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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 11, Issue 3


Volume 13 (2015)

On some properties of Hamel bases and their applications to Marczewski measurable functions

François Dorais
  • Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, MI, 48109-1043, USA
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/ Rafał Filipów / Tomasz Natkaniec
Published Online: 2012-12-22 | DOI: https://doi.org/10.2478/s11533-012-0144-1


We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.

MSC: 28A20; 26A15; 03E35; 26A51

Keywords: Linear function; Additive function; (s)-measurable set; Marczewski measurable set; (s)-measurable function; Marczewski measurable function; The intermediate value property; Darboux function; Connectivity function; Extendable function; Covering Property Axiom

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About the article

Published Online: 2012-12-22

Published in Print: 2013-03-01

Citation Information: Open Mathematics, Volume 11, Issue 3, Pages 487–508, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0144-1.

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