Absolute continuity and convolution

Question

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt $\nu$? As far as I understand, this is a consequence of the fact that $\chi_e\mu\ast\mu\le\mu\ast\mu$ for any measurable set $e$. What can be said about the density $\frac{d(f\mu\ast g\mu)}{d\nu}$? Can this fact be found in a textbook?