Let $f:[0,1]\rightarrow [0,1]$ be a function. Let $E\subseteq [0,1]$, such that $\mu(E)>0$. What conditions on $f$ would guarantee that $f(E)$ is also measurable, and $\mu(f(E))>0$?
I guess that $f$ should be measurable and continuous bijection, but this is not the correct condition.