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A function whose graph has positive doubling measure


Authors: Tuomo Ojala and Tapio Rajala
Journal: Proc. Amer. Math. Soc. 144 (2016), 733-738
MSC (2010): Primary 28A12; Secondary 30L10
DOI: https://doi.org/10.1090/proc12748
Published electronically: June 30, 2015
MathSciNet review: 3430849
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Abstract: We show that a doubling measure on the plane can give positive measure to the graph of a continuous function. This answers a question by Wang, Wen and Wen posed in a 2013 paper. Moreover, we show that the doubling constant of the measure can be chosen to be arbitrarily close to the doubling constant of the Lebesgue measure.


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Tuomo Ojala
Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
Email: tuomo.j.ojala@jyu.fi

Tapio Rajala
Affiliation: Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014 University of Jyvaskyla, Finland
Email: tapio.m.rajala@jyu.fi

DOI: https://doi.org/10.1090/proc12748
Keywords: Doubling measure, thin set, fat set
Received by editor(s): January 28, 2015
Received by editor(s) in revised form: February 3, 2015
Published electronically: June 30, 2015
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society