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Dynkin system

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Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set \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.[1] These set families have applications in measure theory and probability.

The primary relevance of λ-systems are their use in applications of the π-λ theorem.

Contents

  • Definitions 1
  • Dynkin's π-λ theorem 2
  • Notes 3
  • References 4

Definitions

Let Ω be a nonempty set, and let D be a collection of subsets of Ω (i.e., D is a subset of the power set of Ω). Then D is a Dynkin system if

  1. Ω ∈ D,
  2. if A, BD and AB, then B \ AD,
  3. if A1, A2, A3, ... is a sequence of subsets in D and AnAn+1 for all n ≥ 1, then \bigcup_{n=1}^\infty A_n\in D.

Equivalently, D is a Dynkin system if

  1. Ω ∈ D,
  2. if AD, then AcD,
  3. if A1, A2, A3, ... is a sequence of subsets in D such that AiAj = Ø for all ij, then \bigcup_{n=1}^\infty A_n\in D.

The second definition is generally preferred as it usually is easier to check.

An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersection) is a σ-algebra. This can be verified by noting that condition 3 and closure under finite intersection implies closure under countable unions.

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

Dynkin's π-λ theorem

If P is a π-system and D is a Dynkin system with P\subseteq D, then \sigma\{P\}\subseteq D. In other words, the σ-algebra generated by P is contained in D.

One application of 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, λ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let μ be another measure on Ω satisfying μ[(a,b)] = b − a, and let D be the family of sets S such that μ[S] = λ[S]. Let I = { (a,b),[a,b),(a,b],[a,b] : 0 < ab < 1 }, and observe that I is closed under finite intersections, that ID, 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 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.

Additional applications are in the article on π-systems.

Notes

  1. ^ Charalambos Aliprantis, Kim C. Border (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide, 3rd ed. Springer. Retrieved August 23, 2010. 

References

  • Gut, Allan (2005). Probability: A Graduate Course. New York: Springer.  
  • Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.  
  • David Williams (2007). Probability with Martingales. Cambridge University Press. p. 193.  

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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