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

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.

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.

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     title = {On {Riesz} product measures; mutual absolute continuity and singularity},
     journal = {Annales de l'Institut Fourier},
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Kilmer, Shelby J.; Saeki, Sadahiro. On Riesz product measures; mutual absolute continuity and singularity. Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 63-93. doi : 10.5802/aif.1135. https://aif.centre-mersenne.org/articles/10.5802/aif.1135/

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

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

[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

[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

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

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

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

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

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

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

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

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

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

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

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

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