Dynkin system

Short description: Family closed under complements and countable disjoint unions

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $\mathrm{\Omega }$ 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 and probability.

A major application of 𝜆-systems is the π-𝜆 theorem, see below.

Definition

Let $\mathrm{\Omega }$ be a nonempty set, and let $D$ be a collection of subsets of $\mathrm{\Omega }$ (that is, $D$ is a subset of the power set of $\mathrm{\Omega }$). Then $D$ is a Dynkin system if

1. $\mathrm{\Omega }\in D;$
2. $D$ is closed under complements of subsets in supersets: if $A,B\in D$ and $A\subseteq B,$ then $B\setminus A\in D;$
3. $D$ is closed under countable increasing unions: if $A_1\subseteq A_2\subseteq A_3\subseteq \cdots$ is an increasing sequence of sets in $D$ then $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in D.$

It is easy to check that any Dynkin system $D$ satisfies:

4. $\varnothing \in D;$
5. $D$ is closed under complements in $\mathrm{\Omega }$: if $A\in D,$ then $\mathrm{\Omega }\setminus A\in D;$
   • Taking $A:=\mathrm{\Omega }$ shows that $\varnothing \in D.$
6. $D$ is closed under countable unions of pairwise disjoint sets: if $A_1,A_2,A_3,\dots$ is a sequence of pairwise disjoint sets in $D$ (meaning that $A_i\cap A_j=\varnothing$ for all $i\ne j$) then $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in D.$
   • To be clear, this property also holds for finite sequences $A_1,\dots ,A_n$ of pairwise disjoint sets (by letting $A_i:=\varnothing$ for all $i>n$).

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.

An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.

Given any collection ${𝒥}$ of subsets of $\mathrm{\Omega },$ there exists a unique Dynkin system denoted $D\{{𝒥}\}$ which is minimal with respect to containing ${𝒥}.$ That is, if $\tilde{D}$ is any Dynkin system containing ${𝒥},$ then $D\{{𝒥}\}\subseteq \tilde{D}.$ $D\{{𝒥}\}$ is called the Dynkin system generated by ${𝒥}.$ For instance, $D\{\varnothing \}=\{\varnothing ,\mathrm{\Omega }\}.$ For another example, let $\mathrm{\Omega }=\{1,2,3,4\}$ and ${𝒥}=\{1\}$; then $D\{{𝒥}\}=\{\varnothing ,\{1\},\{2,3,4\},\mathrm{\Omega }\}.$

Sierpiński–Dynkin's π-λ theorem

Sierpiński-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 $D.$ Thus a Dynkin system contains a π-system if and only if it contains the 𝜎-algebra generated by that π-system.

One application of Sierpiński-Dynkin's π-𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):

Let $(\mathrm{\Omega },{ℬ},\ell )$ be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let $m$ be another measure on $\mathrm{\Omega }$ satisfying $m[(a,b)]=b-a,$ and let $D$ be the family of sets $S$ such that $m[S]=\ell [S].$ Let $I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\le b<1\},$ and observe that $I$ is closed under finite intersections, that $I\subseteq D,$ 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 Sierpiński-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 ${ℬ}$.

Application to probability distributions

Pi system

See also

• Algebra of sets – Identities and relationships involving sets
• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Σ-algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Ring closed under countable unions

Further reading

• Gut, Allan (2005). Probability: A Graduate Course. Springer Texts in Statistics. 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 978-0-521-40605-5. https://books.google.com/books?id=e9saZ0YSi-AC&pg=PA193. 

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