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.


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