Dynkin system

From Wikipedia, the free encyclopedia

A Dynkin system on $Ω$ is a set $\mathcal{D}$ consisting of certain subsets of $Ω$ such that

the set $Ω$ itself is in $\mathcal{D}$
if $A,B \in \mathcal{D}$ and $A \subseteq B$ then $B \setminus A \in \mathcal{D}$
if $A n$ is a sequence of sets in $\mathcal{D}$ which is increasing in the sense that $A_n \subseteq A_{n+1},\ n \ge 1$ , then the union $\bigcup_{k=1}^{\infty}A_k$ also lies in $\mathcal{D}.$

If $\mathcal{J}$ is any set of subsets of $Ω$ , then the intersection of all the Dynkin systems containing $\mathcal{J}$ is itself a Dynkin system. It is called the Dynkin system generated by $\mathcal{J}$ . It is the smallest Dynkin system containing $\mathcal{J}$ .

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

Dynkin systems are named after the Russian mathematician Eugene Dynkin.

The Dynkin system theorem (monotone class theorem, Dynkin's lemma) states:

Let $\mathcal{C}$ be a π-system; that is, a collection of subsets of $Ω$ which is closed under pairwise intersections. If $\mathcal{D}$ is a Dynkin system containing $\mathcal{C}$ , then $\mathcal{D}$ also contains the $σ$ -algebra $\sigma(\mathcal{C})$ generated by $\mathcal{C}$ .

One application of Dynkin's lemma is the uniqueness of the Lebesgue measure:

Let (Ω, B, λ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let μ be another measure on Ω satisfying μ[(a,b)] = b - a, and let D be the family of sets such that μ[D] = λ[D]. Let I = { (a,b),[a,b),(a,b],[a,b] : 0 < a ≤ b < 1 }, and observe that I is closed under finite intersections, that I ⊂ D, and that B is the σ-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.