Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T05:01:21.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Some inequalities that arise in measure theory

Part of: Abstract harmonic analysis Inequalities

Published online by Cambridge University Press:  09 April 2009

Gavin Brown
Gavin Brown
Affiliation:
School of MathematicsUniversity of New South WalesKensington, N.S.W. 2033, Australia
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.

The Lebesgue measure, λ (E + F), of the algebraic sum of two Borel sets, E, F of the classical “middle-thirds’ Cantor set on the circle can be estimated by evaluating the Cantor meaure, μ of the summands. For example log λ (E + F) exceeds a fixed scalar multiple of log μ (E)+ log μ (F). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

MSC classification

Secondary: 26D20: Other analytical inequalities 43A05: Measures on groups and semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 83 - 94
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Beckenbach, E. F. and Bellman, R., Inequalities (Springer-Verlag, Berlin, 1971).Google Scholar
[2]Brown, G., Keane, M. S., Moran, W. and Pearce, C. E. M., ‘An inequality, with applications to Cantor measures and normal numbers’, preprint.Google Scholar
[3]Brown, G. and Moran, W., ‘Raikov systems and radicals in convolution measure algebras’, J. London Math. Soc. (2), 28 (1983), 531–542.CrossRefGoogle Scholar
[4]Brown, G. and Shepp, L., ‘A convolution inequality’, preprint.Google Scholar
[5]Brown, G., ‘Measures of algebraic sums of sets’, in preparation.Google Scholar
[6]Graham, C. C. and McGehee, O. C., Essays in commutative harmonic analysis (Springer-Verlag, Berlin, 1979).CrossRefGoogle Scholar
[7]Hajela, D. and Seymour, P., ‘Counting points in hypercubes, isoperimetric inequalities and convolution measure algebras’, preprint.Google Scholar
[8]Hall, R. R., ‘A problem in combinatorial geometry’, J. London Math. Soc. (2) 12 (1976), 315–319.CrossRefGoogle Scholar
[9]Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalilties (Cambridge University Press, London, 1951).Google Scholar
[10]Mitrinovic, D. S., Analytic inequalities (Springer-Verlag, Berlin, 1970).CrossRefGoogle Scholar
[11]Oberlin, D. M., ‘The size of sum sets, II’, preprint.Google Scholar
[12]Talagrand, M., ‘Solution d'un probléme de R. Haydon’, Publ. du Dept. de Math. Lyon 12-2 (1975), 43–46.Google Scholar
[13]Woodall, D. R., ‘A theorem on cubes’, Mathematika 24 (1977), 60–62.CrossRefGoogle Scholar