# Signed topological measures on locally compact spaces

## TL;DR

This paper introduces and analyzes signed topological measures on locally compact spaces, extending measure theory concepts and providing new representation results for these measures.

## Contribution

It defines signed deficient and signed topological measures, explores their properties, and extends classical measure results to this broader context.

## Key findings

- Signed deficient topological measures are τ-smooth on open and compact sets.
- Signed topological measures include differences of Radon measures but form a larger family.
- Finite norm signed topological measures can be represented as differences of two topological measures under certain conditions.

## Abstract

In this paper we define and study signed deficient topological measures and signed topological measures (which generalize signed measures) on locally compact spaces. We prove that a signed deficient topological measure is $\tau$-smooth on open sets and $\tau$-smooth on compact sets. We show that the family of signed measures that are differences of two Radon measures is properly contained in the family of signed topological measures, which in turn is properly contained in the family of signed deficient topological measures. Extending known results for compact spaces, we prove that a signed topological measure is the difference of its positive and negative variations if at least one variation is finite; we also show that the total variation is the sum of the positive and negative variations. If the space is locally compact, connected, locally connected, and has the Alexandroff one-point compactification of genus 0, a signed topological measure of finite norm can be represented as a difference of two topological measures.