Dynkin's theorem

First we define a structure named $\lambda$-system.

A class of subsets $\mathcal{L}$ of $\Omega$ is called a $\lambda$-system if it satisfies the following postulates

1.  $\Omega\in \mathcal{L}$
2. $A\in\mathcal{L} \Rightarrow A^c \in \mathcal{L}$
3. $n\neq m, A_nA_m = \emptyset, A_n \in \mathcal{L} \Rightarrow \cup_n A_n \in \mathcal{L}$

It is clear that a $\sigma$-field is always a $\lambda$-system.

Next a $\pi$-system is a class of sets closed under finite intersections.

Dynkin's theorem

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

b) If $\mathcal{P}$ is a $\pi$-system,

$\sigma(\mathcal{P})=\mathcal{L}(\mathcal{P})$

that is, the minimal $\sigma$-field over $\mathcal{P}$ equals the minimal $\lambda$-system over $\mathcal{P}$