When the integral is the limit
type: Item
instance of: scholarly article
title: When the integral is the limit
zbMATH Open document ID: 0873.28011
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-10-26T00:00:00Z
review text: A sequence \((\mu_n)\) of probability measures on \(\mathbb{Z}\) is a \textit{norm good averaging method} if \[ \lim_{n\to\infty}\sum^\infty_{k=-\infty}\mu_n(k) f\circ\tau^k \] exists in \(L^2_m\) for all invertible measure-preserving transformations \(\tau\) of a probability space \((X,{\mathcal B},m)\) and for all \(f\in L^2_m\). A transformation \(\tau\) on \([0,1]\) leaving Lebesgue measure \(\lambda\) invariant (this class is denoted by \(\mathcal T\)) is \textit{good} if this limit equals \(\int f d\lambda\) for all \(f\in L^2_\lambda\). An averaging method is \textit{uniformly dissipative} if \(\lim_{n\to\infty}\sup_{k\in\mathbb{Z}} \mu_n(k)=0\). The following results are proved: 1) Given a uniformly dissipative, norm good averaging method, the set of good transformations of \([0,1]\) is a dense \(G_\delta\)-set in the weak topology on \(\mathcal T\). 2) The set of transformations in \(\mathcal T\) which are good for all uniformly dissipative, norm good averaging methods are of first category. This continues the author's work from Can. J. Math. 46, No. 1, 184-199 (1994; Zbl 0796.28011).
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 1005676
zbMATH Keywords: ergodic theorem
zbMATH Keywords: measure-preserving transformations
zbMATH Keywords: good averaging method
zbMATH Keywords: good transformations
author: J. Rosenblatt
reviewed by: G. Keller
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5241
OpenAlex ID: W2044935466
cites work: Q3955608
cites work: Q3234755
cites work: Q3895740
cites work: Norm Convergence in Ergodic Theory and the Behavior of Fourier Transforms
cites work: Q3980096
DOI: 10.1006/JMAA.1997.5241
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description: scientific article; zbMATH DE number 1005676