Amenable Groups and Invariant Means

Banach showed in 1923 that Lebesgue measure is not the unique rotation invariant finitely additive probability measure on the measurable subsets of S1. Margulis and Sullivan (for n ≥ 4) and Drinfield (for n = 2, 3) independently showed that Lebesgue measure is the unique isometry invariant finitely additive probability measure on Sn. These results all used special properties of the group action. Rosenblatt asked whether an amenable group can uniquely determine an invariant mean. Using techniques from set theory we obtain information on this question and give a complete solution in the case of locally finite groups acting on the integers.