Measure space: Difference between revisions
imported>Roffaduft m a probability measure is an example of a finite measure. All finite measures are \sigma-finite |
imported>AromaticPolygon m Added oxford comma to distinguish sigma algebra from measure when talking about what the space contains. |
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{{short description|Set on which a generalization of volumes and integrals is defined}} | {{short description|Set on which a generalization of volumes and integrals is defined}} | ||
A '''measure space''' is a basic object of [[measure theory]], a branch of [[mathematics]] that studies generalized notions of [[volume]]s. It contains an underlying set, the [[subset]]s of this set that are feasible for measuring (the [[σ-algebra|{{mvar|σ}}-algebra]]) and the method that is used for measuring (the [[Measure (mathematics)|measure]]). One important example of a measure space is a [[probability space]]. | A '''measure space''' is a basic object of [[measure theory]], a branch of [[mathematics]] that studies generalized notions of [[volume]]s. It contains an underlying set, the [[subset]]s of this set that are feasible for measuring (the [[σ-algebra|{{mvar|σ}}-algebra]]), and the method that is used for measuring (the [[Measure (mathematics)|measure]]). One important example of a measure space is a [[probability space]]. | ||
A [[measurable space]] consists of the first two components without a specific measure. | A [[measurable space]] consists of the first two components without a specific measure. | ||
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* <math>\mathcal A</math> is a [[σ-algebra|{{mvar|σ}}-algebra]] on the set <math>X</math> | * <math>\mathcal A</math> is a [[σ-algebra|{{mvar|σ}}-algebra]] on the set <math>X</math> | ||
* <math>\mu</math> is a [[Measure (mathematics)|measure]] on <math>(X, \mathcal{A})</math> | * <math>\mu</math> is a [[Measure (mathematics)|measure]] on <math>(X, \mathcal{A})</math> | ||
* <math>\mu</math> must satisfy countable additivity. That is, if <math>(A_{n})_{n=1}^{\infty}</math> are pair-wise disjoint then <math>\mu(\cup_{n=1}^{\infty}A_{n}) =\sum_{n=1}^{\infty}\mu(A_{n})</math> | |||
In other words, a measure space consists of a [[measurable space]] <math>(X, \mathcal{A})</math> together with a [[Measure (mathematics)|measure]] on it. | In other words, a measure space consists of a [[measurable space]] <math>(X, \mathcal{A})</math> together with a [[Measure (mathematics)|measure]] on it. | ||
Latest revision as of 19:40, 27 October 2025
Template:Short description A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the [[σ-algebra|Template:Mvar-algebra]]), and the method that is used for measuring (the measure). One important example of a measure space is a probability space.
A measurable space consists of the first two components without a specific measure.
Definition
A measure space is a triple where[1][2]
- is a set
- is a [[σ-algebra|Template:Mvar-algebra]] on the set
- is a measure on
- must satisfy countable additivity. That is, if are pair-wise disjoint then
In other words, a measure space consists of a measurable space together with a measure on it.
Example
Set . The -algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by Sticking with this convention, we set
In this simple case, the power set can be written down explicitly:
As the measure, define by so (by additivity of measures) and (by definition of measures).
This leads to the measure space It is a probability space, since The measure corresponds to the Bernoulli distribution with which is for example used to model a fair coin flip.
Important classes of measure spaces
Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:
- Probability spaces, a measure space where the measure is a probability measure[1]
- Finite measure spaces, where the measure is a finite measure[3]
- -finite measure spaces, where the measure is a -finite measure[3]
Another class of measure spaces are the complete measure spaces.[4]