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Entropy of probability kernels from the backwards tail boundary

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 $(X,μ)$. 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.