# 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

- is any basic set and
- 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.

## 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

and then defines the two σ-algebras

- , that is, the power set of , and
- ,

then and measuring rooms, but the amount is measurable with respect and not respect .

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 .

## 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.

## 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 .

### 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.

### Separate measuring rooms

A measuring room is called a **separate measuring room** if the set of functions

a point-dividing crowd is on . The characteristic function of the set denotes .

This is exactly the case when, for any two points a lot there, so however .

### 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

- ^ 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 .