Maximal ergodic theorem

Last updated July 20, 2022

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

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

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

Then the maximal ergodic theorem states that

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

for any λ ∈ R.

This theorem is used to prove the point-wise 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", 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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