# Dynkin system

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

1. 1.

$\Omega\in\mathcal{D}$

2. 2.

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

3. 3.

$A_{n}\in\mathcal{D},\ A_{n}\subset A_{n+1},\ n\geq 1\Rightarrow\bigcup_{k=1}^{% \infty}A_{k}\in\mathcal{D}$.

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

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

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

 $\mathcal{D}(F):=\bigcap_{X\in\Gamma}X$ (2)

One can easily verify that $\mathcal{D}(F)$ is itself a Dynkin system and that it contains $F$. We call $\mathcal{D}(F)$ 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).

Title Dynkin system DynkinSystem 2013-03-22 12:21:19 2013-03-22 12:21:19 mathwizard (128) mathwizard (128) 9 mathwizard (128) Definition msc 03E20 msc 28A60 DynkinsLemma