Rectifiable set
2020 Mathematics Subject Classification: Primary: 49Q15 [MSN][ZBL]
Also called countably 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. As such they have a dimension. In what follows we will use the terminology $m$-dimensional rectifiable set. Some authors prefer the terminology countably $m$-rectifiable set or, briefly, $m$-rectifiable.
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 $\alpha$-dimensional Hausdorff measure.
Definition 1 (cp. with Definition 15.3 of [Ma]) 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 (cp. with Definition 4.1 of [De]) 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 $E$ (a $k$-dimensional Lipschitz graph is a subset $G$ of $\mathbb R^n$ such that there is a system of orthonormal coordinates $x_1, \ldots, x_n$ and a Lipschitz map $(f^{k+1}, \ldots, f^n)=f:\mathbb R^k\to\mathbb R^{n-k}$ with \[ G=\{(x_1,\ldots x_k, f^{k+1} (x_1, \ldots , x_k),\ldots , f^n (x_1, \ldots , x_k))\}\, \Big)\, . \]
Definition 3 (cp. with Lemma 11.1 of [Si]) 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 (see Lemma 11.1 of [Si]). 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 (consider for instance a $C^1$ embedding $\gamma: [0,1]\to \mathbb R^2$ and the intersection $V$ of the usual Vitali set of $\mathbb R$ with $[0,1]$; the set $E:= \gamma (V)$ has Hausdorff dimension $1$, it can be covered by a single $C^1$ submanifold but it is not $\mathcal{H}^1$ measurable). In what follows we might assume that $E$ is $\mathcal{H}^k$ measurable: $\sigma$-finite $\mathcal{H}^k$-measurable sets can be decomposed into the union of a Borel set and an $\mathcal{H}^k$-null set.
Remark If a set can be covered $\mathcal{H}^k$-almost all with a countable family of $k$-dimensional $C^1$ submanifolds, then its Hausdorff dimension is at most $k$. Thus the requirement that the set $E$ has dimension $k$ in the definitions given above is meant to exclude sets of smaller dimension. On the other hand, according to the definitions above, a $\mathcal{H}^k$-null set of Hausdorff dimension $k$ is a $k$-dimensional rectifiable set.
A Borel set of Hausdorff dimension $k$ which is not rectifiable is called unrectifiable.
Definition 4(cp. with Definition 5.6 of [De]) 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 (see Theorem 5.7 of [De]) If $E\subset \mathbb R^n$ is a Borel set with $\sigma$-finite $\mathcal{H}^k$-measure, then there is a rectifiable set $R$ and a purely unrectifiable set $P$ such that $E= R\cup P$. The decomposition is unique up to $\mathcal{H}^k$-null sets.
Remark We recall that a Borel set of Hausdorff dimension $k$ might not have $\sigma$-finite $\mathcal{H}^k$ measure: in that case the set is necessarily unrectifiable. On the other hand Theorem 5 might fail for such sets.
A useful decomposition of rectifiable sets is the following (cp. again with Lemma 11.1 of [Si]).
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 $\mathbb R^n$ 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 (A) = \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 (cp. with Corollary 4.4 of [De] and Theorem 11.6 of [Si]):
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 $\mu$ (cp. with Definition 11.4 of [Si]), 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 (see Theorem 4.8 of [De]):
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 (cp. with Theorem 4.6 of [De]) 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 (cp. with Theorem 18.1 of [Ma]) 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 $\mathcal{H}^k (P_\pi (E)) = 0$.
B. White in [Wh] has shown how the higher-dimensional version of this theorem follows via an inductive argument from the planar version.
One-dimensional rectifiable sets
The theory of one-dimensional rectifiable sets is somewhat special since much stronger theorems can be proved which fail for higher dimensions. Perhaps the most useful one is the following (cp. with Theorem 3.14 of [Fa]):
Theorem 11 A continuum, i.e. a compact connected set, $E\subset\mathbb R^n$ of finite $\mathcal{H}^1$ measure is always rectifiable and arcwise connected. Indeed it is always the image of a rectifiable curve.
We refer to [Fa] for a comprehensive account of the theory of rectifiable one-dimensional sets.
References
[AFP] | L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 |
[De] | C. De Lellis, "Rectifiable sets, densities and tangent measures" Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008. MR2388959 Zbl 1183.28006 |
[Fa] | K. J. Falconer. "The geometry of fractal sets". Cambridge Tracts in Mathematics, 85. Cambridge University Press, Cambridge, 1986. MR0867284 Zbl 0587.28004 |
[Fe] | H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001 |
[Ma] | P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005 |
[Wh] | B. White, "A new proof of Federer's structure theorem for $k$-dimensional subsets of $\mathbb R^n$ J. Amer. Math. Soc. , 11 : 3 (1998) pp. 693–701 |
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