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Universally measurable subgroups of countable index

Published online by Cambridge University Press:  12 March 2014

Christian Rosendal
Christian Rosendal*
Affiliation:
Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St. Chicago, Il 60607-7045, USA. E-mail: [email protected], URL: http://www.math.uic.edu/~rosendal
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Abstract

It is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S∞ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.

Keywords

Universally measurable subgroupautomatic continuityHaar null sets
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 1081 - 1086
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1]Christensen, J.P.R., Borel structures in groups and semigroups, Mathematica Scandinavica, vol. 28 (1971), pp. 124–128.CrossRefGoogle Scholar
[2]Christensen, J.P.R., Topology and Borel structure. Descriptive topology and set theory with applications to functional analysis and measure theory, North-Holland Mathematics Studies Vol. 10. (Notas de Matemática, No. 51), North-Holland, 1974.Google Scholar
[3]Kechris, A.S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society, vol. 94 (2007), no. 2, pp. 302–350.CrossRefGoogle Scholar
[4]Mycielski, J., Algebraic independence and measure, Fundamenta Mathematical vol. 61 (1967), pp. 165–169.CrossRefGoogle Scholar
[5]Rosendal, C., Automatic continuity of group homomorphisms. The Bulletin of Symbolic Logic, vol. 15 (2009), no. 2, pp. 184–214.CrossRefGoogle Scholar
[6]Solecki, S., Amenability, free subgroups, and Haar null sets in non-locally compact groups. Proceedings of the London Mathematical Society, vol. 93 (2006), no. 3, pp. 693–722.CrossRefGoogle Scholar