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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Automatic continuity via analytic thinning


Authors: N. H. Bingham and A. J. Ostaszewski
Journal: Proc. Amer. Math. Soc. 138 (2010), 907-919
MSC (2000): Primary 26A03
Published electronically: November 5, 2009
MathSciNet review: 2566557
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Abstract: We use Choquet's analytic capacitability theorem and the Kestelman-Borwein-Ditor theorem (on the inclusion of null sequences by translation) to derive results on `analytic automaticity' - for instance, a stronger common generalization of the Jones/Kominek theorems that an additive function whose restriction is continuous/bounded on an analytic set $ T$ spanning $ \mathbb{R}$ (e.g., containing a Hamel basis) is continuous on $ \mathbb{R}$. We obtain results on `compact spannability' - the ability of compact sets to span $ \mathbb{R}$. From this, we derive Jones' Theorem from Kominek's. We cite several applications, including the Uniform Convergence Theorem of regular variation.


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Additional Information

N. H. Bingham
Affiliation: Department of Mathematics, Imperial College London, South Kensington, London SW7 2AZ, United Kingdom
Email: n.bingham@ic.ac.uk

A. J. Ostaszewski
Affiliation: Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom
Email: a.j.ostaszewski@lse.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-09-09984-5
PII: S 0002-9939(09)09984-5
Keywords: Jones' theorem, Kominek's theorem, analytic set, Choquet capacity, Hamel basis, uniform convergence theorem, regular variation, automatic continuity
Received by editor(s): June 28, 2008
Received by editor(s) in revised form: April 3, 2009
Published electronically: November 5, 2009
Dedicated: To Roy Davies on the occasion of his 80th birthday
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2009 American Mathematical Society