Dynkin system
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 π-algebra. Dynkin systems are sometimes referred to as π-systems (Dynkin himself used this term) or d-system.[2] These set families have applications in measure theory and probability.
A major application of π-systems is the Ο-π theorem, see below.
Definition
Let Β be a nonempty set, and let Β be a collection of subsets of Β (that is, Β is a subset of the power set of Β ). Then Β is a Dynkin system if
- Β
- Β is closed under complements of subsets in supersets: if Β and Β then Β
- Β is closed under countable increasing unions: if Β is an increasing sequence[note 1] of sets in Β then Β
It is easy to check[proof 1] that any Dynkin system Β satisfies:
- Β
- Β is closed under complements in Β : if Β then Β
- Taking Β shows that Β
- Β is closed under countable unions of pairwise disjoint sets: if Β is a sequence of pairwise disjoint sets in Β (meaning that Β for all Β ) then Β
- To be clear, this property also holds for finite sequences Β of pairwise disjoint sets (by letting Β for all Β ).
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 Ο-system (that is, closed under finite intersections) is a π-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 Β of subsets of Β there exists a unique Dynkin system denoted Β which is minimal with respect to containing Β That is, if Β is any Dynkin system containing Β then Β Β is called the Dynkin system generated by Β For instance, Β For another example, let Β and Β ; then Β
SierpiΕskiβDynkin's Ο-Ξ» theorem
SierpiΕski-Dynkin's Ο-π theorem ([3]: If Β is a Ο-system and Β is a Dynkin system with Β then Β
In other words, the π-algebra generated by Β is contained in Β Thus a Dynkin system contains a Ο-system if and only if it contains the π-algebra generated by that Ο-system.
One application of SierpiΕski-Dynkin's Ο-π theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):
Let Β be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let Β be another measure on Β satisfying Β and let Β be the family of sets Β such that Β Let Β and observe that Β is closed under finite intersections, that Β and that Β is the π-algebra generated by Β It may be shown that Β satisfies the above conditions for a Dynkin-system. From SierpiΕski-Dynkin's Ο-π Theorem it follows that Β in fact includes all of Β which is equivalent to showing that the Lebesgue measure is unique on Β
Application to probability distributions
The Ο-π theorem motivates the common definition of the probability distribution of a random variable Β in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as
A similar result holds for the joint distribution of a random vector. For example, suppose Β and Β are two random variables defined on the same probability space Β with respectively generated Ο-systems Β and Β The joint cumulative distribution function of Β is
However, Β and Β Because
In the theory of stochastic processes, two processes Β are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all Β
The proof of this is another application of the Ο-π theorem.[4]
See also
- Algebra of setsΒ β Identities and relationships involving sets
- Ξ΄-ringΒ β Ring closed under countable intersections
- Field of setsΒ β Algebraic concept in measure theory, also referred to as an algebra of sets
- Monotone class
- Ο-systemΒ β Family of sets closed under intersection
- Ring of setsΒ β Family closed under unions and relative complements
- Ο-algebraΒ β Algebric structure of set algebra
- π-idealΒ β Family closed under subsets and countable unions
- π-ringΒ β Ring closed under countable unions
Notes
- ^ A sequence of sets Β is called increasing if Β for all Β
Proofs
- ^ Assume Β satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using Β The following lemma will be used to prove (6). Lemma: If Β are disjoint then Β Proof of Lemma: Β implies Β where Β by (5). Now (2) implies that Β contains Β so that (5) guarantees that Β which proves the lemma. Proof of (6) Assume that Β are pairwise disjoint sets in Β For every integer Β the lemma implies that Β where because Β is increasing, (3) guarantees that Β contains their union Β as desired. Β
- ^ Assume Β satisfies (4), (5), and (6). proof of (2): If Β satisfy Β then (5) implies Β and since Β (6) implies that Β contains Β so that finally (4) guarantees that Β is in Β Proof of (3): Assume Β is an increasing sequence of subsets in Β let Β and let Β for every Β where (2) guarantees that Β all belong to Β Since Β are pairwise disjoint, (6) guarantees that their union Β belongs to Β which proves (3).Β
- ^ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
- ^ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (ThirdΒ ed.). Springer. Retrieved August 23, 2010.
- ^ Sengupta. "Lectures on measure theory lecture 6: The Dynkin Ο β Ξ» Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
- ^ Kallenberg, Foundations Of Modern probability, p.Β 48
References
- Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBNΒ 0-387-22833-0.
- Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBNΒ 0-471-00710-2.
- Williams, David (2007). Probability with Martingales. Cambridge University Press. p.Β 193. ISBNΒ 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 Β of sets over Β | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of Β or, is Β closed under: | Directed by Β | Β | Β | Β | Β | Β | Β | Β | Β | F.I.P. |
Ο-system | Β | Β | Β | Β | Β | Β | Β | Β | Β | Β |
Semiring | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
Semialgebra (Semifield) | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
Monotone class | Β | Β | Β | Β | Β | only if Β | only if Β | Β | Β | Β |
π-system (Dynkin System) | Β | Β | Β | only if Β | Β | Β | only if Β or they are disjoint | Β | Β | Never |
Ring (Order theory) | Β | Β | Β | Β | Β | Β | Β | Β | Β | Β |
Ring (Measure theory) | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
Ξ΄-Ring | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
π-Ring | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
Algebra (Field) | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
π-Algebra (π-Field) | Β | Β | Β | Β | Β | Β | Β | Β | Β | Never |
Dual ideal | Β | Β | Β | Β | Β | Β | Β | Β | Β | Β |
Filter | Β | Β | Β | Never | Never | Β | Β | Β | Β | Β |
Prefilter (Filter base) | Β | Β | Β | Never | Never | Β | Β | Β | Β | Β |
Filter subbase | Β | Β | Β | Never | Never | Β | Β | Β | Β | Β |
Open Topology | Β | Β | Β | Β | Β | Β | Β (even arbitrary Β ) | Β | Β | Never |
Closed Topology | Β | Β | Β | Β | Β | Β (even arbitrary Β ) | Β | Β | Β | Never |
Is necessarily true of Β or, is Β closed under: | directed downward | finite intersections | finite unions | relative complements | complements in Β | countable intersections | countable unions | contains Β | contains Β | Finite Intersection Property |
Additionally, a semiring is a Ο-system where every complement Β is equal to a finite disjoint union of sets in Β |