If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$.
But is it enough to conclude whether $f\in L_2(\mu)$ or $\lim_{n \to \infty}\int_{R}{|f_n(x)-f(x)|}^2<\infty$
What about the assumption change to $\sup\int_{R}{|f_n(x)|}^2d(\mu)<\infty$