Rectifiable set

2020 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]

Also called countable rectifiable set. A central concept in Geometric measure theory, first introduced by Besicovitch for $1$-dimensional sets in the plane. Rectifiable sets of the euclidean space can be thought as measure-theoretic generalizations of $C^1$ submanifolds.

Definitions

Rectifiable subsets of the Euclidean space $\mathbb R^n$ can be defined in several ways. In what follows we denote by $\mathcal{H}^\alpha$ the Hausdorff $\alpha$-dimensional measure.

Definition 1 A Borel set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if it has Hausdorff dimension $k$ and there is a countable family of Lipschitz maps $f_i: \mathbb R^k \to \mathbb R^n$ such that their images cover $\mathcal{H}^k$-almost all $E$.

Definition 2 A Borel set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if it has Hausdorff dimension $k$ and there is a countable family of Lipschitz $k$-dimensional graphs of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.

Definition 3 A Borel set $E\subset \mathbb R^n$ is a rectifiable subset of dimension $k$ if it has Hausdorff dimension $k$ and there is a countable family of $C^1$ $k$-dimensional submanifolds of $\mathbb R^n$ which cover $\mathcal{H}^k$-almost all $E$.

All these definitions are equivalent. The first one can be easily generalized to define rectifiable subsets in metric spaces. The assumption that $E$ is a Borel set might be dropped. In that case, however, the set might not be $\mathcal{H}^k$-measurable. In what follows we might assume that $E$ is $\mathcal{H}^k$ measurable: $\mathcal{H}^k$-measurable sets can be decomposed into the union of a Borel set and an $\mathcal{H}^k$-null set.

A Borel set of Hausdorff dimension $k$ which is not rectifiable is called unrectifiable.

Definition 4 An unrectifiable $k$-dimensional set $E\subset \mathbb R^n$ is called purely unrectifiable if its intersection with any $k$-dimensional rectifiable set is an $\mathcal{H}^k$-null set.

It follows from the equivalence of the first three definitions that an unrectifiable set is purely unrectifiable if and only if its intersection with the image of an arbitrary Lipschitz map $f:\mathbb R^k\to \mathbb R^n$ (resp. with an arbitrary Lipschitz $k$-dimensional graph or with an arbitrary $C^1$ $k$-dimensional submanifold) is an $\mathcal{H}^k$-null set.

Properties

It follows from the definition that a rectifiable set $E$ has $\sigma$-finite $\mathcal{H}^k$ measure. A simple argument gives the following decomposition theorem.

Theorem 5 If $E\subset \mathbb R^n$ is a Borel set, then there is a rectifiable set $R$ and a purely unrectifiable set $R$ such that $E= R\cup P$. The decomposition is unique up to $\mathcal{H}^k$-null sets.

A useful decomposition of rectifiable sets is the following.

Theorem 6 If $E\subset \mathbb R^n$ is a rectifiable $k$-dimensional set, then there are

• An $\mathcal{H}^k$-null set $E_0$
• Countably many $C^1$ $k$-dimensional submanifolds $\Gamma_i$ ($i\geq 1$) of $\mathbb R^n$
• Compact subsets $E_i$ of $\Gamma_i$

such that the collection $\{E_i\}_{i\in\mathbb N}$ is a partition of $E$ (i.e. the sets are pairwise disjoint and their union is $E$).

Approximate tangent planes

Let $E$ be a rectifiable $k$-dimensional subset of $E\mathbb R^n$ and $f$ be a nonnegative Borel function $f: E\to \mathbb R$ such that $\int_E f\, d\mathcal{H}^k <\infty$. Consider the Radon measure $\mu$ defined through \begin{equation}\label{e:misura} \mu (E) = \int_{E\cap A} f\, d\mathcal{H}^k \, . \end{equation} Then the measure $\mu$ has approximate tangent planes at $\mu$--a.e. point $x$, in the following sense:

Proposition 7 For $\mu$-a.e. $x\in\mathbb R^n$ there is a $k$-dimensional plane $\pi$ such that the rescaled measures $\mu_{x,r}$ given by \begin{equation}\label{e:rescaled} \mu_{x,r} (A) = r^{-k} \mu (x+rA) \end{equation} converge, as $r\downarrow 0$ to the measure $\mu_{x,0}$ given by \begin{equation}\label{e:app_tangent} \mu_{x, 0} (A) = f(x_0) \mathcal{H}^k (A\cap \pi) \end{equation} in the weak$^\star$ topology (see Convergence of measures).

The plane $\pi$ of the above proposition is called approximate tangent plane of the measure $\mu$, but it is related to the geometry of the set $E$ and it generalizes the classical notion of tangent plane for $C^1$ submanifolds of the euclidean space. Indeed it can be proved that $\pi$ coincides with the classical tangent plane of the submanifold $\Gamma_i$ of Proposition 7 at $\mathcal{H}^k$-a.e. $x\in E_i$.

The following converse of Proposition 7 holds:

Theorem 8 Let $\mu$ be a Radon measure on $\mathbb R^n$ and $k$ be an integer. Assume that for $\mu$-a.e. $x\in \mathbb R^n$ there is a positive real $f(x_0)$ and a $k$-dimensional plane such that the measures $\mu_{x,r}$ as in \ref{e:rescaled} converge in the weak$^\star$ topology to the measure $\mu$ of \ref{e:app_tangent} as $r\downarrow 0$. Then $f$ coincides with a Borel function $\mu$-almost everywhere and there is a rectifiable $k$-dimensional set $E$ such that \ref{e:misura} holds.

Criteria of rectifiability

There are several ways to prove that a set is rectifiable or purely unrectifiable. We list here the best known criteria.

Through tangent measures

This has already been discussed in Proposition 7 and Theorem 8: A Borel set of dimension $k$ and positive $\mathcal{H}^k$ measure is rectifiable if and only there is a nonnegative Borel function $f:E \to \mathbb R$ such that the measure $\mu$ of \ref{e:misura} has approximate tangents $\mu$-almost everywhere.

Through cones

This criterion needs the concept of $k$-dimensional lower density of a set: see Density of a set for the relevant definition. In what follows, given a $k$-dimensional plane of $\mathbb R^n$ we denote by $P_\pi$ the orthogonal projection onto $\pi$ and $Q_\pi$ the orthogonal projection on the orthogonal complement of $\pi$.

Proposition 9 Let $E\subset \mathbb R^n$ be a Borel set with $0<\mathcal{H}^k (E)<\infty$. The set $E$ is rectifiable if and only if for $\mathcal{H}^k$-a.e. $x\in E$ the following properties hold:

• The lower $k$-dimensional density $\theta^k_* (E,x)$ is positive
• There is a $k$-dimensional plane $\pi$ and a real number $\alpha$ such that, if $C (x,\pi,\alpha)$ denotes the cone

\[ C (x,\pi,\alpha) := \{ y\in\mathbb R^n: |Q_\pi (y-x)|\leq \alpha |P_\pi (y-x)|\}\, , \] then \[ \lim_{r\downarrow 0} \frac{\mathcal{H}^k (B_r (x)\setminus C (V,\pi, x))}{r^k} = 0\, . \]

Through densities

Recitifiable sets can be characterized through the existence of $k$-dimensional density: for the relevant statement, due to Besicovitch for $1$-dimensional sets and generalzed by Preiss to all dimensions, we refer to Density of a set.

Through projections

Purely unrectifiable sets can be characterized as those sets which are hidden through most projections. The following theorem was proved by Besicovitch for one-dimensional sets and generalized by Federer. Authors often refer to it as Besicovitch-Federer projection theorem. In order to state it we need to consider the standard uniform measure on the Grassmanian $G(k,n)$ of $k$-dimensional planes in $\mathbb R^n$, which we will denote by $\nu$ (observe that $\nu$ enters in the statement through $\nu$-null sets: therefore $\nu$ might be substituted by any $\mu$ which is the volume measure for some Riemannian structure on the manifold $G (k,n)$).

Theorem 10 Let $E\subset \mathbb R^n$ be a Borel set with $0<\mathcal{H}^k (E)<\infty$. $E$ is purely unrectifiable if and only if for $\nu$-a.e. $k$-dimensional plane $\pi$ we have $\mathcal{H}^k (P_\pi (E)) = 0$.