Measurability of Fixed Point Sets of Multivalued Random Operators

Let M, d.be a complete separable metric space, V, S.a measurable space with S a s-algebra of subsets of V, and T: V=MªCB M. a multivalued random operator. The measurability of the function F of fixed point sets of T defined by F v.[ xgM: xgT v, x.4is studied. In particular, it is proved that F is measurable provided T is a random contraction, or M is a weakly compact convex separable subset of a Banach space and T is a random multivalued nonexpansive mapping such that IyT v, ?.is demiclosed at 0 for every v gV. The same result is also verified for a single-valued random nonexpansive mapping in a uniformly smooth Banach space.