While I was summarizing the relations between different modes of convergence for random variables, I got stuck at the convergence of the densities... My question:
$X_n \to X$ a.s. (or in $L_1$/ in probability) $\quad \overset{?}{\Rightarrow} \quad f_{X_n} \to f_X$ a.e. (or in any other sense)
where $X,X_n: (\Omega,\mathcal{F},P)\to (\mathbb{R},\mathcal{B}(\mathbb{R}))$ measurable, with laws $\mu_{X_n},\mu_X << \lambda$ [Lebesgue measure] (in order to admit density functions $f_{X_n}, f_X$ on $\mathbb{R}$)