$L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$

doi:10.1007/s10473-020-0211-9

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (2): 457-469.doi: 10.1007/s10473-020-0211-9

$L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$

郭铁信¹, 张二鑫¹, 王亚超¹, 袁先智²

1. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
2. Centre for Financial Engineering, Soochow University, Suzhou 215006, China

收稿日期:2018-11-19 出版日期:2020-04-25 发布日期:2020-05-26
通讯作者: Tiexin GUO E-mail:tiexinguo@csu.edu.cn
作者简介:Erxin ZHANG,E-mail:zhangerxin6666@163.com;Yachao WANG,E-mail:wychao@csu.edu.cn;George YUAN,E-mail:george_yuan@yahoo.com
基金资助:
This work was supported by National Natural Science Foundation of China (11571369).

$L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$

Tiexin GUO¹, Erxin ZHANG¹, Yachao WANG¹, George YUAN²

1. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
2. Centre for Financial Engineering, Soochow University, Suzhou 215006, China

Received:2018-11-19 Online:2020-04-25 Published:2020-05-26
Contact: Tiexin GUO E-mail:tiexinguo@csu.edu.cn
Supported by:
This work was supported by National Natural Science Foundation of China (11571369).

摘要/Abstract

摘要： 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 operator theory. 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 $V$. The central purpose of this article 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.

关键词: Complete random normed modules, fixed point theorem, $L^0$-convex compactness, random normal structure, 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 operator theory. 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 $V$. The central purpose of this article 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.

Key words: Complete random normed modules, fixed point theorem, $L^0$-convex compactness, random normal structure, random nonexpansive operators

中图分类号:

46A16

郭铁信, 张二鑫, 王亚超, 袁先智. $L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$[J]. 数学物理学报(英文版), 2020, 40(2): 457-469.

Tiexin GUO, Erxin ZHANG, Yachao WANG, George YUAN. $L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$[J]. Acta mathematica scientia,Series B, 2020, 40(2): 457-469.

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$L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$

$L^0$-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN $L^0(\mathcal{F},B)$

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