How to prove the following exercise:
Let $(f_n)$ be a sequence in $M^+(\mathbb{R})$ proove that $ \displaystyle{\int \sum_{n=1}^{\infty}f_n = \sum_{n=1}^{\infty}\int f_n}$
it reminds me the Lebesgue's Monotone Convergence Theorem:
Let $\{f_n ,n =1,2,...,\}$be a sequence of non-negative measurable fucntions such that $\{f_n(x)\}$ is monotone increasing for each $x$. Let $ f = \lim f_n$. Then $\int f= \lim \int f_n$
we can define a sequence $s_k$ of partial sums such that $\displaystyle{s_k = \sum_{n=1}^{k}f_n}$ so $s_k$ satisfies the theorem hipothesis. But how to proceed with $f = \lim f_n$? In this case $\displaystyle{ \lim s_k = \sum_{n=1}^{\infty}f_n } = s$ (is that correct?)
and applying the theorem:
\begin{array} {lcl} \int s & = & \lim\int{s_k} \\ \displaystyle{\int \sum_{n=1}^{\infty} f_n} & = & \displaystyle{\lim \int \sum_{n=1}^{k} f_n} \\ & = & \displaystyle{\lim\int (f_1+f_2+\ldots+f_k)} \\ &=& \displaystyle{\lim(\int f_1 +\int f_2+\ldots+\int f_k)} \\ & = & \displaystyle{ \sum_{n=1}^{\infty}\int f_n}&\end{array}
Is that correct?