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 mass and probability of events. These seemingly distinct concepts have many simil ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a finite measure or totally finite measure is a special

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

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 probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...

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

\mu

measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...

(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 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 i ...

(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 vector ...

signed measure In mathematics, signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values. Definition There are two slightly different concepts of a signed measure, depending on whether or not o ...

s with the

total variation In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'' ...

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 convex ...

, 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 unit b ...

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 ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...

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 settin ...

and the

\mathcal A

is again the Borel

\sigma

-algebra, the

weak convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...

can be defined. The corresponding topology is called weak topology and is the

initial topology In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' tha ...

of all bounded continuous functions on

X

. The weak topology corresponds to the

weak* topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...

in functional analysis. If

X

is also separable, the weak convergence is metricized by the

Lévy–Prokhorov metric In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (mathematics), metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the F ...

Polish spaces

X

is a

Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...

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 topologi ...

and therefore a

Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...

. If

X

is Polish, then the set of all finite measures with the weak topology is Polish too.

References

{{mathanalysis-stub Measures (measure theory)