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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Notes
- 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.
- 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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Cohn, D.L. (2013). Differentiation. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_6
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