# Besicovitch covering theorem

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: Given any Besicovitch cover F of a bounded set E, there are cN subcollections of balls A1 = {Bn1}, …, AcN = {BncN} contained in F such that each collection Ai consists of disjoint balls, and {displaystyle Esubseteq bigcup _{i=1}^{c_{N}}bigcup _{Bin A_{i}}B.} 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 (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).

There exists a constant bN depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most bN different balls from the subcover G.

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, {displaystyle mathbf {1} _{E}leq S_{mathbf {G} }:=sum _{Bin mathbf {G} }mathbf {1} _{B}leq b_{N}.} 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 {displaystyle f^{*}} by setting for every x (using the convention {displaystyle infty times 0=0} ) {displaystyle f^{*}(x)=sup _{r>0}{Bigl (}mu (B(x,r))^{-1}int _{B(x,r)}|f(y)|,dmu (y){Bigr )}.} This maximal function is lower semicontinuous, hence measurable. The following maximal inequality is satisfied for every λ > 0 : {displaystyle lambda ,mu {bigl (}{x:f^{*}(x)>lambda }{bigr )}leq b_{N},int |f|,dmu .} Proof.

The set Eλ of the points x such that {displaystyle f^{*}(x)>lambda } clearly admits a Besicovitch cover Fλ by balls B such that {displaystyle int mathbf {1} _{B},|f| dmu =int _{B}|f(y)|,dmu (y)>lambda ,mu (B).} 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 {displaystyle {begin{aligned}lambda ,mu (E')&leq lambda ,sum _{Bin mathbf {G} }mu (B)\&leq sum _{Bin mathbf {G} }int mathbf {1} _{B},|f|,dmu =int S_{mathbf {G} },|f|,dmu leq b_{N},int |f|,dmu ,end{aligned}}} 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).

See also Vitali covering lemma References Besicovitch, A. S. (1945), "A general form of the covering principle and relative differentiation of additive functions, I", Proceedings of the Cambridge Philosophical Society, 41 (02): 103–110, doi:10.1017/S0305004100022453. "A general form of the covering principle and relative differentiation of additive functions, II", Proceedings of the Cambridge Philosophical Society, 42: 205–235, 1946, doi:10.1017/s0305004100022660. DiBenedetto, E (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5. Füredi, Z; Loeb, P.A. (1994), "On the best constant for the Besicovitch covering theorem", Proceedings of the American Mathematical Society, 121 (4): 1063–1073, doi:10.2307/2161215, JSTOR 2161215. Categories: Covering lemmasTheorems in analysis

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