Maximal ergodic theorem

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2020 Mathematics Subject Classification: Primary: 37A30 [MSN][ZBL]

If $ T $ is an endomorphism of a measure space $ ( X , \mu ) $, if $ f \in L _ {1} ( X , \mu ) $ and if $ E $ is the set of $ x \in X $ for which

$$ \sup _ {n \geq 0 } \sum_{i=0} ^ { n } f ( T ^ { i } x ) \geq 0 , $$


$$ \int\limits _ { E } f d \mu \geq 0. $$

The maximal ergodic theorem is due to K. Yosida and S. Kakutani [YK], who showed that it can play a central role in the proof of the Birkhoff ergodic theorem (G.D. Birkhoff himself, instead of the maximal ergodic theorem, used somewhat different arguments). In later proofs of generalizations of Birkhoff's theorem (and also in related questions on the decomposition of the phase space into conservative and dissipative parts under conditions such that these generalizations make sense) a generalized maximal ergodic theorem is used in a similar way. There is a generalization of the maximal ergodic theorem due to E. Hopf and a simple proof of this generalization was given by A. García (see [N]). See also [VY] and the references in Birkhoff ergodic theorem.


[YK] K. Yosida, S. Kakutani, "Birkhoff's ergodic theorem and the maximal ergodic theorem" Proc. Imp. Acad. Tokyo , 15 (1939) pp. 165–168 MR355
[N] J. Neveu, "Mathematical foundations of the calculus of probability" , Holden-Day (1965) (Translated from French) MR0198505 Zbl 0137.11301
[VY] A.M. Vershik, S.A. Yuzvinskii, "Dynamical systems with invariant measure" Progress in Math. , 8 (1970) pp. 151–215 Itogi Nauk. Anal. (1967) pp. 133–187 MR0286981 Zbl 0252.28006


A variety of ergodic theorems (including historical remarks) can be found in [K].


[K] U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Zbl 0575.28009
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Maximal ergodic theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article