Abstract. We study the structure of the equivalence relations induced by the orbits 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. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed. This paper is a contribution to the study of Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the hyperfinite ones. These are by definition the increasing unions of sequences of Borel equivalence relations with finite equivalence classes but equivalently they can be also described as the ones induced by the orbits of a single Borel automorphism.

Abstract.We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0,oo) -» (0,oo) with h(0+) = 0 , all the Hausdorff /¡-measure 0 closed sets F C E are sets of uniqueness (for trigonometric series). In conjunction with Körner's result on the existence of Helson sets of multiplicity, this implies the existence of closed sets of multiplicity ( M-sels) within which Hausdorff /i-measure 0 implies uniqueness, for some h . This is contrasted with the case of closed sets of strict multiplicity ( A/o-sets), where results of Ivashev-Musatov and Kaufman establish the opposite.A Hausdorff determining function is any function h : (0 , oo) -► (0, oo) which is nondecreasing and satisfies h(0+) = 0. A subset E of T (the unit circle, viewed here as R/2nZ) has Hausdorff h-measure 0 if for every e > 0 there is a sequence {In} of open intervals (arcs) in T with E ç \Jn En and Z)^(l^sl) < e • (Here |/J = (arc length of I"/2n.) It is a well-known theorem of Ivashev-Musatov [2] that metric thinness in the form of Hausdorff «-measure 0 cannot imply uniqueness for trigonometric series. More precisely, for any « as above there is a closed set E of Hausdorff «-measure 0 which is of restricted multiplicity (we will give a review of terminology below). Kaufman [3; see also 4, VIII.3.3] extended this result by proving that, for any « as above, any closed set of strict multiplicity has a closed subset of Hausdorff «-measure 0 which is still of restricted multiplicity. Later Kaufman asked (in a private conversation) whether the same holds for ordinary multiplicity. The main result of this paper gives a negative answer to Kaufman's question, and provides a characterization of those closed sets which, for some « , contain no closed subsets of Hausdorff «-measure 0 which are of multiplicity.We now give a brief review of the notation, terminology, and results we need; this material all appears in Kechris-Louveau [4]. Let K(T) be the collection of closed subsets of T. Denote by U the collection of sets E e K(T) which are of uniqueness (i.e., every trigonometric series converging to 0 on T\E is identically 0), and let M = K(T)\U (the collection of closed sets of multiplicity). This is