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 ?
Asked
Active
Viewed 29 times
0
Martin Sleziak
- 53,687
mathemather
- 2,959
-
What happens if you add more than $nM$ numbers together, each of which is at least $1/n$? – Mees de Vries Jan 12 '17 at 13:29
-
1This seems to be basically the same question as The sum of an uncountable number of positive numbers. See also other questions linked there. – Martin Sleziak Jan 12 '17 at 16:14
-
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.) – Martin Sleziak Jan 12 '17 at 16:15