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Equivalence singularity dichotomies for a class of ergodic measures

Published online by Cambridge University Press:  24 October 2008

Marek Kanter
Marek Kanter
Affiliation:
Sir George Williams Campus, Concordia University, Montreal
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Let µ be a probability measure on the Borel subsets of R∞. If D is a countable subgroup of R∞ we say that µ is D-ergodic if (1) for any D invariant Borel subset A of R we have µ(A) = 0 or 1 and (2) if µ*δx ≈ µ for all x ∈ D (where δx stands for unit mass at x while the equivalence relation ≈ signifies that the two measures have the same null sets.) We say that x is an admissible translate for µ if µ*δx ≈ µ. We say that µ is D-smooth if sx is an admissible translate for µ for all x ∈ D and all s ∈ R. We say that µ is a smooth ergodic measure if µ is D-ergodic and D-smooth for some countable subgroup D as above. In this paper we show that any two smooth ergodic probability measures µl, µ2 are either equivalent or singular (where the latter means that there exist disjoint Borel sets Al, A2 ⊂ R∞ such that µi(Ai) = 1 and is signified by µ1 ┴ µ2). It is important to note that the countable subgroup D1 associated with µl need not be the same as the subgroup D2 associated with µ2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 2 , March 1977 , pp. 249 - 252
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Breiman, L.Probability (Reading, Mass., Addison Wesley, 1968).Google Scholar
(2)Feldman, J.Equivalence and perpendicularity of Gaussian processes. Pacific J. Math. 8 (1958), 699–708.CrossRefGoogle Scholar
(3)Feller, W.An Introduction to Probability Theory and its Applications, vol. 2 (New York, Wiley, 1966).Google Scholar
(4)Hajek, J. (1958). On a property of normal distributions of any stochastic process. Selected Translations in Mathematics, Statistics and Probability, vol. 1 (Providence, R. I., Amer. Math. Soc., 1961), pp. 245–252.Google Scholar
(5)Halmos, P.Measure Theory (Princeton, New Jersey, Van Nostrand, 1950).CrossRefGoogle Scholar
(6)Kanter, M. Equivalence and singularity dichotomies for a class of infinitely divisible measures (preprint, 1975).Google Scholar
(7)Nemetz, T. O. H.On the orthogonality of probability measures. Studia Scientiarum Mathematicarum Hungarica 7 (1972), 111–115.Google Scholar
(8)Skorokhod, A. V.Integration in Hilbert Space (Berlin, Springer-Verlag, 1974).CrossRefGoogle Scholar
(9)Yosida, K.Functional Analysis (Berlin, Springer-Verlag, 1966).Google Scholar
(10)Zinn, J.Zero-one laws for non-Gaussian measures. Proc. Amer. Math. Soc. 44 (1974), 179–185.CrossRefGoogle Scholar