$F_σ$ ideals of perfectly bounded sets

Abstract: Let ${\bf x}=(x_n)n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $|\sum{n\in F}x_n|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $\varphi$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(\varphi)={A\subseteq \mathbb{N}: \;\varphi(A)<\infty}$. We address the questions of when $FIN(\varphi)$ is a tall ideal and has a Borel selector. We show that in $c_0$ the ideal $B({\bf x})$ is tall iff $(x_n)_n$ is weakly null, in which case, it also has a Borel selector.