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 where^{[1]}^{[2]}

## Example

Set . The -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 . Sticking with this convention, we set

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

As the measure, define by

so (by additivity of measures) and (by definition of measures).

This leads to the measure space . It is a probability space, since . The measure corresponds to the Bernoulli distribution with , 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

- Probability spaces, a measure space where the measure is a probability measure
^{[1]} - Finite measure spaces, where the measure is a finite measure
^{[3]} - -finite measure spaces, where the measure is a -finite measure
^{[3]}

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

## References

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