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Open Mathematics

formerly Central European Journal of Mathematics

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

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IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

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2391-5455
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Volume 11, Issue 3

Issues

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

Abstract

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

  • [1] Bartoszyński T., Judah H., Set Theory, A K Peters, Wellesley, 1995 Google Scholar

  • [2] Brown J.B., Negligible sets for real connectivity functions, Proc. Amer. Math. Soc., 1970, 24(2), 263–269 http://dx.doi.org/10.1090/S0002-9939-1970-0249545-9CrossrefGoogle Scholar

  • [3] Cichoń J., Jasiński A., A note on algebraic sums of subsets of the real line, Real Anal. Exchange, 2002/03, 28(2), 493–499 Google Scholar

  • [4] Cichoń J., Kharazishvili A., Węglorz B., Subsets of the Real Line, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 1995 Google Scholar

  • [5] Cichoń J., Szczepaniak P., Hamel-isomorphic images of the unit ball, MLQ Math. Log. Q., 2010, 56(6), 625–630 http://dx.doi.org/10.1002/malq.200910113CrossrefGoogle Scholar

  • [6] Ciesielski K., Jastrzębski J., Darboux-like functions within the classes of Baire one, Baire two, and additive functions, Topology Appl., 2000, 103(2), 203–219 http://dx.doi.org/10.1016/S0166-8641(98)00169-2CrossrefGoogle Scholar

  • [7] Ciesielski K., Pawlikowski J., The Covering Property Axiom, CPA, Cambridge Tracts in Math., 164, Cambridge University Press, Cambridge, 2004 http://dx.doi.org/10.1017/CBO9780511546457CrossrefGoogle Scholar

  • [8] Ciesielski K., Pawlikowski J., Nice Hamel bases under the covering property axiom, Acta Math. Hungar., 2004, 105(3), 197–213 http://dx.doi.org/10.1023/B:AMHU.0000049287.44877.2cCrossrefGoogle Scholar

  • [9] Ciesielski K., Recław I., Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange, 1995/96, 21(2), 459–472 Google Scholar

  • [10] Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472 Google Scholar

  • [11] Erdős P., Stone A.H., On the sum of two Borel sets, Proc. Amer. Math. Soc., 1970, 25(2), 304–306 CrossrefGoogle Scholar

  • [12] Filipów R., Recław I., On the difference property of Borel measurable and (s)-measurable functions, Acta Math. Hungar., 2002, 96(1–2), 21–25 http://dx.doi.org/10.1023/A:1015661511337CrossrefGoogle Scholar

  • [13] Gibson R.G., Natkaniec T., Darboux like functions, Real Anal. Exchange, 1996/97, 22(2), 492–533 Google Scholar

  • [14] Gibson R.G., Natkaniec T., Darboux-like functions. Old problems and new results, Real Anal. Exchange, 1998/99, 24(2), 487–496 Google Scholar

  • [15] Gibson R.G., Roush F., The restrictions of a connectivity function are nice but not that nice, Real Anal. Exchange, 1986/87, 12(1), 372–376 Google Scholar

  • [16] Kechris A.S., Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4612-4190-4CrossrefGoogle Scholar

  • [17] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser, Basel, 2009 http://dx.doi.org/10.1007/978-3-7643-8749-5CrossrefGoogle Scholar

  • [18] Kysiak M., Nonmeasurable algebraic sums of sets of reals, Colloq. Math., 2005, 102(1), 113–122 http://dx.doi.org/10.4064/cm102-1-10CrossrefGoogle Scholar

  • [19] Miller A.W., Popvassilev S.G., Vitali sets and Hamel bases that are Marczewski measurable, Fund. Math., 2000, 166(3), 269–279 Google Scholar

  • [20] Mycielski J., Independent sets in topological algebras, Fund. Math., 1964, 55, 139–147 Google Scholar

  • [21] Natkaniec T., On extendable derivations, Real Anal. Exchange, 2008/09, 34(1), 207–213 Google Scholar

  • [22] Natkaniec T., Covering an additive function by < c-many continuous functions, J. Math. Anal. Appl., 2012, 387(2), 741–745 http://dx.doi.org/10.1016/j.jmaa.2011.09.035CrossrefGoogle Scholar

  • [23] Natkaniec T., Recław I., Universal summands for families of measurable functions, Acta Sci. Math. (Szeged), 1998, 64(3–4), 463–471 Google Scholar

  • [24] Natkaniec T., Wilczyński W., Sums of periodic Darboux functions and measurability, Atti Sem. Mat. Fis. Univ. Modena, 2003, 51(2), 369–376 Google Scholar

  • [25] Rogers C.A., A linear Borel set whose difference set is not a Borel set, Bull. London Math. Soc., 1970, 2(1), 41–42 http://dx.doi.org/10.1112/blms/2.1.41CrossrefGoogle Scholar

  • [26] Sierpiński W., Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math., 1920, 1, 105–111 Google Scholar

  • [27] Sierpiński W., Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1920, 1, 132–141 Google Scholar

  • [28] Szpilrajn E., Sur une classe de fonctions de M. Sierpiński et la classe correspondante d’ensembles, Fund. Math., 1935, 24, 17–34 Google Scholar

  • [29] Taylor A.D., Partitions of pairs of reals, Fund. Math., 1978, 99(1), 51–59 Google Scholar

  • [30] Walsh J.T., Marczewski sets, measure and the Baire property, Fund. Math., 1988, 129(2), 83–89 Google Scholar

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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