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 calledthe 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 thatif ${\mu }_n$ 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 $n$.