# 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 .