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Nonpathological submeasures and Fσ ideals
Abstract
We study the relationship between nonpathological lower semicontinuous submeasures, Fσ ideals on ω, and matrix summability ideals. First, we construct in ZFC a non-analytic ideal on ω which is representable as an intersection of matrix summability ideals. Then, we show that an Fσ 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.
Related
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BibTeX
@misc{leonetti2026nonpathological,
title = {{Nonpathological submeasures and $F_σ$ ideals}},
author = {Paolo Leonetti},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02266},
eprint = {2611.02266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02266},
}
RIS
TY - UNPB
TI - Nonpathological submeasures and F_σ ideals
AU - Paolo Leonetti
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02266
UR - https://arxiv.org/abs/2611.02266
AB - We study the relationship between nonpathological lower semicontinuous submeasures, F_σ ideals on ω, and matrix summability ideals. First, we construct in 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.
ER -