Finitely-additive, countably-additive and internal probability measures

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.