Dynkin system

A Dynkin system,^[1] named after Eugene Dynkin, is a collection of subsets of another universal set ${\displaystyle \Omega }$ 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.

Definition

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.

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

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]

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
• 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

Notes

1. A sequence of sets ${\displaystyle A_{1},A_{2},A_{3},\ldots }$ is called increasing if ${\displaystyle A_{n}\subseteq A_{n+1}}$ for all ${\displaystyle n\geq 1.}$

Proofs

1. Assume ${\displaystyle {\mathcal {D}}}$ satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using ${\displaystyle B:=\Omega .}$ The following lemma will be used to prove (6). Lemma: If ${\displaystyle A,B\in {\mathcal {D}}}$ are disjoint then ${\displaystyle A\cup B\in {\mathcal {D}}.}$ Proof of Lemma: ${\displaystyle A\cap B=\varnothing }$ implies ${\displaystyle B\subseteq \Omega \setminus A,}$ where ${\displaystyle \Omega \setminus A\subseteq \Omega }$ by (5). Now (2) implies that ${\displaystyle {\mathcal {D}}}$ contains ${\displaystyle (\Omega \setminus A)\setminus B=\Omega \setminus (A\cup B)}$ so that (5) guarantees that ${\displaystyle A\cup B\in {\mathcal {D}},}$ which proves the lemma. Proof of (6) Assume that ${\displaystyle A_{1},A_{2},A_{3},\ldots }$ are pairwise disjoint sets in ${\displaystyle {\mathcal {D}}.}$ For every integer ${\displaystyle n>0,}$ the lemma implies that ${\displaystyle D_{n}:=A_{1}\cup \cdots \cup A_{n}\in {\mathcal {D}}}$ where because ${\displaystyle D_{1}\subseteq D_{2}\subseteq D_{3}\subseteq \cdots }$ is increasing, (3) guarantees that ${\displaystyle {\mathcal {D}}}$ contains their union ${\displaystyle D_{1}\cup D_{2}\cup \cdots =A_{1}\cup A_{2}\cup \cdots ,}$ as desired. ${\displaystyle \blacksquare }$
2. Assume ${\displaystyle {\mathcal {D}}}$ satisfies (4), (5), and (6). proof of (2): If ${\displaystyle A,B\in {\mathcal {D}}}$ satisfy ${\displaystyle A\subseteq B}$ then (5) implies ${\displaystyle \Omega \setminus B\in {\mathcal {D}}}$ and since ${\displaystyle (\Omega \setminus B)\cap A=\varnothing ,}$ (6) implies that ${\displaystyle {\mathcal {D}}}$ contains ${\displaystyle (\Omega \setminus B)\cup A=\Omega \setminus (B\setminus A)}$ so that finally (4) guarantees that ${\displaystyle \Omega \setminus (\Omega \setminus (B\setminus A))=B\setminus A}$ is in ${\displaystyle {\mathcal {D}}.}$ Proof of (3): Assume ${\displaystyle A_{1}\subseteq A_{2}\subseteq \cdots }$ is an increasing sequence of subsets in ${\displaystyle {\mathcal {D}},}$ let ${\displaystyle D_{1}=A_{1},}$ and let ${\displaystyle D_{i}=A_{i}\setminus A_{i-1}}$ for every ${\displaystyle i>1,}$ where (2) guarantees that ${\displaystyle D_{2},D_{3},\ldots }$ all belong to ${\displaystyle {\mathcal {D}}.}$ Since ${\displaystyle D_{1},D_{2},D_{3},\ldots }$ are pairwise disjoint, (6) guarantees that their union ${\displaystyle D_{1}\cup D_{2}\cup D_{3}\cup \cdots =A_{1}\cup A_{2}\cup A_{3}\cup \cdots }$ belongs to ${\displaystyle {\mathcal {D}},}$ which proves (3).${\displaystyle \blacksquare }$

1. Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
2. Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. ISBN 978-3-540-29587-7. Retrieved August 23, 2010.
3. Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
4. Kallenberg, Foundations Of Modern Probability, p. 48

References

• 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 ${\displaystyle {\mathcal {F}}}$ of sets over ${\displaystyle \Omega }$ v t e
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 where every complement ${\displaystyle \Omega \setminus A}$ is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$ ${\displaystyle A,B,A_{1},A_{2},\ldots }$ are arbitrary elements of ${\displaystyle {\mathcal {F}}}$ and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$