Vague and weak convergence of signed measures

TL;DR

This paper explores the conditions under which signed measures converge weakly or vaguely, clarifying their relationship and linking vague convergence to distribution function convergence on real spaces.

Contribution

It provides a comprehensive analysis of weak and vague convergence for signed measures and connects vague convergence with distribution function convergence on the real line.

Findings

Characterizes necessary and sufficient conditions for convergence of signed measures.

Clarifies the relationship between weak and vague convergence.

Links vague convergence to pointwise convergence of distribution functions.

Abstract

Necessary and sufficient conditions for weak and vague convergence of measures are important for a diverse host of applications. This paper aims to give a comprehensive description of the relationship between the two modes of convergence when the measures are signed, which is largely absent from the literature. Furthermore, when the underlying space is $\mathbb{R}$, we study the relationship between vague convergence of signed measures and the pointwise convergence of their distribution functions.