Let $u \in L^2(\Omega, \lambda; \mathbb{R}^n)$, represented by a Borel map $u: \Omega \to \mathbb{R}^n$. Assume that $u$ is nondegenerate in the sense that for every Lebesgue null set $N \subset \mathbb{R}^n$,
Set $\nu := u_\#\lambda$. Then there exist a $\lambda$-measure-preserving Borel map $s: \Omega \to \Omega$, a proper lower semicontinuous convex function $\varphi: \mathbb{R}^n \to (-\infty,\infty]$, and a Borel map $T: \mathbb{R}^n \to \mathbb{R}^n$ such that $T = \nabla \varphi$ $\lambda$-a.e. on the set where $\varphi$ is differentiable,
$\lambda$-a.e. on $\Omega$. The monotone factor $T$ is unique $\lambda$-a.e. as the Brenier map from $\lambda$ to $\nu$. If there exists a Borel map $R: \mathbb{R}^n \to \mathbb{R}^n$ which is the reverse Brenier map from $\nu$ to $\lambda$ and is an essential inverse for $T$, then the canonical choice of rearrangement factor is