weak-* topology of the space of Radon measures

Let $X$ be a locally compact Hausdorff space. Let $M(X)$ denote the space of complex Radon measures on $X$, and $C_0{(X)}^{*}$ denote the dual of the $C_0(X)$, the complex-valued continuous functions on $X$ vanishing at infinity, equipped with the uniform norm. By the Riesz Representation Theorem, $M(X)$ is isometric to $C_0{(X)}^{*}$, The isometry maps a measure $\mu$ into the linear functional $I_{\mu }(f)={\int }_{X}f𝑑\mu$.

The weak-* topology (also called the vague topology) on $C_0{(X)}^{*}$, is simply the topology of pointwise convergence of $I_{\mu }$: $I_{{\mu }_{\alpha }}\to I_{\mu }$ if and only if $I_{{\mu }_{\alpha }}(f)\to I_{\mu }(f)$ for each $f\in C_0(X)$.

The corresponding topology on $M(X)$ induced by the isometry from $C_0{(X)}^{*}$ is also called the weak-* or vague topology on $M(X)$. Thus one may talk about “weak convergence” of measures ${\mu }_n\to \mu$. One of the most important applications of this notion is in probability theory: for example, the central limit theorem is essentially the statement that if ${\mu }_n$ are the distributions for certain sums of independent random variables. then ${\mu }_n$ converge weakly to a normal distribution, i.e. the distribution ${\mu }_n$ is “approximately normal” for large $n$.

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