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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
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/1997222
Page Count: 10
You can always find the topics here!Topics: Poisson process, Mathematical functions, Probability theory, Nonstandard analysis, Algebra, Monads, Markov processes, Probabilities, Real numbers, Logical givens
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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.
Transactions of the American Mathematical Society © 1975 American Mathematical Society