π system

This article deals with a special set system in mathematics. In chemistry, the “π system” denotes conjugated multiple bonds .

A π system , including average amount stable system or short cut stable system called, is a specific quantity system , which the axiomatic construction of the theory of probability and measure theory can be used.

definition

A set system is given , i.e. a subset of the power set of a basic set . is called a π-system , an average-stable system of sets or an intersection-stable system , if for any two sets in the system that is. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle A,B}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle A\cap B\in {\mathcal {S}}}$

Examples

For any basic set, let the set system be ${\displaystyle \Omega }$

${\displaystyle {\mathcal {S}}:=\{A\subset \Omega \mid |A|<\infty \}}$

of all finite subsets. For any two is now , the intersection of finite sets is always finite. So it is , it is a stable system. ${\displaystyle A,B\in {\mathcal {S}}}$ ${\displaystyle |A\cap B|\leq \min(|A|,|B|)}$ ${\displaystyle A\cap B\in {\mathcal {S}}}$

properties

• If the system of sets is stable with the formation of complement, then it is stable on average precisely when it is stable in union. This follows directly from de Morgan's laws .
• If the set system is stable with the formation of difference sets , then it is also a π-system. This follows from . ${\displaystyle {\mathcal {S}}}$ ${\displaystyle A\cap B=A\setminus (A\setminus B)\in {\mathcal {S}}}$

use

Set systems with average stability appear at some points in probability theory and stochastics . The average stability is an important prerequisite for the generator of a σ-algebra in order to have to check the stochastic independence of the random variables only on this generator .

The most important application is the so-called Dynkin π-λ theorem . If there is a π-system, then the σ-algebra generated by and the Dynkin system generated agree, so it is true ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$

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

See also

• Σ-algebra
• Dynkin system
• Monotonous class

literature

• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , pp. 20 , doi : 10.1007 / 978-3-663-01244-3 .
• Heinz Bauer : Measure and integration theory . 2nd, revised edition. de Gruyter, Berlin et al. 1992, ISBN 3-11-013626-0 .
• 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 .