TITLE: use of $\sum $ for uncountable indexing set
QUESTION [35 upvotes]: I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say
$$
\sum_{a \in A} a \quad \text{where A is some uncountable indexing set e.g. some $A \subset \mathbb{R}$ }
$$
Would it be better to avoid the above notation in general for uncountable indexing sets ? Any help in making better sense of this would be very appreciated.

REPLY [5 votes]: For sums of real numbers, I have always thought of all sums as integrals with respect to a suitable counting measure (which is somewhat limiting, I know, since I am handling everything in the Lebesgue sense, which messes up conditional convergence). At the very least, that interpretation reduces to something most graduate students have seen a well-developed theory for.
As an aside, it is not uncommon to see sums over possibly uncoutnable index sets with interpretation in the above sense.  One of the classical examples (you can find it in Rudin's Real and Complex Analysis) arises in the study of Fourier series over a Hilbert Space and if you would like an example in both the statement and proof of the Parseval identity.  Consider $\mathcal{l}^2 (\mathbb{R})$ (that is, the Lebesgue space of $\mathbb{R}$ equipped with the counting measure).  Appealing to Zorn's Lemma we know this space has a maximal Hilbert Space basis (uncountable) $e_\alpha$ and that for any $x \in \mathcal{l}^2 (\mathbb{R})$ we have $||x||^2$ = $\sum_\alpha |\langle e_\alpha, x\rangle|^2$