Difference between revisions of "Dynkin system"

Revision as of 23:26, 2 August 2015

Note: a Dynkin system is also called a "$d$-system"^[1] and the page d-system just redirects here.

Definition

First Definition

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ Given a set $X$ and a family of subsets of $X$, which we shall denote $\mathcal{D}\subseteq\mathcal{P}(X)$ is a Dynkin system^[2] if:

• $X\in\mathcal{D}$
• For any $D\in\mathcal{D}$ we have $D^c\in\mathcal{D}$
• For any $(D_n)_{n=1}^\infty\subseteq\mathcal{D}$ is a sequence of pairwise disjoint sets we have $\udot_{n=1}^\infty D_n\in\mathcal{D}$

Second Definition

Given a set $X$ and a family of subsets of $X$ we denote $\mathcal{D}\subseteq\mathcal{P}(X)$ is a Dynkin system^[3] on $X$ if:

• $X\in\mathcal{D}$
• $\forall A,B\in\mathcal{D}[B\subseteq A\implies A-B\in\mathcal{D}]$
• Given a sequence $(A_n)_{n=1}^\infty\subseteq\mathcal{D}$ that is increasing^{[Note 1]} and has $\lim_{n\rightarrow\infty}(A_n)=A$ we have $A\in\mathcal{D}$

Immediate results

See also

• Dynkin system generated by
• Types of set algebras
• $p$-system
• Conditions for a $d$-system to be a $\sigma$-algebra

Notes

1. Jump up ↑ Recall this means $A_{n}\subseteq A_{n+1}$

References

1. Jump up ↑ Probability and Stochastics - Erhan Cinlar
2. Jump up ↑ Measures, Integrals and Martingales - René L. Schilling
3. Jump up ↑ Probability and Stochastics - Erhan Cinlar