# On Riesz product measures ; mutual absolute continuity and singularity

Shelby J. Kilmer; Sadahiro Saeki

Annales de l'institut Fourier (1988)

- Volume: 38, Issue: 2, page 63-93
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topKilmer, Shelby J., and Saeki, Sadahiro. "On Riesz product measures ; mutual absolute continuity and singularity." Annales de l'institut Fourier 38.2 (1988): 63-93. <http://eudml.org/doc/74803>.

@article{Kilmer1988,

abstract = {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.},

author = {Kilmer, Shelby J., Saeki, Sadahiro},

journal = {Annales de l'institut Fourier},

keywords = {absolute continuity; singularity; Riesz product measures; locally compact abelian groups; random Riesz product measures},

language = {eng},

number = {2},

pages = {63-93},

publisher = {Association des Annales de l'Institut Fourier},

title = {On Riesz product measures ; mutual absolute continuity and singularity},

url = {http://eudml.org/doc/74803},

volume = {38},

year = {1988},

}

TY - JOUR

AU - Kilmer, Shelby J.

AU - Saeki, Sadahiro

TI - On Riesz product measures ; mutual absolute continuity and singularity

JO - Annales de l'institut Fourier

PY - 1988

PB - Association des Annales de l'Institut Fourier

VL - 38

IS - 2

SP - 63

EP - 93

AB - 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.

LA - eng

KW - absolute continuity; singularity; Riesz product measures; locally compact abelian groups; random Riesz product measures

UR - http://eudml.org/doc/74803

ER -

## References

top- [1] G. BROWN and W. MORAN, On orthogonality for Riesz products, Proc. Cambridge Philos. Soc., 76 (1974), 173-181. Zbl0282.43001MR50 #2812
- [2] C. C. GRAHAM and O. C. MCGEHEE, Essays in Commutative Harmonic Analysis, Springer-Verlag (1979), New York, Heidelberg-Berlin. Zbl0439.43001MR81d:43001
- [3] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis I, Springer-Verlag (1963), New York, Heidelberg-Berlin. Zbl0115.10603
- [4] E. HEWITT and K. A. ROSS, Abstract Harmonic Analysis II, Springer-Verlag (1970), New York, Heidelberg-Berlin. Zbl0213.40103
- [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. Zbl0148.38101
- [6] S. KAKUTANI, On equivalence of infinite product measures, Ann. of Math., 49 (1948), 214-224. Zbl0030.02303MR9,340e
- [7] M. LOÈVE, Probability Theory I, 4th ed., Springer-Verlag (1977), New York, Heidelberg-Berlin. Zbl0359.60001MR58 #31324a
- [8] J. M. LÓPEZ and K. A. ROSS, Sidon Sets, Marcel Dekker (1975), New York. Zbl0351.43008MR55 #13173
- [9] J. PEYRIÈRE, Sur les produits de Riesz, C. R. Acad. Sci., Paris Ser., A-B 276 (1973), 1417-1419. Zbl0258.43002MR47 #5512
- [10] L. PIGNO and S. SAEKI, Constructions of singular measures with remarkable convolution properties, Studia Math., 69 (1980), 133-141. Zbl0477.43003MR82h:43001
- [11] G. RITTER, Unendliche Produkte unkorrelierte Funktionen auf kompakten Abelschen Gruppen, Math. Scand., 42 (1978), 251-270. Zbl0398.42013MR80a:43001
- [12] G. RITTER, On Kakutani's theorem for infinite products of not necessarily independent functions, Math. Ann., 239 (1979), 35-53. Zbl0377.28004MR80g:60042
- [13] W. RUDIN, Fourier Analysis on Groups. Interscience Tract n° 12, Wiley (1962). New York. Zbl0107.09603MR27 #2808
- [14] J. L. TAYLOR, Measure Algebras. Regional Conference Series in Math., n° 16, Providence, R. I. Amer. Math. Soc. (1972). Zbl0269.28001
- [15] A. ZYGMUND, Trigonometric Series, 2 vols. Cambridge University Press (1959). Zbl0085.05601
- [16] A. PEYRIÈRE, Étude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier, Grenoble, 25-2 (1975), 127-169. Zbl0302.43003MR53 #8771

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.