Israel Journal of Mathematics

, Volume 63, Issue 1, pp 79–97 | Cite as

Almost sure convergence and bounded entropy

  • J. Bourgain
Article

Abstract

It is shown that the almost sure convergence property for certain sequences of operators {Sn{ implies a uniform bound on the metrical entropy of the sets {Snf|n=1, 2, ...{, wheref is taken in theL2-unit ball. This criterion permits one to unify certain counterexamples due to W. Rudin [Ru] and J.M. Marstrand [Mar] and has further applications. The theory of Gaussian processes is crucial in our approach.

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References

  1. [Be]
    A. Bellow,Two problems, Proc. Oberwolfach Conference on Measure Theory (June 1987), Springer Lecture Notes in Math.945, 1987.Google Scholar
  2. [Du]
    R. M. Dudley,Sample functions of the Gaussian process, Ann. Probab.1 (1973), 66–103.MATHMathSciNetGoogle Scholar
  3. [Er]
    P. Erdös, Lecture, Louisiana State University, November 1987.Google Scholar
  4. [J]
    B. Jessen,On the approximation of Lebesgue integrals by Riemann sums, Ann. of Math. (2)35 (1934), 248–251.CrossRefMathSciNetGoogle Scholar
  5. [Kh]
    A. Khintchine,Ein Satz über Kettenbruche mit arithmetischen Anwendungen, Math. Z.18 (1923), 289–306.CrossRefMathSciNetGoogle Scholar
  6. [Ko]
    J. F. Koksma,A diophantine property of some summable functions, J. Indian Math. Soc. (N.S.)15 (1951), 87–96.MathSciNetGoogle Scholar
  7. [M-Z]
    J. Marcinkiewicz and A. Zygmund,Mean values of trigonometrical polynomials, Fund. Math.28 (1937), 131–166.Google Scholar
  8. [Mar]
    J. M. Marstrand,On Khintchine’s conjecture about strong uniform distribution, Proc. London Math. Soc.21 (1970), 540–556.MATHCrossRefMathSciNetGoogle Scholar
  9. [Ru]
    W. Rudin,An arithmetic property of Riemann sums, Proc. Am. Math. Soc.15 (1964), 321–324.MATHCrossRefMathSciNetGoogle Scholar
  10. [St]
    E. M. Stein,On limits of sequences of operators, Ann. Math.74 (1961), 140–170.CrossRefGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1988

Authors and Affiliations

  • J. Bourgain
    • 1
  1. 1.Department of MathematicsIHESBures-sur-YvetteFrance

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