A density version of a theorem of Banach

TL;DR

This paper uses nonstandard analysis to extend a measure-theoretic intersection result and applies it to develop a density-based version of Banach's representation theorem.

Contribution

It introduces a density version of Banach's theorem using the S-measure construction from nonstandard analysis.

Findings

Extended measure intersection results in finitely-additive spaces

Developed a density-limit version of Banach's representation theorem

Applied nonstandard analysis techniques to measure theory

Abstract

The S-measure construction from nonstandard analysis is used to prove an extension of a result on the intersection of sets in a finitely-additive measure space. This is then used to give a density-limit version of a representation theorem of Banach.