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.

countabilityof this subfamily of disjoint open balls? But any family of disjoint nonempty open sets of a separable space must be countable (there is a countable set that meets each member of the family). $\endgroup$ – Pietro Majer Jul 13 '18 at 14:53