Abstract
One of the (if not the) most important theorems of classical analysis is the Newton–Leibniz formula:
allowing us to compute many integrals. The purpose of this chapter is to extend its validity to Lebesgue integrable functions.
If Newton and Leibniz had thought that continuous functions need not have derivatives, and this is the general case, the differential calculus would not have been born.—É. Picard
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Notes
- 1.
- 2.
- 3.
- 4.
The identity map of \(\mathbb{R}\) shows that this is not necessarily true for unbounded intervals.
- 5.
We may assume that E does not contain the right endpoint of I.
- 6.
- 7.
Fubini [165].
- 8.
- 9.
- 10.
Lebesgue [290].
- 11.
- 12.
See, e.g., an example of F. Riesz in Sz.-Nagy [448].
- 13.
Lebesgue [295, pp. 232–249].
- 14.
de la Vallée-Poussin [465, p. 467].
- 15.
- 16.
A perfect set is a closed set with no isolated points. A set is nowhere dense if its closure has no interior points.
- 17.
A set A is of the first category (Baire [17]) if it is the countable union of nowhere dense sets.
- 18.
A set A is of the second category (Baire [17]) if it is not of the first category. Baire’s theorem (see p. 32) states that every complete metric space and every compact Hausdorff space is of the second category.
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Komornik, V. (2016). Generalized Newton–Leibniz Formula. In: Lectures on Functional Analysis and the Lebesgue Integral. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6811-9_6
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