Let $X$ be any set (could be uncountable). Let $f:X \to \Bbb R$ (real numbers) be any function. If the series $\sum f(x)$ is absolutely convergent for all $x \in X$, then the set $A=\{x\in X: f(x)\not=0\}$ is at most countable.

I am having 2 cases: a) if $X$ is countable. b) if $X$ is uncountable.

I am completely fine with a. But I am getting stock with b

What I did, I distinguish 2 cases; first/ If $f(x)=0$ for all $x\in X$ (since $f$ is any function), then $A=\emptyset$ which is countable.

Second/if there is $x$ such that $f(x)=0$, then $|A|\le|X|$. Now we have $X= A \cup \{x\}$ which makes $A$ uncountable (here I could not continue)

I would appreciate any help or hint. Thank you.

  • 1
    $\begingroup$ It is unclear what $\displaystyle \sum_{x \in X} |f(x)|$ would even mean if we had $|f(x)| > 0$ for an uncountable quantity of $x \in X$. The way sums are defined, you need a bijection between $\mathbb{N}$ and the set to be summed. $\endgroup$
    – Kaj Hansen
    Oct 18 '17 at 7:15
  • 2
    $\begingroup$ @KajHansen There is a general notion of a sum $$\sum_{x \in X} f(x)$$ for arbitrary $X$ and $f : X \to \mathbb{R}$. Let $\mathcal{F} \subseteq \mathcal{P}(X)$ be the family of finite subsets of $X$ ordered by inclusion. Then $\mathcal{F}$ is a directed set and we define $$\sum_{x \in X} f(x) = \lim_{F \in \mathcal{F}} \sum_{x \in F} f(x).$$ See limits of nets. $\endgroup$
    – Adayah
    Oct 18 '17 at 7:18
  • $\begingroup$ Thanks for that information @Adayah ! I was unaware of this $\endgroup$
    – Kaj Hansen
    Oct 18 '17 at 7:24

No cases are necessary. Let

$$\sum_{x \in X} |f(x)| = L.$$

Then for each $n$ the set $A_n = \{ x \in X : |f(x)| \geqslant \frac{L}{n} \}$ is finite, because it has at most $n$ elements. Now just notice that

$$A = \bigcup_{n=1}^{\infty} A_n$$

is countable as a countable union of finite sets.


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