“…Indeed, the work of Gács, Galatolo, Hoyrup, and Rojas [68,67], together with the results of [8], shows that the Schnorr random reals are exactly the ones at which the ergodic theorem holds with respect to every ergodic measure and computable measure-preserving transformation (see also the discussion in [166]). Bienvenu, Day, Hoyrup, and Shen [15] and, independently, Franklin, Greenberg, Miller, and Ng [59] have extended V'yugin's result to the characteristic function of a computably open set, in the case where the measure is ergodic.…”