Let $(X,d)$ be a Polish [metric space](/page/Metric%20Space), let $p \in [1,\infty)$, and let $x_0 \in X$. Let $(\mu_N)_{N \in \mathbb{N}}$ and $(\nu_N)_{N \in \mathbb{N}}$ be sequences in $\mathcal{P}_p(X)$, and let $\mu,\nu \in \mathcal{P}_p(X)$. Suppose that $\mu_N \to \mu$ narrowly and $\nu_N \to \nu$ narrowly in $\mathcal{P}(X)$, and that

\begin{align*} \int_X d(x,x_0)^p \, d\mu_N(x) \to \int_X d(x,x_0)^p \, d\mu(x) \end{align*}

and

\begin{align*} \int_X d(x,x_0)^p \, d\nu_N(x) \to \int_X d(x,x_0)^p \, d\nu(x). \end{align*}

Then

\begin{align*} W_p(\mu_N,\nu_N) \to W_p(\mu,\nu). \end{align*}

Moreover, suppose that for each $N \in \mathbb{N}$, $\pi_N \in \Pi(\mu_N,\nu_N)$ is an optimal transport plan for the cost function $c:X \times X \to [0,\infty)$ defined by $c(x,y)=d(x,y)^p$. If $(N_j)_{j \in \mathbb{N}}$ is a strictly increasing sequence of natural numbers and $\pi_{N_j} \to \pi$ narrowly in $\mathcal{P}(X \times X)$, then $\pi \in \Pi(\mu,\nu)$ and $\pi$ is an optimal transport plan from $\mu$ to $\nu$ for the cost $d^p$.