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.

中文翻译：

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L 0（F，B）中的L 0-凸紧致性和随机正态结构

令（B，∥·∥）为Banach空间，（Ω，F，P）为概率空间，L ⁰（F，B）为（Ω）强随机元素（或强可测函数）的等价类的集合，F，P）至（B，∥·∥）。众所周知，L ⁰（F，B）成为一个完整的随机范数模块，在将随机范数模块应用于Lebesgue-Bochner函数空间理论和随机算符理论的过程中发挥了重要作用。令V为的闭合凸子集B和L ⁰（F，V）从（Ω，F，P）到V的强随机元素的等价类的集合。本文的主要目的是证明以下两个结果：（1）L ⁰（F，V）是且仅当V是弱紧致时才是L ^0-凸紧致；（2）如果V满足，则L ⁰（F，V）具有随机的正态结构体积小巧，结构正常。作为一个应用，给出了一个强随机非扩张算子的一般随机不动点定理，它推广和改进了一些众所周知的结果。我们希望我们的新方法，即巧妙地结合可测量的选择定理，随机范数模块的理论和Banach空间技术，可以应用于其他相关方面。