Dynkin system

Let $\mathrm{\Omega }$ be a set, and $𝒫(\mathrm{\Omega })$ be the power set of $\mathrm{\Omega }$. A Dynkin system on $\mathrm{\Omega }$ is a set $𝒟\subset 𝒫(\mathrm{\Omega })$ such that

1. $\mathrm{\Omega }\in 𝒟$

2. $A,B\in 𝒟\text{ and }A\subset B\Rightarrow B\setminus A\in 𝒟$

3. $A_n\in 𝒟,A_n\subset A_{n+1},n\ge 1\Rightarrow {\bigcup }_{k=1}^{\mathrm{\infty }}{A}_k\in 𝒟$.

Let $F\subset 𝒫(\mathrm{\Omega })$, and consider

$$\mathrm{\Gamma }=\{X:X\subset 𝒫(\mathrm{\Omega })\text{ is a Dynkin system and }F\subset X\}.$$(1)

We define the intersection of all the Dynkin systems containing $F$ as

$$𝒟(F):=\bigcap _{X\in \mathrm{\Gamma }}X$$(2)

One can easily verify that $𝒟(F)$ is itself a Dynkin system and that it contains $F$. We call $𝒟(F)$ the Dynkin system generated by $F$. It is the “smallest” Dynkin system containing $F$.

A Dynkin system which is also $\pi$-system (http://planetmath.org/PiSystem) is a $\sigma$-algebra (http://planetmath.org/SigmaAlgebra).