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

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Integration can (informally) be considered a form of this. – meh Mar 13 '13 at 2:29
A sum of uncountably many positive terms is always infinite (this is a nice exercise by itself). So summation over an uncountable index set isn't generally very interesting. – mjqxxxx Mar 13 '13 at 2:51
up vote 12 down vote accepted

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 $0$.

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