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)


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 we define new set . , Than one can show that every iff , every is π-system, .

Ok the motivation. I want to generate from P. So you can do it with transfinite induction that in ever step you add new sets in form and . 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 and but insted you use to generate new sets.

This is obvious.

every is π-system. . This is again in form because every previous are already pi-systems.

Can be show thanks to that every previous are pi-systems. You can than convert sum of sets to sum of disjoint sets.

This is quite easy. But you have to use fact that cofinality of is

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)

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