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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).

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    This is really $c$lose to being a duplicate of the last two questions linked by Jonas Meyer.2012-02-02

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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?

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    @Shahab: That is exactly it, you have shown that an uncountable set of positive numbers always has an infinite sum.2012-02-02
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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.)

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    You mean uncountable number of additions, not infinite.2012-02-02
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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.

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    Dear @Martin: Thanks for your reply. Yes, I think (like you, if I understand your comments correctly) that the best is to write $f=f_+-f_-$ with $f_+,f_-\ge0$ (in the usual way), and to define $\sum f(x)$ only if we do **not** have $\sum f_+(x)=\infty=\sum f_-(x).$2012-02-03