# identically distributed random variables and measure-preserving transformations

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.

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.
• Thank you. What if if I do not insist on $T$ being invertible? – Keivan Karai Jul 19 '13 at 23:10
• @KeivanKarai: I believe in that case you can get a counterexample by letting $X(\omega) = 2\omega \mod 1$ and $Y(\omega) = 3\omega\mod 1$, though I haven't checked the details. – Noah Stein Jul 22 '13 at 1:21