Here is the statement of the problem, from Royden and Fitzpatrick's Real Analysis, 4 ed:
For a set $X$, let $\mathcal{M}$ be the $\sigma$-algebra of all subsets of $X$. Let $\eta$ be the counting measure of $\mathcal{M}$. Characterize the real-valued functions $f$ on $X$, which are integrable over $X$ with respect to $\eta$ and the value of $\int_{X} f d\eta$ for such functions.
Now, this is what I have done:
Suppose $\int_{X} \left|f\right| d\eta \lt \infty$. Note that $\int_{X} \left|f\right| d\eta = \int_{X}f^{+}d\eta + \int_{X}f^{-}d\eta$. Since $f^{+}$ and $f^{-}$ are nonnegative, by the Simple Approximation Theorem, there exist increasing sequences of simple functions $\{\varphi_{k}\}$ and $\{\psi_{k}\}$ such that $\varphi_{k}\to f^{+}$ and $\psi_{k}\to f^{-}$ pointwise.
Now for each $k$, let $\varphi_{k}$ and $\psi_{k}$ have canonical representations $$\varphi_{k}=\sum_{i=1}^{n_{k}}a_{i}\chi_{A_{i}}\;,\;\psi_{k}=\sum_{i=1}^{m_{k}}b_{i}\chi_{B_{i}}.$$
So we have $$\int_{X}\left|f\right|d\eta=\int_{X}f^{+}d\eta+\int_{X}f^{-}d\eta=\int_{X}\left[\lim_{k\to\infty}\varphi_{k}\right]d\eta +\int_{X}\left[\lim_{k\to\infty}\psi_{k}\right]d\eta.$$
Using the Monotone Convergence Theorem, this becomes $$\lim_{k\to\infty}\int_{X}\varphi_{k}\,d\eta+\lim_{k\to\infty}\int_{X}\psi_{k}\,d\eta=\lim_{k\to\infty}\left[\int_{X}\sum_{i=1}^{n_{k}}a_{i}\chi_{A_{i}}+\int_{X}\sum_{i=1}^{m_{k}}b_{i}\chi_{B_{i}} \right]=\sum_{i=1}^{\infty}\left[a_{i}\eta(A_{i})+b_{i}\eta(B_{i})\right]. $$
So $f$ is integrable iff this last series converges.
Am I missing something for this to be considered a "characterization"? How much more specific should I be? I think there is more to this as if I leave it that way, there's not much I can say about "the value of $\int_{X}f\,d\eta$". I appreciate any input.
You have to give a more concrete characterization. If f is integrable the $\{x: f(x) \neq0\}$ is at most countable because $\{x: |f(x)| > \frac 1 n\}$ has at most $n\int |f|$ points. Now $\sum |f(x)|$ makes sense. [ In general we can define $\sum |f(x)|$ as the supremum of all finite sums of the numbers $|f(x)|,x \in X$. With this understanding of $\sum |f(x)|$ the class of integrable functions are precisely those for which $\sum |f(x)| < \infty$. The value of $\int f$ is precisely the sum $\sum f(x)$ (which is the sum of an ordinary absolutely convergent series obtained by ignoring all terms with $f(x)=0$.