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 Katětov 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.