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New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem. (English) Zbl 0551.28018

The maximal ergodic theorem has been stated as follows: Let \(f\in L^ 1(X)\) and define \(f^*(x)=\sup_{n}\sum^{n-1}_{k=0}f(T^ kx).\) Then \(\int_{\{f^*\geq 0\}}f(x)dm(x)\geq 0.\) A new proof has been presented and the method used in the proof has been used to study the Hardy-Littlewood maximal function.
Reviewer: Sh.Singh

MSC:

28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
42B25 Maximal functions, Littlewood-Paley theory
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References:

[1] Paul R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956. · Zbl 0073.09302
[2] P. C. Shields, A simple, direct proof of Birkhoff’s ergodic theorem, preprint.
[3] A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968.
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