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Maximal functions and the additivity of various families of null sets


Author: Juris Steprāns
Journal: Trans. Amer. Math. Soc. 364 (2012), 3555-3584
MSC (2010): Primary 03E17, 42B25
DOI: https://doi.org/10.1090/S0002-9947-2012-05402-X
Published electronically: March 8, 2012
MathSciNet review: 2901224
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown to be consistent with set theory that every set of reals of size $ \aleph _1$ is null yet there are $ \aleph _1$ planes in Euclidean 3-space whose union is not null. Similar results are obtained for circles in the plane as well as other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain maximal operators and a measure-theoretic pigeonhole principle.


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Additional Information

Juris Steprāns
Affiliation: Department of Mathematics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Email: steprans@yorku.ca

DOI: https://doi.org/10.1090/S0002-9947-2012-05402-X
Keywords: Maximal operator, Besicovitch set, Kakeya set, cardinal invariant, proper forcing
Received by editor(s): January 12, 2005
Received by editor(s) in revised form: September 8, 2006, August 4, 2009, May 17, 2010, and June 16, 2010
Published electronically: March 8, 2012
Additional Notes: Research for this paper was partially supported by NSERC of Canada.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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