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:


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