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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 Dunford–Pettis theorem ： ウィキペディア英語版 Uniform integrability Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability. ==Measure theoretic definition== Textbooks on real analysis and measure theory often use the following definition. Let $(X,\backslash mathfrak,\; \backslash mu)$ be a positive measure space. A set $\backslash Phi\backslash subset\; L^1(\backslash mu)$ is called uniformly integrable if to each $\backslash epsilon>0$ there corresponds a $\backslash delta>0$ such that whenever $f\; \backslash in\; \backslash Phi$ and ==Probability definition== In the theory of probability, the following definition applies. * A class $\backslash mathcal$ of random variables is called uniformly integrable (UI) if given $\backslash epsilon>0$, there exists $K\backslash in[0,\backslash infty)$ such that $E(|X|I\_)\backslash le\backslash epsilon\backslash \; \backslash text\; \backslash in\; \backslash mathcal$, where $I\_$ is the indicator function . * An alternative definition involving two clauses may be presented as follows: A class $\backslash mathcal$ of random variables is called uniformly integrable if: * * There exists a finite $M$ such that, for every $X$ in $\backslash mathcal$, $E(|X|)\backslash leq\; M$. * * For every $\backslash epsilon\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that, for every measurable $A$ such that $P(A)\backslash leq\; \backslash delta$ and every $X$ in $\backslash mathcal$, $E(|X|:A)\backslash leq\backslash epsilon$. The two probabilistic definitions are equivalent. 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「Uniform integrability」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.