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 μn → μ.
One application of this is to probability theory: for example, the central limit theorem is essentially a statement that if μn are the probability measures for certain sums of independent random variables, then μn converge weakly to a normal distribution, i.e. the measure μn is "approximately normal" for large n.
Famous quotes containing the word vague:
“One of the most significant effects of age-segregation in our society has been the isolation of children from the world of work. Whereas in the past children not only saw what their parents did for a living but even shared substantially in the task, many children nowadays have only a vague notion of the nature of the parent’s job, and have had little or no opportunity to observe the parent, or for that matter any other adult, when he is fully engaged in his work.”
—Urie Bronfenbrenner (b. 1917)