Amenable group

Remarks

Example: a finite group $G$ is amenable, with $\mu(A) = \abs{A} / \abs{G}$; left translation is a bijection, so $\abs{gA} = \abs{A}$.

Non-example: the free group $F_2$ on $a, b$ is not amenable. Let $W(x)$ be the set of reduced words that begin with the letter $x$. Then $F_2 = W(a) \sqcup a W(a^{-1})$ and $F_2 = W(b) \sqcup b W(b^{-1})$. An invariant $\mu$ would give $\mu(W(a)) + \mu(W(a^{-1})) = 1$ and $\mu(W(b)) + \mu(W(b^{-1})) = 1$, but these four disjoint sets lie in $F_2$, so their total is at most $1$, a contradiction.

Equivalently, there is a left-invariant mean on $\ell^\infty(G)$, a positive linear functional $m$ with $m(1) = 1$ and $m(g \cdot f) = m(f)$. For a locally compact group, one asks instead for a left-invariant mean on $L^\infty(G)$; for discrete groups the two notions agree. For a countable group, amenability is equivalent to the Følner condition. Subgroups, quotients and extensions of amenable groups are amenable, and abelian and solvable groups are amenable.