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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 Hahn-Kolmogorov theorem ： ウィキペディア英語版 Hahn–Kolmogorov theorem In mathematics, the Hahn–Kolmogorov theorem characterizes when a finitely additive function with non-negative (possibly infinite) values can be extended to a ''bona fide'' measure. It is named after the Austrian mathematician Hans Hahn and the Russian/Soviet mathematician Andrey Kolmogorov. ==Statement of the theorem== Let $\backslash Sigma\_0$ be an algebra of subsets of a set $X.$ Consider a function :$\backslash mu\_0\backslash colon\; \backslash Sigma\_0\; \backslash to()$ which is ''finitely additive'', meaning that : $\backslash mu\_0\backslash left(\backslash bigcup\_^N\; A\_n\backslash right)=\backslash sum\_^N\; \backslash mu\_0(A\_n)$ for any positive integer ''N'' and $A\_1,\; A\_2,\; \backslash dots,\; A\_N$ disjoint sets in $\backslash Sigma\_0$. Assume that this function satisfies the stronger ''sigma additivity'' assumption :$\backslash mu\_0\backslash left(\backslash bigcup\_^\backslash infty\; A\_n\backslash right)\; =\; \backslash sum\_^\backslash infty\; \backslash mu\_0(A\_n)$ for any disjoint family $\backslash$ of elements of $\backslash Sigma\_0$ such that $\backslash cup\_^\backslash infty\; A\_n\backslash in\; \backslash Sigma\_0$. (Functions $\backslash mu\_0$ obeying these two properties are known as pre-measures.) Then, $\backslash mu\_0$ extends to a measure defined on the sigma-algebra $\backslash Sigma$ generated by $\backslash Sigma\_0$; i.e., there exists a measure :$\backslash mu\; \backslash colon\; \backslash Sigma\; \backslash to()$ such that its restriction to $\backslash Sigma\_0$ coincides with $\backslash mu\_0.$ If $\backslash mu\_0$ is $\backslash sigma$-finite, then the extension is unique. 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「Hahn–Kolmogorov 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.