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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0906814-1
MathSciNet review: 906814
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1987-0906814-1
Keywords: Loeb-measures, Baire- and Borel-measures, representation and extension of measures, $ K$-saturated models
Article copyright: © Copyright 1987 American Mathematical Society

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