Besicovitch-Federer projection theorem for measures

Abstract

In this paper we prove a Besicovitch-Federer type projection theorem for general finite Borel measures in ${\mathbb{R}}^n$. Assuming only that the conditional measures along typical affine $(n-m)$-planes are purely atomic, we characterize concentration on a purely $m$-unrectifiable set in terms of the singularity of the projected measure and the $\mu$-almost everywhere injectivity of typical orthogonal projections. No regularity or absolute-continuity assumption on the projected measures is imposed, and the result is new even for measure of the form $\mu ={\mathcal{H}}^m└E$. In this extended version, we further develop applications of the projection theorem to currents. First, we obtain a rectifiability criterion for Radon measures in terms of atomic disintegrations. We then use this criterion to prove that a finite-mass classical Federer-Fleming current satisfying the intrinsic integer-valued push-forward condition is rectifiable, and hence a real flat chain, without assuming any a priori flat-chain or metric-current structure. We also show that a Euclidean metric current is rectifiable whenever its slices are atomic along a set of projections of positive Grassmannian measure, with no assumption on the mass of its boundary. Finally, the projection theorem extends to separable locally compact metric spaces.