# Dynkin System

## 1 Definitions #

$$\mathcal{P}$$ is a $$\pi$$-system if: $A,B \in \mathcal{P} \Longrightarrow A\cap B \in \mathcal{P}$

Example

$$\left\{ (-\infty,a]: a\in \RR \right\}$$ is a $$\pi$$-system.

$$\left\{ (a,b]:-\infty<a\leq b<\infty \right\}$$ is a $$\pi$$-system. Note that here $$a$$ may equal to $$b$$, so that $$\emptyset$$ is also included, otherwise this is not a $$\pi$$-system.

$$\mathcal{L}$$ is a $$\lambda$$-system if:

1. $$\Omega\in \mathcal{L}$$
2. If $$A,B\in \mathcal{L}$$ and $$A \subset B$$, then $$B \setminus A \in \mathcal{L}$$
3. If $$A_1, A_2, \dots \in \mathcal{L}$$ and $$A_i \uparrow A$$, then $$A \in \mathcal{L}$$

Theorem: $$\mathcal{F}$$ is both a $$\pi$$-system and a $$\lambda$$-system, if and only if $$\mathcal{F}$$ is a $$\sigma$$-field.

Proof

(=>)

The first two conditions in the definition of $$\sigma$$-field is trivial. We only need to verify the third: If $$A_i \in \mathcal{F}$$, for $$i=1,2,3, \dots$$, then $$\bigcup_{i=1}^{\infty} A_i\in \mathcal{F}$$. This follows by observing that:

$\bigcup_{i=1}^{\infty} A_i = A_1 \cup (A_1 \cup A_2) \cup (A_1 \cup A_2 \cup A_3) \cup \dots$ $$A_1 \cup A_2 = \left( A_1^c \cap A_2^c \right)^c$$, and using the definitions of $$\lambda$$-system and $$\pi$$-system.

(<=)

Since $$A\cap B = (A^c \cup B^c)^c$$, by definition, it is a $$\pi$$-system.

The first and the third condition in the definition of $$\lambda$$-system is easy.

For the second condition, since $$B\setminus A = (\Omega \setminus A) \cap B$$, by definition this concludes the proof.

## 2 Dynkin’s $$\pi$$-$$\lambda$$ Theorem #

If $$\mathcal{P}$$ is a $$\pi$$-system, $$\mathcal{L}$$ is a $$\lambda$$-system, and $$\mathcal{P}\subset \mathcal{L}$$, then $$\sigma(\mathcal{P}) \subset \mathcal{L}$$.