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]

General case edit

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.

Topological spaces edit

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.

Metric spaces edit

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]

Polish spaces edit

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]