Let G be a group of transformations of X , and let A be a -field of

Let G be a group of transformations of X , and let A be a σ-field of subsets of X , and µ a measure over (X , A). Then a set A ∈ A is said to be almost invariant if its indicator function is almost invariant.

(i) The totality of almost invariant sets forms a σ-field A0, and a critical function is almost invariant if and only if it is A0-measurable.

(ii) Let P = {Pθ, θ ∈ Ω} be a dominated family of probability distributions over (X , A), and suppose that ¯gθ = θ for all ¯g ∈ G, θ ¯ ∈ Ω. Then the σ-field A0 of almost invariant sets is sufficient for P.

[Let λ = ciPθi , be equivalent to P. Then dPθ

dλ (gx) = 

dPg−1θ

ci dPg−1θi

(x) = dPθ

dλ (x) (a.e. λ), so that dPθ/dλ is almost invariant and hence A0-measurable.]