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.
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
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
- Finite measure spaces, where the measure is a finite measure
- -finite measure spaces, where the measure is a -finite measure
- 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) , "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.