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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Besicovitch covering theorem ： ウィキペディア英語版 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: * Given any Besicovitch cover F of a bounded set ''E'', there are ''c''''N'' subcollections of balls ''A''1 = , …, ''A''''c''''N'' =  contained in F such that each collection ''A''i consists of disjoint balls, and :$E\; \backslash subseteq\; \backslash bigcup\_^\; \backslash bigcup\_\; B.$ Let G denote the subcollection of F consisting of all balls from the ''c''''N'' disjoint families ''A''1,...,''A''''c''''N''. 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). * There exists a constant ''b''N 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'' ∈ R''N'' belongs to at most ''b''''N'' different balls from the subcover G. 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'', :$\backslash mathbf\_E\; \backslash le\; S\_\}\; \backslash mathbf\_B\; \backslash le\; b\_N.$ ==Application to maximal functions and maximal inequalities== 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 $f^$ * by setting for every ''x'' (using the convention $\backslash infty\; \backslash times\; 0\; =\; 0$) :$f^$ *(x) = \sup_ \Bigl( \mu(B(x, r))^ \int_ |f(y)| \, d\mu(y) \Bigr). This maximal function is upper semicontinuous, hence measurable. The following maximal inequality is satisfied for every λ > 0 : :$\backslash lambda\; \backslash ,\; \backslash mu\; \backslash bigl(\; \backslash \; \backslash bigr)\; \backslash le\; b\_N\; \backslash ,\; \backslash int\; |f|\; \backslash ,\; d\backslash mu.$ ;Proof. The set ''E''λ of the points ''x'' such that $f^$ *(x) > \lambda clearly admits a Besicovitch cover Fλ by balls ''B'' such that :$\backslash int\; \backslash mathbf\_B\; \backslash ,\; |f|\; \backslash \; d\backslash mu\; =\; \backslash int\_\; |f(y)|\; \backslash ,\; d\backslash mu(y)\; >\; \backslash lambda\; \backslash ,\; \backslash mu(B).$ For every bounded Borel subset ''E''´ of ''E''λ, one can find a subcollection G extracted from Fλ that covers ''E''´ and such that ''S''G ≤ ''b''''N'', hence :$\backslash begin$ \lambda \, \mu(E') &\le \lambda \, \sum_} \int \mathbf_B \, |f| \, d\mu = \int S_ which implies the inequality above. When dealing with the Lebesgue measure on R''N'', 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Besicovitch covering theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.