## Finitely-additive, countably-additive and internal probability measures.(English)Zbl 1474.60008

This article contributes to the well-established field of nonstandard measure theory. There are two main nonstandard theorems with one main standard application.
The first theorem states that if $$(X,d)$$ is a bounded, $$\sigma$$-compact metric space and $$\nu$$ is an internal probability measure on $$(X^*,\mathcal{B}(X)^*)$$, then the (internal) Wasserstein distance between $$\nu$$ and $$(\nu^p)^*$$ is infinitesimal. Here, $$\nu^p$$ is the standard finitely additive probability measure on $$(X,\mathcal{B}(X))$$ defined by $$\nu^p(A):=\operatorname{st}(\nu(A^*))$$. Also, $$\mathcal{B}(X)$$ is the Borel $$\sigma$$-algebra on $$X$$.
The second theorem is that, under the same assumptions, the (internal) Wasserstein distance between $$\nu$$ and $$(\nu_p)^*$$ is infinitesimal, provided $$\nu_p$$ is a countably additive probability measure on $$(X,\mathcal{B}(X))$$. Here, $$\nu_p(A)$$ is the Loeb measure of $$\operatorname{st}^{-1}(A)$$.
A standard application of the above results is the following: if $$X$$ is a totally bounded, separable metric space and $$P$$ is a finitely additive probability measure on $$(X,\mathcal{B}(X))$$, then there is a sequence of finitely supported probability measures $$P_n$$ on $$(X,\mathcal{B}(X))$$ such that $$P_n$$ converges to $$P$$ weakly in the sense that $$\int f dP_n\to\int fdP$$ for all bounded, uniformly continuous real-valued functions $$f$$ on $$X$$.
The paper discusses various generalizations of these results and comparisons with related results in the literature.

### MSC:

 60B10 Convergence of probability measures 03H05 Nonstandard models in mathematics 26E35 Nonstandard analysis 28E05 Nonstandard measure theory
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