Let $(X,d)$ be a Polish [metric space](/page/Metric%20Space), let $1 \le p < \infty$, and define $\mathcal{P}_p(X)$ to be the set of Borel probability measures $\mu$ on $X$ such that, for some equivalently every $x_0 \in X$,
where $\Pi(\mu,\nu)$ is the set of Borel probability measures on $X \times X$ with first marginal $\mu$ and second marginal $\nu$. Then $W_p$ is a metric on $\mathcal{P}_p(X)$.