Let $X$ and $Y$ be Polish spaces, equipped with their Borel $\sigma$-algebras. Let $\mu \in \mathcal{P}(X)$ and $\nu \in \mathcal{P}(Y)$ be Borel probability measures, and define
where $\pi_X: X \times Y \to X$ and $\pi_Y: X \times Y \to Y$ are the coordinate projections. Then $\Pi(\mu,\nu)$ is tight as a family of probability measures on $X \times Y$; that is, for every $\varepsilon > 0$, there exists a compact set $K \subset X \times Y$ such that