Fix $\epsilon$ small. By Lemma 3.3, for each $x\in\Sigma_{v}$ we can choose an arbitrarily small $r$ such that

$$Mv(B_{r}(x))>\frac{1}{\epsilon}r^{n-1}.$$

Cover $\Sigma_{v}\cap B_{1/2}$ with such balls, and choose a Vitali subcover $\{B_{r_{i}}(x_{i})\}_{i=1}^{N}$, i.e. a disjoint subcollection such that $B_{3r_{i}}(x_{i})$ cover $\Sigma\cap B_{1/2}$. Then

	$\displaystyle\sum_{i=1}^{N}(3r_{i})^{n-1}$	$\displaystyle\leq C\epsilon\sum_{i=1}^{N}Mv(B_{r_{i}}(x_{i}))$
		$\displaystyle\leq C\epsilon,$

since $v$ is locally Lipschitz and the $B_{r_{i}}$ are disjoint. This means exactly that

$$\mathcal{H}^{n-1}(\Sigma_{v}\cap B_{1/2})=0.$$

∎