  The original definition of amenability given by von Neumann in the highly
non-constructive terms of means was later recast by Day using approximately
invariant probability measures. Moreover, as it was conjectured by Furstenberg
and proved by Kaimanovich-Vershik and Rosenblatt, the amenability of a locally
compact group is actually equivalent to the existence of a single probability
measure on the group with the property that the sequence of its convolution
powers is asymptotically invariant. In the present article we extend this
characterization of amenability to measured groupoids. It implies, in
particular, that the amenability of a measure class preserving group action is
equivalent to the existence of a random environment on the group parameterized
by the action space, and such that the tail of the random walk in almost every
environment is trivial.
