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.
be a locally compact Hausdorff space.
be the space of complex Radon measures on
denote the dual of
the Banach space of complex continuous functions on
vanishing at infinity equipped with the uniform norm.
By the Riesz representation theorem
The isometry maps a measure
to a linear functional
μ
The vague topology is the weak-* topology on
induced by the isometry from
is also called the vague topology on
Thus in particular, a sequence of measures
converges vaguely to a measure
whenever for all test functions
It is also not uncommon to define the vague topology by duality with continuous functions having compact support
that is, a sequence of measures
converges vaguely to a measure
whenever the above convergence holds for all test functions
This construction gives rise to a different topology.
In particular, the topology defined by duality with
can be metrizable whereas the topology defined by duality with
One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if
are the probability measures for certain sums of independent random variables, then
converge weakly (and then vaguely) to a normal distribution, that is, the measure
is "approximately normal" for large
This article incorporates material from Weak-* topology of the space of Radon measures on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.