Let $f_n,f\in\mathcal{L}^1$. Assume that
$$\sum_n\int|f_n-f|<\infty$$
I am trying to show that $f_n\rightarrow f$ almost everywhere. I have been trying to relate the set $$\{x\in X\mid f_n(x)\text{ does not converge to } f(x)\}$$ with sets of the form $$A_{n,k}=\{x\in X\mid |f_n(x)-f(x)|\geq \frac{1}{k}\}$$ since those can, somewhat, be related to the sum by Chebychev's theorem $$\mu(A_{n,k})\leq k\int|f-f_n|$$
Although I could not get much further this way, am I going in the right direction?