Ferenczi studied measure-theoretic complexity using α-names of a partition and the Hamming distance. He proved that if a measure preserving system is ergodic, then the complexity function is bounded if and only if the system has discrete spectrum. We show that this result holds without the assumption of ergodicity.

Measure complexity with respect to a function is also introduced. For a function f, it is shown that f is an almost periodic function if and only if f is measure-theoretic mean equicontinuous if and only if the measure-theoretic complexity with respect to f is bounded.

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