## 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

Manifold Radon## Preview

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## Copyright information

© Springer Science+Business Media New York 2001