Can you tell me where the mistake is?
If $(f_n) \in L^1$ is a sequence of functions such that $\sum_n \|f_n\|_1 < \infty$ I can prove that $f(x) = \sum_{n=1}^\infty f_n(x) < \infty$ for all $x \in X$.
To this end, let $n \in \mathbb N$, $x \in X$. Then $ \sum_{k=1}^n f_k (x) \leq \sum_{k=1}^n |f_k (x) | \leq \sum_{k=1}^n \int_X |f_k (x) | d \mu$
Hence $ f(x) = \lim_{n \to \infty} \sum_{k=1}^n f_k (x) \leq \lim_{n \to \infty} \sum_{k=1}^n \int_X |f_k (x) | d \mu < \infty$
Since $x$ was arbitrary we have $f(x) < \infty$ for all $x \in X$.
This proof must be wrong since we should have $f(x) < \infty$ almost everywhere.