Dynkin's π-λ theorem

The Dynkin π-λ theorem (after Eugene Dynkin ) is a theorem from measure theory , a branch of mathematics . It makes a statement about the conditions under which two systems of sets coincide. For example, it serves as an aid in the principle of good quantities .

statement

Let be a set system as well as the σ-algebra generated by the set system and the Dynkin system generated by the set system . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \sigma ({\mathcal {M}})}$ ${\displaystyle \delta ({\mathcal {M}})}$

The statement now reads: If there is an average stable set system , then the σ-algebra it generates and the Dynkin system it generates agree. It then applies ${\displaystyle {\mathcal {M}}}$

${\displaystyle \sigma ({\mathcal {M}})=\delta ({\mathcal {M}})}$.

Naming

The naming of the theorem follows from the fact that Dynkin systems are also called λ systems and sets systems with average stability are also called π systems. The theorem can thus also be formulated as follows: The generated σ-algebra of a π-system is equal to the generated λ-system of the π-system.

Evidence sketch

It is because every σ-algebra is a Dynkin system and the smallest Dynkin system that contains. ${\displaystyle \delta ({\mathcal {M}})\subset \sigma ({\mathcal {M}})}$ ${\displaystyle \delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}}$

It then remains to be shown that . One shows that the Dynkin system is a σ-algebra. Then the Dynkin system contains the σ-algebra, since the Dynkin system contains the generator and is the smallest σ-algebra that contains the generator. ${\displaystyle \delta ({\mathcal {M}})\supset \sigma ({\mathcal {M}})}$ ${\displaystyle \delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \sigma ({\mathcal {M}})}$

A Dynkin system is a σ-algebra if and only if it is stable on the average. So the average stability has to be shown. To do this, you define the system of auxiliary amounts

${\displaystyle {\mathcal {D}}_{1}:=\{A\in \delta ({\mathcal {M}})\,|\,A\cap B\in \delta ({\mathcal {M}}),\,B\in {\mathcal {M}}\}\supset \{A\in {\mathcal {M}}\,|\,A\cap B\in \delta ({\mathcal {M}}),\,B\in {\mathcal {M}}\}={\mathcal {M}}}$,

since is an average stable set system. Now you can show that there is also a Dynkin system. But since is and is true, then is . ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {D}}_{1}}$ ${\displaystyle {\mathcal {D}}_{1}\subset \delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {D}}_{1}}$ ${\displaystyle {\mathcal {D}}_{1}=\delta ({\mathcal {M}})}$

Now one forms the second auxiliary set system of the average stable sets of the Dynkin system

${\displaystyle {\mathcal {D}}_{2}:=\{A\in \delta ({\mathcal {M}})\,|\,A\cap B\in \delta ({\mathcal {M}}),\,B\in \delta ({\mathcal {M}})\}}$.

By definition is then , but since according to the above statement, is also applies . Now it can be shown that a Dynkin system is also, so is and therefore also . Since, according to the definition, is stable to the mean, is also stable to the mean, i.e. a σ-algebra, which was to be shown. ${\displaystyle {\mathcal {D}}_{2}\subset \delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {D}}_{1}=\delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}\subset {\mathcal {D}}_{2}}$ ${\displaystyle {\mathcal {D}}_{2}}$ ${\displaystyle {\mathcal {D}}_{2}\supset \delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {D}}_{2}=\delta ({\mathcal {M}})}$ ${\displaystyle {\mathcal {D}}_{2}}$ ${\displaystyle \delta ({\mathcal {M}})}$

literature

• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , pp. 21-22 , doi : 10.1007 / 978-3-663-01244-3 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .