Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T05:54:23.450Z Has data issue: false hasContentIssue false
  • English
  • Français

Paradoxical decompositions using Lipschitz functions

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

M. Laczkovich
M. Laczkovich
Affiliation:
Department of Analysis, Eötvös Loránd University, Múzeum krt. 6–8, Budapest, Hungary.
Get access

Extract

§1. Introduction and main results. A map f: A → R (A ⊂ R) is called piecewise contractive if there is a finite partition A = A1∪ … ∪ An such that the restriction f| Ai is a contraction for every i = 1, …, n. According to a theorem proved by von Neumann in [3], every interval can be mapped, using a piecewise contractive map, onto a longer interval. This easily implies that whenever A, B are bounded subsets of R with nonempty interior, then A can be mapped, using a piecewise contractive map, onto B (see [6], Theorem 7.12, p. 105). Our aim is to determine the range of the Lebesgue measure of B, supposing that the number of pieces in the partition of A is given. The Lebesgue outer measure will be denoted by λ. If I is an interval then we write |I| = λ(I).

MSC classification

Secondary: 28A99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 216 - 222
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Hall, M. Jr. Distinct representatives of subsets. Bull. Amer. Math. Soc., 54 (1948), 922–926.CrossRefGoogle Scholar
2.Laczkovich, M.. Von Neumann's paradox with translations. Fund. Math., 131 (1988), 1–12.CrossRefGoogle Scholar
1.Neumann, J. von. Zur allgemeinen Theorie des Masses. Fund. Math., 13 (1929), 73–116.CrossRefGoogle Scholar
4.Nevanlinna, R. and Paatero, V.. Introduction to Complex Analysis (Chelsea, 1969).Google Scholar
5.Rado, R.. Factorization of even graphs. Quart. J. Math., Oxford Ser., 20 (1949), 95–104.CrossRefGoogle Scholar
6.Wagon, S.. The Banach-Tarski Paradox (Cambridge Univ. Press, 1986).Google Scholar