On the book Dynamical System and Ergodic theory by Pollicott and Yuri there is a proof of the Rudolph x2 x3 theorem (page 153). It looks very clean in comparison with the original proof but I didn't understand when they calculated the conditional expectation $\mathbb{E}(f|T^{-n}\mathcal{B})$. They used the notation $$T'(x) = \frac{d\mu T}{d\mu} (x)$$ for the Radon-Nykodim derivative but $\mu T$ is not even a measure because $T$ is not invertible. So, I didn't understand the notations or I'am missing something.
Any help will be appreciated.
The notation $T'$ suggests that $\mu T$ is the subadditive functional $B\mapsto\mu(T(B))$. Under some continuity hypothesis, versions of Radon-Nykodym theorem still hold in this context.
See the paper A Radon-Nikodym theorem for capacities, Siegfried Graf, Journal für die Reine und Angewandte Mathematik 320 (1980) 192-214.