Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 22, 2012

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

  • François Dorais EMAIL logo , Rafał Filipów and Tomasz Natkaniec
From the journal Open Mathematics


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.

[1] Bartoszyński T., Judah H., Set Theory, A K Peters, Wellesley, 1995 10.1201/9781439863466Search in Google Scholar

[2] Brown J.B., Negligible sets for real connectivity functions, Proc. Amer. Math. Soc., 1970, 24(2), 263–269 in Google 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 10.14321/realanalexch.28.2.0493Search in Google Scholar

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

[5] Cichoń J., Szczepaniak P., Hamel-isomorphic images of the unit ball, MLQ Math. Log. Q., 2010, 56(6), 625–630 in Google 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 in Google Scholar

[7] Ciesielski K., Pawlikowski J., The Covering Property Axiom, CPA, Cambridge Tracts in Math., 164, Cambridge University Press, Cambridge, 2004 in Google Scholar

[8] Ciesielski K., Pawlikowski J., Nice Hamel bases under the covering property axiom, Acta Math. Hungar., 2004, 105(3), 197–213 in Google Scholar

[9] Ciesielski K., Recław I., Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange, 1995/96, 21(2), 459–472 10.2307/44152660Search in Google Scholar

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

[11] Erdős P., Stone A.H., On the sum of two Borel sets, Proc. Amer. Math. Soc., 1970, 25(2), 304–306 10.2307/2037209Search in Google 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 in Google Scholar

[13] Gibson R.G., Natkaniec T., Darboux like functions, Real Anal. Exchange, 1996/97, 22(2), 492–533 10.2307/44153937Search in 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 10.2307/44152975Search in 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 10.2307/44151804Search in Google Scholar

[16] Kechris A.S., Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer, New York, 1995 in Google Scholar

[17] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser, Basel, 2009 in Google Scholar

[18] Kysiak M., Nonmeasurable algebraic sums of sets of reals, Colloq. Math., 2005, 102(1), 113–122 in Google Scholar

[19] Miller A.W., Popvassilev S.G., Vitali sets and Hamel bases that are Marczewski measurable, Fund. Math., 2000, 166(3), 269–279 10.4064/fm-166-3-269-279Search in Google Scholar

[20] Mycielski J., Independent sets in topological algebras, Fund. Math., 1964, 55, 139–147 10.4064/fm-55-2-139-147Search in Google Scholar

[21] Natkaniec T., On extendable derivations, Real Anal. Exchange, 2008/09, 34(1), 207–213 10.14321/realanalexch.34.1.0207Search in Google Scholar

[22] Natkaniec T., Covering an additive function by < c-many continuous functions, J. Math. Anal. Appl., 2012, 387(2), 741–745 in Google Scholar

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

[26] Sierpiński W., Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math., 1920, 1, 105–111 10.4064/fm-1-1-105-111Search in Google Scholar

[27] Sierpiński W., Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1920, 1, 132–141 10.4064/fm-1-1-132-141Search in 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 10.4064/fm-24-1-17-34Search in Google Scholar

[29] Taylor A.D., Partitions of pairs of reals, Fund. Math., 1978, 99(1), 51–59 10.4064/fm-99-1-51-59Search in Google Scholar

[30] Walsh J.T., Marczewski sets, measure and the Baire property, Fund. Math., 1988, 129(2), 83–89 10.4064/fm-129-2-83-89Search in Google Scholar

Published Online: 2012-12-22
Published in Print: 2013-3-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 8.2.2023 from
Scroll Up Arrow