Parametrizing doubly stochastic measures

It has been of traditional interest in probability theory to produce a random vector (or metric space element), which has a given distribution and is defined on a standard space, such as [0,1] endowed with Lebesgue measure. In a classic work, Levy (1937, section 23) used an approach based on conditioning. For the purpose of the Skorokhod representation, Billingsley (1971, Theorem 3.2) considered the case of random elements of a general metric space. Whitt (1976, Lemma 2.7) considered general measures on Rn and employed a Borei isomorphism to treat questions of extremal correlation and minimal variance. R?schendorf (1983) used a similar approach to consider a general class of optimization problems.