Admissible and Singular Translates of Measures on Vector Spaces

We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on D[0, 1]. Further, we show that in many cases convex, balanced, bounded, and complete subsets of the admissible translates are compact. In addition, we generalize Sudakov's theorem on the characterization of certain quasi-invariant sets to separable reflexive spaces which have the Central Limit Property. Introduction. It is the purpose of this paper to provide a general setting for studying admissible and singular translates of measures on linear spaces. In [9] Sudakov studied admissible translates of measures on a separable Hilbert space by exploiting Hilbert-Schmidt operators. These operators were also considered by Dudley [1] but are not available in a more general context. However, by considering maps between spaces one can obtain analogues of those operators. Based on our results we would conjecture the following (cf. Corollary 1.8 below): Let AT be a locally convex linear topological space. Let p be a positive Borel measure on X for which the set of admissible translates forms a subspace of X. Then there exists auxiliary spaces Ya and "compact" maps Ta: Ya—+ X, such that the set of admissible translates of p is given by "lara(ya). The organization of the paper is as follows. The first section contains general theorems in which properties of certain auxiliary spaces and maps are discussed. In §2 we show that in many cases a convex, balanced, complete, and bounded subset of the admissible translates is compact. In §3 we extend Sudakov's theorem on the characterization of certain quasi-invariant sets to separable reflexive Banach spaces with the Central Limit Property. These include / and separable ¿p(m)-spaces, 2 < p < °°. In §4 we apply the techniques of §§1 and 2 to obtain similar results in the nontopological vector space D[0, 1]. To the best of our knowledge these are the first general results on admissible translates for measures on D[0, 1]. Received by the editors March 8, 1974 and, in revised form, September 9, 1974 and June 23, 1975. AMS (MOS) subject classifications (1970). Primary 28A40; Secondary 46A05, 60B05.