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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 Frostman lemma ： ウィキペディア英語版 Frostman lemma In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' denotes the ''s''-dimensional Hausdorff measure. *There is an (unsigned) Borel measure ''μ'' satisfying ''μ''(''A'') > 0, and such that ::$\backslash mu(B(x,r))\backslash le\; r^s$ :holds for all ''x'' ∈ R''n'' and ''r''>0. Otto Frostman proved this lemma for closed sets ''A'' as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin sets. A useful corollary of Frostman's lemma requires the notions of the ''s''-capacity of a Borel set ''A'' ⊂ R''n'', which is defined by :$C\_s(A):=\backslash sup\backslash Bigl\backslash :\backslash mu\backslash text\backslash mu(A)=1\backslash Bigr\backslash \}.$ (Here, we take inf ∅ = ∞ and  = 0. As before, the measure $\backslash mu$ is unsigned.) It follows from Frostman's lemma that for Borel ''A'' ⊂ R''n'' :$\backslash mathrm\_H(A)=\; \backslash sup\backslash .$ ==References== * 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「Frostman lemma」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.