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## Summary

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

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

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

## 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.

## 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.