Let $X_1, X_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $f_1, f_2, \dots, f$ be those density functions $\mathbb{R} \to \mathbb{R}$:
$$X_i(E) = \int_E f_i d\mu$$
Does the convergence of $X_1, X_2, \dots$ in $L^p(\Omega)$, i.e. $\int_\Omega|X - X_i|^pdP \xrightarrow{n\to \infty}0$, mean the convergence of $f_1,f_2,\dots$ in $L^p(\mathbb{R})$?