Let $(\mathcal{X},d)$ be a Polish metric space, and let $1\le p<\infty$. For any two Probability Measures $\mu,\nu$ on $\mathcal{X}$ the Wasserstein Metric of order $p$ between $\mu$ and $\nu$ is defined by the formula $$\begin{align*} W_{p}(\mu,\nu)&=\left( \inf_{\pi \in\Pi(\mu,\nu)}\int\limits _{\mathcal{X}\times \mathcal{X}}d(x,y)^{p} \, d\pi(x,y) \right)^{1/p}\\ &=\inf\left \{ [E_{}\left[ d(X,Y)^{p} \right]^{1/p},\,\text{law}(X)=\mu,\,\text{law}(Y)=\nu \right\} \end{align*}$$

The Wasserstein metric of order 1 for any two measures $\mu,\nu \in\mathcal{P}(\mathbb{X})$ is defined as $$W_{1}(\mu,\nu)=\inf_{\eta \in\mathcal{H}(\mu,\nu)}\int\limits _{\mathbb{X}\times \mathbb{X}}\eta(dx,dy)|x-y|$$where $\mathcal{H}(\mu,\nu)$ denotes the set of probability measures on $\mathbb{X}\times \mathbb{X}$ with the first marginal $\mu$ and second marginal $\nu$. Furthermore we equivalently have $$W_{1}(\mu,\nu)=\sup_{\lVert f \rVert _{Lip}\le 1}\left|\int\limits f \, d\mu -\int\limits f \, d\nu \right|$$where $$\lVert f \rVert _{Lip}:=\sup_{x\not=y} \frac{f(x)-f(y)}{d_{\mathbb{X}}(x,y)}$$and $d_{\mathbb{X}}$ is the metric on $\mathbb{X}$.