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Differentiation

  • Edwin Hewitt
  • Karl Stromberg

Abstract

This chapter contains first a brief but reasonably complete treatment of the theory of differentiation for complex-valued functions defined on intervals of the line. Section 17 is severely classical, containing examples and Lebesgue’s famous theorem on differentiation of functions of finite variation. In § 18, we explore the conditions under which the classical equality
$$f(b) - f(a) = \int\limits_a^b {f'(t) \ dt}$$
is valid. This exploration leads to interesting and perhaps unexpected measure-theoretic ideas, which have little to do with differentiation and which have applications in extraordinarily diverse fields. The main result in this direction is the Lebesgue-Radon-Nikodým theorem, which we examine thoroughly in § 19 and apply to the decomposition of measures on R. In § 20, we present several other applications of the Lebesgue-Radon-Nikodým theorem to problems in abstract analysis. Sections 17 and 18 are important, and should be studied by all readers. The same is true of § 19, up to and including (19.24). The remainder of § 19 may be omitted by readers pressed for time. Of § 20, (20.1)–(20.5) and (20.41)–(20.52) are topics important for every student. The remainder of § 20 is in our opinion interesting but less vital, and it too may be omitted by readers pressed for time.

Keywords

Measure Space Pairwise Disjoint Complex Measure Compact Hausdorff Space Finite Variation 
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.

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Copyright information

© Springer-Verlag Berlin · Heidelberg 1965

Authors and Affiliations

  • Edwin Hewitt
    • 1
  • Karl Stromberg
    • 2
  1. 1.The University of WashingtonUSA
  2. 2.The University of OregonUSA

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