Given a real random variable $X$, suppose $T:\mathbb{R}\to\mathbb{R}$ is non-decreasing. Define $Y=T\left(X\right)$.
Let $Q_{X}$, $Q_{Y}$ be the corresponding right-continuous quantile functions.
Let $\mu_{X},\mu_{Y}:\mathcal{B}\left(\mathbb{R}\right)\to\left[0,\infty\right)$ be the measures induced by $Q_{X},Q_{Y}$, respectively, via the extension to a Borel measure of $$\mu_{X} :(a,b] \to [Q_{X}(b)-Q_{X}(a)]$$ defined for all half-open intervals $\left(a,b\right]$. These are called Stieltjes measures.
1. What is an equivalent condition for $\mu_{Y}$ to be absolutely continuous w.r.t. $\mu_{X}$?
2. Under the previous sufficient and necessary condition, is there always a closed form solution for the Radon-Nikodym derivative $\frac{d\mu_{Y}}{d\mu_{X}}$?
Context:
An alternative characterization of the right-continuous quantile function is as a right-continuous generalized inverse to the distribution function of $X$.
1. I know that continuity of $T$ would suffice but I don't know whether you can weaken this condition or not. Specifically it can be shown that $Q_{Y}=T\circ Q_{X}$ even for right-continuous $T$ but with only right-continuity there can be contradictions to the absolute continuity of $\mu_{Y}$ w.r.t $\mu_{X}$
2. I know that if one assume $Q_{X}$ is absolutely continuous and $T$ is continuous then it is not very hard to get that $\frac{d\mu_{x}}{d\mu_{Y}}=T^{'}\circ Q_{X}$ where $T^{'}$ is the a.e derivative of $T$ . However I'm not sure if this is still true if one forgoes the assumption that $Q_{X}$ is absolutely continuous.