Dynkin’s lemma

Dynkin’s lemma

Dynkin’s lemma is a result in

measure theory

showing that the

σ

-algebra (

http://planetmath.org/SigmaAlgebra

)

generated by

any given

π

-system (

http://planetmath.org/PiSystem

) on a set

X

coincides with the

Dynkin system

generated the

π

-system. The result can be used to prove that

measures

are uniquely determined by their values on

π

-systems generating the required

σ

-algebra. For example, the Borel

σ

-algebra on

ℝ

is generated by the

π

-system of

open intervals

(

a

,

b

)

for

a

<

b

and consequently the

Lebesgue measure

μ

is uniquely determined by the property that

μ

⁢

(

(

a

,

b

)

)

=

b

-

a

.

Note that this lemma generalizes the statement that a Dynkin system which is also a

π

-system is a

σ

-algebra.

Lemma

(Dynkin).

Let

A

be a

π

-system on a set

X

. Then

D

⁢

(

A

)

=

σ

⁢

(

A

)

. That is, the smallest Dynkin system containing

A

coincides with the

σ

-algebra generated by

A

.

Proof.

As

A

is a

π

-system, the set

𝒟

1

≡

{

S

⊆

X

:

S

∩

T

∈

𝒟

⁢

(

A

)

⁢

for every

⁢

T

∈

A

}

contains

A

. We show that

𝒟

1

is also a Dynkin system.

First, for every

T

∈

A

,

X

∩

T

=

T

∈

A

so

X

is in

𝒟

1

. Second, if

S

1

⊆

S

2

are in

𝒟

1

and

T

∈

A

then

(

S

2

∖

S

1

)

∩

T

=

(

S

2

∩

T

)

∖

(

S

1

∩

T

)

is in

𝒟

⁢

(

A

)

showing that

S

2

∖

S

1

∈

𝒟

1

.
Finally. if

S

n

∈

𝒟

1

is a

sequence

increasing

to

S

⊆

X

and

T

∈

A

then

S

n

∩

T

is a sequence in

𝒟

⁢

(

A

)

increasing to

S

∩

T

. As Dynkin systems are

closed under

limits of increasing sequences

this shows that

S

∩

T

∈

𝒟

⁢

(

A

)

and therefore

S

∈

𝒟

1

. So

𝒟

1

is indeed a Dynkin system. In particular,

𝒟

⁢

(

A

)

⊆

𝒟

1

and

S

∩

T

∈

𝒟

⁢

(

A

)

for all

S

∈

𝒟

⁢

(

A

)

and

T

∈

A

.

We now set

𝒟

2

≡

{

S

⊆

X

:

S

∩

T

∈

𝒟

⁢

(

A

)

⁢

for every

⁢

T

∈

𝒟

⁢

(

A

)

}

which, as shown above, contains

A

. Also, as in the

argument

above for

𝒟

1

,

𝒟

2

is a Dynkin system. Therefore,

𝒟

⁢

(

A

)

is

contained

in

𝒟

2

and it follows that

S

∩

T

∈

𝒟

⁢

(

A

)

for any

S

,

T

∈

𝒟

⁢

(

A

)

. So

𝒟

⁢

(

A

)

is both a

π

-system and a Dynkin system.

We can now show that

𝒟

⁢

(

A

)

is a

σ

-algebra. As it is a Dynkin system,

S

c

=

X

∖

S

∈

𝒟

⁢

(

A

)

for every

S

∈

𝒟

⁢

(

A

)

and, as it is also a

π

-system, this shows that

𝒟

⁢

(

A

)

is an

algebra of sets

on

X

. Finally, choose any sequence

A

n

∈

𝒟

⁢

(

A

)

. Then,

⋃

m

=

1

n

A

m

is a sequence in

𝒟

⁢

(

A

)

increasing to

⋃

n

A

n

which, as

𝒟

⁢

(

A

)

is Dynkin system, must be in

𝒟

⁢

(

A

)

. So,

𝒟

⁢

(

A

)

is a

σ

-algebra and must contain

σ

⁢

(

A

)

.

Conversely

, as

σ

⁢

(

A

)

is a Dynkin system (as it is a

σ

-algebra) containing

A

, it must also contain

𝒟

⁢

(

A

)

.
∎

References

1

David Williams,

Probability with martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

Title

Dynkin’s lemma

Canonical name

DynkinsLemma

Date of creation

2013-03-22 18:33:05

Last modified on

2013-03-22 18:33:05

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

11

Author

gel (22282)

Entry type

Theorem

Classification

msc 28A12

Synonym

pi-system d-system lemma

Related topic

PiSystem

Related topic

DynkinSystem

Related topic

UniquenessOfMeasuresExtendedFromAPiSystem

Generated on Fri Feb 9 20:06:40 2018 by

LaTeXML