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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1973, Volume 18, Issue 1, Pages 66–77 (Mi tvp2681)

Some general questions of the theory of probability measures in linear spaces.

D. Kh. Mushtari

Kazan

Abstract: In § 1, some questions of the theory of cylindrical measures are considered connected to Sazonov's theorem [1]. $\mathrm B$-space $E$ is said to possess the $\mathrm M-\mathrm O$-property if, for any a.s. converging series $\sum r_n(t)x_n$ (where $r_n(t)$ are the Rademacher functions, $x_n\in E$), the series $\sum\|x_n\|^2$ is also converging. The main result of $§ 1$ is: For the existence of such topology $L_E$ in a separable $\mathrm B$-space $E$ that the class of continuous in $L_E$ characteristic functionals would coincide with the class of Fourier transforms of Radon measures in $E'$, it is necessary (Theorem 1 (B)) that the adjoint space $E'$ would possess the $\mathrm M-\mathrm O$-property, and it is sufficient (Theorem 1 (C)), that $E$ would be realizable as a space of random variables and there would exist a Schauder basis in $E$.
§ 2 deals with some generalizations of converse Minlos' theorem [2] on nuclearity of a countably-Hilbert space on which every continuous characteristic functional is associated with a Radon measure (condition $M$). This theorem is generalized for Frechet spaces. We give also examples of locally convex non-nuclear spaces, separable or not, satisfying the condition $M$; in the separable case the construction is based on the continuum hypothesis and choice axiom. These examples answer in the affirmative the question of Pietsch [12] about existence of non-nuclear locally convex separable spaces every bilinear form on which is nuclear.

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English version:
Theory of Probability and its Applications, 1973, 18:1, 64–75

Bibliographic databases:

Citation: D. Kh. Mushtari, “Some general questions of the theory of probability measures in linear spaces.”, Teor. Veroyatnost. i Primenen., 18:1 (1973), 66–77; Theory Probab. Appl., 18:1 (1973), 64–75

Citation in format AMSBIB
\Bibitem{Mus73} \by D.~Kh.~Mushtari \paper Some general questions of the theory of probability measures in linear spaces. \jour Teor. Veroyatnost. i Primenen. \yr 1973 \vol 18 \issue 1 \pages 66--77 \mathnet{http://mi.mathnet.ru/tvp2681} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=344409} \zmath{https://zbmath.org/?q=an:0304.60003} \transl \jour Theory Probab. Appl. \yr 1973 \vol 18 \issue 1 \pages 64--75 \crossref{https://doi.org/10.1137/1118005} 

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• http://mi.mathnet.ru/eng/tvp/v18/i1/p66

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. D. Kh. Mushtari, “Nekotorye voprosy teorii dualno-radonovskikh otobrazhenii”, Konstr. teor. funkts. i funkts. anal., 1, Izd-vo Kazanskogo un-ta, Kazan, 1977, 51–53
2. D. Kh. Mushtari, “O prostranstvakh, v kotorykh vypolnyayutsya kriterii tipa Bokhnera i Levi”, Konstr. teor. funkts. i funkts. anal., 2, Izd-vo Kazanskogo un-ta, Kazan, 1979, 69–82
3. D. Kh. Mushtari, “Dostatochnye bokhnerovskie topologii”, Konstr. teor. funkts. i funkts. anal., 3, Izd-vo Kazanskogo un-ta, Kazan, 1981, 64–72