A random version of Mazur's lemma

TL;DR

This paper extends Mazur's lemma to locally $L^0$-convex modules, incorporating a countable concatenation condition, and demonstrates its necessity through a counterexample.

Contribution

It introduces a generalized version of Mazur's lemma within the framework of locally $L^0$-convex modules, including a crucial countable concatenation condition.

Findings

The generalized Mazur's lemma holds under the countable concatenation condition.

A counterexample shows the condition cannot be omitted.

The extension broadens the applicability of Mazur's lemma in convex analysis.

Abstract

The purpose of this paper is to generalize the classical Mazur's lemma from the classical convex analysis to the framework of locally $L^0$-convex modules. In this version an extra condition of countable concatenation is included. We provide a counterexample showing that this condition cannot be removed.