Duality and perfect probability spaces

Abstract:

Given probability spaces $(X_i,\mathcal {A}_i,P_i), i=1,2,$ let $\mathcal {M}(P_1,P_2)$ 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 $(X_1 \times X_2,\mathcal {A}_1 \otimes \mathcal {A}_2).$ Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubinštein (1958) for the case of compact metric spaces are concerned with the validity of the duality \begin{align*} &\sup \{ \int h dP: P \in \mathcal {M}(P_1,P_2) \} \ &\qquad = \: \inf \{ \sum _{i=1}^{2} \int h_i dP_i : h_i \in \mathcal {L}^1 (P_i) \; \; and \; \; h \leq {\oplus }_i h_i\} \end{align*} (where $\mathcal {M}(P_1,P_2)$ is the collection of all probability measures on $(X_1 \times X_2, \mathcal {A}_1 \otimes \mathcal {A}_2)$ with $P_1$ and $P_2$ 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.