2025/11/16 by Emanuele Tasso, Tasso, Emanuele
Mathematics · #Advanced Banach Space Theory #Point processes and geometric inequalities #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2511.12636
In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let μ be a finite Borel measure on ℝn and let 0 < m < n be an integer. We show that, under the sole assumption that the slice μ∩ W is atomic for a typical (n-m)-plane W ⊂ ℝn, pure unrectifiability can be characterized simultaneously by the μ-almost everywhere injectivity of the orthogonal projection πV \colon ℝn → V and by the singularity of the projected measure for a typical m-plane V. In particular, no assumption on πVμ 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.