# Finitely-additive, countably-additive and internal probability measures

Commentationes Mathematicae Universitatis Carolinae (2018)

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

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topDuanmu, 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 -

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