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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Conversion from nonstandard to standard measure spaces and applications in probability theory


Author: Peter A. Loeb
Journal: Trans. Amer. Math. Soc. 211 (1975), 113-122
MSC: Primary 28A10; Secondary 02H25, 60J99
MathSciNet review: 0390154
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Abstract: Let $ (X,\mathcal{A},\nu )$ be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard $ \sigma $-algebra $ \mathfrak{M}$ containing the algebra $ \mathcal{A}$, where the extended real-valued measure $ \mu $ on $ \mathfrak{M}$ is generated by the standard part of $ \nu $. If f is $ \mathcal{A}$-measurable, then its standard part $ ^0f$ is $ \mathfrak{M}$-measurable on X. If f and $ \mu $ are finite, then the $ \nu $-integral of f is infinitely close to the $ \mu $-integral of $ ^0f$. Applications include coin tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the jth event and a construction of standard sample functions for Poisson processes.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0390154-8
PII: S 0002-9947(1975)0390154-8
Keywords: Measure space, probability space, standard integral, coin tossing, Poisson processes, sample function, strong Markov property
Article copyright: © Copyright 1975 American Mathematical Society