# Measure space

A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. Measure spaces contain information about the underlying set, the subsets of said set that are feasible for measuring (the ${\displaystyle \sigma }$-algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space.

Measure space should not be confused with the related measurable spaces.

## Definition

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

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

## 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 ${\textstyle {\mathcal {P}}(\cdot )}$. Sticking with this convention, we set

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

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

${\displaystyle {\mathcal {P}}(X)=\{\emptyset ,\{0\},\{1\},X\}.}$

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

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

so ${\textstyle \mu (X)=1}$ (by additivity of measures) and ${\textstyle \mu (\emptyset )=0}$ (by definition of measures).

This leads to the measure space ${\textstyle (X,{\mathcal {P}}(X),\mu )}$. 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

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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
4. ^ Klenke, Achim (2008). Probability Theory. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.