Duality Theorems for Assignments with upper Bounds

Abstract

Let (X _i, A _i, P _i), i = 1, 2 be two probability spaces. A probability µ on (X ₁ × X ₂, A ₁⊗A ₂) is said to have marginals P ₁ and P ₂ if

$$P(A_1 \times X_2) = P_1 (A_1) \textup{for all} A_1 \in \mathcal {A}_1 \\ \textup{and} \\ P(X_1 \times A_2) = P_2 (A_2) \textup{for all} A_2 \in \mathcal {A}_2 \\$$

Let $\mathcal{M} = \left\{\mu \ \textup{on} \ \mathcal{A}_1 \otimes \mathcal{A}_2: \mu\ \textup<Emphasis Type="ItalicSmallcaps"> a probability with marginals</Emphasis> \ P_1 \ \textup{and} \ P_2 \right\}$.