Amenable discrete group

Definition
A discrete group $$G$$ is termed an amenable discrete group if there exists a function that assigns to each subset of $$G$$ a real number in $$[0,1]$$ such that:


 * The number associated with the whole of $$G$$ is $$1$$
 * The function is finitely additive on subsets. In other words, the number associated with a disjoint union of finitely many subsets is the sum of the numbers associated with each of the subsets
 * The function is left-invariant, viz., the value associated with a subset $$S$$ is the same as the value associated with $$gS$$

Often, when dealing with an abstract group, we use the term amenable group for amenable discrete group (in other words, we assume that the discrete topology is put over the group).

Stronger properties

 * Finite group, equipped with the counting measure. This is the measure associating to every subset the ratio of its cardinality to the cardinality of the group
 * Solvable group:

Metaproperties
A finite direct product of amenable discrete groups is again an amenable discrete group.

A subgroup of an amenable group is amenable.

A quotient of an amenable group is amenable.