I have a question about a specific version of the Vitali covering Lemma. I understand the first part and how to prove it; it all makes sense, however, the author claims that the second part follows obviously from the first, which I honestly don't see.
Statement of the theorem: Given a subset $E$ of $\mathbb{R}^d$ with $0 < m_*(E) < \infty$, suppose that $F$ is a family of a balls which covers $E$ in the sense of Vitali. Then there exists a countable subset $\{B_n\}$ of disjoint balls such that, given any $\epsilon >0$ $$ m_*( E \setminus \cup_n B_n) = 0 \quad \text{ and } \quad \sum_{n=1}^{\infty} m(B_n) \leq (1+\epsilon) m_*(E). $$
Here $m$ represented the $d$-dimensional Lebesgue measure, and $m_*$ represented the outer measure.
Any suggestions for understanding the second part would be greatly appreciated! : )