Vague Topology

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, one may refer to vague convergence of measure μ_n → μ.

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 to a normal distribution, i.e. the measure μ_n is "approximately normal" for large n.