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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Analytic mappings on hyperfinite sets


Authors: C. Ward Henson and David Ross
Journal: Proc. Amer. Math. Soc. 118 (1993), 587-596
MSC: Primary 03H05; Secondary 03E15, 04A15, 28A12, 28E05
MathSciNet review: 1126195
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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{{\vert S\vert}} {{\vert T\vert}} \approx 1$.

(ii) There is a bijection from $ \frac{{\vert S\vert}} {{\vert T\vert}} \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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1126195-9
PII: S 0002-9939(1993)1126195-9
Article copyright: © Copyright 1993 American Mathematical Society