In mathematics, an amenable group is a locally compact topological group G carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive measure (or mean) on subsets of G, was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "mean".[a]
The critical step in the Banach–Tarski paradox construction is to find inside the rotation group SO(3) a free subgroup on two generators. Amenable groups cannot contain such groups, and do not allow this kind of paradoxical construction.
Amenability has many equivalent definitions. In the field of analysis, the definition is in terms of linear functionals. An intuitive way to understand this version is that the support of the regular representation is the whole space of irreducible representations.
In discrete group theory, where G has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of G any given subset takes up. For example, any subgroup of the group of integers is generated by some integer . If then the subgroup takes up 0 proportion. Otherwise, it takes up of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
If a group has a Følner sequence then it is automatically amenable.
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