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Generalizations of Gross's and Minlos's theorems

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Generalizations of Gross's and Minlos's theorems SÉMINAIRE DE PROBABILITÉS (STRASBOURG) JIA-AN YAN Generalizations of Gross’s and Minlos’s theorems Séminaire de probabilités (Strasbourg), tome 23 (1989), p. 395-404. <> © Springer-Verlag, Berlin Heidelberg New York, 1989, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma., implique l’accord avec les conditions gé- nérales d’utilisation ( Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques GENERALIZATIONS OF GROSS’ AND MINLOS’ THEOREMS by Jia An YAN Institute of Applied Mathematics Academia Sinica, Beijing The purpose of this note is to give simple proofs, with some extensions, of the well known theorems of Gross, Dudley-Feldman-LeCam and Minlos, and also of the general version of Gross’ theorem given by Lindstr~m. 1. Introduction Let X be a Banach space (or more generally any locally convex space) and X’ be its dual. Denote by x, y> the natural pairing between X and X~ . Let J~(X~~ , or simply x if the meaning is clear, be the collection of all finite dimensional subspaces of X’ Given K E k we denote by S(K) the u-algebra of all cylinder sets based on K , s. e. of all sets of the following form ~ (x~ J Yl >, .. , , y E ~} a where belong to K and E is a Borel set in . Let R(X) denote the algebra UK~K S(K). A non-negative set function p defined on is called a cylinder (probability) measure if p(X) = 1 and is u-additive on each u-algebra S(K) . A function f defined on X is called a cylinder function if there exists some KE x such that f is S(K~-measurable. The value of a cylinder measure on a boun

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