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 [math]\displaystyle{ X }[/math] be a locally compact Hausdorff space. Let [math]\displaystyle{ M(X) }[/math] be the space of complex Radon measures on [math]\displaystyle{ X, }[/math] and [math]\displaystyle{ C_0(X)^* }[/math] denote the dual of [math]\displaystyle{ C_0(X), }[/math] the Banach space of complex continuous functions on [math]\displaystyle{ X }[/math] vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem [math]\displaystyle{ M(X) }[/math] is isometric to [math]\displaystyle{ C_0(X)^*. }[/math] The isometry maps a measure [math]\displaystyle{ \mu }[/math] to a linear functional [math]\displaystyle{ I_\mu(f) := \int_X f\, d\mu. }[/math]

The vague topology is the weak-* topology on [math]\displaystyle{ C_0(X)^*. }[/math] The corresponding topology on [math]\displaystyle{ M(X) }[/math] induced by the isometry from [math]\displaystyle{ C_0(X)^* }[/math] is also called the vague topology on [math]\displaystyle{ M(X). }[/math] Thus in particular, a sequence of measures [math]\displaystyle{ \left(\mu_n\right)_{n \in \N} }[/math] converges vaguely to a measure [math]\displaystyle{ \mu }[/math] whenever for all test functions [math]\displaystyle{ f \in C_0(X), }[/math]

[math]\displaystyle{ \int_X f d\mu_n \to \int_X f d\mu. }[/math]

It is also not uncommon to define the vague topology by duality with continuous functions having compact support [math]\displaystyle{ C_c(X), }[/math] that is, a sequence of measures [math]\displaystyle{ \left(\mu_n\right)_{n \in \N} }[/math] converges vaguely to a measure [math]\displaystyle{ \mu }[/math] whenever the above convergence holds for all test functions [math]\displaystyle{ f \in C_c(X). }[/math] This construction gives rise to a different topology. In particular, the topology defined by duality with [math]\displaystyle{ C_c(X) }[/math] can be metrizable whereas the topology defined by duality with [math]\displaystyle{ C_0(X) }[/math] is not.

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

See also

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

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.