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 answered negatively for n=1. However, it turns out to be true when n1, and was solved independently by Margulis and Sullivan for n3, and by Drinfeld for n=2,3. Their proofs are based on Rosenblatt’s reformulation of the problem: if there exists a finitely generated dense subgroup in SO(n+1), whose regular representation on $L^2(S^n)$ has a spectral gap, then the Lebesgue integral is the unique rotation invariant state on $L^\infty(S^n)$. In this talk I will discuss a generalization of this result in the setting of group actions on Von Neumann algebras. As an application we will show that a finitely generated ICC group is inner amenable iff there are more than one inner invariant states on its group Von Neumann algebra.No prior knowledge on operator algebras will be assumed. This is a joint work with Chi-Keung Ng.