Dynkin system

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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[edit]

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

  1. Ω ∈ ,
  2. if A, B and AB, then B \ A,
  3. if A1, A2, A3, ... is a sequence of subsets in and AnAn+1 for all n ≥ 1, then .

Equivalently, is a Dynkin system if

  1. Ω ∈ ,
  2. if AD, then AcD,
  3. if A1, A2, A3, ... is a sequence of subsets in such that AiAj = Ø for all ij, 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[edit]

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 < ab < 1 }, and observe that I is closed under finite intersections, that ID, 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[edit]

  1. ^ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010. 

References[edit]

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.