Functions and Integrals

  • Donald L. Cohn
Chapter
Part of the Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT)

Abstract

Chapter 2 is devoted to the definition and basic properties of the Lebesgue integral. functions, the functions that are simple enough that the integral can be defined for them, if their values are not too large (Section 2.1). After a brief look in Section 2.2 at properties that hold almost everywhere (that is, that may fail on some set of measure zero, as long as they hold everywhere else), we turn to the definition of the Lebesgue integral and to its basic properties (Sections 2.3 and 2.4). The chapter ends with a sketch of how the Lebesgue integral relates to the Riemann integral (Section 2.5) and then with a few more details about measurable functions (Section 2.6).

Keyword

Measurable function Borel function Cantor set Cantor function Almost everywhere Integral Integrable function Lebesgue integral Monotone convergence theorem Fatou’s lemma Dominated convergence theorem Riemann integrability Riemann integral 

References

  1. 32.
    Daniell, P.J.: A general form of integral. Ann. of Math. (2) 19, 279–294 (1917–1918)Google Scholar
  2. 54.
    Halmos, P.R.: Measure Theory. Van Nostrand, Princeton (1950). Reprinted by Springer, 1974Google Scholar
  3. 114.
    Stone, M.H.: Notes on integration. Proc. Nat. Acad. Sci. U.S.A. 34, 336–342, 447–455, 483–490 (1948); Stone, M.H.: Notes on integration. Proc. Nat. Acad. Sci. U.S.A. 35, 50–58 (1949)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Donald L. Cohn
    • 1
  1. 1.Department of Mathematics and Computer ScienceSuffolk UniversityBostonUSA

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