# Countable-to-one factors of measure preserving systems do not change entropy

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.