Analytic mappings on hyperfinite sets

Henson, C. Ward; Ross, David

doi:10.1090/S0002-9939-1993-1126195-9

Analytic mappings on hyperfinite sets
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by C. Ward Henson and David Ross

Proc. Amer. Math. Soc. 118 (1993), 587-596

DOI: https://doi.org/10.1090/S0002-9939-1993-1126195-9

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Abstract:

Let $S$ and $T$ be hyperfinite sets in an ${\aleph _1}$-saturated nonstandard universe. The following are equivalent: (i) $\frac {{|S|}} {{|T|}} \approx 1$. (ii) There is a bijection from $\frac {{|S|}} {{|T|}} \approx 1.$ onto $T$ whose graph is Borel (over the internal subsets of $S \times T$). This follows from somewhat more general results about analytic partial functions on hyperfinite sets, the proofs of which use Choquet’s theorem on the capacitibility of analytic sets. This paper includes: proofs of the above results; an elementary direct construction for extensions of internal set functions to capacities; and a surprising corollary asserting the nonexistence of ergodic Borel transformations of a Loeb probability space.

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