L0-Convex Compactness and Random Normal Structure in L0 (F, B)

Abstract

Let (B, ∥ · ∥) be a Banach space, (Ω, F, P) a probability space, and L0 (F, B) the set of equivalence classes of strong random elements (or strongly measurable functions) from (Ω, F, P) to (B, ∥ · ∥). It is well known that L0 (F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L0 (F, V) the set of equivalence classes of strong random elements from (Ω, F, P) to V. The central purpose of this article is to prove the following two results: (1) L0 (F, V) is L0-convexly compact if and only if V is weakly compact; (2) L0 (F, V) has random normal structure if V is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.