Measure space

Example

Set ${\displaystyle X=\{0,1\}}$. 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 $${\displaystyle \mathcal{A}=\mathrm{\wp }(X)}$$

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

As the measure, define $\mu$ by $${\displaystyle \mu (\{0\})=\mu (\{1\})=\frac{1}{2},}$$ 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
• ${\displaystyle \sigma }$-finite measure spaces, where the measure is a ${\displaystyle \sigma }$-finite measure

Another class of measure spaces are the complete measure spaces.