I was reading a work and an apparently simple proof is causing me technical troubles. The lemma is about differential entropy and change of measures.
Assume random variables with continuous and positive densities.
\begin{equation} h(T(X)) = h(X) + E[\log(T'(X))], \end{equation} where $T$ is an increasing function (transportation map) such that $T(X^*)$ and $X$ have the same distribution.
The proof is quite short and goes like this:
We have that $p_{T(X)}(T(x))dT(x)= p_X(x)dx$ and thus: $$h(T(X))= - E[\log(p_{T(X)}(T(X)))] = -E\left[\log\left(\frac{p_X(X)}{T'(X)}\right)\right].$$
I tried to fill in the gaps, and the missing step would be: $$ P_{T(X)}(T(X))= P_X(X)\frac{dX}{dT(X)}, $$ but then, is it true that $$ \frac{dX}{dT(X)} = \frac{1}{T'(X)}?$$ Why? Does, in this case, the Radon-Nikodym derivative coincide with the usual derivative? Which technical steps am I missing?