The Lévy continuity theorem for nuclear groups

W. Banaszczyk

Studia Mathematica (1999)

  • Volume: 136, Issue: 2, page 183-196
  • ISSN: 0039-3223

Abstract

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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].

How to cite

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Banaszczyk, W.. "The Lévy continuity theorem for nuclear groups." Studia Mathematica 136.2 (1999): 183-196. <http://eudml.org/doc/216666>.

@article{Banaszczyk1999,
abstract = {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].},
author = {Banaszczyk, W.},
journal = {Studia Mathematica},
keywords = {Lévy continuity theorem; convergence of probability measures; nuclear groups; nuclear group; probability measures; positive definite functions; locally compact abelian group},
language = {eng},
number = {2},
pages = {183-196},
title = {The Lévy continuity theorem for nuclear groups},
url = {http://eudml.org/doc/216666},
volume = {136},
year = {1999},
}

TY - JOUR
AU - Banaszczyk, W.
TI - The Lévy continuity theorem for nuclear groups
JO - Studia Mathematica
PY - 1999
VL - 136
IS - 2
SP - 183
EP - 196
AB - 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].
LA - eng
KW - Lévy continuity theorem; convergence of probability measures; nuclear groups; nuclear group; probability measures; positive definite functions; locally compact abelian group
UR - http://eudml.org/doc/216666
ER -

References

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  1. [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.). Zbl0905.22001
  2. [B1] W. Banaszczyk, Additive Subgroups of Topological Vector Spaces, Lecture Notes in Math. 1466, Springer, Berlin, 1991. 
  3. [B2] W. Banaszczyk, Inequalities for convex bodies and polar reciprocal lattices in ℝ n , Discrete Comput. Geom. 13 (1995), 217-231. Zbl0824.52011
  4. [B3] W. Banaszczyk, The Minlos lemma for positive-definite functions on additive subgroups of ℝ n , Studia Math. 126 (1997), 13-25. Zbl0894.43004
  5. [BT] W. Banaszczyk and V. Tarieladze, The Lévy continuity theorem and related questions for nuclear groups, in preparation. 
  6. [Bo] P. Boulicaut, Convergence cylindrique et convergence étroite d'une suite de probabilités de Radon, Z. Wahrsch. Verw. Gebiete 28 (1973), 43-52. Zbl0276.60003
  7. [G] J. Galindo, Structure and analysis on nuclear groups, preprint, 1997. 
  8. [H] J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9 (1959), 544-562. Zbl0132.18202
  9. [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, Berlin, 1970. 
  10. [M] A. Mądrecki, Minlos' theorem in non-Archimedean locally convex spaces, Comment. Math. Prace Mat. 30 (1990), 101-111. Zbl0754.43002
  11. [VTCh] N. N. Vakhaniya, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, D. Reidel, Dordrecht, 1987. 
  12. [Y] Y. Yang, On a generalization of Minlos theorem, Fudan Xuebao 20 (1981), 31-37 (in Chinese). Zbl0565.28011

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