Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-18T14:16:11.406Z Has data issue: false hasContentIssue false
  • English
  • Français

CONTINUITY OF MEASURABLE HOMOMORPHISMS

Part of: Connections with other structures, applications Topological and differentiable algebraic systems

Published online by Cambridge University Press:  01 August 2008

JANUSZ BRZDȨK
JANUSZ BRZDȨK*
Affiliation:
Department of Mathematics, Pedagogical University, Podchorążych 2, 30-084 Kraków, Poland (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

Keywords

Christensen measurabilityBaire propertyuniversal measurabilityHaar measurabilitymeasurable homomorphism of groups

MSC classification

Secondary: 22A05: Structure of general topological groups 39B52: Equations for functions with more general domains and/or ranges 54H11: Topological groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 1 , August 2008 , pp. 171 - 176
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Beck, A., Corson, H. H. and Simon, A. B., ‘The interior points of the product of two sets of a locally compact group’, Proc. Amer. Math. Soc. 9 (1958), 648–652.Google Scholar
[2]Brzdȩk, J., ‘On the Cauchy difference on normed spaces’, Abh. Math. Sem. Univ. Hamburg 66 (1996), 143–150.CrossRefGoogle Scholar
[3]Brzdȩk, J., ‘On almost additive functions’, Bull. Austral. Math. Soc. 54 (1996), 281–290.Google Scholar
[4]Christensen, J. P. R., ‘Borel structures in groups and semigroups’, Math. Scand. 28 (1971), 124–128.CrossRefGoogle Scholar
[5]Christensen, J. P. R., ‘On sets of Haar measure zero in abelian Polish groups’, Israel J. Math. 13 (1972), 255–260.CrossRefGoogle Scholar
[6]Fisher, P. and Słodkowski, Z., ‘Christensen zero sets and measurable convex functions’, Proc. Amer. Math. Soc. 79 (1980), 449–453.CrossRefGoogle Scholar
[7]Kominek, Z. and Kuczma, M., ‘Theorems of Bernstein-Doetsch, Piccard and Mehdi, and semilinear topology’, Arch. Math. (Basel) 52 (1989), 595–602.CrossRefGoogle Scholar
[8]Kleppner, A., ‘Measurable homomorphisms of locally compact groups’, Proc. Amer. Math. Soc. 106 (1989), 391–395.CrossRefGoogle Scholar
[9]Kleppner, A., ‘Correction to “Measurable homomorphisms of locally compact groups” 106 (1989), 391–395’, Proc. Amer. Math. Soc. 111 (1991), 1199.Google Scholar
[10]Noll, D., ‘Souslin measurable homomorphisms of topological groups’, Arch. Math. (Basel) 59 (1992), 294–301.CrossRefGoogle Scholar
[11]Oxtoby, J. C., Measure and Category, Graduate Texts in Mathematics (Springer, Berlin, 1971).CrossRefGoogle Scholar
[12]Pettis, B. J., ‘On continuity and openness of homomorphisms in topological groups’, Ann. of Math. (2) 52 (1950), 293–308.CrossRefGoogle Scholar
[13]Sander, W., ‘Verallgemeinerungen eines Satzes von S. Piccard’, Manuscripta Math. 16 (1975), 11–25.Google Scholar
[14]Sander, W., ‘A generalization of a theorem of S. Piccard’, Proc. Amer. Math. Soc. 73 (1979), 281–282.Google Scholar
[15]Sander, W., ‘Ein Beitrag zur Baire-Kategorie-Theorie’, Manuscripta Math. 34 (1981), 71–83.Google Scholar
[16]Sasvári, Z., ‘On measurable local homomorphisms’, Proc. Amer. Math. Soc. 112 (1991), 603–604.Google Scholar
[17]Solecki, S., ‘Amenability, free subgroups, and Haar null sets in non-locally compact groups’, Proc. London Math. Soc. (3) 93 (2006), 693–722.CrossRefGoogle Scholar
[18]Stromberg, K., ‘An elementary proof of Steinhaus’s theorem’, Proc. Amer. Math. Soc. 36 (1972), 308.Google Scholar