Dynkin system

A Dynkin system on $$\Omega$$ is a set $$\mathcal{D}$$ consisting of certain subsets of $$\Omega$$ such that


 * the set $$\Omega$$ itself is in $$\mathcal{D}$$
 * if $$A,B \in \mathcal{D}$$ and $$A \subseteq B$$ then $$B \setminus A \in \mathcal{D}$$
 * if $$A_n$$ is a sequence of sets in $$\mathcal{D}$$ which is increasing in the sense that $$A_n \subseteq A_{n+1},\ n \ge 1$$, then the union $$\bigcup_{k=1}^{\infty}A_k$$ also lies in $$\mathcal{D}.$$

If $$\mathcal{J}$$ is any set of subsets of $$\Omega$$, then the intersection of all the Dynkin systems containing $$\mathcal{J}$$ is itself a Dynkin system. It is called the Dynkin system generated by $$\mathcal{J}$$. It is the smallest Dynkin system containing $$\mathcal{J}$$.

A Dynkin system which is also a &pi;-system is a &sigma;-algebra.

Dynkin systems are named after the Russian mathematician Eugene Dynkin.

The Dynkin system theorem (monotone class theorem, Dynkin's lemma) states:

Let $$\mathcal{C}$$ be a &pi;-system; that is, a collection of subsets of $$\Omega$$ which is closed under pairwise intersections. If $$\mathcal{D}$$ is a Dynkin system containing $$\mathcal{C}$$, then $$\mathcal{D}$$ also contains the $$\sigma$$-algebra $$\sigma(\mathcal{C})$$ generated by $$\mathcal{C}$$.

One application of Dynkin's lemma is the uniqueness of the Lebesgue measure:

Let (&Omega;, B, &lambda;) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let &mu; be another measure on &Omega; satisfying &mu;[(a,b)] = b - a, and let D be the family of sets such that &mu;[D] = &lambda;[D]. Let I = { (a,b),[a,b),(a,b],[a,b] : 0 < a &le; b < 1 }, and observe that I is closed under finite intersections, that I &sub; D, and that B is the &sigma;-algebra generated by I. One easily shows D satisfies the above conditions for a Dynkin-system. From Dynkin's lemma it follows that D is in fact all of B, which is equivalent to showing that the Lebesgue measure is unique.

Dynkin-System Dynkinsysteem ディンキン族 Λ-układ