The problem is :
For every function $f : X \rightarrow\ \mathbb C$, the equality $\displaystyle \sum_{x\in X}|f(x)|=\sup\left\{\sum_{x\in F}|f(x)|:F \subset X\text{ is finite set} \right\}$ holds.
I tried to solve this problem, using $(X,P(X)(\text{Power set of } X),c)$ $P(X)$ is a Power set of $X$ $c$ is counting measure : $c(A) = \text{the number of elements of }A$
Then, $\displaystyle \sum_{x\in X}|f(x)| =\int_X |f|dc$
So, the problem can be changed into
$\displaystyle \int_{x\in X} |f(x)|dc = \sup \left\{\int_{x\in F}|f(x)| : F \subset X\text{ is finite set}\right\}$
But, here's my question. In $\mathbb R$, It was able to make $F \subset \mathbb R$ just like $[-N,N]$ or $E\cap[-N,N]$ to make some bounded set which can go to $\mathbb R$ when $N \rightarrow \infty$.
How to get a seq of finite set $F_n$ in arbitrary X? Is it okay to get any aribitrary of it? Please let me know the full solution of this problem. Is there another way to solve this problem using $\sigma -$algebra which I wrote?