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 sub sets of S-1. Margulis and Sullivan (for n greater than or equal to 4) and Drinfield (for n = 2, 3) independently showed that Lebesgue measur e is the unique isometry invariant finitely additive probability measu re on S-n. These results al used special properties of the group actio n. Rosenblatt asked whether an amenable group can uniquely determine a n invariant mean. Using techniques from set theory we obtain informati on on this question and give a complete solution in the case of locall y finite groups acting on the integers. (C) 1994 Academic Press, Inc.