#jsDisabledContent { display:none; } My Account | Register | Help

# Maximal ergodic theorem

Article Id: WHEBN0010050297
Reproduction Date:

 Title: Maximal ergodic theorem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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). Define f^* by

f^* = \sup_{N\geq 1} \frac1N \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

• Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Institute of Mathematical Statistics Lecture Notes - Monograph Series, Institute of Mathematical Statistics Lecture Notes - Monograph Series 48: 248–251, .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.