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Vague topology

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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 be a locally compact Hausdorff space. Let be the space of complex Radon measures on and 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 is isometric to The isometry maps a measure to a linear functional

The vague topology is the weak-* topology on The corresponding 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 is not.

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

See also

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References

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  • Dieudonné, Jean (1970), "§13.4. The vague topology", Treatise on analysis, vol. II, Academic Press.
  • G. B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.

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.