Egoroff’s theorem and the distribution of standard points in a nonstandard model

Abstract:

We study the relationship between the Loeb measure ${}^0({}^*\mu )$ of a set $E$ and the $\mu$-measure of the set $S(E) = \{ x | {}^* x \in E \}$ of standard points in $E$. If $E$ is in the $\sigma$-algebra generated by the standard sets, then ${}^0({}^ * \mu )(E) = \mu (S(E))$. This is used to give a short nonstandard proof of Egoroff’s Theorem. If $E$ is an internal, * measurable set, then in general there is no relationship between the measures of $S(E)$ and $E$. However, if ${}^ * X$ is an ultrapower constructed using a minimal ultrafilter on $\omega$, then ${}^ * \mu (E) \approx 0$ implies that $S(E)$ is a $\mu$-null set. If, in addition, $\mu$ is a Borel measure on a compact metric space and $E$ is a Loeb measurable set, then \[ \underline \mu (S(E)) \leqslant {}^0({}^ * \mu )(E) \leqslant \overline \mu (S(E))\] where $\underline \mu$ and $\overline \mu$ are the inner and outer measures for $\mu$.