Duality and perfect probability spaces

Abstract

Given probability spaces ( X i , A i , P i ) , i = 1 , 2 , (X_i,\mathcal {A}_i,P_i), i=1,2, let M ( P 1 , P 2 ) \mathcal {M}(P_1,P_2) denote the set of all probabilities on the product space with marginals P 1 P_1 and P 2 P_2 and let h h be a measurable function on ( X 1 × X 2 , A 1 ⊗ A 2 ) . (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 a m p ; sup { ∫ h d P : P ∈ M ( P 1 , P 2 ) } a m p ; = inf { ∑ i = 1 2 ∫ h i d P i : h i ∈ L 1 ( P i ) a n d h ≤ ⊕ i h i } \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 M ( P 1 , P 2 ) \mathcal {M}(P_1,P_2) is the collection of all probability measures on ( X 1 × X 2 , A 1 ⊗ A 2 ) (X_1 \times X_2, \mathcal {A}_1 \otimes \mathcal {A}_2) with P 1 P_1 and P 2 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.