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)