The Stein Str\"omberg Covering Theorem in metric spaces

TL;DR

This paper adapts the Stein-Str"omberg covering theorem to metric spaces, providing a new proof, weakening conditions, and refining measure estimates, thus extending classical Euclidean results to more general settings.

Contribution

It offers a novel adaptation of the Stein-Str"omberg argument to metric spaces, broadening the theorem's applicability and improving measure bounds.

Findings

New proof of Stein-Str"omberg theorem in metric spaces

Weakened hypotheses for the covering theorem

Sharpened measure estimates for Lebesgue measure

Abstract

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.