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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 Doob–Dynkin lemma ： ウィキペディア英語版 Doob–Dynkin lemma In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the $\backslash sigma$-algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the $\backslash sigma$-algebra generated by the other. The lemma plays an important role in the conditional expectation in probability theory, where it allows to replace the conditioning on a random variable by conditioning on the $\backslash sigma$-algebra that is generated by the random variable. ==Statement of the lemma== Let $\backslash Omega$ be a sample space. For a function $f:\backslash Omega\; \backslash rightarrow\; R^n$, the $\backslash sigma$-algebra generated by $f$ is defined as the family of sets $f^(S)$, where $S$ are all Borel sets. Lemma Let $X,Y:\; \backslash Omega\; \backslash rightarrow\; R^n$ be random elements and $\backslash sigma(X)$ be the $\backslash sigma$ algebra generated by $X$. Then $Y$ is $\backslash sigma(X)$-measurable if and only if $Y=g(X)$ for some Borel measurable function $g:R^n\backslash rightarrow\; R^n$. The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one. By definition, $Y$ being $\backslash sigma(X)$-measurable is the same as $Y^(S)\backslash in\; \backslash sigma(X)$ for any Borel set $S$, which is the same as $\backslash sigma(Y)\; \backslash subset\; \backslash sigma(X)$. So, the lemma can be rewritten in the following, equivalent form. Lemma Let $X,Y:\; \backslash Omega\; \backslash rightarrow\; R^n$ be random elements and $\backslash sigma(X)$ and $\backslash sigma(Y)$ the $\backslash sigma$ algebras generated by $X$ and $Y$, respectively. Then $Y=g(X)$ for some Borel measurable function $g:R^n\backslash rightarrow\; R^n$ if and only if $\backslash sigma(Y)\; \backslash subset\; \backslash sigma(X)$. 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「Doob–Dynkin lemma」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.