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 \( f : X \rightarrow \mathbb{R} \) 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 \( \infty \), 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?].