Let $X$ and $Y$ be Polish spaces, and let $c:X \times Y \to (-\infty,\infty]$ be lower semicontinuous and bounded below. Let $(\mu_k)_{k \ge 1} \subset \mathcal{P}(X)$ and $(\nu_k)_{k \ge 1} \subset \mathcal{P}(Y)$ satisfy $\mu_k \to \mu$ narrowly in $\mathcal{P}(X)$ and $\nu_k \to \nu$ narrowly in $\mathcal{P}(Y)$. For each $k \ge 1$, let $\pi_k \in \Pi(\mu_k,\nu_k)$ be an optimal transport plan for the cost $c$, and assume that the sequence $(\pi_k)_{k \ge 1}$ is tight in $\mathcal{P}(X \times Y)$.

If $(k_j)_{j \ge 1}$ is a strictly increasing sequence of natural numbers and $\pi_{k_j} \to \pi$ narrowly in $\mathcal{P}(X \times Y)$, then $\pi \in \Pi(\mu,\nu)$ and

\begin{align*} \int_{X \times Y} c(x,y)\,d\pi(x,y) \leq \liminf_{j \to \infty} \int_{X \times Y} c(x,y)\,d\pi_{k_j}(x,y). \end{align*}

Assume in addition that for every $\gamma \in \Pi(\mu,\nu)$ with

\begin{align*} \int_{X \times Y} c(x,y)\,d\gamma(x,y) < \infty, \end{align*}

there exists a sequence $(\gamma_k)_{k \ge 1}$ with $\gamma_k \in \Pi(\mu_k,\nu_k)$, $\gamma_k \to \gamma$ narrowly in $\mathcal{P}(X \times Y)$, and

\begin{align*} \lim_{k \to \infty} \int_{X \times Y} c(x,y)\,d\gamma_k(x,y) = \int_{X \times Y} c(x,y)\,d\gamma(x,y). \end{align*}

Then every narrow subsequential limit $\pi$ of $(\pi_k)_{k \ge 1}$ is an optimal transport plan in $\Pi(\mu,\nu)$. Moreover, if

\begin{align*} V_k := \inf_{\sigma \in \Pi(\mu_k,\nu_k)} \int_{X \times Y} c(x,y)\,d\sigma(x,y) \end{align*}

and

\begin{align*} V := \inf_{\sigma \in \Pi(\mu,\nu)} \int_{X \times Y} c(x,y)\,d\sigma(x,y), \end{align*}

with both infima understood in $(-\infty,\infty]$, then $V_k \to V$ in the extended real sense.