# Vague topology

In mathematics, particularly in the area of functional analysis and topological vector spaces, the **vague topology** is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces.

Let *X* be a locally compact Hausdorff space. Let *M*(*X*) be the space of complex Radon measures on *X*, and *C*_{0}(*X*)^{*} denote the dual of *C*_{0}(*X*), the Banach space of complex 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 *μ* to a linear functional

The **vague topology** is the weak-* topology on *C*_{0}(*X*)^{*}. The corresponding topology on *M*(*X*) induced by the isometry from *C*_{0}(*X*)^{*} is also called the vague topology on *M*(*X*). Thus, in particular, one may refer to vague convergence of measure *μ*_{n} → *μ*.

One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if *μ*_{n} are the probability measures for certain sums of independent random variables, then *μ*_{n} converge weakly (and then vaguely) to a normal distribution, i.e. the measure *μ*_{n} is "approximately normal" for large *n*.

## References

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

*This article incorporates material from Weak-* topology of the space of Radon measures on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*