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 ${P_n}_{nin mathbb{N}}$ in the sense that $int f ,{m d} P=lim_{no infty} int f, {m d} P_n$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.
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机译：我们讨论了从内部概率测度构建标准概率测度（称为下推测度）的两种方法。我们证明了内部概率测度与其下推测度之间的Wasserstein距离是无限的。作为对标准概率论的一种应用，我们证明了在可分离度量空间上的每个有限加性Borel概率度量$ P $是可加性Borel概率度量$ {P_n } _ {n in mathbb {N}} $表示$ int f ，{ rm d} P = lim_ {n to infty} int f ，{ rm d} P_n $对于所有有界一致连续实值函数$ f $当且仅当空间完全有界时。
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