Nonpathological submeasures and $F_σ$ ideals

Abstract

We study the relationship between nonpathological lower semicontinuous submeasures, $F_σ$ ideals on $ω$, and matrix summability ideals. First, we construct in $\mathsf{ZFC}$ a non-analytic ideal on $ω$ which is representable as an intersection of matrix summability ideals. Then, we show that an $F_σ$ ideal is nonpathological if and only if it can be represented as an intersection of summable ideals if and only if it can be represented as an intersection of matrix summability ideals. We also prove that the random graph ideal is pathological. Finally, we compute the exact degree of pathology of a certain submeasure related to a construction of Topsøe.