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On measurable multifunctions with applications to random multivalued equations. (English) Zbl 0634.28005

Let (${\Omega }$,$𝒜,\mu \right)$ be a complete, finite measure space, let X be a separable, reflexive Banach space and let $F:{\Omega }×X\to {2}^{X}\setminus \left\{\varnothing \right\}$ be a set-valued function with closed and convex values such that $\left(i\right)\phantom{\rule{1.em}{0ex}}F\left(·,x\right)$ is measurable, $\left(ii\right)\phantom{\rule{1.em}{0ex}}F\left(\omega ,·\right)$ is upper semicontinuous (with respect to the weak topology) and $\left(iii\right)\phantom{\rule{1.em}{0ex}}F\left(·,·\right)$ is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that $F\left(·,·\right)$ is $𝒜×ℬ\left(X\right)$- measurable, where $ℬ\left(X\right)$ are the Borel-sets on X.

Now suppose that (${\Omega }$,$𝒜\right)$ is a measurable space with ${𝒜}^{a}$Souslin family and X is a Souslin metric space. Then the measurability of ${lim sup}_{n}{F}_{n}$ and ${lim inf}_{n}{F}_{n}$ is proved for a sequence of closed valued, measurable set-valued functions ${F}_{n}:{\Omega }\to {2}^{X}\setminus \left\{\varnothing \right\}·$ 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 49J27 Optimal control problems in abstract spaces (existence)