Semisolid sets and topological measures

TL;DR

This paper develops a new method for constructing topological measures on locally compact, connected, and locally connected spaces using semisolid sets, simplifying existing approaches especially for compact spaces.

Contribution

It introduces a novel approach to build topological measures from solid-set functions, providing simpler methods for compact spaces and tools for measure comparison.

Findings

New construction method for topological measures from semisolid sets

Simplified approach for compact spaces

Tools for identifying measure equivalence

Abstract

This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is a set function which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. We examine semisolid sets and give a way of constructing topological measures from solid-set functions on locally compact, connected, locally connected spaces. For compact spaces our approach produces a simpler method than the current one. We give examples of finite and infinite topological measures on locally compact spaces and present an easy way to generate topological measures on spaces whose one-point compactification has genus 0. Results of this paper are necessary for various methods for constructing topological measures, give additional properties of topological measures, and provide a tool for determining whether two topological measures or quasi-linear functionals are the same.