Measure space

Not to be confused with Measurable 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 $(X,𝒜,\mu ),$ where

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

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

Example

Set $X=\{0,1\}$. 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 $\mathrm{\wp }(\cdot ).$ Sticking with this convention, we set $$𝒜=\mathrm{\wp }(X)$$

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

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

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

Another class of measure spaces are the complete measure spaces.