On the integrability of the maximal ergodic function

Author:
Nghiêm Đăng-Ngọc

Journal:
Proc. Amer. Math. Soc. **79** (1980), 565-570

MSC:
Primary 28D10

DOI:
https://doi.org/10.1090/S0002-9939-1980-0572303-9

MathSciNet review:
572303

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Abstract | References | Similar Articles | Additional Information

Abstract: Let or and consider an ergodic measure-preserving action of *G* on a probability space , let and *Mf* be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: if is integrable then *Mf* is integrable. For the particular case this result was already obtained by D. Ornstein whose proof is based on induced transformations and seems to be specific to Z, our proof is based on a result of E. M. Stein on the Hardy-Littlewood maximal function on and its analogue on .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0572303-9

Keywords:
Ergodic measure-preserving action,
maximal ergodic function,
ergodic theorem,
integrability

Article copyright:
© Copyright 1980
American Mathematical Society