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If you sum an expression over an uncountable set $\sum_{x\in \mathbb{R}}f(x)$, then do we need $f(x)=0$ on all but a countable subset in order for the sum to have a finite value?

If not can you give an example of a function everywhere nonzero that has a transfinite sum with a finite value?

Possible keywords: Transseries, Écalle–Borel Summation, analyzable function

Transseries for beginners, GA Edgar, 2009

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    I realise odds are you'll never see this, but by any chance do you happen to be poster from this [thread](https://www.physicsforums.com/threads/uncountable-sum.328751/)? Be seeing you...2016-07-28

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