Abstract
This is the central chapter of the book. In the first four sections, we define and study elementary properties of the integral. In Sect. 4.5, we prove the countable additivity and absolute continuity of the integral. In Sects. 4.6 and 4.7, we discuss the properties of the integral for functions of one and several variables separately. In the next section, we thoroughly study conditions under which the passage to the limit under the integral sign is possible and establish very important sufficient conditions for the possibility of such a passage (the Lebesgue majorant convergence theorem), and also subtler conditions (the Vitali and Vallée Poussin theorems). Section 4.9 is devoted to differentiation of the integral with respect to a set. The results established here are obtained by the maximal function theorem also proved in this section.
The last two sections are devoted to generalizations of the integral with respect to the Lebesgue measure, the Stieltjes integrals.
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Notes
- 1.
The quotation is borrowed from [Lus, p. 499].
- 2.
Gottfried Wilhelm Leibniz (1646–1716)—German philosopher and mathematician.
- 3.
Jacob Bernoulli (1654–1705)—Swiss mathematician.
- 4.
Beppo Levi (1875–1961)—Italian mathematician.
- 5.
Pafnuty L’vovich Chebyshev (1821–1894)—Russian mathematician.
- 6.
Ludwig Otto Hölder (1859–1937)—German mathematician.
- 7.
Viktor Yakovlevich Bunyakovsky (1804–1889)—Russian mathematician.
- 8.
Isaac Barrow (1630–1677)—English mathematician.
- 9.
Following tradition, we often denote the integral over an infinite interval (a,+∞) by \(\int_{a}^{\infty} f(x)\,dx\), omitting the plus sign in front of the symbol ∞.
- 10.
Recall that n!! stands for the product of all positive integers less than or equal to n and having the same parity as n.
- 11.
John Wallis (1616–1703)—English mathematician.
- 12.
Leonhard Euler (1707–1783)—Swiss mathematician.
- 13.
Siméon Denis Poisson (1781–1840)—French mathematician.
- 14.
Augustin-Jean Fresnel (1788–1827)—French physicist.
- 15.
Johann Peter Gustav Lejeune Dirichlet (1805–1859)—German mathematician.
- 16.
Niels Henrik Abel (1802–1829)—Norwegian mathematician.
- 17.
Giuliano Frullani (1795–1834)—Italian mathematician.
- 18.
Georg Friedrich Bernhard Riemann (1826–1866)—German mathematician.
- 19.
Pierre Joseph Louis Fatou (1878–1929)—French mathematician.
- 20.
Charles-Jean Étienne Gustave Nicolas de La Vallée Poussin (1866–1962)—Belgian mathematician.
- 21.
Godfrey Harold Hardy (1877–1947)—English mathematician.
- 22.
John Edensor Littlewood (1885–1977)—English mathematician.
- 23.
In some books, the term “Lebesgue set” refers to the set of Lebesgue points of a function. We draw the reader’s attention to this terminological ambiguity.
- 24.
Thomas Joannes Stieltjes (1856–1894)—Dutch mathematician.
- 25.
It is instructive to compare this proof with that of the countable additivity of the ordinary volume (Theorem 2.1.1).
References
Bogachev, V.I.: Measure Theory, vols. 1, 2. Springer, Berlin (2007). 1.1.3, 1.5.1, 4.8.7
Lebesgue, H.: Lectures on Integration and Analysis of Primitive Functions. Cambridge University Press, Cambridge (2009). Chap. 4
Luzin, N.N.: Collected Works, vol. 2. Academy of Sciences, Moscow (1958) [in Russian]. Chap. 4
Matsuoka, Y.: An elementary proof of the formula \(\sum_{k=1}^{\infty}\frac{1}{k^{2}}=\frac{\pi^{2}}{6}\). Am. Math. Mon. 68, 486–487 (1961). 4.6.2
Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar, New York (1955/1961) 1.4.1, 2.4.3, Chap. 4
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Makarov, B., Podkorytov, A. (2013). The Integral. In: Real Analysis: Measures, Integrals and Applications. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-5122-7_4
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