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Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory

Peter A. Loeb
Transactions of the American Mathematical Society
Vol. 211 (Oct., 1975), pp. 113-122
DOI: 10.2307/1997222
Stable URL: http://www.jstor.org/stable/1997222
Page Count: 10
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Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory
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Abstract

Let (X, a, ν) be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard σ-algebra M containing the algebra A, where the extended real-valued measure μ on M is generated by the standard part of ν. If f is a-measurable, then its standard part 0 f is M-measurable on X. If f and μ are finite, then the ν-integral of f is infinitely close to the μ-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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