Guo Tiexin, Zhao Shien
Published 2011-03-16Version 1
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.
A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module
Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules
A counterexample shows that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms
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