Convergence in measure under Finite Additivity

TL;DR

This paper explores replacing the topology of convergence in probability with convergence in L^1 under finite additivity, and characterizes continuous linear functionals on measurable functions.

Contribution

It introduces a new perspective on convergence in measure under finite additivity and provides a characterization of continuous linear functionals.

Findings

Convergence in measure can be replaced by convergence in L^1 under finite additivity.

A characterization of continuous linear functionals on measurable functions is established.

The results extend the understanding of convergence and functional analysis in non-additive measure spaces.

Abstract

We investigate the possibility of replacing the topology of convergence in probability with convergence in $L^1$. A characterization of continuous linear functionals on the space of measurable functions is also obtained.