Measures in banach spaces are determined by their values on balls

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

The exact tail behavior of stable measures in Banach spaces and measures in their domains of attraction is given. Conditions for a p.m. to be in the domain of attraction of a stable p.m. of order α \alpha are derived which are sufficient in type p spaces, p > α p > \alpha , and necessary in general. This paper also contains a short proof of the Lévy-Khinchin formula in Banach spaces.

We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.

Multipliers with respect to spectral measures in banach spaces and approximation: II. One-dimensional Fourier multipliers

A characterization of type $p$ and cotype $p$ separable Banach spaces is given in terms of integrability properties of Levy measures. The following consequences are derived: (i) a separable Banach space is isomorphic to Hilbert space if and only if the set of Levy measures on it coincides with the set of Borel measures which integrate the function $\min (1, \|x\|^2)$; and (ii) the classical Levy-Khintchine representation of characteristic functions of infinitely divisible distributions holds in separable Banach spaces of cotype 2, in particular, in the separable $L_p$ spaces for $p \in \lbrack 1,2\rbrack$.

Inverse problem for potentials of measures in Banach spaces

We prove results on exact asymptotics of the probabilities $$P\left\{ {\int\limits_0^1 {e^{\varepsilon \xi (t)} dt > b} } \right\},P\left\{ {\int\limits_0^1 {e^{\varepsilon |\xi (t)|} dt > b} } \right\},\varepsilon \to 0,$$ where b > 1, for two Gaussian processes ?(t), namely, a Wiener process and a Brownian bridge. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type with the use of Legendre functions.

Existence and Convergence of Probability Measures in Banach Spaces

We introduce notions of ordinary and standard products of σ-finite measures and prove their existence. This approach allows us to construct invariant extensions of ordinary and standard products of Haar measures. In particular, we construct translation-invariant extensions of ordinary and standard Lebesgue measures on ℝ∞ and Rogers-Fremlin measures on l∞, respectively, such that topological weights of quasi-metric spaces associated with these measures are maximal (i.e., 2c). We also solve some Fremlin problems concerned with an existence of uniform measures in Banach spaces.

Asymptotic analysis of probability measures in banach spaces

Radon–Nikodym theorem and Vitali–Hahn–Saks theorem on fuzzy number measures in Banach spacers

Mappings of Gaussian Cylindrical Measures in Banach Spaces

Previous article Next article Remarks on Compact Families of Probability Measures in Banach SpacesD. Kh. MushtariD. Kh. Mushtarihttps://doi.org/10.1137/1129054PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. V. Sazonov, A remark on characteristic functionals, Theory Prob. Appl., 3 (1959), 188–192 LinkGoogle Scholar[2] Nguen Zuy Tien, , V. I. Tarieladze and , S. A. Chobanyan, On compactness of families of second-order measures in a Banach space, Theory Prob. Appl., 22 (1977), 805–810 0399.60010 LinkGoogle Scholar[3] D. Kh. Mushtari, Probabilities and Topologies in Banach Spaces, Kazan', 1979, Manuscript dep. in VINITI, 12/6/79, Kazan, 1979. (In Russian.) Google Scholar[4] A. V. Skorokhod, Studies in the theory of random processes, Translated from the Russian by Scripta Technica, Inc, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965viii+199 32:3082b 0146.37701 Google Scholar[5] V. V. Buldygin, Convergence of random elements in Banach spaces, Limit Theorems for Random Processes, Kiev, 1977, 20–27, (In Russian.) 0394.60006 Google Scholar[6] S. A. Chobanyan and , V. I. Tarieladze, Gaussian characterizations of certain Banach spaces, J. Multivariate Anal., 7 (1977), 183–203 10.1016/0047-259X(77)90038-0 55:6545 CrossrefGoogle Scholar[7] D. Kh. Mushtari, Cylindrical probabilities, criteria for countable additivity and supportsStudies in Applied Mathematics, izd-vo Kazanskogo un-ta, Kazan', 1979, 46–59, (In Russian.) Google Scholar[8] Alejandro de Acosta, Stable measures and seminorms, Ann. Probability, 3 (1975), 865–875 52:12023 0434.60005 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 29, Issue 2| 1985Theory of Probability & Its Applications205-425 History Submitted:19 May 1981Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1129054Article page range:pp. 407-410ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Tensor product of stable measures in Banach spaces of stable type