Monotone rearrangement
= 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.