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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.

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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 $$m(\bigcup B_i)\leq m(U) \leq m(A)+\eta.$$

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