Measure-preserving parametrization of one-dimensional transport maps ($T=G^{-1}\circ S\circ F$)

When source and target have continuous strictly increasing cumulative distributions, all transport maps are obtained by choosing a Lebesgue-measure-preserving map $S$ on the unit interval. The maps $F$ and $G$ turn the two measures into uniform measure. The choice $S=\mathrm{id}$ gives the monotone rearrangement.