Let $1 \leq p < \infty$. Let $\mu,\nu \in \mathcal{P}_p(\mathbb{R})$, where $\mathcal{P}_p(\mathbb{R})$ denotes the Borel probability measures on $\mathbb{R}$ with finite $p$-th moment. Let $F_\mu,F_\nu:\mathbb{R}\to[0,1]$ be their distribution functions. Define the generalized inverse functions $F_\mu^{-1},F_\nu^{-1}:(0,1)\to\mathbb{R}$ by

\begin{align*} F_\mu^{-1}(s)=\inf\{x\in\mathbb{R}:F_\mu(x)\geq s\}. \end{align*}

\begin{align*} F_\nu^{-1}(s)=\inf\{x\in\mathbb{R}:F_\nu(x)\geq s\}. \end{align*}

Let $\Pi(\mu,\nu)$ denote the set of Borel probability measures on $\mathbb{R}^2$ with first marginal $\mu$ and second marginal $\nu$, and define

\begin{align*} W_p(\mu,\nu)^p=\inf_{\pi\in\Pi(\mu,\nu)}\int_{\mathbb{R}^2}|x-y|^p\,d\pi(x,y). \end{align*}

Then

\begin{align*} W_p(\mu,\nu)^p=\int_0^1 |F_\mu^{-1}(s)-F_\nu^{-1}(s)|^p\,d\mathcal{L}^1(s). \end{align*}