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 on measurable space is called a finite measure iff it satisfies
By the monotonicity of measures, this implies
If is a finite measure, the measure space 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 is a Hausdorff space and contains the Borel -algebra then every finite measure is also a locally finite Borel measure.

Metric spaces

If is a metric space and the is again the Borel -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. The weak topology corresponds to the weak* topology in functional analysis. If is also separable, the weak convergence is metricized by the Lévy–Prokhorov metric.

Polish spaces

If is a Polish space and is the Borel -algebra, then every finite measure is a regular measure and therefore a Radon measure.
If is Polish, then the set of all finite measures with the weak topology is Polish too.