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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 Kolmogorov automorphism ： ウィキペディア英語版 Kolmogorov automorphism In mathematics, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero-one law.〔Peter Walters, ''An Introduction to Ergodic Theory'', (1982) Springer-Verlag ISBN 0-387-90599-5〕 All Bernoulli automorphisms are ''K''-automorphisms (one says they have the ''K''-property), but not vice versa. Many ergodic dynamical systems have been shown to have the ''K''-property, although more recent research has shown that many of these are in fact Bernoulli automorphisms. Although the definition of the ''K''-property seems reasonably general, it stands in sharp distinction to the Bernoulli automorphism. In particular, the Ornstein isomorphism theorem does not apply to ''K''-systems, and so the entropy is not sufficient to classify such systems – there exist uncountably many non-isomorphic ''K''-systems with the same entropy. In essence, the collection of ''K''-systems is large, messy and uncategorized; whereas the ''B''-automorphisms are 'completely' described by Ornstein theory. ==Formal definition== Let $(X,\; \backslash mathcal,\; \backslash mu)$ be a standard probability space, and let $T$ be an invertible, measure-preserving transformation. Then $T$ is called a ''K''-automorphism, ''K''-transform or ''K''-shift, if there exists a sub-sigma algebra $\backslash mathcal\backslash subset\backslash mathcal$ such that the following three properties hold: :$\backslash mbox\backslash mathcal\backslash subset\; T\backslash mathcal$ :$\backslash mbox\backslash bigvee\_^\backslash infty\; T^n\; \backslash mathcal=\backslash mathcal$ :$\backslash mbox\backslash bigcap\_^\backslash infty\; T^\; \backslash mathcal\; =\; \backslash$ Here, the symbol $\backslash vee$ is the join of sigma algebras, while $\backslash cap$ is set intersection. The equality should be understood as holding almost everywhere, that is, differing at most on a set of measure zero. 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「Kolmogorov automorphism」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.