David Meza-Alcántara

arXiv2024

Pathology of submeasures and $F_σ$ ideals

We address some phenomena about the interaction between lower semicontinuous submeasures on $\mathbb{N}$ and $F_σ$ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $F_σ$ ideals. We give a partial answers to the question of whether every nonpathological tall $F_σ$ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $F_σ$ ideals using sequences in Banach spaces.

math.FA↗

arXiv2022

$F_σ$ ideals of perfectly bounded sets

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 $φ$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(φ)=\{A\subseteq \mathbb{N}: \;φ(A)<\infty\}$. We address the questions of when $FIN(φ)$ 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.

math.LO↗