$X$ is a set

${\mathcal {A}}$ is a on the set $X$

$σ$ -algebra

$\mu$ is a on $(X,{\mathcal {A}})$

measure

A measure space is a triple $(X,{\mathcal {A}},\mu ),$ where^[1]^[2]

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

a measure space where the measure is a probability measure^[1]

Probability spaces

Finite measure spaces, where the measure is a ^[3]

finite measure

$\sigma$ -finite measure spaces, where the measure is a ^[3]

$\sigma$ -finite measure

Most important classes of measure spaces are defined by the properties of their associated measures. This includes, in order of increasing generality:

Another class of measure spaces are the complete measure spaces.^[4]

σ-algebra

measure

Probability spaces

finite measure

σ {\displaystyle \sigma } -finite measure

$σ$ -algebra

$\sigma$ -finite measure