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 $n$ .

Bibliography

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G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.