Finitely-additive, countably-additive and internal probability measures

Haosui Duanmu; William Weiss

Commentationes Mathematicae Universitatis Carolinae (2018)

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

Abstract

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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 { P n } n ∈ ℕ in the sense that ∫ f d P = lim n → ∞ ∫ f d P n for all bounded uniformly continuous real-valued function f if and only if the space is totally bounded.

How to cite

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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},
title = {Finitely-additive, countably-additive and internal probability measures},
url = {http://eudml.org/doc/294738},
volume = {59},
year = {2018},
}

TY - JOUR
AU - Duanmu, Haosui
AU - Weiss, William
TI - Finitely-additive, countably-additive and internal probability measures
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

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