What is this sum? 
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The sum of an uncountable number of positive numbers 

Consider the following question:
For each real number $x$, let $\epsilon_x>0$ be an associated positive number. Is the sum $\sum_{x\in \mathbb{R}} \epsilon_x$ infinite? 
I have been puzzling over this for some time. Can someone help? Also I am not sure whether this qualifies as a series, (and if not, what is it called).
 A: Remark:  I am giving hints because I think this is a fun problem, and worth working out.
Hint 1:  Consider $$E_n:= \left\{ x\in \mathbb{R}: \epsilon_x >\frac{1}{n}\right\}.$$  
Hint 2: Since each $\epsilon_x>0$ we know that every real number $x$ must lie within some $E_n$.  However, there are countably many $E_n$, yet uncountably many real numbers.  What can you conclude from this, and what does it tell us about the original series?
A: I was going to say no: Say $\epsilon_x=e^{-x^2}$ then $\int_{-\infty}^{\infty} \epsilon_x dx$ is finite.
But $\sum_{x\in \mathbb{R}} \epsilon_x$, or even $\sum_{x\in \mathbb{P}} \epsilon_x$ where $\mathbb{P}$ is a non-empty open subset of $\mathbb{R}$, involves an uncountable number of additions, and if every $\epsilon_x$ is positive, no matter how small, you'll always reach infinity eventually.
Of course if $\mathbb{P}$ was a countable infinite subset of $\mathbb{R}$, it would be isomorphic to the integers and so finite series are then possible again.
(Edited as suggested to specify uncountable instead of just infinite.)
A: Edit: Below is an [excellent] example of something that is definitely false.
It is not a series. It's sort of one of the ideas behind calculus, actually. 
If you have certain conditions on how the real numbers $x$ determine the [positive] real numbers $\epsilon_x$, then what you wrote is really $\int_{-\infty}^\infty\epsilon_x dx$. 
Whether or not the sum (=integral) is finite is something you need calculus to sort out.
