A topological commutative monoid is a commutative monoid $(X,+)$ with a topology $\cal O$ such that $+$ is a continuous function. There is a notion of infinite sum on topological commutative monoids, where we say $\sum_{i\in I}a_i\to a$ if for any neighborhood $U\ni a$ there is a finite subset $A\subseteq I$ such that for any finite $A\subseteq B\subseteq I$, $\sum_{i\in B}a_i\in U$ (where the sum over $B$ is a regular finite sum within the monoid).
The theorem I would like to prove is a variation of Fubini's theorem:
For any topological commutative monoid $(X,+,\cal O)$ such that $\sum_{j\in J}a_{ij}\to b_i$ for each $i\in I$, $$\sum_{i\in I}b_i\to a\iff\sum_{(i,j)\in I\times J}a_{ij}\to a.$$
This is likely the most general context in which a statement like this makes sense, but it may not be true in this generality. Is it true in only one direction? Is it true for topological groups? If anyone has a reference dealing with this kind of infinite sum that would also be useful.
As requested, a proof that in a complete abelian topological group, the summability of a family of elements implies the summability of every subfamily, and for a partition of the family, the family of sums of parts is again summable to the sum of the whole family. Formally: $\newcommand{\Iota}{\mathrm{I}} \newcommand{\Kappa}{\mathrm{K}}$
If we don't assume the group is Hausdorff, the use of the equality sign in the statement is not strictly justified. If we don't want to make special provisions for the case of finite families, where the sum is uniquely defined by the group axioms, we can interpret it as meaning both sides represent the same coset in $G/\overline{\{0\}}$. If we want to distinguish between the case of finite families and the case of infinite families, where the sum of a summable family is only determined modulo $\overline{\{0\}}$ anyway, we need a more complicated formulation.
We assume that $\Iota$ is infinite, otherwise there is nothing to show.
Proof that the subfamilies are summable:
This is where the completeness of $G$ plays a role. We consider the directed set $\mathscr{F}(\Iota) = \{ F \subset \Iota : F\text{ is finite}\}$, and the net $\sigma\colon \mathscr{F}(\Iota) \to G$ given by
$$\sigma_F = \sum_{\iota \in F} a_\iota.$$
For a summable family this is a Cauchy net, and that implies that the nets $\sigma^{(\kappa)} \colon \mathscr{F}(\Iota_\kappa) \to G$ are also Cauchy nets.
Let $V \in \mathscr{V}(0)$, and pick a symmetric $W \in \mathscr{V}(0)$ with $W + W \subset V$. By definition of summability, there is an $F_W \in \mathscr{F}(\Iota)$ such that for all $F\in \mathscr{F}(\Iota)$ with $F_W \subset F$ we have $s - \sigma_F \in W$. Thus, for any two $F_1,F_2 \in \mathscr{F}(\Iota)$ with $F_W \subset F_1 \cap F_2$, we have $\sigma_{F_1} - \sigma_{F_2} \in W - W \subset V$. This shows that $\sigma$ is a Cauchy net.
In view of the goal to treat sums in commutative monoids, it is better to characterise families giving rise to Cauchy nets without using subtraction:
We skip the straightforward proof, and use this criterion to deduce that if $\sigma^{(\Gamma)}$ is a Cauchy net, then so is $\sigma^{(\Delta)}$ for all $\Delta \subset \Gamma$. For, given $V \in \mathscr{V}(0)$, choose $F^{(\Gamma)}_V \in \mathscr{F}(\Gamma)$ according to the criterion, and set $F^{(\Delta)}_V = F^{(\Gamma)}_V \cap \Delta$. Since $F\in \mathscr{F}(\Delta)$ and $F \cap F^{(\Delta)}_V = \varnothing$ implies $F\in \mathscr{F}(\Gamma)$ and $F \cap F^{(\Gamma)}_V = \varnothing$, the conclusion is immediate.
Thus we have seen that the $\sigma^{(\kappa)}$ are all Cauchy nets, and the completeness of $G$ guarantees the existence of $s_\kappa = \lim \sigma^{(\kappa)}$ for all $\kappa \in \Kappa$.
Without completeness, the summability of the subfamilies corresponding to the parts of the partition needs to be explicitly required.
Proof that the family $(s_\kappa)_{\kappa \in \Kappa}$ of partial sums is summable with sum $s$, if the original family is summable with sum $s$:
Given $V \in \mathscr{V}(0)$, pick $W\in \mathscr{V}(0)$ with $W + W\subset V$. By summability of $(a_\iota)$, there is an $F_W \in \mathscr{F}(\Iota)$ such that $\sigma_F \in s + W$ for all $F \in \mathscr{F}(\Iota)$ with $F_W \subset F$. Let $H = \{ \kappa \in \Kappa : \Iota_\kappa \cap F_W \neq \varnothing\}$. Then we have
$$\sum_{\kappa \in M} s_\kappa \in s + W + W$$
for each finite $H \subset M \subset \Kappa$.
Let $n = \operatorname{card} M$, and choose $U \in \mathscr{V}(0)$ such that $\underbrace{U + \dotsc + U}_{n \text{ terms}} \subset W$. For each $\kappa \in M$, there is a finite $F_W \cap \Iota_\kappa \subset F_U^{(\kappa)} \subset \Iota_\kappa$ such that
$$\sum_{\iota \in F} a_\iota \in s_\kappa - U$$
for all finite $F_U^{(\kappa)} \subset F \subset \Iota_\kappa$. Hence
$$\sum_{\kappa \in M} s_\kappa \in \sum_{\kappa \in M} \Biggl(U + \sum_{\iota \in F^{(\kappa)}_U} a_\iota\Biggr) = \Biggl(\sum_{\iota \in \bigcup_{\kappa \in M} F^{(\kappa)}_U} a_\iota\Biggr) + \underbrace{U + \dotsc + U}_{n\text{ terms}} \subset s + W + W.$$
since $F_W \subset \bigcup_{\kappa \in M} F^{(\kappa)}_U$. Thus $(s_\kappa)_{\kappa \in \Kappa}$ is summable with sum $s$.
If all we require for a topological monoid is the continuity of addition, I'm not sure whether the implication holds. In the proof, we used the existence and continuity of negation (at $0$) to write a neighbourhood of $s_\kappa$ as $s_\kappa - U$, so that we could conclude
$$s_\kappa \in \Biggl(\sum_{\iota \in F^{(\kappa)}_U} a_\iota\Biggr) + U.$$
I don't see how we can get rid of that. On the other hand, so far I have not yet come up with an example in a monoid with continuous addition where the implication fails.