Covering Theorems

Abstract

All covering theorems are based on the same idea: from an arbitrary cover of a set in a metric space, one tries to select a subcover that is, in a certain sense, as disjointed as possible. In order to have such a result, one needs to assume that the covering sets are somehow nice, usually balls. In applications, the metric space normally comes with a measure μ, so that if F = {B} is a covering of a set A by balls, then always

$$ \mu (A) \leqslant \sum\limits_\mathcal{F} {\mu (B)} $$

(with proper interpretation of the sum if the collection F is not countable). What we often would like to have, for instance, is an inequality in the other direction,

$$ \mu (A) \geqslant C \sum\limits_{\mathcal{F}'} {\mu (B)} , $$

for some subcollection F′ ⊂ F that still covers A and for some positive constant C that is independent of A and the covering F. There are many versions of this theme.