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On critical pairs of product sets in a certain matrix group

Published online by Cambridge University Press:  24 October 2008

M. McCrudden
M. McCrudden
Affiliation:
University of Birmingham
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Extract

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) by

and, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 67 , Issue 3 , May 1970 , pp. 569 - 581
Copyright
Copyright © Cambridge Philosophical Society 1970

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References

REFERENCES

(1)Halmos, P.Measure Theory. (New York, 1959).Google Scholar
(2)Henstock, R. and Macbeath, A. M.On the measure of sum-sets, 1. Proc. London Math. Soc. 3 (1953), 182–194.Google Scholar
(3)Kneser, M.Summenmengen in lokalkompakten Abelschen Gruppen. Math. Z. 66 (1956), 88–110.Google Scholar
(4)Macbeath, A. M.On the measure of product sets in a topological group. J. London Math. Soc. 35 (1960), 403–407.Google Scholar
(5)McCrudden, M.On product sets of a unimodular group. Proc. Cambridge Philos. Soc. 64 (1968), 1001–1007.Google Scholar
(6)McCrudden, M.On the Brunn-Minkowski coefficient of a locally compact unimodular group. Proc. Cambridge Philos. Soc. 65 (1969), 33–45.Google Scholar