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 {Pn}n(Formula presented) in the sense that (Formula presented) f dP = lmi _n→∞ (Formula presented) f dPn for all bounded uniformly continuous real-valued function f if and only if the space is totally bounded.