Let be a measure space with measure
; let
be a measure-preserving transformation. Last time we looked at how the averages
behave in . But, now we want pointwise convergence.
The pointwise ergodic theorem
We consider the pointwise ergodic theorem of Garrett George Birkhoff:
Theorem 1 (Birkhoff) Let
. Then the averages
converge almost everywhere to a function
with
a.e. (more…)