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Measures Read chapter Read first chapter

2013 | OriginalPaper | Chapter

6. Differentiation

Author : Donald L. Cohn

Published in: Measure Theory

Publisher: Springer New York

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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.

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next chapter Measures on Locally Compact Spaces
Footnotes
1
Since T  − 1(U) and T(U) are open and hence Borel whenever U is a open subset of \({\mathbb{R}}^{d}\), the measurability of T and T  − 1 follows from Proposition 2.6.2.
 
2
See Bartle [4], Hoffman [60], Loomis and Sternberg [85], Rudin [104], or Thomson, Bruckner, and Bruckner [117]​.
 
3
Here we are dealing with the components of the matrices of these operators with respect to the usual ordered basis of \({\mathbb{R}}^{d}\).
 
4
The symbols x j and x 0, j in (11) refer to the jth components of the vectors x and x 0.
 
5
Thus μ n, a  ≪ λ and μ n, s  ⊥ λ.
 
6
It is easy to modify the definition of absolute continuity for functions on \(\mathbb{R}\) to make it apply to functions on [a, b].
 
7
Some authors use the condition \(\lim _{h\rightarrow {0}^{+}} \frac{1} {h}\int _{0}^{h}\vert f(x + t) + f(x - t) - 2f(x)\vert \,dt = 0\) as the defining condition for being a Lebesgue point; of course each point that satisfies (2) is also a Lebesgue point in this sense.
 
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21.
Bruckner, A.M.: Differentiation of integrals. Herbert Ellsworth Slaught Memorial Papers. Amer. Math. Monthly 78 (Suppl.) (1971)
22.
Bruckner, A.M.: Differentiation of Real Functions. Lecture Notes in Mathematics, vol. 659. Springer, Berlin (1978)
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Kölzow, D.: Differentiation von Massen. Lecture Notes in Mathematics, vol. 65. Springer, Berlin (1968)
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Loomis, L.H., Sternberg, S.: Advanced Calculus. Addison-Wesley, Reading (1968)MATH
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Munroe, M.E.: Measure and Integration, 2nd edn. Addison-Wesley, Reading (1971)MATH
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Riesz, F., Sz-Nagy, B.: Functional Analysis. Ungar, New York (1955). Reprinted by Dover, 1990
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Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)MATH
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Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)MATH
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Saks, S.: Theory of the Integral, 2nd revised edn. Dover, New York (1964). Reprint of 2nd revised edition, 1937
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Thomson, B.S., Bruckner, J.B., Bruckner, A.M.: Elementary Real Analysis, 2nd edn. ClassicalRealAnalysis.com (2008)
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Wheeden, R.L., Zygmund, A.: Measure and Integral. Monographs and Textbooks in Pure and Applied Mathematics, vol. 43. Marcel Dekker, New York (1977)
Metadata
Title
Differentiation
Author
Donald L. Cohn
Copyright Year
2013
Publisher
Springer New York
Book
Measure Theory

Print ISBN: 978-1-4614-6955-1

Electronic ISBN: 978-1-4614-6956-8

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-6956-8_6

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