Measure space

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 ${\displaystyle (X,{\mathcal {A}},\mu ),}$ where

• ${\displaystyle X}$ is a set
• ${\displaystyle {\mathcal {A}}}$ is a σ-algebra on the set ${\displaystyle X}$
• ${\displaystyle \mu }$ is a measure on ${\displaystyle (X,{\mathcal {A}})}$
• ${\displaystyle \mu }$ must satisfy countable additivity. That is, if ${\displaystyle (A_{n})_{n=1}^{\infty }}$ are pair-wise disjoint then ${\displaystyle \mu (\cup _{n=1}^{\infty }A_{n})=\sum _{n=1}^{\infty }\mu (A_{n})}$

In other words, a measure space consists of a measurable space ${\displaystyle (X,{\mathcal {A}})}$ together with a measure on it.

Example

Set ${\displaystyle X=\{0,1\}}$. The ${\textstyle \sigma }$-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 ${\textstyle \wp (\cdot ).}$ Sticking with this convention, we set $${\displaystyle {\mathcal {A}}=\wp (X)}$$

In this simple case, the power set can be written down explicitly: $${\displaystyle \wp (X)=\{\varnothing ,\{0\},\{1\},\{0,1\}\}.}$$

As the measure, define ${\textstyle \mu }$ by $${\displaystyle \mu (\{0\})=\mu (\{1\})={\frac {1}{2}},}$$ so ${\textstyle \mu (X)=1}$ (by additivity of measures) and ${\textstyle \mu (\varnothing )=0}$ (by definition of measures).

This leads to the measure space ${\textstyle (X,\wp (X),\mu ).}$ It is a probability space, since ${\textstyle \mu (X)=1.}$ The measure ${\textstyle \mu }$ corresponds to the Bernoulli distribution with ${\textstyle p={\frac {1}{2}},}$ 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
• ${\displaystyle \sigma }$-finite measure spaces, where the measure is a ${\displaystyle \sigma }$-finite measure

Another class of measure spaces are the complete measure spaces.