REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS IN A NON-METRIC SETTING

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1 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS IN A NON-METRIC SETTING DIEGO AVERNA Abstract. Si stabilisce l esistenza di una regolarizzazione per multifunzioni Φ : T Z e F : T X Y, ove T è uno spazio topologico misurabile e X, Y e Z sono spazi topologici a base numerabile (Teoremi 1 e 3). Utilizzando il Teorema di selezione di Sainte- Beuve ([6]), si forniscono anche teoremi di unicità (Teoremi 2 e 4). I risultati conseguiti generalizzano quelli di Rzeżuchowski in [5]. 1. Introduction Scorza Dragoni type theorems for multifunctions F : T X Y of Carathéodory type are useful for the study of the set of solutions of Cauchy problems associated with the differential inclusion ẋ F (t, x). (1) This occurs because the separated regularity of F with respect to t and x (i.e. the Carathéodory type property) implies (through the Scorza Dragoni type theorem) an almost regularity with respect to (t, x). For example, if T = [0, 1], X = Y = R, F : T X Y has closed values, F (., x) is weakly measurable and F (t,.) is continuous, then for each ε > 0 there exists a closed set C ε [0, 1], whose Lebesgue measure is > 1 ε, such that F Cε is lower semicontinuous and has closed graph. A tool for the study of the solutions-set of (1), when F is not of Carathéodory type, can be the regularization of F. Roughly speaking, a regularization of F is a multifunction G : T X Y, which has the following properties: Date: January 20, Mathematics Subject Classification. Primary 28B20, 54C60; Secondary 26E25. Key words and phrases. Multifunctions. Regularization. Measurability. Semicontinuity. This research was supported by 60% MURST.. 1

2 2 D.AVERNA i 1 ) G(t, x) F (t, x), i 2 ) q(t) F (t, p(t)) q(t) G(t, p(t)) whenever p : T X and q : T Y are measurable functions, plus some Scorza Dragoni type property. In virtue of the properties i 1 ) and i 2 ), the solutions-set of (1) is the same as that of ẋ G(t, x). (2) Existence and uniqueness theorems for regularizations G of certain multifunctions F have been given in [1], [4], [5]. In particular, Rzeżuchowski proved in [5] an existence theorem for regularizations of a given closed-valued multifunction F : T X Y such that F (t,.) has closed graph, when T is a locally compact metric space endowed with a Borel, σ-finite, regular and complete measure µ, and X and Y are separable metric spaces (Theorem 1 of [5]). An uniqueness theorem was also presented when X and Y are even complete (Theorem 4 of [5]). The aim of this paper is essentially to show that the Rzeżuchowski existence and uniqueness theorems above remain still valid without assuming that T, X and Y are metric and without strengthening of other hypotheses; that is, existence and uniqueness theorems for regularizations proved here extend in a more general framework the results of [5]. The main idea for doing this, in existence theorems, is to substitute the point-set distance function in the range-space (which plays a key rule in the Rzeżuchowski existence proof) with a function δ, taking only two values, which flags when two suitable sets, one coming from a basis of the topology of the range-space, the other from the values of the multifunction, intersect. This idea have been tested also in a previous paper ([3]), concerning with Lusin and Scorza Dragoni type theorems; some results of [3] are also useful here. The extension in uniqueness theorems essentially carries out in virtue of Sainte-Beuve s selection theorem. Moreover, the results of this paper not only extend but also improve those of [5], establishing indeed further properties for the regularization G. 2. Preliminaries Let S be a non-empty set and (Z, τ Z ) be a topological space. P(Z) (resp. Cl(Z)) denotes the family of all subsets (resp. closed subsets) of

3 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS... 3 Z, while B(Z) denotes the Borel σ-algebra on Z. Let Φ : S Z be a multifunction, i.e. a function from S into the family P(Z). If the values of Φ are closed subsets of Z we write Φ : S Cl(Z). Gr(Φ) denotes the graph of Φ, i.e. the set {(s, z) S Z : z Φ(s)}. If E S, we call Φ E the restriction of Φ to E. If W Z, we put Φ (W ) = {s S : Φ(s) W } and Φ + (W ) = {s S : Φ(s) W }. We have the fundamental relations Φ (W ) = S Φ + (Z W ) = proj S (Gr(Φ) (S W )), where proj S denotes the projection map of S Z onto S, and, for each family {W α : α A} P(Z), Φ ( α A W α ) = α A Φ (W α ). If Φ 1, Φ 2 : S Z are two multifunctions, we denote by Φ 1 Φ 2 the simmetric difference multifunction, that is the multifunction defined by (Φ 1 Φ 2 )(s) = Φ 1 (s) Φ 2 (s) for each s S. If (S, τ S ) is a topological space, we say that Φ is lower (resp. upper) semicontinuous at s 0 S if for each W τ Z such that s 0 Φ (W ) (resp. s 0 Φ + (W )) there exists an open neighbourhood I of s 0 such that I Φ (W ) (resp. I Φ + (W )). We say that Φ is lower (resp. upper) semicontinuous if it is lower (resp. upper) semicontinuous at every s S, or, equivalently, if for each W τ Z the set Φ (W ) (resp. Φ + (W )) is open in S. We say that Φ is continuous if it is simultaneously lower and upper semicontinuous. If (S, Σ S, µ) is a measure space, we denote with Σ S the completion of Σ S with respect to µ and with µ the completion measure. (S, Σ S, µ ) is a complete measure space. Recall that E Σ S iff there exist E, E Σ S such that E E E and µ(e ) = µ (E) = µ(e ). If Σ is a σ-algebra of subsets of S, we say that Φ is Σ-weakly measurable (resp. Σ-measurable) [resp. Σ-B measurable] if for each W τ Z (resp. W Cl(Z)) [resp. W B(Z)] Φ (W ) Σ S. The definitions of lower and upper semicontinuity for real valued functions and those of measurability and continuity for functions with values in a topological space are the usual ones. If (S, τ) is a topological space, (S, Σ) is a measurable space and E S, then τ E = τ E and Σ E = Σ E denote respectively the induced topology and the induced σ-algebra on E. If E S and E has the induced topology, then B(S) E = B(E). If E Σ, then Σ E = {A Σ : A E}; so we speak of Σ-weak measurability (resp. Σ-measurability) of a multifunction (resp. function), instead of Σ E - weak measurability (resp. Σ E -measurability), whenever the multifunction (resp. function) is defined on E. If S is a structured space and we want a structure on E S, if no precisation is made we refer to the induced structure; i.e., if S is a topological (resp. measurable) space, then E is a topological (resp. measurable) space with the induced topology (resp. σ-algebra).

4 4 D.AVERNA If S and S are two sets and E S S, then, for s S, E s = {s S : (s, s ) E} denotes the s-section of E and, for s S, E s = {s S : (s, s ) E} denotes the s -section of E. If S and S are two structured spaces and we want a structure on S S, if no precisation is made we refer to the product structure; i.e., if S and S are topological (resp. measurable) spaces, then S S is a topological (resp. measurable) space with the product topology (resp. product σ-algebra). We notice that, if S and S are topological spaces, in general B(S) B(S ) B(S S ) and the inclusion can be proper; B(S) B(S ) = B(S S ) if, for example, S and S are second-countable topological spaces or Suslin spaces. Moreover, when in the sequel we deal with the product of three sets S, S and S, we always identify S (S S ) with S S S, even if the structure is essentially that of S (S S ). So, for example, if Σ S and Σ S S are σ-algebras on S and S S respectively, when we say that E S S S lies in Σ S Σ S S we mean that {(s, (s, s )) S (S S ) : (s, s, s ) E} Σ S Σ S S. As in [7], we say that a topological space is Polish if it is separable and metrizable by a complete metric, Suslin if it is Hausdorff and a continuous image of a Polish space. 3. Regularization of closed-valued multifunctions We begin with the following proposition in measure theory. Lemma 1. Let (T, Σ T ) be a measurable space and µ be a σ-finite measure on Σ T. For each subset E of T there exists M Σ T such that: α 1 ) M E; α 2 ) for each L Σ T such that L E Σ T and µ (L E) = 0, then µ (L M) = 0. Proof. We prove Lemma 1 for µ(t ) < +, because it is obvious how to extend it to the σ-finite case. Let α = sup{µ(a) : A Σ T, A E} < +. Then there exists a sequence (A n ) n of sets in Σ T such that, for each n N, A n E and µ(a n ) > α 1/n. The set M = n A n is the requested set. In fact, obviously, M Σ T and satisfies α 1 ). Moreover let L Σ T be such that L E Σ T and µ (L E) = 0; then (L M) E = (L M) (L E) Σ T and µ ((L M) E) = µ (L M). Obviously ((L M) E) M Σ T ; so let L Σ T be such that L ((L M) E) M and µ(l ) = µ (((L M) E) M). We

5 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS... 5 have L E and α µ(l ) = µ (L M) + µ(m) α, from which µ (L M) = 0. Hence M verifies α 2 ). We need, for the sequel, to reformulate Lemma 1 in terms of functions with only two values. Corollary 1. Let (T, Σ T ) be a measurable space, µ be a σ-finite measure on Σ T and {0, 1} endowed with the discrete topology. If ϕ : T {0, 1} is a function, then there exists a Σ T -measurable function ψ : T {0, 1} such that: β 1 ) ψ(t) ϕ(t) for each t T ; β 2 ) for each Σ T -measurable function ϑ : T {0, 1} such that ϑ(t) ϕ(t) a.e. in T, then ϑ(t) ψ(t) a.e. in T. From now on, unless otherwise stated, (T, τ T ) is a topological space, Σ T is a σ-algebra of subsets of T such that τ T Σ T (equivalently B(T ) Σ T ), µ is a σ-finite measure on Σ T such that for every A Σ T and every ε > 0 there exists a closed set C ε A, with µ(a C ε ) < ε. Obviously Σ T and µ have also these properties. Lemma 2. Let Z be a topological space and B(T Z) = B(T ) B(Z). Let Ψ : T Z be a multifunction such that for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(Ψ Cε ) is closed in a Σ T B(Z)-measurable set Ω which contains Gr(Ψ). Then there exists T 0 Σ T, with µ(t 0 ) = 0, such that Gr(Ψ T T0 ) Σ T B(Z). Proof. For each k N there exists a closed set C k T, with µ(t C k ) < 1/k, such that Gr(Ψ Ck ) = Gr(Ψ Ck ) Ω where Gr(Ψ Ck ) denotes the closure of Gr(Ψ Ck ) in T Z. But B(T Z) = B(T ) B(Z), so Gr(Ψ Ck ) Σ T B(Z). Put T 0 = k (T C k ); then µ(t 0 ) = 0 and Gr(Ψ T T0 ) = k Gr(Ψ Ck ) Σ T B(Z). We need also the following proposition, whose proof we give for completeness. Lemma 3. Let Z be a Suslin space. If Ψ : T Z is a multifunction such that Gr(Ψ) Σ T B(Z), then for every ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Ψ Cε is lower semicontinuous. Proof. Let g be a continuous function from a Polish space Z onto Z. Define the multifunction Ψ : T Z by putting Ψ (t) = g 1 (Ψ(t)) for all t T.

6 6 D.AVERNA We claim that Gr(Ψ ) Σ T B(Z ). Indeed, it is easily seen that (1 T, g) : T Z T Z, defined by (1 T, g)(t, z ) = (t, g(z )) for all (t, z ) T Z, is continuous and (1 T, g) 1 (Ω) Σ T B(Z ) for each Ω Σ T B(Z). Thus the claim follows from the fact that Gr(Ψ ) = (1 T, g) 1 (Gr(Ψ)). By Sainte-Beuve s projection Theorem ([6], Theorem 4), Ψ is Σ T - B measurable, since Ψ (W ) = proj T (Gr(Ψ ) (T W )) for each W Z. Hence, a fortiori, Ψ is Σ T -weakly measurable. Now, by Theorem 1 of [3], for every ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Ψ Cε is lower semicontinuous. Since g is surjective, Ψ(t) = g(ψ (t)) for every t T, and so Ψ (W ) = Ψ (g 1 (W )) for each W Z; thus Ψ Cε is lower semicontinuous. As in [3], if B, B Z, we define δ(b, B ) = { 1 if B B 0 if B B =. The following theorem is the key result of this paper. Theorem 1. Let Z be a second-countable topological space and Φ : T Cl(Z) a multifunction. Then there exists a multifunction Ψ : T Cl(Z) such that: γ 1 ) Ψ(t) Φ(t) for each t T ; γ 2 ) for each Σ T and for each Σ T -weakly measurable multifunction Θ : Z such that Θ(t) Φ(t) a.e. in, then Θ(t) Ψ(t) a.e. in ; γ 3 ) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(Ψ Cε ) is closed in T Z; γ 4 ) Gr(Ψ) Σ T B(Z). If, moreover, we assume that Z is also a Suslin space, then: γ 5 ) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Ψ Cε is lower semicontinuous; γ 6 ) for each Σ T and for each multifunction Θ : Z, with Gr(Θ) Σ T B(Z), such that Θ(t) Φ(t) a.e. in, then Θ(t) Ψ(t) a.e. in. Proof. Let B = {B n : n N} be a countable basis for τ Z. For each n N define ϕ n : T {0, 1} by putting, for each t T, ϕ n (t) = δ(b n, Φ(t)).

7 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS... 7 By Corollary 1, there exists a Σ T -measurable function ψ n : T {0, 1} such that: 1) ψ n (t) ϕ n (t) for each t T ; 2) for each Σ T -measurable function ϑ : T {0, 1} such that ϑ(t) ϕ n (t) a.e. in T, then ϑ(t) ψ n (t) a.e. in T. Let us define Ψ : T Cl(Z) by putting, for each t T, Ψ(t) = {Z B n : ψ n (t) = 0}. Ψ verifies γ 1 ). In fact, for each z Ψ(t) and each n N such that ϕ n (t) = 0, it follows that z Z B n, in virtue of 1). Thus we obtain z Φ(t), taking into account that, Φ(t) being closed, Φ(t) = {Z Bn : ϕ n (t) = 0}. Ψ verifies γ 2 ). Let Σ T and Θ : Z be a Σ T -weakly measurable multifunction such that Θ(t) Φ(t) a.e. in. For each n N define ϑ n : T {0, 1} by putting: { δ(bn, Θ(t)) if t ϑ n (t) = 0 if t. Then, by using 3)( ) of Lemma 2 of [3], it follows that ϑ n is Σ T - measurable; moreover, since Θ(t) Φ(t) a.e. in, ϑ n (t) ϕ n (t) a.e. in T. Hence, by 2), ϑ n (t) ψ n (t) a.e. in T and thus Θ(t) Ψ(t) a.e. in. Ψ verifies γ 3 ). Fix ε > 0. By Lusin s Theorem (see also Lemma 1 of [3]) and by a standard argument which takes into account the countability of the family {ψ n : n N}, we can find a closed set C ε T, with µ(t C ε ) < ε, such that ψ n Cε is continuous for each n N. We claim that Gr(Ψ Cε ) is closed in T Z. Indeed, if z 0 Ψ(t 0 ), t 0 C ε, then there is n N such that ψ n (t 0 ) = 0 and z 0 B n. By the upper semicontinuity of ψ n Cε at t 0, there exists an open neighbourhood I of t 0 such that ψ n (t) = 0 for each t I C ε. Thus ((I C ε ) B n ) Gr(Ψ Cε ) =. Ψ verifies γ 4 ). In fact, (T Z) Gr(Ψ) = n (ψn 1 ({0}) B n ). Finally, under the additional hypothesis on Z, γ 5 ) is a direct consequence of γ 4 ), by Lemma 3, while γ 6 ) is a consequence of γ 2 ), because, using the equality Θ (W ) = proj T (Gr(Θ) ( W )) for W τ Z, it follows that Θ is Σ T -weakly measurable by Sainte-Beuve s projection Theorem. Remark 1 Obviously γ 1 ) and γ 2 ) of Theorem 1 imply respectively the following γ 1) Ψ(t) Φ(t) a.e. in T ;

8 8 D.AVERNA γ 2) for each Σ T and for each Σ T -measurable function θ : Z such that θ(t) Φ(t) a.e. in, then θ(t) Ψ(t) a.e. in. Hence Theorem 1 extends and improves Theorem 2 of [5], in which γ 1), γ 2) and γ 3 ) are proved when T is a locally compact metric space, µ is a Borel, σ-finite, regular and complete measure on T and Z is a separable metric space. Moreover, if Ω is a Σ T B(Z)-measurable set, with Gr(Ψ) Ω, then γ 3 ) of Theorem 1 implies the following γ 3) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(Ψ Cε ) is closed in Ω. Now we prove the following uniqueness result, whose part 1) extends Theorem 5 of [5]. Theorem 2. Let Z be a Suslin space and Φ, Ψ 1, Ψ 2 : T Z be three multifunctions. Let us consider the following properties for i = 1, 2: γ 1) Ψ i (t) Φ(t) a.e. in T ; γ 2) for each Σ T and for each Σ T -measurable function θ : Z such that θ(t) Φ(t) a.e. in, then θ(t) Ψ i (t) a.e. in ; γ 3) for some Ω Σ T B(Z), with Gr(Φ) Ω, and for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(Ψ i Cε ) is closed in Ω; γ 4) there is T 0 Σ T, with µ(t 0 ) = 0, such that Gr(Ψ i T T0 ) Σ T B(Z); γ 5 ) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Ψ i Cε is lower semicontinuous. Then: 1) γ 1), γ 2) and γ 4) imply that Ψ 1 (t) = Ψ 2 (t) a.e. in T. 2) If B(T Z) = B(T ) B(Z), then γ 1), γ 2), and γ 3) imply that Ψ 1 (t) = Ψ 2 (t) a.e. in T. 3) If Z is also a second-countable topological space and Ψ 1 and Ψ 2 are closed-valued, then γ 1), γ 2) and γ 5 ) imply that Ψ 1 (t) = Ψ 2 (t) a.e. in T. Proof. 1). Gr((Ψ 1 Ψ 2 ) T T0 ) = Gr(Ψ 1 T T0 ) Gr(Ψ 2 T T0 ) Σ T B(Z). Put = proj T (Gr((Ψ 1 Ψ 2 ) T T0 )); thus Σ T, by Sainte- Beuve s projection Theorem, and T T 0. Define Γ : T Z by putting { (Ψ1 Ψ Γ(t) = 2 )(t) if t Z if t.

9 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS... 9 Gr(Γ) = Gr((Ψ 1 Ψ 2 ) T T0 ) ((T ) Z) Σ T B(Z); thus, by Saint-Beuve s selection Theorem ([6], Theorem 3), there exists a Σ T -measurable selection θ of Γ. By γ 1), θ(t) Φ(t) a.e. in, thus, by γ 2), θ(t) Ψ 1 (t) Ψ 2 (t) a.e. in. It follows that µ ( ) = 0. 2). By Lemma 2, γ 3) implies γ 4). So the conclusion follows by 1). 3). Ψ 1 and Ψ 2 are Σ T -weakly measurable. In fact, for i = 1, 2 and for each k N there exists a closed set C k T, with µ(t C k ) < 1/k, such that Ψ i Ck is lower semicontinuous, thus Σ Ck -weakly measurable (Σ Ck = {A Σ T : A C k }). Hence, for W τ Z, we have Ψ i (W ) = k (Ψ i Ck (W )) N, where µ (N) = 0, so Ψ i (W ) Σ T. Then, thanks to Theorem 2.4 of [2] (see also Remarks 2.1 and 2.4 in [2]), we have that Gr(Ψ 1 ), Gr(Ψ 2 ) Σ T B(Z); thus the conclusion follows again by 1). The following Theorem 3 is the two-variables version of Theorem 1. Theorem 3. Let X and Y be two second-countable topological spaces and D T X. If F : D Cl(Y ) is a multifunction such that there is a T 0 Σ T, with µ(t 0 ) = 0, such that Gr(F (t,.)) is closed in D t Y for each t proj T (D) T 0, then there exists a multifunction G : D Cl(Y ) such that: i 0 ) Gr(G(t,.)) is closed in D t Y for each t proj T (D); i 1 ) G(t, x) F (t, x) for each (t, x) D; i 2 ) for each Σ T, for each Σ T -weakly measurable multifunction Q : Y and for each Σ T -measurable function p : X such that (t, p(t)) D and Q(t) F (t, p(t)) a.e. in, then Q(t) G(t, p(t)) a.e. in ; i 3 ) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(G D (Cε X)) is closed in D Y ; i 4 ) Gr(G) (Σ T B(X Y )) D Y. If, moreover, we assume that Y is also a Suslin space and that B(T Y ) = B(T ) B(Y ), then i 5 ) for each Σ T, for each Σ T -measurable function p : X, with (t, p(t)) D a.e. in, and for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that G(., p(.)) Cε is lower semicontinuous. Finally, if X and Y are also two Suslin spaces and D Σ T B(X), then i 6 ) for each Σ T and for each multifunction H : D ( X) Y, with Gr(H) Σ T B(X Y ), such that H(t, x) F (t, x) for almost

10 10 D.AVERNA all t proj T (D) and for each x D t, then H(t, x) G(t, x) for almost all t proj T (D) and for each x D t. Proof. First suppose D = T X. Consider the multifunction Φ : T Cl(X Y ) defined by { Gr(F (t,.)) if t T T0 Φ(t) = if t T 0. By Theorem 1, there exists a multifunction Ψ : T Cl(X Y ) satisfying γ 1 ), γ 2 ), γ 3 ), γ 4 ) and γ 6 ). We claim that the multifunction G : T X Cl(Y ) defined by G(t, x) = (Ψ(t)) x is the requested multifunction. In fact, it is easily seen that G verifies i 0 ), i 1 ), i 3 ) and i 4 ). G verifies i 2 ). Let Σ T, Q : Y be a Σ T -weakly measurable multifunction and p : X be a Σ T -measurable function such that Q(t) F (t, p(t)) a.e. in. The multifunction Θ : X Y defined by Θ(t) = {(p(t), y) : y Q(t)} is Σ T -weakly measurable, because, for U τ X and V τ Y, Θ (U V ) = p 1 (U) Q (V ) Σ T. Moreover Θ(t) Φ(t) a.e. in, thus, by γ 2 ), Θ(t) Ψ(t) a.e. in, from which it follows that Q(t) G(t, p(t)) a.e. in. G verifies i 5 ). Let Σ T and p : X be a Σ T -measurable function. Extend p to the Σ T -measurable function ˆp : T X defined by putting ˆp(t) = { p(t) if t constant if t. If we show that G(., ˆp(.)) is Σ T -weakly measurable, then, by Theorem 1 of [3], it follows that for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that G(., ˆp(.)) Cε, and thus also G(., p(.)) Cε, is lower semicontinuous. To prove this, it suffices to show that for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(G(., ˆp(.)) Cε ) is closed in T Y. In fact, this last condition being verified, by Lemma 2 there exists T 0 Σ T, with µ(t 0 ) = 0, such that Gr(G(., ˆp(.)) T T0 ) Σ T B(Y ); so, from the equality G(., ˆp(.)) (V ) = proj T (Gr(G(., ˆp(.)) T T0 ) ((T T 0 ) V )) N, where N T 0, and by Sainte-Beuve s projection Theorem, it follows that G(., ˆp(.)) is Σ T -weakly measurable. So fix ε > 0. By i 3 ) and using Theorem 1 of [3], there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(G Cε X) is closed in T X Y and ˆp Cε is continuous. Gr(G(., ˆp(.)) Cε ) is closed in T Y. Indeed, if we take (t 0, y 0 ) Gr(G(., ˆp(.)) Cε ), t 0 C ε, then (t 0, ˆp(t 0 ), y 0 )

11 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS Gr(G Cε X); hence, by this and by the continuity of ˆp Cε, there are two open neighbourhoods I and V of t 0 and y 0 respectively such that, for t I C ε and y V, (t, y) Gr(G(., ˆp(.)) Cε ). Finally we prove i 6 ). Define Θ : X Y by putting, for each t, Θ(t) = Gr(H(t,.)). Gr(Θ) = Gr(H) Σ T B(X Y ). Moreover Θ(t) Φ(t) a.e. in ; then, by γ 6 ) (X Y is Suslin), Θ(t) Ψ(t) a.e. in, hence H(t, x) G(t, x) for almost all t and for each x X. Now we sketch tre proof when D T X. Define ˆF : T X Cl(Y ) by putting ˆF (t, x) = { (Gr(F (t,.))x if t T T 0 if t T 0, where the closure is taken in X Y. Gr( ˆF (t,.)) = Gr(F (t,.)) for each t T T 0 and, taking into account that Gr(F (t,.)) = Gr(F (t,.)) (D t Y ) for each t proj T (D) T 0, then we obtain that ˆF (t, x) = F (t, x) for all (t, x) D (T 0 X). Let Ĝ : T X Y be as in the first part of the proof with respect to ˆF ; it is no difficult to verify that G = Ĝ D is the requested multifunction. Remark 2 Obviously i 1 ) and i 2 ) of Theorem 3 imply respectively the following i 1) G(t, x) F (t, x) for almost every t proj T (D) and for each x D t ; i 2) for each Σ T and for all Σ T -measurable functions q : Y and p : X such that (t, p(t)) D and q(t) F (t, p(t)) a.e. in, then q(t) G(t, p(t)) a.e. in. Hence Theorem 3 extends and improves Theorem 1 of [5], in which i 1), i 2) and i 3 ) are proved when T is a locally compact metric space, µ is a Borel, σ-finite, regular and complete measure on T, X and Y are two separable metric spaces. Moreover, if Ω is a Σ T B(X Y )-measurable set, with Gr(G) Ω, then i 3 ) in Theorem 3 implies the following i 3) for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(G D (Cε X)) is closed in (D Y ) Ω.

12 12 D.AVERNA The following is an uniqueness theorem for the two-variables case; its part 2) extends Theorem 4 of [5]. Theorem 4. Let X a topological space, Y be a Suslin space, D Σ T B(X) and F, G 1, G 2 : D Y be three multifunctions. Let us consider the following properties for i = 1, 2: i 1) G i (t, x) F (t, x) for almost every t proj T (D) and for each x D t ; i 2) for each Σ T, for all Σ T -measurable functions q : Y and p : X such that (t, p(t)) D and q(t) F (t, p(t)) a.e. in, then q(t) G i (t, p(t)) a.e. in ; i 3) for some Σ T B(X Y )-measurable set Ω, with Gr(F ) Ω, and for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that Gr(G i D (Cε X)) is closed in (D Y ) Ω; i 4) there is T 0 Σ T, with µ(t 0 ) = 0, such that Gr(G i D ((T T0 ) X)) Σ T B(X Y ); i 5 ) for each Σ T, for each Σ T -measurable function p : X, with (t, p(t)) D a.e. in, and for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that G i (., p(.)) Cε is lower semicontinuous. Then: 1) If X is a Suslin space, then i 1), i 2) and i 4) imply that G 1 (t, x) = G 2 (t, x) for almost every t proj T (D) and for each x D t. 2) If X is a Suslin space and B(T X Y ) = B(T ) B(X Y ), then i 1), i 2) and i 3) imply that G 1 (t, x) = G 2 (t, x) for almost every t proj T (D) and for each x D t. 3) If Y is also a second-countable topological space and G 1 and G 2 are closed-valued, then i 1), i 2) and i 5 ) imply that for each Σ T and for each Σ T -measurable function p : X, with (t, p(t)) D a.e. in, it is G 1 (t, p(t)) = G 2 (t, p(t)) a.e. in. Sketch of the proof. First we prove the assertion 1) for D = T X. The multifunctions Φ, Ψ 1, Ψ 2 : T X Y defined respectively by Φ(t) = Gr(F (t,.)), Ψ 1 (t) = Gr(G 1 (t,.)) and Ψ 2 (t) = Gr(G 2 (t,.)) satisfy 1) of Theorem 2; then Ψ 1 (t) = Ψ 2 (t) a.e. in T, from which G 1 (t, x) = G 2 (t, x) for almost every t T and for each x X. The assertion 2) can be proved as above, taking into account 2) of Theorem 2. To prove 3) when D = T X, extend p to all of T, by putting p(t) = constant outside of. Then apply 3) of Theorem 2 to Φ(.) = F (., p(.)), Ψ 1 (.) = G 1 (., p(.)) and Ψ 2 (.) = G 2 (., p(.)); so we obtain Ψ 1 (t) = Ψ 2 (t) a.e. in T. Now return to the original p defined in, so we obtain G 1 (t, p(t)) = G 2 (t, p(t)) a.e. in.

13 REGULARIZATION OF CLOSED-VALUED MULTIFUNCTIONS For the general case D T X, extend F, G 1 and G 2 to all of T X, by putting their values empty outside of D; then apply the already proved unicity theorem for the case D = T X and finally return to D. References [1] Artstein Z., Weak convergence of set-valued functions and control, SIAM Control, 13 (1975), pp [2] Averna D., Separation properties in X and 2 X. Measurable multifunctions and graphs, Math. Slovaca, 41 (1991), pp [3] Averna D., Lusin type theorems for multifunctions, Scorza Dragoni s property and Carathéodory selections, Boll. Un. Mat. It., to appear. [4] Jarnik J. and Kurzweil J., On conditions on right-hand sides of differential relations, Čas. Pest. Mat., 102 (1977) pp [5] Rzeżuchowski T., Scorza-Dragoni type theorem for upper semicontinuous multivalued functions, Bull. Acad. Polon. Sci., 28 (1980), pp [6] Sainte-Beuve M.F., On the extension of Von Neumann-Aumann s theorem, J. Funct. Anal., 17 (1974), pp [7] Schwartz L., Radon measures on arbitrary topological spaces and cylindrical mesures, Oxford U. P., Oxford (1973). Dipartimento di Matematica ed Applicazioni, Facoltà di Ingegneria, Viale delle Scienze, Palermo (Italy) address: AVERNA@IPAMAT.CRES.IT