# 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 $I_{\mu}(f)=\int_{X}f\,d\mu$.

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 $n$.

## References

• 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 WeakTopologyOfTheSpaceOfRadonMeasures 2013-03-22 15:22:58 2013-03-22 15:22:58 stevecheng (10074) stevecheng (10074) 4 stevecheng (10074) Example msc 46A03 msc 28A33 LocallyCompactHausdorffSpace