Israel Journal of Mathematics

, Volume 63, Issue 1, pp 79–97

Almost sure convergence and bounded entropy


  • J. Bourgain
    • Department of MathematicsIHES

DOI: 10.1007/BF02765022

Cite this article as:
Bourgain, J. Israel J. Math. (1988) 63: 79. doi:10.1007/BF02765022


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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© The Weizmann Science Press of Israel 1988