Sum of Integrals and $L^{p}$ Spaces

Question

Let $(X,\mathfrak{M},\mu)$ be a finite measure space and let $\{f_{n}\}$ be a sequence of real-valued measurable functions on $X$. Assume that

1. $F(x)=\sum_{n=1}^{\infty}f_{n}(x)$ exists for almost every $x\in X$;
2. $G(x)=\sum_{n=1}^{\infty}|f_{n}(x)|$ belongs to $L^{p}(\mu)$ for some $p\geq 1$.

True or False: $F\in L^{1}(\mu)$ and $\int_{X}F~d\mu=\sum_{n=1}^{\infty}\int_{X}f_{n}~d\mu$?

I think that the problem is true, since $G$ belongs to some $L^{p}$ space. And functions are generally well-behaved on finite measure spaces. However, I'm note sure if this is right, and if it is I'm not sure how to prove it if it is. Any help is appreciated.

Answer 1

Let $F_N\colon x\mapsto \sum_{n=1}^Nf_n(x)$ and $g_N\colon x\mapsto \sum_{n=1}^N\left\lvert f_n(x)\right\rvert$. By the monotone convergence theorem, $g_N\to g$ in $\mathbb L^1$ and $\sum_{n=1}^{+\infty}\left\lVert f_n\right\rVert_1$ is finite hence we can apply the dominated convergence theorem to the sequence $\left(F_N\right)_{N\geqslant 1}$.