Initial σ algebra

An initial σ-algebra is a term from measure theory , a branch of mathematics . It is used to define σ-algebras on spaces that previously had no structure and has the product σ-algebra and the trace σ-algebra as special cases . It is closely linked to the initial topology . The counterpart to the initial σ algebra is the final σ algebra . It is the largest system of sets, so that a given set of functions can be measured . The initial σ-algebra is also called the σ-algebra generated (by the functions ) . However, this naming is not unique, since σ-algebras can also be generated by set systems . ${\displaystyle f_{i}}$

definition

Let images and a family of measurement spaces be given for a non-empty index set . Then it is called σ-algebra ${\displaystyle f_{i}\colon \Omega \to \Omega _{i}}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle I}$

${\displaystyle {\mathcal {I}}(f_{i},\,i\in I):=\sigma \left(\bigcup _{i\in I}f_{i}^{-1}({\mathcal {A}}_{i})\right)}$

on the initial σ-algebra of the mappings or the σ-algebra generated by the mappings . ${\displaystyle \Omega }$ ${\displaystyle (f_{i})_{i\in I}}$ ${\displaystyle (f_{i})_{i\in I}}$

properties

• The initial σ-algebra is by definition the smallest σ-algebra in terms of set- theoretical inclusion , with respect to which all functions are measurable . ${\displaystyle \Omega }$ ${\displaystyle (f_{i})_{i\in I}}$
• Are producers of , then is a producer of . ${\displaystyle {\mathcal {E}}_{i}}$ ${\displaystyle {\mathcal {A}}_{i}}$ ${\displaystyle \bigcup _{i\in I}f_{i}^{-1}({\mathcal {E}}_{i})}$ ${\displaystyle {\mathcal {I}}(f_{i},\,i\in I)}$

Examples

• There is already a σ-algebra for a single mapping in a measurement space , so it applies . For example, if a constant function is the trivial σ-algebra . The following applies to the indicator function of a subset . ${\displaystyle f\colon \Omega \to \Omega '}$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle f^{-1}({\mathcal {A}}')}$ ${\displaystyle {\mathcal {I}}(f)=f^{-1}({\mathcal {A}}')}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {I}}(f)}$ ${\displaystyle \{\emptyset ,\Omega \}}$ ${\displaystyle \chi _{A}}$ ${\displaystyle A\subset \Omega }$ ${\displaystyle {\mathcal {I}}(\chi _{A})=\{\emptyset ,A,A^{\mathsf {c}},\Omega \}}$
• Is and a measurement space and the natural embedding , the initial σ-algebra is exactly the track σ-algebra : . ${\displaystyle X\subset Y}$ ${\displaystyle (Y,{\mathcal {A}})}$ ${\displaystyle i\colon X\to Y,\,i(x)=x}$ ${\displaystyle {\mathcal {I}}(i)={\mathcal {A}}|_{X}}$
• Is the Cartesian product of sets for a non-empty index set and measurement spaces. If the projections on the -th component are selected as the map , then the initial σ-algebra of the projections is exactly the product σ-algebra of : ${\displaystyle \textstyle \Omega =\prod _{i\in I}\Omega _{i}}$ ${\displaystyle \Omega _{i}}$ ${\displaystyle I}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle \pi _{i}\colon \Omega \to \Omega _{i},\,\pi _{i}(\omega )=\omega _{i}}$ ${\displaystyle i}$ ${\displaystyle {\mathcal {A}}_{i}}$

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

use

Initial σ algebras are used, for example, in probability theory to define the stochastic independence of random variables . Two random variables are independent if and only if their initial σ-algebras are independent set systems .

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 .