Let $\mathcal{B}$ be a collection (not necessarily finite) of open balls in $\mathbb{R}^d$ with the property $| \cup_{B\in \mathcal{B}} B|<\infty$. Prove that there exists a countable subcollection $\{B_i\}_{i\in I}$ of pairwise disjoint balls in $\mathcal{B}$ such that the following two conditions are satisfied:
- For every $B\in \mathcal{B}$, there is $i\in I$ such that $B\subset 5B_i$;
- There exists a positive constant $C_d$ depending only on $d$ such that
$$\left|\bigcup_{B\in \mathcal{B}}B\right|\leq C_d \sum_{i\in I}|B_i| .$$
Here $|U|$ denotes the Lebesgue measure of the set $U$.
Attempted Solution
Using Zorn's lemma it is possible to to find a subcollection that satisfies condition 1 (such a proof is well known, and can be found on the wikipedia page http://en.wikipedia.org/wiki/Vitali_covering_lemma). Given property $1$, property $2$ can easily be proven using subadditivity:
$$\left|\bigcup_{B\in \mathcal{B}}B\right|\leq \left|\bigcup_{i\in I}5B_i\right|\leq\sum_{i\in I}|5B_i|=5^d\sum_{i\in I} |B_i|. $$
The only difficulty I am having is showing that the resulting collection can indeed be taken to be countable, any advice on this would be greatly appreciated.
Best Answer
Not only can it be taken to be countable, it must be countable!
Since $\mathbb{R}^d$ is separable (has a countable dense set, namely $\mathbb{Q}^d$), any family of pairwise disjoint (nonempty) open sets is countable. (Each open set in the family has nonempty intersection with $\mathbb{Q}^d$, no two distinct sets in the family can contain the same point of $\mathbb{Q}^d$.)