Dynkin system

A Dynkin system,^[1] named after Eugene Dynkin, is a collection of subsets of another universal set ${\displaystyle \Omega }${\displaystyle \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.^[2] These set families have applications in measure theory and probability.

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

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

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

It is easy to check^{[note 2]} that any Dynkin system ${\displaystyle D}${\displaystyle D} satisfies:

4. ${\displaystyle \varnothing \in D;}${\displaystyle \varnothing \in D;}
5. ${\displaystyle D}${\displaystyle D} is closed under complements in ${\displaystyle \Omega }${\displaystyle \Omega }: if ${\textstyle A\in D,}${\textstyle A\in D,} then ${\displaystyle \Omega \setminus A\in D;}${\displaystyle \Omega \setminus A\in D;}
   • Taking ${\displaystyle A:=\Omega }${\displaystyle A:=\Omega } shows that ${\displaystyle \varnothing \in D.}${\displaystyle \varnothing \in D.}
6. ${\displaystyle D}${\displaystyle D} is closed under countable unions of pairwise disjoint sets: if ${\displaystyle A_{1},A_{2},A_{3},\ldots }${\displaystyle A_{1},A_{2},A_{3},\ldots } is a sequence of pairwise disjoint sets in ${\displaystyle D}${\displaystyle D} (meaning that ${\displaystyle A_{i}\cap A_{j}=\varnothing }${\displaystyle A_{i}\cap A_{j}=\varnothing } for all ${\displaystyle i\neq j}${\displaystyle i\neq j}) then ${\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}${\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}
   • To be clear, this property also holds for finite sequences ${\displaystyle A_{1},\ldots ,A_{n}}${\displaystyle A_{1},\ldots ,A_{n}} of pairwise disjoint sets (by letting ${\displaystyle A_{i}:=\varnothing }${\displaystyle A_{i}:=\varnothing } for all ${\displaystyle i>n}${\displaystyle i>n}).

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.^{[note 3]} 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 ${\displaystyle {\mathcal {J}}}${\displaystyle {\mathcal {J}}} of subsets of ${\displaystyle \Omega ,}${\displaystyle \Omega ,} there exists a unique Dynkin system denoted ${\displaystyle D\{{\mathcal {J}}\}}${\displaystyle D\{{\mathcal {J}}\}} which is minimal with respect to containing ${\displaystyle {\mathcal {J}}.}${\displaystyle {\mathcal {J}}.} That is, if ${\displaystyle {\tilde {D}}}${\displaystyle {\tilde {D}}} is any Dynkin system containing ${\displaystyle {\mathcal {J}},}${\displaystyle {\mathcal {J}},} then ${\displaystyle D\{{\mathcal {J}}\}\subseteq {\tilde {D}}.}${\displaystyle D\{{\mathcal {J}}\}\subseteq {\tilde {D}}.} ${\displaystyle D\{{\mathcal {J}}\}}${\displaystyle D\{{\mathcal {J}}\}} is called the Dynkin system generated by ${\displaystyle {\mathcal {J}}.}${\displaystyle {\mathcal {J}}.} For instance, ${\displaystyle D\{\varnothing \}=\{\varnothing ,\Omega \}.}${\displaystyle D\{\varnothing \}=\{\varnothing ,\Omega \}.} For another example, let ${\displaystyle \Omega =\{1,2,3,4\}}${\displaystyle \Omega =\{1,2,3,4\}} and ${\displaystyle {\mathcal {J}}=\{1\}}${\displaystyle {\mathcal {J}}=\{1\}}; then ${\displaystyle D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.}${\displaystyle D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.}

Sierpiński-Dynkin's π-λ theorem:^[3] If ${\displaystyle P}${\displaystyle P} is a π-system and ${\displaystyle D}${\displaystyle D} is a Dynkin system with ${\displaystyle P\subseteq D,}${\displaystyle P\subseteq D,} then ${\displaystyle \sigma \{P\}\subseteq D.}${\displaystyle \sigma \{P\}\subseteq D.}

In other words, the σ-algebra generated by ${\displaystyle P}${\displaystyle P} is contained in ${\displaystyle D.}${\displaystyle 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 ${\displaystyle (\Omega ,{\mathcal {B}},\ell )}${\displaystyle (\Omega ,{\mathcal {B}},\ell )} be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let ${\displaystyle m}${\displaystyle m} be another measure on ${\displaystyle \Omega }${\displaystyle \Omega } satisfying ${\displaystyle m[(a,b)]=b-a,}${\displaystyle m[(a,b)]=b-a,} and let ${\displaystyle D}${\displaystyle D} be the family of sets ${\displaystyle S}${\displaystyle S} such that ${\displaystyle m[S]=\ell [S].}${\displaystyle m[S]=\ell [S].} Let ${\displaystyle I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\leq b<1\},}${\displaystyle I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\leq b<1\},} and observe that ${\displaystyle I}${\displaystyle I} is closed under finite intersections, that ${\displaystyle I\subseteq D,}${\displaystyle I\subseteq D,} and that ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} is the σ-algebra generated by ${\displaystyle I.}${\displaystyle I.} It may be shown that ${\displaystyle D}${\displaystyle D} satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-λ Theorem it follows that ${\displaystyle D}${\displaystyle D} in fact includes all of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}, which is equivalent to showing that the Lebesgue measure is unique on ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}.

The π-λ theorem motivates the common definition of the probability distribution of a random variable ${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} } in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as $${\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}$${\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,} whereas the seemingly more general law of the variable is the probability measure $${\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}$${\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),} where ${\displaystyle {\mathcal {B}}(\mathbb {R} )}${\displaystyle {\mathcal {B}}(\mathbb {R} )} is the Borel σ-algebra. The random variables ${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} } and ${\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} }${\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} } (on two possibly different probability spaces) are equal in distribution (or law), denoted by ${\displaystyle X,円{\stackrel {\mathcal {D}}{=}},円Y,}${\displaystyle X,円{\stackrel {\mathcal {D}}{=}},円Y,} if they have the same cumulative distribution functions; that is, if ${\displaystyle F_{X}=F_{Y}.}${\displaystyle F_{X}=F_{Y}.} The motivation for the definition stems from the observation that if ${\displaystyle F_{X}=F_{Y},}${\displaystyle F_{X}=F_{Y},} then that is exactly to say that ${\displaystyle {\mathcal {L}}_{X}}${\displaystyle {\mathcal {L}}_{X}} and ${\displaystyle {\mathcal {L}}_{Y}}${\displaystyle {\mathcal {L}}_{Y}} agree on the π-system ${\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}}${\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}} which generates ${\displaystyle {\mathcal {B}}(\mathbb {R} ),}${\displaystyle {\mathcal {B}}(\mathbb {R} ),} and so by the example above: ${\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}${\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}

A similar result holds for the joint distribution of a random vector. For example, suppose ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} are two random variables defined on the same probability space ${\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),}${\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),} with respectively generated π-systems ${\displaystyle {\mathcal {I}}_{X}}${\displaystyle {\mathcal {I}}_{X}} and ${\displaystyle {\mathcal {I}}_{Y}.}${\displaystyle {\mathcal {I}}_{Y}.} The joint cumulative distribution function of ${\displaystyle (X,Y)}${\displaystyle (X,Y)} is $${\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}$${\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}

However, ${\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}}${\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}} and ${\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.}${\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.} Because $${\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}}$${\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}} is a π-system generated by the random pair ${\displaystyle (X,Y),}${\displaystyle (X,Y),} the π-λ theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of ${\displaystyle (X,Y).}${\displaystyle (X,Y).} In other words, ${\displaystyle (X,Y)}${\displaystyle (X,Y)} and ${\displaystyle (W,Z)}${\displaystyle (W,Z)} have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes ${\displaystyle (X_{t})_{t\in T},(Y_{t})_{t\in T}}${\displaystyle (X_{t})_{t\in T},(Y_{t})_{t\in T}} are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all ${\displaystyle t_{1},\ldots ,t_{n}\in T,,円n\in \mathbb {N} ,}${\displaystyle t_{1},\ldots ,t_{n}\in T,,円n\in \mathbb {N} ,} $${\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right),円{\stackrel {\mathcal {D}}{=}},円\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}$${\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right),円{\stackrel {\mathcal {D}}{=}},円\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}

The proof of this is another application of the π-λ theorem.^[4]

• 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
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• σ-algebra – Algebraic structure of set algebra
• σ-ideal – Family closed under subsets and countable unions
• σ-ring – Family of sets closed under countable unions

• 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.

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

• Families of sets
• Lemmas
• Probability theory

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