A remark on the Birkhoff ergodic theorem
Donald S. Ornstein · Illinois Journal of Mathematics · 1971
In this note we will prove the following theorem" THEOREM.Let T be a 1-1, invertable, measure-preserving, ergodic transformation of a measure space X onto itself.Let f* (x) sup, (1/n )f T' (x ).(a) Assume X has finite measure.Then for f >_ O, f* (x ) is integrable if and only if If (x) log (x)]+ is integrable (g+ is the positive part of g).(b) Assume Z has infinite measure.Then for f >_ O, f* (x is not integrable.The "if" part of (a) is well known and is only stated here for the sake of completeness.This paper has as its starting point the following theorem of Burkholder"Let X be a sequence of independent identically distributed, non-negative random variables.Then sup, (l/n)X() is integrable if and only if [X () log (X ())]+ has finite expectation.Gundy, in an unpublished paper, proves a reverse maximal inequality from which he deduces the above theorem.(This is generalized in Proposition 1.) Gundy also suggested that his theorem holds in the more general case of an ergodic transformation, and that is what we prove here.This seems to be the natural setting for the