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 and probability.
A major application of λ-systems is the π-λ theorem, see below.
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 ∈ ,
- if A, B ∈ and A ⊆ B, then B \ A ∈ ,
- if A_{1}, A_{2}, A_{3}, ... is a sequence of subsets in and A_{n} ⊆ A_{n+1} for all n ≥ 1, then .
Equivalently, is a Dynkin system if
- Ω ∈ ,
- if A ∈ D, then A^{c} ∈ D,
- if A_{1}, A_{2}, A_{3}, ... is a sequence of subsets in such that A_{i} ∩ A_{j} = Ø for all i ≠ j, then .
The second definition is generally preferred as it usually is easier to check.
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 a measure that evaluates the length of an interval (known as 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. It may be shown that D satisfies the above conditions for a Dynkin-system. From Dynkin's π-λ Theorem it follows that D in fact includes all of B, which is equivalent to showing that the Lebesgue measure is unique on B.
Additional applications are in the article on π-systems.
Notes
- ^ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010.
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.
- Williams, David (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.