Israel Journal of Mathematics

, Volume 63, Issue 1, pp 79-97

First online:

Almost sure convergence and bounded entropy

  • J. BourgainAffiliated withDepartment of Mathematics, IHES

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It is shown that the almost sure convergence property for certain sequences of operators {S n{ implies a uniform bound on the metrical entropy of the sets {S nf|n=1, 2, ...{, wheref is taken in theL 2-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.