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

Authors:
D. Landers and L. Rogge

Journal:
Trans. Amer. Math. Soc. **304** (1987), 229-243

MSC:
Primary 28E05; Secondary 03H05

MathSciNet review:
906814

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Abstract: It is shown in this paper that for -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 -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 -smooth Borel-measures, the decomposition theorems for -smooth Baire-measures and -smooth Borel-measures and Kakutani's theorem for product measures.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0906814-1

Keywords:
Loeb-measures,
Baire- and Borel-measures,
representation and extension of measures,
-saturated models

Article copyright:
© Copyright 1987
American Mathematical Society