Orderings and dilations of measures

Abstract

If X is a compact convex subset of a locally convex space E, and if μ, λ are nonnegative measures on X, we have defined μ ≻ λ to mean that μ(f) ≧ λ(f) for each continuous convex function f on X. For finite dimensional spaces E, this ordering has long been of interest in statistics; it is used to defined “comparison of experiments.” A characterization in terms of dilations (defined below) has been given by Hardy, Littlewood, and Polya for one dimensional spaces, and by Blackwell [10], C. Stein, and S. Sherman for finite dimensional spaces. The general case has been proved by P. Cartier [15], based in part on the work of Fell and Meyer; this is the proof we present below. An entirely different approach has been carried out by Strassen [75].