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The '''maximal ergodic theorem''' is a [[theorem]] in [[ergodic theory]], a discipline within [[mathematics]].
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Suppose that <math>(X, \mathcal{B}, \mu)</math> is a [[probability space]], that <math>T : X \to X</math> is a (possibly noninvertible) [[measure-preserving transformation]], and that <math>f \in L^1(\mu)</math>. Define <math>f^*</math> by
:<math>f^* = \sup_{N\geq 1} \frac1N \sum_{i=0}^{N-1} f \circ T^i. </math>
Then the maximal ergodic theorem states that
:<math> \int_{f^* > \lambda} f \,d\mu \ge \lambda \cdot \mu\{ f^* > \lambda\} </math>
for any λ ∈ '''R'''.
 
This theorem is used to prove the point-wise [[ergodic theorem]].
 
== References ==
* {{citation | first1=Michael | last1=Keane | first2=Karl | last2=Petersen | year=2006 | title=Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem | journal=Institute of Mathematical Statistics Lecture Notes - Monograph Series | volume=48 | pages=248–251 | doi=10.1214/074921706000000266 | series=Institute of Mathematical Statistics Lecture Notes - Monograph Series | isbn=0-940600-64-1}}.
 
[[Category:Probability theorems]]
[[Category:Ergodic theory]]
[[Category:Theorems in dynamical systems]]
 
{{probability-stub}}

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