Exchangeable Random Orders and Almost Uniform Distributions

Random orders on N invariant under permutations are called exchangeable. The compact and convex set of all random total orders is shown to be a Bauer simplex whose set of extreme points, the socalled totally ordered paintbox processes, is homeomorphically parametrized by "almost uniform" distributions on the unit interval, i.e. by probability measures w on [0, 1] whose distribution functions are w-almost surely the identity.