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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 ail 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 . . dlrectlon. Moreover, the standard part map can be used as a measure preserving transformation for all -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 -smooth Baire-measures to T-smooth Borel-measures, the decomposition theorems for -smooth Baire-measures and -smooth Borel-measures and Kakutanis theorem for product measures.

Universal Loeb-measurability of sets and of the standard part map with applications