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

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 $(X,{\mathcal {A}})$

Example

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 ${\textstyle {\mathcal {P}}(\cdot )}$ . Sticking with this convention, we set

{\mathcal {A}}={\mathcal {P}}(X)

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

{\mathcal {P}}(X)=\{\emptyset ,\{0\},\{1\},X\}.

As the measure, define ${\textstyle \mu }$ by

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

so ${\textstyle \mu (X)=1}$ (by additivity of measures) and ${\textstyle \mu (\emptyset )=0}$ (by definition of measures).

This leads to the measure space ${\textstyle (X,{\mathcal {P}}(X),\mu )}$ . 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

Most important classes of measure spaces are defined by the properties of their associated measures. This includes

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

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.
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.

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[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.