Monotone rearrangement

For an atomless measure $\mu$ on $\mathbb{R}$ with cumulative distribution function $F$, the monotone transport to a target with quantile function $G^{-1}$ is $T=G^{-1}\circ F$, defined $\mu$-almost everywhere. It minimizes well-defined costs $d(x-y)$ for convex continuous $d$. With atoms in the source, the common-quantile transport plan $(F^{-1},G^{-1}{)}_{\#}\mathcal{U}(0,1)$ remains available but need not be induced by a map.

For atomless source and target probabilities with continuous strictly increasing cumulative distributions, the squared-distance optimal coupling is unique. Two-point optimality forbids crossed support pairs. This makes every pair of source and target lower-tail events nested up to null sets and forces the displayed joint distribution, which is the common-quantile coupling and is induced by $G^{-1}F$.