As undergraduates, we learn that conditional convergence of infinite series is highly sensitive to the order structure on $\mathbb N$: if $\sum_{n=0}^\infty x_n$ conditionally converges and $x \in \mathbb R$ then there exists a bijection $\psi: \mathbb N \to \mathbb N$ such that $\sum_{n=0}^\infty a_{\psi(n)} = x$.
On the other hand, it is often useful to discuss sums over arbitrary infinite sets, not just $\mathbb N$, and to my knowledge this is usually made rigorous using nets. Let $A$ be an infinite set, and let $[A]^{<\omega}$ be the set of finite subsets of $A$. The limit of a function $f: [A]^{<\omega} \to \mathbb R$ is characterized by: for every $\varepsilon > 0$ there exists $b \in [A]^{<\omega}$ such that for every $c \supseteq b$, $$\left|f(c) - \lim_{a \in [A]^{<\omega}} f(a)\right| \leq \varepsilon.$$ Therefore, the "natural" way to define the sum of an indexed set $(x_\alpha)_{\alpha \in A}$ of real numbers is $$\sum_{\alpha \in A} x_\alpha = \lim_{a \in [A]^{<\omega}} \sum_{\alpha \in a} x_\alpha.$$ However, this notion of summation is much closer to absolute convergence than conditional convergence, because $A$ has no order structure. Indeed, the above construction is preserved by bijections $A \to A$, so if $A = \mathbb N$ this notion corresponds only to absolute convergence.
Now when $A = \mathbb N$ there are lots of ways to "renormalize" divergent series to make them converge in a weaker sense of the word. This suggests that mathematicians may have studied notions of conditional convergence, in some weak sense, over infinite sets. What is the literature on conditionally convergent sums over infinite sets?
My motivation (and the reason why I include the math.GT tag): In Bonahon's paper "Transverse Hölder distributions for geodesic laminations", he proves the Gap Lemma (Lemma 3), which involves a sum over an infinite ordered set $A$ of intervals (called gaps). It is not too hard to show that in general, the ordering on $A$ is not isomorphic to $\mathbb N$ (though it is at least isomorphic to a subset of $\mathbb Q$), and that the sum cannot converge absolutely (ie, in the net-based sense I linked above). In the proof of the Gap Lemma, Bonahon seems to use the following definition of conditional convergence: given a sequence $(a_n)$ of finite subsets of $A$ which grow to $A$, $$\sum_{\alpha \in A} x_\alpha = \lim_{n \to \infty} \sum_{\alpha \in a_n} x_\alpha.$$ (In Bonahon's notation, $a_n$ is the set of gaps arising from an open cover $(I_i)$ such that $\sum_i \ell_i^\nu \leq 2^{-n}$.) This seems extremely counterintuitive to me, because the $a_n$s are not canonically defined (for each one we have to choose an open cover) and because the notation $\sum_{d' > d}$ that Bonahon uses for what I called $\sum_{\alpha \in A}$ seems to suggest that there is a way to interpret the sum using the ordering of the gaps. So I wonder if there is a better way to interpret this sum.
