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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

$ p$-extremal length and $ p$-measurable curve families


Author: Joseph Hesse
Journal: Proc. Amer. Math. Soc. 53 (1975), 356-360
MSC: Primary 28A10; Secondary 30A44, 31B15
MathSciNet review: 0385052
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Abstract: It is well known that the reciprocal of $ p$-extremal length, considered as a set function, is an outer measure. We show that if a curve family in euclidean $ n$-space is measurable with respect to this outer measure, then the $ p$-extremal length of the curve family is zero or infinite.


References [Enhancements On Off] (What's this?)

  • [1] Bent Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219. MR 0097720 (20 #4187)
  • [2] H. Renggli, Extremallängen und eine konform invariante Massfunktion für Kurvenscharen, Comment. Math. Helv. 41 (1966/1967), 10–17 (German). MR 0204650 (34 #4489)
  • [3] J. Väisälä, Lectures on $ n$-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin and New York, 1971.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0385052-5
PII: S 0002-9939(1975)0385052-5
Keywords: $ p$-extremal length, $ p$-measurable curve families
Article copyright: © Copyright 1975 American Mathematical Society