Let $X$ and $Y$ be Polish spaces, and equip $X \times Y$ with the [product topology](/page/Product%20Topology) and its Borel $\sigma$-algebra. Let
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\begin{align*}
c: X \times Y \to (-\infty,\infty]
\end{align*}
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be lower semicontinuous, and suppose that there exists a constant $m \in \mathbb{R}$ such that $c(x,y) \geq m$ for every $(x,y) \in X \times Y$. If $(\pi_k)_{k \in \mathbb{N}} \subset \mathcal{P}(X \times Y)$ converges narrowly to $\pi \in \mathcal{P}(X \times Y)$, then