Measure space

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 edit

A measure space is a triple $(X,{\mathcal {A}},\mu ),$ where^[1]^[2]

$X$ 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 edit

Set $X=\{0,1\}$ . The ${\textstyle \sigma }$ -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 ${\textstyle \wp (\cdot ).}$ 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 ${\textstyle \mu }$ by

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

so

{\textstyle \mu (X)=1}

(by additivity of measures) and

{\textstyle \mu (\varnothing )=0}

(by definition of measures).

This leads to the measure space ${\textstyle (X,\wp (X),\mu ).}$ It is a probability space, since ${\textstyle \mu (X)=1.}$ The measure ${\textstyle \mu }$ corresponds to the Bernoulli distribution with ${\textstyle p={\frac {1}{2}},}$ which is for example used to model a fair coin flip.

Important classes of measure spaces edit

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]
$\sigma$ -finite measure spaces, where the measure is a $\sigma$ -finite measure^[3]

Another class of measure spaces are the complete measure spaces.^[4]

References edit

^ ^a ^b Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.
^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
^ ^a ^b Anosov, D.V. (2001) [1994], "Measure space", Encyclopedia of Mathematics, EMS Press
^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[Kosorok83-1] Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.

[Klenke18-2] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

[eommeasurespace-3] Anosov, D.V. (2001) [1994], "Measure space", Encyclopedia of Mathematics, EMS Press

[Klenke33-4] Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.

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