The ergodic theorem with time compression
type: Item
instance of: scholarly article
title: The ergodic theorem with time compression
zbMATH Open document ID: 0662.28011
DOI: 10.1007/BF02791124
published in: Journal d'Analyse Mathématique
publication date: 1988-01-01T00:00:00Z
review text: Classical ergodic theorems deal mainly with Cesàro-averages. Recently, averages taken with a nonlinear time scale or along subsequences have received considerable attention. E.g., Bourgain showed that if \(\tau\) is measure-preserving in a probability space, f bounded measurable and \(k_ j=j^ 2\), then \(f_ n=\frac{1}{n}\sum^{n}_{j=1}f\circ \tau^{k_ j}\) converges a.e. The present author presents a very novel counterpart of this result. He considers a measure-preserving flow \(\{\tau^ t,- \infty <t<\infty \}\) and shows that \(f_ n\) converges a.e. for \(k_ j=j^{\rho}\) with \(0<\rho <1\) even for all \(f\in L_ 1\). The result extends to more general concave functions of t of the kind used to define Orlicz spaces. There also is a maximal inequality.
Mathematics Subject Classification ID: 28D10
zbMATH DE Number: 4081886
zbMATH Keywords: ergodic theorem
zbMATH Keywords: measure-preserving flow
zbMATH Keywords: concave functions
zbMATH Keywords: maximal inequality
author: Karl Petersen
reviewed by: Ulrich Krengel
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02791124
OpenAlex ID: W4249478186
description: scientific article; zbMATH DE number 4081886