The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
Suppose that is a probability space, that is a (possibly noninvertible) measurepreserving transformation, and that . Define by
Then the maximal ergodic theorem states that
for any λ ∈ R.
This theorem is used to prove the pointwise ergodic theorem.
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References
 Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Institute of Mathematical Statistics Lecture Notes  Monograph Series, Institute of Mathematical Statistics Lecture Notes  Monograph Series, 48: 248–251, arXiv:math/0004070, doi:10.1214/074921706000000266, ISBN 0940600641.