For instance, we know that if $\nu\ll\mu$ then there exists $f\in L^1$ s.t. $\nu = \int f d\mu$.
I wonder if, under extra conditions, we can say more about $f$, like $f\in \mathcal{C}$ or similar.
$f$ is not uniquely determined (for example, if $\mu$ is absolutely continuous it can have countably many arbitrary removable discontinuities), so I doubt you will be able to get a condition like $f \in \mathcal C$. – Jonathan Christensen Dec 14 '12 at 16:30
Yes, I know, that is precisely my question. If I can add more conditions maybe to $\mu$ or $\nu$ or anything else in order to get $f\mathcal{C}$. – user53215 Dec 15 '12 at 17:18