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

In other words, a measure space consists of a measurable space $$(X, \mathcal{A})$$ 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 $\wp(\cdot).$ Sticking with  this convention, we set $$\mathcal{A} = \wp(X)$$

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

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 $(X, \wp(X), \mu).$ 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.