Summary
The principal aim of the paper is an analogue of Lévy's theorem in the following way: Let \(\mathfrak{X}\) be a locally compact space denumerable in the infinity and consider the space \(\mathfrak{M}(\mathfrak{X})\) of all positive Radon measures on \(\mathfrak{X}\) with the “topologie vague” in the sense of Bourbaki. The Fouriertransform of a tight measure P on \(\mathfrak{M}(\mathfrak{X})\) is the functional
where \(\mathfrak{L}(\mathfrak{X})\) is the space of all continuous functions with compact support.
Consider a sequence P 1, P 2,... of tight measures on \(\mathfrak{M}(\mathfrak{X})\), with the property that \(\hat P_1 (\varphi ),\hat P_2 (\varphi ),...\) converges to F(ϕ) for fixed ϕ ε \(\mathfrak{L}(\mathfrak{X})\), and let F(λϕ) → F(0) forϕ ≧ 0, λ → 0. Then P 1, P 2,... is uniformely tight and converges to a tight measure P on \(\mathfrak{M}(\mathfrak{X})\) for each continuous bounded function on \(\mathfrak{M}(\mathfrak{X})\) and \(F = \hat P\). An application of this theorem is given.
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Literatur
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Von Waldenfels, W. Charakteristische Funktionale zufÄlliger Ma\e. Z. Wahrscheinlichkeitstheorie verw Gebiete 10, 279–283 (1968). https://doi.org/10.1007/BF00531849
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DOI: https://doi.org/10.1007/BF00531849