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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 : with the following structure: * is a set, * is a σ-algebra over , * is a probability measure, so that μ(''X'') = 1, and μ(∅) = 0, * is a measurable transformation which preserves the measure , i.e., . 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: * , the identity function on ''X''; * , whenever all the terms are well-defined; * , 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
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