## Haosui Duanmu, William Weiss

*Finitely-additive, countably-additive and internal probability measures*

Comment.Math.Univ.Carolin. 59,4 (2018) 467-485.**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 $\{P_n\}_{n\in \mathbb{N}}$ in the sense that $\int f \,{\rm d} P=\lim_{n\to \infty} \int f\, {\rm d} P_n$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.

**Keywords:** nonstandard model in mathematics; nonstandard analysis; nonstandard measure theory; convergence of probability measures

**DOI:** DOI 10.14712/1213-7243.2015.270

**AMS Subject Classification:** 03H05 26E35 28E05 60B10

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