martingale proof of the Radon-Nikodym theorem

martingale proof of the Radon-Nikodym theorem

We apply the

martingale convergence theorem

to prove the

Radon-Nikodym theorem

, which states that if

μ

and

ν

are

σ

-finite

measures

on a

measurable space

(

Ω

,

ℱ

)

and

ν

is

absolutely continuous

with respect to

μ

then there exists a non-negative and

measurable

f

:

Ω

→

ℝ

such that

ν

⁢

(

A

)

=

∫

A

f

⁢

𝑑

μ

for all

measurable sets

A

.

As any

σ

-finite measure is

equivalent

to a

probability measure

(

http://planetmath.org/AnySigmaFiniteMeasureIsEquivalentToAProbabilityMeasure

), it is enough to prove the result in the case where

μ

and

ν

are probability measures. Furthermore, by the

Jordan decomposition

, the result generalizes to the case where

ν

is a

signed measure

. So, we just need to prove the following.

Theorem

(Radon-Nikodym).

Let

P

and

Q

be probability measures on the measurable space

(

Ω

,

F

)

, such that

Q

is absolutely continuous with respect to

P

.
Then, there exists a non-negative

random variable

X

such that

E

P

⁢

[

X

]

=

1

and

Q

⁢

(

A

)

=

E

P

⁢

[

1

A

⁢

X

]

for every

A

∈

F

.

Here,

X

is called the

Radon-Nikodym derivative

of

ℚ

with respect to

ℙ

.

More generally, for any sub-

σ

-

algebra

𝒢

of

ℱ

we can restrict the measures

ℙ

and

ℚ

to

𝒢

and ask if the Radon-Nikodym derivative of

ℚ

|

𝒢

with respect to

ℙ

|

𝒢

exists. If it does we shall denote it by

X

𝒢

, which by definition is a non-negative

𝒢

-measurable random variable satisfying

ℚ

⁢

(

A

)

=

𝔼

ℙ

⁢

[

1

A

⁢

X

𝒢

]

for all

A

∈

𝒢

.

We note that if

X

𝒢

does exist, then it is uniquely defined (

ℙ

-almost everywhere). Suppose that

X

~

𝒢

also satisfied the required properties, then

𝔼

ℙ

⁢

[

max

⁡

(

X

𝒢

-

X

~

𝒢

,

0

)

]

=

𝔼

ℙ

⁢

[

X

𝒢

⁢

1

{

X

𝒢

>

X

~

𝒢

}

]

-

𝔼

ℙ

⁢

[

X

~

𝒢

⁢

1

{

X

𝒢

>

X

~

𝒢

}

]

=

ℚ

(

X

𝒢

>

X

~

𝒢

)

-

ℚ

(

X

𝒢

>

X

~

𝒢

)

=

0

so

X

𝒢

≤

X

~

𝒢

almost surely. Similarly,

X

~

𝒢

≤

X

𝒢

and therefore

X

~

𝒢

=

X

𝒢

(almost surely).

First, the easy case. For a finite

σ

-algebra, the Radon-Nikodym derivative can be written out explicitly.

Lemma 1

.

If

G

is a finite sub-

σ

-algebra of

F

then the Radon-Nikodym derivative

X

G

exists.

Proof.

Let

A

1

,

A

2

,

…

,

A

n

be the

minimal

non-empty elements of

𝒢

. These are

pairwise disjoint

subsets of

Ω

such that every set in

𝒢

is a union of a subcollection of the

A

k

. Set

X

𝒢

=

∑

k

=

1

n

ℚ

⁢

(

A

k

)

ℙ

⁢

(

A

k

)

⁢

1

A

k

Note that whenever

ℙ

⁢

(

A

k

)

=

0

then

ℚ

⁢

(

A

k

)

=

0

, and we adopt the convention that

0

0

=

0

. Clearly,

X

𝒢

is

𝒢

-measurable, and

𝔼

ℙ

⁢

[

1

A

k

⁢

X

𝒢

]

=

ℚ

⁢

(

A

k

)

ℙ

⁢

(

A

k

)

⁢

𝔼

ℙ

⁢

[

1

A

k

]

+

∑

j

≠

k

ℚ

⁢

(

A

j

)

ℙ

⁢

(

A

j

)

⁢

𝔼

ℙ

⁢

[

1

A

k

∩

A

j

]

=

ℚ

⁢

(

A

k

)

.

Here, we have used

𝔼

ℙ

⁢

[

1

A

k

]

=

ℙ

⁢

(

A

k

)

and

1

A

k

∩

A

j

=

0

. By linearity, this equality remains true if both sides are replaced by any union of the

A

k

, and therefore

X

𝒢

is the required Radon-Nikodym derivative.
∎

Next,

martingale

convergence is used to prove the

existence

of the Radon-Nikodym derivative in the case where the

σ

-algebra

𝒢

is

separable

. By separable, we mean that there is a

countable

sequence

of sets

A

1

,

A

2

,

…

generating

𝒢

. Note that if we let

𝒢

n

be the

σ

-algebra

generated by

A

1

,

A

2

,

…

,

A

n

, then

𝒢

n

is an

increasing sequence

of finite sub-

σ

-algebras such that

⋃

n

𝒢

n

generates

𝒢

. The following result is general enough to apply in many useful cases, such as with the Boral

σ

-algebra on

ℝ

n

.

Lemma 2

.

Let

G

be a separable sub-

σ

-algebra of

F

. Then, the Radon-Nikodym derivative

X

G

exists.
If furthermore,

G

n

is an increasing sequence of finite

σ

-algebras satisfying

G

=

σ

⁢

(

⋃

n

G

n

)

then

E

P

⁢

[

|

X

G

-

X

G

n

|

]

→

0

as

n

→

∞

.

Proof.

Let us set

X

n

≡

X

𝒢

n

.
If

m

<

n

then the

conditional expectation

𝔼

ℙ

[

X

n

∣

𝒢

m

]

is

𝒢

m

-measurable, and for every

A

∈

𝒢

m

,

𝔼

ℙ

[

1

A

𝔼

ℙ

[

X

n

∣

𝒢

m

]

]

=

𝔼

ℙ

[

1

A

X

n

]

=

ℚ

(

A

)

.

This equality just uses the definition of the conditional expectation and then the definition of

X

n

as the Radon-Nikodym derivative restricted to

𝒢

n

. So,

𝔼

ℙ

[

X

n

∣

𝒢

m

]

is the Radon-Nikodym derivative restricted to

𝒢

m

, and equals

X

m

(almost-surely).

Therefore,

X

n

is a martingale and the martingale convergence theorem implies that the limit

X

𝒢

=

lim

n

→

∞

⁡

X

n

(1)

exists almost surely. We now show that the sequence

X

n

is

uniformly integrable

. Choose any

ϵ

>

0

. As

ℚ

is absolutely continuous with respect to

ℙ

, there exists a

δ

>

0

such that

ℚ

⁢

(

A

)

<

ϵ

whenever

ℙ

⁢

(

A

)

<

δ

. Using

ℙ

(

X

n

>

K

)

=

𝔼

ℙ

[

1

{

X

n

>

K

}

]

≤

𝔼

ℙ

[

X

n

K

]

=

1

K

we see that

ℙ

(

X

n

>

K

)

<

δ

whenever

K

>

δ

-

1

and, therefore,

ℚ

(

X

n

>

K

)

<

ϵ

. So

𝔼

ℙ

[

X

n

1

{

X

n

>

K

}

]

=

ℚ

(

X

n

>

K

)

<

ϵ

for every

n

, showing that

X

n

is a uniformly integrable sequence with respect to

ℙ

.
Therefore, convergence in (

1

) is in

L

1

, and

𝔼

ℙ

⁢

[

|

X

n

-

X

𝒢

|

]

→

0

as

n

→

∞

. So, for any

A

∈

⋃

n

𝒢

n

,

𝔼

ℙ

⁢

[

X

𝒢

⁢

1

A

]

=

lim

m

→

∞

⁡

𝔼

ℙ

⁢

[

X

m

⁢

1

A

]

=

ℚ

⁢

(

A

)

.

(2)

By linearity and the

monotone convergence theorem

, the

collection

of sets

A

satisfying (

2

) is a

Dynkin system

containing the

π

-system

⋃

n

𝒢

n

so, by Dynkin’s lemma, is satisfied for every

A

∈

σ

⁢

(

⋃

n

𝒢

n

)

=

𝒢

and, by definition,

X

𝒢

is the Radon-Nikodym derivative restricted to

𝒢

.
∎

Finally, by approximating by finite

σ

-algebras we can prove the Radon-Nikodym theorem for arbitrary inseparable

σ

-algebras

ℱ

.

Proof of the Radon-Nikodym theorem:

First, we use

contradiction

to show that for any

ϵ

>

0

there exists a finite

σ

-algebra

𝒢

⊆

ℱ

satisfying

𝔼

ℙ

⁢

[

|

X

𝒢

-

X

ℋ

|

]

<

ϵ

for every finite

σ

-algebra

ℋ

with

𝒢

⊆

ℋ

⊆

F

. If this were not the case, then by

induction

we could find an increasing sequence of finite sub-

σ

-algebras of

ℱ

satisfying

𝔼

ℙ

⁢

[

|

X

𝒢

n

-

X

𝒢

m

|

]

≥

ϵ

.
However, letting

𝒢

=

σ

⁢

(

⋃

n

𝒢

n

)

, Lemma

2

shows that

X

𝒢

exists and

ϵ

≤

lim

n

→

∞

⁡

𝔼

ℙ

⁢

[

|

X

𝒢

n

-

X

𝒢

n

+

1

|

]

≤

lim

n

→

∞

⁡

𝔼

ℙ

⁢

[

|

X

𝒢

n

-

X

𝒢

|

]

+

lim

n

→

∞

⁡

𝔼

ℙ

⁢

[

|

X

𝒢

n

+

1

-

X

𝒢

|

]

=

0

— a contradiction.

So, there exists a sequence of finite sub-

σ

-algebras

𝒢

n

of

ℱ

such that

𝔼

ℙ

⁢

[

|

X

𝒢

n

-

X

ℋ

|

]

<

2

-

n

for every finite sub-

σ

-algebra

ℋ

of

ℱ

containing

𝒢

n

. Let

𝒢

be the (separable)

σ

-algebra generated by

⋃

n

𝒢

n

, and set

𝒢

~

n

=

σ

⁢

(

⋃

k

=

1

n

𝒢

k

)

.
By Lemma

2

, the Radon-Nikodym derivative restricted to

𝒢

,

X

𝒢

, exists, and we show that it is the required

derivative

of

ℚ

with respect to

ℙ

.

Choose any set

A

∈

ℱ

and let

ℋ

n

be the (finite)

σ

-algebra generated by

𝒢

n

∪

{

A

}

. Then,

X

ℋ

n

exists and

satisfies

𝔼

ℙ

⁢

[

X

ℋ

n

⁢

1

A

]

=

ℚ

⁢

(

A

)

and,

|

𝔼

ℙ

⁢

[

X

𝒢

⁢

1

A

]

-

ℚ

⁢

(

A

)

|

=

lim

n

→

∞

⁡

|

𝔼

ℙ

⁢

[

X

𝒢

~

n

⁢

1

A

]

-

ℚ

⁢

(

A

)

|

=

lim

n

→

∞

⁡

|

𝔼

ℙ

⁢

[

X

𝒢

~

n

⁢

1

A

]

-

𝔼

ℙ

⁢

[

X

ℋ

n

⁢

1

A

]

|

≤

lim

n

→

∞

⁡

𝔼

ℙ

⁢

[

|

X

𝒢

~

n

-

X

𝒢

n

|

]

+

lim

n

→

∞

⁡

𝔼

ℙ

⁢

[

|

X

ℋ

n

-

X

𝒢

n

|

]

≤

lim

n

→

∞

⁡

(

2

-

n

+

2

-

n

)

=

0

.

So,

𝔼

ℙ

⁢

[

X

𝒢

⁢

1

A

]

=

ℚ

⁢

(

A

)

as required.

References

1

David Williams,

Probability

with martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

Title

martingale proof of the Radon-Nikodym theorem

Canonical name

MartingaleProofOfTheRadonNikodymTheorem

Date of creation

2013-03-22 18:34:10

Last modified on

2013-03-22 18:34:10

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

9

Author

gel (22282)

Entry type

Proof

Classification

msc 60G42

Classification

msc 28A15

Related topic

RadonNikodymTheorem

Related topic

MartingaleConvergenceTheorem

Generated on Sat Feb 10 11:59:17 2018 by

LaTeXML