Dynkin system

A Dynkin system (sometimes also called the λ system ) is a term from measure theory , a branch of mathematics . It is named after the Russian mathematician Eugene Dynkin . In combination with Dynkin's π-λ theorem, they are an important tool for deriving uniqueness statements in measure theory and stochastics (see measure uniqueness theorem )

definition

A subset of the power set of a non-empty basic set is called a Dynkin system over if it has the following properties: ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {P}}(\Omega )}$ ${\displaystyle \Omega }$ ${\displaystyle \Omega }$

• The system contains the basic quantity:

${\displaystyle \Omega \in {\mathcal {D}}}$.

• The system is completed with the formation of complements :

${\displaystyle A\in {\mathcal {D}}\implies A^{c}\in {\mathcal {D}}}$.

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

${\displaystyle \{A_{n}\}_{n\in \mathbb {N} }\subset {\mathcal {D}}}$disjoint . ${\displaystyle \implies \bigcup _{n\in \mathbb {N} }A_{n}\in {\mathcal {D}}}$

δ operator

Any averages of Dynkin systems over result in a Dynkin system. Therefore, if there is a system of sets, then through ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {E}}\subseteq {\mathcal {P}}(\Omega )}$

${\displaystyle \delta ({\mathcal {E}}):=\bigcap _{{\mathcal {E}}\subseteq {\mathcal {S}} \atop {\mathcal {S}}{\text{ Dynkin-System}}}{\mathcal {S}}.}$

a Dynkin system defined called by generated Dynkin system . It is the smallest Dynkin system that contains. is called producer of . ${\displaystyle \delta ({\mathcal {E}})}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle \delta ({\mathcal {E}})}$

The δ operator is an envelope operator . In some cases it is noted as an -operator according to the naming system . Other alternative names are or . ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda (\cdot )}$ ${\displaystyle d({\mathcal {E}})}$ ${\displaystyle {\mathcal {D}}({\mathcal {E}})}$

The Dynkin System Argument

With Dynkin systems, statements about σ-algebras can be proven relatively easily in many cases. Be a statement that either applies to sets or not. Furthermore, let us be a σ-algebra with an average stable generator , for whose elements one can show. Following the principle of good quantities , one now looks at the quantity system and shows that it is a Dynkin system. Then follows because of the average stability on the one hand, but on the other hand it also applies and therefore already . ${\displaystyle \alpha }$ ${\displaystyle A\subseteq \Omega }$ ${\displaystyle \Sigma }$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle \alpha }$ ${\displaystyle {\mathcal {D}}:=\{A\in \Sigma \colon A{\mbox{ erfüllt }}\alpha \}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle \delta ({\mathcal {E}})=\sigma ({\mathcal {E}})}$ ${\displaystyle {\mathcal {E}}\subseteq {\mathcal {D}}\subseteq \Sigma }$ ${\displaystyle \Sigma =\sigma ({\mathcal {E}})=\delta ({\mathcal {E}})\subseteq {\mathcal {D}}}$ ${\displaystyle \Sigma ={\mathcal {D}}}$

The defining properties of a Dynkin system are often easier to prove, because in the case of closeness to countable union only sequences of pairwise disjoint individual sets have to be considered, while this additional property is not available for σ-algebras.

Connection with other quantity systems

Hierarchy of the quantity systems used in measure theory

σ-algebras

Every σ-algebra is always a Dynkin system. Conversely, every average stable Dynkin system is also a σ-algebra. An example of a Dynkin system that is not a σ-algebra is

${\displaystyle {\mathcal {M}}=\{\emptyset ,\{1,2\},\{3,4\},\{1,4\},\{2,3\},\{1,2,3,4\}\}}$

on the basic set . The set system is a Dynkin system, but not an algebra (since it is not stable in terms of cuts) and therefore also not a σ-algebra. ${\displaystyle \Omega =\{1,2,3,4\}}$

In addition , Dynkin's π-λ theorem applies : If the system of sets is stable on average, then the σ-algebra generated by and the Dynkin system generated by coincide. ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}}$

Monotonous classes

Dynkin systems are also monotone classes define: A system of exactly then a Dynkin system, when a monotonous class is that the superset contains, and in for arbitrary sets with true that is. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \Omega }$ ${\displaystyle A,B\in {\mathcal {D}}}$ ${\displaystyle B\subset A}$ ${\displaystyle A\backslash B\in {\mathcal {D}}}$

literature

• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , pp. 20-21 , doi : 10.1007 / 978-3-663-01244-3 .
• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Heinz Bauer : Measure and integration theory. Walter de Gruyter, Berlin a. a. 1992, ISBN 3-11-013626-0 .

Individual evidence

1. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 4 , doi : 10.1007 / 978-3-642-36018-3 .