Here's a problem that I'm stuck with for a while.
If $E \subset \mathbb{R}^d$ is Lebesgue measurable, and $T:\mathbb{R}^d \to \mathbb{R}^d$ is a linear transformation, then:
$(i)$ Show that $T(E)$ is Lebesgue measurable
$(ii)$ Show that $m(T(E))=~|\mbox{det}(T)| \times m(E)$
$(iii)$ Give a counter example that if $T:\mathbb{R}^d \to \mathbb{R}^{d'}$ is a linear map to a space $\mathbb{R}^d$ of strictly smaller dimension than $\mathbb{R}^d$, then $T(E)$ need not be Lebesgue measurable.
I think that the problem can be solved without involving the Lebesgue integration as the book I'm following introduced Lebesgue integration after the section on Lebesgue measure (and hence, this exercise). So to avoid any kind of circularity, I'm looking for a solution without relying on Lebesgue integration. Any help would be greatly appreciated!