# Finite measure

In measure theory, a branch of mathematics, a finite measure or totally finite measure 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.

## Definition

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

$\mu (X)<\infty .$

By the monotonicity of measures, this implies

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

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

## Properties

### General case

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, that 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

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

### Metric spaces

If $X$  is a metric space and the ${\mathcal {A}}$  is again the Borel $\sigma$ -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 $X$ . The weak topology corresponds to the weak* topology in functional analysis. If $X$  is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.

### Polish spaces

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