Beyond Lebesgue and Baire III: Steinhausʼ Theorem and its descendants

We generalize from the real line to normed groups (i.e., the invariantly metrizable, right topological groups) a result on infinite combinatorics, valid for ‘large’ subsets in both the category and measure senses, which implies both Steinhausʼs Theorem and many of its descendants. We deduce the inhe...

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Bibliographic Details
Published in:Topology and its applications 2013-06, Vol.160 (10), p.1144-1154
Main Author: Ostaszewski, A.J.
Format: Article
Language:English
Subjects:
Analytic Baire and Cantor theorems
Automatic continuity
Group-norm
Measure-category duality
Non-commutative groups
Shift-compactness
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Summary:We generalize from the real line to normed groups (i.e., the invariantly metrizable, right topological groups) a result on infinite combinatorics, valid for ‘large’ subsets in both the category and measure senses, which implies both Steinhausʼs Theorem and many of its descendants. We deduce the inherent measure-category duality in this setting directly from properties of analytic sets. We apply the result to extend the Pettis Theorem and the Continuous Homomorphism Theorem to normed groups, i.e., beyond metrizable topological groups, and in a companion paper deduce automatic joint-continuity results for group operations.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2013.04.005