I want to prove:
Let $U$ be an open subset of $\mathbb{R}^n$ and $F:U\to \mathbb{R}^n$ a $C^\infty$ function. Let $S$ be a subset of $U$ of zero-measure. Then $F(S)$ has zero-measure.
Proof
We can find a countable open cover of $U$, say $\mathfrak{B}$, such that for every $V\in \mathfrak{B}$ we have $\overline{V}\subseteq U$, and $V$ is an open ball.
Since countable union of subsets of zero-measure has zero-measure, then it is sufficient to prove that $F(S\cap V)$ has zero-measure for every $V \in \mathfrak{B}$.
We can suppose that $S \subseteq V$.
Since $S$ has zero-measure, for every $\epsilon >0$ there is $\{R_j\}$ countable family of $n-$cubes such that $$S\subseteq\bigcup_j R_j, \quad \text{and} \quad\sum_j \text{Vol}(R_j)<\epsilon$$
Let $l_j>0$ be the length of the side of the $n-$cube $R_j$. We can inscribe $R_j$ in an open ball $B_j$ of radius $r_j=\sqrt{n}\frac{l_j}{2}$.
Since $F$ is $C^1$ in $\overline{V}$ and $\overline{V}$ is compact, then $F$ is Lipschitz in $\overline{V}$, i.e. there is $c>0$ such that $$\lVert F(x)-F(y)\rVert \leq c \lVert x-y \rVert$$ for every $x,y \in \overline{V}$.
My book says that we can inscribe $F(B_j)$ in an open ball, say $B'_j$, of radius $r'_j=cr_j$.
My objections are:
1) I don't know if $B_j\subseteq U$ (right?), so I should write $F(B_j\cap U)$.
2) If I want to use the Lipschitz of $F$ than I should write $F(B_j \cap \overline{V})$, right?
3) I don't know if the center of $B_j$ lies in $\overline{V}$ (right?), so shouldn't be $r'_j=2cr_j$?
Take $x \in B_j\cap \overline{V}$ and let $B'_j$ be the $n-$open ball of center $F(x)$ and radius $r'_j=2cr_j$. If $y$ is any point in $B_j\cap \overline{V}$ then we have $\lVert x-y \rVert <2 r_j$ and then $$\lVert F(x)-F(y)\rVert \leq c \lVert x-y \rVert <2cr_j$$ so $F(y)\in B_j'$, so $F(B_j \cap \overline{V}) \subseteq B_j'$. Or is there any way in which it is true that we can inscribe $F(B_j)$ (or, better, $F(B_j \cap \overline{V})$) in an open ball of radius $r'_j=cr_j$?
For the sake of completeness I will finish the proof.
We can inscribe $B'_j$ in an open $n-$ cube $R'_j$ with length of the side $l'_j=2r'_j=4cr_j=2cl_j\sqrt{n}$. Then we have $$F(S)\subseteq \bigcup_j R'_j \quad \text{and} \quad \sum_J \text{Vol} (R'_j)<2^n c^n n^{n/2}\epsilon $$