Hausdorff measure

Hausdorff measure

Introduction

Given a

real number

α

≥

0

we are going to define a Borel external

measure

ℋ

α

on

ℝ

n

with values in

[

0

,

+

∞

]

which will comprehend and generalize the concepts of length (for

α

=

1

), area (

α

=

2

) and volume (

α

=

3

) of sets in

ℝ

n

.
In particular if

M

⊂

ℝ

n

is an

m

-dimensional regular surface then one will show that

ℋ

m

⁢

(

M

)

is the

m

-dimensional area of

M

.
However, being an external measure,

ℋ

m

is defined not only on regular surfaces but on every subset of

ℝ

n

thus generalizing the concepts of length, area and volume. In particular, for

m

=

n

, it turns out that the

Hausdorff measure

ℋ

n

is nothing else than the

Lebesgue measure

of

ℝ

n

.

Given any fixed set

E

⊂

ℝ

n

one can consider the measures

ℋ

α

⁢

(

E

)

with

α

varying in

[

0

,

+

∞

)

. We will see that for a fixed set

E

there exists at most one value

α

such that

ℋ

α

⁢

(

E

)

is finite and positive; while for every other value

β

one will have

ℋ

β

⁢

(

E

)

=

0

if

β

>

α

and

ℋ

β

⁢

(

E

)

=

+

∞

if

β

<

α

.
For example, if

E

is a regular

2

-dimensional surface then only

ℋ

2

⁢

(

E

)

(which is the area of the surface) may possibly be finite and different from

0

while, for example, the volume of

E

will be

0

and the length of

E

will be

infinite

.

This can be used to define the dimension of a set

E

(this is called the

Hausdorff dimension

). A very interesting fact is the existence of sets with dimension

α

which is not integer, as happens for most

fractals

.

Also, the measure

ℋ

α

is naturally defined on every metric space

(

X

,

d

)

, not only on

ℝ

n

.

Definition

Let

(

X

,

d

)

be a metric space.
Given

E

⊂

X

we define the diameter of

E

as

diam

⁢

(

E

)

:=

sup

x

,

y

∈

E

⁡

d

⁢

(

x

,

y

)

.

Given a real number

α

we consider the conventional constant

ω

α

=

π

α

/

2

Γ

⁢

(

α

/

2

+

1

)

where

Γ

⁢

(

x

)

is the

gamma function

.

For all

δ

>

0

,

α

≥

0

and

E

⊂

X

let us define

ℋ

δ

α

⁢

(

E

)

:=

inf

⁡

{

∑

j

=

0

∞

ω

α

⁢

(

diam

⁢

(

B

j

)

2

)

α

:

B

j

⊂

X

,

⋃

j

=

0

∞

B

j

⊃

E

,

diam

⁢

(

B

j

)

≤

δ

⁢

∀

j

=

0

,

1

,

…

}

.

(1)

The

infimum

is taken over all possible enumerable families of sets

B

0

,

B

1

,

…

,

B

j

,

…

which are sufficiently small (

diam

⁢

B

j

≤

δ

) and which cover

E

.

Notice that the

function

ℋ

δ

α

⁢

(

E

)

is

decreasing

in

δ

. In fact given

δ

′

>

δ

the family of sequences

B

j

considered in the definition of

ℋ

δ

′

α

contains the family of sequences considered in the definition of

ℋ

δ

α

and hence the infimum is smaller.
So the limit in the following definition exists:

ℋ

α

⁢

(

E

)

:=

lim

δ

→

0

+

⁡

ℋ

δ

α

⁢

(

E

)

.

(2)

The number

ℋ

α

⁢

(

E

)

∈

[

0

,

+

∞

]

is called

α

-dimensional Hausdorff measure

of the set

E

⊂

X

.

Title

Hausdorff measure

Canonical name

HausdorffMeasure

Date of creation

2013-03-22 14:27:26

Last modified on

2013-03-22 14:27:26

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

8

Author

paolini (1187)

Entry type

Definition

Classification

msc 28A78

Related topic

HausdorffDimension

Related topic

LebesgueMeasure

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