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.
A measure space is a triple where
- is a set
- is a ?-algebra on the set
- is a measure on
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
Another class of measure spaces are the complete measure spaces.