$L^0$--convex compactness and random normal structure in $L^0(\mathcal{F},B)$

TL;DR

This paper investigates the properties of $L^0(, B)$ spaces, establishing conditions for $L^0$-convex compactness and random normal structure based on the weak compactness of subsets in Banach spaces, and applies these results to fixed point theorems.

Contribution

It proves that $L^0$-convex compactness and random normal structure in $L^0(, B)$ are characterized by weak compactness and normal structure in the Banach space $V$, introducing new methods combining measurable selection and Banach space techniques.

Findings

$L^0$-convexly compact if and only if $V$ is weakly compact.

$L^0(,V)$ has random normal structure if $V$ is weakly compact and has normal structure.

Established a general random fixed point theorem for strong random nonexpansive operators.

Abstract

Let $(B,\|\cdot\|)$ be a Banach space, $(\Omega,\mathcal{F},P)$ a probability space and $L^0(\mathcal{F},B)$ the set of equivalence classes of strong random elements (or strongly measurable functions) from $(\Omega,\mathcal{F},P)$ to $(B,\|\cdot\|)$. It is well known that $L^0(\mathcal{F},B)$ 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 functional analysis. Let $V$ be a closed convex subset of $B$ and $L^0(\mathcal{F},V)$ the set of equivalence classes of strong random elements from $(\Omega,\mathcal{F},P)$ to $(B,\|\cdot\|)$, the central purpose of this paper is to prove the following two results: (1). $L^0(\mathcal{F},V)$ is $L^0$--convexly compact if and only if $V$ is weakly compact; (2). $L^0(\mathcal{F},V)$ 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.