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 $\backslash mu$ to a linear functional $I\_\backslash mu(f)\; :=\; \backslash int\_X\; f\backslash ,\; d\backslash mu.$

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 $\backslash left(\backslash mu\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}$ converges vaguely to a measure $\backslash mu$ whenever for all test functions $f\; \backslash in\; C\_0(X),$

:$\backslash int\_X\; f\; d\backslash mu\_n\; \backslash to\; \backslash int\_X\; f\; d\backslash mu.$

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 $\backslash left(\backslash mu\_n\backslash right)\_\{n\; \backslash in\; \backslash N\}$ converges vaguely to a measure $\backslash mu$ whenever the above convergence holds for all test functions $f\; \backslash in\; C\_c(X).$ 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 $\backslash mu\_n$ are the probability measures for certain sums of independent random variables, then $\backslash mu\_n$ converge weakly (and then vaguely) to a normal distribution, that is, the measure $\backslash mu\_n$ is "approximately normal" for large $n.$