On Riesz product measures; mutual absolute continuity and singularity
Annales de l'Institut Fourier, Volume 38 (1988) no. 2, p. 63-93

We give some criteria for mutual absolute continuity and for singularity of Riesz product measures on locally compact abelian groups. The first section gives the definition of such a measure which is more general than the usual definition. The second section provides three sufficient conditions for one Riesz product measure to be absolutely continuous with respect to another. One of our results contains a theorem of Brown-Moran-Ritter as a special case. The final section deals with random Riesz product measures. We shall establish a certain probabilistic dichotomy for such measures.

Nous donnons des critères pour la continuité absolue mutuelle et pour la singularité des produits de Riesz sur les groupes abéliens localement compacts. La première partie définit une telle mesure d’une manière plus générale que la définition usuelle. La deuxième partie fournit trois conditions suffisantes pour qu’un produit de Riesz soit absolument continu par rapport à un autre. Un de nos résultats contient un théorème de Brown-Moran-Ritter comme cas spécial. La dernière partie traite des produits de Riesz aléatoires. Nous établissons une certaine dichotomie probabiliste pour de telles mesures.

@article{AIF_1988__38_2_63_0,
     author = {Kilmer, Shelby J. and Saeki, Sadahiro},
     title = {On Riesz product measures; mutual absolute continuity and singularity},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {38},
     number = {2},
     year = {1988},
     pages = {63-93},
     doi = {10.5802/aif.1135},
     mrnumber = {949011},
     zbl = {0633.43001},
     mrnumber = {90a:42006},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_2_63_0}
}
Kilmer, Shelby J.; Saeki, Sadahiro. On Riesz product measures; mutual absolute continuity and singularity. Annales de l'Institut Fourier, Volume 38 (1988) no. 2, pp. 63-93. doi : 10.5802/aif.1135. http://www.numdam.org/item/AIF_1988__38_2_63_0/

[1] G. Brown and W. Moran, On orthogonality for Riesz products, Proc. Cambridge Philos. Soc., 76 (1974), 173-181. | MR 50 #2812 | Zbl 0282.43001

[2] C. C. Graham and O. C. Mcgehee, Essays in Commutative Harmonic Analysis, Springer-Verlag (1979), New York, Heidelberg-Berlin. | MR 81d:43001 | Zbl 0439.43001

[3] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, Springer-Verlag (1963), New York, Heidelberg-Berlin.

[4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag (1970), New York, Heidelberg-Berlin. | Zbl 0213.40103

[5] E. Hewitt and H. S. Zuckerman, Singular measures with absolutely continuous convolution squares, Proc. Cambridge Philos. Soc., 62 (1966), 399-420 ; Corrigendum, ibid., 63 (1967), 367-368. | Zbl 0148.38101

[6] S. Kakutani, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224. | MR 9,340e | Zbl 0030.02303

[7] M. Loève, Probability Theory I, 4th ed., Springer-Verlag (1977), New York, Heidelberg-Berlin. | MR 58 #31324a | Zbl 0359.60001

[8] J. M. López and K. A. Ross, Sidon Sets, Marcel Dekker (1975), New York. | MR 55 #13173 | Zbl 0351.43008

[9] J. Peyrière, Sur les produits de Riesz, C. R. Acad. Sci., Paris Ser., A-B 276 (1973), 1417-1419. | MR 47 #5512 | Zbl 0258.43002

[10] L. Pigno and S. Saeki, Constructions of singular measures with remarkable convolution properties, Studia Math., 69 (1980), 133-141. | MR 82h:43001 | Zbl 0477.43003

[11] G. Ritter, Unendliche Produkte unkorrelierte Funktionen auf kompakten Abelschen Gruppen, Math. Scand., 42 (1978), 251-270. | MR 80a:43001 | Zbl 0398.42013

[12] G. Ritter, On Kakutani's theorem for infinite products of not necessarily independent functions, Math. Ann., 239 (1979), 35-53. | MR 80g:60042 | Zbl 0377.28004

[13] W. Rudin, Fourier Analysis on Groups. Interscience Tract n° 12, Wiley (1962). New York. | MR 27 #2808 | Zbl 0107.09603

[14] J. L. Taylor, Measure Algebras. Regional Conference Series in Math., n° 16, Providence, R. I. Amer. Math. Soc. (1972). | Zbl 0269.28001

[15] A. Zygmund, Trigonometric Series, 2 vols. Cambridge University Press (1959). | Zbl 0085.05601

[16] A. Peyrière, Étude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier, Grenoble, 25-2 (1975), 127-169. | Numdam | MR 53 #8771 | Zbl 0302.43003