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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 Lévy–Prokhorov metric ： ウィキペディア英語版 Lévy–Prokhorov metric In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric. ==Definition== Let $(M,\; d)$ be a metric space with its Borel sigma algebra $\backslash mathcal\; (M)$. Let $\backslash mathcal\; (M)$ denote the collection of all probability measures on the measurable space $(M,\; \backslash mathcal\; (M))$. For a subset $A\; \backslash subseteq\; M$, define the ε-neighborhood of $A$ by :$A^\; :=\; \backslash \; =\; \backslash bigcup\_\; B\_\; (p).$ where $B\_\; (p)$ is the open ball of radius $\backslash varepsilon$ centered at $p$. The Lévy–Prokhorov metric $\backslash pi\; :\; \backslash mathcal\; (M)^\; \backslash to\; [0,\; +\; \backslash infty)$ is defined by setting the distance between two probability measures $\backslash mu$ and $\backslash nu$ to be :$\backslash pi\; (\backslash mu,\; \backslash nu)\; :=\; \backslash inf\; \backslash left\backslash \; \backslash \; \backslash nu\; (A)\; \backslash leq\; \backslash mu\; (A^)\; +\; \backslash varepsilon\; \backslash \; \backslash text\; \backslash \; A\; \backslash in\; \backslash mathcal(M)\; \backslash right\backslash \}.$ For probability measures clearly $\backslash pi\; (\backslash mu,\; \backslash nu)\; \backslash le\; 1$. Some authors omit one of the two inequalities or choose only open or closed $A$; either inequality implies the other, and $(\backslash bar)^\backslash varepsilon\; =\; A^\backslash varepsilon$, but restricting to open sets may change the metric so defined (if $M$ is not Polish). 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「Lévy–Prokhorov metric」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.