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

 Teor. Veroyatnost. i Primenen., 1983, Volume 28, Issue 4, Pages 700–714 (Mi tvp2218)

Sufficient topologies and norms

D. H. Muštaria, A. N. Čuprunov

a Kazan'

Abstract: 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 $n^{th}$ 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 $B'$.

Full text: PDF file (1055 kB)

English version:
Theory of Probability and its Applications, 1984, 28:4, 736–751

Bibliographic databases:

Citation: D. H. Muštari, A. N. Čuprunov, “Sufficient topologies and norms”, Teor. Veroyatnost. i Primenen., 28:4 (1983), 700–714; Theory Probab. Appl., 28:4 (1984), 736–751

Citation in format AMSBIB
\Bibitem{MusChu83}
\by D.~H.~Mu{\v s}tari, A.~N.~{\v C}uprunov
\paper Sufficient topologies and norms
\jour Teor. Veroyatnost. i Primenen.
\yr 1983
\vol 28
\issue 4
\pages 700--714
\mathnet{http://mi.mathnet.ru/tvp2218}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=726896}
\zmath{https://zbmath.org/?q=an:0544.60009|0522.60005}
\transl
\jour Theory Probab. Appl.
\yr 1984
\vol 28
\issue 4
\pages 736--751
\crossref{https://doi.org/10.1137/1128072}