The Integral

  • Boris Makarov
  • Anatolii Podkorytov
Part of the Universitext book series (UTX)


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.


Absolute Continuity Maximal Function Theorem Lebesgue Stieltjes Measure Admissible Partition Lebesgue Point 
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.


  1. [Bo]
    Bogachev, V.I.: Measure Theory, vols. 1, 2. Springer, Berlin (2007). 1.1.3, 1.5.1, 4.8.7 CrossRefGoogle Scholar
  2. [L]
    Lebesgue, H.: Lectures on Integration and Analysis of Primitive Functions. Cambridge University Press, Cambridge (2009). Chap. 4 zbMATHGoogle Scholar
  3. [Lus]
    Luzin, N.N.: Collected Works, vol. 2. Academy of Sciences, Moscow (1958) [in Russian]. Chap. 4 Google Scholar
  4. [M]
    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 Google Scholar
  5. [N]
    Natanson, I.P.: Theory of Functions of a Real Variable. Frederick Ungar, New York (1955/1961) 1.4.1, 2.4.3, Chap. 4 Google Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Boris Makarov
    • 1
  • Anatolii Podkorytov
    • 1
  1. 1.Mathematics and Mechanics FacultySt Petersburg State UniversitySt PetersburgRussia

Personalised recommendations