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.

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

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

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

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

In this simple case, the power set can be written down explicitly: $${\displaystyle \wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.}$${\displaystyle \wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.}

As the measure, define ${\textstyle \mu }${\textstyle \mu } by $${\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2}},}$${\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2}},} so ${\textstyle \mu (X)=1}${\textstyle \mu (X)=1} (by additivity of measures) and ${\textstyle \mu (\varnothing )=0}${\textstyle \mu (\varnothing )=0} (by definition of measures).

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

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

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

• Measure theory
• Space (mathematics)

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