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Immediately following the commentary below, this previously published article is reprinted in its entirety: D. Ornstein and B. Weiss, “Ergodic theory of amenable group actions, I. The Rohlin Lemma”, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. Up to a small error, an ergodic measure preserving action of a group on a probability space can be approximated by a periodic action. For Z-actions this was proven by Kakutani [14] and Rohlin [23], and this statement is generally referred to as the Rohlin Lemma. More precisely, assume that (X,X , μ, T ) is a measure preserving system, meaning that (X,X , μ) is a probability space and T : X → X is a measurable, measure preserving, and invertible transformation. Thus the transformation T acting on the space (X,X , μ) gives rise to a Z-action. Further assume that this system is ergodic, meaning that the only measurable sets A ∈ X such that TA = A either have measure 0 or have measure 1. The Rohlin Lemma states that for an ergodic measure preserving system (X,X , μ, T ), any ε > 0, and N ∈ N, there exists a set F ∈ X such that T, TF, T F, . . . , TN−1F are pairwise disjoint and such that

Commentary on “Ergodic theory of amenable group actions”: Old and new