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.

$\Omega\in\mathcal{D}$
2.

$A,B\in\mathcal{D}\text{ and }A\subset B\Rightarrow B\setminus A\in\mathcal{D}$
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
Canonical name	DynkinSystem
Date of creation	2013-03-22 12:21:19
Last modified on	2013-03-22 12:21:19
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	9
Author	mathwizard (128)
Entry type	Definition
Classification	msc 03E20
Classification	msc 28A60
Related topic	DynkinsLemma