On a geometric property of the set of invariant means on a group

If G is a discrete group andx∈G then x^˜denotes the homeomorphism of βG onto βG induced by left multiplication by x.A subset K of βG is said to be invariant if it is closed, nonempty and x^˜K⊂K for each x∈G. Let ML(G) denote the set of leftinvariant means on G. (They can be considered as measures on βG.)Theorem. Let G be a countably infinite amenable group and let K be an invariant subset of βG. Then the nonempty w*-compact convex set M{G, K) = {ϕ∈ML(G): suppt ϕ⊂K) has no exposed points (with respect to w*-topology). Therefore, it is infinite dimensional.