Suppose I have a compact set $K \subset B_1(0) \subset \mathbb{R}^n$. Can I always find a family of open balls $\{B_{r_j}(x_j)\}$ such that

$x_j \in K$ and $B_{r_j}(x_j) \subset B_1(0)$ for each $j$;

$K \subset \bigcup_j B_{r_j}(x_j)$; and

The collection $\{B_{2r_j}(x_j)\}$ has bounded overlap, that is to say that there is some number $N = N(n)$ such that each point of $\bigcup_j B_{2r_j}(x_j)$ lies in at most $N$ balls from the collection. In other words, $$ \sum_j \mathbf{1}_{B_{2r_j}(x_j)} \leq N $$

The motivation behind a question like this is a fairly common situation where at each point $y \in K$ I can prove an estimate like $$ \int_{B_{r/2}(y)} f \leq C\int_{B_r(y)} g, $$ but what I really want is an estimate of the form $$ \int_{\{x\ :\ \mathrm{dist} (x,K) < r/10\}} f \leq C \int_{\{x\ :\ \mathrm{dist} (x,K) < 10r\}} g $$ The numbers 1/2 and 10 are not important, but to do this it is natural to try to find a collection of balls as described. In specific cases I have constructed this collection myself but now I wonder if there is a general lemma that I happen not to have heard of.