Let $X$ and $Y$ be Polish spaces, let $\mu \in \mathcal{P}(X)$, and let $\nu \in \mathcal{P}(Y)$. Let $\Pi(\mu,\nu)$ denote the set of Borel probability measures on $X \times Y$ whose first marginal is $\mu$ and whose second marginal is $\nu$. If $(\gamma_n)_{n=1}^{\infty}$ is a sequence in $\Pi(\mu,\nu)$ and $\gamma_n \rightharpoonup \gamma$ weakly in $\mathcal{P}(X \times Y)$, then $\gamma \in \Pi(\mu,\nu)$.