Maximal ergodic theorem

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

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

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

Then the maximal ergodic theorem states that

${\displaystyle \underset{f^{*}>\lambda }{\int }fd\mu \ge \lambda \cdot \mu \{f^{*}>\lambda \}}$

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, 48, pp. 248–251, doi:10.1214/074921706000000266, ISBN 0-940600-64-1 .

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