Removable Sets and Hausdorff Measure
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!
KeywordsHausdorff Dimension Hausdorff Measure Arclength Measure Besicovitch Dimension Hausdorff Content
- [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)
- [PAIN]P. Painlevé, Sur les lignes singulières des fonctions analytiques, Annales de la Faculté des Sciences de Toulouse (1888). (Preface and Section 2.2)
- [KK]R. Kannan and C. K. Krueger, Advanced Analysis on the Real Line, Springer-Verlag (1996). ( Section 2.1)
- [RUD]W. Rudin, Real and Complex Analysis, 3rd Edition, McGraw-Hill Book Company (1987). (Preface and Many Sections)Google Scholar