In fact, sequences of continuous piecewise linear transformations of $[0,1]$ suffice to approximate a.e. any measure preserving transformation of $[0,1]$ as proven in this paper (therefore in measure, by Severini-Egorov theorem).

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 (therefore in measure, by Severini-Egorov theorem; but in fact in that proof convergence in measure is first proved).