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
A measure space is a triple where is a set is a -algebra on the set is a measure on
In other words, a measure space consists of a measurable space together with a measure on it.
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, in order of increasing generality: 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
Another class of measure spaces are the complete measure spaces.
References