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 be an internal measure space in a denumerably comprehensive enlargement. The set *X* is a standard measure space when equipped with the smallest standard -algebra containing the algebra , where the extended real-valued measure on is generated by the standard part of . If *f* is -measurable, then its standard part is -measurable on *X*. If *f* and are finite, then the -integral of *f* is infinitely close to the -integral of . 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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Additional Information

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

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

Article copyright:
© Copyright 1975
American Mathematical Society