I have seen in more than one place statements like: Let $P$ be a fixed measure and let $Q$ be the measure such that $\frac{dQ}{dP}= \phi$ for some function $\phi$. I just found this in a work: let $Q$ be such that $\frac{dQ}{dP}= \phi^2$ where $\phi:\mathbb{R}\to\mathbb{R}$ is a smooth function.
My question is then, which regularity conditions do one need to impose in order to have a valid Radon-Nikodym derivative? Smoothness, in what sense?