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**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.

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.

Reviewer: Isaac Goldbring (Irvine)

### MSC:

60B10 | Convergence of probability measures |

03H05 | Nonstandard models in mathematics |

26E35 | Nonstandard analysis |

28E05 | Nonstandard measure theory |

### Keywords:

nonstandard model in mathematics; nonstandard analysis; nonstandard measure theory; convergence of probability measures
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\textit{H. Duanmu} and \textit{W. Weiss}, Commentat. Math. Univ. Carol. 59, No. 4, 467--485 (2018; Zbl 1474.60008)

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