This is the version of Beppo Levi's theorem I have:
If $f_{1}(x) \leq f_{2}(x) \leq f_{3}(x) \leq ...$ in all points of $A$, $f_{n}$ are Lebesgue-measurables and Lebesgue-integrables, $\forall n \in \mathbb{N}$ and $|\int_{A} fd\mu \leq K < \infty$ Then $f(x)=\lim_{n \rightarrow \infty} f_{n}(x)$ is a finite number in most point of $A$, $f$ is Lebesgue-integrable in $A$ and $\int_{A}fd\mu=\lim_{n \rightarrow \infty} \int_{A} f_{n} d \mu$
I need to use but I don't know where in this problem, becuase how can I relate the limit what these sums.
Problem: If $\phi_{n} \geq 0$ are Lebesgue-measurable and Lebesgue-integrable and $\sum_{n=1}^{\infty} \int _{A} \phi_{n}d\mu < \infty$. Then $\sum_{n=1}^{\infty} \phi_{n}$ are Lebesgue-integrable and $\int_{A} (\sum_{n=1}^{\infty} \phi_{n})d\mu=\sum_{n=1}^{\infty} \int _{A} \phi_{n} d\mu$
Just take $f_n=\phi_1+\phi_2+...+\phi_n$.