Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set ${\displaystyle \Omega }$ 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.

A major application of λ-systems is the π-λ theorem, see below.

Definitions

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

1. Ω ∈ ${\displaystyle D}$,
2. if A, B${\displaystyle D}$ and AB, then B \ A${\displaystyle D}$,
3. if A1, A2, A3, ... is a sequence of subsets in ${\displaystyle D}$ and AnAn+1 for all n ≥ 1, then ${\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D}$.

Equivalently, ${\displaystyle D}$ is a Dynkin system if

1. Ω ∈ ${\displaystyle D}$,
3. if A1, A2, A3, ... is a sequence of subsets in ${\displaystyle D}$ such that AiAj = Ø for all ij, then ${\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D}$.

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

Dynkin's π-λ theorem

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

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