Universal Loeb-Measurability of Sets and of the Standard Part Map with Applications

Abstract

It is shown in this paper that for $K$-saturated models many important external sets of nonstandard analysis—such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points—are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction. Moreover, the standard part map can be used as a measure preserving transformation for all $\tau$-smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of $\tau$-smooth Baire-measures to $\tau$-smooth Borel-measures, the decomposition theorems for $\tau$-smooth Baire-measures and $\tau$-smooth Borel-measures and Kakutani’s theorem for product measures.