TITLE: Does uncountable summation, with a finite sum, ever occur in mathematics?
QUESTION [25 upvotes]: Obviously, “most” of the terms must cancel out with opposite algebraic sign.
You can contrive examples such as the sum of the members of R being 0, but does an uncountable sum, with a finite sum, ever occur naturally as part of a larger scenario (eg, proof of a theorem, or preparation for a definition)?

REPLY [5 votes]: One application of uncountable sums (or, to be more precise, sums along arbitrary index set) I am aware of is the definition of the Hilbert space $\ell_2(A)$. 
A very basic example of a Hilbert space is the space 
$\ell_2=\ell_2(\mathbb N)$. The elements of this space are sequences such that $\sum\limits_{i\in\mathbb N} x(i)^2<\infty$. It is endowed  with the
inner product is given by $\langle x,y \rangle =\sum\limits_{i\in\mathbb N} x(i)y(i)$. 
If we allow summation over
arbitrary sets, then we can define $\ell_2(A)$ using almost the same
construction; in this case, we take all functions $x\colon A\to\mathbb R$ such that 
$$\sum_{i\in A} x(i)^2 < \infty$$
and the inner product will be $$\langle x,y
\rangle = \sum\limits_{i\in A} x(i)y(i).$$ 
It can be shown that this is indeed a Hilbert space and that every Hilbert space $X$ is isomorphic to
$\ell_2(A)$ for some set $A$. Cardinality of $A$ is precisely the "Hilbert dimension", i.e. cardinality 
of orthonormal basis for $X$.
This result is, in some sense, a
classification of all Hilbert spaces. 
These results can be found, for example, in:

Chapter 13 Roman's Advanced Linear Algebra;
Chapter IX of Dixmier's General Topology;
Chapter II of Retherford's Hilbert space;
Corollary 1.4.19 in Tao's Epsilon of Room...

And there are probably many other places to look. Just google for some reasonable phrases, for example:

"every hilbert space" isomorphic "l2" Google Books, Google, Google Scholar
"every hilbert space" isomorphic cardinality Google Books, Google, Google Scholar 
"every hilbert space" "linearly isometric" Google Books, Google, Google Scholar

See also: Is every Hilbert space an $L^2$ space?