Confusion on the meaning of the Vitali- Covering Lemma

Question

In Folland, Lemma 3.15 gives a version of the Vitali-Covering Lemma:

Lemma 3.15: Let $\mathscr{C}$ be a collection of open balls in $\mathbb{R}^{n}$, and let $U= \bigcup_{B \in \mathscr{C}} B$. If $c<m(U)$, there exists disjoint $B_{1}, \ldots, B_{k} \in \mathscr{C}$ such that $\sum_{j=1}^{k} m(B_{j})> 3^{-n} c$.

I am very unclear on the significance of this result.

Is it true that $U \subset \bigcup_{j=1}^{k} 3 B_{j}$? If so, how is this implied by this lemma?

Answer 1

This Lemma is not quite easy to understand just by reading the terse proof.

However, the following picture from wiki is much helpful:

1. Green balls represent $B_j$
2. purple balls represent the balls $A_i$ with $A_i \cap Bj \neq \emptyset$

Anwser: YES

To explain it, notice the following:

1. No isolated purple balls.
2. Purple balls will no larger than the green balls if they intersect.
3. If two balls are of the same size and both are almost disjoint. Then choose one of them to be green ball $B$ and the other, purple $A$. Then $3B \supset A $ (Verify it just by drawing circle !)

So the result indeed imply that $U \subset \bigcup_{j=1}^{k} 3 B_{j}$ (No purple balls are isolated from the covering.)

Also 1.~ 3. are implied by the proof of this Lemma. So the Lemma indeed implies this result and this picture.