Measure and integration Geometric measure theory and fractal measures Theorem

Besicovitch–Federer projection theorem

Builds on Rectifiable set Hausdorff measure

Statement

Let $1 \le m < n$ be integers, and let $\mathcal{H}^m$ be the $m$ -dimensional Hausdorff measure on $\R^n$ . Let $G(n, m)$ be the Grassmannian of $m$ -dimensional linear subspaces of $\R^n$ , with its unique probability measure $\gamma_{n,m}$ that is invariant under the orthogonal group. For $V \in G(n, m)$ , let $P_V \colon \R^n \to V$ be the orthogonal projection onto $V$ . Let $E \subseteq \R^n$ be $\mathcal{H}^m$ -measurable with $\mathcal{H}^m(E) < \infty$ .

$E$ is purely $m$ -unrectifiable if and only if $\mathcal{H}^m(P_V(E)) = 0$ for $\gamma_{n,m}$ -almost every $V \in G(n, m)$ .
If $E$ is $m$ -rectifiable and $\mathcal{H}^m(E) > 0$ , then $\mathcal{H}^m(P_V(E)) > 0$ for $\gamma_{n,m}$ -almost every $V \in G(n, m)$ .

Remarks

Part 2 is the easier half. A rectifiable set of positive measure has an approximate tangent plane $W$ at almost every point, and for almost every $V$ the projection $P_V$ is injective on $W$ . Then $P_V$ is bi-Lipschitz on a piece of $E$ of positive measure, so the projection has positive measure. Part 1 is the deep half; Besicovitch proved it for $m = 1$ and $n = 2$ , and Federer proved the general case. Example: the four-corner Cantor set in $\R^2$ has positive and finite $\mathcal{H}^1$ measure and is purely $1$ -unrectifiable, so almost all of its projections onto lines have length zero.

Going deeper

For a student

A set $E$ of finite measure splits as $E = R \cup P$ with $R$ rectifiable and $P$ purely unrectifiable. The theorem says that almost every projection of $E$ sees only $R$ . It is the model for many later results that characterise rectifiability by densities, tangent measures or singular integrals. It is used in the study of removable sets for bounded analytic functions, through the analytic capacity of planar sets.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Hausdorff measure concept
Rectifiable set concept
Besicovitch–Federer projection theorem statement

Tag	`8J8AW0TG9M`
URI	`m2://8J8AW0TG9M`
Kind	statement, theorem
Writing	W0 stub
MSC	28A75, 28A78
Status	This copy of the site has no build data. Run `m2 status 8J8AW0TG9M`.
About	Rectifiable set
Uses	Rectifiable set, Hausdorff measure
Changed	2 November 2026