19. Let G be a group of transformations of !!E, and let SiI be a a-field of subsets of !!E, and !J. a measure over (!!E, SiI). Then a set A E SiI is said to be almost invariant if its indicator function is almost invariant. (i) The totality of almost invariant sets forms a a-field Silo, and a critical function is almost invariant if and only if it is Silo-measurable. (ii) Let 9 = {Po, 8 EO} be a dominated family of probability distributions over (!!E, SiI), and suppose that g8 = 8 for all g E G, 8 E O. Then the a-field Silo of almost invariant sets is sufficient for 9 . [Let "A = L.C;Po, be equivalent to 9 . Then dPo dPg - .o dPo --;n:(gx) = cdP - 10 (x) = --;n:(x) "-,, g j (a.e. "A), so that dPold"A is almost invariant and hence Silo-measurable.) Section 7