  In this paper, we study discrete spectrum of invariant measures for countable
discrete amenable group actions.
  We show that an invariant measure has discrete spectrum if and only if it has
bounded measure complexity. We also prove that, discrete spectrum can be
characterized via measure-theoretic complexity using names of a partition and
the Hamming distance, and it turns out to be equivalent to both mean
equicontinuity and equicontinuity in the mean.
