Vague topology

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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 C0(X)* denote the dual of C0(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 C0(X)*. The isometry maps a measure μ to a linear functional

${\displaystyle I_{\mu }(f):=\int _{X}f\,d\mu .}$

The vague topology is the weak-* topology on C0(X)*. The corresponding topology on M(X) induced by the isometry from C0(X)* is also called the vague topology on M(X). Thus, in particular, one may refer to vague convergence of measure μnμ.

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