proof of Carathéodory’s extension theorem

proof of Carathéodory’s extension theorem

The first step is to extend the

set function

μ

0

to the

power set

P

⁢

(

X

)

. For any subset

S

⊆

X

the value of

μ

*

⁢

(

S

)

is defined by taking

sequences

S

i

in

A

which cover

S

,

μ

*

⁢

(

S

)

≡

inf

⁡

{

∑

i

=

1

∞

μ

0

⁢

(

S

i

)

:

S

i

∈

A

,

S

⊆

⋃

i

=

1

∞

S

i

}

.

(1)

We show that this is an

outer measure

(

http://planetmath.org/OuterMeasure2

). First, it is clearly non-negative. Secondly, if

S

=

∅

then we can take

S

i

=

∅

in (

1

) to obtain

μ

*

⁢

(

S

)

≤

∑

i

μ

0

⁢

(

∅

)

=

0

, giving

μ

*

⁢

(

∅

)

=0. It is also clear that

μ

*

is

increasing

, so that if

S

⊆

T

then

μ

*

⁢

(

S

)

≤

μ

*

⁢

(

T

)

. The only remaining property to be proven is subadditivity. That is, if

S

i

is a sequence in

P

⁢

(

X

)

then

μ

*

⁢

(

⋃

i

S

i

)

≤

∑

i

μ

*

⁢

(

S

i

)

.

(2)

To prove this

inequality

, choose any

ϵ

>

0

and, by the definition (

1

) of

μ

*

, for each

i

there exists a sequence

S

i

,

j

∈

A

such that

S

i

⊆

⋃

j

S

i

,

j

and,

∑

j

=

1

∞

μ

0

⁢

(

S

i

,

j

)

≤

μ

*

⁢

(

S

i

)

+

2

-

i

⁢

ϵ

.

As

⋃

i

S

i

⊆

⋃

i

,

j

S

i

,

j

, equation (

1

) defining

μ

*

gives

μ

*

⁢

(

⋃

i

S

i

)

≤

∑

i

,

j

μ

0

⁢

(

S

i

,

j

)

=

∑

i

∑

j

μ

0

⁢

(

S

i

,

j

)

≤

∑

i

(

μ

*

⁢

(

S

i

)

+

2

-

i

⁢

ϵ

)

=

∑

i

μ

*

⁢

(

S

i

)

+

ϵ

.

As

ϵ

>

0

is arbitrary, this proves subadditivity (

2

). So,

μ

*

is indeed an outer measure.

The next step is to show that

μ

*

agrees with

μ

0

on

A

. So, choose any

S

∈

A

. The inequality

μ

*

⁢

(

S

)

≤

μ

0

⁢

(

S

)

follows from taking

S

1

=

S

and

S

i

=

∅

in (

1

), and it remains to prove the reverse inequality. So, let

S

i

be a sequence in

A

covering

S

, and set

S

i

′

=

(

S

∩

S

i

)

∖

⋃

j

=

1

i

-

1

S

j

∈

A

.

Then,

S

i

′

are

disjoint sets

satisfying

⋃

j

=

1

i

S

j

′

=

S

∩

⋃

j

=

1

i

S

j

and, therefore,

⋃

i

S

i

′

=

S

. By the

countable additivity

of

μ

0

,

∑

i

μ

0

⁢

(

S

i

)

=

∑

i

(

μ

0

⁢

(

S

i

′

)

+

μ

0

⁢

(

S

i

∖

S

i

′

)

)

≥

∑

i

μ

0

⁢

(

S

i

′

)

=

μ

0

⁢

(

S

)

.

As this inequality hold for any sequence

S

i

∈

A

covering

S

, equation (

1

) gives

μ

*

⁢

(

S

)

≥

μ

0

⁢

(

S

)

and, by combining with the reverse inequality, shows that

μ

*

does indeed agree with

μ

0

on

A

.

We have shown that

μ

0

extends to an outer measure

μ

*

on the power set of

X

. The final step is to apply Carathéodory’s lemma on the

restriction

of outer measures. A set

S

⊆

X

is said to be

μ

*

-

measurable

if the inequality

μ

*

⁢

(

E

)

≥

μ

*

⁢

(

E

∩

S

)

+

μ

*

⁢

(

E

∩

S

c

)

(3)

is satisfied for all subsets

E

of

X

. Carathéodory’s lemma then states that the

collection

ℱ

of

μ

*

-

measurable sets

is a

σ

-

algebra

(

http://planetmath.org/SigmaAlgebra

) and that the restriction of

μ

*

to

ℱ

is a

measure

.
To

complete

the proof of the

theorem

it only remains to be shown that every set in

A

is

μ

*

-measurable, as it will then follow that

ℱ

contains

𝒜

=

σ

⁢

(

A

)

and the restriction of

μ

*

to

𝒜

is a measure.

So, choosing any

S

∈

A

and

E

⊆

X

, the proof will be complete once it is shown that (

3

) is satisfied.
Given any

ϵ

>

0

, equation (

1

) says that there is a sequence

E

i

in

A

such that

E

⊆

⋃

i

E

i

and

∑

i

μ

0

⁢

(

E

i

)

≤

μ

*

⁢

(

E

)

+

ϵ

.

As

E

∩

S

⊆

⋃

i

(

E

i

∩

S

)

and

E

∩

S

c

⊆

⋃

i

(

E

i

∩

S

c

)

,

μ

*

⁢

(

E

∩

S

)

+

μ

*

⁢

(

E

∩

S

c

)

≤

∑

i

μ

0

⁢

(

E

i

∩

S

)

+

∑

i

μ

0

⁢

(

E

i

∩

S

c

)

=

∑

i

μ

0

⁢

(

E

i

)

≤

μ

*

⁢

(

E

)

+

ϵ

.

Since

ϵ

is arbitrary, this shows that (

3

) is satisfied and

S

is

μ

*

-measurable.

Title

proof of Carathéodory’s extension theorem

Canonical name

ProofOfCaratheodorysExtensionTheorem

Date of creation

2013-03-22 18:33:28

Last modified on

2013-03-22 18:33:28

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 28A12

Related topic

CaratheodorysLemma

Related topic

Measure

Related topic

OuterMeasure2

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