Loeb空间的遍历定理与动力系统的超有限逼近
Ergodic theorem for a Loeb space and hyperfinite approximations of dynamical systems
摘要：
虽然G.Birkhoff遍历定理（BET）对于有限空间是平凡的，但这对于超有限Loeb空间并没有帮助。T. Kamae，实际上，适用于任意概率空间，如Y所示。Katznelson和B.外斯本文讨论了通常的方法，即利用非标准分析技术，将任意大的有限空间的一些简单事实转移到无限空间上，对BET不起作用的原因。我们证明了超有限空间的BET可以解释为非常大的有限空间的一些定性结果。引入了动力系统超有限逼近的概念，并证明了超有限逼近的存在性。用有限动力系统的序列形式给出了所得结果的标准形式。
英文摘要：
Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces, this does not help in proving it for hyperfinite Loeb spaces. The proof of the BET for this case, suggested by T. Kamae, works, actually, for arbitrary probability spaces, as it was shown by Y. Katznelson and B. Weiss. In this paper we discuss the reason why the usual approach, based on transfer of some simple facts about arbitrary large finite spaces on infinite spaces using nonstandard analysis technique, does not work for the BET. We show that the the BET for hyperfinite spaces may be interpreted as some qualitative result for very big finite spaces. We introduce the notion of a hyperfinite approximation of a dynamical system and prove the existence of such an approximation. The standard versions of the results obtained in terms of sequences of finite dynamical systems are formulated.
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