• Source: 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
,


A


,
μ
)
,


{\displaystyle (X,{\mathcal {A}},\mu ),}

where




X


{\displaystyle X}

is a set






A




{\displaystyle {\mathcal {A}}}

is a σ-algebra on the set



X


{\displaystyle X}





μ


{\displaystyle \mu }

is a measure on



(
X
,


A


)


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


In other words, a measure space consists of a measurable space



(
X
,


A


)


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

together with a measure on it.




Example


Set



X
=
{
0
,
1
}


{\displaystyle X=\{0,1\}}

. The



σ


{\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






A


=
℘
(
X
)


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


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




℘
(
X
)
=
{
∅
,
{
0
}
,
{
1
}
,
{
0
,
1
}
}
.


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


As the measure, define



μ


{\textstyle \mu }

by




μ
(
{
0
}
)
=
μ
(
{
1
}
)
=


1
2


,


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


so



μ
(
X
)
=
1


{\textstyle \mu (X)=1}

(by additivity of measures) and



μ
(
∅
)
=
0


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

(by definition of measures).
This leads to the measure space



(
X
,
℘
(
X
)
,
μ
)
.


{\textstyle (X,\wp (X),\mu ).}

It is a probability space, since



μ
(
X
)
=
1.


{\textstyle \mu (X)=1.}

The measure



μ


{\textstyle \mu }

corresponds to the Bernoulli distribution with



p
=


1
2


,


{\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, 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




σ


{\displaystyle \sigma }

-finite measure spaces, where the measure is a



σ


{\displaystyle \sigma }

-finite measure
Another class of measure spaces are the complete measure spaces.




References

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