Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities

Authors: P. D. Humke and M. Laczkovich
Journal: Trans. Amer. Math. Soc. 357 (2005), 31-44
MSC (2000): Primary 03E35; Secondary 28A20, 26A03
Published electronically: August 19, 2004
MathSciNet review: 2098085
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using the continuum hypothesis, Sierpinski constructed a nonmeasurable function $f$ such that $\{ h: f(x+h)\ne f(x-h)\}$ is countable for every $x.$ Clearly, such a function is symmetrically approximately continuous everywhere.

Here we to show that Sierpinski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable.

References [Enhancements On Off] (What's this?)

  • 1. K. Ciesielski, Generalized continuities. In: Encyclopedia of General Topology (J.I. Nagata, J.E. Vaughan, and K.P. Hart, eds.), Elsevier, to appear.
  • 2. C. Freiling, Axioms of symmetry: throwing darts at the real number line, J. Symb. Logic 51 (1986), 190-200. MR 0830085 (87f:03148)
  • 3. Chris Freiling, A converse to a theorem of Sierpiński on almost symmetric sets, Real Anal. Exchange 15 (1989/90), no. 2, 760–767. MR 1059437
  • 4. D. H. Fremlin, Measure Theory, Vol. 2. Torres Fremlin, Colchester, 2001.
  • 5. P. D. Humke and M. Laczkovich, Parametric semicontinuity implies continuity, Real Anal. Exchange 17 (1991/92), no. 2, 668–680. MR 1171407
  • 6. Masaru Kada and Yoshifumi Yuasa, Cardinal invariants about shrinkability of unbounded sets, Proceedings of the International Conference on Set-theoretic Topology and its Applications (Matsuyama, 1994), 1996, pp. 215–223. MR 1425940,
  • 7. Miklós Laczkovich, Two constructions of Sierpiński and some cardinal invariants of ideals, Real Anal. Exchange 24 (1998/99), no. 2, 663–676. MR 1704742
  • 8. M. Laczkovich and Arnold W. Miller, Measurability of functions with approximately continuous vertical sections and measurable horizontal sections, Colloq. Math. 69 (1995), no. 2, 299–308. MR 1358935,
  • 9. I. N. Pesin, The measurability of symmetrically continuous functions, Teor. Funkciĭ Funkcional. Anal. i Priložen. Vyp. 5 (1967), 99–101 (Russian). MR 0227331
  • 10. David Preiss, A note on symmetrically continuous functions, Časopis Pěst. Mat. 96 (1971), 262–264, 300 (English, with Czech summary). MR 0306411
  • 11. Wacław Sierpiński, Oeuvres choisies, PWN—Éditions Scientifiques de Pologne, Warsaw, 1974 (French). Tome I: Bibliographie, théorie des nombres et analyse mathématique; Publié par les soins de Stanisław Hartman et Andrzej Schinzel; Contenant des articles sur la vie et les travaux de Sierpiński, par Kazimierz Kuratowski, Andrzej Schinzel et Stanisław Hartman; Comité de rédaction: Stanisław Hartman, Kazimierz Kuratowski, Edward Marczewski, Andrzej Mostowski, Andrzej Schinzel, Roman Sikorski, Marceli Stark. MR 0414302
  • 12. Brian S. Thomson, Symmetric properties of real functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 183, Marcel Dekker, Inc., New York, 1994. MR 1289417
  • 13. J. Uher, Symmetric semicontinuity implies continuity, Trans. Amer. Math. Soc. 293 (1986), no. 1, 421-429. MR 0814930 (87b:26005)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E35, 28A20, 26A03

Retrieve articles in all journals with MSC (2000): 03E35, 28A20, 26A03

Additional Information

P. D. Humke
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450 – and – Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057

M. Laczkovich
Affiliation: Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C, 1117 Hungary – and – Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, England

Keywords: Fubini, symmetrically approximately continuous, covering number, shrinking number
Received by editor(s): March 10, 2003
Published electronically: August 19, 2004
Additional Notes: The second author’s research was supported by the Hungarian National Foundation for Scientific Research Grant No. T032042.
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society