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

DOI:
https://doi.org/10.1090/S0002-9947-1975-0390154-8

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

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Keywords:
Measure space,
probability space,
standard integral,
coin tossing,
Poisson processes,
sample function,
strong Markov property

Article copyright:
© Copyright 1975
American Mathematical Society