Measurability of Fixed Point Sets of Multivalued Random Operators

AbstractLet (M,d) be a complete separable metric space, (Ω,Σ) a measurable space with Σ a σ-algebra of subsets of Ω, andT: Ω×M→CB(M) a multivalued random operator. The measurability of the functionFof fixed point sets ofTdefined byF(ω)≔{x∈M:x∈T(ω,x)} is studied. In particular, it is proved thatFis measurable providedTis a random contraction, orMis a weakly compact convex separable subset of a Banach space andTis a random multivalued nonexpansive mapping such thatI−T(ω,·) is demiclosed at 0 for every ω∈Ω. The same result is also verified for a single-valued random nonexpansive mapping in a uniformly smooth Banach space