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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.

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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.

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  • $\begingroup$ Thank you. What if if I do not insist on $T$ being invertible? $\endgroup$
    – Keivan Karai
    Commented Jul 19, 2013 at 23:10
  • $\begingroup$ @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. $\endgroup$
    – Noah Stein
    Commented Jul 22, 2013 at 1:21

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