On the nonstandard representation of measures

Abstract:

In this paper it is shown that every finitely additive probability measure $\mu$ on $S$ which assigns 0 to finite sets can be given a nonstandard representation using the counting measure for some $^ \ast$-finite subset $F$ of $^ \ast S$. Moreover, if $\mu$ is countably additive, then $F$ can be chosen so that \[ \int {fd\mu } = {\text {st( }}\frac {1}{{||F||}}\sum _{p \in F} ^\ast f(p))\] for every $\mu$-integrable function $f$. An application is given of such representations. Also, a simple nonstandard method for constructing invariant measures is presented.