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# 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 βsmallestβ Dynkin system containing $F$.

A Dynkin system which is also $\pi$-system is a $\sigma$-algebra.

## Mathematics Subject Classification

03E20*no label found*28A60

*no label found*

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