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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629501.png" /> is an [[Endomorphism|endomorphism]] of a measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629502.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629503.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629504.png" /> is the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629505.png" /> for which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629506.png" /></td> </tr></table>
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{{MSC|37A30}}
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[[Category:Ergodic theorems, spectral theory, Markov operators]]
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If  $  T $
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is an [[Endomorphism|endomorphism]] of a measure space  $  ( X , \mu ) $,
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if  $  f \in L _ {1} ( X , \mu ) $
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and if  $  E $
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is the set of  $  x \in X $
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for which
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$$
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\sup _ {n \geq  0 } \sum_{i=0} ^ { n } f ( T ^ { i } x )  \geq  0 ,
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$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062950/m0629507.png" /></td> </tr></table>
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$$
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\int\limits _ { E } f  d \mu  \geq  0.
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$$
  
The maximal ergodic theorem is due to K. Yosida and S. Kakutani [[#References|[1]]], who showed that it can play a central role in the proof of the [[Birkhoff ergodic theorem|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 [[#References|[2]]]). See also [[#References|[3]]] and the references in [[Birkhoff ergodic theorem|Birkhoff ergodic theorem]].
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The maximal ergodic theorem is due to K. Yosida and S. Kakutani {{Cite|YK}}, who showed that it can play a central role in the proof of the [[Birkhoff ergodic theorem|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 {{Cite|N}}). See also {{Cite|VY}} and the references in [[Birkhoff ergodic theorem|Birkhoff ergodic theorem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,   S. Kakutani,   "Birkhoff's ergodic theorem and the maximal ergodic theorem" ''Proc. Imp. Acad. Tokyo'' , '''15''' (1939) pp. 165–168</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Neveu,   "Mathematical foundations of the calculus of probability" , Holden-Day (1965) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"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</TD></TR></table>
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{|
 
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|valign="top"|{{Ref|YK}}|| K. Yosida, S. Kakutani, "Birkhoff's ergodic theorem and the maximal ergodic theorem" ''Proc. Imp. Acad. Tokyo'' , '''15''' (1939) pp. 165–168 {{MR|355}} {{ZBL|}}
 
+
|-
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|valign="top"|{{Ref|N}}|| J. Neveu, "Mathematical foundations of the calculus of probability" , Holden-Day (1965) (Translated from French) {{MR|0198505}} {{ZBL|0137.11301}}
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|-
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|valign="top"|{{Ref|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 {{MR|0286981}} {{ZBL|0252.28006}}
 +
|}
  
 
====Comments====
 
====Comments====
A variety of ergodic theorems (including historical remarks) can be found in [[#References|[a1]]].
+
A variety of ergodic theorems (including historical remarks) can be found in {{Cite|K}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Krengel,   "Ergodic theorems" , de Gruyter (1985)</TD></TR></table>
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{|
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|valign="top"|{{Ref|K}}|| U. Krengel, "Ergodic theorems" , de Gruyter (1985) {{MR|0797411}} {{ZBL|0575.28009}}
 +
|}

Latest revision as of 08:12, 9 January 2024


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

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

then

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

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.

References

[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

Comments

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

References

[K] U. Krengel, "Ergodic theorems" , de Gruyter (1985) MR0797411 Zbl 0575.28009
How to Cite This Entry:
Maximal ergodic theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_ergodic_theorem&oldid=17718
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article