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 C₀(X)^* denote the dual of C₀(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 C₀(X)^*. The isometry maps a measure μ to a linear functional I μ ( f ) := ∫ X f d μ .

The vague topology is the weak-* topology on C₀(X)^*. The corresponding topology on M(X) induced by the isometry from C₀(X)^* is also called the vague topology on M(X). Thus in particular, a sequence of measures (μ_n)_n∈ℕ converges vaguely to a measure μ whenever for all test functions f ∈ C₀(X),

∫ X f d μ n → ∫ X f d μ .

It is also not uncommon to define the vague topology by duality with continuous functions having compact support C_c(X), i.e. a sequence of measures (μ_n)_n∈ℕ converges vaguely to a measure μ whenever the above convergence holds for all test functions f ∈ 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 C₀(X) is not.

One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if μ_n are the probability measures for certain sums of independent random variables, then μ_n converge weakly (and then vaguely) to a normal distribution, i.e. the measure μ_n is "approximately normal" for large n.