Maximal ergodic theorem

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

Suppose that ${\displaystyle (X,{\mathcal {B}},\mu )}${\displaystyle (X,{\mathcal {B}},\mu )} is a probability space, that ${\displaystyle T:X\to X}${\displaystyle T:X\to X} is a (possibly noninvertible) measure-preserving transformation, and that ${\displaystyle f\in L^{1}(\mu ,\mathbb {R} )}${\displaystyle f\in L^{1}(\mu ,\mathbb {R} )}. Define ${\displaystyle f^{*}}${\displaystyle f^{*}} by

${\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.}${\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.}

Then the maximal ergodic theorem states that

${\displaystyle \int _{f^{*}>\lambda }f,円d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}}${\displaystyle \int _{f^{*}>\lambda }f,円d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}}

for any λ ∈ R.

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

• 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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