Covering Theorems

  • Juha Heinonen
Part of the Universitext book series (UTX)


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.


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

Authors and Affiliations

  • Juha Heinonen
    • 1
  1. 1.Mathematics Department East HallUniversity of MichiganAnn ArborUSA

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