Vague and basic convergence of signed measures

TL;DR

This paper explores various convergence types of finite signed measures, establishing conditions for vague convergence and introducing the concepts of basic and almost basic convergence, with implications for measure theory.

Contribution

It introduces the concept of almost basic convergence and characterizes vague convergence in terms of local boundedness and distribution function convergence.

Findings

Vague convergence occurs when measures are locally bounded and distribution functions converge in specific ways.

Introduces and formalizes the concept of almost basic convergence.

Provides conditions linking measure convergence with distribution function behavior.

Abstract

We study the relationship between different kinds of convergence of finite signed measures and discuss their metrizability. In particular, we study the concept of basic convergence recently introduced by Khartov [arXiv:2204.13667] and introduce the related concept of almost basic convergence. We discover that a sequence of finite signed measures converges vaguely if and only if it is locally uniformly bounded in variation and the corresponding sequence of distribution functions either converges in Lebesgue measure up to constants, converges basically, or converges almost basically.