Radon-Nikodýmification of arbitrary measure spaces

Abstract

We study measurable spaces equipped with a σ-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; 𝒜 𝜇;) to a strictly localizable version (X̂ Â, 𝜇̂; ), so that the dual of L₁(X; 𝒜, 𝜇) is L∞( X̂ ;Â; 𝜇̂). Corresponding to this duality is a generalized Radon-Nikoým 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 unrecti_able sets in a given dimensión.