Geodesic
Finitely-additive, countably-additive and internal probability measures
Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 4, pp. 467-485.

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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.
DOI : 10.14712/1213-7243.2015.270
Classification : 03H05, 26E35, 28E05, 60B10
Keywords: nonstandard model in mathematics; nonstandard analysis; nonstandard measure theory; convergence of probability measures 
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Duanmu, Haosui; Weiss, William. Finitely-additive, countably-additive and internal probability measures. Commentationes Mathematicae Universitatis Carolinae, Tome 59 (2018) no. 4, pp. 467-485. doi : 10.14712/1213-7243.2015.270. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2015.270/

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