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.

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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 ${\textstyle \wp (\cdot ).}$ 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.

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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:

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