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Showing content from https://en.wikipedia.org/wiki/Vague_topology below:

Vague topology - Wikipedia

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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 X {\displaystyle X} be a locally compact Hausdorff space. Let M ( X ) {\displaystyle M(X)} be the space of complex Radon measures on X , {\displaystyle X,} and C 0 ( X ) ∗ {\displaystyle C_{0}(X)^{*}} denote the dual of C 0 ( X ) , {\displaystyle C_{0}(X),} the Banach space of complex continuous functions on X {\displaystyle X} vanishing at infinity equipped with the uniform norm. By the Riesz representation theorem M ( X ) {\displaystyle M(X)} is isometric to C 0 ( X ) ∗ . {\displaystyle C_{0}(X)^{*}.} The isometry maps a measure μ {\displaystyle \mu } to a linear functional I μ ( f ) := ∫ X f d μ . {\displaystyle I_{\mu }(f):=\int _{X}f\,d\mu .}

The vague topology is the weak-* topology on C 0 ( X ) ∗ . {\displaystyle C_{0}(X)^{*}.} The corresponding topology on M ( X ) {\displaystyle M(X)} induced by the isometry from C 0 ( X ) ∗ {\displaystyle C_{0}(X)^{*}} is also called the vague topology on M ( X ) . {\displaystyle M(X).} Thus in particular, a sequence of measures ( μ n ) n ∈ N {\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }} converges vaguely to a measure μ {\displaystyle \mu } whenever for all test functions f ∈ C 0 ( X ) , {\displaystyle f\in C_{0}(X),}

∫ X f d μ n → ∫ X f d μ . {\displaystyle \int _{X}fd\mu _{n}\to \int _{X}fd\mu .}

It is also not uncommon to define the vague topology by duality with continuous functions having compact support C c ( X ) , {\displaystyle C_{c}(X),} that is, a sequence of measures ( μ n ) n ∈ N {\displaystyle \left(\mu _{n}\right)_{n\in \mathbb {N} }} converges vaguely to a measure μ {\displaystyle \mu } whenever the above convergence holds for all test functions f ∈ C c ( X ) . {\displaystyle f\in C_{c}(X).} This construction gives rise to a different topology. In particular, the topology defined by duality with C c ( X ) {\displaystyle C_{c}(X)} can be metrizable whereas the topology defined by duality with C 0 ( X ) {\displaystyle 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 μ n {\displaystyle \mu _{n}} are the probability measures for certain sums of independent random variables, then μ n {\displaystyle \mu _{n}} converge weakly (and then vaguely) to a normal distribution, that is, the measure μ n {\displaystyle \mu _{n}} is "approximately normal" for large n . {\displaystyle n.}

List of topologies – List of concrete topologies and topological spaces

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.

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