Dynkin system
A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himself used this term) or d-system.[1] These set families have applications in measure theory.
Definitions
Let Ω be a nonempty set, and let be a collection of subsets of Ω (i.e., is a subset of the power set of Ω). Then is a Dynkin system if
- Ω ∈ ,
- A, B ∈ and A ⊆ B implies B \ A ∈ ,
- A1, A2, A3, ... is a sequence of subsets in and An ⊆ An+1 for all n ≥ 1, then .
Equivalently, is a Dynkin system if
- Ω ∈ ,
- A ∈ D implies Ac ∈ ,
- A1, A2, A3, ... is a sequence of subsets in such that Ai ∩ Aj = Ø for all i ≠ j implies .
An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra. This can be verified by noting that condition 3 and closure under finite intersection implies closure under countable unions.
Given any collection of subsets of , there exists a unique Dynkin system denoted which is minimal with respect to containing . That is, if is any Dynkin system containing , then . is called the Dynkin system generated by . Note . For another example, let and ; then .
Dynkin's π-λ Theorem
If is a π-system and is a Dynkin system with , then . In other words, the σ-algebra generated by is contained in .
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 S such that μ[S] = λ[S]. 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.
Notes
- ^ Infinite Dimensional Analysis: a Hitchhiker's Guide, 3rd ed. Springer. 2006. Retrieved August 23, 2010.
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References
- 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.
- David Williams (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6.
This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.