Dynkin system

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

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

It is easy to check[proof 1] that any Dynkin system ${\displaystyle D}$ satisfies:

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

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.[proof 2] For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system as they are easier to verify.

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

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

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

Application to probability distributions

The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable ${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }$ in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as

$${\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}$$

whereas the seemingly more general law of the variable is the probability measure

$${\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}$$

where ${\displaystyle {\mathcal {B}}(\mathbb {R} )}$ is the Borel 𝜎-algebra. The random variables ${\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} }$ and ${\displaystyle Y:({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\to \mathbb {R} }$ (on two possibly different probability spaces) are equal in distribution (or law), denoted by ${\displaystyle X\,{\stackrel {\mathcal {D}}{=}}\,Y,}$ if they have the same cumulative distribution functions; that is, if ${\displaystyle F_{X}=F_{Y}.}$ The motivation for the definition stems from the observation that if ${\displaystyle F_{X}=F_{Y},}$ then that is exactly to say that ${\displaystyle {\mathcal {L}}_{X}}$ and ${\displaystyle {\mathcal {L}}_{Y}}$ agree on the π-system ${\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}}$ which generates ${\displaystyle {\mathcal {B}}(\mathbb {R} ),}$ and so by the example above: ${\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}$

A similar result holds for the joint distribution of a random vector. For example, suppose ${\displaystyle X}$ and ${\displaystyle Y}$ are two random variables defined on the same probability space ${\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),}$ with respectively generated π-systems ${\displaystyle {\mathcal {I}}_{X}}$ and ${\displaystyle {\mathcal {I}}_{Y}.}$ The joint cumulative distribution function of ${\displaystyle (X,Y)}$ is

$${\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all }}a,b\in \mathbb {R} .}$$

However, ${\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}}$ and ${\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.}$ Because

$${\displaystyle {\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}}$$

is a π-system generated by the random pair ${\displaystyle (X,Y),}$ the π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of ${\displaystyle (X,Y).}$ In other words, ${\displaystyle (X,Y)}$ and ${\displaystyle (W,Z)}$ have the same distribution if and only if they have the same joint cumulative distribution function.

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

$${\displaystyle \left(X_{t_{1}},\ldots ,X_{t_{n}}\right)\,{\stackrel {\mathcal {D}}{=}}\,\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).}$$

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 – Algebric structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions
• 𝜎-ring – Ring closed under countable unions

Proofs

• 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.
• Williams, David (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.

Families ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$ or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$)			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$)				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$ or, is ${\displaystyle {\mathcal {F}}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ A semialgebra is a semiring that contains ${\displaystyle \Omega .}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$