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Lorentz and Gale–Ryser theorems on general measure spaces

Part of: Inequalities Linear function spaces and their duals Classical measure theory

Published online by Cambridge University Press:  09 August 2021

Santiago Boza Open the ORCID record for Santiago Boza [Opens in a new window] ,
Martin Křepela Open the ORCID record for Martin Křepela [Opens in a new window]  and
Javier Soria Open the ORCID record for Javier Soria [Opens in a new window]
Santiago Boza
Affiliation:
Department of Mathematics, EETAC, Polytechnical University of Catalonia, 08860 Castelldefels, Spain [email protected]
Martin Křepela
Affiliation:
Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic [email protected]
Javier Soria
Affiliation:
Interdisciplinary Mathematics Institute (IMI), Department of Analysis and Applied Mathematics, Complutense University of Madrid, 28040 Madrid, Spain [email protected]
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Abstract

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

Keywords

Cross-sectionsNonincreasing rearrangementHardy-Littlewood-Pólya relation

MSC classification

Primary: 26D15: Inequalities for sums, series and integrals
Secondary: 28A35: Measures and integrals in product spaces 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 857 - 878
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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