Let ${X}$ be a measure space with measure ${\mu}$; let ${T: X \rightarrow X}$ be a measure-preserving transformation. Last time we looked at how the averages

$\displaystyle A_N := \frac{1}{N} \sum_{i=0}^{N-1} f \circ T^i$

behave in ${L^2}$. But, now we want pointwise convergence.

The pointwise ergodic theorem

We consider the pointwise ergodic theorem of Garrett George Birkhoff:

Theorem 1 (Birkhoff) Let ${f \in L^1(\mu)}$. Then the averages ${A_N}$ converge almost everywhere to a function ${f^* \in L^1(\mu)}$ with ${f^* \circ = f^*}$ a.e. (more…)