We state and prove the Ergodic theorems for almost everywhere convergence and for convergence in Lp, which are attributed to Birkhoff and Von Neumann, respectively. These require the notion of a measure preserving map and ergodicity.
My typed lecture notes can be found at https://sites.ualberta.ca/~kashlak/kashTeaching.html
For example, see https://en.wikipedia.org/wiki/Ergodic_theory#Probabilistic_formulation:_Birkhoff%E2%80%93Khinchin_theorem