Variant of the Vitali Covering Lemma

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.

Solutions Collecting From Web of "Variant of the Vitali Covering Lemma"

See Theorem 2.2 here on p.26. Your set $E$ is called $A$ in there. Follow the proof and notice that $\bigcup_i B_i \subset U$ and that $U$ is an open set containing $A$ with measure $$m(U)\leq (1+7^{-n})m(A).$$ Here $n$ is the dimension of the space.

For your $\eta$ you could start with $U$ (containing $A$) such that $$m(U) \leq (1+7^{-(n+k)})m(A),$$ where $k$ is such that $7^{-(n+k)}m(A)<\eta$. Then you will obtain $$m(\bigcup B_i)\leq m(U) \leq m(A)+\eta.$$