Suppose first that $D=[0,1]$ is equipped with the usual Lebesgue measure, and that $\varphi$ is a measure-preserving transformation $\varphi:D\to D$ that is bijective and whose inverse is also measure preserving (called automorphism).


Is it true that there exists a sequence of continuous measure-preserving transformations $\varphi\_n:D\to D$ converging in measure to $\varphi$?



automorphisms of a measurable space can be approximated by continuous measure preserving maps?
In fact, sequences of continuous piecewise linear transformations of $[0,1]$ suffice to approximate a.e. any measure preserving transformation of $[0,1]$ as proved [in thm 2.1 in this paper](https://www.emis.de/journals/HOA/IJMMS/Volume13_2/708257.pdf) (therefore in measure, by Severini-Egorov theorem; but in fact in that proof convergence in measure is first proved). 


