Maximal ergodic theorem

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

Suppose that $(X,{ℬ},\mu )$ 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

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

Then the maximal ergodic theorem states that

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

for any λ ∈ R.

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

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