Uncountable Sum as Series/Corollary

Theorem

Let $X$ be an uncountable set.

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

Let $f: X \to \closedint 0 {+\infty}$ have uncountably infinite support.

Then:

$\ds \sum_{x \mathop \in X} \map f x = +\infty$

Proof

This is the first case of Uncountable Sum as Series.

$\blacksquare$

Sources

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