Consider a set x together with a σ-algebra B of subsets of x. Let G be a family of B-measurable transformations on x, let p(X) be the convex set of all prbability measures on B and let I be the convex set of all G-invariant probablity measures in p(X). For μµ p(X) we define Bµ = {A ∈ B: µ(g A Δ A) = 0 for all g ∈ G} and we define B0 = {A ∈B: gA = A for all g ∈ G}. Then B0 ⊆ Bµ and both are σ-subalgebras of B. G is said to act transitively on X if for x ∈ X, y ∈ X, gx = y for some g ∈ G.