The Algebraic Structure of Finitely Generated L⁰(F,K)-Modules and the Helly Theorem in Random Normed Modules

2010/09/27 by Tiexin Guo, Guang Shi, Guo, Tiexin +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1009.5170

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arxiv created 2011/03/29 · arxiv updated 2011/03/30

Abstract

Let K be the scalar field of real numbers or complex numbers and L⁰(F,K) the algebra of equivalence classes of K-valued random variables defined on a probability space (Ω,F,P). In this paper, we first characterize the algebraic structure of finitely generated L⁰(F,K)-modules and then combining the recently developed separation theorem in random locally convex modules we prove the Helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property. By the way,we also give an application to the existence problem of the random solution of a system of random linear functional equations.

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