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