Dynkin system

A Dynkin system, 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. 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 Ω 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

1. $\Omega \in D,$
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 $A_{1},A_{2},A_{3},\ldots$ is a sequence of subsets in $D$ and $A_{n}\subseteq A_{n+1}$ for all $n\geq 1,$ then $\bigcup _{n=1}^{\infty }A_{n}\in D.$

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

4. $\Omega \in D,$
5. $D$ is closed under complements in $\Omega$: if ${\textstyle A\in D,}$ then $\Omega \setminus A\in D$,
6. $D$ is closed under countable unions of pairwise disjoint sets: if $A_{1},A_{2},A_{3},\ldots$ is a sequence of subsets in $D$ such that $A_{i}\cap A_{j}=\varnothing$ for all $i\neq j,$ then $\bigcup _{n=1}^{\infty }A_{n}\in D.$

Conversely, it is easy to check that a family of sets that satisfy (4-6) is a Dynkin class. 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 a Dynkin system which 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 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

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 $D.$

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 $(\Omega ,B,\lambda )$ be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let $\mu$ be another measure on $\Omega$ satisfying $\mu [(a,b)]=b-a,$ and let $D$ be the family of sets $S$ such that $\mu [S]=\lambda [S].$ 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 $B$ 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

The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable $X\colon (\Omega ,{\mathcal {F}},\operatorname {P} )\rightarrow \mathbb {R}$ 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. We say that the random variables $X\colon (\Omega ,{\mathcal {F}},\operatorname {P} ),$ and $Y\colon ({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\rightarrow \mathbb {R}$ (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, $F_{X}=F_{Y}.$ 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 $\left\{(-\infty ,a]\colon a\in \mathbb {R} \right\}$ 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

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

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.

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
• 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

References

• 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 ${\mathcal {F}}$ of sets over $\Omega$
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 unions)			Never
Closed Topology						(even arbitrary intersections)				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 that contains $\Omega .$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$