Ergodic theory of amenable group actions. I: The Rohlin lemma

Classically, ergodic theory began with the study of flows or actions of R. Later, for technical reasons, much of the theory was first developed for actions of Z.More recently, there has been interest in extending the theory to actions of more general groups such as Z d , R d , abelian groups etc. (see e.g.[G], [K], [KW], [L]).The natural setting seems to be amenable groups and wë have been able to do much of the theory in that generality.We report here on the discrete case, so henceforth G will always be countable, and focus on, what has turned out to be a very basic tool in the modern developments, the Rohlin lemma.Let G denote a countable amenable group, (X, B, ju) a finite nonatomic Lebesgue measure space, and T: G x X -> X & measure-preserving action of G on X.We will usually suppress T and write simply gx for T(g, x).Recall that the action is free if for fx a.e.x, gx J= x for g =£ 1 G G. Rohlin's lemma is valid for a finite set F C G, and a free action of G on (X, 8, M) if ft> r all e > 0 there is a set BEB such that (i) {ƒ»: feF} are disjoint, (Ü) n(FB) = KÖ/GF f B ) > 1 -e.We say that F tiles G, or that F is a tiling set if there is a set of centers C C G such that {Fc: c G C} is a partition of G.PROPOSITION 1.If Rohlin's lemma is valid for a finite set F for some free action of G then F tiles G.THEOREM 2. If F tiles G, and G is amenable, then Rohlin's lemma is valid for F and any nonsingular free action of G.