On some Problem of Sierpiński and Ruziewicz Concerning the Superposition of Measurable Functions. Microscopic Hamel Basis

Aleksandra Karasińska; Elżbieta Wagner-Bojakowska

doi:10.2478/tmmp-2014-0008

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On some Problem of Sierpiński and Ruziewicz Concerning the Superposition of Measurable Functions. Microscopic Hamel Basis

Tatra Mountains Mathematical Publications

Volume 58 (2014): Issue 1 (March 2014)

By: Aleksandra Karasińska and Elżbieta Wagner-Bojakowska

Open Access

|Aug 2014

Abstract

S. Ruziewicz and W. Sierpiński proved that each function f : ℝ → ℝ can be represented as a superposition of two measurable functions. Here, a strengthening of this theorem is given. The properties of Lusin set and microscopic Hamel bases are considered

References

[1] APPELL, J.-D’ANIELLO, E.-VATH, M.: Some remarks on small sets, Ric. Mat. 50 (2001), 255-274.
Search in Google Scholar
[2] GER, R.-KUCZMA, M.: On the boundedness and continuity of convex functions and additive functions, Aequationes Math. 4 (1970), 157-162.10.1007/BF01817756
Search in Google Scholar
[3] HORBACZEWSKA, G.-KARASIŃSKA, A.-WAGNER-BOJAKOWSKA, E.: Properties of the σ-ideal of microscopic sets, Chapter 20 in Traditional and Present-Day Topics in Real Analysis, Łódź University Press, Łódź, 2013, pp. 325-344.10.18778/7525-971-1.20
Search in Google Scholar
[4] KARASIŃSKA,A.-WAGNER-BOJAKOWSKA,E.:Homeomorphisms of linear and planar sets of the first category into microscopic sets, Topology Appl. 159 (2012), 1894-1898.10.1016/j.topol.2011.11.055
Search in Google Scholar
[5] KARASIŃSKA, A.-POREDA, W.-WAGNER-BOJAKOWSKA, E.: Duality principle for microscopic sets, in: Real Functions, Density Topology and Related Topics (M. Filipczak et al., eds.), Łódź University Press, Łódź, 2011, pp. 83-87.
Search in Google Scholar
[6] KUCZMA, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality (2nd ed.), Birkh¨auser, Basel, 2009.10.1007/978-3-7643-8749-5
Search in Google Scholar
[7] KUCZMA, M.: On some analogies between measure and category and their applications in the theory of additive functions, Ann. Math. Sil. 1 (1985), 155-162.
Search in Google Scholar
[8] OXTOBY, J. C.: Measure and Category. A survey of the analogies between topological and measure spaces (2nd ed.), in: Grad. Texts in Math., Vol. 2, Springer-Verlag, New York, 1980.
Search in Google Scholar
[9] RUZIEWICZ, S.-SIERPIŃSKI, W.: Un théoréme sur les familles de fonctions, Mathematica (Cluj) 7 (1933), 89-91.
Search in Google Scholar
[10] SIERPIŃSKI, W.: Sur un ensemble non dénombrable, dont toute image continue est de mesure nulle, Fund. Math. 11 (1928), 302-304.10.4064/fm-11-1-302-303
Search in Google Scholar
[11] SIERPIŃSKI, W.: Sur un probléme de M. Ruziewicz concernant les superpositions de functions jouissant de la propriété de Baire, Fund. Math. 24 (1935), 12-16.10.4064/fm-24-1-12-16
Search in Google Scholar
[12] SIERPIŃSKI, W.: Oeuvres Choisies, T. III, PWN, Warszawa, 1975.
Search in Google Scholar
[13] SIERPIŃSKI, W.: Sur la question de la mesurabilité de la base de Hamel, Fund. Math. 1 (1920), 105-111.10.4064/fm-1-1-105-111
Search in Google Scholar
[14] SIERPIŃSKI, W.: La base de M. Hamel et la propriété de Baire, Publ. Math. Univ. Beograd 4 (1935), 221-225.
Search in Google Scholar
[15] SZPILRAJN, E.: Remarques sur les fonctions complétement additives d’ensemble et sur les ensembles joissant de la propriété de Baire, Fund. Math. 22 (1934), 303-311. 10.4064/fm-22-1-303-311
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2014-0008 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 91 - 99

Submitted on: Nov 28, 2013

Published on: Aug 15, 2014

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

microscopic sets,

nullsets,

sets of the first category,

sets having the Baire property,

Related subjects:

© 2014 Aleksandra Karasińska, Elżbieta Wagner-Bojakowska, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 58 (2014): Issue 1 (March 2014)