Definition:Uncountable Sum

Definition
Let $X$ be an uncountable set.

Let $f: X \to \closedint 0 \to$ be an extended real-valued function.

The uncountable sum of $f$ over $X$ is defined to be the supremum of the finite sums:


 * $\ds \sum_{x \mathop \in X} \map f x := \sup \set {\sum_{x \mathop \in F} \map f x: F \subseteq X, F \text{ finite} }$

Also see

 * Uncountable Sum as Series