Representing abstract measures by Loeb measures: a generalization of the standard part map

Aldaz, J. M.

doi:10.1090/s0002-9939-1995-1260159-1

Proc. Amer. Math. Soc.

1995

DOI: 10.1090/s0002-9939-1995-1260159-1

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Representing abstract measures by Loeb measures: a generalization of the standard part map

J. M. Aldaz¹

Abstract: Loeb measures have been utilized to represent Radon and r-smooth measures on topological spaces via the standard part map. The purpose of this paper is to show how to extend these results to a nontopological setting.

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2020

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Finitely-additive, countably-additive and internal probability measures

Duanmu¹,

Weiss²

2020

Preprint

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

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Finitely-additive, countably-additive and internal probability measures

Duanmu¹,

Weiss²

2020

Preprint

View full text Add to dashboard Cite

show abstract

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Representing abstract measures by Loeb measures: a generalization of the standard part map

Abstract: Loeb measures have been utilized to represent Radon and r-smooth measures on topological spaces via the standard part map. The purpose of this paper is to show how to extend these results to a nontopological setting.

Cited by 1 publication

References 14 publications

Finitely-additive, countably-additive and internal probability measures

Finitely-additive, countably-additive and internal probability measures

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