Published Paper

Inserted: 29 oct 2024
Last Updated: 29 oct 2024

Journal: Exctracta Math.
Volume: 38
Number: 2
Year: 2023
Links: journal

Abstract:

We study measurable spaces equipped with a $\sigma$-ideal of negligible sets. We find conditions under which they admit a localizable locally determined version – a kind of fiber space that locally describes their directions – defined by a universal property in an appropriate category that we introduce. These methods allow to promote each measure space $(X,\mathcal{A},\mu)$ to a strictly localizable version $(\hat{X},\hat{\mathcal{A}},\hat{\mu})$, so that the dual of $L_1(X,\mathcal{A},\mu)$ is $L_\infty(\hat{X},\hat{\mathcal{A}},\hat{\mu})$. Corresponding to this duality is a generalized Radon-Nikodym 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.