rectifiable set

rectifiable set

Let us denote with

ℋ

m

the

m

-dimensional

Hausdorff measure

in

ℝ

n

.

A set

S

⊂

ℝ

n

is said to be

countably

m

-

rectifiable

if there exists a

countable

sequence

of

Lipschitz continuous

functions

f

k

:

ℝ

m

→

ℝ

n

such that

S

⊂

⋃

k

f

k

⁢

(

ℝ

m

)

.

A set

S

⊂

ℝ

n

is said to be

countably

(

H

m

,

m

)

-rectifiable

if
there exists a set

S

′

which is countable

m

-rectifiable and such that

ℋ

m

⁢

(

S

∖

S

′

)

=

0

.

A set

S

⊂

ℝ

n

is said to be

(

H

m

,

m

)

-rectifiable

or simply

m

-rectifiable

if it is

(

ℋ

m

,

m

)

-rectifiable and

ℋ

m

⁢

(

S

)

<

+

∞

.

If

S

is any

Borel subset

of

ℝ

n

and

x

∈

ℝ

n

is given, one can define the

density

of

S

in

x

as

Θ

m

⁢

(

S

,

x

)

=

lim

ρ

→

0

⁡

ℋ

m

⁢

(

S

∩

B

ρ

⁢

(

x

)

)

ω

m

⁢

ρ

m

where

ω

m

is the

Lebesgue measure

of the

unit ball

in

ℝ

m

. Notice that an

m

-dimensional plane

Π

has density

1

in all points

x

∈

Π

and density

0

in all points

x

∉

Π

.

It turns out that if

S

is rectifiable, then in

ℋ

m

-a.e. point

x

∈

S

the density

Θ

m

⁢

(

S

,

x

)

exists and is equal to

1

. Moreover in

ℋ

m

-a.e. point

x

∈

S

there exists an

approximate tangent plane

to

S

as defined below.

Given a point

x

∈

S

⊂

ℝ

n

and a vector

v

∈

ℝ

n

we say that

v

is

tangent

to

S

in

x

if there exists a sequence of points

x

k

∈

S

,

x

k

→

x

and a sequence of positive numbers

λ

k

such that

lim

k

→

∞

⁡

λ

k

⁢

(

x

k

-

x

)

=

v

.

If

S

is an

m

-dimensional manifold, then the set of

tangent vectors

to a point

x

∈

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

′

of

S

such that

Θ

m

⁢

(

S

∖

S

′

,

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

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

λ

⁢

v

is also a tangent vector, for all

λ

>

0

).
If the approximate tangent cone is an

m

-dimensional vector subspace of

ℝ

n

, it is called the

approximate tangent plane

.

Notice that if

S

⊂

ℝ

n

is any

m

-dimensional regular surface,
and

Q

is the set of all points of

ℝ

n

with

rational

coordinates

, then the set

S

∪

Q

is an

m

-rectifiable set since

ℋ

1

⁢

(

Q

)

=

0

. Notice, however, that

S

∪

Q

¯

=

ℝ

n

and consequently every vector

v

is tangent to

S

∪

Q

in every point

x

∈

S

∪

Q

. On the other hand the approximately tangent vectors to

S

∪

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

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