Function that sends every Lebesgue measureable sets to a lebesgue measurable set. Then it sends measure zero sets to measure zero sets.

Question

I want to prove:

If $f: \mathbb R \to \mathbb R $ is a function that send every Lebesgue measureable sets to Lebesgue measurable sets then it send measure zero sets to measure zero.

I do not know how to start to think. Can someone help me. Thanks

Answer 1

Suppose that $N \subset \mathbb R$ is a set of Lebesgue measure zero with the property that $f(N)$ has positive measure. Then $f(N)$ contains a nonmeasurable set $Z$. If you let $Y = N \cap f^{-1}(Z)$ then $Y$ has Lebesgue measure zero, so it is measurable, but $f(Y) = Z$ is not.