Let $(X,d)$ be a Polish [metric space](/page/Metric%20Space) equipped with its Borel $\sigma$-algebra $\mathcal{B}(X)$. Let $\mathbb{N}=\{1,2,3,\dots\}$ and let $\mathbb{R}$ denote the [real numbers](/page/Real%20Numbers). Let $1 \le p < \infty$, let $x_0 \in X$, and let $(\mu_n)_{n \in \mathbb{N}}$ and $\mu$ be elements of $\mathcal{P}_p(X)$, where $\mathcal{P}_p(X)$ denotes the set of Borel probability measures $\nu$ on $X$ satisfying

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

For $\nu,\eta \in \mathcal{P}_p(X)$, let $W_p$ denote the $p$-Wasserstein distance

\begin{align*} W_p(\nu,\eta) := \inf_{\pi \in \Pi(\nu,\eta)} \left(\int_{X \times X} d(x,y)^p\, d\pi(x,y)\right)^{1/p}, \end{align*}

where $\Pi(\nu,\eta)$ is the set of Borel probability measures on $X \times X$ with first marginal $\nu$ and second marginal $\eta$. Then

\begin{align*} W_p(\mu_n,\mu) \to 0 \end{align*}

as $n \to \infty$ if and only if $\mu_n$ converges weakly to $\mu$, meaning

\begin{align*} \int_X f(x)\, d\mu_n(x) \to \int_X f(x)\, d\mu(x) \end{align*}

for every bounded continuous map $f: X \to \mathbb{R}$, and

\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*}

as $n \to \infty$.