Dynkin system

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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 An 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.

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the GFDL.

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