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 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, a sequence of measures n)n∈ℕ converges vaguely to a measure μ whenever for all test functions f ∈ C0(X),

X f dμn → ∫X f dμ.

It is also not uncommon to define the vague topology by duality with continuous functions having compact support Cc(X), i.e. a sequence of measures n)n∈ℕ converges vaguely to a measure μ whenever the above convergence holds for all test functions f ∈ Cc(X). This construction gives rise to a different topology. In particular, the topology defined by duality with Cc(X) can be metrizable whereas the topology defined by duality with C0(X) is not.

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

• Dieudonné, Jean (1970), "§13.4. The vague topology", Treatise on analysis, II, Academic Press.
• G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.