Pathology of submeasures and $$F_{\sigma }$$ ideals

Archive for Mathematical Logic:1-27 (forthcoming)
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Abstract

We address some phenomena about the interaction between lower semicontinuous submeasures on $${\mathbb {N}}$$ N and $$F_{\sigma }$$ F σ ideals. We analyze the pathology degree of a submeasure and present a method to construct pathological $$F_{\sigma }$$ F σ ideals. We give a partial answers to the question of whether every nonpathological tall $$F_{\sigma }$$ F σ ideal is Katětov above the random ideal or at least has a Borel selector. Finally, we show a representation of nonpathological $$F_{\sigma }$$ F σ ideals using sequences in Banach spaces.

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Katětov order on Borel ideals.Michael Hrušák - 2017 - Archive for Mathematical Logic 56 (7-8):831-847.
Analytic ideals and their applications.Sławomir Solecki - 1999 - Annals of Pure and Applied Logic 99 (1-3):51-72.
Ramsey type properties of ideals.M. Hrušák, D. Meza-Alcántara, E. Thümmel & C. Uzcátegui - 2017 - Annals of Pure and Applied Logic 168 (11):2022-2049.
Bases and borel selectors for tall families.Jan Grebík & Carlos Uzcátegui - 2019 - Journal of Symbolic Logic 84 (1):359-375.

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