Amenable actions and almost invariant sets

In this paper, we study the connections between properties of the action of a countable group $\Gamma$ 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 $\Gamma \curvearrowright M^X$ has almost invariant sets.