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).
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Notes
- 1.
Recall that \(0 \cdot (+\infty ) = 0\) and that if x≠ −∞, then \(x + (+\infty ) = (+\infty ) + x = +\infty \). See Appendix B.
- 2.
An extended real-valued function is, of course, a \([-\infty, +\infty ]\)-valued function.
- 3.
There are also cases of functions defined on \(\mathbb{R}\) that are not Lebesgue integrable over \(\mathbb{R}\) but for which the corresponding improper integral exists. For instance, define \(f : \mathbb{R} \rightarrow \mathbb{R}\) by f(x) = 0 if x < 1 and \(f(x) = {(-1)}^{n}/n\) if n ≤ x < n + 1, where n = 1, 2, ….
- 4.
Another notation for μf − 1 is μ ∘ f − 1.
References
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Halmos, P.R.: Measure Theory. Van Nostrand, Princeton (1950). Reprinted by Springer, 1974
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)
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Cohn, D.L. (2013). Functions and Integrals. In: Measure Theory. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6956-8_2
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