Let G be a group of measurable transformations of (X , A) leaving P = {Pθ, θ ∈ } invariant, and let T (x) be a measurable transformation to

(T , B). Suppose that T (x1) = T (x2) implies T (gx1) = T (gx2) for all g ∈ G, so that G induces a group G∗ on T through g∗T (x) = T (gx), and suppose further that the induced transformations g∗ are measurable B. Then G∗ leaves the family PT = {PT

θ , θ ∈ } of distributions of T invariant.

Section 6.2