Localizable locally determined measurable spaces with negligibles
Philippe Bouafia, Thierry De Pauw

TL;DR
This paper develops a framework for transforming measurable spaces with negligible sets into localizable, fiber space structures, enabling duality results and a generalized Radon-Nikodym theorem, with applications to integral geometric measures.
Contribution
It introduces conditions and methods to obtain localizable, locally determined versions of measure spaces with negligibles, extending duality and Radon-Nikodym theorems.
Findings
Established conditions for localizable locally determined spaces.
Constructed a universal fiber space with a categorical property.
Extended duality and Radon-Nikodym theorems to these spaces.
Abstract
We study measurable spaces equipped with a -ideal of negligible sets. We find conditions under which they admit a localizable locally determined version -- a kind of fiber space that describes locally their directions -- defined by a universal property in an appropriate category that we introduce. These methods allow to promote each measure space to a strictly localizable version , so that the dual of is . Corresponding to this duality is a generalized Radon-Nikod\'ym theorem. We also provide a characterization of the strictly localizable version in special cases that include integral geometric measures, when the negligibles are the purely unrectifiable sets in a given dimension.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Fuzzy and Soft Set Theory
