Definition

Let $(\mathcal{X},d)$ be a Polish space, and $1\le p<\infty$. Let $(\mu_{k})_{k\in\mathbb{N}}\subset P_{p}(\mathcal{X})$ and $\mu \in P_{p}(\mathcal{X})$ where $P_{p}(\mathcal{X})$ is the Wasserstein space. Then $\mu_{k}$ is said to converge weakly in $P_{p}(\mathcal{X})$ if any one of the following properties is satisfied for any $x_{0}\in\mathcal{X}$: 1. $\mu_{k}\to \mu$ and $$\int\limits d(x_{0},x)^{p} \, d\mu_{k}(x)\to \int\limits d(x_{0},x)^{p} \, d\mu(x)$$ 2. $\mu_{k}\to \mu$ and $$\limsup_{ k \to \infty } \int\limits d(x_{0},x)^{p} \, d\mu_{k}(x)\le \int\limits d(x_{0},x)^{p} \, d\mu(x)$$ 3. $\mu_{k}\to \mu$ and $$\lim_{ R \to \infty } \limsup_{ k \to \infty } \int\limits_{d(x_{0},x)\ge R} d(x_{0},x)^{p} \, d\mu_{k}(x)=0$$ 4. For all continuous functions $\varphi$ with $|\varphi(x)|\le C(1+d(x_{0},x)^{p})$, $C\in\mathbb{R}$, one has $$\int\limits \varphi(x) \, d\mu_{k}(x)\to \int\limits \varphi(x) \, d\mu(x)$$