My textbook does not provide much about counting measures and integration. So I decided to set up integration on space $(N , P(N) , \mu_c ,R)$ myself, imitating the construction of the Lebesgue integral. (All functions are measurable since we are considering the power set.) I started with simple functions, which are basically sequences with finite range set.
$$\int f\,d\mu \;=\; \sum_{i=1}^{n} a_i \mu (A_i) \;=\; \sum_{k\in N} f(k).$$
Then I defined integration for $[0,\infty]$-valued sequences:
$$\int f\,d\mu \;=\; \sup\left\{\int g\,d\mu: \text{$g \leq f$ and $g$ is simple and $[0,\infty]$-valued }\right\}.$$
The above expression is simply summation of all terms since $f$ can be approached by a sequence of simple non-decreasing functions $f_n=fX_{(-n,n)}$.
Then for $[-\infty,\infty]$-valued sequences we define
$$\int f\,d\mu \;=\; \int f^+-\int f^- \;=\; \sum_{k\in N}f(k).$$
So I deduced that the integral of any function on $N$ is just summation of the terms of the sequence of its image. $f$ is said to be integrable if $\int f^+$ and $\int f^-$ are finite. Hence $\sum_{k\in N} f(k)$ must converge.
Now here I got confused. We know that every measurable function $f$ is integrable if and only if $|f|$ is integrable. So from the above construction, does it imply that every convergent sequence is absolutely convergent? (We know this ain't true)
Where did I make a fault in my above imitation?