Cylinder quantity

A cylinder set , sometimes called fringe events , is a special set that is used in measure theory , a branch of mathematics . A special case of a cylinder set is a rectangular cylinder . Systems of cylinder sets are used to define product σ-algebras , which in turn form the basis for the definition of product dimensions and product models .

definition

An arbitrary index set , a basic set, is given ${\displaystyle I}$

${\displaystyle \Omega :=\prod _{i\in I}\Omega _{i}}$

as well as the canonical projection for a subset ${\displaystyle J\subset I}$

${\displaystyle \pi _{J}:\Omega \to \prod _{i\in J}\Omega _{i},\quad \pi _{J}(\omega )=\omega |_{J}}$,

where the restriction to the components is referred to in. Then a lot is called the form ${\displaystyle \omega |_{J}}$ ${\displaystyle J}$

${\displaystyle \pi _{J}^{-1}(M)\subset \Omega {\text{ für }}M\in \Omega _{J}:=\prod _{i\in J}\Omega _{i}}$

a set of cylinders with a base . ${\displaystyle J}$

Derived terms

System of cylinder quantities

If a σ-algebra is given on the set , the set system is called ${\displaystyle \Omega _{J}}$ ${\displaystyle {\mathcal {A}}_{J}}$

${\displaystyle {\mathcal {Z}}_{J}:=\{\pi _{J}^{-1}(A_{J})\,|\,A_{J}\in {\mathcal {A}}_{J}\}}$

the quantity system of cylinder quantities.

Rectangular cylinder

An element of the σ-algebra can be as Cartesian product of sets in the σ-algebra to write, so ${\displaystyle {\mathcal {A}}_{J}}$ ${\displaystyle {\mathcal {A}}_{j}}$ ${\displaystyle \Omega _{i}}$

${\displaystyle A_{J}=\prod _{i\in J}A_{i}{\text{ für }}A_{i}\in {\mathcal {A}}_{i}}$,

this is what one calls a rectangular cylinder with a base . Then you define ${\displaystyle A_{J}}$ ${\displaystyle J}$

${\displaystyle {\mathcal {Z}}_{J}^{R}:=\{\pi _{J}^{-1}(A_{J})\,|\,A_{J}{\text{ ist Rechteckszylinder }}\}}$

as a quantity system of all rectangular cylinders.

properties

One defines the set system

${\displaystyle {\mathcal {Z}}:=\bigcup _{J\subseteq I \atop J{\text{ endlich}}}{\mathcal {Z}}_{J}}$,

so this is a producer of the product σ-algebra of , so it's ${\displaystyle {\mathcal {A}}_{i}}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}=\sigma ({\mathcal {Z}})}$.

Likewise, the system of sets that arises from the union of all finite rectangular cylinders is

${\displaystyle {\mathcal {Z}}^{R}:=\bigcup _{J\subseteq I \atop J{\text{ endlich}}}{\mathcal {Z}}_{J}^{R}}$

a producer of the product σ-algebra , so it is ${\displaystyle {\mathcal {A}}_{i}}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}=\sigma ({\mathcal {Z}}^{R})}$.

literature

• 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 .
• 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 .