A measure space is a special mathematical structure that plays an essential role in measure theory and the axiomatic structure of stochastics .
The triple is called dimensional space, if
is any non-empty set. is then also called the basic amount.
is a σ-algebra over the basic set .
is a measure defined upon .
Alternatively, a measurement space can also be defined as a measurement space provided with a measurement .
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 .
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.
In the probability theory used probability spaces are all measure spaces. They consist of the result set , the event algebra and the probability measure .
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.
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 ).
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 .
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.
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 .