The Journal of Symbolic Logic

Research Article

Amenable versus hyperfinite Borel equivalence relations

Alexander S. Kechris

Department of Mathematics, California Institute of Technology, Pasadena, California 91125, E-mail: kechris@caltech.bitnet, E-mail: kechris@romeo.caltech.edu

Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [x]E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.

The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is called hyperfinite if it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relation E induced by a Borel action of a countable amenable group G on X and for every (Borel) probability measure μ on X, there is a Borel invariant set YX with μ(Y) = 1 such that EY (= the restriction of E to Y) is hyperfinite. (Recall that a countable group G is amenable if it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether every E induced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true for G = ℤn. However, the problem is still open even for abelian G. Our main purpose here is to provide a weaker affirmative answer for general amenable G (and more—see below). We need a definition first. Given two standard Borel spaces X, Y, a universally measurable isomorphism between X and Y is a bijection ƒ: XY such that both ƒ, ƒ-1 are universally measurable. (As usual, a map g : ZW, with Z and W standard Borel spaces, is called universally measurable if it is μ-measurable for every probability measure μ on Z.) Notice now that to assert that a countable Borel equivalence relation on X is hyperfinite is trivially equivalent to saying that there is a standard Borel space Y and a hyperfinite Borel equivalence relation F on Y, which is Borel isomorphic to E, i.e., there is a Borel bijection ƒ: XY with xEy ⇔ ƒ(x)F ƒ(y). We have the following theorem.

(Received April 01 1992)

(Revised August 20 1992)