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 -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 $(X, \mathcal A, \mu),$ where
* $X$ is a set
* $\mathcal A$ is a -algebra on the set $X$
* $\mu$ is a measure on $(X, \mathcal)$

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

Example

Set $X = \$. 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
$\mathcal = \wp(X)$

In this simple case, the power set can be written down explicitly:
$\wp(X) = \.$

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

This leads to the measure space $(X, \wp(X), \mu).$ It is a probability space, since $\mu(X) = 1.$ The measure $\mu$ corresponds to the Bernoulli distribution with $p = \frac,$ 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.

References

{{Lp spaces

Measure theory
Space (mathematics)