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Math and science::Analysis::Tao::08. Infinite sets

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:

\[ \sum_{x \in X}f(x) := [...] \]

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?].