Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $\Omega$ 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. These set families have applications in measure theory.

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. A, B ∈ $D$ and A ⊆ B implies B \ A ∈ $D$,
3. A₁, A₂, A₃, ... is a sequence of subsets in $D$ and A_n ⊆ A_n+1 for all n ≥ 1, then $\cup _{n=1}^{\infty }A_{n}\in D$.

Equivalently, $D$ is a Dynkin system if

1. Ω ∈ $D$,
2. A ∈ D implies A^c ∈ $D$,
3. A₁, A₂, A₃, ... is a sequence of subsets in $D$ such that A_i ∩ A_j = Ø for all i ≠ j implies $\cup _{n=1}^{\infty }A_{n}\in D$.

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 ${\mathcal {J}}$ of subsets of $\Omega$, 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 $\Omega =\{1,2,3,4\}$ and ${\mathcal {J}}=\{1\}$; then $D\{{\mathcal {J}}\}=\{\emptyset ,\{1\},\{2,3,4\},\Omega \}$.

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 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.

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.