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Path of pathology

Rafał Filipów, Jacek Tryba

TL;DR

This work presents a two-part program to analyze pathology in submeasures and ideals. It develops a precise framework around nonpathological vs pathological submeasures using hat-operations, studies degrees of pathology, and provides explicit finite and infinite examples; it then transfers these insights to ideals, proving a characterization of ideals that are intersections of matrix summability ideals and establishing substantial results for generalized density ideals, Van der Waerden, and the Josefson-Nissenzweig properties. A key contribution is showing the Solecki ideal is pathological and giving consistency results for non-Borel intersections of matrix ideals, thereby addressing longstanding questions about the limits of nonpathological representations and the complexity of these ideals. The findings have implications for Katětov-order analyses, topological representations of ideals, and the structural understanding of density-type and combinatorially defined ideals in set theory and analysis.

Abstract

We present a few results about (non)pathology of submeasures and ideals.

Path of pathology

TL;DR

This work presents a two-part program to analyze pathology in submeasures and ideals. It develops a precise framework around nonpathological vs pathological submeasures using hat-operations, studies degrees of pathology, and provides explicit finite and infinite examples; it then transfers these insights to ideals, proving a characterization of ideals that are intersections of matrix summability ideals and establishing substantial results for generalized density ideals, Van der Waerden, and the Josefson-Nissenzweig properties. A key contribution is showing the Solecki ideal is pathological and giving consistency results for non-Borel intersections of matrix ideals, thereby addressing longstanding questions about the limits of nonpathological representations and the complexity of these ideals. The findings have implications for Katětov-order analyses, topological representations of ideals, and the structural understanding of density-type and combinatorially defined ideals in set theory and analysis.

Abstract

We present a few results about (non)pathology of submeasures and ideals.
Paper Structure (18 sections, 45 theorems, 97 equations, 1 figure, 1 table)

This paper contains 18 sections, 45 theorems, 97 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ideals
  4. Submeasures
  5. (Non)pathological submeasures
  6. Definition of (non)pathological submeasures
  7. Properties of (non)pathological submeasures
  8. Examples of nonpathological submeasures
  9. Examples of pathological submeasures
  10. Degree of pathology of submeasures
  11. (Non)pathological ideals
  12. (Non)pathological ideals
  13. Intersections of matrix ideals
  14. Generalized density ideals
  15. F-sigma ideals and the degree of pathology
  16. ...and 3 more sections

Key Result

Proposition 3.2

For every submeasure $\phi$ on $X$ and every $A\subseteq X$ there exists a measure $\mu$ on $X$ such that $\mu\leq\phi$ and $\widehat{\phi}(A)=\mu(A)$. In particulary, if $\phi$ is nonpathological, then for every $A\subseteq X$ there exists a measure $\mu$ on $X$ such that $\mu\leq\phi$ and $\phi(A)

Figures (1)

  • Figure 1: Relationships between (non)lsc and (non)pathological nonzero submeasures on $\omega$ (in gray regions there are no submeasures).

Theorems & Definitions (84)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Remark
  • Proposition 3.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • ...and 74 more