rectifiable set
Let us denote with ${\mathscr{H}}^{m}$ the $m$-dimensional Hausdorff measure^{} in ${\mathbb{R}}^{n}$.
A set $S\subset {\mathbb{R}}^{n}$ is said to be countably $m$-rectifiable if there exists a countable^{} sequence of Lipschitz continuous functions ${f}_{k}:{\mathbb{R}}^{m}\to {\mathbb{R}}^{n}$ such that
$$S\subset \bigcup _{k}{f}_{k}({\mathbb{R}}^{m}).$$ |
A set $S\subset {\mathbb{R}}^{n}$ is said to be countably $\mathrm{(}{\mathrm{H}}^{m}\mathrm{,}m\mathrm{)}$-rectifiable if there exists a set ${S}^{\prime}$ which is countable $m$-rectifiable and such that ${\mathscr{H}}^{m}(S\setminus {S}^{\prime})=0$.
A set $S\subset {\mathbb{R}}^{n}$ is said to be $\mathrm{(}{\mathrm{H}}^{m}\mathrm{,}m\mathrm{)}$-rectifiable or simply $m$-rectifiable if it is $({\mathscr{H}}^{m},m)$-rectifiable and $$.
If $S$ is any Borel subset of ${\mathbb{R}}^{n}$ and $x\in {\mathbb{R}}^{n}$ is given, one can define the density of $S$ in $x$ as
$${\mathrm{\Theta}}^{m}(S,x)=\underset{\rho \to 0}{lim}\frac{{\mathscr{H}}^{m}(S\cap {B}_{\rho}(x))}{{\omega}_{m}{\rho}^{m}}$$ |
where ${\omega}_{m}$ is the Lebesgue measure^{} of the unit ball^{} in ${\mathbb{R}}^{m}$. Notice that an $m$-dimensional plane $\mathrm{\Pi}$ has density $1$ in all points $x\in \mathrm{\Pi}$ and density $0$ in all points $x\notin \mathrm{\Pi}$.
It turns out that if $S$ is rectifiable, then in ${\mathscr{H}}^{m}$-a.e. point $x\in S$ the density ${\mathrm{\Theta}}^{m}(S,x)$ exists and is equal to $1$. Moreover in ${\mathscr{H}}^{m}$-a.e. point $x\in S$ there exists an approximate tangent plane to $S$ as defined below.
Given a point $x\in S\subset {\mathbb{R}}^{n}$ and a vector $v\in {\mathbb{R}}^{n}$ we say that $v$ is tangent^{} to $S$ in $x$ if there exists a sequence of points ${x}_{k}\in S$, ${x}_{k}\to x$ and a sequence of positive numbers ${\lambda}_{k}$ such that
$$\underset{k\to \mathrm{\infty}}{lim}{\lambda}_{k}({x}_{k}-x)=v.$$ |
If $S$ is an $m$-dimensional manifold, then the set of tangent vectors to a point $x\in S$ is nothing else than the usual tangent plane^{} to $S$ in $x$.
We say that a vector $v$ is approximately tangent to $S$ in $x$ if it is a tangent vector to every subset ${S}^{\prime}$ of $S$ such that ${\mathrm{\Theta}}^{m}(S\setminus {S}^{\prime},x)=0$. Notice that every tangent vector is also an approximately tangent vector while the converse^{} is not always true, as it is shown in an example below. The point, here, is that being the set $S$ defined ${\mathscr{H}}^{m}$-almost everywhere, we need a stronger definition for tangent vectors.
The approximate tangent cone to $S$ in $x$ is the set of all approximately tangent vectors to $S$ in $x$ (notice that if $v$ is a tangent vector then $\lambda v$ is also a tangent vector, for all $\lambda >0$). If the approximate tangent cone is an $m$-dimensional vector subspace of ${\mathbb{R}}^{n}$, it is called the approximate tangent plane.
Notice that if $S\subset {\mathbb{R}}^{n}$ is any $m$-dimensional regular surface, and $Q$ is the set of all points of ${\mathbb{R}}^{n}$ with rational coordinates, then the set $S\cup Q$ is an $m$-rectifiable set since ${\mathscr{H}}^{1}(Q)=0$. Notice, however, that $\overline{S\cup Q}={\mathbb{R}}^{n}$ and consequently every vector $v$ is tangent to $S\cup Q$ in every point $x\in S\cup Q$. On the other hand the approximately tangent vectors to $S\cup Q$ are only the tangent vectors to $S$, because the set $Q$ has density $0$ everywhere.
References
- 1 Frank Morgan: Geometric Measure Theory: A Beginner’s Guide.
Title | rectifiable set |
---|---|
Canonical name | RectifiableSet |
Date of creation | 2013-03-22 14:28:12 |
Last modified on | 2013-03-22 14:28:12 |
Owner | paolini (1187) |
Last modified by | paolini (1187) |
Numerical id | 11 |
Author | paolini (1187) |
Entry type | Definition |
Classification | msc 49Q15 |
Defines | density |
Defines | tangent vector |
Defines | approximate tangent plane |