Let $X$ and $Y$ be Polish spaces, let $\mu \in \mathcal{P}(X)$ and $\nu \in \mathcal{P}(Y)$, and let $c: X \times Y \to (-\infty,\infty]$ be lower semicontinuous. Assume that $c$ is bounded from below, meaning that there exists $m \in \mathbb{R}$ such that $c(x,y) \geq m$ for all $(x,y) \in X \times Y$.
paragraph
admin
Let $\pi_X: X \times Y \to X$ and $\pi_Y: X \times Y \to Y$ denote the coordinate projections, and define the set of Kantorovich plans from $\mu$ to $\nu$ by