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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's decomposition theorem ： ウィキペディア英語版 Lebesgue's decomposition theorem In mathematics, more precisely in measure theory, Lebesgue's decomposition theorem states that for every two σ-finite signed measures $\backslash mu$ and $\backslash nu$ on a measurable space $(\backslash Omega,\backslash Sigma),$ there exist two σ-finite signed measures $\backslash nu\_0$ and $\backslash nu\_1$ such that: * $\backslash nu=\backslash nu\_0+\backslash nu\_1\backslash ,$ * $\backslash nu\_0\backslash ll\backslash mu$ (that is, $\backslash nu\_0$ is absolutely continuous with respect to $\backslash mu$) * $\backslash nu\_1\backslash perp\backslash mu$ (that is, $\backslash nu\_1$ and $\backslash mu$ are singular). These two measures are uniquely determined by $\backslash mu$ and $\backslash nu$. ==Refinement== Lebesgue's decomposition theorem can be refined in a number of ways. First, the decomposition of the singular part of a regular Borel measure on the real line can be refined: :$\backslash ,\; \backslash nu\; =\; \backslash nu\_\}\; +\; \backslash nu\_\{\backslash mathrm\{pp\}\}$ where * ''ν''cont is the absolutely continuous part * ''ν''sing is the singular continuous part * ''ν''pp is the pure point part (a discrete measure). Second, absolutely continuous measures are classified by the Radon–Nikodym theorem, and discrete measures are easily understood. Hence (singular continuous measures aside), Lebesgue decomposition gives a very explicit description of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of a singular continuous 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「Lebesgue's decomposition theorem」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.