Dynkin's $\pi-\lambda$ theorem

Contents

The Dynkin's π-λ theorem is very useful result in a measure theory which allows to replace σ-algebra with π-system in numerous applications.

The core insight behind Dynkin's

\pi-\lambda

theorem is that

\sigma

-algebras are 'difficult', but

\pi

-systems are 'easy', prompting us to primarily work with the latter. For example, this theorem significantly simplifies the proof of the uniqueness aspect of Carathéodory's Theorem.

A class

\mathcal{P}

of subsets of space

\Omega

is called a $\pi$ -system if it is closed under the formation of finite intersections:

$(\pi_1)$
if $A,B \in \mathcal{P} \:$ then $\: A \cap B \in \mathcal{P}$

A class

\mathcal{L}

of subsets is a $\lambda$ -system if

$(\lambda_1)$
$\Omega \in \mathcal{L}$ ;
$(\lambda_2)$
if $A,B \in \mathcal{L}$ , $A \subset B \:$ then $\: B \setminus A \in \mathcal{L}$
$(\lambda_3)$
if $A_n \in \mathcal{L}$ and $A_n \uparrow A$ then $A \in \mathcal{L}$

Because of the monotonicity condition in

(\lambda_3)

, the definition of

\lambda

-system is weaker than that of

\sigma

-algebra. Although a

\sigma

-algebra is a

\lambda

-system, the reverse is not true. But if the class is both a

\pi

-system and a

\lambda

-system is a

\sigma

-algebra.

Lemma . A class that is both a

\pi

-system and a

\lambda

-system is a

\sigma

-algebra.

\quad

Proof. The class includes

\Omega

, as indicated by

(\lambda_1)

. Furthermore, it is closed under the operation of taking complements, because

(\lambda_2)

implies that if

A \in \mathcal{L}

then

A^{c} = \Omega \setminus A \in \mathcal{L}

. Moreover, the class is closed under finite unions, exemplified by the fact that for any sets

A

and

B

in the class,

A \cup B

can be expressed as

(A^{c} \cap B^{c})^{c}

due to

(\pi_1)

. Extending this principle, the closure under countable unions is demonstrated as

\bigcup_{i=1}^{n} A_i \uparrow \bigcup_{i=1}^{\infty} A_i

, and the condition

(\lambda_3)

ensures that

\bigcup_{i=1}^{\infty} A_i

is contained within the class. Therefore, the class fulfills all the criteria to be considered a

\sigma

-algebra.

\Box

Now we can formulate the main theorem of the article.

Theorem
If

\mathcal{P}

is a

\pi

-system and

\mathcal{L}

\lambda

-system that contains

\mathcal{P}

then we have implication

\begin{equation*}\mathcal{P} \subset \mathcal{L} \Longrightarrow \sigma(\mathcal{P}) \subset \mathcal{L}\end{equation*}

The illustration of Dynkin's Theorem

Proof Dynkin's $\pi-\lambda$ theorem

The foolowing proof is a combination of [Durrett2019] p. 395 and [Billingsley2012] p. 43

Proof Dynkin's $\pi-\lambda$ theorem

The foolowing proof is a combination of [Durrett2019] p. 395 and [Billingsley2012] p. 43

The proof strategy involves considering the smallest

\lambda

-system

\mathcal{L}_0

, that includes

\mathcal{P}

. We will demonstrate that

\mathcal{L}_0

is also a

\pi

-system. From this, it follows by Lemma 1 that

\mathcal{L}_0

is a

\sigma

-algebra as well. Given that

\sigma(\mathcal{P})

(the smallest

\sigma

-algebra containing

\mathcal{P}

) is included in

\mathcal{L}_0

and consequently we establish the following relationship:

\begin{equation*}\mathcal{P} \subset \sigma(\mathcal{P}) \subset \mathcal{L}_0 \subset \mathcal{L}\end{equation*}

Thus, the critical task at hand is to verify that

\mathcal{L}_0

is indeed a

\pi

-system.

The smallest $\lambda$ -system $\mathcal{L}_0$

Let

\mathcal{L}_0

denote the

\lambda

-system generated by

\mathcal{P}

. By definition,

\mathcal{L}_0

is the intersection of all

\lambda

-systems that contain

\mathcal{P}

\begin{equation*}\mathcal{L}_0 = \bigcap_{\alpha} \mathcal{L}_{\alpha}\end{equation*}

As such,

\mathcal{L}_0

itself is a

\lambda

-system, incorporates

\mathcal{P}

, and is simultaneously a subset of any

\lambda

-system that includes

\mathcal{P}

. This establishes the relationship:

\mathcal{P} \subset \mathcal{L}_0 \subset \mathcal{L}

$G_A$ is a $\lambda$ -system

Define

G_A = \{ B : A \cap B \in \mathcal{L}_0 \}

for

A \in \mathcal{L}_0

then

G_A

is a

\lambda

-system as it satisfies the following conditions:

$\Omega \in G_A$ since $A \cap \Omega = A \in \mathcal{L}_0$ , fulfilling $(\lambda_1)$ ;
if $B, C \in G_A$ and $C \subset B$ then the $\lambda$ -system $\mathcal{L}_0$ contains $A \cap B$ and $A \cap C$ and hence contains the proper difference $(A \cap B) - (A \cap C) = A \cap (B - C)$ , meeting $(\lambda_2)$ ;
if $B_n \in G_A$ and $B_n \uparrow B$ then $A \cap B_n \uparrow A \cap B \in \mathcal{L}_0$ since $A \cap B_n \in \mathcal{L}_0$ and $\mathcal{L}_0$ is a $\lambda$ -system, so $(\lambda_3)$ is satisfied.

$\mathcal{L}_0$ is a $\pi$ -system

A \in \mathcal{P}

, then

\mathcal{P} \subset G_A

(since

\mathcal{P}

\pi

-system). Since

\mathcal{L}_0

is the minimal

\lambda

-system containing

\mathcal{P}

, it follows that

\mathcal{L}_0 \subset G_A

. Hence, if

A \in \mathcal{P}

and

B \in \mathcal{L}_0

, then

A \cap B \in \mathcal{L}_0

. Interchanging

A

and

B

in the last sentence: if

A \in \mathcal{L}_0

and

B \in \mathcal{P}

then

A \cap B \in \mathcal{L}_0

. But this implies that if

A \in \mathcal{L}_0

then

\mathcal{P} \subset G_A

and

\mathcal{L}_0 \subset G_A

.This conclusion implies that if

A, B \in \mathcal{L}_0

, then

A \cap B \in \mathcal{L}_0

, thereby completing the proof that

\mathcal{L}_0

is indeed a

\pi

-system.

\Box

Applications

Application 1. Uniqueness of Carathéodory's Theorem

The foolowing lemma is taken from [Durrett2019] p. 396

Theorem (Carathéodory's Extension Theorem)
Let

\mu

be a probability measure on an algebra

\mathcal{A}

. Then

\mu

has a unique extension to

\sigma(\mathcal{A})

= the smallest

\sigma

-algebra containing

\mathcal{A}

To prove that the extension in the Carathéodory's Theorem is unique, we need to the following lemma.

Lemma . Let

P

be a

\pi

-system. If

v_1

and

v_2

are probability measures (on

\sigma

-algebras

F_1

and

F_2

) that agree on

\mathcal{P}

and there is a sequence

A_n \in P

with

A_n \uparrow \Omega

, then

v_1

and

v_2

agree on

\sigma(P)

\quad

Proof. Let

A \in \mathcal{P}

have

v_1(A) = v_2(A) < \infty

. Let

\begin{equation}\mathcal{L} = \{ B \in \sigma(P) : \: \: v_1(A \cap B) = v_2(A \cap B)\}\end{equation}

We will now show that

\mathcal{L}

is a

\lambda

-system. Since

A \in P

\:v_1(A) = v_2(A)

and

\Omega \in \mathcal{L}

, so

(\lambda_1)

is satisfied. If

B, C \in \mathcal{L}

with

C \subset B

then

\begin{align}v_1(A \cap (B - C)) &= v_1(A \cap B) - v_1(A \cap C) = \\&= v_2(A \cap B) - v_2(A \cap C) = v_2(A \cap (B - C))\end{align}

(\lambda_2)

is verified. Finally, if

B_n \in \mathcal{L}

and

B_n \uparrow B

, then the continuity of probability measure implies

\begin{equation}v_1(A \cap B) = \lim_{n \to \infty} v_1(A \cap B_n) = \lim_{n \to \infty} v_2(A \cap B_n) = v_2(A \cap B)\end{equation}

Since

\mathcal{P}

\pi

-system, the

\pi - \lambda

theorem implies

\sigma(\mathcal{P}) \subset \mathcal{L}

, i.e., if

A \in \mathcal{P}

with

v_1(A) = v_2(A) < \infty

and

B \in \sigma(\mathcal{P})

then

v_1(A \cap B) = v_2(A \cap B)

. Letting

A_n \in \mathcal{P}

with

A_n \uparrow \Omega

and using the continuity of probability measure, we have the desired conclusion.

\Box

Application 2. Independence of $\sigma$ -algebras

The foolowing theorem is partially taken from [Durrett2019] p. 39

The illustration of Theorem about Independence of

\sigma

-algebras

Theorem
Suppose

\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_n

are independent

\pi

-systems. Then the minimal

\sigma

-algebras

\sigma(\mathcal{A}_1), \sigma(\mathcal{A}_2), \ldots, \sigma(\mathcal{A}_n)

are independent.

\quad

Proof. It is sufficient to show that if one of the classes i.e.

\mathcal{A}_1

is replaced by

\sigma(\mathcal{A}_1)

, then the new sequence of classes will also be independent.
Let

A_2, \ldots, A_n

be sets with

A_i \in \mathcal{A}_i

, let

F = A_2 \cap \ldots \cap A_n

and let

\mathcal{L} = \{A : P(A \cap F) = P(A)P(F) \}

is the set of all events which do not depend on

\mathcal{A}_2, \ldots, \mathcal{A}_n

. By definition,

\mathcal{A}_1 \subset \mathcal{L}

and we only need to prove that

\mathcal{L}

is a

\lambda

-system in order to use Dynkin's

\pi-\lambda

theorem.
Since

P(\Omega \cap F) = P(\Omega)P(F)

, so

\Omega \in \mathcal{L}

and

(\lambda_1)

is fulfilled. To check

(\lambda_2)

we note that if

A, B \in \mathcal{L}

with

A \subset B

then

(B - A) \cap F = (B \cap F) - (A \cap F)

\begin{align*}P((B - A) \cap F) &= P(B \cap F) - P(A \cap F) \\&= P(B)P(F) - P(A)P(F) \\&= (P(B) - P(A))P(F) \\&= P(B - A)P(F)\end{align*}

and we have

B - A \in \mathcal{L}

. To check

(\lambda_3)

let

B_k \in \mathcal{L}

with

B_k \uparrow B

and

(B_k \cap F) \uparrow (B \cap F)

then using the continuity property of probability measure we get:

\begin{equation}P(B \cap F) = \lim_{k \to \infty} P(B_k \cap F) = \lim_{k \to \infty} P(B_k)P(F) = P(B)P(F)\end{equation}

Thus

\mathcal{L}

\lambda

-system and with the

\pi-\lambda

theorem now gives

\sigma(A_1) \subset \mathcal{L}

. It follows that if

A_1 \in \sigma(A_1)

and

A_i \in \mathcal{A}_i

for

2 \leq i \leq n

then

\begin{equation}P\left(\bigcap_{i=1}^n A_i\right) = P(A_1)P\left(\bigcap_{i=2}^n A_i\right) = \prod_{i=1}^n P(A_i)\end{equation}

The theorem is thus proved.

\Box

References

[Durrett2019]
Rick Durrett. Probability Theory and Examples. Fifth edition. 2019.
[Billingsley2012]
Billingsley, Patrick. Probability and measure. Anniversary edition. 2012.

Dynkin's π−λ\pi-\lambdaπ−λ theorem

Proof Dynkin's π−λ\pi-\lambdaπ−λ theorem

Applications

Proof Dynkin's π−λ\pi-\lambdaπ−λ theorem

Proof Dynkin's π−λ\pi-\lambdaπ−λ theorem

The smallest λ\lambdaλ-system L0\mathcal{L}_0L0​

GAG_AGA​ is a λ\lambdaλ-system

L0\mathcal{L}_0L0​ is a π\piπ-system

Applications

Applications

Application 1. Uniqueness of Carathéodory's Theorem

Application 2. Independence of σ\sigmaσ-algebras

References

References

Dynkin's $\pi-\lambda$ theorem

Proof Dynkin's $\pi-\lambda$ theorem

Proof Dynkin's $\pi-\lambda$ theorem

The smallest $\lambda$ -system $\mathcal{L}_0$

$G_A$ is a $\lambda$ -system

$\mathcal{L}_0$ is a $\pi$ -system

Application 2. Independence of $\sigma$ -algebras