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

: $\int_{f^{*} > \lambda} f \, d\mu \ge \lambda \cdot \mu\{ f^{*} > \lambda\}$

for any λ ∈ R.

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

References

{{citation | first1=Michael | last1=Keane | first2=Karl | last2=Petersen | year=2006 | title= Dynamics & Stochastics| volume=48 | pages=248–251 | doi=10.1214/074921706000000266 | series=Institute of Mathematical Statistics Lecture Notes - Monograph Series | isbn=0-940600-64-1| arxiv=math/0004070 | chapter=Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem }}.