\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \newcommand{\multiBetaReduction}[0] {\twoheadrightarrow_{\beta}} \newcommand{\betaReduction}[0] {\rightarrow_{\beta}} \newcommand{\betaEq}[0] {=_{\beta}} \newcommand{\string}[1] {\texttt{"}\mathtt{#1}\texttt{"}} \newcommand{\symbolq}[1] {\texttt{`}\mathtt{#1}\texttt{'}} \newcommand{\groupMul}[1] { \cdot_{\small{#1}}} \newcommand{\groupAdd}[1] { +_{\small{#1}}} \newcommand{\inv}[1] {#1^{-1} } \newcommand{\bm}[1] { \boldsymbol{#1} } \require{physics} \require{ams} \require{mathtools} \)

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?].