Prove that summation of nonnegative and measurable $f_k$'s is summation of each integral of $f_k$. Prove.
If $f_k$, $k=1, 2, \cdots,$ are nonnegative, measurable, and defined on $E\subset\mathbb{R}^n$, then $$\int_E{\left(\sum_{k=1}^{\infty}f_k\right)}=\sum_{k=1}^{\infty}\int_E{f_k}$$
Proof.


*

*Let $F_N$ be a function defined by $\displaystyle F_N=\sum_{k=1}^{N}f_k$

*Then, $F_N$ are nonnegative and measurable, and increase to $\displaystyle\sum_{k=1}^{\infty}f_k$

*Hence $$\int_E{\left(\sum_{k=1}^{\infty}f_k\right)}=\lim_{N\to\infty}\int_EF_N=\lim_{N\to\infty}\sum_{k=1}^{N}\int_Ef_k=\sum_{k=1}^{\infty}\int_Ef_k$$



I think the fact that $\displaystyle F_N\nearrow{}F_\infty$ is required, in order to be $$\int_E{\left(\sum_{k=1}^{\infty}f_k\right)}=\int_E{F_\infty}=\int_E\lim_{N\to\infty}F_N=\lim_{N\to\infty}\int_EF_N$$
since Monotone Convergence Theorem gives a sufficient condition for interchanging the operations of integration and passage to the limit.
Where and how can I get the fact that $\displaystyle F_N\nearrow{}F_\infty$.
 A: Background information: $L^+$ is the space of all measurable functions from $X$ to $[0,\infty]$. 
Theorem 2.13 (Real Analysis, Folland) - Let $\phi$ and $\psi$ be simple functions in L^+$
b.) $\int (\phi + \psi) = \int \phi + \int \psi$

If $\{f_n\}$ is a finite or infinite sequence in $L^{+}$ and $f = \sum_{n}f_n$, then $\int f = \sum_{n}\int f_n$

Proof - Let $f,g\in L^{+}$, and suppose $\phi_n\rightarrow f, \psi_n\rightarrow g$ monotonically. Applying theorem 2.13b we get
$$\int f + \int g = \lim_{n\rightarrow \infty}\int \phi_n + \lim_{n\rightarrow \infty}\int \psi_n = \lim_{n\rightarrow \infty}\int \phi_n + \int \psi_n$$
$$= \lim_{n\rightarrow \infty}\int (\phi_n + \psi_n) = \int (f+g)$$ Hence by induction, $\int\sum_{1}^{n}f_n = \sum_{1}^{n}\int f_n$ for any finite $n$. Let $n\rightarrow \infty$, by the monotone convergence theorem $$\int \sum_{1}^{\infty}f_n = \sum_{1}^{\infty}\int f_n$$
I am not 100% sure this is entirely relevant to your question. I will delete it if that happens to be the case.
