The Wasserstein distance on $\mathbb{R}$

Question

I was reading "On choosing and bounding probability metrics" and in it there was a remark, that I couldn't quite understand. Take the Wasserstein distance, defined by

$W(\mu,\nu):=\inf\limits_{\pi\in\Pi(\mu,\nu)}\int\limits_{\mathcal{X\times{}X}} d(x,y)d\pi(x,y)$

for two probability measures $\mu$ and $\nu$ on a Polish space. In this case $\Pi(\mu,\nu)$ denotes the space of all couplings of $\mu$ and $\nu$. Now if we let that Polish space be $\mathbb{R}$ with the regular metric $d$, then we obtain

$W(\mu,\nu)= \int\limits_{-\infty}^{\infty}|F(x)-G(x)|dx$,

whereby $F$ and $G$ are the respective distribution functions of $\mu$ and $\nu$.

I was trying to prove this, but have no clue where to start. Does anyone have an idea?

Answer 1

See "Topics in Optimal Transport" by Cedric Villani section 2.2 page 75-77