On the existence of probability measures with given marginals

Abstract

Let X be a compact ordered space and let μ , ν be two probabilities on X such that μ ( f ) ≤ ν ( f ) for every increasing continuous function f : X → R . Then we show that there exists a probability θ on X × X such that (i) θ ( R ) = 1 , where R is the graph of the order in X , (ii) the projections of θ onto X are μ and ν . This theorem is generalized to the completely regular ordered spaces of Nachbin and also to certain infinite products. We show how these theorems are related to certain results of Nachbin, Strassen and Hommel.