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A dichotomy for infinite convolutions of discrete measures

Published online by Cambridge University Press:  24 October 2008

G. Brown  and
W. Moran
G. Brown
Affiliation:
The University of Liverpool
W. Moran
Affiliation:
The University of Liverpool
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Extract

Measures, μ which can be realized as an infinite convolution

where each measure μn is a discrete measure, arise naturally in many parts of analysis and number theory (see (15)). The basic property of these measures is ‘purity’; i.e. such a measure μ 1must be absolutely continuous, continuous and singular, or discrete.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 2 , March 1973 , pp. 307 - 316
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Brown, G. and Moran, W.Translation and power independence for Bernoulli convolutions. Colloq. Math. 27 (1973).Google Scholar
(2)Brown, G. and Moran, W.In general, Bernoulli convolutions have independent powers. Studia Math. 47 (1973).CrossRefGoogle Scholar
(3)Ellis, R.Locally compact transformation groups. Duke Math. J. 24 (1957), 119–125.CrossRefGoogle Scholar
(4)Garsia, A. M.Arithmetic properties of Bernoulli convolutions. Trans. Amer. Math. Soc. 102 (1962), 409–432.Google Scholar
(5)Garsia, A. M.Entropy and singularity of infinite convolutions. Pacific J. Math. 13 (1963), 1159–1169.Google Scholar
(6)Halmos, P. R.Measure theory (Van Nostrand, Princeton, 1950).CrossRefGoogle Scholar
(7)Hewitt, E. and Kakutani, S.Some multiplicative linear functional on M(G). Ann. of Math. 79 (1964), 489–505.Google Scholar
(8)Kahane, J. P. and Salem, R.Sur la convolution d'une infinité de distributions de Bernoulli. Colloq. Math. 6 (1958), 193–202.Google Scholar
(9)Kahane, J. P. and Salem, R.Ensembles parfaits et séries trigonométriques (Hermann, Paris 1963).Google Scholar
(10)Kaufman, R.Some measures determined by mappings of the Cantor set. Colloq. Math. 19 (1968), 77–83.Google Scholar
(11)Pakshirajan, R. P.An analogue of Kolmogorov's Three Series Theorem for abstract random variables. Pacific J. Math. 13 (1963), 639–646.Google Scholar
(12)Salem, R.Algebraic numbers and Fourier analysis (Heath, Boston, 1963).Google Scholar
(13)ŠReider, Yu. A.The structure of maximal ideals in rings of measures with convolution. Mat. Sb. 27 (1950), 297–318; Amer. Math. Soc. Transl. no. 81.Google Scholar
(14)Taylor, J. L. Inverse, logarithms and idempotents in M(G). (To appear.)Google Scholar
(15)Wintner, A.Lectures on asymptotic distributions and infinite convolutions (Institute for Advanced Study, Princeton, 1938).Google Scholar
(16)Wintner, A.The Fourier transforms of probability distributions (The Johns Hopkins University, Baltimore, 1947).Google Scholar