The Stein Str\"omberg Covering Theorem in metric spaces

In \cite{NaTa} Naor and Tao extended to the metric setting the $O(d\log d)$ bounds given by Stein and Str"omberg for Lebesgue measure in ${\mathbb{R}}^d$, deriving these bounds first from a localization result, and second, from a random Vitali lemma. Here we show that the Stein-Str"omberg original argument can also be adapted to the metric setting, giving a third proof. We also weaken the hypotheses, and additionally, we sharpen the estimates for Lebesgue measure.