Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets
Johanna N. Y. Franklin, Noam Greenberg, Joseph S. Miller, Keng Meng Ng · Proceedings of the American Mathematical Society · 2012
We show that if a point in a computable probability space $X$ satisfies the ergodic recurrence property for a computable measure-preserving $T\colon X\to X$ with respect to effectively closed sets, then it also satisfies Birkhoff’s ergodic theorem for $T$ with respect to effectively closed sets. As a corollary, every Martin-Löf random sequence in the Cantor space satisfies Birkhoff’s ergodic theorem for the shift operator with respect to $\Pi ^0_1$ classes. This answers a question of Hoyrup and Rojas.