Let $(X,Z,\mu)$ be a standard borel measure, I will like to know if there is an isomorphism mod zero preserving the measures form $([0,1],B,\lambda)$ where $\lambda$ is absolutely continuous with respect to the Lebesgue measure in the case where $X$ is uncountable. Look that when $X$ is countable we can find such an isomorphism into the counting measure. If you know such result please give me a reference.

Classical Descriptive Set Theory, Section 17.F (in the continuous case, the countable case is trivial). The result goes back to Hausdorff and was stated in this form by Rokhlin in his thesis, von Neumann should also be mentioned. If I'm not mistaken, Ch III of Arveson'sAn invitation to $C^{\ast}$-algebrasalso contains a proof (at least it can easily be extracted from the results in there). – Theo Buehler Jul 6 '11 at 0:23