Measuring room (mathematics)

Measurement space or measurable space is a term in measurement theory , a sub-area of ​​mathematics that deals with the generalization of volume concepts. Measurement spaces here form an analogue to the domain of definition , they indicate the quantities about which a statement can be made.

definition

A tuple is called a measuring space or measurable space, if ${\displaystyle (\Omega ,{\mathcal {A}})}$

• ${\displaystyle \Omega }$ is any basic set and
• ${\displaystyle {\mathcal {A}}}$is a σ-algebra on this basic set.

In stochastics, measuring rooms are also called event rooms . A quantity is called a measurable quantity , if is. ${\displaystyle A}$ ${\displaystyle A\in {\mathcal {A}}}$

Differentiation from other measurability terms

It is important for the term measurable quantity used here that no measure has to be defined, but only a measuring space. This is why one sometimes speaks of measurability in relation to a measuring room.

This is to be distinguished from the measurability according to Carathéodory of quantities with regard to an external measure . Here, too, no measure is required, only an external measure .

Examples

Consider the basic room as an example

${\displaystyle \Omega =\{1,2,3,4\}}$

and then defines the two σ-algebras

${\displaystyle {\mathcal {A}}_{1}={\mathcal {P}}(\Omega )}$, that is, the power set of , and ${\displaystyle \Omega }$

${\displaystyle {\mathcal {A}}_{2}=\{\emptyset ,\{1,2\},\{3,4\},\Omega \}}$,

then and measuring rooms, but the amount is measurable with respect and not respect . ${\displaystyle M_{1}=(\Omega ,{\mathcal {A}}_{1})}$ ${\displaystyle M_{2}=(\Omega ,{\mathcal {A}}_{2})}$ ${\displaystyle \{1\}}$ ${\displaystyle M_{1}}$ ${\displaystyle M_{2}}$

In general, every set with its power set forms a measurement space. The measurement space of Borel's σ-algebra is often used, especially in probability theory . ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

Isomorphism of measuring spaces

Two measurement spaces and are called isomorphic if there is a bijective function from to that is - -measurable and whose inverse mapping - -measurable. ${\displaystyle (\Omega _{1},{\mathcal {A}}_{1})}$ ${\displaystyle (\Omega _{2},{\mathcal {A}}_{2})}$ ${\displaystyle f}$ ${\displaystyle \Omega _{1}}$ ${\displaystyle \Omega _{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle f^{-1}}$ ${\displaystyle {\mathcal {A}}_{2}}$ ${\displaystyle {\mathcal {A}}_{1}}$

Classes of measuring rooms

Borelian spaces

A measuring space is called a Borel space or Borel space if there is a measurable set such that and are Borel isomorphic . ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle B\in {\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle (B,{\mathcal {B}}(B))}$

Decision-making spaces

A decision space is a measurement space in which the σ-algebra contains all single-element sets, i.e. if the set is for each . is for example a decision space. ${\displaystyle \omega \in \Omega }$ ${\displaystyle \{\omega \}\in {\mathcal {\mathcal {A}}}}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

Separate measuring rooms

A measuring room is called a separate measuring room if the set of functions ${\displaystyle (\Omega ,{\mathcal {A}})}$

${\displaystyle M:=\{\chi _{A}\,|\,A\in {\mathcal {A}}\}}$

a point-dividing crowd is on . The characteristic function of the set denotes . ${\displaystyle \Omega }$ ${\displaystyle \chi _{A}}$ ${\displaystyle A}$

This is exactly the case when, for any two points a lot there, so however . ${\displaystyle x,y\in \Omega }$ ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle x\in A}$ ${\displaystyle y\notin A}$

Countably generated measuring rooms

A measurement space is called a countably generated measurement space if the σ-algebra of the measurement space is a countably generated σ-algebra , i.e. has a countable generator.

use

There are numerous applications for measurement rooms in probability theory and measurement theory. On the one hand, they can be expanded to a dimensional space by choosing a dimension , on the other hand they correspond to the range of values ​​when constructing image dimensions using measurable functions .

In stochastics , the measuring spaces are also sometimes called event spaces , and the measurable quantities are then called events . After choosing a probability measure , it is then a probability space .

literature

• Jürgen Elstrodt: Measure and integration theory . 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .

Individual evidence

1. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 10 , doi : 10.1515 / 9783110215274 .