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 exist a sequence of continuous automorphisms $\varphi_n:D\to D$ converging in measure to $\varphi$?