L0-Convex Compactness and Random Normal Structure in L0 (F, B)
L0-Convex Compactness and Random Normal Structure in L0 (F, B)
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DOI：
10.1007/s10473-020-0211-9
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发表时间：
2020-03
影响因子：
1
通讯作者：
T. Guo;Erxin Zhang;Yachao Wang;G. Yuan
中科院分区：
文献类型：
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作者：
T. Guo;Erxin Zhang;Yachao Wang;G. Yuan
设(B,∥·∥)为Banach空间，(Ω,F,P)为概率空间，L0(F,B)为从(Ω,F,P)到(B,∥·∥)的强随机元素(或强可测函数)的等价类集合。众所周知，L0(F,B)成为一个完整的随机赋范模，在随机赋范模应用于Lebesgue-Bochner函数空间理论和随机算子理论的过程中发挥了重要作用。设V是BandL0(F,V)的闭凸子集，从(Ω,F,P)到V的强随机元素等价类的集合。本文的中心目的是证明以下两个结果： (1)L0(F,V)是L0凸紧当且仅当Vi是弱紧的； (2)如果L0(F,V)是弱紧致的且具有正规结构，则L0(F,V)具有随机正规结构。作为应用，给出了强随机非扩张算子的一般随机不动点定理，该定理推广并改进了几个众所周知的结果。我们希望我们的新方法，即巧妙地结合可测选择定理、随机范模理论和Banach空间技术，可以应用于其他相关方面。
Let (B, ∥ · ∥) be a Banach space, (Ω,F,P) a probability space, andL0(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 thatL0(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. LetVbe a closed convex subset ofBandL0(F,V) the set of equivalence classes of strong random elements from (Ω,F,P) toV.The central purpose of this article is to prove the following two results: (1)L0(F,V) isL0-convexly compact if and only ifVis weakly compact; (2)L0(F,V) has random normal structure ifVis 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.