Admissible and singular translates of measures on vector spaces

Gleit, Alan; Zinn, Joel

doi:10.1090/S0002-9947-1976-0436244-3

Admissible and singular translates of measures on vector spaces

Authors: Alan Gleit and Joel Zinn
Journal: Trans. Amer. Math. Soc. 221 (1976), 199-211
MSC: Primary 60B05; Secondary 28A40, 60G30
DOI: https://doi.org/10.1090/S0002-9947-1976-0436244-3
MathSciNet review: 0436244
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Abstract: We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on . 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.

[1] R. M. Dudley, Singular translates of measures on linear spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 128–137 (1964). MR 0169277, https://doi.org/10.1007/BF00535972
[2] R. Fortet and E. Mourier, Les fonctions aléatoires comme éléments aléatoires dans les espaces de Banach, Studia Math. 15 (1955), 62–79 (French). MR 0093052, https://doi.org/10.4064/sm-15-1-62-79
[3] J. Hoffmann-Jørgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces (preprint).
[4] J. L. Kelley and Isaac Namioka, Linear topological spaces, With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR 0166578
[5] J. Kuelbs, Gaussian measures on a Banach space, J. Functional Analysis 5 (1970), 354–367. MR 0260010
[6] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Lecture Notes in Mathematics, Vol. 338, Springer-Verlag, Berlin-New York, 1973. MR 0415253
[7] A. Pietsch, Absolut 𝑝-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/1967), 333–353 (German). MR 0216328, https://doi.org/10.4064/sm-28-3-333-353
[8] A. P. Robertson and Wendy Robertson, Topological vector spaces, 2nd ed., Cambridge University Press, London-New York, 1973. Cambridge Tracts in Mathematics and Mathematical Physics, No. 53. MR 0350361
[9] V. N. Sudakov, On the characterization of the quasi-invariance of measures in Hilbert space, Uspehi Mat. Nauk 18 (1963), no. 1, 188–190 (Russian). MR 0149250
[10] Dao Xing Xia, Measure and integration theory on infinite-dimensional spaces. Abstract harmonic analysis, Academic Press, New York-London, 1972. Translated by Elmer J. Brody; Pure and Applied Mathematics, Vol. 48. MR 0310179
[11] Joel Zinn, Admissible translates of stable measures, Studia Math. 54 (1975/76), no. 3, 245–257. MR 0400376, https://doi.org/10.4064/sm-54-3-245-257