The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
Suppose that ( X , B , μ ) {\displaystyle (X,{\mathcal {B}},\mu )} is a probability space, that T : X → X {\displaystyle T:X\to X} is a (possibly noninvertible) measure-preserving transformation, and that f ∈ L 1 ( μ , R ) {\displaystyle f\in L^{1}(\mu ,\mathbb {R} )} . Define f ∗ {\displaystyle f^{*}} by
f ∗ = sup N ≥ 1 1 N ∑ i = 0 N − 1 f ∘ 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
∫ f ∗ > λ f d μ ≥ λ ⋅ μ { f ∗ > λ } {\displaystyle \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.
- 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.