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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$ \Pi\sp 1\sb 1$ functions are almost internal


Author: Boško Živaljević
Journal: Trans. Amer. Math. Soc. 347 (1995), 2621-2632
MSC: Primary 03H05; Secondary 03E15, 54H05, 54J05
DOI: https://doi.org/10.1090/S0002-9947-1995-1277144-0
MathSciNet review: 1277144
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Abstract: In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets $ S$ and $ T$ does there exists a bijection $ f$ from $ S$ onto $ T$ whose graph is a projective subset of $ S \times T$? In particular, when is there a $ \Pi _1^1$ bijection from $ S$ onto $ T$? In this paper we prove that given an internal, bounded measure $ \mu $, any $ \Pi _1^1$ function is $ L(\mu )$ a.e. equal to an internal function, where $ L(\mu )$ is the Loeb measure associated with $ \mu $. It follows that if two $ \Pi _1^1$ subsets $ S$ and $ T$ of a hyperfinite set $ X$ are $ \Pi _1^1$ bijective, then $ S$ and $ T$ have the same measure for every uniformly distributed counting measure $ \mu $. When $ S$ and $ T$ are internal it turns out that any $ \Pi _1^1$ bijection between them must already be Borel. We also prove that if a $ \Pi _1^1$ graph in the product of two hyperfinite sets $ X$ and $ Y$ is universal for all internal subsets of $ Y$, then $ \vert X\vert \geqslant {2^{\vert Y\vert}}$, which is a partial answer to Henson and Ross's Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper $ K$-analytic subsets of the product of two completely regular Hausdorff topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of $ \Pi _1^1$ subsets of the products $ X \times Y$ of two internal sets $ X$ and $ Y$, all of whose $ Y$-sections are $ \Sigma _1^0(\kappa )$ sets.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1277144-0
Article copyright: © Copyright 1995 American Mathematical Society