Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-26T20:56:13.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Absolutely non-measurable and singular co-analytic set

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

A. J. Ostaszewski
A. J. Ostaszewski
Affiliation:
Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE
Get access

Extract

Davies and Rogers [5] constructed a compact metric space Ω which is singular for a certain Hausdorff measure μh, in the sense that all subsets of Ω have μh-measure zero or infinity and μh(Ω) = ∞. (For a further study of this example see Boardman [3]). The interest lies in its extremely good descriptive character, which was lacking in the earlier examples given by Besicovitch [2] (a plane set singular for linear measure) and Choquet [4] (a plane set singular for any Hausdorff measure for which a segment has positive measure).

MSC classification

Secondary: 28A05: Classes of sets (Borel fields, $sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets
Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975 , pp. 161 - 163
Copyright
Copyright © University College London 1975

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.Addison, J. W.. “Some consequences of the axiom of constructibility”, Fund. Math., 46 (1959), 123–135.CrossRefGoogle Scholar
2.Besicovitch, A. S.. “Concentrated and rarified sets of points”, Acta Math., 62 (1934), 289–300.Google Scholar
3.Boardman, E.. “The problem of subsets of finite positive measure in the space of compact subsets of [0,1]”, Quart. J. Math. Oxford, to appear.Google Scholar
4.Choquet, G.. “Ensembles singuliers et structure des ensembles mesurables pour les mesures de Hausdorff”, Bull. Soc Math. France, 74 (1946), 1–14.Google Scholar
5.Davies, R. O. and Rogers, C. A.. “The problem of subsets of finite positive measure”, Bull. London Math. Soc, 1 (1969), 47–54.CrossRefGoogle Scholar
6.Eggleston, H. G.. “Concentrated sets”, Proc. Camb. Phil. Soc, 63 (1967), 931–933.CrossRefGoogle Scholar
7.Fenstad, J. E. and Normann, D.. “On absolutely measurable sets”, Fund. Math., 81 (1974), 91–98.Google Scholar
8.Godel, K.. The consistency of the continuum hypothesis (Annals of Math. Studies, Princeton, 1940).Google Scholar
9.Kechris, A. S.. “Measure and category in effective descriptive set theory”, Ann. Math. Logic, 5 (1973), 337–384.CrossRefGoogle Scholar
10.Kondỗ, M.. “Sur l'uniformisation des complémentaires analytiques et des ensembles projectifs de la seconde classe”, Japanese J. Math., 15 (1939), 197–230.CrossRefGoogle Scholar
11.Kuratowski, K.. Topologie (PWN, Warszawa, 1958).Google Scholar
12.Kuratowski, K.. “Ensembles projectifs et ensembles singuliers”, Fund. Math., 35 (1948), 131–140.CrossRefGoogle Scholar
13.Rogers, C. A.. Hausdorff measures (Cambridge University Press, 1970).Google Scholar
14.Saks, S.. Theory of the integral (Dover Publications, New York, 1964).Google Scholar