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

François Dorais; Rafał Filipów; Tomasz Natkaniec

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 487-508
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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François Dorais, Rafał Filipów, and Tomasz Natkaniec. "On some properties of Hamel bases and their applications to Marczewski measurable functions." Open Mathematics 11.3 (2013): 487-508. <http://eudml.org/doc/269437>.

@article{FrançoisDorais2013,
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.},
author = {François Dorais, Rafał Filipów, Tomasz Natkaniec},
journal = {Open Mathematics},
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; -measurable set; -measurable function; connectivity function; extendable function; covering property axiom},
language = {eng},
number = {3},
pages = {487-508},
title = {On some properties of Hamel bases and their applications to Marczewski measurable functions},
url = {http://eudml.org/doc/269437},
volume = {11},
year = {2013},
}

TY - JOUR
AU - François Dorais
AU - Rafał Filipów
AU - Tomasz Natkaniec
TI - On some properties of Hamel bases and their applications to Marczewski measurable functions
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 487
EP - 508
AB - 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.
LA - eng
KW - 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; -measurable set; -measurable function; connectivity function; extendable function; covering property axiom
UR - http://eudml.org/doc/269437
ER -

References

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  1. [1] Bartoszyński T., Judah H., Set Theory, A K Peters, Wellesley, 1995 
  2. [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[Crossref] Zbl0189.53703
  3. [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 Zbl1052.28001
  4. [4] Cichoń J., Kharazishvili A., Węglorz B., Subsets of the Real Line, Wydawnictwo Uniwersytetu Łódzkiego, Łódź, 1995 
  5. [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[Crossref] Zbl1221.28001
  6. [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[Crossref] Zbl0951.26003
  7. [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[Crossref] Zbl1066.03047
  8. [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[Crossref] Zbl1083.26001
  9. [9] Ciesielski K., Recław I., Cardinal invariants concerning extendable and peripherally continuous functions, Real Anal. Exchange, 1995/96, 21(2), 459–472 Zbl0879.26005
  10. [10] Császár Á., Laczkovich M., Discrete and equal convergence, Studia Sci. Math. Hungar., 1975, 10(3–4), 463–472 Zbl0405.26006
  11. [11] Erdős P., Stone A.H., On the sum of two Borel sets, Proc. Amer. Math. Soc., 1970, 25(2), 304–306 [Crossref] Zbl0192.40304
  12. [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[Crossref] Zbl1006.28006
  13. [13] Gibson R.G., Natkaniec T., Darboux like functions, Real Anal. Exchange, 1996/97, 22(2), 492–533 Zbl0942.26004
  14. [14] Gibson R.G., Natkaniec T., Darboux-like functions. Old problems and new results, Real Anal. Exchange, 1998/99, 24(2), 487–496 Zbl0969.26003
  15. [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. [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[Crossref] 
  17. [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[Crossref] 
  18. [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[Crossref] Zbl1072.28002
  19. [19] Miller A.W., Popvassilev S.G., Vitali sets and Hamel bases that are Marczewski measurable, Fund. Math., 2000, 166(3), 269–279 Zbl0968.03051
  20. [20] Mycielski J., Independent sets in topological algebras, Fund. Math., 1964, 55, 139–147 Zbl0124.01301
  21. [21] Natkaniec T., On extendable derivations, Real Anal. Exchange, 2008/09, 34(1), 207–213 
  22. [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[Crossref] 
  23. [23] Natkaniec T., Recław I., Universal summands for families of measurable functions, Acta Sci. Math. (Szeged), 1998, 64(3–4), 463–471 Zbl0913.04003
  24. [24] Natkaniec T., Wilczyński W., Sums of periodic Darboux functions and measurability, Atti Sem. Mat. Fis. Univ. Modena, 2003, 51(2), 369–376 Zbl1221.26004
  25. [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[Crossref] Zbl0193.30902
  26. [26] Sierpiński W., Sur la question de la mesurabilité de la base de M. Hamel, Fund. Math., 1920, 1, 105–111 Zbl47.0180.03
  27. [27] Sierpiński W., Sur les suites transfinies convergentes de fonctions de Baire, Fund. Math., 1920, 1, 132–141 Zbl47.0237.01
  28. [28] Szpilrajn E., Sur une classe de fonctions de M. Sierpiński et la classe correspondante d’ensembles, Fund. Math., 1935, 24, 17–34 Zbl0010.19901
  29. [29] Taylor A.D., Partitions of pairs of reals, Fund. Math., 1978, 99(1), 51–59 Zbl0377.04010
  30. [30] Walsh J.T., Marczewski sets, measure and the Baire property, Fund. Math., 1988, 129(2), 83–89 Zbl0652.28001

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