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Where can I find the proof of Von Neumann's ergodic theorem? Please, give me references or write names of books where I can find it.

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The original proof (pdf) is in

J. von Neumann, "Proof of the quasi-ergodic hypothesis" Proc. Nat. Acad. Sci. USA , 18 (1932) pp. 70–82

A simpler proof is due to Halmos, found in

P.R. Halmos, "Lectures on ergodic theory" , Math. Soc. Japan (1956)

I believe this book is a reprinting of those lectures.

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    $\begingroup$ An alternative name for the same result is the mean ergodic theorem and there's a proof in almost every book on ergodic theory. Terry Tao has a blog post on it. $\endgroup$
    – Martin
    Jan 24, 2013 at 1:25
  • $\begingroup$ Due to riesz but it is mentiond in Halmos's lectures. $\endgroup$ Jan 2, 2019 at 20:16
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If you want more modern books I recommend P. Walters or K. Petersen books on Ergodic Theory.

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Einsiedler. Ward. Ergodic Theory with a view towards Number Theory. P. 32.

Petersen. Ergodic Theory. P. 23.

Brin. Sturk. An Introduction to Dynamical Systems. P. 80.

Krengel. Ergodic Theorems. P. 4.

Halmos. Lectures on Ergodic Theory. P. 16.

Coudene. Ergodic Theory and Dynamical Systems. P. 4.

Eisner. Farkas. Haase. Nagel. Operator Theoretic Aspects of Ergodic Theory. P. 139.

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