Construction of Regular Non-Atomic Strictly-Positive Measures in Second-Countable Non-Atomic Locally Compact Hausdorff Spaces

TL;DR

This paper provides a constructive method to establish the existence of a regular, non-atomic, strictly-positive measure on second-countable, non-atomic, locally compact Hausdorff spaces, using recursive set functions.

Contribution

It introduces a new constructive approach to build such measures, leveraging the space's non-atomicity to ensure the measure's properties.

Findings

Existence of regular non-atomic strictly-positive measures proven constructively.

Method applicable to all second-countable non-atomic locally compact Hausdorff spaces.

Recursive construction of measures via finitely-additive set functions.

Abstract

This paper presents a constructive proof of the existence of a regular non-atomic strictly-positive measure on any second-countable non-atomic locally compact Hausdorff space. This construction involves a sequence of finitely-additive set functions defined recursively on an ascending sequence of rings of subsets with a premeasure limit that is extendable to a measure with the desired properties. Non-atomicity of the space provides a non-trivial way to ensure that the limit is a premeasure.