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1. Introduction. In this paper we announce the introduction of a new notion of amenability for ergodic group actions and ergodic equivalence relations. Amenable ergodic actions occupy a position in ergodic theory parallel to that of amenable groups in group theory and one can therefore expect this notion to be useful in diverse circumstances. Here we announce applications to hyperfinite factor von Neumann algebras, skew products, Poincaré flows, and Poisson boundaries of random walks. We remark that from Mackey's virtual group viewpoint, we are considering amenable virtual subgroups of locally compact groups. The author wishes to thank J. Feldman and C. C. Moore for discussion on the results of this paper. In particular, in discussion of a preliminary version of these results, C. C. Moore suggested the possibility of proving Theorem 4.1. Full details and related results will appear elsewhere.

Amenable ergodic actions, hyperfinite factors, and Poincaré flows