Maximal ergodic theorem

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.

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.