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 $\mu$ to a linear functional $I_{\mu }(f):=\int _{X}f\,d\mu .$

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, a sequence of measures $\left(\mu _{n}\right)_{n\in \mathbb {N} }$ converges vaguely to a measure $\mu$ whenever for all test functions $f\in C_{0}(X),$

$\int _{X}fd\mu _{n}\to \int _{X}fd\mu .$

It is also not uncommon to define the vague topology by duality with continuous functions having compact support $C_{c}(X),$ that is, a sequence of measures $\left(\mu _{n}\right)_{n\in \mathbb {N} }$ converges vaguely to a measure $\mu$ whenever the above convergence holds for all test functions $f\in C_{c}(X).$ This construction gives rise to a different topology. In particular, the topology defined by duality with $C_{c}(X)$ can be metrizable whereas the topology defined by duality with $C_{0}(X)$ is not.

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

See also

• List of topologies – List of concrete topologies and topological spaces

References

• Dieudonné, Jean (1970), "§13.4. The vague topology", Treatise on analysis, vol. II, Academic Press.
• 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.