Hostname: page-component-5c6d5d7d68-7tdvq Total loading time: 0 Render date: 2024-08-16T08:23:42.413Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear measures of some sets of the Cantor type

Published online by Cambridge University Press:  24 October 2008

Trevor J. Mcminn
Trevor J. Mcminn
Affiliation:
University of CaliforniaBerkeley
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. Let 0 < λ < 1 and remove from the closed unit interval the open interval of length λ concentric with the unit interval. From each of the two remaining closed intervals of length ½(1 − λ) remove the concentric open interval of length ½λ(1 − λ). From each of the four remaining closed intervals of length ¼λ(1 − λ)2 remove the concentric open interval of length ¼λ(l − λ)2, etc. The remaining set is a perfect non-dense set of Lebesgue measure zero and is the Cantor set for λ = ⅓. Let Tλr be the Cartesian product of this set with the set similar to it obtained by magnifying it by a factor r > 0. Letting L be Carathéodory linear measure (1) and letting G be Gillespie linear square(2), Randolph(3) has established the following relations:

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 53 , Issue 2 , April 1957 , pp. 312 - 317
Copyright
Copyright © Cambridge Philosophical Society 1957

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

REFERENCES

(1)Carathéodory, C.Über das lineare Mass von Punktmengen eine Verallgemeinerung das Längenbegriffs. Nachr. Ges. Wiss. Göttingen (1914), 404–26.Google Scholar
(2)Morse, A. P. and Randolph, J. F.Gillespie measure. Duke Math. J. 6 (1940), 408–19.CrossRefGoogle Scholar
(3)Randolph, J. F.Some properties of sets of the Cantor type. J. Lond. Math. Soc. 16 (1941), 38–42.CrossRefGoogle Scholar