Sufficient topologies and norms

D. H. Muštari ; A. N. Čuprunov

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 700-714

Résumé

On the conjugate space $B'$ of the Banach space $B$ we consider norms and topologies such that the continuity of the characteristic functional of cylindrical probability $\mu$ (with respect to this norms and topologies) is sufficient for $\mu$ to be countably additive. In the case when $B$ is realizable as a space of random variables we introduce the notion of measurability of the norm on $B'$ which guarantees its sufficiency. In the case when $B=H$ is a Hilbert space we prove that different notions of measurability of the norm are not equivalent; a family of necessary and sufficient topologies $\tau_\alpha$ on $H$ is introduced and the connection between the $\tau_n$-differentiability of the characteristic functional $\mu$ and the integrability of the $n^{\text{th}}$ power of the norm with respect to $\mu$ is found. It is proved also that for the infinite-dimensional Banach space $B$ there are not a strongest locally convex sufficient topology in $B'$.