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 R^N 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_N depending only on the dimension N with the following property:

Let G denote the subcollection of F consisting of all balls from the c_N disjoint families A₁,...,A_cN. The less precise following statement is clearly true: every point x ∈ R^N belongs to at most c_N different balls from the subcollection G, and G remains a cover for E (every point y ∈ 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_G equal to the sum of the indicator functions of the balls in G is larger than 1_E and bounded on R^N by the constant b_N,

Let μ be a Borel non-negative measure on R^N, finite on compact subsets and let f be a μ-integrable function. Define the maximal function ${\displaystyle f^{*}}$ by setting for every x (using the convention ${\displaystyle \infty \times 0=0}$)

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