TITLE: Continuity criteria for Radon-Nikodym derivative
QUESTION [6 upvotes]: I have been looking for results or theorems which give me regularity conditions of the Radon-Nikodym derivative, but I have not found any :(
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.
Are there results? Thank you very much for any help!

REPLY [3 votes]: As noted above, the lack of pointwise definedness of RN-derivatives means that this question should maybe be stated, "when does a version of the RN-derivative exist that has property $P$" where $P$ might be membership in $C^{k}$ or something.  Here's an example of this sort of result (for $k=0$):
https://mathscinet.ams.org/mathscinet-getitem?mr=712855
M. Piccioni, "Continuous versions of Radon—Nikodym derivatives as likelihood ratios"
Here, the author shows that the likelihood ratio $\ell_{P,Q}(x)$ between two distributions $P$ and $Q$ (essentially, the limit of $Q(A)/P(A)$ as a net of open sets about $x$ converges) exists at $x$ iff the RN derivative $dQ/dP$ is continuous at $x$.