Amenable actions and almost invariant sets

In this paper, we study the connections between properties of the action of a countable group G on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Gamma on M-X, where M is a measure space. In particular, we show that the action of Gamma on X is amenable iff the shift Gamma curved right arrow MX has almost invariant sets.