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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the integrability of the maximal ergodic function


Author: Nghiêm Đăng-Ngọc
Journal: Proc. Amer. Math. Soc. 79 (1980), 565-570
MSC: Primary 28D10
MathSciNet review: 572303
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Abstract: Let $ G = {{\mathbf{R}}^d}$ or $ {{\mathbf{Z}}^d}$ and consider an ergodic measure-preserving action of G on a probability space $ (X,\mathfrak{A},P)$, let $ f \in {L^1}(X,P)$ and Mf be its maximal ergodic function. Our purpose is to prove the converse of the following theorem of N. Wiener: if $ \vert f\vert{\log ^ + }\vert f\vert$ is integrable then Mf is integrable. For the particular case $ G = {\mathbf{Z}}$ 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 $ {{\mathbf{R}}^d}$ and its analogue on $ {{\mathbf{Z}}^d}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0572303-9
PII: S 0002-9939(1980)0572303-9
Keywords: Ergodic measure-preserving action, maximal ergodic function, ergodic theorem, integrability
Article copyright: © Copyright 1980 American Mathematical Society