12 GREG HJORTH AND ALEXANDER S. KECHRIS that the measure /i is E-invariant (resp., quasi-invariant) if /i is T-invariant (resp., quasi-invariant) for some (equivalently any) countable group Y and Borel action of r on X with E* = E. In the case of quasi-invariance, it is easy to see that this is equivalent to saying that the ^-saturation of any null Borel set is null. Finally, if /i is a measure on X and we have a Borel action of a countable group r on I , we say that the measure /i is T-ergodic or that the T-action is ergodic (relative to /x) if every T-invariant Borel set A is either null or conull. Similarly we say that a countable Borel equivalence relation E on X is ergodic (with respect to /i) or that fi is E- ergodic, if every ^-invariant Borel set is either null or conull. OF. Amenabilit y We follow here [JKL, Section 2]. Given a countable set C, a finitely additive probability measure (f.a.p.) on C is a map (p : {A : A C C} [0,1] such that p(C) = 1, ip(A U B) = p(A) + £(#), if A n £ = 0. A mean on C is a positive linear functional (p on £QO(C), the Banach space of bounded real functions on C, with (^(1) = 1. Means and f.a.p.'s are the same thing via the identification: cp -• £, where p(f) = f fd(p and (f(A) = ^(1A)5 with 1A the characteristic function of A. We will not distinguish between tp, (p from now on. A countable group Y is amenable if there is a left-invariant mean ^ o n T (i.e., a mean p on Y such that p(f) = ^ ( 7 - / ) , with 7./(£ ) = /(S_17), V7 G T , / G 4 c ( r ) ) - We also say that a mean y? on N is shift-invariant if (/?(/) = ^ ( / s ) , when / a (n ) = / ( n + l ) f o r / e 4 o ( N ) . By a result of Christensen [C], Mokobodzki, for each measure \i on [—1,1]N, N admits a /i-measurable, shift-invariant mean (p (i.e., a shift-invariant mean p such that (p\[— 1,1]N is /i-measurable), and similarly for any amenable group Y and measure /i on [—1, l ] r there is left-invariant mean p on Y such that (^?|[—1, l ] r is /x- measurable. Moreover, assuming the Continuum Hypothesis (CH), /?, in both cases, can be taken to be universally measurable, i.e., /i-measurable for any \i as before. Finally, in all the above, "left-invariant" can be replaced by "right-invariant" or even "(two-sided)-invariant". A basic result, due to F0lner, is that every countable amenable group Y admits a sequence (Fn) of nonempty finite subsets with the property l7 ,^ . * -^ 0, V7 G T, where \A\ = card (A) is the cardinality of A. Such a sequence is called a F0lner sequence. If now E is a countable Borel equivalence relation on a standard Borel space X and ji is a measure on A , we say that E is n~amenable if there is a map C *— » ipc assigning to each ^-class C a mean ipc on C, such that if / : E [—1,1] is Borel, then x 1— » £[£]£(/a) is /i-measurable, where fx(y) = f(x,y). We call E measure amenable if x \-^ ip[x]E(fx) is universally measurable. If T is amenable and acts in a Borel way on a standard Borel space A , then, for any measure \i on is /i-amenable. Since the hyperfinite equivalence rela- tions are exactly those of the form E% , every hyperfinite equivalence relation is /1- amenable for each /i. Finally, by the result of [CFW], \i-amenability is equivalent to
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