Let $\mu$ and $\nu$ be Borel probability measures on $\mathbb R$, and assume that $\mu$ has no atoms. Write $F_\mu(x)=\mu((-\infty,x])$ and $F_\nu(x)=\nu((-\infty,x])$, and use the left-continuous quantile convention \begin{align*} F_\nu^{-1}(t)=\inf\{y\in\mathbb R:F_\nu(y)\ge t\} \end{align*} for $0<t<1$. Let $T:\mathbb R\to\mathbb R$ be a Borel map satisfying, at every point where $0<F_\mu(x)<1$, \begin{align*} T(x)=F_\nu^{-1}(F_\mu(x)). \end{align*} Assume also that $T$ has arbitrary Borel real values on the $\mu$-null endpoint set where $F_\mu(x)\in\{0,1\}$. Then $T_{\#}\mu=\nu$, and $T$ agrees on a full $\mu$-measure set with a nondecreasing function.