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Israel Journal of Mathematics

, Volume 61, Issue 1, pp 39–72 | Cite as

On the maximal ergodic theorem for certain subsets of the integers

  • J. Bourgain
Article

Abstract

It is shown that the set of squares {n 2|n=1, 2,…} or, more generally, sets {n t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL 2-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL p-functions,p>1.

Keywords

Maximal Function Ergodic Theorem Maximal Inequality Bounded Measurable Function Left Member 
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Copyright information

© The Weizmann Science Press of Israel 1988

Authors and Affiliations

  • J. Bourgain
    • 1
  1. 1.IHESBures-sur-YvetteFrance

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