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 ${\displaystyle X}$ be a locally compact Hausdorff space. Let ${\displaystyle M(X)}$ be the space of complex Radon measures on ${\displaystyle X,}$ and ${\displaystyle C_{0}(X)^{*}}$ denote the dual of ${\displaystyle C_{0}(X),}$ the Banach space of complex continuous functions on ${\displaystyle X}$ vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem ${\displaystyle M(X)}$ is isometric to ${\displaystyle C_{0}(X)^{*}.}$ The isometry maps a measure ${\displaystyle \mu }$ to a linear functional ${\displaystyle I_{\mu }(f):=\int _{X}f\,d\mu .}$

The vague topology is the weak-* topology on ${\displaystyle C_{0}(X)^{*}.}$ The corresponding topology on ${\displaystyle M(X)}$ induced by the isometry from ${\displaystyle C_{0}(X)^{*}}$ is also called the vague topology on ${\displaystyle M(X).}$ Thus in particular, a sequence of measures ${\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }}$ converges vaguely to a measure ${\displaystyle \mu }$ whenever for all test functions ${\displaystyle f\in C_{0}(X),}$

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

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

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