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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 Lebesgue-Stieltjes integration ： ウィキペディア英語版 Lebesgue–Stieltjes integration In measure-theoretic analysis and related branches of mathematics, Lebesgue–Stieltjes integration generalizes Riemann–Stieltjes and Lebesgue integration, preserving the many advantages of the former in a more general measure-theoretic framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be associated to any function of bounded variation on the real line. The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. Lebesgue–Stieltjes integrals, named for Henri Leon Lebesgue and Thomas Joannes Stieltjes, are also known as Lebesgue–Radon integrals or just Radon integrals, after Johann Radon, to whom much of the theory is due. They find common application in probability and stochastic processes, and in certain branches of analysis including potential theory. ==Definition== The Lebesgue–Stieltjes integral :$\backslash int\_a^b\; f(x)\backslash ,dg(x)$ is defined when is Borel-measurable and bounded and is of bounded variation in and right-continuous, or when is non-negative and is monotone and right-continuous. To start, assume that is non-negative and is monotone non-decreasing and right-continuous. Define and (Alternatively, the construction works for left-continuous, and ). By Carathéodory's extension theorem, there is a unique Borel measure on which agrees with on every interval . The measure arises from an outer measure (in fact, a metric outer measure) given by :$\backslash mu\_g(E)\; =\; \backslash inf\backslash left\backslash$ the infimum taken over all coverings of by countably many semiopen intervals. This measure is sometimes called〔Halmos (1974), Sec. 15〕 the Lebesgue–Stieltjes measure associated with . The Lebesgue–Stieltjes integral :$\backslash int\_a^b\; f(x)\backslash ,dg(x)$ is defined as the Lebesgue integral of with respect to the measure in the usual way. If is non-increasing, then define :$\backslash int\_a^b\; f(x)\backslash ,dg(x)\; :=\; -\backslash int\_a^b\; f(x)\; \backslash ,d\; (-g)(x),$ the latter integral being defined by the preceding construction. If is of bounded variation and is bounded, then it is possible to write :$dg(x)=dg\_1(x)-dg\_2(x)$ where is the total variation of in the interval , and . Both and are monotone non-decreasing. Now the Lebesgue–Stieltjes integral with respect to is defined by :$\backslash int\_a^b\; f(x)\backslash ,dg(x)\; =\; \backslash int\_a^b\; f(x)\backslash ,dg\_1(x)-\backslash int\_a^b\; f(x)\backslash ,dg\_2(x),$ where the latter two integrals are well-defined by the preceding construction. 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「Lebesgue–Stieltjes integration」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.