About Vitali covering theorem

Question

In the paper by Nazarov, Treil and Volberg: Weak type estimates and Cotlar inequalities for Calderon-Zygmund ... /1998, Int. Math. Res. Not.,

www.math.brown.edu/~treil/papers/l1/l1-5.pdf

on the page 6, they stated a version of Vitali covering theorem for a seperable metric space $X$ endowed with a measure $\mu$ “not necessarily having the doubling property”:

Claim: Let $E \subset X$ be any set and $(B(x,r_x))_{x\in E}$ be a family of balls with uniformly bounded radii. Then there always exists a “countable” subfamily of disjoint balls whose triple extensions cover $E$.

The fact “countable subfamily” is strange to me. The general covering theorem for metric spaces seems to claim the existence of a subfamily without mentioning its countability. If the space is equiped with a doubling measure then it’s well known that such a countable subfamily exists. Where can I find a reference for this claim of their paper? Do the main ingredient of this countable property come from the hypothesis of “seperable” metric spaces?

Thanks.

Answer 1

The covering lemma as stated here is true in any separable metric space. No measure is needed at all.

Theorem 1. Let $\mathcal{B}$ be a family either of closed balls or open balls from a separable metric space such that $$ \sup\{\operatorname{diam}(B):B\in\mathcal{B}\}<\infty. $$ Then there is a finite of countable sequence $\{ B_i\}_{i\in I}$ of pairwise disjoint balls such that $$ \bigcup_{B\in\mathcal B} B\subset\bigcup_{i\in I} 5 B_i. $$

This is (verbatim) Theorem 2.2 in [2]. For a proof, see page 47 in [1] or almost any book on geometric measure theory.

The "$3r"$ case is true, at least in the case of finite families of balls. The following statement is Theorem 2.1 taken verbatim from [2]:

Theorem 2. Let $\mathcal{B}$ be a finite family either of closed balls or open balls from a metric space. Then there exists a finite subfamily $\{ B_i\}_{i\in I}\subset\mathcal{B} $ of pairwise disjoint balls such that $$ \bigcup_{B\in\mathcal B} B\subset\bigcup_{i\in I} 3 B_i. $$

Without separability there are easy counterexamples to Theorem 1: uncountable space with the discrete metric, covered by balls of radii $1/10$.

I am not sure if Theorem 1 is true with $5$ replaced by $3$ as otherwise Tolsa would state it with $3$ instead of $5$. However, in all applications the actual constant $3$ or $5$ is not important.

In fact Nazarov, Treil and Volberg assume that the space is separable.

[1] http://www.pitt.edu/~hajlasz/Notatki/Analysis%20I.pdf

[2] X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory. Progress in Mathematics, 307. Birkhäuser/Springer, Cham, 2014.