I just read that for $f: I \rightarrow [0, \infty),$ we can define $\sum_{i \in I} f(i)$ as $\sup\{\sum_{i \in F} f(i) \mid F \subset I, F$ finite $\}.$ It makes sense that we can define it in this way. It coincides with my notion for countable sums. However, how can we define such a sum where $f$ may include negative values. It suddenly does not make sense to take the supremum.
That is, if $I$ is an arbitrary set and $x_i \in \mathbb{R},$ how would we define $\sum_{i \in I} x_i?$
Compute the sum over the non-negative values and compute the sum over the non-positive values. If one of them is finite take the sum of the two results. As in the Lebesgue's integral.