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Removable Sets and Hausdorff Measure

  • James J. Dudziak
Chapter
Part of the Universitext book series (UTX)

Abstract

At a fuzzy intuitive level, removable sets have small “size” and nonremovable sets big “size.” A precise notion of “size” applicable to arbitrary subsets of ℂ and appropriate to our problem is given by Hausdorff measure (and Hausdorff dimension). So in this section we will simply introduce Hausdorff measure as a gauge of the smallness of a set and as a necessary preliminary for another such gauge, Hausdorff dimension. Surprisingly, the assertions 2.1 through 2.4 below are enough to get us through to the end of Chapter 4. It is only after, in Section 5.1, that we shall need to take up the fact that Hausdorff measure is indeed a positive measure defined on a σ-algebra containing the Borel subsets of ℂ!

Keywords

Hausdorff Dimension Hausdorff Measure Arclength Measure Besicovitch Dimension Hausdorff Content 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Referneces

  1. [MUR]
    T. Murai, Construction of H 1 functions concerning the estimate of analytic capacity, Bull. London Math. Soc., Vol. 19 (1986), 154–160. ( Section 2.4)MathSciNetCrossRefGoogle Scholar
  2. [JON1]
    P. W. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, Lecture Notes in Math., Vol. 1384, Springer-Verlag (1989), 24–68. ( Section 2.4)
  3. [FALC]
    K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press (1985). (Preface and  Sections 2.1,  4.7,  5.1,  5.2,  5.3,  6.5, and  6.6)
  4. [McM]
    T. J. McMinn, Linear measures of some sets of the Cantor type, Proc. Cambridge Philos. Soc., Vol. 53 (1957), 312–317. ( Sections 2.1 and  2.4)MathSciNetMATHCrossRefGoogle Scholar
  5. [PAIN]
    P. Painlevé, Sur les lignes singulières des fonctions analytiques, Annales de la Faculté des Sciences de Toulouse (1888). (Preface and  Section 2.2)
  6. [KK]
    R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag (1996). ( Section 2.1)
  7. [VIT1]
    A. G. Vitushkin, Example of a set of positive length but zero analytic capacity, Dokl. Akad. Nauk. SSSR, Vol. 127 (1959), 246–249 (Russian). ( Section 2.4)MathSciNetMATHGoogle Scholar
  8. [GAR2]
    J. Garnett, Analytic capacity and measure, Lecture Notes in Math., Vol. 297, Springer-Verlag (1972). (Preface and  Sections 1.2,  2.4, and  3.1)
  9. [GAR1]
    J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc., Vol. 21 (1970), 696–699. ( Section 2.4)MathSciNetCrossRefGoogle Scholar
  10. [MAT1]
    P. Mattila, A class of sets with positive length and zero analytic capacity, Ann. Acad. Sci. Fenn. Ser. A I Math., Vol. 10 (1985), 387–395. ( Section 2.4)MathSciNetMATHGoogle Scholar
  11. [RUD]
    W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill Book Company (1987). (Preface and Many Sections)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Lyman Briggs College, Michigan State UniversityEast LansingUSA

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