Hyperfiniteness and the Halmos-Rohlin theorem for nonsingular Abelian actions

Abstract

Theorem 1. Let the countable abelian group G G act nonsingularly and aperiodically on Lebesgue space ( X , μ ) (X,\mu ) . Then for each finite subset A ⊂ G A \subset G and ε > 0 ∃ \varepsilon > 0\exists finite B ⊂ G B \subset G and F ⊂ X F \subset X with { b F : b ∈ B } \{ bF:b \in B\} disjoint and μ [ ( ∩ a ∈ A B − a ) F ] > 1 − ε \mu [({ \cap _{a \in A}}B - a)F] > 1 - \varepsilon . Theorem 2. Every nonsingular action of a countable abelian group on a Lebesgue space is hyperfinite.