Definition:Uncountable Sum

Definition
Let $X$ be an uncountable set.

Let $f: X \to \left[{0 \,.\,.\, +\infty}\right]$ be an extended real-valued function.

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


 * $\displaystyle \sum_{x \mathop \in X} f \left({x}\right) := \sup \left\{ {\sum_{x \mathop \in F} f \left({x}\right): F \subseteq X, F \text{ finite} }\right\}$

Also see

 * Uncountable Sum as Series