Dynkin system

Short description: Family closed under complements and countable disjoint unions

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

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

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

$\emptyset \in D;$
D is closed under complements in Ω: if A∈D, then Ω∖A∈D;
- Taking $A : = Ω$ shows that $\emptyset \in D .$
D is closed under countable unions of pairwise disjoint sets: if A1,A2,A3,… is a sequence of pairwise disjoint sets in D (meaning that Ai∩Aj=∅ for all i≠j) then ⋃n=1∞An∈D.
- To be clear, this property also holds for finite sequences $A_{1}, \dots, A_{n}$ of pairwise disjoint sets (by letting $A_{i} : = \emptyset$ 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 as they are easier to verify.

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 $𝒥$ of subsets of $Ω,$ there exists a unique Dynkin system denoted $D {𝒥}$ which is minimal with respect to containing $𝒥 .$ That is, if $\tilde{D}$ is any Dynkin system containing $𝒥,$ then $D {𝒥} \subseteq \tilde{D} .$ $D {𝒥}$ is called the Dynkin system generated by $𝒥 .$ For instance, $D {\emptyset} = {\emptyset, Ω} .$ For another example, let $Ω = {1, 2, 3, 4}$ and $𝒥 = {1}$ ; then $D {𝒥} = {\emptyset, {1}, {2, 3, 4}, Ω} .$

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 $σ {P} \subseteq D .$

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

Notes

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

Proofs

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

↑ 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. https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA135. Retrieved August 23, 2010.
↑ Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem". https://www.math.lsu.edu/~sengupta/7360f09/DynkinPiLambda.pdf.

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. https://books.google.com/books?id=e9saZ0YSi-AC&pg=PA193.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Dynkin system. Read more

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

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

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

[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. https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA135. Retrieved August 23, 2010.

[6] Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem". https://www.math.lsu.edu/~sengupta/7360f09/DynkinPiLambda.pdf.

[1]

[2]

[note 1]

[proof 1]

[proof 2]

[3]

Anonymous

Search

Dynkin system

Namespaces

More

Page actions

Contents

Definition

Sierpiński–Dynkin's π-λ theorem

Application to probability distributions

See also

Notes

References

Navigation

Navigation

Help

googletranslator

Navigation

Wiki tools

Wiki tools

Anonymous

Search

Dynkin system

Definition

Sierpiński–Dynkin's π-λ theorem

Application to probability distributions

See also

Notes

References

Navigation

Wiki tools

Page tools

Other projects

Categories