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 ...

, particularly in the area of

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

and

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

s, the vague topology is an example of 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 ...

which arises in the study of measures on

locally compact Hausdorff space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

s. Let

X

be a

. Let

M(X)

be the space of

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

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 ...

s on

X,

and

C_0(X)^*

denote the dual of

C_0(X),

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 ...

of complex

continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

s on

X

vanishing at infinity equipped with the

uniform norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...

. By the

Riesz representation theorem :''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...

M(X)

is isometric to

C_0(X)^*.

The isometry maps a measure

\mu

to a

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...

I_\mu(f) := \int_X f\, d\mu.

The vague topology is the

C_0(X)^*.

The corresponding topology on

M(X)

induced by the isometry from

C_0(X)^*

is also called the vague topology on

M(X).

Thus in particular, a sequence of measures

\left(\mu_n\right)_

converges vaguely to a measure

\mu

whenever for all test functions

f \in C_0(X),

\int_X f d\mu_n \to \int_X f d\mu.

It is also not uncommon to define the vague topology by duality with continuous functions having compact support

C_c(X),

that is, a sequence of measures

\left(\mu_n\right)_

converges vaguely to a measure

\mu

whenever the above convergence holds for all test functions

f \in C_c(X).

This construction gives rise to a different topology. In particular, the topology defined by duality with

C_c(X)

can be metrizable whereas the topology defined by duality with

C_0(X)

is not. One application of this is to

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

: for example, the

central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...

is essentially a statement that if

\mu_n

are the

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 ge ...

s for certain sums of

independent random variables Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...

, then

\mu_n

converge weakly (and then vaguely) to a

normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

, that is, the measure

\mu_n

is "approximately normal" for large

n.

References

* . * G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999. {{Topological vector spaces Real analysis Measure theory Topology of function spaces

See also

References