# Maximal ergodic theorem

Maximal ergodic theorem The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.

Suppose that {displaystyle (X,{mathcal {B}},mu )} is a probability space, that {displaystyle T:Xto X} is a (possibly noninvertible) measure-preserving transformation, and that {displaystyle fin L^{1}(mu ,mathbb {R} )} . Define {displaystyle f^{*}} by {displaystyle f^{*}=sup _{Ngeq 1}{frac {1}{N}}sum _{i=0}^{N-1}fcirc T^{i}.} Then the maximal ergodic theorem states that {displaystyle int _{f^{*}>lambda }f,dmu geq lambda cdot mu {f^{*}>lambda }} for any λ ∈ R.

This theorem is used to prove the point-wise ergodic theorem.

References Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Dynamics & Stochastics, Institute of Mathematical Statistics Lecture Notes - Monograph Series, vol. 48, pp. 248–251, arXiv:math/0004070, doi:10.1214/074921706000000266, ISBN 0-940600-64-1.

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