I want to show that if $E\subset \mathbb R^n$ is measurable then $aE$ is measurable for all $a\in\mathbb R$ and $m(aE)=a^nm(E)$. I don't really know how to do. Notice that $m$ is Lebesgue measure.
My idea was to consider the function $f(x)=ax$ ($a\neq 0$), then $f(E)=aE$ and since $f$ is bijective and $f^{-1}$ is measurable then $f(E)=aE$ is measurable. But my teacher told me that it doesn't work, and I don't understand why.