  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.
