Let (, be a complete, finite measure space, let X be a separable, reflexive Banach space and let be a set-valued function with closed and convex values such that is measurable, is upper semicontinuous (with respect to the weak topology) and is separable (an analogue to the definition of a separable process). Under these assumptions the author shows that is - measurable, where are the Borel-sets on X.
Now suppose that (, is a measurable space with Souslin family and X is a Souslin metric space. Then the measurability of and is proved for a sequence of closed valued, measurable set-valued functions 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.