measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a finite measure or totally finite measure is a special measure that always takes on finite values. Among finite measures are

probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...

s. 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 A \in \mathcal A.

\mu

is a finite measure, the

measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...

(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 (, ) 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 vectors and ...

signed measure In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign. Definition There are two slightly different concepts of a signed measure, de ...

s with the

total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local property, local or global) structure of the codomain of a Function (mathematics), function or a measure (mathematics), measure. For a real ...

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

convex subset In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

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

unit sphere In mathematics, a unit sphere is a sphere of unit radius: the locus (mathematics), set of points at Euclidean distance 1 from some center (geometry), center point in three-dimensional space. More generally, the ''unit -sphere'' is an n-sphere, -s ...

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

Topological spaces

X

is a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

and

\mathcal A

contains the Borel

\sigma

-algebra then every finite measure is also a locally finite

Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...

Metric spaces

X

is a

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

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 In mathematics, a regular measure on a topological space is a measure for which every measurable set can be approximated from above by open measurable sets and from below by compact measurable sets. Definition Let (''X'', ''T'') be a topol ...

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)