Ergodic group actions with nonunique invariant means

Let M(X, G) be the set of G-invariant means on L∞(X, B, P), where G is a countable group acting ergodically as measure preserving transformations on a nonatomic probability space (X, B, P). We show that if there exists μ ∈ M(X, G), μ ≠P, then M(X, G) contains an isometric copy of βN\N, where βN\N is considered as a subset of (l∞)*. This provides an answer to a question raised by J. Rosenblatt in 1981.