THE STRUCTURE OF HYPERFINITE BOREL EQUIVALENCE-RELATIONS

We study the structure of the equivalence relations induced by the orb its of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (Thi s utilizes among other things work of Effros and Weiss.) Using this an d also results of Dye, Varadarajan, and recent work of Nadkarni, we sh ow that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such e quivalence relations. In particular, since the only possible such card inalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorp hism types. Canonical examples of each type are also discussed.