Definition

Let $(\mu_{n})_{n\in\mathbb{N}}\subset \mathcal{P}(\mathbb{X}),\mu \in\mathcal{P}(\mathbb{X})$. We say that $\mu_{n}\to \mu$ in $W_{1}(\mu_{n},\mu)$ (the Wasserstein metric) if $$W_{1}(\mu_{n},\mu)\to0\text{ as }n\to \infty$$