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 if 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.

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, 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
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.