Let $1 \le k \le n-1$. We say $M \subset \mathbb{R}^n$ is countably $\mathcal{H}^k$-rectifiable if $M$ is $\mathcal{H}^k$-measurable and there exist countably many Lipschitz maps $f_h : \mathbb{R}^k \rightarrow \mathbb{R}^n$ such that $$ \mathcal{H}^k(M \setminus \bigcup_{h=1}^\infty f_h(\mathbb{R}^k)) = 0. $$
In the book "Sets of Finite Perimeter and Geometric Variational Problems" on page 97 it is claimed that $M$ is countably $\mathcal{H}^k$-rectifiable if and only if there exist a Borel set $M_0$, countably many Lipschitz maps $f_h : \mathbb{R}^k \rightarrow \mathbb{R}^n$ and Borel sets $F_h \subset \mathbb{R}^k$ such that $$ M = M_0 \cup \bigcup_{h=1}^\infty f_h(F_h), \quad \mathcal{H}^k(M_0) = 0. $$
Assuming that $M$ is a Borel set, it is easy to prove the decomposition. However, I do not see how we can pick $M_0$ and $F_h$ to be Borel sets when $M$ is merely $\mathcal{H}^k$-measurable. Can someone help?
