Wasserstein distance on $\mathbb{R}$

Question

I have the following problem.

Given the Wasserstein distance, defined by $W_1(\mu,\nu)=\inf\limits_{\pi\in\Pi(\mu,\nu)}\int\limits_{\mathbb{R}^2}|x-y|d\pi(x,y)$,

whereby $\mu$ and $\nu$ are probability measures on the Borel $\sigma$-algebra on $\mathbb{R}$ and $\Pi(\mu,\nu)$ is the set of all couplings of $(\mu,\nu)$.

I also know that

$W_1(\mu,\nu) = \int\limits_{-\infty}^{\infty}|F(x)-G(x)|dx$

for the distribution functions $F$ and $G$ of $\mu$ and $\nu$, (i.e $F(x)=\mu((-\infty,x])$).

I now want to prove that

$W_1(\mu,\nu)=\int\limits_{0}^{1}|F^{-1}(z)-G^{-1}(z)|dz.$

Here we of course define $F^{-1}(z):=\inf\lbrace{}x\in\mathbb{R};F(x)\geq{}z\rbrace$

I don't quite know how to aproach this problem and would be quite grateful for any advice. Thanks

Answer 1

Both integrals are equal to the area between the graphs of $F$ and $G$, i.e. of the set $$\{(x,z)\in \mathbb R^2: F(x)\wedge G(x)\leq z\leq F(x)\lor G(x)\}$$ where $\wedge$ and $\lor$ indicate the minimum and the maximum between $F$ and $G$. The first time integrate in $dx$ the height of the vertical sections, the second integrate in $dz$ the width of the horizontal sections.

You need to use the monotonocity of both $F$ and $G$ and maybe the fact that a monotone function on $\mathbb R$ can have at most countably many jumps and values taken more than once.