Showing that a sequence converges almost everywhere

Question

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?

Answer 1

We may and do assume that $f=0.$ We have to proof that $f_n\to 0$ a.e. (or, what's the same, $|f_n|\to0$ a.e.) if $\sum ||f_n||_1<\infty.$ Since the series $\sum |f_n|$ converges in $L_1$ (which is complete), we have $\sum |f_n|<\infty$ a.e.; hence, $|f_n|\to 0$ a.e.