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In mathematics, particularly in the area o … 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 largeeasure is "approximately normal" for large
, Die vage Konvergenz ist eine Konvergenzart … Die vage Konvergenz ist eine Konvergenzart in der Maßtheorie, einem Teilgebiet der Mathematik, das sich mit abstrahierten Volumenbegriffen beschäftigt und die Basis für die Stochastik und die Integrationstheorie bildet. Die vage Konvergenz ist ein Konvergenzbegriff für Folgen von Radon-Maßen und unterscheidet sich dadurch und durch die Wahl einer anderen Klasse von Testfunktionen von der schwachen Konvergenz. Die Topologie, welche die vage Konvergenz beschreibt, heißt die vage Topologie.genz beschreibt, heißt die vage Topologie.
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| rdfs:comment |
In mathematics, particularly in the area o … 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 functionaletry maps a measure to a linear functional
, Die vage Konvergenz ist eine Konvergenzart … Die vage Konvergenz ist eine Konvergenzart in der Maßtheorie, einem Teilgebiet der Mathematik, das sich mit abstrahierten Volumenbegriffen beschäftigt und die Basis für die Stochastik und die Integrationstheorie bildet. Die vage Konvergenz ist ein Konvergenzbegriff für Folgen von Radon-Maßen und unterscheidet sich dadurch und durch die Wahl einer anderen Klasse von Testfunktionen von der schwachen Konvergenz. Die Topologie, welche die vage Konvergenz beschreibt, heißt die vage Topologie.genz beschreibt, heißt die vage Topologie.
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Vage Konvergenz (Maßtheorie)
, Vague topology
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