A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set Ω {\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. 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 {\displaystyle \Omega } be a set, and let D {\displaystyle D} be a collection of subsets of Ω {\displaystyle \Omega } (that is, D {\displaystyle D} is a subset of the power set of Ω {\displaystyle \Omega } ). Then D {\displaystyle D} is a Dynkin system if
Ω ∈ D ; {\displaystyle \Omega \in D;}
D {\displaystyle D} is closed under complements of subsets in supersets: if A , B ∈ D {\displaystyle A,B\in D} and A ⊆ B , {\displaystyle A\subseteq B,} then B ∖ A ∈ D ; {\displaystyle B\setminus A\in D;}
D {\displaystyle D} is closed under countable increasing unions: if A 1 ⊆ A 2 ⊆ A 3 ⊆ ⋯ {\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \cdots } is an increasing sequence of sets in D {\displaystyle D} then ⋃ n = 1 ∞ A n ∈ D . {\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}
It is easy to check that any Dynkin system D {\displaystyle D} satisfies:
∅ ∈ D ; {\displaystyle \varnothing \in D;}
D {\displaystyle D} is closed under complements in Ω {\displaystyle \Omega } : if A ∈ D , {\textstyle A\in D,} then Ω ∖ A ∈ D ; {\displaystyle \Omega \setminus A\in D;}
Taking A := Ω {\displaystyle A:=\Omega } shows that ∅ ∈ D . {\displaystyle \varnothing \in D.}
D {\displaystyle D} is closed under countable unions of pairwise disjoint sets: if A 1 , A 2 , A 3 , … {\displaystyle A_{1},A_{2},A_{3},\ldots } is a sequence of pairwise disjoint sets in D {\displaystyle D} (meaning that A i ∩ A j = ∅ {\displaystyle A_{i}\cap A_{j}=\varnothing } for all i ≠ j {\displaystyle i\neq j} ) then ⋃ n = 1 ∞ A n ∈ D . {\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}
To be clear, this property also holds for finite sequences A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} of pairwise disjoint sets (by letting A i := ∅ {\displaystyle A_{i}:=\varnothing } for all i > n {\displaystyle 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 J {\displaystyle {\mathcal {J}}} of subsets of Ω , {\displaystyle \Omega ,} there exists a unique Dynkin system denoted D { J } {\displaystyle D\{{\mathcal {J}}\}} which is minimal with respect to containing J . {\displaystyle {\mathcal {J}}.} That is, if D ~ {\displaystyle {\tilde {D}}} is any Dynkin system containing J , {\displaystyle {\mathcal {J}},} then D { J } ⊆ D ~ . {\displaystyle D\{{\mathcal {J}}\}\subseteq {\tilde {D}}.} D { J } {\displaystyle D\{{\mathcal {J}}\}} is called the Dynkin system generated by J . {\displaystyle {\mathcal {J}}.} For instance, D { ∅ } = { ∅ , Ω } . {\displaystyle D\{\varnothing \}=\{\varnothing ,\Omega \}.} For another example, let Ω = { 1 , 2 , 3 , 4 } {\displaystyle \Omega =\{1,2,3,4\}} and J = { 1 } {\displaystyle {\mathcal {J}}=\{1\}} ; then D { J } = { ∅ , { 1 } , { 2 , 3 , 4 } , Ω } . {\displaystyle D\{{\mathcal {J}}\}=\{\varnothing ,\{1\},\{2,3,4\},\Omega \}.}
Sierpiński–Dynkin's π-λ theorem Sierpiński-Dynkin's π-𝜆 theorem: If P {\displaystyle P} is a π-system and D {\displaystyle D} is a Dynkin system with P ⊆ D , {\displaystyle P\subseteq D,} then σ { P } ⊆ D . {\displaystyle \sigma \{P\}\subseteq D.} In other words, the 𝜎-algebra generated by P {\displaystyle P} is contained in D . {\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 ( Ω , B , ℓ ) {\displaystyle (\Omega ,{\mathcal {B}},\ell )} be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let m {\displaystyle m} be another measure on Ω {\displaystyle \Omega } satisfying m [ ( a , b ) ] = b − a , {\displaystyle m[(a,b)]=b-a,} and let D {\displaystyle D} be the family of sets S {\displaystyle S} such that m [ S ] = ℓ [ S ] . {\displaystyle m[S]=\ell [S].} Let I := { ( a , b ) , [ a , b ) , ( a , b ] , [ a , b ] : 0 < a ≤ b < 1 } , {\displaystyle I:=\{(a,b),[a,b),(a,b],[a,b]:0<a\leq b<1\},} and observe that I {\displaystyle I} is closed under finite intersections, that I ⊆ D , {\displaystyle I\subseteq D,} and that B {\displaystyle {\mathcal {B}}} is the 𝜎-algebra generated by I . {\displaystyle I.} It may be shown that D {\displaystyle D} satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that D {\displaystyle D} in fact includes all of B {\displaystyle {\mathcal {B}}} , which is equivalent to showing that the Lebesgue measure is unique on B {\displaystyle {\mathcal {B}}} .
Application to probability distributions
The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable X : ( Ω , F , P ) → R {\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
F X ( a ) = P [ X ≤ a ] , a ∈ R , {\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
L X ( B ) = P [ X − 1 ( B ) ] for all B ∈ B ( R ) , {\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all }}B\in {\mathcal {B}}(\mathbb {R} ),}
where B ( R ) {\displaystyle {\mathcal {B}}(\mathbb {R} )} is the Borel 𝜎-algebra. The random variables X : ( Ω , F , P ) → R {\displaystyle X:(\Omega ,{\mathcal {F}},\operatorname {P} )\to \mathbb {R} } and Y : ( Ω ~ , F ~ , P ~ ) → R {\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 X = D Y , {\displaystyle X\,{\stackrel {\mathcal {D}}{=}}\,Y,} if they have the same cumulative distribution functions; that is, if F X = F Y . {\displaystyle F_{X}=F_{Y}.} The motivation for the definition stems from the observation that if F X = F Y , {\displaystyle F_{X}=F_{Y},} then that is exactly to say that L X {\displaystyle {\mathcal {L}}_{X}} and L Y {\displaystyle {\mathcal {L}}_{Y}} agree on the π-system { ( − ∞ , a ] : a ∈ R } {\displaystyle \{(-\infty ,a]:a\in \mathbb {R} \}} which generates B ( R ) , {\displaystyle {\mathcal {B}}(\mathbb {R} ),} and so by the example above: L X = L Y . {\displaystyle {\mathcal {L}}_{X}={\mathcal {L}}_{Y}.}
A similar result holds for the joint distribution of a random vector. For example, suppose X {\displaystyle X} and Y {\displaystyle Y} are two random variables defined on the same probability space ( Ω , F , P ) , {\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),} with respectively generated π-systems I X {\displaystyle {\mathcal {I}}_{X}} and I Y . {\displaystyle {\mathcal {I}}_{Y}.} The joint cumulative distribution function of ( X , Y ) {\displaystyle (X,Y)} is
F X , Y ( a , b ) = P [ X ≤ a , Y ≤ b ] = P [ X − 1 ( ( − ∞ , a ] ) ∩ Y − 1 ( ( − ∞ , b ] ) ] , for all a , b ∈ R . {\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, A = X − 1 ( ( − ∞ , a ] ) ∈ I X {\displaystyle A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}} and B = Y − 1 ( ( − ∞ , b ] ) ∈ I Y . {\displaystyle B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.} Because
I X , Y = { A ∩ B : A ∈ I X , and B ∈ I Y } {\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 ( X , Y ) , {\displaystyle (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 ) . {\displaystyle (X,Y).} In other words, ( X , Y ) {\displaystyle (X,Y)} and ( W , Z ) {\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 ( X t ) t ∈ T , ( Y t ) t ∈ T {\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 t 1 , … , t n ∈ T , n ∈ N , {\displaystyle t_{1},\ldots ,t_{n}\in T,\,n\in \mathbb {N} ,}
( X t 1 , … , X t n ) = D ( Y t 1 , … , Y t 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.
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 – Measure theory and probability theoremPages displaying short descriptions of redirect targets 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
References
Further reading 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.