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


[1]:https://www.emis.de/journals/HOA/IJMMS/Volume13_2/708257.pdf