Measure Theory pp 155-179 | Cite as

Differentiation

  • Donald L. Cohn
Chapter
First Online:
Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

Abstract

In Chapter 6 we look at two aspects of the relationship between differentiation and integration. First, in Section 6.1, we look at changes of variables in integrals on Euclidean spaces. Such changes of variables occur, for example, when one evaluates an integral over a region in the plane by converting to polar coordinates. Then, in Sections 6.2 and 6.3, we look at some deeper aspects of differentiation theory, including the almost everywhere differentiability of monotone functions and of indefinite integrals and the relationship between Radon-Nikodym derivatives and differentiation theory. The Vitali covering theorem is an important tool for this. The discussion of differentiation theory will be resumed when we discuss the Henstock-Kurzweil integral in Appendix H.

Keyword

Change of variable Vitali covering theorem Upper derivate Lower derivate Derivative Almost everywhere differentiability Point of density 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Donald L. Cohn
    • 1
  1. 1.Department of Mathematics and Computer ScienceSuffolk UniversityBostonUSA

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