# Measure space

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 -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

## 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 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

- 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. - Anosov, D.V. (2001) [1994], "Measure space", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Klenke, Achim (2008).
*Probability Theory*. Berlin: Springer. p. 33. doi:10.1007/978-1-84800-048-3. ISBN 978-1-84800-047-6.