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 $$\left(\mu_n\right)_{n \in \N}$$ converges vaguely to a measure $$\mu$$ whenever for all test functions $$f \in C_0(X),$$

$$\int_X f d\mu_n \to \int_X f d\mu.$$

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 $$\left(\mu_n\right)_{n \in \N}$$ 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 $$n.$$