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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 Metric outer measure ： ウィキペディア英語版 Metric outer measure In mathematics, a metric outer measure is an outer measure ''μ'' defined on the subsets of a given metric space (''X'', ''d'') such that :$\backslash mu\; (A\; \backslash cup\; B)\; =\; \backslash mu\; (A)\; +\; \backslash mu\; (B)$ for every pair of positively separated subsets ''A'' and ''B'' of ''X''. ==Construction of metric outer measures== Let ''τ'' : Σ → () be a set function defined on a class Σ of subsets of ''X'' containing the empty set ∅, such that ''τ''(∅) = 0. One can show that the set function ''μ'' defined by :$\backslash mu\; (E)\; =\; \backslash lim\_\; \backslash mu\_\; (E),$ where :$\backslash mu\_\; (E)\; =\; \backslash inf\; \backslash left\backslash \; \backslash tau\; (C\_)\; \backslash right|\; C\_\; \backslash in\; \backslash Sigma,\; \backslash mathrm\; (C\_)\; \backslash leq\; \backslash delta,\; \backslash bigcup\_^\; C\_\; \backslash supseteq\; E\; \backslash right\backslash \},$ is not only an outer measure, but in fact a metric outer measure as well. (Some authors prefer to take a supremum over ''δ'' > 0 rather than a limit as ''δ'' → 0; the two give the same result, since ''μ''''δ''(''E'') increases as ''δ'' decreases.) For the function ''τ'' one can use : $\backslash tau(C)\; =\; \backslash mathrm\; (C)^s,\backslash ,$ where ''s'' is a positive constant; this ''τ'' is defined on the power set of all subsets of ''X''; the associated measure ''μ'' is the ''s''-dimensional Hausdorff measure. More generally, one could use any so-called dimension function. This construction is very important in fractal geometry, since this is how the Hausdorff and packing measures are obtained. 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「Metric outer measure」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.