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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 Hausdorff measure ： ウィキペディア英語版 Hausdorff measure In mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in () to each set in R''n'' or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. The one-dimensional Hausdorff measure of a simple curve in R''n'' is equal to the length of the curve. Likewise, the two dimensional Hausdorff measure of a measurable subset of R2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. ==Definition== Let $(X,\backslash rho)$ be a metric space. For any subset $\backslash scriptstyle\; U\backslash subset\; X$, let $\backslash mathrm\backslash ;U$ denote its diameter, that is :$\backslash mathrm\backslash ;U\; :=\backslash sup\backslash ,\; \backslash quad\; \backslash mathrm\backslash ;\backslash emptyset:=0$ Let ''$S$'' be any subset of ''$X$'', and $\backslash delta>0$ a real number. Define : (The infimum is over all countable covers of ''$S$'' by sets $\backslash scriptstyle\; U\_i\backslash subset\; X$ satisfying .) Note that $\backslash scriptstyle\; H^d\_\backslash delta(S)$ is monotone decreasing in $\backslash delta$ since the larger $\backslash delta$ is, the more collections of sets are permitted, making the infimum smaller. Thus, the limit $\backslash scriptstyle\backslash lim\_H^d\_\backslash delta(S)$ exists but may be infinite. Let :$H^d(S):=\backslash sup\_\; H^d\_\backslash delta(S)=\backslash lim\_H^d\_\backslash delta(S).$ It can be seen that $H^d(S)$ is an outer measure (more precisely, it is a metric outer measure). By general theory, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the $d$-dimensional Hausdorff measure of $S$. Due to the metric outer measure property, all Borel subsets of $X$ are $H^d$ measurable. In the above definition the sets in the covering are arbitrary. However, they may be taken to be open or closed, and will yield the same measure, although the approximations $\backslash scriptstyle\; H^d\_\backslash delta(S)$ may be different . If ''$X$'' is a normed space the sets may be taken to be convex. However, the restriction of the covering families to balls gives a different measure. 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「Hausdorff 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.