Product σ algebra

A product σ-algebra , also known as Kolmogorow's σ-algebra , is a term from measure theory , a branch of mathematics . Product σ-algebras allow the definition of product dimensions that generalize the intuitive concept of volume to higher-dimensional spaces .

definition

A basic set is given , which is the Cartesian product for a non-empty index set . Each of the sets is also provided with a σ-algebra . The product σ-algebra of (or Kolmogorow's σ-algebra ) is then defined as ${\displaystyle \textstyle \Omega =\prod _{i\in I}\Omega _{i}}$ ${\displaystyle I}$ ${\displaystyle \Omega _{i}}$ ${\displaystyle {\mathcal {A}}_{i}}$ ${\displaystyle ({\mathcal {A}}_{i})_{i\in I}}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}:=\sigma \left(\{\pi _{i}^{-1}(A_{i})\,|\,i\in I,\,A_{i}\in {\mathcal {A}}_{i}\}\right)}$,

where denotes the projection onto the -th component. The couple ${\displaystyle \textstyle \pi _{i}\colon \Omega \rightarrow \Omega _{i}}$ ${\displaystyle i}$

${\displaystyle {\biggl (}\prod _{i\in I}\Omega _{i},\bigotimes _{i\in I}{\mathcal {A}}_{i}{\biggr )}}$

forms a measuring room , which is also referred to as the measurable product of the family . ${\displaystyle ((\Omega _{i},{\mathcal {A}}_{i}))_{i\in I}}$

Notation conventions

Is , so one often writes instead . ${\displaystyle I=\{1,2\}}$ ${\displaystyle {\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2}}$ ${\displaystyle \textstyle \bigotimes _{i=1}^{2}{\mathcal {A}}_{i}}$

If for all , one also uses the notation for the corresponding product σ-algebra in some cases . ${\displaystyle {\mathcal {A}}_{i}={\mathcal {A}}}$ ${\displaystyle i\in I}$ ${\displaystyle {\mathcal {A}}^{\otimes I}}$

Alternative definitions

Using measurable functions

The product σ-algebra can also be defined as the smallest σ-algebra with respect to which the projections onto the individual components can be measured . Since measurability only has to be checked on one generator of the σ-algebras , this results in ${\displaystyle {\mathcal {E}}_{i}}$ ${\displaystyle {\mathcal {A}}_{i}}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}=\sigma {\biggl (}\bigcup _{i\in I}\pi _{i}^{-1}({\mathcal {A}}_{i}){\biggr )}=\sigma {\biggl (}\bigcup _{i\in I}\pi _{i}^{-1}({\mathcal {E}}_{i}){\biggr )}}$.

Thus the product σ-algebra is the initial σ-algebra of : ${\displaystyle {\mathcal {A}}_{i}}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle \pi _{i}}$

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

As a product of families

If one understands two σ-algebras as set algebras and forms the product of these algebras , then again an algebra and a generator of the product σ-algebra is: ${\displaystyle {\mathcal {A}}_{1},{\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {C}}:={\mathcal {A}}_{1}\boxtimes {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {C}}}$

${\displaystyle {\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2}=\sigma ({\mathcal {A}}_{1}\boxtimes {\mathcal {A}}_{2})}$.

If one generalizes this to larger index sets, the following applies: If it is countable (or finite ), then applies ${\displaystyle I}$

${\displaystyle \bigotimes _{i\in I}{\mathcal {A}}_{i}=\sigma {\biggl (}\prod '_{i\in I}{\mathcal {A}}_{i}{\biggr )},}$

in which

${\displaystyle \prod '_{i\in I}{\mathcal {A}}_{i}={\biggl \{}\prod _{i\in I}A_{i}\mid (A_{i})_{i\in I}\in \prod _{i\in I}{\mathcal {A}}_{i}{\biggr \}}}$

is the product of the family . Note that it is the product of two σ-algebras and in general not a σ-algebra. However, a half-ring, and in particular, is stable. ${\displaystyle ({\mathcal {A}}_{i})_{i\in I}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{1}\times {\mathcal {A}}_{2}}$ ${\displaystyle \cap }$

Cylinder quantities

Alternatively, for any index sets, the product σ-algebra can also be defined as the σ-algebra generated by the cylinder sets. The cylinder sets are the archetypes of the elements of a σ-algebra under the canonical projection.

Examples

• Be and two measuring rooms. Then the corresponding product σ-algebra is: ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})=(\{K,Z\},\{\emptyset ,\{K\},\{Z\},\{K,Z\}\})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})=(\{a,b\},\{\emptyset ,\{a,b\}\})}$

${\displaystyle {\mathcal {A}}_{1}\otimes {\mathcal {A}}_{2}=\{\emptyset ,\{(K,a),(K,b)\},\{(Z,a),(Z,b)\},\{(K,b),(Z,b),(K,a),(Z,a)\}\}}$

• The Borel σ-algebra on is the same as the product σ-algebra on , so the following applies: ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle ({\mathcal {B}}(\mathbb {R} ))_{i\in \{1,\dotsc ,n\}}}$

${\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})=\bigotimes _{i=1}^{n}{\mathcal {B}}(\mathbb {R} )}$

It is the smallest σ-algebra that contains all sets of the kind . ${\displaystyle \{A_{1}\times A_{2}\times \dots \times A_{n}\,|\,A_{i}\in {\mathcal {B}}(\mathbb {R} )\}}$

Applications

Product σ algebras are the basis for the theory of product measures , which in turn form the basis for Fubini's general theorem .

Product σ-algebras are of fundamental importance for stochastics in order to make statements about the existence of product probability measures and product probability spaces. On the one hand, these are important to describe multi-stage random experiments , and on the other hand, they are fundamental for the theory of stochastic processes .

literature

• Achim Klenke: Probability Theory. 2nd Edition. Springer-Verlag, Berlin Heidelberg 2008, ISBN 978-3-540-76317-8
• Jürgen Elstrodt : Measure and integration theory. Springer, Berlin et al. 1996, ISBN 3-540-15307-1 .

Individual evidence

1. ↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 39 , doi : 10.1007 / 978-3-642-21026-6 .