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Transactions of the American Mathematical Society
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
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03682-7
PII: S 0002-9947(04)03682-7
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