Maximal ergodic theorem

This article relies largely or entirely on a single source. Please help improve this article by introducing citations to additional sources.
Find sources: "Maximal ergodic theorem" – news · newspapers · books · scholar · JSTOR (March 2024)

|

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

Suppose that (X,ℬ,μ) is a probability space, that T:X->X is a (possibly noninvertible) measure-preserving transformation, and that f∈L1(μ,ℝ). Define f* by

f*=supN≥11N∑i=0N−1f∘Ti.

Then the maximal ergodic theorem states that

∫f*>λfdμ≥λ⋅μ{f*>λ}

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 .

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Maximal ergodic theorem. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Maximal_ergodic_theorem&oldid=4108849"