I've found the increasing rearrangement in Villani's book. Thank you very much!

Could you please give more details on this construction of $f_x$?

Could you please specify a bit more for the case of dimension $1$? Actually, let $\rho$ be a standard Gaussian with normal distribution $F$, so we are looking for $f_x(\cdot)$ s.t. $\mathbb P[f_x(G)\le y]=\lambda_x((-\infty,y])$ for all $y\in\mathbb R$. Notice that $\lambda_x((-\infty,y])=\mathbb P[f_x(G)\le y]=\mathbb P[G\le f_x^{-1}(y)]=F\circ f_x^{-1}(y)$, so basically we should take $f_x^{-1}(y)=F^{-1}(\lambda_x((-\infty,y]))$. But how to define properly $f_x$? Thank you very much!

Merci pour la réponse et je vais verifier moi-meme. Je vais peut-être retourner vers vous (car je suis ne suis pas très familier avec la théorie de mesure).

@user95282 I've found the paper "On a representation of random variables". But I didn't see the construction proof for $\mathbb P[|Y_0-X|\ge 2d_0]\le 2d_0$.

@user95282 Yes. Actually we may start by considering the case $n=1$ for the sake of simplicity.

@user95282 Thanks a lot for pointing out this typo. I've denoted the dimension by $n$.

Or maybe we may find some increasing function $\alpha:\mathbb R_+\to\mathbb R_+$ with $\alpha(0)=0$ s.t. $\mathbb E[|X-Y|]\le \alpha(d)$