# Maximal ergodic theorem

The **maximal ergodic theorem** is a theorem in ergodic theory, a discipline within mathematics.

Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation, and that . Define by

Then the maximal ergodic theorem states that

for any λ ∈ **R**.

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

## References[edit]

- 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.