Hyperfinite transversal theory

Abstract

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 UP^k) 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 2^(k) monotone graphs and give an example of a Z^ graph which satisfies Hall's condition and which does not possess an internal a.e. matching.