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URN: urn:nbn:de:bsz:291-scidok-46129
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4612/


The Lebesgue decomposition theorem for arbitrary contents

König, Heinz

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Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer: 144
Sprache: Englisch
Erstellungsjahr: 2005
Publikationsdatum: 15.02.2012
Kurzfassung auf Englisch: The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a \sigma algebra, in final form due to Hahn 1921, and for the bounded contents with respect to some fixed content on an algebra, due to Bochner-Phillips 1941 and Darst 1962. Then came the version for arbitrary measures, due to R.A.Johnson 1967 and N.Y.Luther 1968. The unpleasant fact with these versions is that each one requires its particular notions of absolutely continuous and singular constituents. It seems mysterious how a common roof for all of them could look, and therefore how a universal version for arbitrary contents could be achieved - and all that while several abstract extensions of particular versions appeared in the subsequent decades, for example due to de Lucia-Morales 2003. After these decades now the present article claims to arrive at the final aim in the original context of arbitrary contents. The article will be based on the author's new difference formation for arbitrary contents 1999. This difference formation even furnishes simple explicit formulas for the two constituents.
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