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-Nikodým 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.