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  "body": "\n## Summary\n\nAn action of a group on a set is amenable when the bounded functions on the set carry a [[mean|ZYNY2VFECQ]] that the group leaves invariant.\nThis is one common meaning; in ergodic theory the phrase also often means simply an action of an amenable group.\nThis extends amenability from [[groups|CEAT9BESKK]] to group actions: a group is amenable exactly when its action on itself by left translation is amenable.\nEvery action of an amenable group is amenable, but a non-amenable group can also act amenably, for example with a finite orbit.\nMeans and Følner sets are the tools that let ergodic theorems pass from $\\Z$ to more general groups.\n\n## Explanations\n\n### L1 human\n\nA mean is a way to take an average of every bounded function on a set, even an infinite one.\nIt must be linear, give $1$ to the constant function $1$, and give a nonnegative answer for nonnegative functions.\nThe action is amenable if some mean does not change when the group moves the points of the set.\nIf the group has a finite orbit, averaging over that orbit gives such a mean.\nThe free group on two generators acting on itself has no such mean, which is the root of paradoxical decompositions.\n",
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