Besicovitch covering theorem


In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space RN by balls such that each point of E is the center of some ball in the cover.
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.
The less precise following statement is clearly true: every point x ∈ RN belongs to at most cN different balls from the subcollection G, and G remains a cover for E. This property gives actually an equivalent form for the theorem.
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,

Application to maximal functions and maximal inequalities

Let μ be a Borel non-negative measure on RN, finite on compact subsets and let f be a μ-integrable function. Define the maximal function by setting for every x
This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every λ > 0 :
;Proof.
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 Vitali covering lemma in order to derive the previous maximal inequality.