×

zbMATH — the first resource for mathematics

On some properties of Hamel bases and their applications to Marczewski measurable functions. (English) Zbl 1259.28005
Summary: 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 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
03E35 Consistency and independence results
26A51 Convexity of real functions in one variable, generalizations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bartoszyński T., Judah H., Set Theory, A K Peters, Wellesley, 1995
[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/S0002-9939-1970-0249545-9 · Zbl 0189.53703
[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
[4] Cichoń J., Kharazishvili A., Węglorz B., Subsets of the Real Line, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 1995
[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.200910113 · Zbl 1221.28001
[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)00169-2 · Zbl 0951.26003
[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/CBO9780511546457 · Zbl 1066.03047
[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.0000049287.44877.2c · Zbl 1083.26001
[9] Ciesielski K., Recław I., Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange, 1995/96, 21(2), 459-472 · Zbl 0879.26005
[10] Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3-4), 463-472
[11] Erdős P., Stone A.H., On the sum of two Borel sets, Proc. Amer. Math. Soc., 1970, 25(2), 304-306 · Zbl 0192.40304
[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:1015661511337
[13] Gibson R.G., Natkaniec T., Darboux like functions, Real Anal. Exchange, 1996/97, 22(2), 492-533 · Zbl 0942.26004
[14] Gibson R.G., Natkaniec T., Darboux-like functions. Old problems and new results, Real Anal. Exchange, 1998/99, 24(2), 487-496 · Zbl 0969.26003
[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
[16] Kechris A.S., Classical Descriptive Set Theory, Grad. Texts in Math., 156, Springer, New York, 1995 http://dx.doi.org/10.1007/978-1-4612-4190-4
[17] Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser, Basel, 2009 http://dx.doi.org/10.1007/978-3-7643-8749-5
[18] Kysiak M., Nonmeasurable algebraic sums of sets of reals, Colloq. Math., 2005, 102(1), 113-122 http://dx.doi.org/10.4064/cm102-1-10 · Zbl 1072.28002
[19] Miller A.W., Popvassilev S.G., Vitali sets and Hamel bases that are Marczewski measurable, Fund. Math., 2000, 166(3), 269-279 · Zbl 0968.03051
[20] Mycielski J., Independent sets in topological algebras, Fund. Math., 1964, 55, 139-147 · Zbl 0124.01301
[21] Natkaniec T., On extendable derivations, Real Anal. Exchange, 2008/09, 34(1), 207-213
[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.2011.09.035 · Zbl 1235.26004
[23] Natkaniec T., Recław I., Universal summands for families of measurable functions, Acta Sci. Math. (Szeged), 1998, 64(3-4), 463-471 · Zbl 0913.04003
[24] Natkaniec T., Wilczyński W., Sums of periodic Darboux functions and measurability, Atti Sem. Mat. Fis. Univ. Modena, 2003, 51(2), 369-376 · Zbl 1221.26004
[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/2.1.41 · Zbl 0193.30902
[26] Sierpiński W., Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math., 1920, 1, 105-111
[27] Sierpiński W., Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1920, 1, 132-141 · JFM 47.0237.01
[28] Szpilrajn E., Sur une classe de fonctions de M. Sierpiński et la classe correspondante d’ensembles, Fund. Math., 1935, 24, 17-34 · Zbl 0010.19901
[29] Taylor A.D., Partitions of pairs of reals, Fund. Math., 1978, 99(1), 51-59 · Zbl 0377.04010
[30] Walsh J.T., Marczewski sets, measure and the Baire property, Fund. Math., 1988, 129(2), 83-89 · Zbl 0652.28001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.