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On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑 2 which are 𝑇 2 -negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇 2 denotes the group of all translations of 𝐑 2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇 2 , where 𝑔 is an arbitrary rotation of 𝐑 2 distinct from the identity transformation and all central symmetries of 𝐑 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

On Negligible and Absolutely Nonmeasurable Subsets of the Euclidean Plane

Georgian Mathematical Journal , Volume 11 (3) – Sep 1, 2004

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References (9)

Publisher
de Gruyter
Copyright
© Heldermann Verlag
ISSN
1072-947X
eISSN
1072-9176
DOI
10.1515/GMJ.2004.479
Publisher site
See Article on Publisher Site

Abstract

The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem. In particular, it is demonstrated that there exist subsets of the plane 𝐑 2 which are 𝑇 2 -negligible and, simultaneously, 𝐺-absolutely nonmeasurable. Here 𝑇 2 denotes the group of all translations of 𝐑 2 and 𝐺 denotes the group generated by {𝑔} ∪ 𝑇 2 , where 𝑔 is an arbitrary rotation of 𝐑 2 distinct from the identity transformation and all central symmetries of 𝐑 2 .

Journal

Georgian Mathematical Journalde Gruyter

Published: Sep 1, 2004

Keywords: Sierpiński partition; quasi-invariant measure; uniform set; negligible set; absolutely nonmeasurable set

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