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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
$\S~1$ 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.
Received: 26.06.1971
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