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