construction of outer measures

construction of outer measures

The following theorem is used in

measure theory

to construct

outer measures

(

http://planetmath.org/OuterMeasure2

) on a set

X

, starting with a non-negative

function

on a collection of subsets of

X

. For example, if we take

X

to be the

real numbers

,

𝒞

to be the collection of bounded

open intervals

of

ℝ

and define

p

by

p

⁢

(

(

a

,

b

)

)

=

b

-

a

for real numbers

a

<

b

, then the Lebesgue outer measure is obtained.

Theorem

.

Let

X

be a set,

C

be a family of subsets of

X

containing the empty set and

p

:

C

→

R

∪

{

∞

}

be a function satisfying

p

⁢

(

∅

)

=

0

.
Then the function

μ

*

:

P

⁢

(

X

)

→

R

∪

{

∞

}

defined by

μ

*

⁢

(

A

)

=

inf

⁡

{

∑

i

=

1

∞

p

⁢

(

A

i

)

:

A

i

∈

𝒞

,

A

⊆

⋃

i

=

1

∞

A

i

}

(1)

is an outer measure.

Proof.

The definition of

μ

*

immediately gives

μ

*

⁢

(

A

)

≤

μ

*

⁢

(

B

)

for sets

A

⊆

B

, and if

A

=

∅

then we can take

A

i

=

∅

in (

1

) to obtain

μ

*

⁢

(

∅

)

≤

∑

i

p

⁢

(

∅

)

=

0

, giving

μ

*

⁢

(

∅

)

=

0

. Only the

countable subadditivity

of

μ

*

remains to be shown.
That is, if

A

i

is a sequence in

𝒫

⁢

(

X

)

then

μ

*

⁢

(

⋃

i

A

i

)

≤

∑

i

μ

*

⁢

(

A

i

)

.

(2)

To prove this

inequality

, we may restrict to the case where

μ

*

⁢

(

A

i

)

<

∞

for each

i

so that, choosing any

ϵ

>

0

, equation (

1

) says that there exists a sequence

A

i

,

j

∈

𝒞

such that

A

i

⊆

⋃

j

A

i

,

j

and,

∑

j

=

1

∞

p

⁢

(

A

i

,

j

)

≤

μ

*

⁢

(

A

i

)

+

2

-

i

⁢

ϵ

.

As

⋃

i

A

i

⊆

⋃

i

,

j

A

i

,

j

, equation (

1

) defining

μ

*

gives

μ

*

⁢

(

⋃

i

A

i

)

≤

∑

i

,

j

p

⁢

(

A

i

,

j

)

=

∑

i

∑

j

p

⁢

(

A

i

,

j

)

≤

∑

i

(

μ

*

⁢

(

A

i

)

+

2

-

i

⁢

ϵ

)

=

∑

i

μ

*

⁢

(

A

i

)

+

ϵ

.

As

ϵ

>

0

is arbitrary, this proves subadditivity (

2

).
∎

Although this result is rather general, placing few restrictions on the function

p

, there is no guarantee that the outer measure

μ

*

will agree with

p

for the sets in

𝒞

nor that

𝒞

will consist of

μ

*

-

measurable

(

http://planetmath.org/CaratheodorysLemma

) sets.

For example, if

X

=

ℝ

,

𝒞

consists of the bounded open intervals, and

p

⁢

(

(

a

,

b

)

)

=

(

b

-

a

)

2

for real numbers

a

<

b

, then

μ

*

⁢

(

(

a

,

b

)

)

=

0

≠

p

⁢

(

(

a

,

b

)

)

.

Alternatively if

p

⁢

(

(

a

,

b

)

)

=

b

-

a

for all

a

<

b

then it follows that

μ

*

⁢

(

(

a

,

b

)

)

=

b

-

a

so

μ

*

⁢

(

(

0

,

1

)

)

+

μ

*

⁢

(

[

1

,

2

)

)

=

1

+

1

≠

μ

*

⁢

(

(

0

,

2

)

)

=

2

,

and

(

0

,

1

)

is not

μ

*

-measurable.

Title

construction of outer measures

Canonical name

ConstructionOfOuterMeasures

Date of creation

2013-03-22 18:33:17

Last modified on

2013-03-22 18:33:17

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

10

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Related topic

OuterMeasure

Related topic

LebesgueOuterMeasure

Related topic

CaratheodorysLemma

Generated on Fri Feb 9 20:06:33 2018 by

LaTeXML