Measure space

{{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 volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the σ-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 $(X, \mathcal A, \mu),$ where

$X$ is a set
$\mathcal A$ is a σ-algebra on the set $X$
$\mu$ is a measure on $(X, \mathcal{A})$

In other words, a measure space consists of a measurable space $(X, \mathcal{A})$ together with a measure on it.

Example

Set $X = \{0, 1\}$ . The $\sigma$ -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 $\wp(\cdot).$ Sticking with this convention, we set

$\mathcal{A} = \wp(X)$

In this simple case, the power set can be written down explicitly:

$\wp(X) = \{\varnothing, \{0\}, \{1\}, \{0, 1\}\}.$

As the measure, define $\mu$ by

$\mu(\{0\}) = \mu(\{1\}) = \frac{1}{2},$

so $\mu(X) = 1$ (by additivity of measures) and $\mu(\varnothing) = 0$ (by definition of measures).

This leads to the measure space $(X, \wp(X), \mu).$ It is a probability space, since $\mu(X) = 1.$ The measure $\mu$ corresponds to the Bernoulli distribution with $p = \frac{1}{2},$ 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
Finite measure spaces, where the measure is a finite measure
$\sigma$ -finite measure spaces, where the measure is a $\sigma$ -finite measure

Another class of measure spaces are the complete measure spaces.

References

{{cite book |last1=Kosorok |first1=Michael R. |year=2008 |title=Introduction to Empirical Processes and Semiparametric Inference |location=New York |publisher=Springer |page=83|isbn=978-0-387-74977-8 }}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=18}}

{{cite book |last1=Klenke |first1=Achim |year=2008 |title=Probability Theory |location=Berlin |publisher=Springer |doi=10.1007/978-1-84800-048-3 |isbn=978-1-84800-047-6 |page=33}}

{{SpringerEOM |title=Measure space |id=Measure_space |author-last1=Anosov |author-first1=D.V.}}

Category:Measure theory

Category:Space (mathematics)