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Assume $f : X \to [0,\infty]$
I want to prove $$\sum_{x \in X} f(x)<\infty \Longrightarrow \{x \in X | f(x) >0\} \text { is a countable set}$$

Is it connected with finite property? Give me some help to prove it.

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    Mimic [this](http://math.stackexchange.com/questions/13781/transfinite-series-uncountable-sums)2012-05-08
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    Thank you for your help @Didier. I got this problem. But I wonder if $E_n$ has any connection with $S$?2012-05-08

3 Answers 3