no. 1
Densities on locally compact abelian groups
Annales de l'Institut Fourier, Tome 19 (1969) no. 1, pp. 81-107.

Une densité sur un groupe abélien localement compact G est un système borné de mesures cohérentes sur les quotients compacts de G. Nous étudions l’algèbre de Banach des densités sur G, en utilisant la théorie des fonctions presque périodiques comme moyen d’investigation principal. En particulier, nous caractérisons les groupes G dans lesquels chaque densité est induite par une mesure sur le compactifié semi-périodique de G. On donne des applications à la théorie d’équirépartition.

A density on a locally compact Abelian group G is a bounded system of compatible measures on the compact quotients of G. We study the Banach algebra of densities on G, using the theory of almost periodic functions as a principal tool. In particular, we characterize those groups G in which each density is induced by a measure on the semi-periodic compactification of G. There are applications to the theory of uniform distribution.

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     title = {Densities on locally compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     pages = {81--107},
     publisher = {Institut Fourier},
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     year = {1969},
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Berg, I. D.; Rubel, L. A. Densities on locally compact abelian groups. Annales de l'Institut Fourier, Tome 19 (1969) no. 1, pp. 81-107. doi : 10.5802/aif.308. https://aif.centre-mersenne.org/articles/10.5802/aif.308/

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[2] I.D. Berg, M. Rajagopalan and L.A. Rubel, Uniform distribution on locally compact Abelian groups, Trans. Amer. Math. Soc. 133 (1968) p. 435-446. | MR | Zbl

[3] R.C. Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), p. 560-580. | MR | Zbl

[4] J. Cigler, Folgen normierter Masse auf kompakten Gruppen, Z. Wahrscheinlichkeitstheorie 1 (1962) p. 3-13. | MR | Zbl

[5] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Springer Verlag, Berlin, (1963).

[6] L.A. Rubel, Uniform distribution in locally compact Abelian groups, Comm. Math. Helv. 93 (1965), p. 253-258. | MR | Zbl

[7] W. Rudin, Fourier Analysis in Groups, Interscience Publishers, (1962), New-York. | MR | Zbl

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