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") 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

:$I\_mu(f)\; :=\; int\_X\; f,\; dmu.$

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 weak 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 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.