Localizable locally determined measurable spaces with negligibles

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 $\sigma$-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 $(X, \mathcal{A},\mu)$ 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-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.