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

where

• is a set

• is a σ-algebra

• is a measure

In other words, a measure space consists of a measurable space together with a measure on it.

Example

-algebra on finite sets such as the one above is usually the power set

(by definition of measures).

It is a probability space corresponds to the Bernoulli distribution 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 \sigma -finite measure

Another class of measure spaces are the complete measure spaces.

References