proof of existence of the Lebesgue measure

proof of existence of the Lebesgue measure

First, let

𝒞

be the

collection

of bounded

open intervals

of the

real numbers

. As this is a

π

-system (

http://planetmath.org/PiSystem

), uniqueness of

measures

extended from a

π

-system (

http://planetmath.org/UniquenessOfMeasuresExtendedFromAPiSystem

) shows that any measure defined on the

σ

-

algebra

σ

⁢

(

𝒞

)

is uniquely determined by its values restricted to

𝒞

. It remains to prove the

existence

of such a measure.

Define the length of an interval as

p

⁢

(

(

a

,

b

)

)

=

b

-

a

for

a

<

b

. The

Lebesgue outer measure

μ

*

:

𝒫

⁢

(

X

)

→

ℝ

+

∪

{

∞

}

is defined as

μ

*

⁢

(

A

)

=

inf

⁡

{

∑

i

=

1

∞

p

⁢

(

A

i

)

:

A

i

∈

𝒞

,

A

⊆

⋃

i

=

1

∞

A

i

}

.

(1)

This is indeed an

outer measure

(

http://planetmath.org/OuterMeasure2

) (see

construction of outer measures

) and, furthermore, for any interval of the form

(

a

,

b

)

it agrees with the standard definition of length,

μ

*

⁢

(

(

a

,

b

)

)

=

p

⁢

(

(

a

,

b

)

)

=

b

-

a

(see proof that the outer (Lebesgue) measure of an interval is its length (

http://planetmath.org/ProofThatTheOuterLebesgueMeasureOfAnIntervalIsItsLength

)).

We show that intervals

(

-

∞

,

a

)

are

μ

*

-

measurable

(

http://planetmath.org/CaratheodorysLemma

). Choosing any

ϵ

>

0

and interval

A

∈

𝒞

the definition of

p

gives

p

⁢

(

A

)

=

p

⁢

(

A

∩

(

-

∞

,

a

)

)

+

p

⁢

(

A

∩

(

a

,

∞

)

)

.

So, choosing an arbitrary set

E

⊆

ℝ

and a

sequence

A

i

∈

𝒞

covering

E

,

∑

i

=

1

∞

p

⁢

(

A

i

)

=

∑

i

=

1

∞

p

⁢

(

A

i

∩

(

-

∞

,

a

)

)

+

∑

i

=

1

∞

p

⁢

(

A

i

∩

(

a

,

∞

)

)

≥

μ

*

⁢

(

E

∩

(

-

∞

,

a

)

)

+

μ

*

⁢

(

E

∩

(

a

,

∞

)

)

.

So, from equation (

1

)

μ

*

⁢

(

E

)

≥

μ

*

⁢

(

E

∩

(

-

∞

,

a

)

)

+

μ

*

⁢

(

E

∩

(

a

,

∞

)

)

.

(2)

Also, choosing any

ϵ

>

0

and using the subadditivity of

μ

*

,

μ

*

⁢

(

E

∩

(

a

,

∞

)

)

≥

μ

*

⁢

(

E

∩

(

a

-

ϵ

,

∞

)

)

-

μ

*

⁢

(

E

∩

(

a

-

ϵ

,

a

+

ϵ

)

)

≥

μ

*

⁢

(

E

∩

[

a

,

∞

)

)

-

μ

*

⁢

(

(

a

-

ϵ

,

a

+

ϵ

)

)

=

μ

*

⁢

(

E

∩

[

a

,

∞

)

)

-

2

⁢

ϵ

.

As

ϵ

>

0

is arbitrary,

μ

*

⁢

(

E

∩

(

a

,

∞

)

)

≥

μ

*

⁢

(

E

∩

[

a

,

∞

)

)

and substituting into (

2

) shows that

μ

*

⁢

(

E

)

≥

μ

*

⁢

(

E

∩

(

-

∞

,

a

)

)

+

μ

*

⁢

(

E

∩

[

a

,

∞

)

)

.

Consequently, intervals of the form

(

-

∞

,

a

)

are

μ

*

-measurable. As such intervals generate the Borel

σ

-algebra and, by Caratheodory’s lemma, the

μ

*

-

measurable sets

form a

σ

-algebra on which

μ

*

is a measure, it follows that the

restriction

of

μ

*

to the Borel

σ

-algebra is itself a measure.

Title

proof of existence of the

Lebesgue measure

Canonical name

ProofOfExistenceOfTheLebesgueMeasure

Date of creation

2013-03-22 18:33:14

Last modified on

2013-03-22 18:33:14

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

10

Author

gel (22282)

Entry type

Proof

Classification

msc 26A42

Classification

msc 28A12

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