Series on uncountable sets (a special case)

We have a definition for series on countable sets (defined as a sum remapped over the natural numbers). In general, this definition doesn't extend to uncountable sets; however, there is a special case where we can make this extension.

Note: the following definitions have a flaw that is examined on the flip side. Understanding the flaw is probably the highlight of this card.

Let $X$ be a set (which could be uncountable), and let $f:X\to \mathbb{R}$ be a function. Consider every subset of $X$ and consider the sum over each of these subsets. The set of all of these sums—if its supremum is less than $\mathrm{\infty }$, then the set [$\{x\in X:?\}$] is at most countable*. In this case, we make the definition

$\sum _{x\in X}f(x)$ is absolutely convergent, and the sum is given by:

So, we have found a case where it makes sense to represet the sum over an uncountable set as a sum over a countable one.

*This requires [what axiom?].