Maximal ergodic theorem

Let $(X,ℬ,\mu )$ be a probability space and $T:X\to X$ a measure preserving transformation. Let $f$ be a $L^1(\mu )$ function. Define the averages

$$f^{*}(x)=\underset{N\ge 1}{sup}\frac{1}{N}\sum _{i=0}^{N-1}f(T^i(x))$$

Then, for any $\lambda \in \text{𝐑}$, we have:

$${\int }_{f^{*}>\lambda }fd\mu \ge \lambda \mu (\{f^{*}>\lambda \})$$

This theorem may be used in the proof of the ergodic theorem (also known as Birkhoff ergodic theorem, or pointwise or strong ergodic theorem)

Title	Maximal ergodic theorem
Canonical name	MaximalErgodicTheorem
Date of creation	2014-03-19 22:15:48
Last modified on	2014-03-19 22:15:48
Owner	Filipe (28191)
Last modified by	Filipe (28191)
Numerical id	3
Author	Filipe (28191)
Entry type	Theorem
Related topic	birkhoff ergodic theorem