L^0-CONVEX COMPACTNESS AND RANDOM NORMAL STRUCTURE IN L^0(F,B)  

作　　者：Tiexin GUO Erxin ZHANG Yachao WANG George YUAN 郭铁信;张二鑫;王亚超;袁先智(School of Mathematics and Statistics,Central South University,Changsha 410083,China;Centre for Financial Engineering,Soochow University,Suzhou 215006,China)

机构地区：[1]School of Mathematics and Statistics,Central South University,Changsha 410083,China [2]Centre for Financial Engineering,Soochow University,Suzhou 215006,China

出　　处：《Acta Mathematica Scientia》2020年第2期457-469,共13页数学物理学报（B辑英文版）

基　　金：This work was supported by National Natural Science Foundation of China(11571369)。

摘　　要：Let(B,||·||)be a Banach space,(?,F,P)a probability space,and L^0(F,B)the set of equivalence classes of strong random elements(or strongly measurable functions)from(?,F,P)to(B,||·||).It is well known that L^0(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(F,V)the set of equivalence classes of strong random elements from(?,F,P)to V.The central purpose of this article is to prove the following two results:(1)L^0(F,V)is L^0-convexly compact if and only if V is weakly compact;(2)L^0(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 

分 类 号：O177[理学—数学] O189.11[理学—基础数学]