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Let $\phi$ be a lower semicontinuous submeasure, that is, a function $\mathcal{P}(\mathbf{N}) \to [0,\infty]$ which is monotone, subadditive, and $$ \phi(A)=\lim_{n\to \infty} \phi(A \cap [1,n]) $$ for all subsets $A$ of positive integers. Moreover, it assumed that $\phi(\emptyset)=0$.

Finally, for each $A\subseteq \mathbf{N}$, set $$ \|A\|_\phi=\lim_{n\to \infty}\phi(A\setminus [1,n]). $$

I am going to ask for a refinement of Question 2 of this MO thread:

Question. Fix a set $A\subseteq \mathbf{N}$ such that $0<\|A\|_\phi\le \phi(A)<\infty$. Is it true that there exists a subset $B\subseteq A$ such that $0<\|B\|_\phi < \|A\|_\phi$ ?

Paolo Leonetti
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  • Would a submeasure such that $\phi(A)=1$ for any $A\ne\emptyset$ be a counterexample? Unless I missed something, it seems to be a lsc submeasure and it gives $|A|_\phi=1$ for each non-empty $A$. – Martin Sleziak Aug 07 '17 at 15:14
  • @MartinSleziak if $\phi(A)=1$ for all $A\neq \emptyset$ then $|A|\phi=1$ if and only if $|A|=\infty$, otherwise $|A|\phi=0$. However, you are right, your example still works. – Paolo Leonetti Aug 07 '17 at 15:22

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