  A number of recent works have sought to generalize the Kolmogorov-Sinai
entropy of probability-preserving transformations to the setting of Markov
operators acting on the integrable functions on a probability space $(X,\mu)$.
These have culminated in a proof by Downarovicz and Frej that these definitions
all coincide, and that the resulting quantity is uniquely characterized by
certain properties.
  On the other hand, Makarov has shown that this `operator entropy' is always
dominated by the Kolmogorov-Sinai entropy of a classical system that may be
constructed from a Markov operator, and that these numbers coincide under
certain extra assumptions. This note proves that equality in all cases.
