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Density and covering properties of intervals of ℝn

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Tamás Keleti
Tamás Keleti
Affiliation:
Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary. E-mail: elek@cs.elte.hu
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Abstract

The key result of this paper proves the existence of functions ρn(h) for which, whenever H is a (Lebesgue) measurable subset of the n-dimensional unit cube In with measure |H| > h and ℛ is a class of subintervals (n-dimensional axis-parallel rectangles) of In that covers H, then there exists an interval R∈ℛ in which the density of H is greater than ρn(h); that is, |H∩R|/|R|>ρn (h) (=(h/2n)2). It is shown how to use this result to find 4 points of a measurable subset of the unit square which are the vertices of an axis-parallel rectangle that has quite large intersection with the original set. Density and covering properties of classes of subsets of ℝn are introduced and investigated. As a consequence, a covering property of the class of intervals of ℝn is obtained: if ℛ is a family of n-dimensional intervals with , then there is a finite sequence R1, …, Rm∈ℛ such that and .

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory

Information

Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 229 - 242
Copyright
Copyright © University College London 2000

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