17. Invariance of likelihoodratio. Let the family of distributions 9' = {Po, 8 E Q} be dominated by p., let Po = dPo/dp., let p.g-I be the measure defined by p.g-l(A) = p.[g-I(A)]. and suppose that p, is absolutely continuous with respect to p.g-Ifor all g E G

(i) Then dp. Po(x) = Pgo(gx)-d- I (gx) p,g (ii) Let Q and w be invariant under G, and countable. Then the likelihood ratio sUPoPo( x)/sup",po(x) is almost invariant under G. (iii) Suppose that Po(x) is continuous in 8 for all x, that Q is a separable pseudometric space, and that Q and w are invariant. Then the likelihood ratio is almost invariant under G.