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This is a joint talk with Henry Towsner. In our 2014 paper "Randomness and Non-ergodic Systems", we showed that, given a non-Martin-Loef random point, one can construct a computable measure-preserving (but non-ergodic) transformation and a computable function for which the point fails to satisfy Birkhoff's ergodic theorem. A natural extension is to ask whether, given a weaker assumption, one can do the same with a lower semicomputable function. We summarize the technique used in our paper and then discuss the obstacles we have encountered in extending this approach to a weaker assumption.