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.
In the field of analysis, the definition is in terms of linear functionals.
In this setting, a group is amenable if one can say what proportion of G any given subset takes up.
Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
A locally compact Hausdorff group is called amenable if it admits a left- (or right-)invariant mean.
By identifying Hom(L∞(G), R) with the space of finitely-additive Borel measures which are absolutely continuous with respect to the Haar measure on G (a ba space), the terminology becomes more natural: a mean in Hom(L∞(G), R) induces a left-invariant, finitely additive Borel measure on G which gives the whole group weight 1.
The graph of a typical function f ≥ 0 looks like a jagged curve above a circle, which can be made by tearing off the end of a paper tube.
Left-invariance would mean that rotating the tube does not change the height of the flat top at the end.
Combined with linearity, positivity, and norm-1, this is sufficient to prove that the invariant mean we have constructed is unique.
A discrete group G is amenable if there is a finitely additive measure (also called a mean)—a function that assigns to each subset of G a number from 0 to 1—such that This definition can be summarized thus: G is amenable if it has a finitely-additive left-invariant probability measure.
Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?
(Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely additive.)
Combining these two gives a bi-invariant measure: The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ.
If a countable discrete group contains a (non-abelian) free subgroup on two generators, then it is not amenable.
The converse to this statement is the so-called von Neumann conjecture, which was disproved by Olshanskii in 1980 using his Tarski monsters.
However, in 2002 Sapir and Olshanskii found finitely presented counterexamples: non-amenable finitely presented groups that have a periodic normal subgroup with quotient the integers.
[12] For finitely generated linear groups, however, the von Neumann conjecture is true by the Tits alternative:[13] every subgroup of GL(n,k) with k a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators.