In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, is an open cover of a subset E of the Euclidean space R<sup>N</sup> 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 c<sub>N</sub> depending only on the dimension N with the following property:
Let G denote the subcollection of F consisting of all balls from the c<sub>N</sub> disjoint families A<sub>1</sub>,...,A<sub>c<sub>N</sub></sub>. The less precise following statement is clearly true: every point x â R<sup>N</sup> belongs to at most c<sub>N</sub> different balls from the subcollection G, and G remains a cover for E (every point y â E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant).
In other words, the function S<sub>G</sub> equal to the sum of the indicator functions of the balls in G is larger than 1<sub>E</sub> and bounded on R<sup>N</sup> by the constant b<sub>N</sub>,
Let ü be a Borel non-negative measure on R<sup>N</sup>, finite on compact subsets and let be a -integrable function. Define the maximal function by setting for every (using the convention )
This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every û > 0:
The set E<sub>û</sub> of the points x such that clearly admits a Besicovitch cover F<sub>û</sub> by balls B such that
For every bounded Borel subset Eô of E<sub>û</sub>, one can find a subcollection G extracted from F<sub>û</sub> that covers Eô and such that S<sub>G</sub> ⤠b<sub>N</sub>, hence
which implies the inequality above.
When dealing with the Lebesgue measure on R<sup>N</sup>, 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).