Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities
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- by P. D. Humke and M. Laczkovich PDF
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Abstract:
Using the continuum hypothesis, Sierpiński 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 Sierpiński’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
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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
- Email: humke@stolaf.edu
- 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
- Email: laczk@cs.elte.hu
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 31-44
- MSC (2000): Primary 03E35; Secondary 28A20, 26A03
- DOI: https://doi.org/10.1090/S0002-9947-04-03682-7
- MathSciNet review: 2098085