Dynkin-System

A Dynkin system is a term from the measure theory, a branch of mathematics. It is named after the Russian mathematician Eugene Dynkin.

Definition

A subset of the power set of a non-empty fundamental quantity called on Dynkin system, if it has the following properties:

• The system contains the basic amount:

• The system is closed under the formation of complements:

• The system is closed with respect to countable unions of pairwise disjoint sets:

δ - operator

Any averages of Dynkin systems on yield again a Dynkin system. If, therefore, a quantity system, then by

A Dynkin system defined called the Dynkin system generated by. It is the smallest Dynkin system containing. ie producers.

The δ - operator is a closure operator.

Associated with σ - algebra

• A Dynkin system is a σ - algebra if and only if it is average stable.
• For each subset of mean stable is that the generated Dynkin system coincides with the generated σ algebra: (see σ operator).

Examples

• Every σ ​​- algebra is a Dynkin system.
• Be, then, is a Dynkin system, but no σ - algebra, since it is not stable average.

The Dynkin - system - argument

With Dynkin systems, in many cases, statements of σ - algebras is relatively easy to prove. Be a statement that is true for either quantities or not. Next is a σ - algebra with an average stable producers, whose elements can be shown. According to the principle of good quantities one now considers the quantity system and shows that it is a Dynkin system. Then it follows by the average stability of one hand, but on the other hand, applies and thus for ever.

The defining characteristics of a Dynkin system are often easier to prove, because in isolation from countable union only sequences of pairwise disjoint individual amounts need to be considered, while σ - algebras this additional property is not available.