Solid-set functions and topological measures on locally compact spaces

TL;DR

This paper explores the construction of topological measures on locally compact spaces using solid-set functions, extending the theory beyond compact spaces and providing practical methods and examples.

Contribution

It introduces a new method to construct topological measures from solid-set functions on locally compact, connected, and locally connected spaces, expanding existing frameworks.

Findings

Provides examples of finite and infinite topological measures on non-compact spaces.

Introduces a method to generate measures on spaces with genus 0 in their one-point compactification.

Extends the theory of topological measures beyond compact spaces.

Abstract

A topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. Almost all works devoted to topological measures, corresponding non-linear functionals, and their applications deal with compact spaces. The present paper is one in a series that investigates topological measures and corresponding non-linear functionals on locally compact spaces. Here we examine solid and semi-solid sets on a locally compact space. We then give a method of constructing topological measures from solid-set functions on a locally compact, connected, locally connected space. The paper gives examples of finite and infinite topological measures on locally compact, non-compact spaces and presents an easy way to generate topological measures on spaces whose one-point compactification has genus 0 from existing examples of topological measures on compact spaces.