A continuity theorem for generalised signed measures with an application to Karamata's Tauberian theorem

TL;DR

This paper extends a classical convergence theorem for positive measures to signed Radon measures and introduces a new Tauberian condition that broadens Karamata's theorem, with implications for measure convergence and analysis.

Contribution

It generalizes the continuity theorem to signed measures and proposes a novel Tauberian condition extending Karamata's theorem.

Findings

Extended the classical continuity theorem to signed Radon measures.

Established additional conditions for convergence in the signed case.

Introduced a new Tauberian condition for signed measures.

Abstract

The Laplace transforms of positive measures on $\mathbb{R}_{+}$ converge if and only if their distribution functions converge at continuity points of the limiting measure. We extend this classical continuity theorem to the case of generalised signed Radon measures. The result for the signed case requires some additional conditions, which follow from recent results on vague convergence of signed Radon measures. As an application, we introduce a novel Tauberian condition for generalised signed Radon measures that extends Karamata's Tauberian theorem.