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Dynkin's π−λ theorem

The illustration of Dynkin's Theorem
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 π−λ\pi-\lambda theorem is that σ\sigma-algebras are 'difficult', but π\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 P\mathcal{P} of subsets of space Ω\Omega is called a π\pi-system if it is closed under the formation of finite intersections:

  • (π1)(\pi_1)

    if A,B∈P A,B \in \mathcal{P} \: then  A∩B∈P\: A \cap B \in \mathcal{P}

A class L\mathcal{L} of subsets is a λ\lambda-system if

  • (λ1)(\lambda_1)

    Ω∈L\Omega \in \mathcal{L};

  • (λ2)(\lambda_2)

    if A,B∈LA,B \in \mathcal{L}, A⊂B A \subset B \: then  B∖A∈L \: B \setminus A \in \mathcal{L}

  • (λ3)(\lambda_3)

    if An∈LA_n \in \mathcal{L} and An↑AA_n \uparrow A then A∈LA \in \mathcal{L}

Because of the monotonicity condition in (λ3)(\lambda_3), the definition of λ\lambda-system is weaker than that of σ\sigma-algebra. Although a σ\sigma-algebra is a λ\lambda-system, the reverse is not true. But if the class is both a π\pi-system and a λ\lambda-system is a σ\sigma-algebra.
Lemma
A class that is both a π\pi-system and a λ\lambda-system is a σ\sigma-algebra.
\quad Proof. The class includes Ω\Omega, as indicated by (λ1)(\lambda_1). Furthermore, it is closed under the operation of taking complements, because (λ2)(\lambda_2) implies that if A∈LA \in \mathcal{L} then Ac=Ω∖A∈LA^{c} = \Omega \setminus A \in \mathcal{L}. Moreover, the class is closed under finite unions, exemplified by the fact that for any sets AA and BB in the class, A∪BA \cup B can be expressed as (Ac∩Bc)c(A^{c} \cap B^{c})^{c} due to (π1)(\pi_1). Extending this principle, the closure under countable unions is demonstrated as ⋃i=1nAi↑⋃i=1∞Ai\bigcup_{i=1}^{n} A_i \uparrow \bigcup_{i=1}^{\infty} A_i, and the condition (λ3)(\lambda_3) ensures that ⋃i=1∞Ai\bigcup_{i=1}^{\infty} A_i is contained within the class. Therefore, the class fulfills all the criteria to be considered a σ\sigma-algebra. □\Box
Now we can formulate the main theorem of the article.
Theorem
If P\mathcal{P} is a π\pi-system and L\mathcal{L} is λ\lambda-system that contains P\mathcal{P} then we have implication
P⊂L⟹σ(P)⊂L\begin{equation*}\mathcal{P} \subset \mathcal{L} \Longrightarrow \sigma(\mathcal{P}) \subset \mathcal{L}\end{equation*}

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

The proof strategy involves considering the smallest λ\lambda-system L0\mathcal{L}_0, that includes P\mathcal{P}. We will demonstrate that L0\mathcal{L}_0 is also a π\pi-system. From this, it follows by Lemma 1 that L0\mathcal{L}_0 is a σ\sigma-algebra as well. Given that σ(P)\sigma(\mathcal{P})(the smallest σ\sigma-algebra containing P\mathcal{P}) is included in L0\mathcal{L}_0 and consequently we establish the following relationship:
P⊂σ(P)⊂L0⊂L\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 L0\mathcal{L}_0 is indeed a π\pi-system.
Let L0\mathcal{L}_0 denote the λ\lambda-system generated by P\mathcal{P}. By definition, L0\mathcal{L}_0 is the intersection of all λ\lambda-systems that contain P\mathcal{P}.
L0=⋂αLα\begin{equation*}\mathcal{L}_0 = \bigcap_{\alpha} \mathcal{L}_{\alpha}\end{equation*}
As such, L0\mathcal{L}_0 itself is a λ\lambda-system, incorporates P\mathcal{P}, and is simultaneously a subset of any λ\lambda-system that includes P\mathcal{P}. This establishes the relationship: P⊂L0⊂L\mathcal{P} \subset \mathcal{L}_0 \subset \mathcal{L}.
Define GA={B:A∩B∈L0}G_A = \{ B : A \cap B \in \mathcal{L}_0 \} for A∈L0A \in \mathcal{L}_0 then GA G_A is a λ\lambda-system as it satisfies the following conditions:
  • Ω∈GA \Omega \in G_A since A∩Ω=A∈L0 A \cap \Omega = A \in \mathcal{L}_0, fulfilling (λ1)(\lambda_1);
  • if B,C∈GA B, C \in G_A and C⊂B C \subset B then the λ\lambda-system L0\mathcal{L}_0 contains A∩B A \cap B and A∩CA \cap C and hence contains the proper difference (A∩B)−(A∩C)=A∩(B−C)(A \cap B) - (A \cap C) = A \cap (B - C), meeting (λ2)(\lambda_2);
  • if Bn∈GAB_n \in G_A and Bn↑BB_n \uparrow B then A∩Bn↑A∩B∈L0A \cap B_n \uparrow A \cap B \in \mathcal{L}_0 since A∩Bn∈L0A \cap B_n \in \mathcal{L}_0 and L0\mathcal{L}_0 is a λ\lambda-system, so (λ3)(\lambda_3) is satisfied.
If A∈PA \in \mathcal{P}, then P⊂GA\mathcal{P} \subset G_A (since P\mathcal{P} is π\pi-system). Since L0\mathcal{L}_0 is the minimal λ\lambda-system containing P\mathcal{P}, it follows that L0⊂GA\mathcal{L}_0 \subset G_A. Hence, if A∈PA \in \mathcal{P} and B∈L0B \in \mathcal{L}_0, then A∩B∈L0A \cap B \in \mathcal{L}_0. Interchanging AA and BB in the last sentence: if A∈L0 A \in \mathcal{L}_0 and B∈PB \in \mathcal{P} then A∩B∈L0 A \cap B \in \mathcal{L}_0. But this implies that if A∈L0A \in \mathcal{L}_0 then P⊂GA\mathcal{P} \subset G_A and L0⊂GA\mathcal{L}_0 \subset G_A.
This conclusion implies that if A,B∈L0A, B \in \mathcal{L}_0, then A∩B∈L0A \cap B \in \mathcal{L}_0, thereby completing the proof that L0\mathcal{L}_0 is indeed a π\pi-system. □\Box
Theorem (Carathéodory's Extension Theorem)
Let μ\mu be a probability measure on an algebra A\mathcal{A}. Then μ\mu has a unique extension to σ(A)\sigma(\mathcal{A}) = the smallest σ\sigma-algebra containing A\mathcal{A}.
To prove that the extension in the Carathéodory's Theorem is unique, we need to the following lemma.
Lemma
Let PP be a π\pi-system. If v1v_1 and v2v_2 are probability measures (on σ\sigma-algebras F1F_1 and F2F_2) that agree on P\mathcal{P} and there is a sequence An∈PA_n \in P with An↑ΩA_n \uparrow \Omega, then v1v_1 and v2v_2 agree on σ(P)\sigma(P).
\quad Proof. Let A∈PA \in \mathcal{P} have v1(A)=v2(A)<∞v_1(A) = v_2(A) < \infty. Let
L={B∈σ(P):  v1(A∩B)=v2(A∩B)}\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 L\mathcal{L} is a λ\lambda-system. Since A∈PA \in P,  v1(A)=v2(A) \:v_1(A) = v_2(A) and Ω∈L\Omega \in \mathcal{L}, so (λ1)(\lambda_1) is satisfied. If B,C∈LB, C \in \mathcal{L} with C⊂BC \subset B then
v1(A∩(B−C))=v1(A∩B)−v1(A∩C)==v2(A∩B)−v2(A∩C)=v2(A∩(B−C))\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}
So (λ2)(\lambda_2) is verified. Finally, if Bn∈LB_n \in \mathcal{L} and Bn↑BB_n \uparrow B, then the continuity of probability measure implies
v1(A∩B)=lim⁡n→∞v1(A∩Bn)=lim⁡n→∞v2(A∩Bn)=v2(A∩B)\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 P\mathcal{P} is π\pi-system, the π−λ\pi - \lambda theorem implies σ(P)⊂L\sigma(\mathcal{P}) \subset \mathcal{L} , i.e., if A∈PA \in \mathcal{P} with v1(A)=v2(A)<∞v_1(A) = v_2(A) < \infty and B∈σ(P)B \in \sigma(\mathcal{P}) then v1(A∩B)=v2(A∩B)v_1(A \cap B) = v_2(A \cap B). Letting An∈PA_n \in \mathcal{P} with An↑ΩA_n \uparrow \Omega and using the continuity of probability measure, we have the desired conclusion. □\Box

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

Theorem (Carathéodory's Extension Theorem)
Let μ\mu be a probability measure on an algebra A\mathcal{A}. Then μ\mu has a unique extension to σ(A)\sigma(\mathcal{A}) = the smallest σ\sigma-algebra containing A\mathcal{A}.
To prove that the extension in the Carathéodory's Theorem is unique, we need to the following lemma.
Lemma
Let PP be a π\pi-system. If v1v_1 and v2v_2 are probability measures (on σ\sigma-algebras F1F_1 and F2F_2) that agree on P\mathcal{P} and there is a sequence An∈PA_n \in P with An↑ΩA_n \uparrow \Omega, then v1v_1 and v2v_2 agree on σ(P)\sigma(P).
\quad Proof. Let A∈PA \in \mathcal{P} have v1(A)=v2(A)<∞v_1(A) = v_2(A) < \infty. Let
L={B∈σ(P):  v1(A∩B)=v2(A∩B)}\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 L\mathcal{L} is a λ\lambda-system. Since A∈PA \in P,  v1(A)=v2(A) \:v_1(A) = v_2(A) and Ω∈L\Omega \in \mathcal{L}, so (λ1)(\lambda_1) is satisfied. If B,C∈LB, C \in \mathcal{L} with C⊂BC \subset B then
v1(A∩(B−C))=v1(A∩B)−v1(A∩C)==v2(A∩B)−v2(A∩C)=v2(A∩(B−C))\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}
So (λ2)(\lambda_2) is verified. Finally, if Bn∈LB_n \in \mathcal{L} and Bn↑BB_n \uparrow B, then the continuity of probability measure implies
v1(A∩B)=lim⁡n→∞v1(A∩Bn)=lim⁡n→∞v2(A∩Bn)=v2(A∩B)\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 P\mathcal{P} is π\pi-system, the π−λ\pi - \lambda theorem implies σ(P)⊂L\sigma(\mathcal{P}) \subset \mathcal{L} , i.e., if A∈PA \in \mathcal{P} with v1(A)=v2(A)<∞v_1(A) = v_2(A) < \infty and B∈σ(P)B \in \sigma(\mathcal{P}) then v1(A∩B)=v2(A∩B)v_1(A \cap B) = v_2(A \cap B). Letting An∈PA_n \in \mathcal{P} with An↑ΩA_n \uparrow \Omega and using the continuity of probability measure, we have the desired conclusion. □\Box

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

The illustration of Theorem about Independence of σ\sigma-algebras
Theorem
Suppose A1,A2,…,An\mathcal{A}_1, \mathcal{A}_2, \ldots, \mathcal{A}_n are independent π\pi-systems. Then the minimal σ\sigma-algebras σ(A1),σ(A2),…,σ(An)\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. A1\mathcal{A}_1 is replaced by σ(A1)\sigma(\mathcal{A}_1), then the new sequence of classes will also be independent.
Let A2,…,AnA_2, \ldots, A_n be sets with Ai∈AiA_i \in \mathcal{A}_i, let F=A2∩…∩AnF = A_2 \cap \ldots \cap A_n and let L={A:P(A∩F)=P(A)P(F)}\mathcal{L} = \{A : P(A \cap F) = P(A)P(F) \} is the set of all events which do not depend on A2,…,An\mathcal{A}_2, \ldots, \mathcal{A}_n. By definition, A1⊂L\mathcal{A}_1 \subset \mathcal{L} and we only need to prove that L\mathcal{L} is a λ\lambda-system in order to use Dynkin's π−λ\pi-\lambda theorem.
Since P(Ω∩F)=P(Ω)P(F)P(\Omega \cap F) = P(\Omega)P(F), so Ω∈L\Omega \in \mathcal{L} and (λ1)(\lambda_1) is fulfilled. To check (λ2)(\lambda_2) we note that if A,B∈LA, B \in \mathcal{L} with A⊂BA \subset B then (B−A)∩F=(B∩F)−(A∩F)(B - A) \cap F = (B \cap F) - (A \cap F):
P((B−A)∩F)=P(B∩F)−P(A∩F)=P(B)P(F)−P(A)P(F)=(P(B)−P(A))P(F)=P(B−A)P(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∈LB - A \in \mathcal{L}. To check (λ3)(\lambda_3) let Bk∈LB_k \in \mathcal{L} with Bk↑BB_k \uparrow B and (Bk∩F)↑(B∩F)(B_k \cap F) \uparrow (B \cap F) then using the continuity property of probability measure we get:
P(B∩F)=lim⁡k→∞P(Bk∩F)=lim⁡k→∞P(Bk)P(F)=P(B)P(F)\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 L\mathcal{L} is λ\lambda-system and with the π−λ\pi-\lambda theorem now gives σ(A1)⊂L\sigma(A_1) \subset \mathcal{L}. It follows that if A1∈σ(A1)A_1 \in \sigma(A_1) and Ai∈AiA_i \in \mathcal{A}_i for 2≤i≤n2 \leq i \leq n then
P(⋂i=1nAi)=P(A1)P(⋂i=2nAi)=∏i=1nP(Ai)\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
One of the main consequences of Dynkin's π\pi-λ\lambda theorem is the Monotone Class Theorem, a fundamental result in measure theory that provides a powerful way to extend properties from simple classes of functions to larger, more complex classes. The Monotone Class Theorem simplifies proofs in measure theory by allowing us to verify properties just for simple functions.
Theorem (Monotone class theorem)
Let A\mathcal{A} be a π\pi-system that contains Ω\Omega and let H\mathcal{H} be a collection of real-valued functions that satisfies:
  • (i) If A∈AA \in \mathcal{A}, then 1A∈H1_A \in \mathcal{H}.
  • (ii) If f,g∈Hf, g \in \mathcal{H}, then f+gf + g, and cf∈Hcf \in \mathcal{H} for any real number cc.
  • (iii) If fn∈Hf_n \in \mathcal{H} are nonnegative and increase to a bounded function ff, then f∈Hf \in \mathcal{H}.
Then H\mathcal{H} contains all bounded functions measurable with respect to σ(A)\sigma(\mathcal{A}).
\quad Proof. The assumption Ω∈A\Omega \in \mathcal{A} is equivalent to (λ1)(\lambda_1) in the definition of λ\lambda-system, (ii) is equivalent to (λ2)(\lambda_2), and (iii) is equivalent to (λ3)(\lambda_3), so G={A:1A∈H}\mathcal{G} = \{A : 1_A \in \mathcal{H}\} is a λ\lambda-system. By (i) and the Dynkin's π\pi-λ\lambda theorem, σ(A)⊂G\sigma(\mathcal{A}) \subset \mathcal{G}.
The condition (ii) implies that H\mathcal{H} contains all simple functions, and (iii) implies that H\mathcal{H} contains all bounded measurable functions, because we can approximate it with combinations of simple functions. □\Box

References