Dynkin system

A Dynkin system, named in honor of the Russian mathematician Eugene Dynkin, is a collection of subsets of another universal set $Omega$ satisfying some specific rules. They are also referred to as λ-systems.

Definitions

Let $Omega$ be a nonempty set, and let $D$ be a collection of subsets of $Omega$, i.e. $D$ is a subset of the power set of $Omega$. Then $D$ is a Dynkin system if
* the set $Omega$ itself is in $D$
* $D$ is closed under relative complementation, i.e. $A,Bin\; D$ and $A\; subseteq\; B$ implies $B\; setminus\; A\; in\; D$
* $D$ is closed under the countable union of increasing sequences, i.e. $A\_nin\; D$ and $A\_n\; subseteq\; A\_\{n+1\},\; n\; ge\; 1$ implies $cup\_\{n=1\}^\{infty\}A\_nin\; D$.

$D$ is a λ-system if
* the set $Omega$ itself is in $D$
* $D$ is closed under complementation, i.e. $Ain\; D$ implies $A^cin\; D$
* $D$ is closed under disjoint countable unions, i.e. $A\_nin\; D,\; ngeq1$ with $A\_icap\; A\_j=emptyset$ for all $i\; eq\; j$ implies $cup\_\{n=1\}^infty\; A\_nin\; D$.

It can be shown that these two definitions are logically equivalent, so that Dynkin systems are λ-systems and vice versa.

A Dynkin system which is also a π-system is a σ-algebra.

Given any collection $mathcal\{J\}$ of subsets of $Omega$, there exists a unique Dynkin system denoted $D\{mathcal\; J\}$ which is minimal with respect to containing $mathcal\; J$. That is, if $ilde\; D$ is any Dynkin system containing $mathcal\; J$, then $D\{mathcal\; J\}subseteq\; ilde\; D$. $D\{mathcal\; J\}$ is called the Dynkin system generated by $mathcal\{J\}$. Note $D\{emptyset\}=\{emptyset,Omega\}$. For another example, let $Omega=\{1,2,3,4\}$ and $mathcal\; J=\{1\}$; then $D\{mathcal\; J\}=\{emptyset,\{1\},\{2,3,4\},Omega\}$.

Dynkin's π-λ Theorem

If $P$ is a π-system and $D$ is a Dynkin system with $Psubseteq\; D$, then $sigma\{P\}subseteq\; D$. In other words, the σ-algebra generated by $P$ is contained in $D$.

One application of Dynkin's π-λ theorem is the uniqueness of the Lebesgue measure:

Let (&Omega;, "B", &lambda;) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let &mu; be another measure on &Omega; satisfying &mu; [("a","b")] = "b" - "a", and let "D" be the family of sets such that &mu; [D] = &lambda; [D] . Let "I" = { ("a","b"), ["a","b"),("a","b"] , ["a","b"] : 0 < "a" &le; "b" < 1 }, and observe that "I" is closed under finite intersections, that "I" &sub; "D", and that "B" is the &sigma;-algebra generated by "I". One easily shows "D" satisfies the above conditions for a Dynkin-system. From Dynkin's lemma it follows that "D" is in fact all of "B", which is equivalent to showing that the Lebesgue measure is unique.

Bibliography

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first = Patrick
title = Probability and Measure
publisher = John Wiley & Sons, Inc.
year = 1995
location = New York
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