Let $G:U \to V$ be a continuously differentiable and injective mapping with $U,V \subset \mathbb{R}^n$ Then I have to show first that if $K \subset U$ a compact set has zero content then so does $G(K)$, and secondly that if $T$ is Jordan measurable and $\bar T \subset U$ then $G(T)$ is also measurable.
For the first part I know $K \subset \cup_i B_i$ for some finite collection of boxes whose total volume can be arbitrarily small. Then if I look at the image of each $B_i$ under $G$ I thought I could similarly bound their volume by continuity of $G$. Is this the right idea? For the second part I am not sure because i wanted to use the first part to show that the boundary of $G(T)$ is zero content since the boundary of $T$ is, but I realized the boundary of $T$ can be mapped somewhere in the interior of $G(T)$...
Edit: The definition of Jordan measurable I know is that the boundary has zero content and the set is bounded