Theorem

Let $(\mathcal{X},d)$ be a Polish space, and $1\le p<\infty$; then the Wasserstein metric $W_{p}$ metrizes the weak convergence in $P_{p}(\mathcal{X})$. i.e. if $(\mu_{k})_{k\in\mathbb{N}}\subset P_{p}(\mathcal{X})$ and $\mu \in P_{p}(\mathcal{X})$, then the statements $$\mu_{k} \,converges\,weakly\,in\,P_{p}(\mathcal{X})\,to\,\mu$$and $$W_{p}(\mu_{k},\mu)\to0$$are equivalent.