Let $(X,\mathscr{B},\mu)$ be a finite measure space and let $(f_n)_{n=1}^\infty$ be a sequence of real valued measurable functions on $X$. Suppose that
- $\sum_n f_n(x)=F(x)$ exists for $\mu$-almost every $x\in X$, and
- $G:=\sum_n\lvert f_n \rvert$ is in $L^p(\mu)$ for some $p\ge 1$.
Then, does the following conclusion hold?
- $F\in L^1(\mu)$ and $\int_x F {\rm d} \mu = \sum_{n=1}^\infty \int_x f_n {\rm d}\mu$
Yes. By Holder's inequality and finiteness of the measure space, $L^p \subset L^1$ with continuous inclusion. This means that $G \in L^1$. Now $\lvert {F}\rvert \leq \lvert {G} \rvert$ pointwise a.e., so $F \in L^1$. The formula for $\int F$ follows by dominated convergence, using $G$ as a dominating function. Explicitly, we have $F - \sum_{n=1}^N f_n$ is dominated by $2G \in L^1$ and converges to $0$ pointwise a.e.. Thus $\int F - \sum_{n=1}^N \int f_n$ converges to $0$ as $N \to \infty$