Dynkin system

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

[edit] Definitions

Let Ω be a nonempty set, and let D be a collection of subsets of Ω, i.e. D is a subset of the power set of Ω. Then D is a Dynkin system if

D is a λ-system if

  • the set Ω itself is in D
  • D is closed under complementation, i.e. A\in D implies A^c\in D
  • D is closed under disjoint countable unions, i.e. A_n\in D, n\geq1 with A_i\cap A_j=\emptyset for all i\neq j implies \cup_{n=1}^\infty A_n\in 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 Ω, there exists a unique Dynkin system denoted D\{\mathcal J\} which is minimal with respect to containing \mathcal J. That is, if \tilde D is any Dynkin system containing \mathcal J, then D\{\mathcal J\}\subseteq\tilde D. D\{\mathcal J\} is called the Dynkin system generated by \mathcal{J}. Note D\{\emptyset\}=\{\emptyset,\Omega\}. For another example, let Ω = {1,2,3,4} and \mathcal J=\{1\}; then D\{\mathcal J\}=\{\emptyset,\{1\},\{2,3,4\},\Omega\}.

[edit] Dynkin's π-λ Theorem

If P is a π-system and D is a Dynkin system with P\subseteq 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 (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.

[edit] Bibliography

Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. DOI:10.1007/b138932. ISBN 0-387-22833-0. 

Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.. ISBN 0-471-00710-2. 

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