Vague topology explained

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} 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} 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.

References

• .
• G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.