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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 Vector measure ： ウィキペディア英語版 Vector measure In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. ==Definitions and first consequences== Given a field of sets $(\backslash Omega,\; \backslash mathcal\; F)$ and a Banach space $X$, a finitely additive vector measure (or measure, for short) is a function $\backslash mu:\backslash mathcal\; \backslash to\; X$ such that for any two disjoint sets $A$ and $B$ in $\backslash mathcal$ one has : $\backslash mu(A\backslash cup\; B)\; =\backslash mu(A)\; +\; \backslash mu\; (B).$ A vector measure $\backslash mu$ is called countably additive if for any sequence $(A\_i)\_^$ of disjoint sets in $\backslash mathcal\; F$ such that their union is in $\backslash mathcal\; F$ it holds that : $\backslash mu\backslash left(\backslash bigcup\_^\backslash infty\; A\_i\backslash right)\; =\backslash sum\_^\backslash mu(A\_i)$ with the series on the right-hand side convergent in the norm of the Banach space $X.$ It can be proved that an additive vector measure $\backslash mu$ is countably additive if and only if for any sequence $(A\_i)\_^$ as above one has : $\backslash lim\_\backslash left\backslash |\backslash mu\backslash left(\backslash displaystyle\backslash bigcup\_^\backslash infty\; A\_i\backslash right)\backslash right\backslash |=0,\; \backslash quad\backslash quad\backslash quad\; ($ *) where $\backslash |\backslash cdot\backslash |$ is the norm on $X.$ Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval $[0,\; \backslash infty),$ the set of real numbers, and the set of complex numbers. 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「Vector 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.