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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!

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    $f$ 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 $f \in \mathcal C$.2012-12-14
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    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 $f\mathcal{C}$.2012-12-15
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    I think Daniel refers to a continuous version of $f$. In other words a $g \in \mathcal{C}$ such that $g = f$ $\mu-$a.e.2018-07-02

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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 $x$.