Let C be any class of procedures that is closed under the transformations of a group G in the sense that δ ∈ C implies g∗δg−1 ∈ C for all g ∈ G. If there exists a unique procedure δ0 that uniformly minimizes the risk within the class C, then δ0 is invariant.7 If δ0 is unique only up to sets of measure zero, then it is almost invariant, that is, for each g it satisfies the equation δ(gx) = g∗δ(x) except on a set Ng of measure 0.