The Algebraic Structure of Finitely Generated $L^{0}(\mathcal{F},K)$-Modules and the Helly Theorem in Random Normed Modules

TL;DR

This paper characterizes finitely generated modules over the algebra of random variables and proves a Helly theorem in random normed modules, extending classical results to a stochastic setting with applications to random linear equations.

Contribution

It provides a detailed algebraic structure of finitely generated $L^{0}(,F,K)$-modules and establishes a Helly theorem in the context of random normed modules with the countable concatenation property.

Findings

Characterization of finitely generated $L^{0}(,F,K)$-modules.

Proof of Helly theorem in random normed modules.

Application to solutions of random linear functional equations.

Abstract

Let $K$ be the scalar field of real numbers or complex numbers and $L^{0}(\mathcal{F},K)$ the algebra of equivalence classes of $K-$valued random variables defined on a probability space $(\Omega,\mathcal{F},P)$. In this paper, we first characterize the algebraic structure of finitely generated $L^{0}(\mathcal{F},K)$-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.