Abstract
A simple proof of the individual ergodic theorem is given. The essential tool is the nonstandard measure theory developed by P. Loeb. Any dynamical system on an abstract Lebesgue space can be represented as a factor of a “cyclic” system with a hyperfinite cycle. The ergodic theorem for such a “cyclic” system is almost trivial because of its simple structure. The general case follows from this special case.
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References
M. Davis,Applied Nonstandard Analysis, John Wiley & Sons, New York, 1977.
P. Loeb,Conversion from non-standard to standard measure space and applications to probability theory, Trans. Am. Math. Soc.211 (1975).
J. Ville,Etude Critique de la Notion de Collectif, Paris, 1939.
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Kamae, T. A simple proof of the ergodic theorem using nonstandard analysis. Israel J. Math. 42, 284–290 (1982). https://doi.org/10.1007/BF02761408
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DOI: https://doi.org/10.1007/BF02761408