# The Lévy continuity theorem for nuclear groups

Studia Mathematica (1999)

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

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topBanaszczyk, 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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- [BT] W. Banaszczyk and V. Tarieladze, The Lévy continuity theorem and related questions for nuclear groups, in preparation.
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- [G] J. Galindo, Structure and analysis on nuclear groups, preprint, 1997.
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- [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. Zbl0754.43002
- [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). Zbl0565.28011

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