$F_\sigma$ ideals of perfectly bounded sets

TL;DR

This paper characterizes $F_\sigma$ ideals generated by perfectly bounded sets in Banach spaces, linking them to lower semicontinuous submeasures, and explores their properties such as tallness and Borel selectors.

Contribution

It provides a characterization of $F_\sigma$ ideals of perfectly bounded sets via non-pathological submeasures and analyzes conditions for tallness and existence of Borel selectors.

Findings

Ideal $B({f x})$ is tall iff $(x_n)_n$ is weakly null in $c_0$.

In $c_0$, $B({f x})$ has a Borel selector when tall.

Characterization of $B({f x})$ as $FIN(\varphi)$ for a non-pathological submeasure.

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.