## 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
(with proper interpretation of the sum if the collection
for some subcollection

*μ*, so that if*F*= {*B*} is a covering of a set*A*by balls, then always$$
\mu (A) \leqslant \sum\limits_\mathcal{F} {\mu (B)}
$$

*F*is not countable). What we often would like to have, for instance, is an inequality in the other direction,$$
\mu (A) \geqslant C \sum\limits_{\mathcal{F}'} {\mu (B)} ,
$$

*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.## Keywords

Radon Measure Closed Ball Doubling Measure Borel Regular Measure Covering Theorem
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 2001