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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[1] Aldaz J. M., Representation of Measures via the Standard Part Map, Ph.D. Thesis, University of Illinois at Urbana-Champaign, Champaign, 1991
[2] Aldaz J. M., Representing abstract measures by Loeb measures: a generalization of the standard part map, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2799-2808 · Zbl 0847.28010 · doi:10.1090/S0002-9939-1995-1260159-1
[3] Anderson R. M., Star-finite representations of measure spaces, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667-687 · Zbl 0494.28005 · doi:10.1090/S0002-9947-1982-0654856-1
[4] Anderson R. M.; Rashid S., A nonstandard characterization of weak convergence, Proc. Amer. Math. Soc. 69 (1978), no. 2, 327-332 · Zbl 0393.03047 · doi:10.1090/S0002-9939-1978-0480925-X
[5] Arkeryd L. O.; Cutland N. J.; Henson C. W.; eds., Nonstandard Analysis, Theory and Applications, Proc. of the NATO Advanced Study Institute on Nonstandard Analysis and Its Applications, Edinburgh, 1996, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 493, Kluwer Academic Publishers Group, Dordrecht, 1997
[6] Bhaskara R. K. P. S.; Bhaskara R. M., Theory of Charges, A study of finitely additive measures, Pure and Applied Mathematics, 109, Academic Press, Harcourt Brace Jovanovich Publishers, New York, 1983 · Zbl 0516.28001
[7] Cutland N. J.; Neves V.; Oliveira F.; Sousa-Pinto J.; eds., Developments in Nonstandard Mathematics, Pitman Research Notes in Mathematics Series, 336, Papers from the Int. Col. (CIMNS94), Aveiro, 1994, Longman, Harlow, 1995
[8] Duanmu H.; Rosenthal J. S.; Weiss W., Ergodicity of Markov Processes via Non-standard Analysis, available at http://probability.ca/jeff/ftpdir/duanmu1.pdf
[9] Duanmu H.; Roy D. M., On Extended Admissible Procedures and Their Nonstandard Bayes Risk, available at arXiv:1612.09305v2 [math.ST]
[10] Kadane J. B.; Schervish M. J.; Seidenfeld T., Statistical implications of finitely additive probability, Stud. Bayesian Econometrics Statist., 6, North-Holland, Amsterdam, 1986, pages 59-76 · Zbl 0619.62007
[11] Keisler H. J., An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, 184 pages · Zbl 0529.60062
[12] Lindström T., Pushing Down Loeb-Measures, Pure Mathematics: Preprint Series, Matematisk Institutt Universitetet i Oslo, Oslo, 1981
[13] Loeb P. A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122 · Zbl 0312.28004 · doi:10.1090/S0002-9947-1975-0390154-8
[14] McShane E. J., Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837-842 · Zbl 0010.34606 · doi:10.1090/S0002-9904-1934-05978-0
[15] Render H., Pushing down Loeb measures, Math. Scand. 72 (1993), no. 1, 61-84 · Zbl 0782.28006 · doi:10.7146/math.scand.a-12437
[16] Rosenthal J. S., A First Look at Rigorous Probability Theory, World Scientific Publishing, Hackensack, 2006 · Zbl 1127.60002
[17] Ross D., Compact measures have Loeb preimages, Proc. Amer. Math. Soc. 115 (1992), no. 2, 365-370 · Zbl 0755.28009 · doi:10.1090/S0002-9939-1992-1079898-8
[18] Ross D. A., Pushing down infinite Loeb measures, Math. Scand. 104 (2009), no. 1, 108-116 · Zbl 1160.28005 · doi:10.7146/math.scand.a-15087
[19] Yosida K.; Hewitt E., Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46-66 · Zbl 0046.05401 · doi:10.1090/S0002-9947-1952-0045194-X
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