Transfinite series: Uncountable sums 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
 A: HINT
Given $\varepsilon>0$, can you measure the set $\{x:f(x)>\varepsilon\}$?

Edit: Giving some background to the somewhat short hint.
Suppose $E\subset\mathbb{R}$ is uncountable, and let $f$ be a non-negative function defined on $E$. The usual definition of the expression $\sum_{x\in E}f(x)$ is given by
$$  
\sum_{x\in E}f(x) =\sup_{F\subset E,\,|F|<\infty}\sum_{x\in F}f(x) \tag{1}
$$
i.e. we take supremum over finite sets. 
Now, let us choose $\varepsilon>0$ and consider the set 
$$E_\varepsilon = \{x\in E :f(x)>\varepsilon\}.$$
This set must be finite in order for the sum in (1) to be finite, because otherwise we may for each positive integer $n$ choose subsets $F_n\subset E_\varepsilon$ such that $|F_n|=n$ and then $$\sum_{x\in E}f(x)\ge \sum_{x\in F_n}f(x)>\sum_{F_n}\varepsilon =n\varepsilon.$$
Since $$\bigcup_{\varepsilon>0}
 E_\varepsilon =\bigcup_{n=1}^\infty E_{1/n} =\{x\in E:f(x)>0\}$$
the conclusion follows.
Note: I added the assumption $f\ge0$. In fact I do not think there is a reasonable definition for conditional convergence of this kind. 
