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.
Similar content being viewed by others
References
Bartoszyński T., Judah H., Set Theory, A K Peters, Wellesley, 1995
Brown J.B., Negligible sets for real connectivity functions, Proc. Amer. Math. Soc., 1970, 24(2), 263–269
Cichoń J., Jasiński A., A note on algebraic sums of subsets of the real line, Real Anal. Exchange, 2002/03, 28(2), 493–499
Cichoń J., Kharazishvili A., Węglorz B., Subsets of the Real Line, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 1995
Cichoń J., Szczepaniak P., Hamel-isomorphic images of the unit ball, MLQ Math. Log. Q., 2010, 56(6), 625–630
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
Ciesielski K., Pawlikowski J., The Covering Property Axiom, CPA, Cambridge Tracts in Math., 164, Cambridge University Press, Cambridge, 2004
Ciesielski K., Pawlikowski J., Nice Hamel bases under the covering property axiom, Acta Math. Hungar., 2004, 105(3), 197–213
Ciesielski K., Recław I., Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange, 1995/96, 21(2), 459–472
Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472
Erdős P., Stone A.H., On the sum of two Borel sets, Proc. Amer. Math. Soc., 1970, 25(2), 304–306
Filipów R., Recław I., On the difference property of Borel measurable and (s)-measurable functions, Acta Math. Hungar., 2002, 96(1–2), 21–25
Gibson R.G., Natkaniec T., Darboux like functions, Real Anal. Exchange, 1996/97, 22(2), 492–533
Gibson R.G., Natkaniec T., Darboux-like functions. Old problems and new results, Real Anal. Exchange, 1998/99, 24(2), 487–496
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
Kechris A.S., Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer, New York, 1995
Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser, Basel, 2009
Kysiak M., Nonmeasurable algebraic sums of sets of reals, Colloq. Math., 2005, 102(1), 113–122
Miller A.W., Popvassilev S.G., Vitali sets and Hamel bases that are Marczewski measurable, Fund. Math., 2000, 166(3), 269–279
Mycielski J., Independent sets in topological algebras, Fund. Math., 1964, 55, 139–147
Natkaniec T., On extendable derivations, Real Anal. Exchange, 2008/09, 34(1), 207–213
Natkaniec T., Covering an additive function by < c-many continuous functions, J. Math. Anal. Appl., 2012, 387(2), 741–745
Natkaniec T., Recław I., Universal summands for families of measurable functions, Acta Sci. Math. (Szeged), 1998, 64(3–4), 463–471
Natkaniec T., Wilczyński W., Sums of periodic Darboux functions and measurability, Atti Sem. Mat. Fis. Univ. Modena, 2003, 51(2), 369–376
Rogers C.A., A linear Borel set whose difference set is not a Borel set, Bull. London Math. Soc., 1970, 2(1), 41–42
Sierpiński W., Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math., 1920, 1, 105–111
Sierpiński W., Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1920, 1, 132–141
Szpilrajn E., Sur une classe de fonctions de M. Sierpiński et la classe correspondante d’ensembles, Fund. Math., 1935, 24, 17–34
Taylor A.D., Partitions of pairs of reals, Fund. Math., 1978, 99(1), 51–59
Walsh J.T., Marczewski sets, measure and the Baire property, Fund. Math., 1988, 129(2), 83–89
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Dorais, F.G., Filipów, R. & Natkaniec, T. On some properties of Hamel bases and their applications to Marczewski measurable functions. centr.eur.j.math. 11, 487–508 (2013). https://doi.org/10.2478/s11533-012-0144-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-012-0144-1