Ameasure space is a basic object ofmeasure theory, a branch ofmathematics that studies generalized notions ofvolumes. It contains an underlying set, thesubsets of this set that are feasible for measuring (theσ-algebra), and the method that is used for measuring (themeasure). One important example of a measure space is aprobability space.

Ameasurable space consists of the first two components without a specific measure.

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

• ${\displaystyle X}$ is a set
• ${\displaystyle \mathcal{A}}$ is aσ-algebra on the set${\displaystyle X}$
• ${\displaystyle \mu }$ is ameasure on${\displaystyle (X,\mathcal{A})}$
• ${\displaystyle \mu }$ must satisfy countable additivity. That is, if${\displaystyle (A_n{)}_{n=1}^{\mathrm{\infty }}}$ are pair-wise disjoint then${\displaystyle \mu ({\cup }_{n=1}^{\mathrm{\infty }}{A}_n)=\sum _{n=1}^{\mathrm{\infty }}\mu (A_n)}$

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

Set${\displaystyle X=\{0,1\}}$. The$\sigma$-algebra on finite sets such as the one above is usually thepower set, which is the set of all subsets (of a given set) and is denoted by$\mathrm{\wp }(\cdot ).$ Sticking with this convention, we set$${\displaystyle \mathcal{A}=\mathrm{\wp }(X)}$$

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

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

This leads to the measure space$(X,\mathrm{\wp }(X),\mu ).$ It is aprobability space, since$\mu (X)=1.$ The measure$\mu$ corresponds to theBernoulli distribution with$p=\frac{1}{2},$ 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 aprobability measure^[1]
• Finite measure spaces, where the measure is afinite measure^[3]
• ${\displaystyle \sigma }$-finite measure spaces, where the measure is a${\displaystyle \sigma }$-finite measure^[3]

Another class of measure spaces are thecomplete measure spaces.^[4]

• Measure theory
• Space (mathematics)

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