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Besicovitch-Federer projection theorem for measures

Emanuele Tasso

TL;DR

This work extends the Besicovitch-Federer projection criterion from sets to general finite Borel measures by leveraging a disintegration framework along typical $(n-m)$-planes and a transversal projection family. The authors prove that, under atomicity of planar slices for almost every projection, one obtains a dichotomy between almost-everywhere injectivity and singular projections, which yields a robust rectifiability criterion via slicing in a measure-theoretic setting. The main contribution is a measure-level projection theorem with a decomposition $\mu=\mu_r\oplus\mu_u$ (rectifiable and purely unrectifiable parts) and the observation that projection properties can be characterized without assuming a priori information on $\pi_V\mu$, while extending the theory to general locally compact metric spaces with transversal projection families. The results unify probabilistic injectivity, projection singularity, and slicing-rectifiability in a cohesive framework, and they connect to classical results (Besicovitch-Federer, White) as well as non-Euclidean contexts (e.g., Heisenberg geometry), providing new tools for analyzing the geometric structure of measures through their projections.

Abstract

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $μ$ 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 $μ\cap W$ is atomic for a typical $(n-m)$-plane $W \subset \mathbb{R}^n$, pure unrectifiability can be characterized simultaneously by the $μ$-almost everywhere injectivity of the orthogonal projection $π_V \colon \mathbb{R}^n \to V$ and by the singularity of the projected measure for a typical $m$-plane $V$. In particular, no assumption on $π_Vμ$ 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.

Besicovitch-Federer projection theorem for measures

TL;DR

This work extends the Besicovitch-Federer projection criterion from sets to general finite Borel measures by leveraging a disintegration framework along typical -planes and a transversal projection family. The authors prove that, under atomicity of planar slices for almost every projection, one obtains a dichotomy between almost-everywhere injectivity and singular projections, which yields a robust rectifiability criterion via slicing in a measure-theoretic setting. The main contribution is a measure-level projection theorem with a decomposition (rectifiable and purely unrectifiable parts) and the observation that projection properties can be characterized without assuming a priori information on , while extending the theory to general locally compact metric spaces with transversal projection families. The results unify probabilistic injectivity, projection singularity, and slicing-rectifiability in a cohesive framework, and they connect to classical results (Besicovitch-Federer, White) as well as non-Euclidean contexts (e.g., Heisenberg geometry), providing new tools for analyzing the geometric structure of measures through their projections.

Abstract

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let be a finite Borel measure on and let be an integer. We show that, under the sole assumption that the slice is atomic for a typical -plane , pure unrectifiability can be characterized simultaneously by the -almost everywhere injectivity of the orthogonal projection and by the singularity of the projected measure for a typical -plane . In particular, no assumption on 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.

Paper Structure

This paper contains 18 sections, 19 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Main result.
  3. Injectivity of projections
  4. Besicovitch-Federer projection theorem for measures
  5. Rectifiability via slicing.
  6. Transversal family of maps
  7. Idea of the proof
  8. Organization of the paper
  9. Notation and Preliminaries
  10. A density result for outer measures
  11. Transversality and cones
  12. The Probabilistic Injective Projection Property
  13. Rectifiability via slicing
  14. Proof of Proposition \ref{['p:equivcone']}
  15. Measurability
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $\Sigma \subset \emph{Gr}(n,m)$ be a Borel set. Assume that for $\gamma_{n,m}$-a.e. $V \in \Sigma$ it holds Then, for $\gamma_{n,m}$-a.e. $V \in \Sigma$, if and only if $\mu$ is concentrated on a purely $m$-unrectifiable set.

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3: Injectivity of projections
  • Theorem 1.4: Besicovitch--Federer projection theorem for measures
  • Corollary 1.5: Rectifiability via slicing
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Definition 2.3: Cone 1
  • Definition 2.4: Cone 2
  • ...and 27 more