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

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- by Peter A. Loeb
- Trans. Amer. Math. Soc.
**211**(1975), 113-122 - DOI: https://doi.org/10.1090/S0002-9947-1975-0390154-8
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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

*j*th event and a construction of standard sample functions for Poisson processes.

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## Bibliographic Information

- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**211**(1975), 113-122 - MSC: Primary 28A10; Secondary 02H25, 60J99
- DOI: https://doi.org/10.1090/S0002-9947-1975-0390154-8
- MathSciNet review: 0390154