Dynkin system

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

1. Ω ∈ D ,
2. if A, B ∈ D and A ⊆ B, then B A ∈ D ,
3. if A₁, A₂, A₃, ... is a sequence of subsets in D and A_n ⊆ A_n+1 for all n ≥ 1, then ⋃ n = 1 ∞ A n ∈ D .

Equivalently, D is a Dynkin system if

1. Ω ∈ D ,
2. if A ∈ D, then A^c ∈ D,
3. if A₁, A₂, A₃, ... is a sequence of subsets in D such that A_i ∩ A_j = Ø for all i ≠ j, then ⋃ n = 1 ∞ A n ∈ D .

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 J of subsets of Ω , there exists a unique Dynkin system denoted D { J } which is minimal with respect to containing J . That is, if D ~ is any Dynkin system containing J , then D { J } ⊆ D ~ . D { J } is called the Dynkin system generated by J . Note D { ∅ } = { ∅ , Ω } . For another example, let Ω = { 1 , 2 , 3 , 4 } and J = { 1 } ; then D { J } = { ∅ , { 1 } , { 2 , 3 , 4 } , Ω } .

Dynkin's π-λ theorem

If P is a π-system and D is a Dynkin system with P ⊆ D , then σ { P } ⊆ 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 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.