Removable Sets and Hausdorff Measure

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


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!


Hausdorff Dimension Hausdorff Measure Arclength Measure Besicovitch Dimension Hausdorff Content 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

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

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