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On measurable multifunctions with applications to random multivalued equations. (English) Zbl 0634.28005
Let ($$\Omega$$,$${\mathcal A},\mu)$$ be a complete, finite measure space, let X be a separable, reflexive Banach space and let $$F: \Omega \times X\to 2^ X\setminus \{\emptyset \}$$ be a set-valued function with closed and convex values such that $$(i)\quad F(\cdot,x)$$ is measurable, $$(ii)\quad F(\omega,\cdot)$$ is upper semicontinuous (with respect to the weak topology) and $$(iii)\quad F(\cdot,\cdot)$$ is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that $$F(\cdot,\cdot)$$ is $${\mathcal A}\times {\mathcal B}(X)$$- measurable, where $${\mathcal B}(X)$$ are the Borel-sets on X.
Now suppose that ($$\Omega$$,$${\mathcal A})$$ is a measurable space with $${\mathcal A}^ a$$Souslin family and X is a Souslin metric space. Then the measurability of $$\limsup_ nF_ n$$ and $$\liminf_ nF_ n$$ is proved for a sequence of closed valued, measurable set-valued functions $$F_ n: \Omega \to 2^ X\setminus \{\emptyset \}.$$ Other similar results are shown.
The measurability results are used to obtain new implicit function theorems [cf. J. C. Himmelberg, Fundam. Math. 87, 53-72 (1975; Zbl 0296.28003)] and to prove existence of solutions for implicit random differential inclusions and random functional-differential inclusions.
Reviewer: W.J.A.Stich

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 60H25 Random operators and equations (aspects of stochastic analysis) 49J27 Existence theories for problems in abstract spaces