PRELIMINARIES 11 A Borel equivalence relation E on X is called tame (or smooth) if there is a Borel map p : X Y, where Y is a standard Borel space, such that xiEx2 ^ p(xi) = p(x2), i.e., E B A ( y ) . If E is also countable, this means exactly that E has a Borel transversal. We call E hypertame (or hypersmooth) if E [jnEn, where E\ C E2 C .. . are tame Borel equivalence relations. A countable Borel equivalence relation is called hyperfinite if it can be written as E [jnEni where E\ C E2 C .. . are finite Borel equivalence relations. (It can be shown, see [DJK, 5.1], that hyperfinite is equivalent to being countable and hypersmooth.) Finally, we call a countable Borel equivalence relation E o n a standard Borel space X compressible if there is a 1-1 Borel map p : X X such that xEp{x)1\/x1 and for each £-class C, p{C) ^ C. Our general references for the descriptive set theory of countable Borel equiv- alence relations will be [DJK] and [JKL]. OD. Measure s In this paper, measure in a standard Borel space, always means a probability Borel measure on that space. If /i is a measure on X , a Borel set A C X is called (p)-null if p(A) = 0 and (p)-conull if p(A) 1. The measure class of a measure p is the equivalence class of p under measure equivalence: p ~ v ^ p,v has the same null sets ^ V53e(/x(A) 6 i/(A) (5) & VJ3e(i/(A) e = /i(,4) J), where A varies over Borel sets. If p is a measure on X and i\ : X F is a Borel function, the image measure n*p is the measure on F defined by ^ / i ( 5 ) = / i ( ^ - 1 ( 5 ) ) , for every Borel set B CY. If p is a measure on X , we say that E is p-hyperfinite if for some ^-invariant conull Borel set A, £^|A is hyperfinite. OE. Borel actions and measure s Let T be a countable group and suppose T acts in a Borel way on a standard Borel space X (i.e., for each 7 G r , x ^ ^ - x i s Borel). Then T acts (also in a Borel way) on the standard Borel space A4(X) of measures on X (see [Ke95]) by (Y H)(A) = fib-1 A), for any Borel set A C X. We say that p if T-invariant if 7 p = /i, V7 G I\ We say that /i is T-quasi-invariant if 7 /i ~ /i, V7 G T (in that case one also says that the action is non-singular). We generalize this to countable Borel equivalence relations. By a theorem of Feldman-Moore [FM], if E is a countable Borel equivalence relation on X , there is a countable group T and a Borel action of T on X such that E* = E. We then say
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