A general duality theorem for marginal problems

Summary Given probability spaces (X_i,A_i,P_i),i=1, 2 letM(P₁,P₂) denote the set of all probabilities on the product space with marginalsP₁ andP₂ and leth be a measurable function on (X₁×X₂,A₁ A₂). In order to determine supfh dP where the supremum is taken overP inM(P₁,P₂), 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 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.