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

D. Kh. Mushtari

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 1, pp. 66-77

Résumé

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.