Amenable actions and almost invariant sets

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