*(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 $F:{\Omega}\times X\to {2}^{X}\setminus \left\{\varnothing \right\}$ be a set-valued function with closed and convex values such that $\left(i\right)\phantom{\rule{1.em}{0ex}}F(\xb7,x)$ is measurable, $\left(ii\right)\phantom{\rule{1.em}{0ex}}F(\omega ,\xb7)$ is upper semicontinuous (with respect to the weak topology) and $\left(iii\right)\phantom{\rule{1.em}{0ex}}F(\xb7,\xb7)$ is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that $F(\xb7,\xb7)$ is $\mathcal{A}\times \mathcal{B}\left(X\right)$- measurable, where $\mathcal{B}\left(X\right)$ 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 ${lim\; sup}_{n}{F}_{n}$ and ${lim\; inf}_{n}{F}_{n}$ is proved for a sequence of closed valued, measurable set-valued functions ${F}_{n}:{\Omega}\to {2}^{X}\setminus \left\{\varnothing \right\}\xb7$ 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.

##### 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) |