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).

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