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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 Pettis integral ： ウィキペディア英語版 Pettis integral In mathematics, the Pettis integral or Gelfand–Pettis integral, named after I. M. Gelfand and B. J. Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral. ==Definition== Suppose that $f\backslash colon\; X\backslash to\; V$, where $(X,\backslash Sigma,\backslash mu)$ is a measure space and $V$ is a topological vector space. Suppose that $V$ admits a dual space $V^$ * that separates points. e.g., $V$ a Banach space or (more generally) a locally convex, Hausdorff vector space. We write evaluation of a functional as duality pairing: $\backslash langle\; \backslash varphi,\; x\; \backslash rangle\; =\; \backslash varphi()$. Choose any measurable set $E\; \backslash in\; \backslash Sigma$. We say that $f$ is Pettis integrable (over $E$) if there exists a vector $e\; \backslash in\; V$ so that : $\backslash langle\; \backslash varphi,\; e\backslash rangle\; =\; \backslash int\_E\; \backslash langle\; \backslash varphi,\; f(x)\; \backslash rangle\; \backslash ,\; d\backslash mu(x)\backslash text\backslash varphi\backslash in\; V^$ *. In this case, we call $e$ the Pettis integral of $f$ (over $E$). Common notations for the Pettis integral $e$ include $\backslash int\_E\; f\; \backslash mu$, $\backslash int\_E\; f(t)\; \backslash ,\; d\backslash mu(t)$ and $\backslash mu(1\_E)$. A function is Pettis integrable (over $X$) if the scalar-valued function $\backslash varphi\; \backslash circ\; f$ is integrable for every functional $\backslash varphi\; \backslash in\; X^$ *. 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「Pettis integral」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.