Definition:Uncountable Sum

It has been suggested that this page or section be merged into Definition:Generalized Sum. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code.

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

Sources

• 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S P.5$