Maximal ergodic theorem

The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics. Suppose that [math]\displaystyle{ (X, \mathcal{B},\mu) }[/math] is a probability space, that [math]\displaystyle{ T : X\to X }[/math] is a (possibly noninvertible) measure-preserving transformation, and that [math]\displaystyle{ f\in L^1(\mu,\mathbb{R}) }[/math]. Define [math]\displaystyle{ f^* }[/math] by

[math]\displaystyle{ f^* = \sup_{N\geq 1} \frac{1}{N} \sum_{i=0}^{N-1} f \circ T^i. }[/math]

Then the maximal ergodic theorem states that

[math]\displaystyle{ \int_{f^{*} \gt \lambda} f \, d\mu \ge \lambda \cdot \mu\\{ f^{*} \gt \lambda\\} }[/math]

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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