Finitely-additive, countably-additive and internal probability measures

TL;DR

This paper explores the relationship between internal and standard probability measures, demonstrating how finitely-additive measures can be approximated by countably-additive ones in totally bounded spaces using Wasserstein distance.

Contribution

It introduces methods to construct push-down measures from internal measures and characterizes when finitely-additive measures can be approximated by countably-additive measures.

Findings

Wasserstein distance between internal and push-down measures is infinitesimal.

Finitely-additive measures are limits of countably-additive measures iff the space is totally bounded.

Abstract

We discuss two ways to construct standard probability measures, called push-down measures, from internal probability measures. We show that the Wasserstein distance between an internal probability measure and its push-down measure is infinitesimal. As an application to standard probability theory, we show that every finitely-additive Borel probability measure $P$ on a separable metric space is a limit of a sequence of countably-additive Borel probability measures if and only if the space is totally bounded.