  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.
