Department Of Mathematics and Statistics, California State University, 6000 J Street, Sacramento, California 95819-6051 ; California State University, Sacramento and Universität Freiburg
Abstract:
Given probability spaces let denote the set of all probabilities on the product space with marginals and and let be a measurable function on Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinv{s}tein (1958) for the case of compact metric spaces are concerned with the validity of the duality
(where is the collection of all probability measures on with and as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.