Let $\mathscr{C}$ be a family of balls contained in some bounded region of $\mathbb{R}^n$. Then for any $\alpha > 3$, there is a finite or countable disjoint subcollection $\{ B_i \}$ such that
$\cup_{B \in \mathscr{C}} B \subset \cup_{i} \tilde{B_i}$
where $\tilde{B_i}$ is the closed ball concentric with $B_i$ and of $\alpha$ times the radius.
So what I'm getting from this, is that it's something similar to the Vitali Covering Lemma, but for balls. I think we have $|B_i| \leq \frac{1}{2\alpha} |B|, \forall i$, right? I'm not sure how to prove this.
EDIT: I have found a nice proof of this, that can be found here: https://matthewhr.wordpress.com/2015/03/05/a-differentiation-basis-without-the-vitali-covering-property/