Maximal ergodic theorem

The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that ${\displaystyle (X,\mathcal{B},\mu )}$ 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 ${\displaystyle f^{\ast }}$ by

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

Then the maximal ergodic theorem states that

${\displaystyle {\int }_{f^{\ast }>\lambda }fd\mu \ge \lambda \cdot \mu \{f^{\ast }>\lambda \}}$

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 .

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