Amenable groups and invariant means (Q1340844)

The problem of nonuniqueness of invariant finitely additive probability measures on \(S^ n\) is naturally linked to amenability properties. The general question was raised by J. Rosenblatt. Whereas by set theoretical arguments nonuniqueness was established for nilpotent and solvable groups, the author makes use of analytical methods to prove that for any amenable group of permutations on \(\mathbb{N}\) that is analytic (essentially any group defined without using transfinite induction is analytic) there exist at least two invariant finitely additive probability measures.