Let $X$ and $Y$ be identically distributed bounded random variables defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$. I want to know if there always exists an invertible measure-preserving transformation $T: \Omega \to \Omega$ such that $ X \circ T= Y$ holds almost surely. 


When $X$ and $Y$ are characteristic functions, this is essentially proven in page 74 of Halmos's Lectures on Ergodic theory. Before attempting to prove the general statement myself, I want to ask if this is possibly a standard result. I would very much appreciate any suggestions or references. 



identically distributed random variables and measure-preserving transformations
No. Take for example the case where $\Omega = [0,1]$ with Lebesgue measure, $X$ is the identity, and $Y = 2X\mod 1$. Then you ask for $T = X\circ T = Y$ almost surely, but changing $Y$ on a set of measure zero cannot make it invertible.


