uniqueness of measures extended from a π-system

uniqueness of measures extended from a

π

-system

The following

theorem

allows

measures

to be uniquely defined by specifying their values on a

π

-system (

http://planetmath.org/PiSystem

) instead of having to specify the measure of every possible

measurable set

. For example, the

collection

of

open intervals

(

a

,

b

)

⊆

ℝ

forms a

π

-system generating the Borel

σ

-algebra (

http://planetmath.org/BorelSigmaAlgebra

) and consequently the

Lebesgue measure

μ

is uniquely defined by the equality

μ

⁢

(

(

a

,

b

)

)

=

b

-

a

.

Theorem

.

Let

λ

,

μ

be measures on a

measurable space

(

X

,

A

)

. Suppose that

A

is a

π

-system on

X

generating

A

such that

λ

=

μ

on

A

and that there exists a

sequence

S

n

∈

A

with

⋃

n

=

1

∞

S

n

=

X

and

λ

⁢

(

S

n

)

<

∞

. Then,

λ

=

μ

.

Proof.

Choose any

T

∈

A

such that

λ

⁢

(

T

)

<

∞

and set

ℬ

=

{

S

∈

𝒜

:

λ

⁢

(

S

∩

T

)

=

μ

⁢

(

S

∩

T

)

}

. For any

S

∈

A

,

S

∩

T

∈

A

and the requirement that

λ

,

μ

agree on

A

gives

S

∈

ℬ

, so

ℬ

contains

A

. We show that

ℬ

is a

Dynkin system

in order to apply Dynkin’s lemma.
It is clear that

X

∈

ℬ

. Suppose that

S

1

⊆

S

2

are in

ℬ

. Then, the

additivity

of

λ

and

μ

gives

λ

⁢

(

(

S

2

∖

S

1

)

∩

T

)

=

λ

⁢

(

S

2

∩

T

)

-

λ

⁢

(

S

1

∩

T

)

=

μ

⁢

(

S

2

∩

T

)

-

μ

⁢

(

S

1

∩

T

)

=

μ

⁢

(

(

S

2

∖

S

1

)

∩

T

)

and therefore

S

2

∖

S

1

∈

ℬ

.
Now suppose that

S

n

is an increasing sequence of sets in

ℬ

increasing

to

S

⊆

X

. Then,

monotone

convergence of

λ

and

μ

gives

λ

⁢

(

S

∩

T

)

=

lim

n

→

∞

⁡

λ

⁢

(

S

n

∩

T

)

=

lim

n

→

∞

⁡

μ

⁢

(

S

n

∩

T

)

=

λ

⁢

(

S

∩

T

)

,

so

S

∈

ℬ

and

ℬ

is a Dynkin system containing

A

. By Dynkin’s lemma this shows that

ℬ

contains

σ

⁢

(

A

)

=

𝒜

.

We have shown that

λ

⁢

(

S

∩

T

)

=

μ

⁢

(

S

∩

T

)

for any

S

∈

𝒜

and

T

∈

A

with

λ

⁢

(

T

)

<

∞

. In the particular case where

X

∈

A

and

λ

,

μ

are finite measures then it follows that

λ

⁢

(

S

)

=

μ

⁢

(

S

)

simply by taking

T

=

X

.
More generally, choose a sequence of sets

T

n

∈

A

satisfying

λ

⁢

(

T

n

)

<

∞

and

⋃

n

T

n

=

X

. For any

S

∈

𝒜

,

S

n

≡

(

S

∩

T

n

)

∖

⋃

m

=

1

n

-

1

T

m

is a

pairwise disjoint

sequence of sets in

𝒜

with

S

n

⊆

T

n

and

⋃

n

S

n

=

S

. So,

λ

⁢

(

S

n

)

=

μ

⁢

(

S

n

)

and the

countable additivity

of

λ

and

μ

gives

λ

⁢

(

S

)

=

∑

n

λ

⁢

(

S

n

)

=

∑

n

μ

⁢

(

S

n

)

=

μ

⁢

(

S

)

.

∎

Title

uniqueness of measures extended from a

π

-system

Canonical name

UniquenessOfMeasuresExtendedFromApisystem

Date of creation

2013-03-22 18:33:08

Last modified on

2013-03-22 18:33:08

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

8

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Related topic

LebesgueMeasure

Related topic

DynkinsLemma

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