The Rusiewicz problem asks if the Lebesgue measure is the only finitely additive, rotation invariant measure defined on all the Lebesgue measurable subsets of the n-sphere. Banach showed that for n=1 it is false. However, it turns out to be true when n>1, and was solved independently by Margulis and Sullivan for n>3, and by Drinfeld for n=2, 3. All the proofs are based on Rosenblatt's reformulation of the problem: for an ergodic discrete group action on a standard probability space, the existence of a spectral gap of the underlying unitary representation implies the uniqueness of the invariant means.

In this talk I will discuss a generalization of this result in the setting of group actions on Von Neumann algebras. This is a joint work with Chi-Keung Ng.