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 .

literature

Jürgen Elstrodt: Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .