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It is known that if $\psi$ is a factor map between probability measure preserving systems $(X,\mathscr{X},\mu,T)$ and $(Y,\mathscr{Y},\nu,S)$ is countable-to-one almost everywhere, then $h(\mu,T)=h(\nu,S)$. The only proof I am aware of is in Downarowicz's book ``Entropy in Dynamical Systems'' and relies on disintegration of measures, conditional Shannon-McMillan-Breiman Theorem and affinity of the entropy with respect to the ergodic decomposition (to cover non-ergodic case). I wonder whether a simpler, more direct proof of this fact exists.

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