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…)