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.

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]

For any measurable space, the finite measures form a convex cone in the Banach space of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, which form a convex subset, and the probability measures, which are the intersection of the unit sphere in the normed space of signed measures and the finite measures.

If 👁 {\displaystyle X}
is a Hausdorff space and 👁 {\displaystyle {\mathcal {A}}}
contains the Borel 👁 {\displaystyle \sigma }
-algebra then every finite measure is also a locally finite Borel measure.

If 👁 {\displaystyle X}
is a metric space and the 👁 {\displaystyle {\mathcal {A}}}
is again the Borel 👁 {\displaystyle \sigma }
-algebra, the weak convergence of measures can be defined. The corresponding topology is called weak topology and is the initial topology of all bounded continuous functions on 👁 {\displaystyle X}
. The weak topology corresponds to the weak* topology in functional analysis. If 👁 {\displaystyle X}
is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.^[2]

If 👁 {\displaystyle X}
is a Polish space and 👁 {\displaystyle {\mathcal {A}}}
is the Borel 👁 {\displaystyle \sigma }
-algebra, then every finite measure is a regular measure and therefore a Radon measure.^[3] If 👁 {\displaystyle X}
is Polish, then the set of all finite measures with the weak topology is Polish too.^[4]

• σ-finite measure