for real-valued random variablesand extended later to the case of random variables with Values in a Banachspace, in finite and infinite dimensions.

In the last fifteen years, this theorem has been extended to the casewhere, / is replaced by an integrably bounded multivalued random variableF (alias "random set" or "measurable multifunction") whose values lie in3F c b(X), the set of closed convex bounded subsets of a Banach space X. Letus mention for example the works of Hiai and Umegaki;(34) Choukairi-Dini;(12) Wang and Xue;(5) Piccinini;(4,5) Ezzaki;(20,21) and Dong andWang.(18)

The main arguments in the previously mentioned extensions consist ofrather straightforward applications of the classical LMCT to suitable realor vector valued functions associated with F. More precisely, this result isapplied to the support function of F at some points or to integrable selec-tions. The proof of (*) has been performed assuming that the usual con-vergence concepts for single-valued random variables are replaced by thePainleve-Kuratowski convergence for sequences of closed sets, or by itsinfinite dimensional versions.

Let us also mention that Hiai and Umegaki;(34) and Wang andXue;(51) the almost sure convergence in the Hausdorff metric topology,TH was proved, when the values of F lie in a .H-separable subspace of.^cb(X) (e.g., .WC(X) of convex compact subsets). In this case, the Radstromembedding is available, so that the multivalued case is a direct consequenceof the vector valued case.

On the other hand, more general martingales convergence theoremshave been extended to the bounded multivalued case. For a recent con-tribution along, this line see, e.a., Papageorgiou(44) who considers multi-valued amarts and uniform amarts. Also, some results for continuousparameter multivalued martingales were obtained by Dong and Wang.(18)

In the previously mentioned works, the random set F is assumed tohave bounded values almost surely. In fact, the boundedness is triviallyimplied by the integrable boundedness of F which is assumed systemati-cally. So, one can ask the natural question: what happens when therandom set F is unbounded, that is, when its values are not assumed to bebounded?

Clearly, these techniques are no longer available. One of the problemsis: that the support function of an unbounded random set may take on thevalue + oo, on a nonnegligible subset. To overcome these difficulties, onemust appeal to a more general version of the LMCT, valid for positive(possibly nonintegrable and/or infinite valued) random variables. Such aversion of the latter theorem already exists (see, e.g., Neveu,(42) or Propo-sition 6(i)), but, there, an additional hypothesis is required, so thatappropriate nontrivial modifications of the method must be implemented inorder to treat the unbounded multivalued case. In particular, we shall seethat the consideration of the barrier cones and asymptotic cones associatedwith F is crucial. On the other hand, the introduction of the concept of"countably supported" convex sets or multifunctions, already considered byBarbati and Hess(5) for other purposes, will be most useful to provideefficient convergence criteria (Section 3), and to manage the interplaybetween the probabilistic arguments and the deterministic ones.

Apart the challenging aspect of the mathematical extension from thebounded to the unbounded multivalued case, there is an important motiva-tion for elaborating this kind of result. Indeed, as it is known, thePainleve-Kuratowski convergence (as well as several of its infinite dimen-sional extensions) when considered for sequences of unbounded sets, givesrise to a specific functional convergence often called "epi-convergence" or"/"-convergence" [see, e.g., De Giorgi;(17) Attouch;(2) and Dal Maso(15)],because it is defined by the set convergence of the sequence of epigraphs(or upper graphs) associated with the given sequence of functions.Moreover, epi-convergence which can be viewed as a one-sided analog ofuniform convergence has good variational properties and plays an impor-tant part in various fields such as Stochastic Optimization;(47) GameTheory, statistics (convergence of estimators(30)), Mechanics (study of com-posite materials(16)), etc. Clearly, from their very definition, epigraphs areunbounded sets which explains the interest of relaxing the boundednesshypothesis.

So, the purpose of the present paper is to provide multivalued versionsof the LMCT for random sets with closed values, as well as applications toepi-convergence in a stochastic context. More precisely, if F is a random setwhose values lie in the Banach space X, we prove that the sequence( E ( F / ! % n ) ) converges to E(F/3Aao) in the sense of Painleve-Kuratowski(PK). Its infinite dimensional extensions such as, the Mosco convergence,the Wijsman topology or the slice topology are considered too.

However, even in the framework of finite dimensional spaces, ourresults are new or subsume already existing ones in the literature. In par-ticular, it is worth pointing out that our results cannot be deduced fromrecent multivalued versions of the general martingale convergence theorem

Multivalued Levy Type Theorem 935

such as in Hess(27) or Wang and Xue(51). It will be shown that the difficultpoint is not to prove the convergence (which follows easily from Hess(27)

or Wang and Xue(51)), but rather to show that the limit has the formappearing in the left-hand side of (*) (Remark 6). Let us also mention thatthe more general case where the random set F is replaced by a PK (orMosco) convergent sequence (Fn) was considered by Ezzaki(21) and byCastaing et al.(9) for the case where the sequence (^n) is nonincreasing).But in the situation considered in the present paper, i.e., Fn = F for all n,our assumptions are much less restrictive (see Remark 5.11).

We also remark that, although the probabilistic convergence resultdiscussed in the present paper is not of a very general nature, the methodhas a wide range of applicability. As the reader will see it is mainly basedon the general convergence criteria provided by Corollary 1. In ouropinion, this approach can be used as well for proving several other multi-valued strong limit theorems for unbounded random sets. Arguments of thesame type have already appeared, under a more implicit form, in otherworks of the second author [e.g., Hess(28,29)].

To conclude this introduction, let us say a word about our choice ofthe infinite dimensional setting. It is important to observe that, although itbrings technical complications, it allows us to modelize situations involvingintegral functionals, which is useful in many applications. For example, letQ be an open subset of Rk, 1 be the Lebesgue measure on Q, X be aBanach space of Rm-valued functions defined on Q, such as an Lp-space ora Sobolev space, and \f/: r x R m - > R be a map owning suitable regularityproperties. Then, consider the integral functional (/>: Tx X-> R defined by

This kind of mathematical object appears, for instance, in stochastichomogeneization [see Dal Maso and Modica(16)]. In Section 5, we showthat, provided an appropriate definition of conditional expectation is given,our main results apply to a general class of integrands including thisintegral functional.

Some notations and preliminaries are set in Section 2. In Section 3, theneeded convergence concepts are introduced, along with set convergencecriteria. This is done in connection with the notion of countably supportedsubset (or multifunction). The main results concerning multivalued analogsof the convergence property (*) with respect to the Painleve-Kuratowskiconvergence are stated and discussed in Section 4. In Section 5, we brieflyprovide applications to integrands. Section 6 contains the proof of the mainresults and final remarks.

936 Couvreux and Hess

Multivalued Levy Type Theorem 937

2. NOTATIONS AND PRELIMINARIES

Let X be a separable Banach space. Its norm is denoted by ||.||, thedual space by X* and the closed unit ball of X* by B*. We denote byS f ( X ) the space of all subsets of X, by ^(X) (resp. &C(X)) the space of allnonempty closed (resp. closed convex) subsets, by J^WC(X) (resp. ^ c b ( X ) )the space of nonempty weakly compact (resp. closed bounded) convex sub-sets and by $>(X) the Borel er-field. The distance function and the supportfunction of P e &(X) are respectively defined by

where x e X and y e X*. As it is known, s( ., P) = s( ., cl co P) where "cl co"denotes the closed convex hull operation. The asymptotic cone ofC e ^c(X), denoted by As(C), is defined by

As(C) does not depend on x0, provided C is closed. Further, As(C) is thepolar cone of the effective domain of the support function of C, defined by

The latter cone is also called the barrier cone of C and is denoted bybar(C) [see e.g., Aubin (1) and Castaing and Valadier(10)]. When C is notconvex, we shall use the notation As(C) for As(clco C) and bar(C) forbarfcl co C).

Given a set T, a multifunction F is a map from T into S f ( X ) . Thedomain of F is defined by

A selection of F is a function /: T-> X such that, f(t) is a member of F(t)for all t e dom(F).

Let (T,$f) be a measurable space. A multifunction F: T->3/(X) issaid to be Effros measurable or, in short, measurable, if for every open setU in X, the set

is a member of si [see Castaing and Valadier;(10) Hiai;(31-33) Hess;(23) orBeer(8)]. A measurable multifunction is also called a random set. A measurableselection of the random set F is an (stf, ^(x))-measurable selection of F,

938 Couvreux and Hess

and a Castaing representation of F is a sequence fn: T-> X of measurableselections of .F such that

Assume that a probability measure // is given on (T, ,s/) and consideran arbitrary sub-cr-field 38 of j/. We denote by L1(x, ^) = L1(T, 38, p, X)the space of (equivalence classes of) (3$, ̂ (x))-measurable functions/: T-* X, such that t -> | | f ( t ) | | is integrable. Such an f is said to be Bochnerintegrable. L 1 ( X , jtf) will be simply denoted by L 1 (X) . Recall that, becauseX is separable, the dual space of Ll(X) can be identified with the spaceL°°(X*, w*) of (equivalence classes of) bounded scalarly measurable2 func-tions g: T-> X*. For a short proof of this fact see Meyer,(39) [p. 301] orRaynaud de Fitte(46) where the completeness of the measure space is notneeded.

For every multifunction F: T - > 0 > ( X ) and every sub-cr-field 38 of j/,we set

It is known that S 1 ( F , stf] characterizes F up to ,«-null sets [see forinstance Hiai and Umegaki(34)]. A measurable multifunction F such thatS 1 ( F , £/) is nonempty is declared integrable. Using Hiai and Umegaki(34)

[Thm. 2.2], it is readily seen that F is integrable if and only if d(0, F ( . ) )eL1. Clearly, an integrable random set F satisfies dom F= T a.s.

Now, consider an integrable ^/-measurable multifunction F: T-> ^(X).Following Hiai and Umegaki(34) we define the multivalued conditionalexpectation of F relative to 33 as the ^-measurable random set G = E(F/38)such that S1(G, 3$) = c l {E( f /38) : f e S 1 (F , stf)}, the closure being taken inL 1 ( X ) (where E(f/3$) denotes the usual conditional expectation of aBochner integrable function). In the special case where 3$ = {T, 0],E(F/36) is simply denoted by E(F) and is equal to c\{E(f) : f e S 1 (F , ,a/)}.It is convenient to recall some properties of the multivalued conditionalexpectation [see Hiai(31,33) and Hiai and Umegaki(34)].

Proposition 1. If F, G, T->^(X) are two ^/-measurable integrablerandom sets and 38 a sub-cr-field of s$ we have:

2 That is, for every x e X, the function t-> (x, g( t )> is measurable (this notion is equivalentto the strong measurability when X* is separable).

(a)

Multivalued Levy Type Theorem 939

(b)

(c) If r is a .^-measurable function such that the random set rF isintegrable, then

(d) If g: T'-> X* is a bounded scalarly .^-measurable function, then

(e) If the random set F is convex valued and .^-measurable, and ifr is a positive ts/-measurable function such that the random setrF is integrable, then

In particular E(F/%) = F a.s.(f) If F is convex valued and if 36 and 98' are two sub-a-fields of ,s/

such that % £ ?$ c ,£/, then

(g) If F(t) £ G(t) a.s., then E(F/@) £ E(G/,«) a.s.

3. THE TOPOLOGICAL CONCEPTS: DEFINITIONS ANDCONVERGENCE CRITERIA

In this section, we recall the needed topological and convergenceconcepts, and introduce set convergence criteria, in connection with count-able supportness properties of closed convex sets and closed convex valuedmultifunctions.

Let a be any topology on X. If ( C n ) n > 1 is a sequence in 2 f i ( X ) we set

and

where ( C n ( k ) k > 1 is a subsequence of (Cn) The set er-Li Cn (resp. <r-Ls Cn)is called the lower limit (resp. the upper limit) of (Cn) relatively to topology a.

940 Couvreux and Hess

When a is metrizable, these subsets are both tr-closed. The sequence (Cn)is said to converge to C in the sense of Painleve-Kuratowski (PK), relativelyto a, if one has

[see for example Choquet;(11) Attouch;(2) and Beer(8)]. This is denoted byC = PK(a)-limn Cn or simply C=PK-limn Cn when X is finite dimensionaland a is the usual topology.

Further, if X is infinite dimensional and (Cn) PK-converges to Crelatively to both the strong topology and the weak topology of X(denoted by s and w respectively) then (Cn) is said to be Mosco convergentto C. This is denoted by

From the definitions, it is clear that the above equality is equivalent to thefollowing two inclusions

It was shown by Beer(6) that the Mosco convergence arises from the so-called "Mosco topology" denoted by .TM.

Now, we define the slice topology. The gap between to subsets B andC of X is defined by

A slice of a ball is the intersection of a closed ball and a closed halfspace passing through the interior of the ball. The slice topology on ^(X)is the weak (or initial) topology determined by the family of gap functional(d(B, .): B is a nonempty slice of a ball}. It is dented by ,TS. This topologywas introduced by Sonntag and Zalinescu(48) and Beer,(7) and intensivelystudied by Beer [see e.g., the monograph Beer(8)]. Further, from Beer,(7)

[Thm. 5.2], we know that the slice topology restricted to ^C(X) is theweak topology determined by the family {S(B, •): B e ^cb(X)}.

On the other hand, the Wijsman topology ,Twon f&(X) is the topologyof pointwise convergence of distance functions. It was introduced byWijsman(52,53) when X is finite dimensional. A net (CJ) of closed sets is saidto converge to C in the Wijsman topology if,

Multivalued Levy Type Theorem 941

Let us recall that the space (&(X), yw) is metrizable and separable,and that the Borel cr-field of this topology coincide with the Effros cr-field$ on g ' ( X ) [see Hess(23,27)]. Thus, Effros measurable multifunctions canbe regarded as single-valued measurable maps with values in a separablemetric space.

The following inclusions always hold: ,TM^3~S and &w<=, ,TS. When Xis reflexive we have .^w— ^~M = ^~s- In addition, if the norm is Frechet dif-ferentiable we have !9~w= &~M [see Beer(8)]. When X is finite dimensionalthe above three topologies (convergence) all coincide.

The PK and Mosco convergences, as well as the slice topology haveinteresting variational properties. More precisely, the polarity, regarded asa map from the set of closed convex subsets of X into the set of w*-closedconvex subsets of X*, is bicontinuous with respect to the slice topology.This continuity property had already been shown to hold for the PK con-vergence by Wijsman(51,52) (resp. for the Mosco convergence by Mosco(41)

when X is finite dimensional (resp. is reflexive). In turn, for convex functionsidentified with epigraphs, this implies the continuity of the conjugacy withrespect to epi-convergence (see Section 5), which is most useful for approxi-mating optimization problems [see Mosco;(41) Attouch;(2) Rockafellar andWets;(47) Beer(8)].

If F, Fn (n > 1) are closed valued multifunctions defined on T and if yis a topology on ,^(X), we shall simply write F=&~-limn Fn a.s., ratherthan F( t )= ,T-limnFn(t) a.s., if there is no risk of confusion.

The following equalities, valid for every x e X and B, C e^c(X), willbe useful. They are simple consequences of the Hahn-Banach Theorem.

where B* denotes the closed unit ball of X*.As already mentioned, the multivalued versions of Levy's Martingale

Convergence Theorem that we shall prove are valid for random sets whosevalues are not necessarily bounded. However, we shall require that thevalues of the multivalued conditional expectations of the random set satisfysuitable countability supportness hypotheses to be introduced later. A con-crete instance where these hypotheses hold is provided by the class ofclosed convex sets not containing any line: they are allowed to containhalf-lines only. When X is finite dimensional, no other condition is needed.In the infinite dimensional setting, additional conditions of topologicalnature are necessary.

942 Couvreux and Hess

For proving the second and more difficult "half" of the convergenceproperties, we shall need countability supportness assumptions concerningthe multifunctions appearing in the LMCT. The following definition intro-duces the needed concepts.

Definition 1. Let C be a nonempty closed convex set and D a count-able subset of B*.

(i) C is said to be D-countably supported if

(ii) C is said to be strongly D-countably supported if

(iii) C is said to be very strongly D-countably supported if

It is not difficult to see that the implications (i i i) = > ( i i ) = > ( i ) alwayshold, which justifies the terminology. More generally, a subfamily cf, of3PC(X) is said to be D-countably supported (resp. strongly D-countably sup-ported, very strongly D-countably supported) if this property hold for every

On the other hand, it is helpful to split each convergence property tobe proved into two parts. This corresponds to a general phenomenon thatis well-known among the specialists of set convergences [see e.g., Sonntagand Zalinescu;(48) Beer;(8) and Levi et al.(36)]. The following result concernsthe first and easier "half" of the convergence properties and does notinvolve the abovementioned countability supportness assumptions. Theproof is straightforward and is omitted.

Proposition 2. If C and Cn, for n > 1, are members of T ( X ) then thefollowing three statements are equivalent:

(i) C<s-Li Cn

(ii) lim sup n d(x, Cn) < d(x, C) for all x e X

( i i i ) lim supn S(B, C n )<8(B , C) for all Be F c b ( X ) .

Multivalued Levy Type Theorem 943

C e %, with respect to the same countable subset D. Similarly, a closedconvex valued random set G: T-> <£ will be declared D-coimtably supported(resp. strongly D-countable supported, very strongly D-countably supported).Sufficient conditions for countable supportness are given at the end of thissection. Proposition 3 and Corollary 1 show the interest of these notions inconnection with the desired convergence properties.

Proposition 3. Let ( C n ) n > 1 be a sequence in .F(X), C e /7c(X) and Da countable subset of B*. Assume in addition that

( 1°) If C is .D-countably supported then

(2°) If C is strongly D-countably supported then, for every x e X,

(3°) If C is very strongly D-countably supported then, for every B e.J7cb(X) one has,

Proof.

(1° ) For every v e w-Ls Cn one has v = w-limk vk where vk e Cn(k) and(Cn(k)) is a subsequence of (Cn). Consequently, for every y e D,

which by the countability supportness hypothesis yields thedesired conclusion.

(2°) Elementary calculations along with equality (3.2) yield

944 Couvreux and Hess

whence, using the hypothesis on the sequence of support func-tions,

Corollary 1 is an immediate consequence of Propositions 2 and 3.It is stated for further reference, as well as for its intrinsic interest.

Corollary 1. Let ( C n ) n > 1 be a sequence in &(X), C e ^c(X) and Da countable subset of B*. Assume that C s s-Li Cn and that (3.7) holds.

(3°) Similarly, using (3.3) and the hypothesis on C, we easily obtain

In order to present instances where these countably supportnessproperties hold (Proposition 5), we introduce the following conditions con-cerning separability properties of X* and continuity properties of supportfunctions. As we shall see next, the latter admit an equivalent geometricformulation.

Definition 2. Let X be a separable Banach space and p be a locallyconvex topology on X*, stronger than the weak-star topology w*, butnot necessarily weaker than the Mackey topology T = r(X*, X). Considera subfamily '-€ of 3F C (X) and a countable subset D of B*. We shall saythat the pair ((£, D) satisfies the separability hypothesis with respect to the

(1°) If C is D-countably supported then C=M-limn Cn.

(2°) If C is strongly D-countably supported then C = Tw-limn Cn.

(3°) If C is very strongly D-countably supported then one has C =,Ts-limn Cn.

Multivalued Levy Type Theorem 945

topology p or, in short, that it satisfies (Hp), if the following two conditionshold:

Clearly, ( H s * ) is a special case of (Hp). As to (Hr), recall that theseparability of X implies that condition (5T) is satisfied for a suitable3

countable set D. Condition (CT) is satisfied too, as Proposition 4 shows. Asa consequence, when X is reflexive (and separable) conditions (Hs*) and( H T ) are identical. On the other hand, Proposition 4 provides a geometriccharacterization of Hfwc(X) and ftfsc(X), and shows that in these formula-tions of ( H s * ) and (Hr), the choice of c£ is maximal. The proof concerningtewc(X) (resp. <gsc(X)) can be found [see Castaing and Valadier,(10)

Cor. I.15] (resp. Moreau,(40) Cor. 8.e.)

Proposition 4. Let X be a normed linear space and C a member of&c(X). The following statements are equivalent:

(i) C is a member of %SC(X) (resp. £ewc(X))

(i i) the function s ( . , C) is finite and continuous at some point of X*in the strong topology (resp. in the Mackey topology)

(i i i ) the intersection of C with some closed half-space is bounded(resp. weakly compact).

3 In tact, the separability or X entails the separability of X* with respect to every topologycompatible with the pairing (X, X*), whence with respect to the Mackey topology.

Now, let us provide examples of pairs ( & ' , D ) for which (Hp) holds.When X is an infinite dimensional Banach space, hypothesis (Hp) issatisfied in the following two situations where p is replaced, either by thestrong topology s* or by the Mackey topology T on X*. So, we considerthe following hypotheses ( H s * ) and ( H T ) .

(Hs.) p = s*, X* is s*-separable, D is a countable s*-dense subset ofB* and c = c S C ( X ) defined by

%sc(X) = {Ce^c(X): s( •, C) is s*-continuous at some point of X*}

(Hr) p = r, D is a countable r-dense subset of B* and <& = ywc(X),the class of nonempty weakly locally compact convex sets which containno line.

(Sp) D is a /?-dense subset of B*

(Cp) for every Ce^, the support function s ( - , C) is /^-continuous atsome point of B* (equivalently, the p-interior of bar(C) is nonempty).

Thus, in the special case where X is a finite dimensional Banach spaceendowed with its usual topology, condition (Hp) is satisfied when c& is theset of all closed convex sets not containing any line. In this case weobviously have # = ^SC(X) = &WC(X).

Remark 1. More generally, when X is not assumed to be finitedimensional, it is useful to mention the following inclusions. First, C$SC(X)contains £fwc(X), because the Mackey topology is weaker than s*. ^SC(X)contains &cb(X), because a closed convex set C is bounded if and only ifthe support function s ( . , C) is finite and strongly continuous at each pointof X* (which in turn is equivalent to the continuity at 0). Similarly, it isknown that a closed convex set C is a member of J#"WC(X) if and only if itssupport function s ( - , C) is continuous in the Mackey topology at everypoint of X* [see Moreau,(40) or Castaing and Valadier(10)] so that &WC(X)contains tfwc(X).

The following result shows the relation between hypotheses (Hp) andthe countability supportness hypotheses introduced earlier. It extendsLemma III.34 of Castaing and Valadier,(10) in the framework of Banachspaces.

Proposition 5. If the pair (c(>, D) satisfies (Hp), then c€ is stronglyD-countably supported. Furthermore, if (#, D) satisfies (Hs*) then <& isvery strongly D-countably supported.

Proof. In view of (3.2) and (3.5), in order to prove the first statementit suffices to show that, for every x e X, one has

4. MULTIVALUED VERSIONS OF THE LEVY MARTINGALECONVERGENCE THEOREM

This section contains the main results of the present paper. Let(^n)n>1 be a non decreasing sequence of sub er-fields of .a/ and ^ be the

which in turn is a consequence of Lemma 2.2 in Hess.(28) Similarly, if(V, D) satisfies (H s * ) , we have <# = C#SC(X). From (3.3) and (3.6), we can seethat it is enough to prove that for every B e ^cb(X)

which is done similarly.

946 Couvreux and Hess

Multivalued Levy Type Theorem 947

CT-field generated by (Jn>1 @n . For sake of clarity and easy reference, webegin by recalling the classical versions of the LMCT for a positivemeasurable function and for a Bochner integrable vector-valued function[see e.g., Neveu(42)].

Proposition 6.

(i) If f: T'^> [0, +00] is a positive measurable function, then

for almost every t in the set

(ii) lf f e L 1 ( X ) , then

where the limit holds almost surely in the norm topology of Xand in L 1 ( X ) .

As already mentioned, our goal is to extend this result to the casewhere f is replaced by a random set. So, we consider an integrable randomset F whose values are nonempty closed subsets of X. The possibility ofunbounded values for F will make necessary to consider nonintegrablemeasurable functions, especially support functions of unbounded randomsets, which explains that we have also recalled the extension of the LMCTto nonintegrable functions.

Theorem 1, and Corollaries 2 and 3 next, are the main results of thepresent paper.

Theorem 1. Let F: T^> 3 F ( X ) be an integrable .^/-measurable multi-function satisfying the following two assumptions (A) and (B):

(A) for almost every t e T, E(F/3$ao)(t) is convex

(B) for some countable subset D of B* and for almost every t e Tthere exists an integer n0(t), such that

948 Couvreux and Hess

Then, the following three statements hold:

[ 1°) If E(F/3$ao) is D-countably supported then

(2°) If E(FI3SX) is strongly D-countably supported then

(3°) If E(F/3$a>) is very strongly D-countably supported then

Observe that condition (B) involves subsets of the dual space X*. In thefollowing two corollaries, where the values of the conditional expectationsare assumed to be in £fwc(X), condition (B) is replaced by condition (B')which involves subsets of X and is of a more geometric nature.

Corollary 2. Consider the additional hypotheses (B'), (C) and (D):

(B') for almost every t e T there exists an integer n0(t), such that

Then,

(a) conditions (C) and (D) imply the equivalence of (B) and (B')

(b) conditions (C), (D) and (B') imply properties (4.3)-(4.4) ofTheorem 1.

In the following corollary, where X is assumed to be finite dimen-sional, condition (C) can be replaced by (A) which is weaker.

Corollary 3. Assume that X is finite dimensional.

(C) for almost every t e T, E ( F / B a o ) ( t ) e ^WC(X)

(D) for almost every te T, there exists an integer n0(t) such that

Multivalued Levy Type Theorem 949

(a) conditions (A) and (D) together imply (C)

(b) if (A), (B') and (D) hold, then one has

Remark 2, Assumption (A) is satisfied in the following two situa-tions ( A l ) or (A2):

(A1) F is convex valued

(A2) the probability space (T, .P/, //) has no ,3^-atom

Clearly, condition ( A l ) implies (A). As to (A2), we recall that given a sub-cr-field :^ of ,a/, a subset A e,«/ is called a £$-atom if, for every A' e .a/ suchthat A'sA, there exists B e,^ verifying fj.((A r\B) AA') = 0 where "/I"denotes the symmetric difference operation. In the special case where$ = {$, Q}, a ^-atom is an atom in the usual meaning. If (A2) holds, con-dition (A) holds even if the values of F are not assumed to be convex.Indeed, from Valadier(50) [Thm. 2], we can deduce that if (T, stf, fi) has no^o-atom, then E(F/36^) is almost surely convex valued. Further, thedefinition of a ,^-atom implies that, for every n>1, (T,stf,p.) has no,^,,-atom either, which in turn implies that E(Fj9Sn) is almost surelyconvex valued. Consequently, condition (A2) entails the a.s. convexity ofE(Fj&n)(t) for every « e N*u{oo} . On the other hand, observe that (C)implies (A).

From Proposition 5 we know that the countably supportness condi-tions of Theorem 1 (1° and 2°) (resp. Theorem 1 (1°)-(3°)) are satisfiedwhen E(FISSoa}(t}€^WC(X} a.s. (resp. when X* is strongly separable andE(F/3$aa)(t)e<gsc(X) a.s.). Consequently, it is interesting to mention thefollowing result which displays a general situation where the values of themultifunction E(F/&) almost surely lie in £fwc(X) (resp. ( g s c (X) ) , where 36is any given sub-u-field of «a/.

Proposition 7. Consider:

(i) a positive ,a/-measurable function r

(i i) a closed convex valued multifunction G which is assumed to be.a/-measurable and integrably bounded

(ii i ) a .^-measurable multifunction L whose values lie in £fwc(X)(resp. ^SC(X)) and such that the random set rL is integrable.

950 Couvreux and Hess

In addition, we assume that the values of G are weakly compact (resp.bounded). If the random set F satisfies the inclusion

then the multifunction E(F/3S) almost surely takes its values in <£WC(X)(resp. in ^SC(X)).

Proof. Using (4.7) and the properties of the multivalued conditionalexpectation, as recalled in Proposition 1, we immediately obtain

Consequently, we can find a negligible subset N such that, for every t eT\N and for every y e X*,

Now, first assume that L is £?wc(X)-valued and observe that, for every real 1,every weakly compact convex subset K and every C e ^ f w c ( X ) , one has

The second property follows from the fact that a closed convex set K is amember of 3fwc(X) (resp. <£WC(X)) if and only if its support function s(., K)is finite and continuous at every point of X* (resp. at some point of X*)in the Mackey topology (see Proposition 4 and Remark 1). This yields thedesired conclusion. Similarly, in the case where L is ^sc(X)-valued and Gis ,̂ cb( X)-valued use the properties of the support functions of members of^cb(X) and V s c ( X ) .

Remark 3. Proposition 7 admits the following consequence. Assumethat inclusion (4.7) holds, that G is ,Xwc(X)-va\ued and that L is &WC(X)-valued. If L is ^-measurable (resp. ^-measurable for some n> 1), thencondition (C) (resp. condition (D) ) is satisfied. More general conditionsimplying (D) could also be given.

Now, let us discuss hypothesis (B') appearing in Corollaries 2 and 3.It limits the unboundedness of the multivalued conditional expectation ofF via an inclusion involving the asymptotic cones. For sake of simplicity,we assume that X is finite dimensional, and that (A) and (D) hold. Thisyields the equivalence of conditions (B) and (B'). The important point isthat, in spite of its rather restrictive nature, assumption (B') cannot beremoved as the following example shows.

Example 1. Consider the case where X=R, r=[0, 1], ,c/ = theBorel CT-field of T and /u = the Lebesgue measure. For each n > 1, considerthe sub-ff-field Mn generated by the family of intervals [k/2n, (k + 1 )/2n]( 0 < k < 2 n ) . Here, .^ = .o/ = the cr-field generated by Un>1 -4n. Let/be apositive real valued .^-measurable function such that E(//.#n)(t)= +xa.s. An example of such a function can be found in [Neveu,(42) p. 31].Now, define the random set F by F(t) = [0, f ( t ) ] , for t e T. By Proposi-tion l (e ) , E(F/.j$x)(t) = F(t) a.s. Moreover, we claim that, for every integern>1,

Multivalued Levy Type Theorem 951

Indeed, consider the support function at y = 1 and observe that by Proposi-tion l ( d ) we have

Clearly, hypotheses (A) and (D) are satisfied, since the values of F areintervals of R+, but hypothesis ( B ' ) is not satisfied, because, for everyn > 1, one has

so that r\n>1 As(E(F/.#n)(t)) = R + which is not contained in As(E(F/.#a:.)(t))= As(F(t)) = {0}. Finally, observe that the conclusion of Corollary 3 doesnot hold, because Ls E ( F / . # „ ) ( t ) = R+ is not contained in E(F/.tf^ ) ( t ) =[0 , f ( t ) ] .

Although we are mainly concerned with the case where F has unboundedvalues, it is interesting to give the following corollary which is a particular-ization of our main results to the case where F has bounded values. Recallfirst that a random set F is said to be integrably bounded, if the positivefunction t -> \\F(t)\\ = sup{ ||.Y|| : x e F(t)} is integrable. Clearly, an integrablybounded random set is also integrable (see Section 2) and takes on boundedvalues a.s.

Corollary 4. Let F: T - > / 7 ( X ) be an integrably bounded random setsatisfying hypothesis (A) of Theorem 1 and one of the following two condi-tions ( w k ) or (bo):

(wk) the values of F are weakly compact

(bo) the values of F are bounded and X* is separable.

952 Couvreux and Hess

where 2T = 3~M or 2TW when (wk) holds and y = $~s when (bo) holds.

Proof. We begin by observing that, since F is integrably bounded, sois E(F/38), the conditional expectation with respect to any sub-er-field 88.Thus, condition (B) of Theorem 1 is satisfied in both cases, because thesupport function of a bounded set is finite at every point of X*. Now, wefirst consider the case where F(t) is weakly compact valued. By Krein'stheorem, we know that the closed convex hull of F(t) is weakly compacttoo. Further, by a result of Coste(13) [p. 955] it is known that, for almostevery t, E(cl co F/3§x,)(t) = E(F/&aa)(t) is convex weakly compact. Conse-quently, it is strongly D-countably supported, so that Theorem 1 (1° and 2°)applies. This finishes the proof in the case (wk). Similarly, when (bo)holds it can be easily shown that E(F/.^ao) is very strongly D-countablysupported.

Then we have

Remark 4.

(i) The proof of Coste's aforementioned result on the weak com-pactness of the multivalued conditional expectation E(FJ3$}appeals to a desintegration theorem. A second proof can begiven by relying on a result of Klei(35) [Thm. 3.6], whichcharacterizes the relative weak compactness of S 1 ( F ) in terms ofrelative weak compactness of F(t) a.s. Using this result and theweak continuity of the conditional expectation map f-» E(f/{%),one can show, that E(F/3S)(t) is weakly relatively compact,whence weakly compact, because E(F/3$)(t) is closed andconvex. When j/ is assumed to be countably generated, a thirdproof is also possible by appealing to Valadier,(50) [Thm. 1].

(ii) Results asserting the weak compactness of S 1 ( F ) , assuming thatthe values of F are convex and weakly compact, are older thanKlei's result. [See e.g., Castaing and Valadier,(10) Chap. V.]

(iii) The case of integrably bounded random sets with values in3CC(X) the set of convex strongly compact subsets of X, wastreated by Hia'i and Umegaki(34), the conditional expectationwas shown to be a member of Jfc(X) [Thm. 5.6(1°)] and theconvergence property (4.13) was proved with respect to theHausdorff metric topology 2TH on Jfc(X) [Hiai and Umegaki,(34)

Multivalued Levy Type Theorem 953

Thm. 6.1]. A L1-type convergence was also proved. Recently,Wang and Xue have extended this result to the case where,tfc(X) is replaced by any .^-separable subspace of .?C(X)[Wang and Xue,(51) Thm. 3.1]. The case where (.tfn) is nonin-creasing was treated by Dong and Wang.(18)

Remark 5. For sake of comparison with recent works, it is interestingto mention a problem related to that of Theorem 1. Instead of consideringonly one random set F, consider a sequence (Fn) such that F(t) =M-limn Fn(t) a.s. So, it is natural to ask under which conditions one has

This problem was addressed by several authors. For random sets withconvex weakly compact values, a result resembling our Corollary 4 (case(wk) ) was obtained by Choukairi-Dini,(12) [Thm. 3.5] assuming theintegrability of sup,, \\Fn( .)|| and the reflexivity of X (which is not neededin Corollary 4). Piccinini(45) considered the case of random sets withstrongly compact values, in a more general probabilistic setting and with-out the reflexivity assumption. On the other hand, the case of unboundedrandom sets was treated in Ezzaki,(21) [Thm. 3.3, p. 139; Rem. 2, p. 140]and in Castaing et al.(9) [Thm. 4.7] when the sequence (.;#n) is nonincreas-ing. Although, this situation is a priori more general than that of the pre-sent paper, the results in Ezzaki,(21) are not comparable with Theorem 1.Indeed, if one set Fn = F in Ezzaki(21) [Thm. 3.3], the conclusion ofTheorem 1 is recovered only under an additional stringent hypothesis,namely, multifunction F would be asked to satisfy As(F(t)) — As(Z(t))almost surely, where Z is a ,^1-measurable integrable <fwc( X (-valued multi-function. Anyhow, it should be mentioned that this sort of condition wasshown to be necessary [see Hess(24,28)], where the Mosco convergence ofE ( F n 3 $ ) to E(Fx/.%) was examined. There, the cr-field J# was fixed. In thepresent paper where the multifunction F is held fixed and the sub-cr-field isallowed to vary, this condition is no longer required.

Remark 6. Let us comment between Theorem 1 and multivaluedversions of more general martingale convergence theorems for unboundedrandom sets. First, we need to introduce the following subfamily of . ¥ C ( X ) :

were 5(0, r) denotes the closed ball of radius r centered at 0. It is readilyseen that ,jfwc(X) contains ywc(X). When X is reflexive .#WC(X) coincides

954 Couvreux and Hess

with ^ C ( X ) . The members of !%WC(X) are said to be weak ball-compact,because the intersection of such a set with any closed ball is weakly com-pact. This class of subsets was introduced by Hess(25-27) and has appearedto be useful in several instances [see e.g., Balder and Hess(3,4)]. Now,assume the following hypothesis, (WB) hereafter:

(WB) there exist a multifunction R with values in 3twc(X) such that,for every n >1,

If one is only interested in the existence of the a.s. limit of the sequence(E(F/ .<% n ) ) , but not in the identification of the limit, it is worthwhile tomention that hypotheses of Theorem 1 can be weakened. Indeed, if weassume (WB), Hess,(27) [Thm. 5.12] applied to the multivalued martingaleFn = E(F/@ln) shows that (Fn) admits an almost sure M-limit (in fact a, s - l im i t as shown by Theorem 2). Consequently, there exists a multi-function G such that

Moreover, Corollary 5.13 of Hess(27) shows that, for every n> 1,

Thus, it seems that the role of the additional hypotheses of Theorem 1and Corollaries 2 and 3 is to permit the identification of the limit, namelythe coincidence of G(t) and E(F/38K>)(t) almost surely [see Couvreux,(14)

Chap. V, Section IV.2 for further details]. In fact, for various reasons, theidentification of the limit is the main difficulty encountered in the theory ofgeneral vector valued and multivalued martingales in infinite dimensionalspaces [see e.g., Neveu(42,43) and Egghe(19)].

This discussion leads us to present another variant of Theorem 1 whencondition (HT) holds. This requires the additional hypothesis (R) .

Theorem 2. Let X be a separable Banach space and F:Q-+tF(X) bean integrable random set satisfying conditions (B) and (C). Also assumethe following hypothesis (R) next.

(R) there exists R 0 e , R w c ( X ) such that F(t) s R0 a.s.

Then, E ( F / B x l ) ( t ) = J s - l i m n E ( F / B n ) ( t ) a.s.

Multivalued Levy Type Theorem 955

Proof. Trivially, condition (C) implies (A). By Proposition 5, it alsoimplies that E(F/t%aa) is strongly D-countably supported with respect toany countable r-dense subset D of B*. Therefore, conclusions (1° and 2°)of Theorem 1 hold. On the other hand, from Hess(29) [Prop. 3.3] we knowthat the restriction of the slice topology to the family of closed convex sub-sets of R0 coincides with the Mosco topology, which entails the desiredresult.

Remark 7.

(i) Observe that condition (R) is stronger than (WB). Also, if R0 isassumed to be a member of ywc(X), conditions (C) and (R) areboth satisfied.

(ii) The following example presents a very simple situation whereTheorem 1 applies, although Corollaries 2 and 3 do not. Let y0

be a nonzero member of B*, x e L1 and F be the random setdefined by

Its support function is given by

Clearly, F(t) contains lines. Further, for every sub-u-field y8, it is not dif-ficult to see that E(FJH8) satisfies

so that E(F/;%) is very strongly D-countably supported with respect toD = { y 0 } . Finally, straightforward calculations and Corollary 1 show that

The reader will check that less trivial examples can be constructed inthe same way. For instance, the constant vector y0 could be replaced by asuitable measurable function g with values in a countable subset of B*.

5. APPLICATION TO EPI-CONVERGENCE OF INTEGRANDS

Almost all the results of Section 4 involve conditional expectationsof unbounded random sets. Therefore, they apply to integrands via their

956 Couvreux and Hess

epigraphical multifunctions. First, it is useful to recall some definitions andknown facts. If u is an extended-real valued function defined on X, itsepigraph (or upper graph) is denoted by epi(w) and defined as the subsetof X x R such that

Further, u is said to be proper if it is not identically + oo and if it doesnot take the value — oo. If u is convex, lower semicontinuous (Isc) andproper, the asymptotic function of u, denoted by AS(M), is defined by theequality epi(As(u)) = As(epi(u)).

The function u is said to be inf-weakly compact for a certain slope ifthere exists y e X* such that the function x - » ( u ( x ) — < y, jc) is inf-weaklycompact (i.e, the level sets are weakly compact). By Castaing andValadier,(10) [Thm. 1.14] if u is Isc and proper, this is equivalent toepi(u) e Jwc(X x R), which shows the connection of this notion withhypothesis (Hr).

The conjugate (or polar) function of u is denoted by u* and definedon X* by

Let M, u n ( n > 1 ) be extended-real valued functions defined on X. Given atopology y on ^(X x R), the set of nonempty closed subsets of X x R, thesequence (u n ) n > 1 is said to be 2T-epi-convergent to u if

which is denoted by u = :T-\\meun. We also say that u is the ,T-epigraphi-cal limit of (un) (the index "e" stands for "epigraphical").

Further, a function <j>: T x X -> [ — co, + oo[ such that < / > ( t , •) is an Iscfunction and such that the epigraphical multifunction F: t—> epi <j>(t, •) ismeasurable, is called a normal integrand. When the probability space iscomplete, the latter is equivalent to the 1a/®,^(A')-measurability of </> [seee.g., Castaing and Valadier(10)]. Moreover, if the random set F is integrable(resp. convex valued), such a normal integrand is said to be integrable(resp. convex).

Let 3$ be a sub-<r-field of stf and </> be a normal integrand as above.The conditional expectation of </> relative to 88 is the normal integrand ij/whose epigraphical multifunction G: t -> epi \f/(t, •) is the conditional expec-tation of the random set t -»epi < f > ( t , . ) with respect to .$. Namely, we have

Multivalued Levy Type Theorem 957

G = E(F/3&). The integrand \l/ is also denoted by Ee($/.$) where the indexV recalls that i/> is obtained by an epigraphical sum (i.e., the multivaluedconditional expectation of an epigraph depending on t). Moreover, accord-ing to Castaing and Valadier(10) [Thm. VIII.36], it is possible to charac-terize the conditional expectation of a convex normal integrand by theconjugate integrand. Indeed, a convex normal integrand tj/ is equal toE e(</>/3$) if and only if, for every ,^-scalarly measurable (class of) functionv e L°°(X*, w*) (see Section 2) and for almost every re T, one has

where the conjugate integrand </>*: T x X* -> [ — cc, + oo[ is defined by

The following result is a particularization of Corollary 2.

Theorem 3. Let <j> be an integrable normal integrand defined onT x X satisfying the assumptions (a) and (b):

Then, one has E e ( $ I M a o ) ( t , . ) = F - l i m e E e ( < l > / M n ) ( t , . ) a.s. where ,^~ standseither for the Mosco convergence or the Wijsman topology.

Proof. Consider the epigraphical multifunction F: /->epi <j>(t, •). It isnot difficult to check that (a) implies conditions (C) and (D) of Corollary 2and that (b) implies (B'). Consequently, in view of the results and nota-tions recalled earlier, it suffices to apply Corollary 2 to the epigraphicalmultifunction F: t -> epi (j>( t, •).

Using the continuity property of the conjugacy, it is possible to obtaina result for integrands defined on T x X*.

Corollary 5. Assume the same hypotheses as in Theorem 3 andassume that X is reflexive. For every n e N* u {cc}, we define the integrandyn on T x X* by

(a) for every n e N*u{co} and for almost every t e T, the functionE e ( ( j > / B n ) ( t , •) is inf-weakly compact for a certain slope

(b) for almost every t, there exists an integer n(t) such that

958 Couvreux and Hess

(here, the conditional expectation is the usual one). Then we have

Proof. First, we remark that the integrability condition on <j> yieldsthe existence of v e L 1 ( X ) such that < / > ( • , v ( - ) ) + eL1. By conjugacy thisimplies that, for every y e X*, (/>*(•,y)~ eL1 which shows that the yn arewell defined. To obtain the desired convergence result, it suffices to useTheorem 3, equality (5.1) and the continuity of the conjugacy with respectto Mosco epi-convergence [see e.g, Attouch,(2) Thm. 3.18].

Remark 8. A formulation of Theorems 2 in terms of integrands ispossible. Instead of giving the complete statement, we observe thathypothesis (R) of Theorem 2 admits the following functional translation.

There exists a function g from X into the extended reals such that:

(i) the restriction of g to every closed ball is inf-weakly compact

(ii) for almost every t e T and every x e X one has g(x) ^</>(t, x).

and

for every y e D n bar(E(F/$ao(t)), where D is the countable subset of B*appearing in the statement of Theorem 1. The proof of this theorem will bedone in two steps.

Step 1. Proof of Eq. (6.1). This step is the easiest one and will alsoserve for Corollaries 2 and 3, because only the integrability condition on Fis required. The values of F only need to be closed.

To prove (6.1) the following auxiliary lemma is needed. It provides acharacterization of the inclusion of two multifunctions by means of their

6. PROOF OF THE MAIN RESULTS

This section contains the proofs of Theorem 1, and Corollaries 2 and 3.

Proof of Theorem 1. We begin by observing that in, view ofCorollary 1, it suffices to show the following two relationships, for almostevery t e T,

Multivalued Levy Type Theorem 959

sets of integrable selections. It can be found in Valadier,(49) where (T, ,<s/, n)is assumed to be complete. Here, the completeness hypothesis is shown tobe unnecessary.

Lemma 1. If F, G: T-> ,J7(X) are two integrable random sets, thenthe following statements are equivalent:

where G(t)c denotes the complement of G(t). Assume that (a) does nothold. Then, there exists a member A of ,<?/ such that ju(A) > 0 and F ( t ) nG(t)c^0 for every t e A. Since G(t}c is open and (fn) is a Castaingrepresentation of F, we have A £ (Jn>1 An. This implies the existence of aninteger k> 1 such that f i ( A k ) > 0 . Now, consider an arbitrary member f ofS 1 ( F , ,a/) and define the selection g = ^(Ak) fk + 1(Ac

k) f, where V(Ak)stands for the (probabilistic) indicator of Ak. Clearly, g is a member ofS 1 F, .P/) but not of S1 G, ,rf).

Now, consider an integrable ^-measurable random set F: T - > ( X )and define the multifunction L by L(t)=s-Li E(F/3$n)(t). From Hiai,(33)

[Thm. 2.2] we know that L is measurable with respect to the //-completionof j/ (equivalently, there exists an ,c/-measurable multifunction L' suchthat L ' ( t ) = L(t) /i-almost surely). The following proposition is the preciseformulation of (6.1). It is a Fatou-like inclusion for the strong upper limitin the same spirit as Hiai,(33) [Thm. 2.3].

Proposition 8. As in Theorem 1 we consider a non decreasingsequence (&„) of sub-er-fields of ,s/ and F: T - > ( X ) an integrable .(/-mea-surable random set. Then for almost every t e T one has

Proof. In view of Lemma 1, it is sufficient to show the inclusionS 1 ( E ( F / 3 # 0 0 ) , , r f ) c S 1 ( L , ,rf) where L = s-Li E(F/3$n). First, appealing toProposition 6(ii), we see that, for any selection f e S 1 (F , ,a / ) , one has

Proof. Since (a)^>(b) is obvious, we prove the converse implication.First observe that by Lemma 1.1 in Hiai and Umegaki,(34) the multifunc-tion F has a Castaing representation ( f n ) n > 1 whose members are Bochner-integrable functions. For each n > 1, define the .c/-measurable subset An by

(a) for almost every t e T, F(t) £ G(t)

(b) S*(F, A ) c S 1 ( G , A).

960 Couvreux and Hess

E(f/3$ao)(t) = l i m n E ( f / , ' % n ) ( t ) a.s., whence E(f/^Xl)(t)e L(t) a.s., becauseE(f/3Sn) is a measurable selection of E(F/Mn). Further, if g e S1(E(F/^ao), .&),the definition of the multivalued conditional expectation yields the exist-ence of a sequence ( f k ) k > 1 in S 1 ( F , stf} such that

where the limit holds in L l ( X ) and almost surely (by extracting a sub-sequence). Hence, the conclusion follows from Lemma 1.

Step 2. Proof of Eq. (6.2). It remains to prove inequality (6.2) incases (l°)-(3°). Without loss of generality we can assume that O e F ( t ) a.s.Indeed, consider an integrable selection f of F and define the random setF' by F' = F — f. Clearly, due to elementary properties of the weak upperlimit, the desired inequality holds for F if and only if it holds for F'.Further, the hypotheses are not modified by a translation. Thus, all therandom variables s(y, E(F/3#n)), for n e N* u {oc}, are positive. Now, forevery t e T and y e D we set

Clearly, A(y) is a 38^ -measurable subset of T, because t e A(y) if and onlyif s(y, E ( F / 3 # n } ( t ) ) is finite for some n > 1. Further, one can find a negligiblesubset N of T such that for every t e T\N the following properties (P1 )-(P6)hold:

Indeed, property (P1) follows from Remark 8, properties (P2) and (P3)follow from the hypotheses, (P4) follows from Proposition l(d). Property(P5) is a consequence of (P1), of Proposition 6(i) and of the countabilityof D. We can also assume that (P6) holds because otherwise inclusion (6.2)

(P1) OeF(t)

(P2) E ( F / B o o ) ( t ) is convex

(P3) inclusion (4.2) holds, namely, D n b a r ( E ( F / . B O ) ( t ) ) c Un>n0(t)

bar(£(F/Bn(t))

(P4) for every ye D and n e N * u { o o } , s ( y , E ( F / B n ) ( t ) ) =E ( s ( y , F ) / B n ) ( t )

(P5) for every yeD, E(s(y, F ) / B m ) ( t ) = limn E(s(y, F ) / B n ) ( t ) pro-vided t e A(y)\N

(P6) E(F/Bm)(t)=X.

Multivalued Levy Type Theorem 961

would be trivially satisfied. Thus, since E(F/.JJ00) is D-countably supported,equality (3.4) implies that for every t eT\N, D n bar (E(F .^ K ) ) ( t ) )^0 ,which in view of (P3) yields Z)n5(?)^0.

Now, let us prove statements ( l°) - (3°) . In view of (P3), the proof willbe completed if we show that (6.2) holds, for every t e T\N and everyyeDriB(t). So, consider t eT\N and y e D r\B(t). Since t e A ( y ) , proper-ties ( P 1 ) and (P4)-(P5) imply

which yields (6.2). Thus in view of the results of the first step, an appealto Corollary 1 finishes the proof of Theorem 1.

For proving Corollaries 2 and 3 it is still possible to assume that con-ditions (P1)-(P6) hold.

Proof of Corollary 2. In order to prove Corollary 2, the followinglemma, of a purely deterministic nature, is necessary.

Lemma 2. Let C, Cn (n > 1) be members of .^C(X) and D be a count-able r-dense subset of X*, where r stands for the Mackey topology. Alsoassume the following hypotheses:

where the last equality is a consequence of hypothesis (b) which shows bypolarity that the sequence (ir*-cl b a r ( C n ) ) n > k is nondecreasing, so that theunion of this sequence of convex subsets is still convex. Further, convexity

Then the following two statements are equivalent:

Proof. ( i ) = > ( i i ) : Taking the polar of each side of inclusion (i) entails

962 Couvreux and Hess

also implies that the closure does not change if w* is replaced by theMackey topology, which yields

From hypothesis (a) and Proposition 4 we know that bar(C) and(jn>kbs.r(Cn) have a nonempty -r-interior, whence by the topologicalproperties of convex sets

Consequently, we see that the intersection of D with each side of (6.3) isnonempty and that the following inclusions hold

( i i ) = > ( i ) : Since r-intbar(C) is nonempty, and the w*-closure andr-closure operations are identical for convex subsets of X*, inclusion (ii)implies

so that (i) is immediately obtained by polarity.

Remark 9. When X is reflexive, topologies r and s* coincide andywc(X) = <&sc(X). When X is not reflexive, implication ( i ) = > ( i i ) , that weneed in the present proof, is not true when C and Ck are only assumed tobe members of ( & S C (X) (even if X* is separable). Indeed, in the dual of anonreflexive Banach space, the r-closure of a convex set need not coincidewith its i*-closure.

Now, provided we show that, for almost every t e T, the sequence( A s ( E ( F / ^ n ) ( t ) ) ) n > n 0 ( t ) is nonincreasing, Lemma 2 will imply that, underhypotheses (C) and (D), conditions (B) and (B') are equivalent, which willprove Corollary 2(a). This is the purpose of Lemma 3.

Lemma 3. Assume that hypothesis (D) holds, namely that, foralmost every t e T, there exists an integer n0(t) such that E (F/ !% n ) ( t ) eywc(X) for all n > n 0 ( t ) . Then, for almost every t e T, the sequence( A s ( E ( F / . < % n ) ( t ) ) ) n > n 0 ( t ) is nonincreasing.

Multivalued Levy Type Theorem 963

Proof. We ought to show that, for almost every t e T and for everyinteger k > n 0 ( t ) , the following inclusion holds

Let D denote a countable r-dense subset of B*. Using the fact that thesequence (.£n) is nondecreasing and elementary properties of conditionalexpectations, we get for all y e D and k > n 0 ( t )

which by Proposition l ( d ) yields

This implies the existence of a negligible subset N such that, for everyt e T\N,

Since D is r-dense and, for all n > n 0 ( t ) , bar(E(F/.j$n)(t)) is a convex conewith non empty r-interior, inclusion (6.5) and the topological properties ofconvex sets imply

Convexity also implies that, in (6.6), the r-closure can be replaced by thew*-closure, so that (6.4) is obtained by polarity.

As mentioned earlier, Lemmas 1 and 2 entail that, under assumptions(C) and (D), conditions (B) and (B ' ) are equivalent. Since, by Proposi-tion 5, <£WC(X) is strongly D-countably supported with respect to any count-able r-dense subset D of B*, Corollary 2(b) follows from part (a) and fromTheorem 1 (1° and 2°).

Proof of Corollary 3. For proving this corollary the following twolemmas are needed. The first one shows that, when X is finite dimensional,(D) can be reformulated in a stronger way, which will permit us todeduce (C).

964 Couvreux and Hess

Lemma 4. When X is finite dimensional, hypothesis (D) ofCorollary 2 implies (is equivalent to) the condition (D') hereafter

(D') there exists a partition ( A n ) n > 1 of T such that, for all n>1,An e &n, and for almost all t e An, E(F/38n)(t) contains no line.

Proof. We proceed in two steps.

Step 1. Let SH be an arbitrary sub-er-field of ,a/. Let us show that ifG: T->3?C(X) is a .^-measurable multifunction, then the subset A defined by

The support function is lower semicontinuous. Thus, the intersection onthe right-hand side can be restricted to Dr\B(y, 1 / p ) , which shows that Ais a member of .$.

Step 2. In order to simplify the proof, we assume that the integern0(t) appearing in condition (D) is equal to 1, which does not restrict thegenerality. For every n > 1 we define the multifunction Gn and the subsetBn of T by

Due to condition (D), we have T= \Jn Bn a.s. Further, each Bn is .^-measur-able by the first claim. Consequently, a partition (An) of Tcan be constructedby setting A1 — B1, A2 = B2\B1 and more generally An = Bn\(B1 u 52u • • •u Bn_1) for all n>2 . Clearly, each An is .^-measurable, which yields thedesired result.

The next lemma is needed to show that, when X is finite dimensional,(D') implies (C), i.e., E (F j ^ x , ) ( t ) e ^WC(X) a.s. It also has an obviousintrinsic interest.

is a member of 36. From Proposition 4 we know that

Since the support function is convex and positively homogenous, it is con-tinuous at some point of X* if and only if it is bounded on some closedball whose center can be assumed to be in D, a given countable dense sub-set of B*. Thus, we have

Multivalued Levy Type Theorem 965

Lemma 5. Let F: T-> 3fc(X) be an integrable random set and 3$, -2ft'two sub-<7-fields of ,rf such that 3ft <^ 3ft'. If for almost every t,E(F/3g)(t) e ^wc(X) and E(F/%')(t) is weakly locally compact thenW(F#')(t) e J2wc(X) a.s.

Proof. Since by Proposition 1 ( f ) one has E(E(F/38')/@) = E(F/.%) a.s.,it suffices to prove the following claim: Assume that G: T - > ^ C ( X ) is an«j/-measurable integrable random set whose values are weakly locallycompact and contain 0. If E(GI@)(t)e<fwc(X) a.s. then G(t)e<fwc(X) a.s.

First note that As(£(G/^)(t) n E( -G/<#)(t)) = {0} a.s. becauseE(G/.<%)(t) contains no line. Define the multifunction H by H = G n ( —G) .Using Proposition 1(g) and easy properties of the asymptotic cone wededuce

Taking the multivalued expectation of both sides of (6.7) and using the lastpart of Proposition 1(e), yields E (As(H)) = {0}. Further, applying Lemma 6of Hiai,(32) we obtain A s ( H ( t ) ) = {0} a.s. because As (H(t)) is a closed convexcone. Thus, we have

which shows that G(t) contains no line. The claim follows from thehypothesis of local weak compactness of G(t) [see e.g., Castaing andValadier,(10) Thm. 1.9].

Now, let us complete the proof of Corollary 3. Observe first that, byLemma 4, (D) and (D') are equivalent. Further, Lemma 5 together with aneasy localization argument shows that (D') implies (C), namely E(F/3Sao) eywc(X) a.s. Indeed, let (An) be the partition appearing in condition (D') ofLemma 4. By Proposition 1(b-c) and hypothesis (D) we deduce that for alln>1

which by (A) and Lemma 5 entails

966 Couvreux and Hess

This proves (C) and shows that E(F/^ao) is countably supported withrespect to any countable dense subset D of B*. Thus, part (a) is proved. Asto part (b), observe that by Corollary 2(a), conditions (C) and (D)together yield the equivalence of (B) and (B'). Thus, the convergenceproperty (4.6) follows from statement (1°) of Theorem 1. The proof ofCorollary 3 is now completed.

Remark 10.

(i) Several variants of Theorem 1 could be proved. For example,consider a countable P-dense subset D of 5*, where p = T or s*,and a subset ^ of ^C(X). If we assume that the pair ((£, D)satisfies (Hp) and that E(F/^rx>)(t)e c£ for almost every t e T,then the conclusions of Theorem 1 ( 1 ° and 2°) still hold underthe following weaker version of inclusion (4.2)

Further, if the pair (<^, D) satisfies (Hs*) and if p = s* in (6.8),conclusion (3°) also hold. This follows from Corollary 1 andProposition 5.

(ii) When conditions (C) and (D) hold, inclusion (B') can be equiv-alently formulated as an equality. Indeed, using similar argu-ments as in the proof of Lemma 3, it is not difficult to see that,for all integers k > n 0 ( t ) ,

and

Clearly, under hypothesis (A) of Theorem 1, statements (1°) to (3°) stillhold for almost every t in the ^-measurable subset T0 defined by

(iii) Another formulation of hypothesis (B) of Theorem 1 is possible.Set, for each y e D,

Consequently, in Theorem 1 (1°-3°), assumption (B) can be replaced bythe condition fi(T0) = 1.

ACKNOWLEDGMENT

We are grateful to the referee for several interesting and usefulremarks.

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Multivalued Levy Type Theorem 969