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 (Dynkin himself used this term) 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 (i.e.,  is a subset of the power set of Ω). Then  is a Dynkin system if
 Ω ∈ ,
 if A, B ∈  and A ⊆ B, then B \ A ∈ ,
 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
 Ω ∈ ,
 if A ∈ , then Ac ∈ ,
 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 (i.e., 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 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 (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 < a ≤ b < 1 }, 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

Notes