Pathology of submeasures and $F_\sigma$ ideals

TL;DR

This paper investigates the complex interactions between lower semicontinuous submeasures and $F_\sigma$ ideals, introducing methods to construct pathological examples and exploring their structural properties and representations.

Contribution

It introduces a new approach to constructing pathological $F_\sigma$ ideals and provides partial answers to longstanding questions about their relationship with the random ideal and Borel selectors.

Findings

Developed a method to construct pathological $F_\sigma$ ideals.

Provided partial answers regarding nonpathological tall $F_\sigma$ ideals and their relation to the random ideal.

Presented a representation of nonpathological $F_\sigma$ ideals using Banach space sequences.

Abstract

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