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The Lévy continuity theorem for nuclear groups

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Let G be an abelian topological group. The Lévy continuity theorem says that if G is an LCA group, then it has the following property (PL) a sequence of Radon probability measures on G is weakly convergent to a Radon probability measure μ if and only if the corresponding sequence of Fourier transforms is pointwise convergent to the Fourier transform of μ. Boulicaut [Bo] proved that every nuclear locally convex space G has the property (PL). In this paper we prove that the property (PL) is inherited by nuclear groups, a variety of abelian topological groups containing LCA groups and nuclear locally convex spaces, introduced in [B1].
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Bibliografia
  • [A] L. Außenhofer, Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Ph.D. thesis, Tübingen University, 1998 (to appear in Dissertationes Math.).
  • [B1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991.
  • [B2] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in $ℝ^n$, Discrete Comput. Geom. 13 (1995), 217-231.
  • [B3] W. Banaszczyk, The Minlos lemma for positive-definite functions on additive subgroups of $ℝ^n$, Studia Math. 126 (1997), 13-25.
  • [BT] W. Banaszczyk and V. Tarieladze, The Lévy continuity theorem and related questions for nuclear groups, in preparation.
  • [Bo] P. Boulicaut, Convergence cylindrique et convergence étroite d'une suite de probabilités de Radon, Z. Wahrsch. Verw. Gebiete 28 (1973), 43-52.
  • [G] J. Galindo, Structure and analysis on nuclear groups, preprint, 1997.
  • [H] J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9 (1959), 544-562.
  • [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, Berlin, 1970.
  • [M] A. Mądrecki, Minlos' theorem in non-Archimedean locally convex spaces, Comment. Math. Prace Mat. 30 (1990), 101-111.
  • [VTCh] N. N. Vakhaniya, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, D. Reidel, Dordrecht, 1987.
  • [Y] Y. Yang, On a generalization of Minlos theorem, Fudan Xuebao 20 (1981), 31-37 (in Chinese).
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bwmeta1.element.bwnjournal-article-smv136i2p183bwm
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