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

Abstract

Let ( X , A , ν ) (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 σ \sigma -algebra M \mathfrak {M} containing the algebra A \mathcal {A} , where the extended real-valued measure μ \mu on M \mathfrak {M} is generated by the standard part of ν \nu . If f is A \mathcal {A} -measurable, then its standard part 0 f ^0f is M \mathfrak {M} -measurable on X. If f and μ \mu are finite, then the ν \nu -integral of f is infinitely close to the μ \mu -integral of 0 f ^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.