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 ${\displaystyle (X,{\mathcal {A)),\mu ),}$ where[1][2]

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

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

## Example

Set ${\displaystyle 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

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

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

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

As the measure, define ${\textstyle \mu }$ by

${\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2)),}$
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.[4]

## References

1. ^ a b Kosorok, Michael R. (2008). Introduction to Empirical Processes and Semiparametric Inference. New York: Springer. p. 83. ISBN 978-0-387-74977-8.
2. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 18. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.
3. ^ a b Anosov, D.V. (2001) [1994], "Measure space", Encyclopedia of Mathematics, EMS Press
4. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.