Hyperfinite transversal theory

Živaljević, Boško

doi:10.1090/S0002-9947-1992-1033237-1

Hyperfinite transversal theory
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by Boško Živaljević

Trans. Amer. Math. Soc. 330 (1992), 371-399

DOI: https://doi.org/10.1090/S0002-9947-1992-1033237-1

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Abstract:

A measure theoretic version of a well-known P. Hall’s theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall’s theorem for $\Pi _1^0(\kappa )$ graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall’s theorem in the case of $\Sigma _1^0(\kappa )$ monotone graphs and give an example of a $\Sigma _1^0$ graph which satisfies Hall’s condition and which does not possess an internal a.e. matching.

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Hyperfinite transversal theory