An amenable group is a locally compact topological group that can be equipped with an additional structure of a left (or right) invariant mean. A mean on a locally compact group is a linear functional on (the Banach space of essentially bounded functions from to ) that maps nonnegative functions to nonnegative functions and sends the constant function (valuing everything to 1) to 1.
By left-invariant we mean that the mean is invariant under the action of the group on the space .
We can also define amenability purely in the context of discrete groups, in which case the definition becomes far simpler. Check out amenable discrete group.