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 A \in \mathcal A.

\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 In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...

of signed measures with the total variation norm. Important subsets of the finite measures are the sub-probability measures, which form a

convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a conve ...

, and the probability measures, which are the intersection of the

unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A u ...

in the normed space of signed measures and the finite measures.

Topological spaces

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

X

is a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

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

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.

References

{{mathanalysis-stub Measures (measure theory)