A Dynkin system,[1] named after Eugene Dynkin, is a collection of subsets of another universal set 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.
To be clear, this property also holds for finite sequences of pairwise disjoint sets (by letting for all ).
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 of subsets of there exists a unique Dynkin system denoted which is minimal with respect to containing That is, if is any Dynkin system containing then is called the Dynkin system generated by For instance, For another example, let and ; then
Sierpiński–Dynkin's π-λ theoremedit
Sierpiński-Dynkin's π-𝜆 theorem:[3] If is a π-system and is a Dynkin system with then
In other words, the 𝜎-algebra generated by is contained in 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 be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let be another measure on satisfying and let be the family of sets such that Let and observe that is closed under finite intersections, that and that is the 𝜎-algebra generated by It may be shown that satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that in fact includes all of , which is equivalent to showing that the Lebesgue measure is unique on .
whereas the seemingly more general law of the variable is the probability measure
where is the Borel 𝜎-algebra. The random variables and (on two possibly different probability spaces) are equal in distribution (or law), denoted by if they have the same cumulative distribution functions; that is, if The motivation for the definition stems from the observation that if then that is exactly to say that and agree on the π-system which generates and so by the example above:
A similar result holds for the joint distribution of a random vector. For example, suppose and are two random variables defined on the same probability space with respectively generated π-systems and The joint cumulative distribution function of is
However, and Because
is a π-system generated by the random pair the π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of In other words, and have the same distribution if and only if they have the same joint cumulative distribution function.
In the theory of stochastic processes, two processes are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all
The proof of this is another application of the π-𝜆 theorem.[4]
See alsoedit
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 – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
π-system – Family of sets closed under intersection
Ring of sets – Family closed under unions and relative complements
^A sequence of sets is called increasing if for all
Proofs
^Assume satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using The following lemma will be used to prove (6). Lemma: If are disjoint then Proof of Lemma: implies where by (5). Now (2) implies that contains so that (5) guarantees that which proves the lemma. Proof of (6) Assume that are pairwise disjoint sets in For every integer the lemma implies that where because is increasing, (3) guarantees that contains their union as desired.
^Assume satisfies (4), (5), and (6). proof of (2): If satisfy then (5) implies and since (6) implies that contains so that finally (4) guarantees that is in Proof of (3): Assume is an increasing sequence of subsets in let and let for every where (2) guarantees that all belong to Since are pairwise disjoint, (6) guarantees that their union belongs to 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. Retrieved August 23, 2010.
^Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
^Kallenberg, Foundations Of Modern probability, p. 48
Referencesedit
Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN0-387-22833-0.
Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN0-471-00710-2.
Williams, David (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN0-521-40605-6.
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in are arbitrary elements of and it is assumed that
February 15, 2024
dynkin, system, named, after, eugene, dynkin, collection, subsets, another, universal, displaystyle, omega, satisfying, axioms, weaker, than, those, 𝜎, algebra, sometimes, referred, 𝜆, systems, dynkin, himself, used, this, term, system, these, families, have, . A Dynkin system 1 named after Eugene Dynkin is a collection of subsets of another universal set W 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 2 These set families have applications in measure theory and probability A major application of 𝜆 systems is the p 𝜆 theorem see below Contents 1 Definition 2 Sierpinski Dynkin s p l theorem 2 1 Application to probability distributions 3 See also 4 Notes 5 ReferencesDefinition editLet W displaystyle Omega nbsp be a nonempty set and let D displaystyle D nbsp be a collection of subsets of W displaystyle Omega nbsp that is D displaystyle D nbsp is a subset of the power set of W displaystyle Omega nbsp Then D displaystyle D nbsp is a Dynkin system if W D displaystyle Omega in D nbsp D displaystyle D nbsp is closed under complements of subsets in supersets if A B D displaystyle A B in D nbsp and A B displaystyle A subseteq B nbsp then B A D displaystyle B setminus A in D nbsp D displaystyle D nbsp is closed under countable increasing unions if A 1 A 2 A 3 displaystyle A 1 subseteq A 2 subseteq A 3 subseteq cdots nbsp is an increasing sequence note 1 of sets in D displaystyle D nbsp then n 1 A n D displaystyle bigcup n 1 infty A n in D nbsp It is easy to check proof 1 that any Dynkin system D displaystyle D nbsp satisfies D displaystyle varnothing in D nbsp D displaystyle D nbsp is closed under complements in W displaystyle Omega nbsp if A D textstyle A in D nbsp then W A D displaystyle Omega setminus A in D nbsp Taking A W displaystyle A Omega nbsp shows that D displaystyle varnothing in D nbsp D displaystyle D nbsp is closed under countable unions of pairwise disjoint sets if A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp is a sequence of pairwise disjoint sets in D displaystyle D nbsp meaning that A i A j displaystyle A i cap A j varnothing nbsp for all i j displaystyle i neq j nbsp then n 1 A n D displaystyle bigcup n 1 infty A n in D nbsp To be clear this property also holds for finite sequences A 1 A n displaystyle A 1 ldots A n nbsp of pairwise disjoint sets by letting A i displaystyle A i varnothing nbsp for all i gt n displaystyle i gt n nbsp 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 p 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 nbsp of subsets of W displaystyle Omega nbsp there exists a unique Dynkin system denoted D J displaystyle D mathcal J nbsp which is minimal with respect to containing J displaystyle mathcal J nbsp That is if D displaystyle tilde D nbsp is any Dynkin system containing J displaystyle mathcal J nbsp then D J D displaystyle D mathcal J subseteq tilde D nbsp D J displaystyle D mathcal J nbsp is called the Dynkin system generated by J displaystyle mathcal J nbsp For instance D W displaystyle D varnothing varnothing Omega nbsp For another example let W 1 2 3 4 displaystyle Omega 1 2 3 4 nbsp and J 1 displaystyle mathcal J 1 nbsp then D J 1 2 3 4 W displaystyle D mathcal J varnothing 1 2 3 4 Omega nbsp Sierpinski Dynkin s p l theorem editSierpinski Dynkin s p 𝜆 theorem 3 If P displaystyle P nbsp is a p system and D displaystyle D nbsp is a Dynkin system with P D displaystyle P subseteq D nbsp then s P D displaystyle sigma P subseteq D nbsp In other words the 𝜎 algebra generated by P displaystyle P nbsp is contained in D displaystyle D nbsp Thus a Dynkin system contains a p system if and only if it contains the 𝜎 algebra generated by that p system One application of Sierpinski Dynkin s p 𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval known as the Lebesgue measure Let W B ℓ displaystyle Omega mathcal B ell nbsp be the unit interval 0 1 with the Lebesgue measure on Borel sets Let m displaystyle m nbsp be another measure on W displaystyle Omega nbsp satisfying m a b b a displaystyle m a b b a nbsp and let D displaystyle D nbsp be the family of sets S displaystyle S nbsp such that m S ℓ S displaystyle m S ell S nbsp Let I a b a b a b a b 0 lt a b lt 1 displaystyle I a b a b a b a b 0 lt a leq b lt 1 nbsp and observe that I displaystyle I nbsp is closed under finite intersections that I D displaystyle I subseteq D nbsp and that B displaystyle mathcal B nbsp is the 𝜎 algebra generated by I displaystyle I nbsp It may be shown that D displaystyle D nbsp satisfies the above conditions for a Dynkin system From Sierpinski Dynkin s p 𝜆 Theorem it follows that D displaystyle D nbsp in fact includes all of B displaystyle mathcal B nbsp which is equivalent to showing that the Lebesgue measure is unique on B displaystyle mathcal B nbsp Application to probability distributions edit This section is transcluded from pi system edit history The p 𝜆 theorem motivates the common definition of the probability distribution of a random variable X W F P R displaystyle X Omega mathcal F operatorname P to mathbb R nbsp in terms of its cumulative distribution function Recall that the cumulative distribution of a random variable is defined asF X a P X a a R displaystyle F X a operatorname P X leq a qquad a in mathbb R nbsp 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 nbsp where B R displaystyle mathcal B mathbb R nbsp is the Borel 𝜎 algebra The random variables X W F P R displaystyle X Omega mathcal F operatorname P to mathbb R nbsp and Y W F P R displaystyle Y tilde Omega tilde mathcal F tilde operatorname P to mathbb R nbsp on two possibly different probability spaces are equal in distribution or law denoted by X D Y displaystyle X stackrel mathcal D Y nbsp if they have the same cumulative distribution functions that is if F X F Y displaystyle F X F Y nbsp The motivation for the definition stems from the observation that if F X F Y displaystyle F X F Y nbsp then that is exactly to say that L X displaystyle mathcal L X nbsp and L Y displaystyle mathcal L Y nbsp agree on the p system a a R displaystyle infty a a in mathbb R nbsp which generates B R displaystyle mathcal B mathbb R nbsp and so by the example above L X L Y displaystyle mathcal L X mathcal L Y nbsp A similar result holds for the joint distribution of a random vector For example suppose X displaystyle X nbsp and Y displaystyle Y nbsp are two random variables defined on the same probability space W F P displaystyle Omega mathcal F operatorname P nbsp with respectively generated p systems I X displaystyle mathcal I X nbsp and I Y displaystyle mathcal I Y nbsp The joint cumulative distribution function of X Y displaystyle X Y nbsp isF 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 nbsp However A X 1 a I X displaystyle A X 1 infty a in mathcal I X nbsp and B Y 1 b I Y displaystyle B Y 1 infty b in mathcal I Y nbsp BecauseI 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 nbsp is a p system generated by the random pair X Y displaystyle X Y nbsp the p 𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of X Y displaystyle X Y nbsp In other words X Y displaystyle X Y nbsp and W Z displaystyle W Z nbsp 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 nbsp 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 nbsp 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 nbsp The proof of this is another application of the p 𝜆 theorem 4 See also editAlgebra of sets Identities and relationships involving sets d ring Ring closed under countable intersections Field of sets Algebraic concept in measure theory also referred to as an algebra of sets Monotone class theoremPages displaying wikidata descriptions as a fallback Pages displaying short descriptions with no spaces p system Family of sets closed under intersection Ring of sets Family closed under unions and relative complements s algebra Algebraic structure of set algebra 𝜎 ideal Family closed under subsets and countable unions 𝜎 ring Ring closed under countable unionsNotes edit A sequence of sets A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp is called increasing if A n A n 1 displaystyle A n subseteq A n 1 nbsp for all n 1 displaystyle n geq 1 nbsp Proofs Assume D displaystyle mathcal D nbsp satisfies 1 2 and 3 Proof of 5 Property 5 follows from 1 and 2 by using B W displaystyle B Omega nbsp The following lemma will be used to prove 6 Lemma If A B D displaystyle A B in mathcal D nbsp are disjoint then A B D displaystyle A cup B in mathcal D nbsp Proof of Lemma A B displaystyle A cap B varnothing nbsp implies B W A displaystyle B subseteq Omega setminus A nbsp where W A W displaystyle Omega setminus A subseteq Omega nbsp by 5 Now 2 implies that D displaystyle mathcal D nbsp contains W A B W A B displaystyle Omega setminus A setminus B Omega setminus A cup B nbsp so that 5 guarantees that A B D displaystyle A cup B in mathcal D nbsp which proves the lemma Proof of 6 Assume that A 1 A 2 A 3 displaystyle A 1 A 2 A 3 ldots nbsp are pairwise disjoint sets in D displaystyle mathcal D nbsp For every integer n gt 0 displaystyle n gt 0 nbsp the lemma implies that D n A 1 A n D displaystyle D n A 1 cup cdots cup A n in mathcal D nbsp where because D 1 D 2 D 3 displaystyle D 1 subseteq D 2 subseteq D 3 subseteq cdots nbsp is increasing 3 guarantees that D displaystyle mathcal D nbsp contains their union D 1 D 2 A 1 A 2 displaystyle D 1 cup D 2 cup cdots A 1 cup A 2 cup cdots nbsp as desired displaystyle blacksquare nbsp Assume D displaystyle mathcal D nbsp satisfies 4 5 and 6 proof of 2 If A B D displaystyle A B in mathcal D nbsp satisfy A B displaystyle A subseteq B nbsp then 5 implies W B D displaystyle Omega setminus B in mathcal D nbsp and since W B A displaystyle Omega setminus B cap A varnothing nbsp 6 implies that D displaystyle mathcal D nbsp contains W B A W B A displaystyle Omega setminus B cup A Omega setminus B setminus A nbsp so that finally 4 guarantees that W W B A B A displaystyle Omega setminus Omega setminus B setminus A B setminus A nbsp is in D displaystyle mathcal D nbsp Proof of 3 Assume A 1 A 2 displaystyle A 1 subseteq A 2 subseteq cdots nbsp is an increasing sequence of subsets in D displaystyle mathcal D nbsp let D 1 A 1 displaystyle D 1 A 1 nbsp and let D i A i A i 1 displaystyle D i A i setminus A i 1 nbsp for every i gt 1 displaystyle i gt 1 nbsp where 2 guarantees that D 2 D 3 displaystyle D 2 D 3 ldots nbsp all belong to D displaystyle mathcal D nbsp Since D 1 D 2 D 3 displaystyle D 1 D 2 D 3 ldots nbsp are pairwise disjoint 6 guarantees that their union D 1 D 2 D 3 A 1 A 2 A 3 displaystyle D 1 cup D 2 cup D 3 cup cdots A 1 cup A 2 cup A 3 cup cdots nbsp belongs to D displaystyle mathcal D nbsp which proves 3 displaystyle blacksquare nbsp 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 Retrieved August 23 2010 Sengupta Lectures on measure theory lecture 6 The Dynkin p l Theorem PDF Math lsu Retrieved 3 January 2023 Kallenberg Foundations Of Modern probability p 48References editGut 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 amp 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 F displaystyle mathcal F nbsp of sets over W displaystyle Omega nbsp vteIs necessarily true of F displaystyle mathcal F colon nbsp or is F displaystyle mathcal F nbsp closed under Directedby displaystyle supseteq nbsp A B displaystyle A cap B nbsp A B displaystyle A cup B nbsp B A displaystyle B setminus A nbsp W A displaystyle Omega setminus A nbsp A 1 A 2 displaystyle A 1 cap A 2 cap cdots nbsp A 1 A 2 displaystyle A 1 cup A 2 cup cdots nbsp W F displaystyle Omega in mathcal F nbsp F displaystyle varnothing in mathcal F nbsp F I P p system nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Semiring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverSemialgebra Semifield nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverMonotone class nbsp nbsp nbsp nbsp nbsp only if A i displaystyle A i searrow nbsp only if A i displaystyle A i nearrow nbsp nbsp nbsp nbsp 𝜆 system Dynkin System nbsp nbsp nbsp only ifA B displaystyle A subseteq B nbsp nbsp nbsp only if A i displaystyle A i nearrow nbsp orthey are disjoint nbsp nbsp NeverRing Order theory nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Ring Measure theory nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Neverd Ring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Never𝜎 Ring nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverAlgebra Field nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Never𝜎 Algebra 𝜎 Field nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp NeverDual ideal nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Filter nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Prefilter Filter base nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Filter subbase nbsp nbsp nbsp Never Never nbsp nbsp nbsp F displaystyle varnothing not in mathcal F nbsp nbsp Open Topology nbsp nbsp nbsp nbsp nbsp nbsp nbsp even arbitrary displaystyle cup nbsp nbsp nbsp NeverClosed Topology nbsp nbsp nbsp nbsp nbsp nbsp even arbitrary displaystyle cap nbsp nbsp nbsp nbsp NeverIs necessarily true of F displaystyle mathcal F colon nbsp or is F displaystyle mathcal F nbsp closed under directeddownward finiteintersections finiteunions relativecomplements complementsin W displaystyle Omega nbsp countableintersections countableunions contains W displaystyle Omega nbsp contains displaystyle varnothing nbsp FiniteIntersectionPropertyAdditionally a semiring is a p system where every complement B A displaystyle B setminus A nbsp is equal to a finite disjoint union of sets in F displaystyle mathcal F nbsp A semialgebra is a semiring where every complement W A displaystyle Omega setminus A nbsp is equal to a finite disjoint union of sets in F displaystyle mathcal F nbsp A B A 1 A 2 displaystyle A B A 1 A 2 ldots nbsp are arbitrary elements of F displaystyle mathcal F nbsp and it is assumed that F displaystyle mathcal F neq varnothing nbsp Retrieved from https en wikipedia org w index php title Dynkin system amp oldid 1176405937 Dynkin s p l Theorem, wikipedia, wiki, book, books, library,