# Talk:Dynkin system

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## λ system

The equivalent λ system is defined as follows: Definition. A family L of subsets of C is called a λ-system if

(1) Ω belongs to L,
(2) L is closed under complementation,
(3) L is closed under countable unions of pw disjoint sets.

Given any class C of sets, L(C) denotes the λ-system generated by C. Jackzhp 23:50, 28 October 2006 (UTC)

## asdf

I changed a slight mistake. Sorry, no latex improvement. September 13 / 2006 (USF)

## A λproof for Dynkin's Lemma?

I suggest adding in a proof for Dynkin's Lemma.--A 20:00, 5 October 2007 (UTC)

I wonder, would [1] be an appropriate source?--A 20:04, 5 October 2007 (UTC)

If the site is self-published, as it appears to be, then it would not qualify unless the author is already a recognized (and published) expert on the topic. The author calls himself a derivatives trader. While he cold have a degree in statistics, we really don't know anything else about him or the website. See WP:RS and WP:V for the definitive rules. ·:· Will Beback ·:· 22:41, 5 October 2007 (UTC)

I came up with proof(not sure if it is right) during preparation for exam. It uses transfinite induction. For every ordinal ${\displaystyle \lambda <\omega _{1}}$ we define new set ${\displaystyle A_{\lambda }}$. ${\displaystyle A_{0}=P\cup \emptyset \cup \Omega }$, ${\displaystyle A_{\lambda }=\left\{\bigcup _{\alpha \in \omega _{0}}G_{\alpha }\setminus \bigcup _{\beta \in \omega _{0}}H_{\beta }|G_{i},H_{j}\in \bigcup _{\mu <\lambda }A_{\mu }\right\}}$ Than one can show that every ${\displaystyle A_{\lambda }\subset D,A_{\lambda }\subset \sigma (P),A_{\lambda }\subset A_{\mu }}$ iff ${\displaystyle \lambda <\mu }$, every ${\displaystyle A_{\lambda }}$ is π-system, ${\displaystyle \bigcup _{\lambda \in \omega _{1}}A_{\lambda }=\sigma (P),\bigcup _{\lambda \in \omega _{1}}A_{\lambda }\subset D}$.

Ok the motivation. I want to generate ${\displaystyle \sigma (P)}$ from P. So you can do it with transfinite induction that in ever step you add new sets in form ${\displaystyle \bigcup A_{i}}$ and ${\displaystyle A\setminus B}$. But than it is hard to show that all these new sets are still in Dynkin's system. So you want in every step create pi-system and than it is easy to show that new set generated from pi-system is still in Dynkin's system. So in every step you don't use operation ${\displaystyle \bigcup }$ and ${\displaystyle \setminus }$ but insted you use ${\displaystyle \bigcup \setminus \bigcup }$ to generate new sets.

${\displaystyle A_{\lambda }\subset \sigma (P)}$ This is obvious.

every ${\displaystyle A_{\lambda }}$ is π-system. ${\displaystyle (\bigcup _{j}A_{j}\setminus \bigcup _{j}B_{j})\cap (\bigcup _{j}C_{j}\setminus \bigcup _{j}D_{j})=\bigcup _{i,j}A_{i}\cap C_{j}\setminus (\bigcup _{j}B_{j}\cup \bigcup _{j}D_{j})}$. This is again in form ${\displaystyle \bigcup \setminus \bigcup }$ because every previous ${\displaystyle A_{\mu },\mu <\lambda }$ are already pi-systems.

${\displaystyle A_{\lambda }\subset D}$ Can be show thanks to that every previous ${\displaystyle A_{\mu },\mu <\lambda }$ are pi-systems. You can than convert sum of sets to sum of disjoint sets.

${\displaystyle \bigcup _{\lambda \in \omega _{1}}A_{\lambda }=\sigma (P)}$ This is quite easy. But you have to use fact that cofinality of ${\displaystyle \omega _{1}}$ is ${\displaystyle \omega _{1}}$

So if anyone would have time a will to check it I would be happy to rewrite it properly and post it.

## Not the Doob-Dynkin lemma

Maybe the "Dynkin's π-λ Theorem" is sometimes called "Dynkin's lemma", but surely it is not the "Doob–Dynkin lemma" (and not related to it). I correct the text accordingly. Boris Tsirelson (talk) 08:21, 7 September 2012 (UTC)