On the Random Conjugate Spaces of a Random Locally Convex Module

TL;DR

This paper investigates the different definitions of random conjugate spaces in random locally convex modules, establishing that only two are universally suitable and revealing that certain random normed modules are totally disconnected under specific topologies.

Contribution

It proves that only two of four possible definitions of random conjugate spaces are suitable and shows that nonatomic random normed modules are totally disconnected with the locally L^0-convex topology.

Findings

Only two of four definitions are suitable for the theory.

Random normed modules over nonatomic bases are totally disconnected.

The paper provides a foundational clarification for the structure of random conjugate spaces.

Abstract

Theoretically speaking, there are four kinds of possibilities to define the random conjugate space of a random locally convex module. The purpose of this paper is to prove that among the four kinds there are only two which are universally suitable for the current development of the theory of random conjugate spaces: in this process we also obtain a somewhat surprising and crucial result that for a random normed module with base $(\Omega,{\cal F},P)$ such that $(\Omega,{\cal F},P)$ is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally $L^{0}-$convex topology.