The Classification of Hyperfinite Borel Equivalence Relations

Abstract

Let X be a standard Borel space and E a Borel equivalence relation on X. We call E hyperfinite if there is a Borel automorphism T of X such that xEy ⇔ ∃ n Є ℤ(T^nx = y). For Borel equivalence relations E,F on X, Y resp. we write E ⊑ F ⇔ 3 ƒ : X → Y(ƒ Borel, injective with E = ƒ^(-1)[F]) E ≈ F ⇔ E ⊑ F and F ⊑ E E ≅ F ⇔ ∃ ƒ :X → Y(ƒ a Borel isomorphism with E= ƒ^(-1)[F]) A Borel equivalence relation E on X is called smooth if there is a Borel map ƒ: X → Y (Y some standard Borel space) with xEy ⇔ ƒ(x) = ƒ(y).