I'm trying to prove that if holomorphic sequence of functions $f_n$ converge to $f$ on closed disks in the $L_1(\Omega)$ sense on an open $\Omega$ and $f$ is continuous, then we can deduce that $f$ is holomorphic and that the convergence is uniform on compacta. So far, some facts I've been able to deduce is that the $L_1$ convergence on a closed disk $D$ allows us to find a subsequence that converges a.e. to $f$. Convergence a.e. then implies, by Egoroff's theorem, uniform convergence on open subsets of $D$ of measure arbitrarily close to the measure of $D$ and thus we can conclude that $f$ is analytic a.e. in $D$ (and specifically on an open set in $D$ with measure zero complement).
Beyond this, I don't know how to show anything else (e.g. showing convergence of all $f_n$ not a subsequence, convergence on compacta or even analyticity). I tried to use the fact that continuity on a line or at isolated points implies analyticity at those points, but not sure how that extends to some measure zero set in $D$.