Besicovitch-Federer projection theorem for measures

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption that the slice $\mu \cap W$ is atomic for a typical $(n-m)$-plane $W \subset \mathbb{R}^n$, pure unrectifiability can be characterized simultaneously by the $\mu$-almost everywhere injectivity of the orthogonal projection $\pi_V \colon \mathbb{R}^n \to V$ and by the singularity of the projected measure for a typical $m$-plane $V$. In particular, no assumption on $\pi_V\mu$ is required a priori. This yields a new rectifiability criterion via slicing for Radon measures. The result is new even in the classical setting of Hausdorff measures, and it further extends to arbitrary locally compact metric spaces endowed with a generalized family of projections.