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

Abstract:

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