TITLE: Is uncountably summation defined?
QUESTION [6 upvotes]: We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$
Is it defined?

REPLY [12 votes]: If $x_i$ is a non-negative number for every $i$ in some index set $I$, then one can define
$$
\sum_{i\in I} x_i = \sup\left\{ \sum_{i\in J} x_k : J\subset I\ \&\ J\text{ is finite.} \right\}
$$
The sum can be shown to be $\infty$ except when for all but countably many $i\in I$, $x_i=0$.  And sometimes the sum is $\infty$ even when that is the case.
Therefore your proposed sum evaluates to $0$.