Israel Journal of Mathematics

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

Almost sure convergence and bounded entropy

  • J. Bourgain


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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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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