Prove: The pre-image of a null-set on a manifold in $\mathbb{R}^k$ is a null-set. Given a $k$ dimensional manifold, $M$, and a mapping $r: U \rightarrow M$, and a null-set $E \subset M$, prove that $r^{-1}(E \cap r(U))$ is a null-set.
We take $E = \cup A_i$ , s.t. $vol(\cup A_i) = 0$. Then of course, $vol(\cup A_i) \cap r(U)) = 0$, and now I think I have to use the fact that $r$ is a homeomorphism ,but I'm not sure how.
Any help would be appreciated!