View source for Vague topology

In [[mathematics]], particularly in the area of [[functional analysis]] and [[topological vector space]]s, the '''vague topology''' is an example of the [[Weak topology|weak-* topology]] which arises in the study of [[Measure theory|measures]] on [[locally compact Hausdorff space]]s.

Let <math>X</math> be a [[locally compact Hausdorff space]].  Let <math>M(X)</math> be the space of [[Complex numbers|complex]] [[Radon measure]]s on <math>X,</math> and <math>C_0(X)^*</math> denote the dual of <math>C_0(X),</math> the [[Banach space]] of complex [[continuous function]]s on <math>X</math> [[Vanish at infinity|vanishing at infinity]] equipped with the [[uniform norm]].  By the [[Riesz–Markov–Kakutani_representation_theorem|Riesz representation theorem]] <math>M(X)</math> is [[Isometry|isometric]] to <math>C_0(X)^*.</math>  The isometry maps a measure <math>\mu</math> to a [[linear functional]] <math>I_\mu(f) := \int_X f\, d\mu.</math>

The '''vague topology''' is the [[Weak topology|weak-* topology]] on <math>C_0(X)^*.</math>  The corresponding topology on <math>M(X)</math> induced by the isometry from <math>C_0(X)^*</math> is also called the vague topology on <math>M(X).</math>  Thus in particular, a sequence of measures <math>\left(\mu_n\right)_{n \in \N}</math> converges vaguely to a measure <math>\mu</math> whenever for all test functions <math>f \in C_0(X),</math>

<math>\int_X f d\mu_n \to \int_X f d\mu.</math>

It is also not uncommon to define the vague topology by duality with continuous functions having compact support <math>C_c(X),</math> that is, a sequence of measures <math>\left(\mu_n\right)_{n \in \N}</math> converges vaguely to a measure <math>\mu</math> whenever the above convergence holds for all test functions <math>f \in C_c(X).</math> This construction gives rise to a different topology. In particular, the topology defined by duality with <math>C_c(X)</math> can be metrizable whereas the topology defined by duality with <math>C_0(X)</math> is not.

One application of this is to [[probability theory]]: for example, the [[central limit theorem]] is essentially a statement that if <math>\mu_n</math> are the [[probability measure]]s for certain sums of [[independent random variables]], then <math>\mu_n</math> converge weakly (and then vaguely) to a [[normal distribution]], that is, the measure <math>\mu_n</math> is "approximately normal" for large <math>n.</math> 

==See also==

* {{annotated link|List of topologies}}

==References==

{{reflist}}
{{reflist|group=note}}

* {{citation|last=Dieudonné|first=Jean|authorlink=Jean Dieudonné|chapter=§13.4. The vague topology|title=Treatise on analysis|volume=II|publisher=Academic Press|year=1970}}.
* [[Gerald Folland|G. B. Folland]], Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

{{PlanetMath attribution|id=7212|title=Weak-* topology of the space of Radon measures}}

{{Banach spaces}}
{{Hilbert space}}
{{Duality and spaces of linear maps}}
{{Topological vector spaces}}

[[Category:Real analysis]]
[[Category:Measure theory]]
[[Category:Topology of function spaces]]