LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉODORY SELECTIONS

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1 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉODORY SELECTIONS D. AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DELLE SCIENZE, PALERMO - ITALY Abstract. Dopo avere provato due teoremi di tipo Lusin per multifunzioni, presentiamo alcuni teoremi sulla proprieta di Scorza Dragoni. La caratteristica principale di tali teoremi consiste nel fatto che le multifunzioni in esame sono definite e, salvo alcuni casi, hanno valori in un contesto in cui non e definita alcuna metrica. Come applicazione, in particolare, si ottiene un teorema di esistenza di selezioni di Carathéodory. 1. Introduction A Carathéodory selection of a multifunction F : T X Y is a function f(t, x), defined whenever F (t, x) is non-empty, which is measurable in t, continuous in x and such that f(t, x) F (t, x). As Artstein and Prikry pointed out ([1], 3), Scorza Dragoni s theorems for multifunctions (which give almost lower semicontinuity) enable to derive existence of Carathéodory selections in situations when lower semicontinuity implies existence of continuous selections. Many Authors have given Scorza Dragoni type theorems for multifunctions (see, e.g., [1], [3], [4], [6], [12]). In all of these papers the following conditions are usually assumed: T is a Hausdorff topological (or metric) measurable space endowed with a σ-finite (or finite) measure µ that is internally regular by compact sets, X is a separable metric space or a Polish space, Y is a metric space and the range of F is separable. Recently in [8] the metric restriction on X and, except for some cases, also on Y has been dropped. In this paper we remain in a general framework, using proof techniques completely different from the arguments given in [8]. Date: May 10,

2 2D. AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DE Our approach is based on arguments of metric type. Indeed, it depends on the construction of an appropriate function δ defined on couples of subsets of Y, which works well as a point-set distance and which, on the other hand, allows to drop the assumption that both X and Y are metric spaces, by working on an auxiliary multifunction H : T X Y. We first establish two Lusin type theorems for multifunctions, from which we derive some Scorza Dragoni type theorems. Furthermore, a distinction is made up when the internal regularity of µ is by closed sets or by closed and compact sets. In this latter case, the requirements on X can be further relaxed. Some of the results obtained seem to be new, some others extend previous ones (see sections 3 and 4). Finally, an existence theorem for Carathéodory selections in conformity to ([1], 3) is given, as an application of our results. 2. Preliminaries Let S be a non-empty set and (Z, τ Z ) be a topological space. (Z) denotes the Borel σ-algebra on Z. Let Φ : S Z be a multifunction, i.e. a function from S into the family (Z) of all subsets of Z. N Φ denotes the set {s S : Φ(s) = }, Gr(Φ) the graph of Φ, i.e. the set {(s, z) S Z : z Φ(s)}, while Φ denotes the multifunction defined by Φ(s) = Φ(s) for all s S, where Φ(s) is the closure of Φ(s). If A S, we put Φ(A) = s AΦ(s) and call Φ A the restriction of Φ to A. If V Z, we put Φ (V ) = {s S : Φ(s) V } and Φ + (V ) = {s S : Φ(s) V }. We have the fundamental relations Φ (V ) = S Φ + (Z V ) = π S (Gr(Φ) (S V )), where π S denotes the projection map of S Z onto S, and, for each family {V α : α A} (Z), Φ ( α A V α ) = α A Φ (V α ). If (S, τ S ) is a topological space, we say that Φ is lower (resp. upper) semicontinuous in s 0 S if for each V τ Z such that s 0 Φ (V ) (resp. s 0 Φ + (V )) there exists an open neighbourhood U(s 0 ) of s 0 such that U(s 0 ) Φ (V ) (resp. U(s 0 ) Φ + (V )). We say that Φ is lower (resp. upper) semicontinuous if it is lower (resp. upper) semicontinuous in every s S, or, equivalently, if for each V τ Z the set Φ (V ) (resp. Φ + (V )) is open in S. We say that Φ is continuous if it is simultaneously lower and upper semicontinuous. If Σ S is a σ-algebra of subsets of S, we say that Φ is Σ S -weakly measurable (resp. Σ S - measurable) if for each V τ Z (resp. V (Z)) Φ (V ) Σ S.

3 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉO 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. The characteristic function of the set E is denoted by χ E. As in [11], we say that a topological space is Polish if it is separable and metrizable by a complete metric, Lusin if it is Hausdorff and a continuous and bijective image of a Polish space, Suslin if it is Hausdorff and a continuous image of a Polish space. If (Z, τ Z ) and (X, τ X ) are topological spaces, a surjective function p : Z X is said to be an identification whenever U τ X if and only if p 1 (U) τ Z (see [5]). From now on, unless otherwise stated, (S, τ S ) is a topological space, (S, Σ S ) is a measurable space, (T, τ T ) is a topological space, Σ T is a σ-algebra of subsets of T such that τ T Σ T, µ is a measure on Σ T such that for every A Σ T and ε > 0 there exists a closed (or closed and compact) set C ε A, with µ(a C ε ) < ε, (X, τ X ) and (Y, τ Y ) are two topological spaces. The completion of Σ T with respect to µ is denoted by Σ T. 3. Lusin type theorems We need to express the requested internal regularity of µ in terms of the following particular and trivial formulation of Lusin s Theorem: Lemma 1. If f : T R takes only the values 0, 1 and is Σ T - measurable, then for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that f Cε is continuous. Definition 1. If A, B Y define { 1 if A B δ(a, B) = 0 if A B =. We have the following Lemma 2. Let Y be second-countable and F : S Y be a multifunction. If B = {B n : n N} is a countable basis for τ Y, then: 1) F is lower semicontinuous in s 0 S iff δ(b n, F (.)) is lower semicontinuos in s 0 for each n N. 2) If δ(b n, F (.)) is upper semicontinuous for each n N, then Gr(F ) is closed in S Y. 3) F is Σ S -weakly measurable iff δ(b n, F (.)) is Σ S -measurable for each n N. Proof. 1)( ). Fix n N. Obviously, δ(b n, F (.)) is lower semicontinuous in each s 0 such that δ(b n, F (s 0 )) = 0. Let δ(b n, F (s 0 )) = 1.

4 4D. AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DE This means that F (s 0 ) B n ; thus, by the lower semicontinuity of F in s 0, there exists an open neighbourhood U(s 0 ) of s 0 such that F (s) B n for every s U(s 0 ), that is δ(b n, F (s)) = 1 for every s U(s 0 ). 1)( ). Fix V τ Y such that F (s 0 ) V. Let y 0 F (s 0 ) V and B n B such that y 0 B n V. Then δ(b n, F (s 0 )) = 1, thus, by the lower semicontinuity of δ(b n, F (.)) in s 0, there exists an open neighbourhood U(s 0 ) of s 0 such that δ(b n, F (s)) = 1 for every s U(s 0 ), that is F (s) B n for every s U(s 0 ). Hence, a fortiori, F (s) V for every s U(s 0 ). 2). We shall prove that S Y Gr(F ) τ S τ Y. Let (s 0, y 0 ) S Y Gr(F ), that is y 0 F (s 0 ). Then there exists B n B such that y 0 B n and F (s 0 ) B n =, that is δ(b n, F (s 0 )) = 0. Then, by the upper semicontinuity of δ(b n, F (.)) in s 0, it follows that there exists an open neighbourhood U(s 0 ) of s 0 such that δ(b n, F (s)) = 0, that is F (s) B n =, for every s U(s 0 ). Consequently also F (s) B n = for every s U(s 0 ), so, in definitive, [U(s 0 ) B n ] Gr(F ) =. 3)( ). This follows from the fact that F (B n ) Σ S and δ(b n, F (.)) = χ F (B n) for each n N. 3)( ). As Y is second-countable, taking into account that F ( i I A i ) = i I F (A i ), it sufficies to show that F (B n ) Σ S for each n N. Obviously this occurs since F (B n ) = {s S : δ(b n, F (s)) = 1}. Theorem 1. Let Y be second-countable. If F : T Y is a Σ T -weakly measurable multifunction, then for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that: a) F Cε is lower semicontinuous; b) Gr(F Cε ) is closed in C ε Y. Theorem 1 improves Théorème 3 of [4], in which Y is a Polish space, Lemma 2.2 of [1], in which b) is proved when Y is a separable metric space, and the necessary part of Corollary 2.2 of [7], in which a) is proved when Y is a separable and locally compact metric space. Furthermore, in [1], [4] and [7] the assumptions on T are more restrictive than ours and µ is a finite measure. Proof. Let B = {B n : n N} be a countable basis for τ Y. For each n N, by Lemma 2, part 3)( ), δ(b n, F (.)) is Σ T -measurable. Hence, by Lemma 1, we can find a closed (or closed and compact) set C ε,n T, with µ(t C ε,n ) < ε 2 n, such that δ(b n, F (.)) Cε,n is continuous. Put C ε = n NC ε,n. Then C ε is closed (or closed and compact), µ(t C ε ) < ε and δ(b n, F (.)) Cε is continuous for each n N,.

5 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉO Thus the conclusions follow by 1)( ) and 2) of Lemma 2 respectively. Corollary 1. Let Y be second-countable. If f : T Y is a Σ T - measurable function, then for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that f Cε is continuous. Theorem 2. Let µ be σ-finite and Y be a Suslin space. If F : T Y is a multifunction such that Gr(F ) Σ T (Y ), then for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that F Cε is lower semicontinuous. Proof. Let p be a continuous function from a Polish space P onto Y. Define the multifunction G : T P by putting G(t) = p 1 (F (t)) for all t T. We claim that Gr(G) Σ T (P ). Indeed, it is easily seen that (1 T, p) : T P T Y, defined by (1 T, p)(t, y) = (t, p(y)) for all (t, y) T P, is continuous and (1 T, p) 1 (Ω) Σ T (P ) for each Ω Σ T (Y ). Thus the claim follows from the fact that Gr(G) = {(t, y) T P : y G(t) = p 1 (F (t))} = {(t, y) T P : p(y) F (t)} = {(t, y) T P : (t, p(y)) Gr(F )} = (1 T, p) 1 (Gr(F )). By Sainte-Beuve s projection Theorem ([10], Theorem 4), G is Σ T - measurable, since G (V ) = π T (Gr(G) (T V )) for each V P. Hence, a fortiori, G is Σ T -weakly measurable. Now, by Theorem 1, for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that G Cε is lower semicontinuous. Since p is surjective, F (t) = p(g(t)) for every t T, and so F (U) = G (p 1 (U)) for each U Y ; thus F Cε is lower semicontinuous. 4. Scorza Dragoni s property Theorem 3. Let X, Y be two second-countable spaces and F : T X Y be a multifunction such that F (t,.) has closed graph for each t T. Let us suppose, furthermore, that one of the following groups 1), 2) or 3) of hypotheses holds: 1) µ is σ-finite, X and Y are Suslin spaces and F is Σ T (X)-weakly measurable; 2) F (t,.) is lower semicontinuous for each t T and {x X : F (., x) is Σ T -weakly measurable} is dense in X; 3) for every ε > 0 there exists T ε Σ T, with µ(t T ε ) < ε, such that F (., x) Tε is lower semicontinuous for each x X.

6 6D. AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DE Then, for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that Gr(F Cε X) is closed in C ε X Y. Proof. We can suppose F = F, because F (V ) = F (V ) for each V τ Y. Let us define H : T X Y by putting H(t) = Gr(F (t,.)) for each t T. Obviously H has closed values. In the case 1), since F is Σ T (X)-weakly measurable and Y is second-countable, it follows that Gr(F ) Σ T (X) (Y ) (see, e.g., [2], Corollary 2.4); hence Gr(H) = Gr(F ) Σ T (X) (Y ). But the product space X Y is Suslin, so, by Sainte-Beuve s projection Theorem ([10], Theorem 4), H (Ω) = π T (Gr(H) (T Ω)) Σ T for each Ω (X Y ). Therefore, H is Σ T - measurable and, a fortiori, Σ T -weakly measurable. We note in passing that, for U τ X, V τ Y : H (U V ) = {t T : x U, y V with y F (t, x)} = = {t T : F (t, x) V }. x U In the case 2), by the lower semicontinuity of F (t,.) for each t T, we obtain H (U V ) = x U D{t T : F (t, x) V }, where D is a countable dense subset of X such that F (., x) is Σ T -weakly measurable for each x D; thus H (U V ) Σ T. This proves that H is Σ T -weakly measurable, since X Y is second-countable. In the case 3), for fixed ε > 0, take T ε/2 as in 3). Since (H Tε/2 ) (U V ) = H (U V ) T ε/2 = x U{t T ε/2 : F (t, x) V }, it follows that H Tε/2 is lower semicontinuous, hence Σ Tε/2 -weakly measurable (Σ Tε/2 = {A Σ T : A T ε/2 }). In each of these cases, since X Y is second-countable, we can apply Theorem 1; thus for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that Gr(H Cε ) = Gr(F Cε X) is closed in C ε X Y. The first part of the assertion of Theorem 4.1 in [6], in which T is a compact Hausdorff space with a positive Radon measure µ, X is a Polish space and Y is a separable and locally compact metric space is a special case of Theorem 3, part 2). Corollary 2. Let X be a second countable space and f : T X R be a function such that f(t,.) is lower semicontinuous for each t T. Moreover, assume that one of the following groups of hypotheses holds: 1) µ is σ-finite, X is a Suslin space and f is Σ T (X)-measurable;

7 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉO 2) f(t,.) is continuous for each t T and {x X : f(., x) is Σ T - measurable} is dense in X; 3) for every ε > 0 there exists T ε Σ T, with µ(t T ε ) < ε, such that f(., x) Tε is upper semicontinuous for each x X. Then, for every ε > 0 there exists a closed (or closed and compact) set C ε T, with µ(t C ε ) < ε, such that {(t, x, r) C ε X R : f(t, x) r} is closed in C ε X R. In particular, for each r R the set {(t, x) C ε X : f(t, x) r} is closed in C ε X, that is, f Cε X is lower semicontinuous. Proof. It is not difficult to see that the multifunction F : T X R, defined by F (t, x) = [f(t, x), + [ for each (t, x) T X, satisfies Theorem 3. The conclusion follows from Gr(F Cε X) = {(t, x, r) C ε X R : f(t, x) r}. In the following Theorems 4 9 we assume that T is Hausdorff and that µ is such that for every A Σ T and ε > 0 there exists a compact set K ε A, with µ(a K ε ) < ε. Theorem 4. Let µ be σ-finite, Z be a second-countable Suslin space and p : Z X be an identification. If f : T X R is a function such that: a ) f(t,.) is lower (upper) semicontinuous, for each t T, b 1 ) f is Σ T (X)-measurable, then, for every ε > 0 there exists a compact set K ε T, with µ(t K ε ) < ε, such that f Kε X is lower (upper) semicontinuous. Proof. It sufficies to prove the theorem for the lower semicontinuous case. It is easily seen that (1 T, p) : T Z T X, defined by (1 T, p)(t, x) = (t, p(x)) for all (t, x) T Z, is continuous and (1 T, p) 1 (Ω) Σ T (Z) for each Ω Σ T (X); then g = f (1 T, p) : T Z R verifies the assumptions of Corollary 2, part 1). Fix ε > 0; there exists a compact set K ε T, with µ(t K ε ) < ε, such that g Kε Z is lower semicontinuous. Now g Kε Z= f Kε X (1 T, p) Kε Z and (1 T, p) Kε Z is an identification ([5], pg.262, Theorem 4.1). Thus, for each α R, (f Kε X ) 1 (]α, + [) is open in K ε X because (g Kε Z) 1 (]α, + [) is open in K ε Z. Hence f Kε X is lower semicontinuous. Analogously we have the following two theorems, the latter of which extends Proposition 1.4 of [3].

8 8D. AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DE Theorem 5. Let Z be a second-countable space and p : Z X be an identification. If f : T X R is a function such that: a ) f(t,.) is continuous, for each t T, b 2 ) f(., x) is Σ T -measurable, for each x X, then, for every ε > 0 there exists a compact set K ε T, with µ(t K ε ) < ε, such that f Kε X is continuous. Theorem 6. Let Z be a second-countable space and p : Z X be an identification. If f : T X R is a function such that: a ) f(t,.) is lower (upper) semicontinuous, for each t T, b 3 ) for every ε > 0 there exists T ε Σ T, with µ(t T ε ) < ε, such that f(., x) Tε is upper (lower) semicontinuous for each x X, then, for every ε > 0 there exists a compact set K ε T, with µ(t K ε ) < ε, such that f Kε X is lower (upper) semicontinuous. The following theorem extends Theorem 2.1 of [1]. Theorem 7. Let µ be σ-finite, Z be a second-countable Suslin space, p : Z X be an identification and Y be a second-countable space. If F : T X Y is a multifunction such that: α ) F (t,.) is lower semicontinuous, for each t T, β 1 ) F is Σ T (X)-weakly measurable, then, for every ε > 0 there exists a compact set K ε T, with µ(t K ε ) < ε, such that F Kε X is lower semicontinuous. Proof. Let B = {B n : n N} be a countable basis for τ Y. Let n N and define f n (t, x) = δ(b n, F (t, x)) for all (t, x) T X. By assumption α) and by virtue of Lemma 2, part 1)( ), it follows that δ(b n, F (t,.)) is lower semicontinuous; on the other hand, by assumption β 1 ) and by Lemma 2, part 3)( ), δ(b n, F (.,.)) is Σ T (X)- measurable. Thus, given ε > 0, by Theorem 4 there exists a compact K ε,n T, with µ(t K ε,n ) < ε, such that f 2 n n Kε,n X is lower semicontinuous. Put K ε = n N K ε,n. Then K ε is compact, µ(t K ε ) < ε and f n Kε X is lower semicontinuous for each n N. Hence the conclusion follows by Lemma 2, part 1)( ). We omit the proofs of the following two theorems, since they can be obtained by modifying respectively those of Theorem 1 and Theorem 2 in [3], thanks to Theorems 5 and 6. Theorem 8. Let Z be a second-countable space, p : Z X an identification and Y metrizable. If F : T X Y is a multifunction such that:

9 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉO α ) F (t,.) is continuous, for each t T, β 2 ) F (., x) is Σ T -weakly measurable and F (T, x) is separable, for each x X, then for every ε > 0 there exists a compact K ε T, with µ(t K ε ) < ε, such that F Kε X is lower semicontinuous and Gr(F Kε X) is closed in K ε X Y. Theorem 9. Let Z be a second-countable space, p : Z X an identification and Y metrizable. If F : T X Y is a multifunction such that: α ) F (t,.) is lower semicontinuous, for each t T, β 3 ) for every ε > 0 there exists T ε Σ T, with µ(t T ε ) < ε, such that, for each x X, F (., x) Tε is upper semicontinuous and the set F (T ε, x) is separable, then, for every ε > 0 there exists a compact set K ε T, with µ(t K ε ) < ε, such that F Kε X is lower semicontinuous. Obviously, some formulations of Scorza Dragoni type theorems for functions f : T X Y follow as corollaries of Theorems 7 9. Remark As an example given in [8] shows, in Corollary 2, and consequently also in Theorem 3, the second axiom of countability on X cannot be dropped, even if X is a normal Lusin space. Hence the identification p of the above theorems cannot be replaced by a continuous and bijective map. Moreover, the existence of a closed set C ε T, with µ(t C ε ) < ε, such that Gr(F Cε ) is closed in C ε Y cannot be deduced by Theorem 2, for otherwise the part 1) of Theorem 3 could be proved when X is a Suslin space, possibly not second-countable, contradicting the example of [8] quoted above. An existence theorem for Carathéodory selections in conformity to ([1], 3) is now given as an application of the previous results. We start by saying the following definitions. Definition 2. Let F : T X Y be a multifunction and let T 0 Σ T be such that µ(t T 0 ) = 0. We say that f : T 0 X N F Y is an a.e.-carathéodory selection of F if: 1) f(t, x) F (t, x) for each (t, x) T 0 X N F, 2) f(., x) is Σ T0 -measurable for each x π X (T 0 X N F ), 3) f(t,.) is continuous for each t π T (T 0 X N F ).

10 D. 10 AVERNA DIPARTIMENTO DI MATEMATICA ED APPLICAZIONI FACOLTA DI INGEGNERIA - VIALE DE Definition 3. A multifunction F : T X Y is said an M-mapping if for each closed set C T, such that F C X is lower semicontinuous, F C X NF has a continuous selection. F is said to be a quasi M-mapping if for each ε > 0 there exists a closed set C ε T, with µ(t C ε ) < ε, such that F Cε X is an M-mapping. For example, if T is Hausdorff, if µ is such that for every ε > 0 there exists a compact set K ε A, with µ(t K ε ) < ε, if X is paracompact and if Y is a Banach space, then F is a quasi M-mapping if its values are non-empty closed convex subsets of Y. This follows by Michael s continuous selection Theorem ([9], Theorem 3.2 ), since K ε X is paracompact. Theorem 10. Under the assumptions of one of Theorems 7 9, if F is also a quasi M-mapping, then it admits an a.e.-carathéodory selection. If, furthermore, µ is complete and for each t T F (t,.) X {x X:(t,x) NF } has a continuous selection, then every a.e.-carathéodory selection can be extended to a Carathéodory selection. Proof. For each n N there exist two closed sets C 1,n, C 2,n T, with µ(t C i,n ) < 1 for i = 1, 2, such that F n C 1,n X is lower semicontinuous and F C2,n X is an M-mapping. Put C n = C 1,n C 2,n. F Cn X N F admits a continuous selection f n. Put T 0 = n N C n ; then T 0 Σ T and µ(t T 0 ) = 0. Now define f : T 0 X N F Y by putting: { f1 (t, x) if t C f(t, x) = 1 f n (t, x) if t C n n 1 i=1 C i, n 2. Since for every x X the set {t T 0 : (t, x) N F } Σ T, it is easily seen that f is an a.e.-carathéodory selection of F. The second part is trivial. Theorem 10 extends Theorem 3.2 of [1] if the hypotheses of Theorem 7 are fulfilled, while if the hypotheses of Theorem 9 hold then it extends the corresponding formulation of Theorem 3.1 in [3]. References [1] Artstein Z. and Prikry K., Carathéodory selections and the Scorza Dragoni property, J. Math. Anal. Appl., 127 (1987), pp [2] Averna D., Separation properties in X and 2 X. Measurable multifunctions and graphs, Math. Slovaca, 41 (1991), pp [3] Bonanno G., Two theorems on the Scorza Dragoni property for multifunctions, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 83 (1989), pp

11 LUSIN TYPE THEOREMS FOR MULTIFUNCTIONS, SCORZA DRAGONI S PROPERTY AND CARATHÉO [4] Castaing C., Une nouvelle extension du théorème de Dragoni-Scorza, C. R. Acad. Sci. Paris, 271 (1970), pp [5] Dugundji J., Topology, Allyn and Bacon, Boston (1966). [6] Himmelberg C.J. and Van Vleck F.S, An extension of Brunovsky s Scorza Dragoni type theorem for unbounded set-valued functions, Math. Slovaca, 26 (1976), pp [7] Jacobs M.Q., Measurable multivalued mappings and Lusin s theorem, Trans. Amer. Math. Soc., 134 (1968), pp [8] Kucia A., Scorza Dragoni type theorems, Fund. Math., 138 (1991), pp [9] Michael E., Continuous selections. I, Ann. of Math., 63 (1956), pp [10] Sainte-Beuve M.F., On the extension of Von Neumann-Aumann s theorem, J. Funct. Anal., 17 (1974), pp [11] Schwartz L., Radon measures on arbitrary topological spaces and cylindrical mesures, Oxford U. P., Oxford (1973). [12] Zygmunt W., The Scorza-Dragoni s type property and product measurability of a multifunction of two variables, Rend. Acc. Naz. delle Scienze (detta dei XL), 106 (1988), pp