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

Peter A. Loeb
1975 Transactions of the American Mathematical Society  
Let (X, 3, v) be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard o-algebra % containing the algebra a, where the extended real-valued measure p. on % is generated by the standard part of v. If / is fl-measurable, then its standard part / is jR-measurable on X. If / and p. are finite, then the vintegral of / is infinitely close to the /¿-integral of /. Applications include coin tossing and
more » ... tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the ;'th event and a construction of standardsample functions for Poisson processes.
doi:10.1090/s0002-9947-1975-0390154-8 fatcat:ha32s7tpizgnzbjzweovyeboqi