DUALITY AND PERFECT PROBABILITY SPACES

Given probability spades (X(i), A(i), P-i), i = 1,2, let M(P-1, P-2) d enote 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) x A(2)). Continuous versions of linear programming stemming from the w orks of Monge (1781) and Kantorovich-Rubinstein (1958) for the case of compact metric spaces are concerned with the validity of the duality sup{integral h dP : P is an element of M(P-1, P-2)} [GRAPHICS] (where M(P-1, P-2) is the collection of all probability measures on (X(1) x X (2), A(1) x A(2)) with P-1 and P-2 as the marginals). A recently estab lished general duality theorem asserts the validity of the above duali ty whenever at least one of the marginals is a perfect probability spa ce. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterizati on of perfect probability spaces.