Martin-Löf random points satisfy Birkhoff’s ergodic theorem for effectively closed sets

We show that if a point in a computable probability space satisfies the ergodic recurrence property for a computable measure-preserving <!-- MATH $T\colon X\to X$ --> with respect to effectively closed sets, then it also satisfies Birkhoff's ergodic theorem for 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 classes. This answers a question of Hoyrup and Rojas.