Amenable group action

Summary

An action of a group on a set is amenable when the bounded functions on the set carry a mean that the group leaves invariant. This is one common meaning; in ergodic theory the phrase also often means simply an action of an amenable group. This extends amenability from groups to group actions: a group is amenable exactly when its action on itself by left translation is amenable. Every action of an amenable group is amenable, but a non-amenable group can also act amenably, for example with a finite orbit. Means and Følner sets are the tools that let ergodic theorems pass from $\Z$ to more general groups.