I am working on the following problem, which is based on a problem from Stein and Shakarchi:
Prove the following variant of the Vitali Covering Lemma: If E is a set of finite Lebesgue measure in $\mathbb{R}^n$, then for every $\eta > 0$ there exists a disjoint collection of balls $\{B_j \}^{\infty}_{j=1}$ such that $m(E / \bigcup_{j=1}^\infty B_j) = 0$ and $\sum_{j=1}^\infty m(B_j) \leq (1+\eta)m(E)$.
It seems that the best place to start this is to look at Stein and Shakarchi's proof of the Vitali convering lemma (or another proof) and then somehow modify this, although I can't seem to bridge the gap. Any help with this would be greatly appreciated. Thank you.