Given that $f_n$ is an integrable function in $L(X,M,\mu)$ and that $$ \sum^\infty_1 \int |f_n| < \infty, $$ prove that $\sum f_n \to f\in L(X,M,\mu)$ a.e. Also that, $$ \int f = \sum^\infty_1 \int f_n. $$
I found this question in Elements of Integration by Bartles, and this is the proof I got so far. Could any of you guys look over it?
Let $g_n = \sum^n_1 |f_k|$. $g$ is measurable, and by the monotone convergence theorem, $$ \int \lim g_n = \sum^\infty_1 \int |f_n| < \infty. $$ Therefore, $g = \lim g_n$ is an integrable function in $L(X,M,\mu)$. Furthermore, since $\int g$ is finite, we know that $g(x) < \infty$ a.e. Therefore for a.e $x \in X$, $f_n(x)$ converges to a measurable function $f$. Since $f$ is dominated by $g$, we therefore conclude by the Lebesgue Dominated convergence theorem that $$ \int f = \sum^\infty_1 \int f_n. $$