Dynkin system

A Dynkin system, named in honor of the Russian mathematician Eugene Dynkin, is a collection of subsets of another universal set $\Omega$ satisfying some specific rules. They are also referred to as λ-systems.

Definitions

Let $\Omega$ be a nonempty set, and let $D$ be a collection of subsets of $\Omega$, i.e. $D$ is a subset of the power set of $\Omega$. Then $D$ is a Dynkin system if

• the set $\Omega$ itself is in $D$
• $D$ is closed under relative complementation, i.e. $A,B\in D$ and $A\subseteq B$ implies $B\setminus A\in D$
• $D$ is closed under the countable union of increasing sequences, i.e. $A_{n}\in D$ and $A_{n}\subseteq A_{n+1},\ n\geq 1$ implies $\cup _{n=1}^{\infty }A_{n}\in D$.

$D$ is a λ-system if

• the set $\Omega$ itself is in $D$
• $D$ is closed under complementation, i.e. $A\in D$ implies $A^{c}\in D$
• $D$ is closed under disjoint countable unions, i.e. $A_{n}\in D,n\geq 1$ with $A_{i}\cap A_{j}=\emptyset$ for all $i\neq j$ implies $\cup _{n=1}^{\infty }A_{n}\in D$.

It can be shown that these two definitions are logically equivalent, so that Dynkin systems are λ-systems and vice versa.

A Dynkin system which is also a π-system is a σ-algebra.

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 &\sigma;-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 such that μ[D] = λ[D]. 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.

Bibliography

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.

---

Dynkin system at PlanetMath.