Σ-finite measure

In mathematics, given a positive or a signed measure ${\displaystyle \mu }$ on a measurable space ${\displaystyle (X,{\mathcal {F}})}$, a ${\displaystyle \sigma }$-finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure ${\displaystyle \mu }$ is called a ${\displaystyle \sigma }$-finite measure if the set ${\displaystyle X}$ is ${\displaystyle \sigma }$-finite.

A finite measure, for instance a probability measure, is always ${\displaystyle \sigma }$-finite.

A different but related notion that should not be confused with ${\displaystyle \sigma }$-finiteness is s-finiteness.