I have a vague problem in a Measure and Integration book here. They ask me to consider $\mu$ the counting measure in $\mathbb{N}$ and interpret Fatou's lemma, monotone and dominated convergence theorems as statements about infinite series. I thought it would be easy but got stuck really quick...
If we consider a sequence of non negative functions $f_n\in L^1$, Fatou's lemma says that $$\int_\mathbb{N}\liminf f_n \ d\mu \leq \liminf\int_\mathbb{N}f_n \ d\mu.$$
The real problem comes when I try to understand what are this integrals. Starting with a simple function $\Phi =\sum_{i=1}^kx_i\mathcal{X}_{E_i}$, we have that $\int_\mathbb{N}\Phi \ d\mu = \sum_{i=1}^kx_i\mu(E_i) = \sum_{i=1}^kx_i|E_i|$, where $|E_i|$ is the number of elements in $E_i$. Each function $f_n$ is the $\sup$ of simple functions, but it's not clear how I should use all this together. Even if I consider a sequence $0\leq \Phi_1\leq\Phi_2\leq\ldots\leq f_n$ converging to $f_n$, they are not partial sums.
To make things worse, there is infinite $f_n$ to consider and the $\liminf$ after that. Any help is welcome to interpret all this.
Thank you.