Prove that if ($X$ can be uncountable)$$M:=\sup\bigg{\{}\sum_{x\in X}|f(x)|:A\subseteq X, A \ \text{finite}\bigg\} < \infty $$
then the sets $\{x\in X:|f(x)|\geq{1}/{n} \}$ are finite with cardinality at most $nM$ for all positive integers n.
Where do I begin ? Why this result should be true ?
How to prove a result regarding summation on infinite sets.
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analysis
summation
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0What happens if you add *more than* $nM$ numbers together, each of which is at least $1/n$? – 2017-01-12
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1This seems to be basically the same question as [The sum of an uncountable number of positive numbers](http://math.stackexchange.com/q/20661). See also other questions [linked there](http://math.stackexchange.com/questions/linked/20661). – 2017-01-12
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1BTW I guess $\sum\limits_{x\in X}|f(x)|$ (inside the supremum) is a typo and you wanted to write $\sum\limits_{x\in A}|f(x)|$. (In fact, one of possible definitions of $\sum\limits_{x\in X}|f(x)|$ is to definite it as supremum of sums over finite subsets.) – 2017-01-12