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 \(\mathcal{P}\)