Rohlin's Lemma And Its Applications

Rohlin's Lemma which was originally(cf[18]) proved for bi-measurable aperiodic automorphisms T of a Polish space X is a basic tool in ergodic theory.It states that for an automorphism of the above type which is invariant with respect to a Borel probability measureμ,any n∈N⁺ and any∈>0,one can find a measurable set R (a so called(n,∈)- Rohlin set) such that,for j = 0,1,...,n - 1,the sets T-^jR are pairwise disjoint and exhaust X with exception of a remainder set whose mass is smaller than∈.In particular,Rohlin's Lemma is indispensable for the canonical construction of generators.Since the classical proof(cf[11]) quoted in the standard textbooks on ergodic theory (e.g.,[7],[9],[17]) uses a Kakutani tower type construction and thus needs forward measurability,we feel obliged to provide an elementary proof which doesn't rely on this assumption.Moreover,the setting is generalised from Polish space to separable space. Here we improve Heinemann and Schmitt's proof(cf[12]),whose most sophisticated tool is Poincar(?)'s Recurrence Theorem,but we don't need it.Classically,ergodic theory began with the study of flows(i.e.,the action of the group is R).But,for technical reasons,much of the theory was first developed for actions of Z.Later,there has been extending the theory to actions of more general groups such as Z^d,R^d,Abelian groups etc.The natural setting seems to be amenable groups,the Rohlin's Lemma under this assumption is:let G be a countable amenable group,freely acts on a Lebesgue space(X,∑,μ).Suppose T(?) G be a finite set that tiles G.Given∈>0,there exists a measurable set B∈∑such that (â…°) {tB:t∈T} are disjoint;and (â…±)μ(TB)≡μ(∪_∈T TtB)＞1 -∈.Later,Alpern developed and proved Multiple Rohlin Tower Theorem.Eigen and Prasad proved it in a more simple way by using Kakutani's proof(cf[8]).