Suppose that $X,Y$ are random-variables in $L^2(\Omega,\mathcal{F},\mathbb{P})$, for some complete probability space $(\Omega,\mathcal{F},\mathbb{P})$, and suppose that there exists a Borel function $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $$ f(Y)=X. $$ When is $f \in L^p(\mathbb{R},\mathcal{B},\mu)$, where $\mu$ is a Borel measure, equivalent to the Lebesgue measure and $\mathcal{B}$ is the Borel $\sigma$-algebra?
Sketch I believe if I take the push-forward of $\mathbb{P}$ by $Y$, then the answer is affirmative for $p=2$... However, I'm not sure if(/when) this measure is Borel?