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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 Measure-preserving dynamical system ： ウィキペディア英語版 Measure-preserving dynamical system In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. ==Definition== A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail, it is a system :$(X,\; \backslash mathcal,\; \backslash mu,\; T)$ with the following structure: *$X$ is a set, *$\backslash mathcal\; B$ is a σ-algebra over $X$, *$\backslash mu:\backslash mathcal\backslash rightarrow()$ is a probability measure, so that μ(''X'') = 1, and μ(∅) = 0, *$T:X\; \backslash rightarrow\; X$ is a measurable transformation which preserves the measure $\backslash mu$, i.e., $\backslash forall\; A\backslash in\; \backslash mathcal\backslash ;\backslash ;\; \backslash mu(T^(A))=\backslash mu(A)$. This definition can be generalized to the case in which ''T'' is not a single transformation that is iterated to give the dynamics of the system, but instead is a monoid (or even a group) of transformations ''Ts'' : ''X'' → ''X'' parametrized by ''s'' ∈ Z (or R, or N ∪ , or [0, +∞)), where each transformation ''Ts'' satisfies the same requirements as ''T'' above. In particular, the transformations obey the rules: * $T\_0\; =\; id\_X\; :X\; \backslash rightarrow\; X$, the identity function on ''X''; * $T\_\; \backslash circ\; T\_\; =\; T\_$, whenever all the terms are well-defined; * $T\_^\; =\; T\_$, whenever all the terms are well-defined. The earlier, simpler case fits into this framework by defining''Ts'' = ''Ts'' for ''s'' ∈ N. The existence of invariant measures for certain maps and Markov processes is established by the Krylov–Bogolyubov theorem. 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「Measure-preserving dynamical 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.