A GENERAL DUALITY THEOREM FOR MARGINAL PROBLEMS

Given probability spaces (X(i), A(i), P-i), i = 1, 2 let M (P-1, P-2) denote the set of all probabilities on the product space with marginals P-1 and P-2 and let h be a measurable function on (X(1) x X(2), A(1) circle times A(2)). In order to determine sup fh dP where the supremum is taken over P in M (P-1, P-2) a general duality theorem is proved. Only the perfectness of one of the coordinate spaces is imposed without any further topological or tightness assumptions. An example is given to show that the assumption of perfectness is essential. Applications to probabilities with given marginals and given supports, stochastic order and probability metrics are included.