Let $X,Y$, be compact complex manifolds, and let $f:X\to Y$ be a smooth, proper (i.e. for each $y\in Y$, $f^{-1}(y)$ is a compact set) and surjective map. Choose metrics on $X,Y$ and let $\mu_X, \mu_Y$ be the corresponding measures. Define the measure $\nu$ on $Y$ by $\nu(A)= \mu_X(f^{-1}(A))$ for any measurable $A\subset Y$. Then $\mu_Y << \nu$, hence by Radon-Nykodym theorem, there exists a measurable function $g:Y\to [0,\infty)$ so that for any measurable $A\subset Y$, we have $ \mu_Y(A) = \int_A g d\nu $.
My question is, when can we say that $g$ is bounded almost everywhere?(e.g. can we choose metrics on $X,Y$ in a way so that $g$ would be bounded?) What if $X,Y$ are no longer compact? Then what conditions are needed so that $g$ would be bounded?
