# 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 $\Omega$ 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 of sets in $D$ then $\bigcup _{n=1}^{\infty }A_{n}\in D.$
It is easy to check that any Dynkin system $D$ satisfies:
* $\varnothing \in D;$
* $D$ is closed under complements in $\Omega$: if A∈D, {\textstyle A\in D, } then $\Omega \setminus A\in D;$
** Taking $A:=\Omega$ shows that $\varnothing \in D.$
* $D$ is closed under countable unions of pairwise disjoint sets: if $A_{1},A_{2},A_{3},\ldots$ is a sequence of pairwise disjoint sets in $D$ (meaning that $A_{i}\cap A_{j}=\varnothing$ for all $i\neq j$) then $\bigcup _{n=1}^{\infty }A_{n}\in D.$
** 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. 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: 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.