Characterization of Pathological Ideals via Matrix Summability and Katetov Order
A submeasure is pathological when no nontrivial measure lies below it, and an ideal is pathological when generated by such a submeasure. This work organizes the competing notions of (non)pathology for submeasures and ideals on the natural numbers and characterizes the ideals representable as intersections of matrix summability ideals in terms of the Katetov order and the ideal of asymptotic density zero sets. Applications show the Solecki ideal is pathological and settle the status of the exponential density zero ideal.
2501.00503
A submeasure is pathological when no nontrivial measure lies below it, and an ideal is pathological when it is generated by a pathological submeasure; this paper organizes the many notions of (non)pa…