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The Haar measure problem


Authors: Adam J. Przeździecki, Piotr Szewczak and Boaz Tsaban
Journal: Proc. Amer. Math. Soc. 147 (2019), 1051-1057
MSC (2010): Primary 28C10, 28A05, 22C05, 03E17
DOI: https://doi.org/10.1090/proc/14221
Published electronically: December 3, 2018
MathSciNet review: 3896055
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Abstract: An old problem asks whether every compact group has a Haar-nonmeasurable subgroup. A series of earlier results reduced the problem to infinite metrizable profinite groups. We provide a positive answer, assuming a weak, potentially provable, consequence of the Continuum Hypothesis. We also establish the dual, Baire category analogue of this result.


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Additional Information

Adam J. Przeździecki
Affiliation: Warsaw University of Life Sciences—SGGW, Warsaw, Poland
Email: adamp@mimuw.edu.pl

Piotr Szewczak
Affiliation: Faculty of Mathematics and Natural Science College of Sciences, Cardinal Stefan Wyszyński University in Warsaw, Warsaw, Poland — and — Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
Email: p.szewczak@wp.pl

Boaz Tsaban
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel
Email: tsaban@math.biu.ac.il

DOI: https://doi.org/10.1090/proc/14221
Keywords: Haar measurable, Baire property, profinite group, compact group, closed measure zero
Received by editor(s): September 7, 2017
Received by editor(s) in revised form: September 8, 2017
Published electronically: December 3, 2018
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2018 American Mathematical Society