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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 Jordan measure ： ウィキペディア英語版 Jordan measure In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. The Peano-Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.〔G. Peano, "Applicazioni geometriche del calcolo infinitesimale", Fratelli Bocca, Torino, 1887.〕 ==Jordan measure of "simple sets"== Consider the Euclidean space R''n''. One starts by considering products of bounded intervals :$C=[a\_1,\; b\_1)\backslash times\; [a\_2,\; b\_2)\; \backslash times\; \backslash cdots\; \backslash times\; [a\_n,\; b\_n)$ which are closed at the left end and open at the right end (half-open intervals is a technical choice; as we see below, one can use closed or open intervals if preferred). Such a set will be called a ''n''-''dimensional rectangle'', or simply a ''rectangle''. One defines the ''Jordan measure'' of such a rectangle to be the product of the lengths of the intervals: :$m(C)=(b\_1-a\_1)(b\_2-a\_2)\; \backslash cdots(b\_n-a\_n).$ Next, one considers ''simple sets'', sometimes called ''polyrectangles'', which are finite unions of rectangles, : $S=C\_1\backslash cup\; C\_2\backslash cup\; \backslash cdots\; \backslash cup\; C\_k$ for any ''k''≥1. One cannot define the Jordan measure of ''S'' as simply the sum of the measures of the individual rectangles, because such a representation of ''S'' is far from unique, and there could be significant overlaps between the rectangles. Luckily, any such simple set ''S'' can be rewritten as a union of another finite family of rectangles, rectangles which this time are mutually disjoint, and then one defines the Jordan measure ''m''(''S'') as the sum of measures of the disjoint rectangles. One can show that this definition of the Jordan measure of ''S'' is independent of the representation of ''S'' as a finite union of disjoint rectangles. It is in the "rewriting" step that the assumption of rectangles being made of half-open intervals is used. 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「Jordan 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.