# Maximal ergodic theorem

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

 $f^{*}(x)=\sup_{N\geq 1}\frac{1}{N}\sum_{i=0}^{N-1}f(T^{i}(x))$

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

 $\int_{f^{*}>\lambda}fd\mu\geq\lambda\mu(\{f^{*}>\lambda\})$

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 MaximalErgodicTheorem 2014-03-19 22:15:48 2014-03-19 22:15:48 Filipe (28191) Filipe (28191) 3 Filipe (28191) Theorem birkhoff ergodic theorem