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

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.

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.

@article{AIF_1969__19_1_81_0,
     author = {Berg, I. D. and Rubel, L. A.},
     title = {Densities on locally compact abelian groups},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {19},
     number = {1},
     year = {1969},
     pages = {81-107},
     doi = {10.5802/aif.308},
     zbl = {0176.11602},
     mrnumber = {40 \#3178},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1969__19_1_81_0}
}
Berg, I. D.; Rubel, L. A. Densities on locally compact abelian groups. Annales de l'Institut Fourier, Volume 19 (1969) no. 1, pp. 81-107. doi : 10.5802/aif.308. http://www.numdam.org/item/AIF_1969__19_1_81_0/

[1] I.D. Berg, The conjugate space of the space of semi-periodic sequences, Michigan Math. J. 13 (1966), p. 293-297. | MR 33 #6291 | Zbl 0144.16803

[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 37 #3279 | Zbl 0165.34401

[3] R.C. Buck, The measure theoretic approach to density, Amer. J. Math. 68 (1946), p. 560-580. | MR 8,255f | Zbl 0061.07503

[4] J. Cigler, Folgen normierter Masse auf kompakten Gruppen, Z. Wahrscheinlichkeitstheorie 1 (1962) p. 3-13. | MR 26 #7012 | Zbl 0109.10702

[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 31 #3537 | Zbl 0152.03703

[7] W. Rudin, Fourier Analysis in Groups, Interscience Publishers, (1962), New-York. | MR 27 #2808 | Zbl 0107.09603