The Besicovitch covering theorem asserts that there exists a constant cN depending only on the dimension N with the following property: Let G denote the subcollection of F consisting of all balls from the cN disjoint families A1,...,AcN.
In other words, the function SG equal to the sum of the indicator functions of the balls in G is larger than 1E and bounded on RN by the constant bN, Let μ be a Borel non-negative measure on RN, finite on compact subsets and let
The following maximal inequality is satisfied for every λ > 0 : The set Eλ of the points x such that
clearly admits a Besicovitch cover Fλ by balls B such that For every bounded Borel subset E´ of Eλ, one can find a subcollection G extracted from Fλ that covers E´ and such that SG ≤ bN, hence which implies the inequality above.
When dealing with the Lebesgue measure on RN, it is more customary to use the easier (and older) Vitali covering lemma in order to derive the previous maximal inequality (with a different constant).