# Dimensional space

A measure space is a special mathematical structure that plays an essential role in measure theory and the axiomatic structure of stochastics .

## definition

The triple is called dimensional space, if ${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$

• ${\ displaystyle \ Omega}$is any non-empty set. is then also called the basic amount.${\ displaystyle \ Omega}$
• ${\ displaystyle {\ mathcal {A}}}$is a σ-algebra over the basic set .${\ displaystyle \ Omega}$
• ${\ displaystyle \ mu}$is a measure defined upon .${\ displaystyle {\ mathcal {A}}}$

Alternatively, a measurement space can also be defined as a measurement space provided with a measurement . ${\ displaystyle (\ Omega, {\ mathcal {A}})}$${\ displaystyle \ mu}$

## Examples

A simple example of a measure space are the natural numbers as the base set , the power set is chosen as the σ-algebra and the Dirac measure on the 1: as the measure . ${\ displaystyle \ Omega = \ mathbb {N}}$ ${\ displaystyle {\ mathcal {A}} = {\ mathcal {P}} (\ mathbb {N})}$${\ displaystyle \ mu = \ delta _ {1}}$

A well-known measure space is the basic set , provided with Borel's σ-algebra and the Lebesgue measure . This is the canonical measure space in integration theory. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle {\ mathcal {B}} (\ mathbb {R})}$

In the probability theory used probability spaces are all measure spaces. They consist of the result set , the event algebra and the probability measure . ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle \ Omega}$ ${\ displaystyle {\ mathcal {A}}}$ ${\ displaystyle P}$

## Classes of measurement spaces

### Finite dimensional spaces

A measure space is called a finite measure space or limited measure space if the measure of the basic set is finite , i.e. is. ${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$${\ displaystyle \ mu (\ Omega) <\ infty}$

### σ-finite measure spaces

A measure space is called a σ-finite measure space or σ-finite measure space if the measure is σ-finite (with respect to the σ-algebra ). ${\ displaystyle {\ mathcal {A}}}$

### Complete dimensional spaces

A measure space is called complete if every subset of a zero set can be measured again with regard to the measure, i.e. if it lies in σ-algebra.

### Signed dimensional spaces

If there is a σ-algebra above the basic set and a signed measure on this σ-algebra, the triple is called a signed measure space . ${\ displaystyle {\ mathcal {A}}}$${\ displaystyle \ Omega}$${\ displaystyle \ nu}$${\ displaystyle (\ Omega, {\ mathcal {A}}, \ nu)}$

### Separable dimensional spaces

A measure space is a separable measure space if a countable quantity system exists, such that for all and any one there, so that is. ${\ displaystyle (\ Omega, {\ mathcal {A}}, \ mu)}$${\ displaystyle {\ mathcal {S}} \ subset {\ mathcal {A}}}$${\ displaystyle A \ in {\ mathcal {A}}}$${\ displaystyle \ varepsilon> 0}$${\ displaystyle S \ in S}$${\ displaystyle \ mu (A \ triangle S) <\ varepsilon}$

### Collapsible dimensional spaces

Decomposable measure spaces appear if one wants to formulate the Radon-Nikodým theorem in a more general way than only for σ-finite measure spaces.

### Localizable dimension spaces

On localizable measurement spaces, measurable functions that correspond to sets of finite measure can be combined to form a locally measurable function .