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\mathit{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 |
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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 |