Localizable locally determined measurable spaces with negligibles

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.