Radon measure

Radon measure

Let

X

be a

Hausdorff space

. A

Borel measure

μ

on

X

is said to be a

Radon measure

if it is:

1.

finite on

compact sets

,

2.

inner regular

(tight),

μ

⁢

(

A

)

=

sup

⁡

{

μ

⁢

(

V

)

∣

compact

⁢

V

⊂

A

}

.

A finite Radon measure

satisfies

μ

⁢

(

A

)

=

inf

⁡

{

μ

⁢

(

G

)

∣

open

⁢

G

⊃

A

}

.

Radon measures are not necessarily

locally finite

, although this is the case for

locally compact

and

metric spaces

. (

Counterexample

: spaces where only finite subsets are

compact

.)

A

Radon space

is a

topological space

on which every finite Borel measure is a Radon measure, this is the case, e.g. for

Polish spaces

or Hausdorff spaces that are

continuous

images of Polish spaces.

Radon measures are the “most important class of

measures

on arbitrary Hausdorff topological spaces” (König

[

2

]

, p.xiv) and formed the base of the

development

of integration theory by Bourbaki and Schwartz. In particular for locally compact spaces one often

defines

Radon measures as

linear functionals

μ

on the space

C

c

⁢

(

X

)

of

continuous functions

with compact support (‘Riesz representation

definition

’). Berg

et al.

give the following summary

[

1

]

, p. 62

f.

:

Given the finctional

μ

one defines

set functions

, in fact Borel measures, the

outer measure

μ

*

and the

essential outer measure

μ

∙

given by

(

G

)

*

=

sup

⁡

{

μ

⁢

(

f

)

|

f

∈

C

c

⁢

(

X

)

,

0

≤

f

≤

1

G

}

⁢

for open

⁢

G

⊂

X

⁢

and

(1)

μ

*

⁢

(

A

)

=

inf

⁡

{

μ

*

⁢

(

A

∩

K

)

|

K

∈

𝔎

⁢

(

X

)

}

⁢

for

⁢

A

⊂

X

,

(2)

μ

∙

⁢

(

A

)

=

sup

⁡

{

μ

*

⁢

(

A

∩

K

)

|

K

∈

𝔎

⁢

(

X

)

}

⁢

for

⁢

A

⊂

X

.

(3)

μ

∙

is a Radon measure in our sense, while

μ

*

is not always Radon. For locally compact and

σ

-

compact spaces

, however, both coincide (on the Borel algebra) and are

equivalent

to our Radon measure. For general Hausdorff spaces, Bourbaki introduces

W

*

⁢

(

A

)

=

sup

⁡

{

(

W

K

)

⁢

(

A

∩

K

)

|

K

∈

𝔎

⁢

(

X

)

}

, where
W, called a

Radon

premeasure

, associates a Radon measure

W

K

to each compact

K

⊂

X

, with

W

K

|

L

=

W

L

,

L

∈

𝔎

. This is a Radon measure (on

Borel sets

), Bourbaki, however, calls it only so if it is in

addition

locally finite.

Consider now Borel measures

ν

:

𝔅

↦

[

0

,

∞

]

which are

•

finite on compact sets,

ν

|

𝔎

<

∞

,

•

inner regular on the

open sets

,

ν

⁢

(

G

)

=

sup

⁡

{

ν

⁢

(

K

)

|

K

⊂

G

}

for

G

open, and

K

compact,

•

outer regular,

ν

(

B

)

=

inf

{

ν

(

G

)

|

B

⊂

G

for open

G

and Borel

B

.

then the measures

ν

correspond bijectively to locally finite Radon measures

μ

on

X

.

References

1

Christian Berg, Jens Peter Reus, Paul Ressel:

Harmonic analysis

on semigroups. – Berlin, 1984 (Graduate Texts in Mathematics; 100)

2

Heinz König: Measure and integration : an advanced course in basic procedures and applications.– Berlin, 1997.

Title

Radon measure

Canonical

name

RadonMeasure

Date of creation

2013-03-22 15:49:41

Last modified on

2013-03-22 15:49:41

Owner

ptr (5636)

Last modified by

ptr (5636)

Numerical id

16

Author

ptr (5636)

Entry type

Definition

Classification

msc 28C05

Classification

msc 28C15

Related topic

BorelMeasure

Related topic

SigmaFiniteBorelMeasureAndRelatedBorelConcepts

Defines

Radon space

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