In measure theory, a branch of mathematics, a finite measure or totally finite measure^[1] is a special measure that always takes on finite values. Among finite measures are probability measures. The finite measures are often easier to handle than more general measures and show a variety of different properties depending on the sets they are defined on.

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A measure ${\displaystyle \mu }$ on measurable space ${\displaystyle (X,{\mathcal {A}})}$ is called a finite measure if it satisfies

${\displaystyle \mu (X)<\infty .}$

By the monotonicity of measures, this implies

${\displaystyle \mu (A)<\infty {\text{ for all }}A\in {\mathcal {A}}.}$

If ${\displaystyle \mu }$ is a finite measure, the measure space ${\displaystyle (X,{\mathcal {A}},\mu )}$ is called a finite measure space or a totally finite measure space.^[1]