Elliott-Morse measures and Kakutani's dichotomy theorem
type: Item
instance of: scholarly article
title: Elliott-Morse measures and Kakutani's dichotomy theorem
zbMATH Open document ID: 0761.28004
DOI: 10.1007/BF02571429
author: Gunter Ritter
author: Edwin Hewitt
published in: Mathematische Zeitschrift
publication date: 1993-02-02T00:00:00Z
full work available at URL: https://eudml.org/doc/174442
review text: We first extend Lebesgue's decomposition theorem for measures: If \(\mu\) is a \(\sigma\)-finite measure and \(\nu\) is an arbitrary measure on a measurable space \((X,{\mathcal A})\), then there exists a unique decomposition of \(\mu\) w.r.t. \(\nu\) into the sum of a singular part, a part with a density function and a diffuse part in a sense especially defined. We then study this decomposition in the case of two product measures \(\mu=\otimes_{k\in\mathbb{N}}\mu_ k\) and \(\eta=\otimes_{k\in\mathbb{N}}\eta_ k\), where \(\mu_ k\) is a probability measure and \(\eta_ k\) is an arbitrary measure \(\geq 0\) on a measurable space \((X_ k,{\mathcal A}_ k)\) such that each \(\mu_ k\) has a density function with respect to \(\eta_ k\). Here the infinite product \(\otimes_{k\in\mathbb{N}}\eta_ k\) is defined in the sense of \textit{E. O. Elliott} and \textit{A. P. Morse} [Trans. Am. Math. Soc. 110, 245-283 (1964; Zbl 0125.303)]. It turns out that, similarly as in Kakutani's classical theorem [\textit{S. Kakutani}, Ann. Math., II. Ser. 49, 214-224 (1948; Zbl 0030.02303)], \(\mu\) is of pure type with respect to \(\eta\), i.e., \(\mu\) and \(\eta\) are either mutually singular, or \(\mu\) has a density function with respect to \(\eta\), or else \(\mu\) is diffuse w.r.t. \(\eta\). We finally use these results to classify the infinite products \(\otimes_{k\in\mathbb{N}}\eta_ k\) according to their finiteness, \(\sigma\)- finiteness, and non-\(\sigma\)-finiteness.
reviewed by: Gunter Ritter
Mathematics Subject Classification ID: 28A35
zbMATH DE Number: 108922
zbMATH Keywords: non-\(\sigma\)-finiteness
zbMATH Keywords: infinite products of positive measures
zbMATH Keywords: Lebesgue decomposition
zbMATH Keywords: Kakutani's dichotomy
zbMATH Keywords: Elliott-Morse measures
zbMATH Keywords: trichotomy
zbMATH Keywords: diffuseness
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 108922