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 $\sigma$-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 $\pi$-𝜆 theorem, see below.

Definition
Let $$\Omega$$ be a nonempty set, and let $$D$$ be a collection of subsets of $$\Omega$$ (that is, $$D$$ is a subset of the power set of $$\Omega$$). Then $$D$$ is a Dynkin system if
 * 1) $$\Omega \in D;$$
 * 2) $$D$$ is closed under complements of subsets in supersets: if $$A, B \in D$$ and $$A \subseteq B,$$ then $$B \setminus A \in D;$$
 * 3) $$D$$ is closed under countable increasing unions: if $$A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$$ is an increasing sequence of sets in $$D$$ then $$\bigcup_{n=1}^\infty A_n \in D.$$

It is easy to check that any Dynkin system $$D$$ satisfies: $$\varnothing \in D;$$ $$D$$ is closed under complements in $$\Omega$$: if $A \in D,$ then $$\Omega \setminus A \in D;$$ $$D$$ is closed under countable unions of pairwise disjoint sets: if $$A_1, A_2, A_3, \ldots$$ is a sequence of pairwise disjoint sets in $$D$$ (meaning that $$A_i \cap A_j = \varnothing$$ for all $$i \neq j$$) then $$\bigcup_{n=1}^\infty A_n \in D.$$ 
 * Taking $$A := \Omega$$ shows that $$\varnothing \in D.$$
 * To be clear, this property also holds for finite sequences $$A_1, \ldots, A_n$$ of pairwise disjoint sets (by letting $$A_i := \varnothing$$ for all $$i > n$$).

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.

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 $$\mathcal{J}$$ 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 For instance, $$D\{\varnothing\} = \{\varnothing, \Omega\}.$$ For another example, let $$\Omega = \{1,2,3,4\}$$ and $$\mathcal{J} = \{1\}$$; then $$D\{\mathcal{J}\} = \{\varnothing, \{1\}, \{2,3,4\}, \Omega\}.$$

Sierpiński–Dynkin's π-λ theorem
Sierpiński-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 $$D.$$ 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 $$(\Omega, \mathcal{B}, \ell)$$ be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let $$m$$ be another measure on $$\Omega$$ satisfying $$m[(a, b)] = b - a,$$ and let $$D$$ be the family of sets $$S$$ such that $$m[S] = \ell[S].$$ Let $$I := \{ (a, b), [a, b), (a, b], [a, b] : 0 < a \leq b < 1 \},$$ and observe that $$I$$ is closed under finite intersections, that $$I \subseteq D,$$ and that $$\mathcal{B}$$ is the 𝜎-algebra generated by $$I.$$  It may be shown that $$D$$ satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that $$D$$ in fact includes all of $$\mathcal{B}$$, which is equivalent to showing that the Lebesgue measure is unique on $$\mathcal{B}$$.