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 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 converges vaguely to a measure \mu whenever for all test functions 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 converges vaguely to a measure \mu 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 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.