The Lebesgue decomposition theorem for arbitrary contents
type: Item
instance of: scholarly article
title: The Lebesgue decomposition theorem for arbitrary contents
zbMATH Open document ID: 1113.28004
published in: Positivity
publication date: 2007-02-13T00:00:00Z
review text: The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of absolutely continuous and singular set functions with respect to some fixed set function. The traditional versions are for bounded measures with respect to some fixed measure on a \(\sigma\)-algebra. In subsequent decades since Hahn's final form (1921), there have been several versions with particular requirements on the two constituents, as well as abstract extensions of particular versions. The author cites a list of these in the abstract of the paper. In an effort to unify the theory (find a ``common roof'' for partial notions), the present article claims to arrive at the aim in the original context of arbitrary contents, using the difference formation introduced by the author in his 1999 paper. This difference formation also affords explicit formulas for the two constituents.
Mathematics Subject Classification ID: 28A12
zbMATH DE Number: 5125398
zbMATH Keywords: Lebesgue decomposition theorem for contents and measures
zbMATH Keywords: absolutely continuous set functions
zbMATH Keywords: singular set functions
zbMATH Keywords: difference formation for arbitrary contents
author: Heinz König
reviewed by: Pao-sheng Hsu
MaRDI profile type: Publication
full work available at URL: s11117-006-0045-2
OpenAlex ID: W2042307160
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cites work: Q4462179
cites work: Lebesgue decomposition and weakly Borel measures
DOI: 10.1007/S11117-006-0045-2
description: scientific article; zbMATH DE number 5125398