weak-* topology of the space of Radon measures

Let X be a locally compact Hausdorff spacePlanetmathPlanetmath. Let M(X) denote the space of complex Radon measuresMathworldPlanetmath on X, and C0(X)* denote the dual of the C0(X), the complex-valued continuous functionsMathworldPlanetmathPlanetmath on X vanishing at infinity, equipped with the uniform norm. By the Riesz Representation TheoremMathworldPlanetmath, M(X) is isometric to C0(X)*, The isometry maps a measureMathworldPlanetmath μ into the linear functionalMathworldPlanetmath Iμ(f)=Xf𝑑μ.

The weak-* topologyMathworldPlanetmath (also called the vague topology) on C0(X)*, is simply the topology of pointwise convergence of Iμ: IμαIμ if and only if Iμα(f)Iμ(f) for each fC0(X).

The corresponding topology on M(X) induced by the isometry from C0(X)* is also called the weak-* or vague topology on M(X). Thus one may talk about “weak convergence” of measures μnμ. One of the most important applications of this notion is in probability theory: for example, the central limit theoremMathworldPlanetmath is essentially the statement that if μn are the distributionsPlanetmathPlanetmathPlanetmath for certain sums of independent random variablesMathworldPlanetmath. then μn convergePlanetmathPlanetmath weakly to a normal distributionMathworldPlanetmath, i.e. the distribution μn is “approximately normal” for large n.


  • 1 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title weak-* topology of the space of Radon measures
Canonical name WeakTopologyOfTheSpaceOfRadonMeasures
Date of creation 2013-03-22 15:22:58
Last modified on 2013-03-22 15:22:58
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 4
Author stevecheng (10074)
Entry type Example
Classification msc 46A03
Classification msc 28A33
Related topic LocallyCompactHausdorffSpace