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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 {\displaystyle M} be the space of complex Radon measures on X , {\displaystyle X,} and C 0 ∗ {\displaystyle C_{0}^{*}} denote the dual of C 0 , {\displaystyle C_{0},} 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 {\displaystyle M} is isometric to C 0 ∗ . {\displaystyle C_{0}^{*}.} The isometry maps a measure μ {\displaystyle \mu } to a linear functional I μ := ∫ X f d μ . {\displaystyle I_{\mu }:=\int _{X}f\,d\mu.}
The vague topology is the weak-* topology on C 0 ∗ . {\displaystyle C_{0}^{*}.} The corresponding topology on M {\displaystyle M} induced by the isometry from C 0 ∗ {\displaystyle C_{0}^{*}} is also called the vague topology on M . {\displaystyle M.} Thus in particular, a sequence of measures n ∈ N {\displaystyle \left_{n\in \mathbb {N} }} converges vaguely to a measure μ {\displaystyle \mu } whenever for all test functions f ∈ C 0 , {\displaystyle f\in C_{0},}
∫ X f d μ n → ∫ X f d μ . {\displaystyle \int _{X}fd\mu _{n}\to \int _{X}fd\mu.}