• Volume: 59, Issue: 4, page 467-485
• ISSN: 0010-2628

top

## Abstract

top
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 ${\left\{{P}_{n}\right\}}_{n\in ℕ}$ in the sense that $\int f\phantom{\rule{0.166667em}{0ex}}\mathrm{d}P={lim}_{n\to \infty }\int f\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{P}_{n}$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.

## How to cite

top

Duanmu, Haosui, and Weiss, William. "Finitely-additive, countably-additive and internal probability measures." Commentationes Mathematicae Universitatis Carolinae 59.4 (2018): 467-485. <http://eudml.org/doc/294738>.

@article{Duanmu2018,
abstract = {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 $\lbrace P_n\rbrace _\{n\in \mathbb \{N\}\}$ in the sense that $\int f \,\{\rm d\} P=\lim _\{n\rightarrow \infty \} \int f\, \{\rm d\} P_n$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.},
author = {Duanmu, Haosui, Weiss, William},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonstandard model in mathematics; nonstandard analysis; nonstandard measure theory; convergence of probability measures},
language = {eng},
number = {4},
pages = {467-485},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
url = {http://eudml.org/doc/294738},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Duanmu, Haosui
AU - Weiss, William
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2018
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 59
IS - 4
SP - 467
EP - 485
AB - 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 $\lbrace P_n\rbrace _{n\in \mathbb {N}}$ in the sense that $\int f \,{\rm d} P=\lim _{n\rightarrow \infty } \int f\, {\rm d} P_n$ for all bounded uniformly continuous real-valued function $f$ if and only if the space is totally bounded.
LA - eng
KW - nonstandard model in mathematics; nonstandard analysis; nonstandard measure theory; convergence of probability measures
UR - http://eudml.org/doc/294738
ER -

## References

top
1. Aldaz J. M., Representation of Measures via the Standard Part Map, Ph.D. Thesis, University of Illinois at Urbana-Champaign, Champaign, 1991. MR2687095
2. Aldaz J. M., 10.1090/S0002-9939-1995-1260159-1, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2799–2808. MR1260159DOI10.1090/S0002-9939-1995-1260159-1
3. Anderson R. M., 10.1090/S0002-9947-1982-0654856-1, Trans. Amer. Math. Soc. 271 (1982), no. 2, 667–687. MR0654856DOI10.1090/S0002-9947-1982-0654856-1
4. Anderson R. M., Rashid S., 10.1090/S0002-9939-1978-0480925-X, Proc. Amer. Math. Soc. 69 (1978), no. 2, 327–332. MR0480925DOI10.1090/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. MR1603229
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. MR0751777
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. MR0881427
11. Keisler H. J., An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, 184 pages. Zbl0529.60062MR0732752
12. Lindström T., Pushing Down Loeb-Measures, Pure Mathematics: Preprint Series, Matematisk Institutt Universitetet i Oslo, Oslo, 1981.
13. Loeb P. A., 10.1090/S0002-9947-1975-0390154-8, Trans. Amer. Math. Soc. 211 (1975), 113–122. Zbl0312.28004MR0390154DOI10.1090/S0002-9947-1975-0390154-8
14. McShane E. J., 10.1090/S0002-9904-1934-05978-0, Bull. Amer. Math. Soc. 40 (1934), no. 12, 837–842. Zbl0010.34606MR1562984DOI10.1090/S0002-9904-1934-05978-0
15. Render H., 10.7146/math.scand.a-12437, Math. Scand. 72 (1993), no. 1, 61–84. MR1225997DOI10.7146/math.scand.a-12437
16. Rosenthal J. S., A First Look at Rigorous Probability Theory, World Scientific Publishing, Hackensack, 2006. MR2279622
17. Ross D., 10.1090/S0002-9939-1992-1079898-8, Proc. Amer. Math. Soc. 115 (1992), no. 2, 365–370. MR1079898DOI10.1090/S0002-9939-1992-1079898-8
18. Ross D. A., 10.7146/math.scand.a-15087, Math. Scand. 104 (2009), no. 1, 108–116. MR2498374DOI10.7146/math.scand.a-15087
19. Yosida K., Hewitt E., 10.1090/S0002-9947-1952-0045194-X, Trans. Amer. Math. Soc. 72 (1952), 46–66. MR0045194DOI10.1090/S0002-9947-1952-0045194-X

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.