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.

Definition

A measure space is a triple where In other words, a measure space consists of a measurable space together with a measure on it.

Example

Set. The \sigma-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 \wp(\cdot). Sticking with this convention, we set In this simple case, the power set can be written down explicitly: As the measure, define \mu by so \mu(X) = 1 (by additivity of measures) and (by definition of measures). This leads to the measure space It is a probability space, since \mu(X) = 1. The measure \mu 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: Another class of measure spaces are the complete measure spaces.