About the meaning of the sum of elements indexed by an uncountable set is less than infinity

Question

Statement of the main question

Let $ \{a_i\} $ be a set of real/complex numbers indexed by $ I $. When $ I $ is finite or countably infinite, the meaning of $$ \sum_{i\in I} a_i <\infty $$ is quite easy to grasp. But, how about when $ I $ is uncountable?

My own guess

When $ I $ is uncountable, $ \sum_{i\in I} a_i <\infty $ means

1. $ A\equiv\{ i\in I |a_i\neq 0 \} $ is countable.
2. $ \sum_{i\in I} a_i = \sum_{i\in A} a_i $.
3. $ A $ is finite. Or if $ A $ is countably infinite, then we could find a bijection $ f $ between $ A $ and $ \mathbb{N} $, such that $ b_n \equiv a_{f(n)} $ converges to some number in the normal sense of an infinite sequence.

Generalized sum?

Even though I myself take my understanding quite naturally, I fail to find any established text or reference on how to understand this notation. The closest concept I found is the generalized sum. From this website in proofwiki, I get the basic idea that a generalised sum is a mapping which gives all finite "subsum" a number. However, if I understand in this way, I fail to get the idea of how a mapping is less than infinite.

Origin of the notation

I encounter this notation when I read the concept of the direct sum of an indexed family of Hilbert spaces, where nothing is said about the cardinality of the indexing set. See examples on page 12, eq. (1.25) of General Principles of Quantum Field Theory or on page 4 of Operator Algebras: Theory of $c^*$-Algebras and von Neumann Algebras.

Expected replies

Anything related to the above material, like your understanding of the notation, why your understanding is better than mine, the fact that my understanding is equivalent to the concept of a generalized sum, and why one expects such a definition in defining the direct sum of Hilbert spaces, is more than welcome.