Dynkin system


A Dynkin system, 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 or d-system. These set families have applications in measure theory and probability.
A major application of λ-systems is the π-λ theorem, see below.

Definitions

Let Ω be a nonempty set, and let be a collection of subsets of Ω. Then is a Dynkin system if
  1. Ω ∈,
  2. if A, B ∈ and A ⊆ B, then B \ A ∈,
  3. if A1, A2, A3,... is a sequence of subsets in and An ⊆ An+1 for all n ≥ 1, then.
Equivalently, is a Dynkin system if
  1. Ω ∈,
  2. if A ∈, then Ac ∈ ,
  3. if A1, A2, A3,... is a sequence of subsets in such that Ai ∩ Aj = Ø for all i ≠ j, then.
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 is a σ-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply 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. Note. For another example, let and ; then.

Dynkin's π-λ theorem

If is a π-system and is a Dynkin system with, then. In other words, the σ-algebra generated by is contained in.
One application of Dynkin's π-λ theorem is the uniqueness of a measure that evaluates the length of an interval :
Let be the unit interval with the Lebesgue measure on Borel sets. Let μ be another measure on Ω satisfying μ = b − a, and let D be the family of sets S such that μ = λ. Let I =, and observe that I is closed under finite intersections, that I ⊂ 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 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