Pathology of submeasures and $F_\sigma$ ideals

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.