Dynkin system

A Dynkin system,^[1] named after Eugene Dynkin, is a collection of subsets of another universal set $\Omega$ 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 $\Omega$ be a nonempty set, and let $D$ be a collection of subsets of $\Omega$ (that is, $D$ is a subset of the power set of $\Omega$ ). Then $D$ is a Dynkin system if

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

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

$\varnothing \in D;$
$D$ $D$ is closed under complements in $\Omega$ $\Omega$ : if ${\textstyle A\in D,}$ ${\textstyle A\in D,}$ then $\Omega \setminus A\in D;$ $\Omega \setminus A\in D;$
- Taking $A:=\Omega$ shows that $\varnothing \in D.$
$D$ $D$ is closed under countable unions of pairwise disjoint sets: if $A_{1},A_{2},A_{3},\ldots$ $A_{1},A_{2},A_{3},\ldots$ is a sequence of pairwise disjoint sets in $D$ $D$ (meaning that $A_{i}\cap A_{j}=\varnothing$ $A_{i}\cap A_{j}=\varnothing$ for all $i\neq j$ $i\neq j$ ) then $\bigcup _{n=1}^{\infty }A_{n}\in D.$ $\bigcup _{n=1}^{\infty }A_{n}\in D.$
- To be clear, this property also holds for finite sequences $A_{1},\ldots ,A_{n}$ 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.^{[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 ${\mathcal {J}}$ 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 ${\mathcal {J}}.$ For instance, $D\{\varnothing \}=\{\varnothing ,\Omega \}.$ For another example, let $\Omega =\{1,2,3,4\}$ and ${\mathcal {J}}=\{1\}$ ; then $D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.$

Sierpiński–Dynkin's π-λ theorem

Sierpiński-Dynkin's $π$ -𝜆 theorem:^[3] 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 $(\Omega ,{\mathcal {B}},\ell )$ be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let $m$ be another measure on $\Omega$ satisfying $m[(a,b)]=b-a,$ and let $D$ be the family of sets $S$ such that $m[S]=\ell [S].$ Let $I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\leq b<1\},$ and observe that $I$ is closed under finite intersections, that $I\subseteq D,$ and that ${\mathcal {B}}$ 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 ${\mathcal {B}}$ , which is equivalent to showing that the Lebesgue measure is unique on ${\mathcal {B}}$ .

Application to probability distributions

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

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

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

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

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

Notes

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

Proofs

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

^ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
^ 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.
^ Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
^ 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 ${\mathcal {F}}$ of sets over $\Omega$ v t e
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	Directed by $\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	F.I.P.
$π$ -system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if $A_{i}\searrow$	only if $A_{i}\nearrow$
𝜆-system (Dynkin System)				only if $A\subseteq B$			only if $A_{i}\nearrow$ 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				$\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary $\cup$ )			Never
Closed Topology						(even arbitrary $\cap$ )				Never
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $\Omega$	countable intersections	countable unions	contains $\Omega$	contains $\varnothing$	Finite Intersection Property
Additionally, a *semiring* is a $π$ -system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ A *semialgebra* is a semiring where every complement $\Omega \setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$

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

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

[DisjointUGlobalComplementGiveDynkin-5] Assume ${\mathcal {D}}$ satisfies (4), (5), and (6). proof of (2): If $A,B\in {\mathcal {D}}$ satisfy $A\subseteq B$ then (5) implies $\Omega \setminus B\in {\mathcal {D}}$ and since $(\Omega \setminus B)\cap A=\varnothing ,$ (6) implies that ${\mathcal {D}}$ contains $(\Omega \setminus B)\cup A=\Omega \setminus (B\setminus A)$ so that finally (4) guarantees that $\Omega \setminus (\Omega \setminus (B\setminus A))=B\setminus A$ is in ${\mathcal {D}}.$ Proof of (3): Assume $A_{1}\subseteq A_{2}\subseteq \cdots$ is an increasing sequence of subsets in ${\mathcal {D}},$ let $D_{1}=A_{1},$ and let $D_{i}=A_{i}\setminus A_{i-1}$ for every $i>1,$ where (2) guarantees that $D_{2},D_{3},\ldots$ all belong to ${\mathcal {D}}.$ Since $D_{1},D_{2},D_{3},\ldots$ are pairwise disjoint, (6) guarantees that their union $D_{1}\cup D_{2}\cup D_{3}\cup \cdots =A_{1}\cup A_{2}\cup A_{3}\cup \cdots$ belongs to ${\mathcal {D}},$ which proves (3). $\blacksquare$

[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.

[6] Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.

[7] Kallenberg, Foundations Of Modern Probability, p. 48

[1]

[2]

[note 1]

[proof 1]

[proof 2]

[3]

[4]