1. Let G be a group of measurable transformations of (~, JJI) leaving 9 = {Po, (J En} invariant, and let T( x) be a measurable transformation to (ff, !fI). Suppose that T(xl) = T(X2) implies T(gxl) = T(gx2) for all g E G, so that G induces a group G* on ff through g*T(x) = T(gx), and suppose further that the induced transformations g* are measurable !fl. Then G* leaves the family 9 T = {pI, (J En} of distributions of T invariant.