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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 Dynkin system ： ウィキペディア英語版 Dynkin system A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $\backslash Omega$ satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability. The primary relevance of λ-systems are their use in applications of the π-λ theorem. == Definitions == Let Ω be a nonempty set, and let $D$ be a collection of subsets of Ω (i.e., $D$ is a subset of the power set of Ω). Then $D$ is a Dynkin system if # Ω ∈ $D$, # if ''A'', ''B'' ∈ $D$ and ''A'' ⊆ ''B'', then ''B'' \ ''A'' ∈ $D$, # if ''A''1, ''A''2, ''A''3, ... is a sequence of subsets in $D$ and ''A''''n'' ⊆ ''A''''n''+1 for all ''n'' ≥ 1, then $\backslash bigcup\_^\backslash infty\; A\_n\backslash in\; D$. Equivalently, $D$ is a Dynkin system if # Ω ∈ $D$, # if ''A'' ∈ ''D'', then ''A''c ∈ ''D'', # if ''A''1, ''A''2, ''A''3, ... is a sequence of subsets in $D$ such that ''A''''i'' ∩ ''A''''j'' = Ø for all ''i'' ≠ ''j'', then $\backslash bigcup\_^\backslash infty\; A\_n\backslash in\; D$. The second definition is generally preferred as it usually is easier to check. An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra. This can be verified by noting that condition 3 and closure under finite intersection implies closure under countable unions. Given any collection $\backslash mathcal$ of subsets of $\backslash Omega$, there exists a unique Dynkin system denoted $D\backslash$ which is minimal with respect to containing $\backslash mathcal\; J$. That is, if $\backslash tilde\; D$ is any Dynkin system containing $\backslash mathcal\; J$, then $D\backslash \backslash subseteq\backslash tilde\; D$. $D\backslash$ is called the Dynkin system generated by $\backslash mathcal$. Note $D\backslash =\backslash$. For another example, let $\backslash Omega=\backslash$ and $\backslash mathcal\; J=\backslash$; then $D\backslash =\backslash ,\backslash Omega\backslash \}$. 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「Dynkin system」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.