Ultimately I want to prove that for two random variables with cdfs F and G, with inverses $F^{-1}(t)=\inf\{x:F(x)\ge t\}$, we have $\int_{\mathbb{R}}|F(x)-G(x)|dx = \int_0^1 |F^{-1}(t)-G^{-1}(t)|dt$.
This fact seems kind of obvious to me, since we are talking about the area between the graphs.
So firstly I tried to prove that the area between bijections $f$, $g$ is the same as the area between the inverses, provided that the area is finite, and then generalise it to my problem.
I thought of doing some substitution in the integral, but I am stuck. Probably there is an easy way of showing this fact and I would appreciate any help.
Hint: Let $E=\{(x,y): y\,\text{is between }F(x)\,\text {and }G(x)\}.$ Then
$$\int_{\mathbb{R}}|F(x)-G(x)|dx = \int_{\mathbb{R}}\int_0^1 \chi_E(x,y)\, dy\, dx$$ $$= \int_0^1 \int_{\mathbb{R}}\chi_E(x,y)\, dx\, dy.$$
Now verify that in the last double integral, the inner integral equals $|F^{-1}(y)-G^{-1}(y)|.$