Dynkin system

Template:Short description 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 [[Sigma algebra|Template:Sigma-algebra]]. Dynkin systems are sometimes referred to as Template:Lambda-systems (Dynkin himself used this term) or d-system.^[2] These set families have applications in measure theory and probability.

A major application of Template:Lambda-systems is the Template:Pi-Template:Lambda 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^{[note 2]} 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.^{[note 3]} 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 [[Pi system|Template:Pi-system]] (that is, closed under finite intersections) is a [[Sigma algebra|Template:Sigma-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 Template:Em 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 theoremTemplate:Anchor

Sierpiński-Dynkin's Template:Pi-Template:Lambda theorem:^[3] If $P$ is a [[Pi-system|Template:Pi-system]] and $D$ is a Dynkin system with $P \subseteq D,$ then $σ {P} \subseteq D .$

In other words, the Template:Sigma-algebra generated by $P$ is contained in $D .$ Thus a Dynkin system contains a Template:Pi-system if and only if it contains the Template:Sigma-algebra generated by that Template:Pi-system.

One application of Sierpiński-Dynkin's Template:Pi-Template:Lambda 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 Template:Sigma-algebra generated by $I .$ It may be shown that $D$ satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's Template:Pi-Template:Lambda 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

Template:Transcluded section Pi system

Notes

Template:Reflist

References

Template:Reflist

Template:Large Template:Navbar
Template:Nowrap Template:Nowrap	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
Template:Nowrap	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
[[Monotone class\|Template:Nowrap]]	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na
[[Dynkin system\|Template:Nowrap Template:Nowrap]]	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Ya	Never
[[Ring of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na	Template:Na
[[Ring of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Na	Template:Ya	Never
[[Delta-ring\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Na	Template:Na	Template:Ya	Never
[[Sigma-ring\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Ya	Never
[[Field of sets\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Ya	Never
[[σ-algebra\|Template:Nowrap Template:Small]]	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Template:Ya	Never
[[Dual ideal\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Na	Template:Na
[[Filter (set theory)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Never	Never	Template:Na	Template:Ya	Template:Ya	$\emptyset \in̸ ℱ$	Template:Ya
[[Prefilter\|Template:Nowrap Template:Small]]	Template:Ya	Template:Na	Template:Na	Never	Never	Template:Na	Template:Na	Template:Na	$\emptyset \in̸ ℱ$	Template:Ya
[[Filter subbase\|Template:Nowrap]]	Template:Na	Template:Na	Template:Na	Never	Never	Template:Na	Template:Na	Template:Na	$\emptyset \in̸ ℱ$	Template:Ya
[[Topology (structure)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Na	Template:Ya	Template:Ya	Template:Ya	Never
[[Topology (structure)\|Template:Nowrap]]	Template:Ya	Template:Ya	Template:Ya	Template:Na	Template:Na	Template:Ya	Template:Na	Template:Ya	Template:Ya	Never
Template:Nowrap Template:Nowrap	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small	Template:Small
Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

Template:Measure theory

↑ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
↑ Template:Cite book
↑ Template:Cite web

Cite error: <ref> tags exist for a group named "note", but no corresponding <references group="note"/> tag was found

[1] Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959

[2] Template:Cite book

[6] Template:Cite web

[1]

[2]

[note 1]

[note 2]

[note 3]

[3]

Dynkin system

Contents

Definition

Sierpiński–Dynkin's π-λ theoremTemplate:Anchor

Application to probability distributions

See also

Notes

References

Further reading

Navigation menu

Dynkin system

Definition

Sierpiński–Dynkin's π-λ theoremTemplate:Anchor

Application to probability distributions

See also

Notes

References

Further reading

Navigation menu

Search